PRODUCTION SCHEDULING POLICY FOR FLEXIBLE … · Last, in the third part (Chapter 8) the effect of machining parameters on production scheduling for FMS is investigated. The cost

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PRODUCTION SCHEDULING POLICY FOR FLEXIBLE MANUFACTURING SYSTEMS

byZ. Doulgeri

A dissertation submitted to the University of London for the degree ofDoctor of Philosophy

Department of Mechanical Engineering Imperial College

London

June 1987

TO MY PARENTS Konstantinos Doulgeris Olympia Siori - Doulgeri

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ACKNOWLEDGEMENTS

Doing a Ph.D. is nothing like studying or working. It is a lifestyle. It can be exciting and depressing, stimulating and inhibiting. This page is dedicated to all those people who were a great part of i t : and most of them still are.

First, to my supervisor Dr R. Hibberd for his support, advice and encouragement but most of all for giving me the opporunity to do this Ph.D. and thus live through such an unforgettable experience.

To my friend Fotini for without her I would have probably remained a 'foreigner' in this country.

To Heinz, for I lived with him one of the most intense periods; and to David for the nights we used to drown in alcohol our Ph.D. problems.

I would also like to thank Yannis for being a great listener to my problems and an excellent squash partner, Alex for his invaluable support, Androulla for her understanding and Sotiria for her special friendship.

Thanks go also to the Imperial College Greek community for the lengthy coffee and gossip zing sessions.

Lots of credit goes to all those people who put up with my general irritability during the writing-up period and to my personal computer for not repeatedly crashing the diskettes and blowing away entire chapters.

Finally I would like to thank the Greek Scholarship Foundation and Lister- Petter Ltd. for their financial support.

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ABSTRACT

This study deals with the problem of FMS scheduling and control. It consists of three main parts. In the first part (Chapter 3 and Chapter 4), a simulation modelling technique using the activity centered approach is proposed for building FMS simulation models which are flexible to changes in control and scheduling policies. FMS simulation models of two real systems were generated using this technique to investigate the effect of different design and operating parameters on system performance (Chapter 4). A general FMS simulation model was also built and subsequently used as the basis for the investigation of different scheduling policies. Simulation models .designed by the activity centered technique are kept simple, flexible to changes in scheduling and control and easily expandable. Remodelling is avoided and the time for developing and debugging of simulation models is greatly reduced.

In the second part (Chapters 5, 6 and 7) which is the main body of the study, a periodic scheduling policy for FMS is proposed, and its performance investigated. Production ratios are determined by the demand and the system's operational state in order to meet the production target specified. Production is carried out in sets of parts which satisfy the predefined ratios and whose size is determined so that k minimise^ the cost of production expressed as the sum of the inventory cost and the system's idle time cost (Chapter 5). A schedule is calculated for the set of parts under the current demand and operational state. A heuristic algorithm is used to calculate this schedule with the objective of minimising its makespan. The algorithm incorporates the general simulation model in order to prpduce feasible and realistic schedules, given the FMS complexity. At each decision stage it selects the alternative which gives the best performance estimate. Estimates are derived by looking ahead using the simulation model and with future conflicts overcome using a dispatching rule. Each performance estimate comes from a complete feasible schedule. Optimallity is not guaranteed but evaluation of the algorithm on the basis of solutions obtained for a number of problems of different sizes and different processing times show that the algorithm achieves solutions with small deviations from the absolute minimum makespan and that it gives significant improvements over the use of simple dispatching rules (Chapter 6). The overall periodic production policy is illustrated with the use of two examples which cover a multitude of aspects of FMSs that are likely to be encountered in practice (Chapter 7). It is shown that adoption of this policy allows full control of the output and production costs, and fast response (steady state is reached as quickly as possible) in the presence of machine failures or demand changes while achieving good performance in steady state in terms of the makespan and the mean flow time of parts.

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Last, in the third part (Chapter 8) the effect of machining parameters on production scheduling for FMS is investigated. The cost of production, when the machining parameters are taken into accounts minimised for a given schedule. The overall cost is reduced from the savings which occur in the tool cost by running the machines at lower speeds, provided that the schedule constraints allow for extension of the operation durations.

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CONTENTS

Acknowledgements 3Abstract 4List of tables 9List of figures 11Nomenclature 14

CHAPTER 1 INTRODUCTION 17

CHAPTER 2 LITERATURE REVIEW 212.1 FMS scheduling problem-main approaches 212.2 Hierarchical policies for FMS scheduling 222.3 Models used for FMS scheduling 232.4 Analytical models 232.5 Use of queueing models for solving 25

intermediate range operational problems in an FMS2.6 The Markovian approach to FMS control 262.7 Integer programming formulations of intermediate 27

range operating problems in an FMS2.8 The scheduling approach 272.9 Simulation models 29

CHAPTER 3 A SIMULATION FOR FMSs 323.1 Introduction 323.2 Simulation model, its components and logical structures 333.3 A new methodology for building FMS simulation models 37

3.3.1 The FMS model 373.3.2 Entity cycle diagrams- 38

A method for building simulation models3.3.3 Activity centered diagrams - 38

The new building blocks for FMS simulation3.3.4 Linking the activity diagrams 42

to construct the Final model3.3.5 The structure of the control module 453.3.6 Measurements in a simulation model 49

3.4 Summary of FMS simulation model's components 49

p age

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3.5 FMSIM- A computer simulation program for modelling 50FMSs for experimentation on general scheduling and control issues

CHAPTER 4 SIMULATION STUDY OF INDUSTRIAL FMSs 534.1 Introduction 534.2 Simulation study of FMS1 for the manufacture of flywheels 53

4.2.1. Cell description 534.2.2. Aim of the FMS 1 simulation study 554.2.3. The FMS1 simulation model 554.2.4. Input data 564.2.5. System performance in the flywheel production 574.2.6 System performance sensitivity on inspection duration 584.2.7 System performance under different robot cycle times 594.2.8 Conclusions 63

4.3 Simulation study of FMS2 for the manufacture of small turned parts 634.3.1 Cell description 634.3.2 Aim of the FMS2 simulation study 654.3.3 The FMS2 simulation model 654.3.4 System performance under the anticipated workload 724.3.5 System performance under different order allocation policies 764.3.6 System performance under different manning levels 794.3.7 System performance under the updated workload 804.3.8 Effect of lathe breakdown on system performance 824.3.9 System performance under production in small batches 834.3.10 System performance under tool changing due to tool wear 864.3.11 Tool holder utilisation 894.3.12 Conclusions 90

CHAPTER 5 A PERIODIC PRODUCTION POLICY 92FOR FMS SCHEDULING

5.1 Introduction 925. 2 Problem statement 945.3 A mathematical model for the production cost per part 96

when producing in Minimal Part Sets5.4 Nature of the objective function 995.5 Discrete nature of the optimising parameter 1035.6 Solution procedure 1035.7 Dynamic behaviour of the system under the periodic production policy 105

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5.8 Improved periodic releasing for FMS 1065.9 Periodic production in an FMS with failure prone machines 107

CHAPTER 6 THE FMS PART SCHEDULING PROBLEM 1116.1 Introduction 1116.2 A general model for the static job shop scheduling problem 1136.3 Review of combinatorial approaches to job shop scheduling 117

6.3.1 Problem description and solution subsets 1176.3.2 Schedule generation procedures 120

6.4 A new simulation-based algorithm 129for the generation of non-delay schedules for an FMS

6.5 Heuristic approaches for FMS scheduling 1336.6 A new simulation based heuristic algorithm for FMS scheduling 135

6.6.1 Evaluation of the simulation based heuristic algorithm 1376.6.2 Discussion 142

CHAPTER 7 COMPUTATIONAL EXAMPLES FOR THE 144PERIODIC PRODUCTION SCHEDULING POLICY FOR FMSs

7.1 Introduction 1447.2 Example 1 1447.3 Example 2 1577.4 Conclusions 164

CHAPTER 8 MACHINING PARAMETER OPTIMISATION IN FMS 1668.1 The role of machining parameters in FMS scheduling 1668.2 Single stage machining economics for FMSs 1678.3 Machining parameter optimisation under schedule constraints 175

8.3.1 Problem formulation 1768.3.2 Problem solution 184

8.4 Results and discussion 187

CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS 193

REFERENCES 195

APPENDIX General FMS optimisation problem 204

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LIST OF TABLES

Table 3.1: The main entities, queues, activities and attributes 50for a general FMS simulation model

Table 4.1: Entities, queues, activities and attributes of FMS 1 model 55Table 4.2: Activities,their start conditions, end actions and duration in FMS 1 56Table 4.3: Operation durations for the flywheel manufacture in FMS 1 57Table 4.4: Robot movements and their durations for the flywheel production 57Table 4.5: Simulation results on FMS1 system performance 58

for flywheel productionTable 4.6: Entities, queues, activities and attributes of FMS2 model 67Table 4.7: Activities,their start conditions, end actions and duration in FMS 2 69Table 4.8: Performance results with the initial workload 73Table 4.9: System performance under the updated workload 82Table 4.10: Data on tool change occurence and duration 87Table 4.11: System performance with tool changing 87Table 4.12: Tool holder utilisation 89

Table 6.1: Routing and processing times for the example of figure 6.1 116Table 6.2: Tabular representation of a Gantt chart 116Table 6.3: Size of problems and distributions of operation times 138Table 6.4: Computational requirements of the SHA algorithm 140Table 6.5: Average percentage efficiency of the SHA solutions 140

compared with the best non delay schedule and average percentage deviation of solution compared with makespan lower bound.

Table 6.6: Worst case values of solution efficiency and solution deviation 141Table 6.7: Average percentage improvement of makespan value 141

produced by the simulation based heuristic algorithm compared with the value produced by using the simulation based priority dispatching with shortest processing time rule.

Table 7.1: Operation sequence and durations of example 1 145Table 7.2: Machine operations in example 1 145Table 7.3: Schedule results for varying set sizes 146

p a g e

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Table 7.4: Loading and output sequences and times 150for periodic production in sets of eight parts

Table 7.5: Machine utilisation for periodic production in sets of 8 parts 150Table 7.6: Performance of the SPT rule and 151

the periodic scheduling policy for the desired production Table 7.7: Content of parts completed in the middle 151

of the schedule when using the SPT ruleTable 7.8: Schedule results for varying set sizes 153

with one machine broken downTable 7.9: Schedule results for varying set sizes 154

after the repair of the broken down machine.Table 7.10: Input data for example 2 157Table 7.11: Schedule results for varying set sizes 158Table 7.12: Performance of the SPT rule and 160

the periodic policy for the desired production Table 7.13: Schedule results for varying set sizes 162

with one machine broken down

Table 8.1: Typical values for machining constants 172Table 8.2: Single stage optimal performance results 173Table 8.3: Operation sequence and durations 187Table 8.4: Memory requirements 190Table 8.5: Number of variables and constraints 192

for the present and the general formulation of cost minimisation

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LIST OF FIGURES

Figure 3.1 : The three phase logic 37Figure 3.2 : Diagram of the machining activity in a FMS 39Figure 3.3 : Diagram of the tool transport activity in a FMS 39Figure 3.4 : Diagram of the part transport activity in a FMS 40Figure 3.5 : Diagram of the part arrival activity in a FMS 41Figure 3.6 : Diagram of duplicated machining activity 41Figure 3.7: The flow of the machine entity in a FMS 43Figure 3.8 : The flow of the par entity in a FMS 44Figure 3.9 : Flow chart of the FMS simulation model 45

Figure 4.1: Schematic diagram of FMS 1 54Figure 4.2: Gauge utilisation versus inspection time 59Figure 4.3: System performance with varying robot cycle time 60Figure 4.4: Effect of unequal robot speed on system performance 60Figure 4.5: System dynamic performance with robots 61

having unequal speedFigure 4.6: System performance with varying drill time 62Figure 4.7: System performance with increasing lathe time 62

for constant drill timeFigure 4.8: Part allocation to lathe type 72Figure 4.9: System performance with lathes working during inspection 74Figure 4.10: Allocation of parts on lathe type without alternative allocation 75Figure 4.11: System performance for increased GE65 workload 75Figure 4.12: Makespan and last order loading time 76Figure 4.13: Lead time histogram 77Figure 4.14: System performance under different allocation policies, case 3 78Figure 4.15: System performance under different allocation policies, case 2 79Figure 4.16: System performance with varying manning levels 80Figure 4.17: Part allocation with the updated workload 81Figure 4.18: Makespan versus lathe downtime 83Figure 4.19: Number of parts produced from one bar 84Figure 4.20: System performance for batches of different size 85Figure 4.21:System performance with batch sizes of one week's required 86

production with a lower bound on batch size equal to the number of parts produced by one bar

page

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Figure 4.22:System performance with and without tool changing 88for batch sizes of four weeks required production

Figure 4.23:System performance with and without tool changing 88for batch sizes of one week required production

Figure 5.1: Relationship of the percentage system idle time 99to the MPS size

Figure 5.2: Relationship of the MPS's makespan 100and mean part flow time to its size

Figure 5.3: Cost per part for varying cost ratio with set deliveries 101Figure 5.4: Cost per part for varying cost ratio with part deliveries 101Figure 5.5: Cost per part for set and part deliveries 102Figure 5.6: An example MPS schedule 105Figure 5.7: Schedule of two successive MPSs 106Figure 5.8: Improved schedule of two successive MPSs 107

Figure 6.1a: Graphical representation of machine workload 115Figure 6. lb: Gantt chart of a feasible schedule 115Figure 6.2: Gantt chart of active schedule resulting by altering 118

the semi-active schedule of figure 6.1b .Figure 6.3: Non-delay schedule of the example given in table 6.1 119Figure 6.4: The search tree of partial solutions 121Figure 6.5: The search tee of the simulation based heuristic 135

dispatching algorithm using a 'look ahead' rule

Figure 7.1: Cost per part for varying cost ratios for set deliveries 146Figure 7.2: Cost per part for varying cost ratios for part deliveries 147Figure 7.3 : Improved periodic production in sets of eight 148Figure 7.4: Improved periodic production in sets of sixteen 149Figure 7.5: Average part flow time with increasing production target 152

for periodic production and production with the SPT rule Figure 7.6: Periodic production in sets of 4 with machine 3 broken down 153Figure 7.7: Cost per part for varying cost ratio for set and part deliveries 155Figure 7.8: Starting and finishing times of machines in the MPS of size 24 156Figure 7.9: Dynamic periodic production in the presence of machine failure 156 Figure 7.10: Cost per part as a function of the set size 158

for the set delivery policyFigure 7.11: Cost per part as a function of the set size fot the part delivery policy 159

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Figure 7.12: Periodic production in sets of 20 parts 159Figure 7.13: Average part flow time for the periodic production policy 161

and the SPT ruleFigure 7.14: Production makespan against production target 161

for the periodic production policy and the SPT rule Figure 7.15: Periodic production in sets of 10 with machine 2 broken down 163 Figure 7.16: Dynamic periodic production in the presence of machine failure 164

Figure 8.1: Operation cost and production rate 172for high speed steel tool operation

Figure 8.2: Operation cost and production rate 173for cemented carbide tool operation

Figure 8.3: Critical operations 180Figure 8.4: Gantt chart for the operations of an example 183Figure 8.5: Machine workload 187Figure 8.6: Total cell percentage savings for increasing MPS sizes 188

Figure 8.7: Percentage machine cost savings 189Figure 8.8: Machine cost savings dependence on workload 190

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AAd

Bl

CqqCTClcmCJ"dDDb

ix(.)I

NOMENCLATURE

system operational state operational state of machine m set of available partsset of operations, candidates for the partial schedule extension at stage d theoretical lower bound of makespan inventory cost rate for part type j operation costcost of producing a minimal part set average inventory cost ratecost ratio of idle time cost rate to the inventory cost ratetotal production costconstant in Taylor's tool life equationchoice set of machine mchoice set of machine m at decision stage ddecision stage bar diameterpercentage solution deviation from makespan lower bound finishing time of a machine m in the minimal part set production completion time of a part platest completion time of an operation on machine mearliest time some unscheduled operation could begin on machine mmean part flow timeflow time of part pset of machine in conflict over a part a large positive number operation subscript priority rule index number of operationsnumber of operations allocated to machine m number of operations of part p

15

I

k

jJkKl*mKtKdLm*mMM

MAMdM/

“JNNrPPmPdm

PdPPdPPa

Q

set of operations set of critical operationsset of operations allocated to machine m set of operations of part p part typenumber of part typesconstant in Taylor's tool life equationcost per unit idle timemachining cost per time unitaverage cost per toolcartesian product of choice setscutting length machinemachine for which conflict is resolvednumber of machinesset of machinesset of allocated machinesset of idle machines at stage d set of idle machinesnumber of parts type j in a minimal part set number of parts in a minimal part set number of parts per tool change partpart allocated to machine mpart allocated to machine m at stage darray of parts allocated to idle machines at stage dnumber of partspartial schedule until stage dset of partsset of allocated partsinstanteneous production ratios required production target

Qj : production target for part type jrj : production ratio for part type jR : production rateS : feed per revolutionSd : set of schedulable operationst : machining timeti : duration of operation iT : tool lifeT c : time to change a toolT i : total machine idle timeT id : system's idle timeV

: minimal parts set’s makespanU i : unscheduled processing time of operation iV : rank of a periodic set in the production horizonV : cutting speedV c : cutting speed for minimum costVP : cutting speed for maximum production rateWj : cumulative production of part type j

m : unscheduled processing time for machine m

Greek symbols

a : allocation timea i : earliest staring time of operation ixm : starting time for machine mTP : starting time of processing part pî : earliest completion time of operation i

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CHAPTER 1

INTRODUCTION

Demands in today's market place have totally changed from those of one or two decades ago where large batch production runs with low product variety were the 'norm'. Todays' markets demand improved quality, increased variety and smaller batches and reduced lead times. Manufacturing industry has accordingly been forced to reorganise its methods of production to meet today's demand. The development of modem computing and information handling techniques has enabled the design of automated systems with computer based control of the individual machines and material handling,so that production is no longer restricted to the manufacture of a single type of product. Investment in new technology has been accordingly seen as a solution to the problem mentioned and has undertaken at a variety of levels, from investing in stand alone CNC machines to 'leading edge' technology systems.

The terminology used to designate this new type of manufacturing emphasizes the production flexibility. The broadly accepted term is Flexible Manufacturing System or FMS which originated in the U.S but other terms like CMS (computerised manufacturing system), FAS (flexible automation system), CIM (computer integrated manufacturing) etc., are also used.

In general, one can define flexibility as the ability to respond effectively to changing circumstances. In an attempt to clarify what 'flexibility' means in the context of manufacturing system, Buzacott suggests two measures of flexibility. Job flexibility which relates to the mix of parts which the system can process and machine flexibility which relates to the effect on the system of internal disturbances like breakdowns. The achievement of a high degree of flexibility in both measures depends on a complex set of design and operational factors [1].

The number of FMS which exists already is not known with precision due to the lack of consensus of opinion between countries on FMS definition [2, 3]. A description of various FMS that have been installed in American and European plants for machining and assembling is given by Dupont-Gatelmand[2]. The automation of the machines and the diversity of parts which they can process are two common points for all systems, but the term FMS is used to designate very different systems depending on the operation mode and the extent of automation and product diversity.

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Generally speaking, an FMS is a system consisting of a set of machines that may or may not be identical and are capable of performing a number of different operations, and a material handling system to transfer parts from one machine to another . The whole system is under computer control, intended to operate mostly automatically and designed to process simultaneously different kinds of parts [4, 5].

The advantages of the use of FMS are repeatedly reported in the literature. They mainly stem from the flexibility and automation characteristics of the system. Flexibility allows quick response to market changes and a much more diversified production, and automation and computer control guarantee improved quality and higher product consi stency. In general, FMS has the technology to eliminate set up and handling costs and to cut down considerably lead times and inventory costs.

The advantages of the use of FMS are only potential benefits and their realisation is dependent on the effectiveness of the production scheduling policy and control adopted. In general, successful FMS is the outcome of an optimal selection of all FMS components ( the FMS design problem ) coupled with their optimal utilisation ( FMS scheduling and control problem). If we were to analyse the reasons behind what appear to be FMS failures, we would find that they are typically failures in planning (production scheduling and control software) and not in hardware [3].

The problem of production scheduling and control of an FMS is a much more complex problem than that of the classical manufacturing system. This is due to new variables related to the specific resources like the automated material handling system, fixtures etc., and the multi-objectives whose priorities may vary with time. The highly stochastic working environment of an FMS (machine failures, changes in demand, uncertainty on operation times) also differentiates such systems from the classical job-shop. In addition, the role of the machining parameters in the highly automated FMS is much more important than in the classical case for determining production capacity since firing and hiring policies do not apply in an FMS. It is mentioned [6] that systems have been encountered with control software which was not only intuitively appealing but also used statistically optimal approaches which nevertheless produced less than optimal results.

The difficulty in making the most out of FMS may also be attributed to the lack of explicit consideration of the real 'cost drivers' in FMS when devising a production scheduling policy. The cost pattern of automated production has changed radically while costing methods have remained the same thus preventing the assessment of the real production costs, [7]. For example, direct labour has shrunk and allocating overheads to

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a product on the basis of the direct labour costs involved is meaningless. There is therefore a need to identify and control the true cost drivers of automated production which, if left uncontrollable, can wreak enormous damage to a company’s profitability. A way to control effectively 'cost drivers' is to develop a production scheduling policy for FMS which will try to minimise these costs. An example of a 'cost driver' in an FMS is the level of inventory and work in process or WIP." The money tied up in stocks and WIP is treated by accountants as assets to be listed on

the good side of the balance sheet instead of being included in the cost of production"[7]

As a result, inventory costs are often left out of consideration when scheduling production through the system and devising control strategies. Thus, inventory costs are virtually left uncontrollable and may increase inexcusably given FMS's potential for then- elimination. For example, high part arrival rates into an FMS aiming to satisfy the need for high utilisation of the FMS's expensive machinery may cause the parts to pile up in local buffers, waiting to be processed,and thus increaseWIP inventory.

The recognition of the importance of production scheduling and control for FMS has stimulated a lot of research in the area but the inherent difficulties of such a task prevented^ far the development of a suitable production scheduling policy except for trivial systems of two or three machines using rather limited information. Buzacott [8] concludes that:" while there is much fundamental research going on at present on discrete event systems, there still seems to be a large gap between the problems addressed in this research and the problems of controlling automated manufacturing systems."What is conceivable about FMS is that they "..should be controlled in a radically different way to conventional manufacturing systems." [ 8]

This thesis is a contribution in the search for a suitable production scheduling policy and control for FMS. The philosophy which underlines it, recognises the need of a different approach as compared to the traditional ways of scheduling manufacturing systems but it also takes the view that there :is a lot to learn and understand from the old methods of job shop scheduling since FMS can in part be considered as an automated job shop. It also recognises the strengths of simulation,given the system's complexity, and believes in the need for an awareness of the real cost drivers of FMS when devising scheduling and control policies.

A review of the current approaches for scheduling and control is the subject of the second chapter. Weaknesses and strengths of different methods are highlighted. The

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description of a simulation methodology for FMS is given in chapter 3. This methodology is developed to facilitate the modelling of FMS for the investigation of scheduling and control issues. In chapter 4, two real systems are simulated using this methodology to predict their performance, investigate their sensitivity on different parameter changes, improve existing control and scheduling rules and gain insight into the basic factors affecting performance. Chapter 5 proposes a hierarchical periodic production scheduling policy for FMS whose aim is to minimise production costs while meeting production requirements in the long run even when machines are unreliable. Parts in the periodic production set are scheduled with the help of an algorithm which is described in the following chapter, chapter 6. Scheduling algorithms and techniques for the job-shop are initially reviewed in this chapter and a hybrid scheduling and simulation algorithm is then suggested for use in FMS scheduling. The algorithm's efficiency is tested statistically for the simplified case of job shop for comparison purposes. Chapter 7 includes a few examples which cover a wide range of real problems and which are employed to illustrate the production scheduling policy's performance. In chapter 8 production costs of the periodic schedule are minimised.under the schedule constraints, by manipulating the machining parameters which so far have been treated as fixed. This problem is formulated as a non-linear optimisation problem under linear constraints. The thesis finishes with conclusive remarks and recommendations for further research in chapter 9.

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CHAPTER 2

LITERATURE REVIEW

2.1 FMS scheduling problem-main approaches

The production scheduling and control of FMS systems consists of a series of decisions that are related to the operation of FMS. Decisions such as which parts and when should be loaded into the system and what machine particular parts should visit next are taken by the FMS control computer. The objective of the FMS controller is to generate production schedules which satisfy demand requirements and exercise control over the system so that the output conforms to the schedule. The controller’s task is further complicated by random failures,of the machines/whose effects should be anticipated.

A lot of effort has, therefore, been put into the development of scheduling policies, models and algorithms which will allow the FMS controller to meet its tasks.

Since most decisions are implemented directly by the FMS control computer, an FMS is an ideal system for use of real time control algorithms for optimisation of performance. Thus, many real time scheduling methods have been suggested which allow for full flexibility since decisions are taken dynamically based on current system status[9, 10, 11]. However, these methods can not claim to have achieved an optimum, although they employ features which enhance the controller's ability to achieve good system performance even when the system operates in a highly stochastic environment. A highly stochastic environment is characterised by frequent changes in demand, machine failures and changing operating times,due to adaptively controlled machine tools. Examples of enhancing features include first, look ahead techiques using simulation to identify future problems which might arise if the present decision is implemented and second, use of complex objective functions for priority assignments, for example weighted sums of various decision factors [9].

As the problem of optimising FMS operation has proved to be extremely difficult it has been indicated that FMS control,which is a large decision making problem, should have a hierarchical multi-level structure. [12]. There is not,however, a universal agreement at present on what decisions should be assigned to each level and each suggestion has its own merits and disadvantages [8]. In general, each level of the hierarchy is characterised by the length of planning horizon and the kind of data required for the decision making process. Higher levels in the hierarchy have long horizons and use highly aggregated

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data while lower levels have shorter horizons and use more detailed information. Determination of the part mix and the production ratios required is a typical problem of the top level of the hierarchy whereas the control of the flow of parts through the system is a problem considered at the lowest level. Solutions found at each level operate as constraints or inputs to the scheduling and control problem of the lower level. These methods have been reported to produce optimal solutions for some of the scheduling subproblems but optimal solutions of subproblems are not necessarily optimal for the complete problem.

2.2 Hierarchical policies for FMS scheduling

Hierarchical scheduling and control policies were suggested by many researchers in an attempt to decompose the complex FMS scheduling and control problem and thus simplify its solution.The principal works on hierarchical policies is those of Hildebrant and Suri and of Kinemia and Gershwin [13, 14].

In Hildebrant's approach the top level of the hierarchy considers the machines and the pallets as limited resources and determines the aggregate flow of parts through machines during each failure period. The objective of this stage is to schedule parts so as to minimise total completion time. The objectives of the lower stage problems are based on the modelling assumptions made at higher level. At the second level the loading schedules of the parts are determined. This is done using a dynamic programming procedure. Level three problem is to resolve short-time conflicts for resources to minimise the average delay of tasks in operation such that the mean delay of parts,which are waiting for resources, is below certain values. This problem is difficult to solve and is suggested to be handled in real time,by using intuitive dispatching rules.

Hildebrant's approach ignores the transition period from steady state in one configuration to steady state in another configuration. In addition, flow rate decisions do not consider current inventory levels. In contrast, Kinemia and Gershwin's hierarchical policy decides flow rates on the basis of current inventory levels, as well as the current set of working machines,so that production requirements are met without the need for large finished and in process parts inventories. The second level calculates part routing to meet the production rate,dictated by the flow controlle^while minimising congestion and delays in the system and the lowest level uses the flow rates calculated by the route controller to determine time intervals between loading parts on each path. The system's capacity constraints are respected and therefore in-process inventory is fully controlled. The different stages of this approach have been enhanced and elaborated by other

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researchers [15,16]. It is worth noting that the lowest stage of the suggested policies is the part release problem. The sequencing of parts to machines is left to be handled in real time by the use of dispatching rules and output results are used to get feedback information to correct part flow. This situation may however create stability problems if the flow rate changes more frequently than parts are released into the system. [15].

2.3 Models used for FMS scheduling

Decision making procedures,like FMS scheduling and control require that at each decision time a rational choice of an alternative should be made. Effective choice of an alternative requires the existence of a model for predicting the consequences of each alternative.

Traditionally, experience, which can be regarded as an implicit model, is used, but although it may be a good guide for existing systems with adequate history, it has been found completely inadequate for systems like FMS for which no history is available [8,6]. The complex interaction of FMS's individual components,together with the multiple objectives imposed, means that the system as a whole behaves in a way which can not be predicted just from the performance of individual components and that its behaviour is often counter-intuitive [8].

The need to comprehend adequately the system behaviour for optimal design and control has led to the development of both analytical and simulation models.

2.4 Analytical models

The most widely used analytical models are networks of queues. Queueing network models of FMS have been predominantly applied for performance prediction and design optimisation,rather than operating problems associated with FMS. [17,18, 5]

Research in queueing networks was initiated by Jackson [19, 20] by defining general characteristics that permit a network to be decomposed and analysed as a set of independent service stations. He devised the steady-state joint probability distribution of the number of customers in queue at all stations and discussed applications of the model to represent practical aspects of job-shop operations as -control over the releases into the shop -control over the number of servers at each station -use of overtime and subcontracting

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-holding the number of customers in the system constant (closed network)The key assumptions of Jacksonian networks are :1. Service time at each station is exponentially distributed.2. Queue discipline is FCFS3. External arrivals at each station follow a Poisson distribution4. There are no restrictions on queue lengths5. Routing is probabilistic

Research in queueing networks was then extended and exploited by researchers primarily interested in modelling computer systems. In a unified interim, a general model for closed, open and mixed networks was developed and modelling capability was enriched by incorporating distinction of customer class to define arrival rates, service times and routings [21].

Closed and open queueing networks were then employed to model FMS; a closed queueing network model for FMS was developed by Solberg [22] who enriched the queueing network of Baskett [21] and applied an efficient computational procedure [23] to develop CAN-Q, a computer model of FMS. Based on a closed network of queues CAN-Q allows the user to specify:- number and type of machinesnumber of part types- sequence of operations and processing time per operation for each part type- operations like rework with some finite probability and,- transport time from one operation to the next.CAN-Q provides a number of performance measures that can be used to evaluate a system design like production rates, machine utilisation, average time spent by a part in the system and length of queues. Solberg's model was extended for limited in process inventory [24].

An open queueing network model for FMS was also developed [5]. In this model, workstations have a finite buffer capacity and in some cases a central buffer storage is considered. The model was used to compare the production capacity of alternative FMS designs and to determine the characteristics of parts that can be most advantageously processed on each FMS alternative configuration (i.e the number of types and the routing structure-flow shop or job shop).

Queueing network models for FMS have been criticised for the unrealistic assumptions on which they are based. Solberg does, however, cite empirical evidence that results

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using exponentially distributed service times are robust and Suri gives a more rigorous theoretical justification [25]. Queueing network models tend to appear in aggregate levels. Aggregation may be adequate for performance prediction and congestion identification but it is restricted in addressing scheduling and control problems,which are combinatorial in nature, and therefore require a model, which is able to represent time dependent routing and scheduling decisions. Nevertheless, queue models of FMS have been used as part of optimisation procedures for the solution of operational problems occuring at the intermediate levels of hierarchical policies which use aggregated data. These optimisation procedures were almost invariably standard non-linear and linear mathematical programming formulations.

2.5 Use of queueing models for solving intermediate range operational problems in an FMS

A typical intermediate range problem,which was addressed using queueing models of FMS, is the routing problem. If the production mix of parts is specified and the location at which all operations can be carried out is known, a sequence of machine visits or manufacturing paths may be chosen in advance,for the parts dispatched into the system. The a priori determination of part routes allows the problem of part scheduling to be reduced to a problem of lower dimension,since not all alternatives have to be considered. However, a routing decision taken at such an early stage imposes an artificial constraint on scheduling and reduces the flexibility of the manufacturing process [10]. Alternatively, decision rules could be devised to allow part routing to be determined in real time as the part makes its way through the system [26].

The part routing problem has been formulated in an FMS modelled by a network of queues using a network flow optimisation approach [27, 28]. Solution techniques were developed which generate routes for each part type and the rate at which parts should be dispatched on those routes. Others [29, 13], solved the part routing problem for an FMS with failure prone machines. The part routes were found for each failure condition of the system by means of a non-linear optimisation program, with an embedded closed network queueing method called the Mean Value Analysis or MV A. MV A is a recent technique from the network of queues theory for efficiently evaluating the formulas of the performance measures of a system,modelled as a closed queueing network.

Another intermediate range problem, which was addressed using queueing models of FMS,is the operation or tool assignment problem. The operational configuration of the

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machines may be determined in advance in a way consistent with the possibilities and constraints of the machines. This problem arises from the tool magazine capacity constraints or the tool availability constraints and implies that the system has no facility for tool transfer or tool transfer is not desired to be considered. Tool allocation to machines prespecifies the number of different types of operations,which may be performed by the machine;which may be a subset of the machine's possible range of operations given the corresponding tooling. Alternatively, operation assignments to machines prespecify each machine's tooling.given that tool constraints are respected. When a tool transfer facility is present, operation assignment could be left to be decided dynamically,constrained only by the machine's possibilities .

This problem has been formulated as a well behaved non linear optimisation problem and its solution approximated for most practical situations by the solution of a linear-programming problem [30]. Stecke and Solberg term this the loading problem [31]. Using a closed queueing network model,they address the problem of optimally allocating workload in a system of servers and found that balanced workloads are only optimal in a system of groups of servers of equal size. When the groups of servers are of unequal size optimality is achieved with unbalanced workload.

2.6 The Markovian approach to FMS control

There have been attempts at developing optimal scheduling and control rules for FMS using optimal or good approximate techniques on Markov chains or Markov renewal processes [26, 32]. The objective is to devise improved control and scheduling rules which will be functions of the system's state. The system's control is modelled as a Markovian decision process or MDP. The state of the MDP is observed at instants when decisions must be made concerning loading, routing , scheduling etc, and the state description includes information about every resource in the system. The decision space consists of all possible actions which could be taken. The resulting model is generally an extremely large but finite MDP.

Due to the size of the problem, the analysis has so far been restricted to systems with only two to three linked machines [26]. Alternatively, a two stage approach is suggested; first, the MDP is solved off-line and the resulting policy is scrutinised, simplified and tabulated in a convenient way and second, the real time, on-line computer which controls the process employs a look up table to make its decisions after using the actual system state as input to the table [32]. However, it is admitted that until an efficient solution method is found for the MDP and especially for the simplification of the state space via

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surrogate variables to replace detailed workload description together with efficient storage and look up procedures for the table, this approach is not practical.

2.7 Integer programming formulations of intermediate range operatingproblems in an FMS

The FMS scheduling problem may be formulated as an integer programming problem of immense dimension but, due to the lack of efficient algorithms .dealing with the solution of integer programming formulations this approach has not been adopted by FMS analysts.(see Chapter 8). Intermediate range operating problems have however been tackled using this approach. Specifically, a mixed -integer non-linear programming approach of the operation assignment problem has been developed [33]. Tool magazine capacities which limit the machine flexibility are incorporated as problem constraints and solution algorithms exploit special features of the problem. The non-linearity in this formulation is due to the design of tools and tool magazines. But in most of the tool magazines available, the sequence of tools does not influence their capacity and this allows a linear mixed integer formulation of the same problem [34].

2.8 The scheduling approach

The combinatorial approach to FMS scheduling and control problem is more of a working tool than the queueing network approach since scheduling decisions are combinatorial in nature. Combinatorial techniques have been used extensively to solve the flow-shop and job-shop scheduling problem. However, direct application of the classical available job-shop scheduling techniques for FMS is not possible. In fact:

1) It is known that computational requirements involved in the solution of the job-shop scheduling problem grow rapidly with the number of jobs and machines. Thus, exact solutions cannot be obtained for FMS scheduling problems of indust rial proportions (the size of typical FMS is 5-10 machines and 10-20 part types and there are many possible alternative routings and alternate machines) since they would require an unacceptable amount of computation. The use of approximation algorithms is one solution to this problem. [35].

2) The added complication of machine failures means that it does not make sense to schedule parts on the machines beyond a very short horizon. For scheduling in the presence of failures, the only computationally acceptable approach offered by conventional theory is to use various heuristics or decision rules [36]. A comprehensive

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trial of these rules for FMS was conducted by Stecke and Solberg [37].

3) FMS scheduling is a problem involving not only the scheduling of parts,as it was the case in the traditional manufacturing system but also the scheduling of the material handling system, the fixtures and the tools. Such a multi-criteria problem is difficult to solve. A solution could be to consider each of the subproblems separately, even though each of them can be very difficult to solve in practice. Then, the question arises as to which problem should be solved first, since the solution to this problem will have an impact on the solutions to the other three. The part scheduling problem has been so far of the greatest interest, mainly because of the tendency for high machine utilisation. As machine tools are the major cost-in many FMS, the cost of machine tools is about 65% of the overall system- such priority seems to be the right decision. However, as the degree of automation and computerisation increases machine tool cost percentage may become smaller and this in turn will change the priorities of scheduling.

The main work on FMS,using a scheduling approach, was proposed by Hitz for scheduling flexible transfer lines [38, 39]. The periodic scheduling algorithm suggested, is a heuristic combinatorial technique for computing optimal loading sequences in a deterministic context so that:1. the output of finished parts is as large as possible,.subject to the constraint that the different types of parts are produced in the prescribed ratios2. the steady state of maximum production in the various ratios is reached as quickly as possible after start up or a momentary disturbance.

The implication of the periodical loading is that it considers part sequencing for the period time rather than for the whole of the planning horizon. The problem's size is therefore reduced. Hitz also showed that further substantial computational savings can be obtained by replacing the minimisation of makespan with the requirement that the bottleneck machines be fully occupied once they have started working. However,before the periodic schedule can be calculated the routing of parts must be established.

Research in connection with periodic releasing, for the FMS case, concentrated on the analysis of the system's dynamic behaviour as well as the determination of procedures for a good periodic releasing strategy [40, 41, 42]. These works use path algebra to prove that the steady state of the system is periodic.

A combinatorial approach to scheduling ma^regardless its problems, be useful both in the insight it provides and the determination of powerful non-obvious decision rules for

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giving priorities to different jobs.

2.9 Simulation models

Simulation models construct a detail representation of the system operation, generally using simulation languages as GPSS, SIMSCRIPT, SLAM or ECSL. They enable detailed representation of the part characteristics, machine behaviour, workflow, job routing and complex sequencing rules. General simulation models for FMS have been developed by many researchers and they seem to differ in the level of detail incorporated [ 44, 45, 46, 47].

Simulation models have been used both for the design and operational problems associated with a particular FMS [48, 49, 50, 51, 52, 53, 54]. For example, simulation was applied to evaluate the best of several types of material handling systems [50], to compare alternative loading strategies [54], or to analyse a real system [51]. In all simulation studies of real systems, simulation has been proved invaluable in all stages to both analysts and managers,particularly after simulation packages' user friendliness has increased with graphic and interaction facilities [6, 8]. However, simulation models have been criticised for requiring substantial time to develop and adequately validate, and for being expensive and time lengthy to run [8]. The major disadvantage is that in all to many cases the attempt to use simulation to find an optimal solution to a problem rapidly degenerates into a trial and error process. Due to the time and expense involved in a run, analysts are deterred from trying a wide range of parameter values. Techniques for overcoming this disadvantage, i.e for making optimisation and simulation more compatible, have however been developed:

1) Automatic optimum seeking in computer simulation, using search techniques, is known from traditional simulation studies. The response surface methodology is a search technique which was used for constrained and unconstrained optimal parameter seeking;in conjuction with deterministic and Monte - Carlo simulation [55, 56, 57]. Parameters which are optimised using this approach should be quantitative and continuous,or at least approximately so. Automatic optimum seeking, to the author's knowledge, has not been incorporated in the commercially available packages for FMS simulation and there are no reports of FMS simulation studies using this approach for parameter optimisation. The simple, fast and effective use of analytical models in the early FMS design problems is one reason for such a lack. On the other hand, FMS scheduling is a discrete optimisation problem which involves non-quantifiable factors; the above stated method could not therefore be applied for this problem.

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2) Another methodology,which tends to capture the precision of the simulation models and the power of the theoretical analysis, is perturbation analysis. A simple simulation run is made and, based on the realisation of the system in this run, analysis is performed to derive expressions which relate a performance measure with various system parameters. These relationships are developed as a result of a perturbation in the value of a parameter such as a buffer capacity,in the observed realisation. Thus, it is possible to predict the behaviour of the system for various realisations of the system without having to do additional runs [58, 59, 60]. Perturbation analysis may deal successfully with parameter optimisation models,but these are mainly design problems which assume fixed operating and scheduling procedures.

3) A recent attempt to combine simulation and analytic approaches whithin the same model resulted in hybrid models. The framework of this modelling approach is an analytic model but for some panywhere analysis is difficult or impossibly a simulation submodel is constructed. From the results of the simulation model an equivalent component for an analytical model is developed. For example, FMS is modelled as a single server queue with service rate dependent on the number of jobs in the system. Using a single simulation run and discrete state level crossing analysis, service rates are obtained as a function of the number of jobs in the system [61]. Thus, mean inventory levels and production capacity may be efficiently evaluated for varying arrival rates and varying common buffer sizes respectively. Hybrid models seem to be the right approach to FMS modelling for systems for which analytic or acceptable approximate results are not available. However, since they are basically analytical models, they tend to appear at aggregate levels and they therefore tend to address mainly design rather than operating problems.

In the work presented in this thesis, an attempt is made to combine simulation and scheduling approaches. The approach involves the periodic production of sets of parts which satisfy the product mix. A schedule is calculated for the set of parts, under the current demand and operational state,and is then periodically repeated. A heuristic algorithm is used to calculate this schedule,with the objective to minimise its makespan. The algorithm incorporates the general simulation model in order to produce feasible and realistic schedules, given the FMS complexity. At each decision stage it selects the alternative which gives the best performance estimate. Estimates are derived by looking ahead using the simulation model and with future conflicts overcome using a dispatching rule. Each performance estimate comes from a complete feasible schedule. Periodic production is justified not only in terms of external requirements (e.g adjoining assembly operations) but also as this thesis argues by production costs. The size of this set, the

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minimal part set (MPS), is determined by minimising the cost per part. The production cost associated with the MPS is further minimised by manipulating machining speeds without affecting schedule performance.

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CHAPTER 3

A SIMULATION FOR FMSs

3.1 Introduction

Simulation software can largely be classified into two types :i) general purpose packages; like simulation languages. These usually require a reasonable amount of technical knowledge in order to be used.ii) Models of systems. These are usually applicable to a limited area of application and so are often short-lived. However, they require little special knowledge to be used.

Traditionally, there has always been a wealth of simulation software of type (i). General simulation languages can, and have,been used successfully to model FMS in order to investigate some of the design and operational problems of FMS. However, they have certain characteristics which make them difficult to use for the study of the scheduling and control problems of FMSs. Control issues are mainly ’hard wired' into most simulations, resulting in complicated and rigid models. For instance:1) Two machines may have linked activity cycles to ensure the availability of the second when the first is unloaded. In general, activity cycles for machines and handling devices can have complex interactions in a FMS because of the high level of automation and system flexibility.2) Different part types have different routings,which are either built in the model,or they are included in a high degree of conditional branching in the part's activity cycle. Conditional branching occurs also in order to accommodate alternative routings of the same type of part.3) Scheduling/priority rules refer to a choice of different rules for queue discipline, provided by the modeler, (including a user defined rule). However, scheduling/priority rules are likely to require the inclusion of complicated attributes for the parts. It is therefore necessary to consider the status of the complete system to make the best scheduling decision.Therefore, it is necessary to almost rebuild the model when we want to experiment with complex control strategies. The more complex is the control, the more complicated the model becomes and the greater the changes needed for testing different control strategies.

As microprocessors increased in power general simulation packages were transferred to

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them and were being joined by a new generation of software. This new generation generally included the following features:1) A facility for the user to interact with the system while it is running.2) Dynamic graphic output, a feature which mainly accounts for the increased attraction of simulation models to management3) Computer aided program generators which allow models to be formulated inhigh-level languages (DRAFT,' etc) [62].Valuable as these features may be for management, it is debatable whether or not they enhance the ability of the analyst to investigate effectively scheduling and control issues of FMS.

There is a need for a model which allows a more detailed and easily adaptable approach to changes in the control strategy. The need for intermediate generality simulation models for FMS has been identified by Peck [63].

In this chapter a simulation model for FMS is described. It has been developed so that the following criteria are met:1) a variety of systems can be modelled each with sufficient detail to allow a thorough evaluation of possible control strategies.2) Control decisions can be separated from the simulation module and collected in a separate module,so that the problem of rebuilding the model,each time a change in the control is required, is avoided.3) Routing of parts is 'attribute value' controlled,and not 'hard-wiredîn order to avoid the construction of complicated models and reduce the amount of changes induced by new sets of part types.

In the first section of this chapter the concepts of simulation, its components and logical structures are reviewed. Then, a new methodology is developed for the construction of an FMS simulation model in order to meet the criteria previously set. The main original feature in this model is the inclusion of all decisions into a separate software module called the control module,where all control decisions are made.

3.2 Simulation model ; its components and logical structures.

Simulation modelDefinition : A simulation model is a simplified representation of a real life system which mimics the operation of the system over a period of time.

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The components of a simulation model are its entities, queues and activities.

EntitiesDefinition : Entity is this component of a model which can be imagined to retain its identity through time.Entities are the resources in a model, such as a machine, a part, a tool or an operator. Entities are either idle^when they wait in queuesp r active^when they engage alone or with other entities in time consuming activities. Each entity may have a number of members. For example, the entity machine may have as many members as there are machines in a system. Entities may have one or more attributes which are used to describe them in greater detail.

AttributesDefinition : An attribute of an entity is any property or characteristic of the entity which influences or measures its behaviour.Attributes may be fixed or variable. For example, an attribute may indicate the type of a part in which case it is fixed since it does not change through the part's life cycle, or it may indicate the next machine on which the part is to be processed; it may be used to record the time an entity spends in some part of its life cycle or it may be a measure of a part's priority. In fact, any property that can be expressed as a positive integer, can be defined as an attribute. The number of attributes we need to include in the model will depend upon the extent to which they are required to describe the behaviour of the entity.

QueuesDefinition : A queue is an ordered list of entities considered in a model which wait to start an activity.Queues can be either real, when they represent a physical location for waiting or notional. when this is not the case. The local buffer in front of a machine is an example of a real queue, whereas the queue of parts,waiting to be transported, is a notional queue since parts may have diverse physical location.

ActivitiesDefinition : Activity is an action that takes time and involves one or more entities.Activity time may be constant or variable: it may be known or estimated. It is calculated beforehand and once the activity has started it will continue for this duration,unless the logic of the model allows it to be interrupted. For example, a machine and a part may engage for a period of time in a machining activity.

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State of the modelThe current state of the model is described by the position of each entity in a queue or activity.

Discrete event simulation concerns the modelling systems in which state changes occur only at finite number of instants in time.These instants in time are the ones at which an event occurs.

An event is a change in the state of the model which occurs at an instant of time.For example, the end of an activity is an event. The beginning of an activity is a conditional event because its occurrence depends on one or more conditions being true, for example, on the availability of the entities which are involved in the activity in question.

A process is a sequence of events ordered on time.

Time advance in simulation modelsIn all modem discrete simulation models time is advanced using the next event approach. Since the state of the entities remain constant between events there is no need to account for this in-activity time in our modelling. With the next event time advance approach, the simulation clock is initialised to zero and the times of occurrence of future events are determined.The simulation clock is then advanced to the time of occurrence of the most imminent (first) of these events, at which point the state of the system is updated and future events are determined. It should be noted that successive jumps of the simulation clock are generally variable or unequal in size. The next event approach to time advance is not the only method available for advancing time and processing events. However, it is the most commonly used and a standard feature in most simulation programming languages.

Alternative discrete event modelling techniquesThe concepts of event, process and activity give rise to three alternative ways of building discrete event models.

The event scheduling approach emphasises a detailed description of the steps that occur when an individual event takes place. Each type of event naturally has a distinct set of steps associated with it. In the event scheduling approach, the beginning and end of each activity are events and consequently they are scheduled for execution. As the number of

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activities grew so do the number of scheduled events,and the amount of computer time spent in creating records of these events, filing them in the list of scheduled events and destroying them once they have been executed.

The activity scanning approach emphasises a review of all activities whenever time is advanced to the next event in order to determine which can be begun or ended. The activity scanning approach substitutes less time consuming logical checking in the model at each time advance^for the required event scheduling steps.

Last, the process interaction approach emphasises the progress of an entity through the system from its arrival event to its departure event Conceptually, it combines features of the event scheduling approach with the activity scanning approach.

The development of these approaches is related to the development of discrete event simulation programming languages. In particular, SIMSCRIPT and GASP use the event scheduling approach, GPSS and SIMULA the process interaction approach and CSL the activity scanning approach. A description illustration and comparison between these programming languages can be found in [64].

SIMON and the three- phase logical structureThe simulation model described in this chapter is programmed in FORTRAN and utilises SIMON routines. SIMON is a library of subroutines in FORTRAN written to aid programming of simulation models. SIMON in ALGOL was written by Hills and the FORTRAN version is due to Hills and Mathewson. Unlike SIMSCRIPT or SIMULA, SIMON is not bound to any one logical structure. It can be used in an event mode but historically it has been associated with the three phase structure which is an activity based approach. The principal advantage of using this set of subroutines in FORTRAN rather than a simulation language is the ability to tailor the simulation to any machine, its adaptability to complex decision processes and its independence from any specific lo g ic a l structuoTe..

The FMS simulation model described here uses the three phase structure largely because of the robustness of this formulation [65]. The three phase logic is given in figure 3.1.

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Initialise the model- set up the physical model- schedule initial events

A- phase- select the next event- advance the clock- if simulation has ended go to output

B-nhase-complete any action arising solely from the event just identified

C-phase- check each activity which has a state dependent start and test whether it can be scheduled.

Return to the A-phase Output

-terminate the simulation-prepare summary statistics and tables-print out the final analysis

Figure 3.1 : The three phase logic

3.3 A new methodology for building FMS simulation models

A new methodology is presented here for the construction of FMS simulation models. It is used for the development of a general FMS simulation model, named FMSIM, which was first used to experiment with different scheduling rules and later it constituted the basis on which FMS scheduling algorithms were developed. The same methodology was used for the detailed simulation of two industrial flexible manufacturing systems. A description of those systems and models together with the experimental results on different issues are described in the next chapter.

3.3.1 The FMS model

The FMS is modelled as a system consisting of:1) M machines, each capable of performing different kinds of operations2) A common storage of given capacity which also serves as loading/ unloading station.3) Buffer storages for each machine of given capacity4) A material transport system with a given number of transport devices

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5) A central storage for tools of given capacity6) Tool magazines for each machine of given capacity7) A tool transport system with a given number of devices

Further assumptions concerning the operation of the model are:1) There will be constraints reflecting the inability to process a part on two machines simultaneously or process two parts on the same machine simultaneously.2) There can be alternative machine tools which can perform the required operations for each processing stage of the parts.3) Each part is processed in a specific operational sequence based on its own order of processing stages.4) Operation durations and transport times are given or drawn randomly from a known distribution.

3.3.2 Entity cycle diagrams- A method for building simulation models

A methodology using entity cycle diagrams has been proposed for building a simulation model by Poole [65]. Entity cycle diagrams describe the path followed by each entity as a series of alternate queues and activities.Then, life cycles are joined along those parts where interaction between entities takes place. This method has been exploited in several software packages and program generators [ HOCUS , DRAFT ]. It presents many advantages in building models where interactions between entities are relatively simple. However, when complex interactions occur, complicated representations of the total system’s activity cycle is unavoidable. These problems have been identified in the modelling of a robot served cell [66]. They were associated with the different type of machines and the different configuration of service, and demonstrated the need to include a plethora of interlinks to cover all eventualities. The new methodology proposed here is such that problems of this sort are avoided.

3.3.3 Activity centered diagrams - The new building blocks for FMS simulation

The first step in building a model is to list the activities involved and describe the entities which need to be available before the activity can start. We need to list the entities required in each activity together with where they come from (source queues) and where they go to (destination queues). For example, let us describe the entities needed for a machining activity to take place.

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Entity-part-machine-tool

Source queue input buffer wait to run tool magazine

Destination queue output buffer idletool magazine

A useful graphical representation of this logic can be achieved with the use of diagrams.In these diagrams an activity is represented by a □ and a queue by a O . Arrows indicate the direction of the flow of entities which are shown into parenthesis.The previous example for the machining activity can be presented graphically as in figure3.2.

Figure 3.2 : Diagram of the machining activity in a FMS

In a similar way,we can represent graphically the rest of the activities in a FMS. Figure 33, gives the diagram of the tool transport activity and Figure 3.4 the part transport activity.

Figure 3.3 : Diagram of the tool transport activity in a FMS

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Figure 3.4 : Diagram of the part transport activity in a FMS

A new important feature has been introduced in the diagram of figure 3.4. This is that a part can be transported to either the input buffer of a machine for further processing, or to the central storage for waiting until a machine is freed or for unloading i-f it isfinished. Note that the unloading station is identified in this model with the central storage. What this means is that alternative destination queues might exist. Alternative destination is necessary when the length of the queue is restricted due to shortage of space or some other constraint. For example, a machine's input buffer may have limited capacity for holding parts.

Boundaries of model- generatorsWhen building a model we have to decide what we have to include and what to leave out. For example, for an FMS model we might not want to include details of what happens to the part before it arrives to the system for processing or after it leaves. The only relevant information is the time a part arrives and certain details concerning the part's description, its type for instance. Then all we have to model is the arrival pattern of parts coming from the outside world and we have to ensure that they arrive in a pattern similar to the real world. This is the boundary of our model and can be represented by a queue (outside world) and an activity which generates parts in a representative manner. The use of a queue and an activity to describe the interface of a model with the outside world is quite common in simulation; the part arrival activity is described graphically in figure 3.5.

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Figure 3.5 : Diagram of the part arrival activity in a FMS

The entity 'input' is a fictitious entity designed to control the part arrivals. A part arrival is scheduled after the end of the previous part arrival. Therefore, the time for the generating activity is the time between arrivals. When parts leave the system, they are put into the outside world to be generated again. Recycling entities is common in simulation and is used to keep the total number of entities in a model to a minimum.

Duplicated activitiesIn many models we find that instead of a single resource, there are several similar resources which are involved in similar activities. For example, in a system comprising two machines, each machine runs independently of the other but it is involved in a machining activity which can be described by the same diagram given in figure 3.2. Therefore, it does not have to be uniquely identified. The model has to be extended to allow for all machines to run simultaneously. A simple representation of this can be achieved by extending the diagram of figure 3.2 to include the other machine as it is shown in figure 3.6.

Figure 3.6 : Diagram of duplicated machining activity

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Therefore, a machining activity can start if machine 1 waits to run and a part is in its input buffer 1, and/or if machine 2 waits to run and a part is in its input buffer. When machining ends, the part joins the output buffer of machine 1 or 2 depending on the alternate condition,which initiated its StarBand the machine joins the queue of idle machines to wait for allocation. In the computer model, duplicated activities are programmed in a do loop,which checks the starting conditions of the activity as many times as there are machines in the system.

Attribute value conditional start of activitiesSo far we dealt with models involving activities whose conditional start depends on the presence of entities involved, in the source queues. We also examined the case of alternate presence of entities (duplicated activities). However, if we want to model complex systems in a flexible and simple way, we have to consider the case where the start of an activity is conditional upon the value of an attribute. To understand the degree of flexibility that this option involves, let us consider the routing problem for the case of two part types which are to be concurrently produced by the system. When a part is entering the system, it may need a turning operation if it is of type A, or a drilling operation if it is of type B. If the part has an attribute,whose value indicates the next operation to be performed on it, it can join the queue for turning or drilling accordingly. By testing the value of this attribute we achieve the desired routing. The alternative is to model the entity life-cycle of this part, i.e. to 'hard wire' its routing by linking the alternate activities and queues it passes until it is completed. Different part types have different routings ; hence : first, the model would be very complicated something we want to avoid and, second, each time a different range of parts is considered we would have to rebuild the model or include a high degree of branching in the part's cycle which would again result in a complicated model. Routing by attribute value solves the above problem and results in a simple and flexible model. Values held as attributes can also be used to decide the destination queue in the case of alternative destinations and they can also hold the activity duration.

3.3.4 Linking the activity diagrams to construct the final model

After having described the activities, their source queues or attribute value conditional starts and their destination queues,the next step in building a model is to link the activity diagrams together in order to describe the operation of the total system. The linking of activities will result in a description of the flow of entities through queues and activitites . To a great extent the flow of entities through a system is a control issue . That is, when

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and which entity, joins the source queues of activity blocks so that activities may start is a control decision. For example,1) when and which machine from those idle is allocated for the processing of a part and thus joins the queue 'wait to run' ?2) when and which machine joins the queue ’ tool request' so that a tool is transported to it?3) when and which part waiting at the central storage or an output buffer of a machine, joins the queue 'wait for transfer' in order to be transported to a machine for further processing ?

The concept of control moduleIn order to allow for full flexibility in the system's control ( entity flow), control decisions, like the above, are taken in a separate module which we call the CONTROL module . It is useful to think of CONTROL as one of the system's activities. Then questions 1 and 2 above concerning the machine flow are answered by the 'control activity'. The machine flow could then be described by the following diagram (figure 3.7).

Figure 3.7: The flow of the machine entity in a FMS

Question 3#conceming the part flow, is also decided by the CONTROL and is described by the following diagram (Figure 3.8):

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Figure 3.8 : The flow of part entity in a FMS

Having described how control operates in the flow of parts and machines, the last question we have to answer is the time instants CONTROL is required; i.e, using activity terms, what are its starting conditions.

According to Tocher [67], the necessary conditions for a control problem to exist are:1) There must be a specified set of times at which a choice of action is possible.2) At each such time, there must be a specified set of actions from which to choose.3) There must be a criterion or objective on which the choice of action is based.

For the FMS case, there are clearly defined action times corresponding to the instants of time a machine previously involved in a task is released and is ready to start a new task or the time a part has finished an operation and is ready to be sent to a work station for further processing.

The choice of action at these times is limited to the choice of a part to be processed by the released machine, including the case where no part is chosen (then machine stays idle), or the choice of the next machine to process the part on a further operation, including the central storage for waiting. The choice space for these two modes is the same but one should expect different approaches for solving the choice problem by each mode. The first mode, which is a machine rather than a part oriented approach, is used in this model. The three phase flow chart of the FMS simulation model is described in Figure 3.9. The instants of time the CONTROL module is called are clearly shown.

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Figure 3.9 : Flow chart of the FMS simulation model

3.3.5 The structure of the control module

The main task of the control module refers to the choice of a part for allocation to an idle machine . This is a scheduling decision. In order for a good scheduling decision to be made, knowledge of the status of the complete system is required. This is the first task of the control module. It is realised by creating an 'image' of the system's current state. This is achieved by access big the dynamic file, which contains information on the

L.

46

machines which are idle at that instant of time and on the parts which are available, waiting at different system locations for further processing.

Then, for each idle machine, a set is created consisting of those parts from those available whose next operation can be performed on that machine. This is called the choice set. It is worth noting that in general a part may belong to more than one set (alternative routes may be possible).

A part from each choice set is then selected for allocation to each machine, according to decision rule. Decision rules may vary from simple priority rules to complicated ones, involving waited sums of attribute values of parts or machines. For example, a rule may use the part's lateness or tardiness, the number of remaining operations for this part or the amount of remaining processing time.

After part allocation is completed, a conflict may occur whereby the same part is allocated to more than one machine. In that case, a conflict set is created comprising all machines which have been allocated the same part The conflict is resolved by applying a decision rule, which may again be simple or complicated,ranging from random rules to sophisticated rules using machine part attributes. For example, the conflict may be resolved fo r:1) the machine with the minimum workload in an attempt to balance the workload2) the machine whose distance from the part's present location is minimum3) the machine for which all tools necessary for processing the part are available in its magazine4) the machine which would process the part faster than the others if it happens that we have different durations in different machines although they are all able to process the same operation5) the machine with unit choice set so that it is not left idle.

After the conflict is resolved, for the remaining machines in the conflict set (all machines except the machine in favour of which the conflict has been resolved) a new part has to be allocated. This part is chosen from the machine's choice set and it is the next part, after the part which created the conflict in the parts list ordered according to the selection priority rule.

Finally, the output of the CONTROL module is prepared. This is the part of the module which interfaces with the simulator and guarantees the flow of parts to the source queues

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of activities. Allocated parts are put in the ' wait for transfer ' queue in order to be transported to their machine destination. Their destination, which is the machine they have been allocated to, is an attribute whose value is here updated. This attribute value is used in the end of the transport activity to chose the alternative destination queue (alternative input buffer) which the part will have to join. Tool availability is checked and then machines join the 'wait to run' queues or the 'tool request' queue, ready to process the allocated part as soon as it arrives to the input buffer or waiting for the necessary tool to arrive before start processing. Machines without a part allocated to them stay in the idle queue to wait until the next allocation time. Last, parts 'waiting for transfer', are ordered according to transport priorities. The simulation is ready to run until the next action time where a new set of decisions will be required.

Since the main task of the CONTROL module is allocation of parts to machines, from now on we refer to action times as allocation times.

A diagram of the CONTROL module is presented next where all its tasks are summarised. First, the notation used in the diagram is given.

Let us denote by:

Mj : the set of idle machinesA the set of available partsm a the set of allocated machinesCP1 the choice set of machine mpm : the part allocated to machine mPa ■ the set of allocated partsF : the set of machines in conflict over a part*m : the machine for which the conflict is resolvedix(.) :: a priority index

CONTROL inout

1. Create the system image- Create Mr and A

2. If A = 0 Exit

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Allocation- Create Cm- V p g Cm calculate ix(p)- Find pm = argmax {ix(p)} & put elements of Cm in order based on ix- If two or more ix(p) have the same value the conflict is resolved in random or by using another priority rule.

4. If there is at least one part pm allocated to more than one machines then- Create F- V m g F calculate ix(m)-Find m = argmax(ix(m)}- Vm g F - { m*} set Cm = Cm - {pm}-Find pm g Cm with max{ix(p)}- go to step 4

end if

3.

V m g Mj do

5. - Create the set of allocated machines - Create the set of allocated parts

CONTROL output6. VpmG P£ -Update part destination

do - if part's location * part's destination then put pm in ' waitfor transfer'

7. Assign transport priorities to all parts waiting in the ' wait for transfer queue'

8. remove m from queue 'idle'Vm g M^ do

- check tool availabilityif tools available then

put m in queue ' wait to run' else

put m in queue 'tool request' endif

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9. exit the CONTROL module

3.3.6 Measurements in a simulation model

In most models we want to know how long entities have spent in certain parts of the system. These measurements of elapsed time may be used either as a base for subsequent control decisions or they may serve as indicators of the system performance and,as such, they are reported in the end of simulation. For example, the time a part has spent waiting after its entrance in the system may be used as a priority rule for its allocation or the time a machine has spent processing parts is a measure of its utilisation.

Elapsed times are measured with clocks. Each entity, we are interested to measure, has a clock associated with it. The clock is switched on and off at the start or the end of any activity. The difference in time the clock is switched off from the time is switched on gives the duration the entity has spent in that part of the system. This duration held in an attribute as a positive number can be manipulated or reported whenever it is necessary according to our needs. For example, in order to measure the time a machine works we can switch on the machine’s clock when activity machining starts ( note that this is not done when conditions for the start of machining are checked but when the activity has been scheduled) and switch it off at the end of this activity. After the clock has been switched off, the duration of the machine’s working time is found and may be added to the contents of the machine's attribute,which holds the total working time of the machine. In the end of the simulation run, the working time over the total simulation time gives the machine's utilisation.

3.4 Summary of FMS simulation model's components

The principles of the new method just described are open ended, in the sense that activities, queues, entities and attributes can be added as and when required; a summary of the most common model components which have been used for the general FMS simulation are summarised in table 3.1.

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ENTITY QUEUES A c n v m E S ATTRIBUTES

part worldcentral storage input buffer output buffer wait for transfer conflict over a m achine

part transport

m achining

part typelocationdestinationnum ber o f com pleted operations

next operation part status

m achine idlew ait for allocation tool request w ait to run downcon flict over a part

m achining tool transfer repair

m achine type

tool tool store tool m agazine

m achining tool transfer

part transfer device

free part transfer

tool transfer device

free tool transfer

Table 3.1: The main entities, queues, activities and attributes for a general FMS simulation model

3.5 FMSIM- A computer simulation program for modelling FMSs for experimentation on general scheduling and control issues

The model of a general FMS described in the previous sections was programmed in the computer using FORTRAN and utilising SIMON routines.

SIMON is a set of subroutines which can be called to perform tasks like:1) Initialise a common area to store the dynamic system information and define the set

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which will be used to store the scheduled events2) Create a queue, reference its members or manipulate them. Creation of a queue refers to the addressing of the queue storage area. The head, the tail and the size of the queue may be referenced. Queue members may be displaced or removed and new members may be added in the queue.3) Create entities. This is done by addressing the dynamic storage area in which information relevant to the entity is found. This information refers to the current event time the entity is associated with and information bound to it.4) Recover the event time and the bound data stored with an entity.5) Schedule an entity to perform an activity for a prespecified time duration alone or with other entities. The clock time of the end of the activity is recorded in the set of future events and also in the entity's dynamic information storage area together with information on the bound entity.6) Collect information in histograms and produce statistics on different entity measures.7) Record time series information (necessary to study the system's dynamic output), plots and analyses them using autoregression analysis, regenerative processes and analysis of batch means.8) Monitor the model activity.

The program requires the following as its input data:1) Number of different part types2) Total number of parts to be produced3) Number of machines in the system4) Number of part transport devices & tool transport devices5) Operation sequences of part types6) Operation durations7) Operations which each machine is able to perform8) Transport times between machines and machines and central storage

The program's output reports include:1) analytical event by event report of the system's operation2) Summary reports of system performance including

machine utilisation transport devices utilisation production rate lead time statistics

The analytical report can be suppressed or activated with the help of a switch which may

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be switched on and off at any point in the system or conditionally after a the elapse of a time period. This feature was very helpful in the debugging stage of the program.

The program provides for controlled measures of machine utilisation in order to avoid distortion of the measurement figures by transient phenomena (for example, until the shop is fully loaded). Controlled measurement is realised with the use of switches which start and stop measurements being taken after a number of parts has been produced or at particular time instants.

This program has been first validated with a real system; it was then used to experiment with different scheduling/priority rules [68], and later it was used as an essential part of a simulation based algorithm which was developed to solve the complete FMS scheduling problem (chapter 6).

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CHAPTER 4

SIMULATION STUDY OF INDUSTRIAL FMSs

4.1 Introduction

The need to validate the simulation described in the previous chapter against a real system led to a collaboration with Lister-Petter Ltd. Lister-Petter Ltd., a member of the Hawker Siddeley group, has been involved in new technology investments since 1982 to help with the production of its air and water cooled diesel engines. Lister was awarded a grant by the DTI and in May 1983 the capital equipment for the first flexible manufacturing system, known as FMS1, was ordered with the delivery and installation achieved by May 1984. In August '85 initial production tests began. The next flexible manufacturing system, known as FMS2, is a totally different system and by the time this thesis is written the first production runs will have begun. The two simulation exercises presented in this chapter involve these systems. FMS1 was used to validate the FMSEM simulation program and simulation of FMS2 which is a more complicated system, showed the advantages of using the activity centered methodology and the control module approach when simulating flexible manufacturing systems.

4.2. Simulation study of FMS1 for the manufacture of flywheels

4.2.1. Cell description

The FMS1 cell is a flexible manufacturing system installed at one of the company's factories to help with the production of flywheels, bearing housings and gear wheel blanks. Before the installation of the FMS1 all the production of these parts was subcontracted out. The system is able to manufacture all these parts from castings to finished components and consists of two robots, five machine tools and a vision and a computer system.

The cell consists of the following equipment:1. Two Cincinnati Milacron 776 electric robots with 68 kg. load carrying capacity at the wrist plate.2. One SMT Swedtum 12, 4-axis lathe3. One Tesa Metrology gauge4. A RS4/1000 Rausch 2-speed vertical pull broach5. A Beaver VC35 machining centre with a specially fitted Fanuc 6MB control

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6. A Schneck 200 WBAE used for flywheel balancing and7. A BRSL vision system consisting of 3 separate stationsThe managment computer is the IBM Series 1 and has been programmed to monitor and control all the activities in the FMS and is linked to all the machine tools and robots in the cell using digital signals and RS232 links.

A schematic diagram of the cell's layout is given in figure 4.1.

TURNOVER

BROACH

ORIENTATION

O C□

ROBOT 1 ROBOT 2

CENTRALISATION PALLET PALLETIN OUT

Figure 4.1: Schematic diagram of FMS 1

At the first stage of production, priority was given to the production of flywheels because it was thought to be more effective and this study refers to the production of flywheels. A flywheel casting arrives in a standard pallet and a camera is used to inform the first robot on the castings coordinates so that it can be picked up. To ensure the accurate positioning of the part on the lathe's chuck jaws, information from the camera is used again and the part is regripped. The robot will then load the flywheel into the lathe so that the first side is machined, then remove the component, turn it over and reload it so that the second side can be machined. The gauge measures the dimensions and provides correction data so that the quality of the component is maintained. Then, the first robot transfers the flywheel from the gauge to the orientation table where, with the camera's help the part is positioned for the next sequence of operations. Robot number 2 collects the part and loads it into the broach for the keyway cutting. It is then transferred to the vertical machining centre where drilling and tapping of both faces is carried out. The robot transfers the part to the balancing machine and on the completion of balancing

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the component is stacked into the output pallet.

4.2.2. Aim of the FMS1 simulation study

The aim of this simulation study is1) To provide information on the system's performance for the production of flywheels2) Identify problem areas .3) Suggest possible solutions4) This simulation exercise was also used to validate the FMSIM simulation model with a real working system and thus to build confidence in the model and its capabilities in order to use it for the extensive simulation study of FMS2, a system which was on the course of being purchased at the time.

4.2.3. The FMS1 simulation model

The system is simulated using the activity centered methodology and the simulator control module srtructure introduced in chapter 3. Table 4.1 summarises the entities , queues, activities and attributes used in the model.

ENTITY Q U E U ES ACTIVITIES ATTRIBUTES

part pallet in transfer part typeworkstation locationoutput buffer operation destination

wait for transfer by robot 1

number o fcom pleted operations next operation

w ait for transfer by robot 2

routingprocessing tim es

m achine idle operation type o f operationw ait to run

robot 1 free robot 1

robot 2 free robot 2

camera free

Table 4.1 : Entities, queues, activities and attributes of FMS1 model

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Table 4.2 gives the start conditions and end actions of each activity.

ACTIVITY ST A R T C O N D ITIO N S E N D A CTIO NS D U R A TIO N

transfer 1. there is a part in'wait for tranfer by robot r'2 . robot r is free3. destination ’output b u ffe r '

is em pty

1. put part in 'workstation' according to destination

2. se t robot 'r ' free

travel tim e from part's location to the part's destination

operation 1. there is a part in the 'workstation'

2 . the m achine is in the 'w ait to run' queue

3. IF operation is v is ion , the cam era is free

1. put part in 'outputbuffer'2 . put m achine in 'idle queue3 . for a v is ion operation set the camera free

operation duration

control 1. there is a m achine idle2. there are parts in 'pallet in' or output buffer

1. put m achine in 'wait to run' queue

2. put part in 'wait for transfer

by robot r ’ queue

Table 4.2 : Activities.their start conditions, end actions and duration in FMS 1

The system operates without buffers ( the output buffer queue in table 4.1 is used to hold the component which has just finished processing on a machine and does not denote a physical location where parts wait) and therefore) a component can proceed to the next operation only if the destination machine is not occupied. This is realised in the model through the third start condition of activity transfer (see table 4.2) where a part can not be transfered if the next machine is not empty. Transfer activity is simulated as a duplicated activity for each robot .which serves a different set of workstations.

4.2.4. Input data

Data for the durations of operations and transfer times are given in tables 4.3 and 4.4 below. These values were extracted from a computer printout which reported the real time events during a trial run of the system. Whenever slight differences occured in the values of the duration of an operation, the average value is taken. The gauge was not operational at the time data was acquired; thus, it was decided to use a duration of 30

57

seconds and to investigate the effect of different gauging times in the range of 30 to 60 seconds. The lathe to gauge and the gauge to orientation transfer times are induced from the lathe to orientation transfer time which it was found to be 57 seconds.

Operation1. crude picture2. centralisation3. first side turning4. turn over5. second side turning6. inspection7. orientation8. key way cutting9. drilling and tapping10. balancing

Duration in seconds 18 9

1823

204303439

55162

Table 4.3 : Operation durations for the flywheel manufacture in FMS1

Robot movements Duration in secondsRobot 1 1. pallet in - centralisation 39

2. centralisation - lathe 453. lathe - turn over 364. turn over - lathe 545. lathe - gauge 276. gauge - orientation 20

Robot 2 7. orientation - broach 218. broach - machine centre 459. machine centre - balance 3010. balance - pallet out 63

Table 4.4 : Robot movements and their durations for the flywheel production

Simulation run time : 50 hours = 180000 sec.

4.2.5. System perform ance in the flywheel production

Results of the simulation run are given table 4.5

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T O TA L O U T PU T 2 8 6 fly w h eels

T H R O U G H PU T TIM E 81.95 m inutes ( 2 7 parts transient)

U TIL ISA TIO N IN %

LATHE 62.1 R O B O T 1 3 5 .6G A U G E 4 .8 R O B O T 2 2 5 .3BR O A C H 6 .2M A C H IN IN G C EN TER 87.5B A L A N C E 9.8

Table 4.5 : Simulation results on FMS1 system performance for flywheel production

The machining centre is the workstation which is most highly utilised at 87.5 % whereas the rest of machines are underutilised with the exception of the lathe whose utilisation is moderate. As a result, the machining centre operates as a bottleneck station for the system. Thus, throughput time for the components is very high (81.9 min) as compared with their production time without delays which is not more than 25.2 minutes. The big delays which occur in the system may be explained better if we think of the system as consisting of two subsystems each comprising the stations served by each robot.The orientation table is their common station. A fast flow through the system would require a balance between the two subsystems in the sense that the flow of parts out of the first subsystem should be equal to the input flow in the second, something which is not true here since the part cycle without delays is 701 seconds (including orientation) at the first subsystem and 845 seconds (including orientation) at the second. Such a difference results in big delays of parts and therefore high throughput time.

4.2.6 System performance sensitivity on inspection duration

The variation of inspection time from 30 seconds to 60 seconds had no effect on the system's performance other than to increase the gauge's utilisation as it is shown in figure 4.2 . Total production, throughput time and machine and robot utilisations are the same as in table 4.5. The reason for the system's insensitivy on changes in inspection duration at this range , lies in the delays occured in the transfer of the components because the robot is busy with another task. Indeed, the throughput time of the first flywheel entering the empty system is 26.21 minutes rather than 25.2 minutes.The

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lminute delay may only be attributed to the busy robot and this time is double the greatest change in inspection duration which has been investigated.

Figure 4.2 : Gauge utilisation versus inspection time

4.2.7 System performance under different robot cycle times

The robot cycle time includes the motion from one point to another and the gripping, releasing and placing of parts. It could be possible to adjust either of those parts such that the robot cycle time becomes faster or slower. It is unlikely that huge savings in cycle time can be effected but small variations may reasonably be considered.

The first set of runs includes changes of ±10% and ±25% for both robots and the second set of runs considers unequal changes for each robot speed in an attempt to balance the system and improve on the total production, lead time and lead time dynamic response. Results are summarised in figures 4.3 and 4.4.

As expected, decreasing robot cycle time results in increased total output, decreased throughput time and faster response (figure 4.3).

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total output lead time

Figure 4.3 : System performance with varying robot cycle time

TOca<D

-O- total output lead time

Figure 4.4 : Effect of unequal robot speed on system performance

In order to balance the system, the cycle time of the two subsystems should be as close as possible. This is attempted here by manipulating the robot speed so that the first robot moves slower than the second. ThecWtfotjof robot 2 was first decreased to 75% and the thecbiYaitioflof robot 1 was increased gradually from 100% to 135%. Figure 4.4, shows the system performance for an increasing t i me difference between the robots.

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The throughput time dynamic response of the system is shown in Figure 4.5.

90

-o- 0 cycle differ.25% cycle diff.

-o- 35%cycle diff -o- 50% cycle diff

60% cycle diff.

200 20 40 60 80 100

number of parts

Figure 4.5 : System dynamic performance with robots having unequal speed

From figure 4.4, we note that as the two subsystem cycles become more balanced ( as the robot cycles difference increases ), the throughput time decreases. The response of the system however is slower but it settles to lower throughput times (see figure 4.5). The lowest throughput and the faster response is achieved when the difference of the robot's speed is at its maximum and therefore the system is more balanced. The throughput gain is almost half an hour less than the original and the response (i.e the throughput time becomes stable) is at the 4th part which goes out of the system as compared to the 27th at the original system. Total output is affected solely by the initial decrease of robot's 2 cycle time where the increase in the total output is almost the same as when both robot’s cycle is reduced to 75% (see figure 4.4 and figure 4.5).

The manipulation of robot speed which was used in this analysis in order to achieve the above mentioned improvements demonstrates only a possible course of action. Alternatively, running the drill faster may also be a possibility in order to decrease the second cycle time. Figure 4.6 gives the system performance for decreasing drill times. The effect is similar to the one presented in figure 4.3. Production output is increased and lead time is slightly reduced but it is still very high. For 530 seconds drill time, the production output and lead time achieved is the same as when the second robot's cycU was reduced to 75% of its original (figure 4.4).

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320

CO■e (0Sr 3100L.

1 300 c _c3■3 290o "c5 o

280490 500 510 520 530 540 550 560

drill processing time in seconds

t o t a l o u tp u t - * • le a d t im e

Figure 4.6 : System performance with varying drill time

Reducing the lead time further requires a balancing of the cycles which involves an increase in the first cycle's time. This may be alternatively achieved by running the lathe in lower speeds in which case tool gains would also be realised. Figure 4.7 shows the effect of system performance for increasing the lathe processing time when the drill time is 530 seconds. Results of the original lathe and drill times are also shown for comparison reasons.

380 400 420 440 460lathe processing time in seconds

-oCO0)

(drill time:530sec.)

H3- to t a l o u tp u t -*■ le a d t im e

Figure 4.7 : System performance with increasing lathe time for constant drill time

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For a range of lathe times from 386 to 441 seconds production output remains stable at 295 parts and lead time decreases to a minimum of 54.13 minutes which is achieved after the exit of the fourth part from the system. Results are the same as in figure 4.4 where increase of the first cycle was achieved by slowing down robot 1. If, however, lathe time is increased further, the two subsystem exchange roles with the first subsystem becoming the bottleneck and thus controlling production output. As a result, output decreases and lead time shows a increasing tendency (encircled area).

4.2.8 Conclusions

Total output may be increased by decreasing the cycle of the second subsystem.The throughput time and the system response may be controlled by maintaining the two subsystem cycles, in balance. The cycle times of the two subsystems and their relative values play an important role in determining the position of the system's bottleneck which in turn controls the production output, and in achieving low lead times . The best system performance is achieved for balanced cycles and the system shows a great sensitivity in this area (encircled area in figure 4.7).

The best way to implement the desired cycle times to achieve the best performance is dependent on other feasibility and economical factors. For example, the improvement achieved by increasing the second robot's speed may only be acquired at the expense of other variables not measured by the model. Possible examples are the increased maintenance costs due to the robot working harder and increased possibility of poor positioning of parts due to increase in speed. A more extensive analysis would be required to determine the best possible action which should be taken.

4.3 Simulation study of FMS2 for the manufacture of small turned parts

4.3.1 Cell description

The FMS2 cell has been designed to manufacture a range of small turned parts previously manufactured on traditional bar auto CAM machines. Initially 162 parts have been selected to be manufactured within the cell.

The cell consists of the following equipment:1. Bar store - Remmert automatic high density bar store and a bar distribution crane

gantry.2. Bar loader - Index MBL

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3. Lathes - GE65 (2) 4-axis CNC Auto turret lathesGS30 (4) 4-axis CNC Auto turret lathes

4. Conveyor - SKF flex-link5. Gauge - Haln & Kolb View 1200

Bar stock is held in cassettes in a high density automated bar store from which selection of the required bar is computer controlled. The bar is transported by an automated gantry crane to the bar magazines on the CNC auto turret lathes. The lathes have the capability to complete all operations except grinding and heat treatment Finished components are collected from the synchronous spindle mounted on turret 1 by a work handling unit, washed in a stream of high pressure coolant and placed in a bin. When the bin is filled with a predetermined quantity of components, it is transported by a conveyor network to a temporary storage area with maximum capacity of 20 bins. From there, components are unloaded into an empty container by an operator and then taken to stores.

Empty bins are returned to the lathes via an overhead conveyor. A queue of empty bins is held on the overhead conveyor. Each lathe has a small queue of one to four empty bins held in the feeder chute. When the number of bins in the chute is reduced to one, the conveyor controller requests three empty bins to be despatched to the lathe from the overhead conveyor queue.

Inspection and gauging is on a first off and sample basis using a post process non-contact gauge and with machine mounted probes. Component test pieces are manufactured for inspection and are not included in the batch. An operator takes the test piece from the lathe to the gauge where inspection is performed. Dimensions measured by the gauge are fed back to the lathe, to allow the tool offsets to be adjusted automatically to maintain component production within the required tolerances.

Lathe set up is performed by an operator. Whenever a bar size is to be changed the bar feeder guides, the spindle sleeve and the chuck collets are changed. Tools are changed to enable the new component to be manufactured. Set up also includes standard housekeeping tasks.

Each lathe type uses approximately 200 different toolholder/consumable tool combinations. Both types of lathes have a total of 24 tool positions including the synchronous collet and the work handling unit gripper. A seven digit code identifies each tool assembly, 3 digits for the tool holder and 4 for the consumable tool. Tool assemblies are changed as complete units.

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A cell host computer is interfaced with the cell's CNC machines, equipment and company mainframe system. The cell computer schedules the cell's production and monitors and controls all stock movements and component manufacture within the cell.

4.3.2 Aim of the FMS2 simulation study

The purpose of the simulation exercise was to provide information on the following issues:

1. The cell's performance under the anticipated workload. (At the time of my involvement FMS2 was being installed)

. production time from the start of the firstoYder to the end of the last

. lathe, gauge, crane and operator utilisations

. delays in lathe set up and in bar deliveries

. tool holder utilisation2. Investigation on the right policy to be adopted by the cell scheduler.3. The number of operators necessary to complete the tasks for which they are needed.4. The effect of lathe breakdown on the cell's performance.5. The system performance under production in different batch sizes.6. The effect of tool changing (due to tool wear) on the cell's performance.

4.3.3 The FMS2 simulation model

The system is simulated using the activity oriented methodology and the simulator control module srtructure introduced in chapter 3.

Model assumptions

1. Bar availability is guaranteed2. Tool availability is guaranteed

The assumptions made are not enforced by the model's inefficiencies but they serve the purpose of collecting information on the bar and tool requirements for the production of a particular workload so that simulation can be used as a planning tool.

Model boundaries

1. From the raw material storage and distribution end, the issuing of bars from the bar store to the bar loading station with the help of a gantry crane.

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2. At the end of finished components removal , the unloading of full bins from the temporary storage area to the empty container.Therefore, detailed activities concerning the bar stores and the transport of finished components in the stores are not included in the model.

Model description

The entities, activities and entity attributes are summarised in table 4.6. The model's activity diagrams are given descriptively in table 4.7. Start conditions of activities are controlled by the availability of entities in the activity's input queues as well as by attribute values. Attribute values may also control the choice of the destination queue.

Manufactured components and tool assemblies are not defined as entities in the model. Instead, production orders are defined as entities and include through attribute values all the necessary information on the type of component and its manufacturing requirements such as tooling, bar type, size and quantity, inspection quantity and maximum bin capacity. For the components, a counter is used to count the number of components produced by the lathe until the quantity required is reached. Tool assemblies are also lathe attributes used to describe the current tooling situation at each lathe. Thus, the low number of entities in the system results in reduced memory requirements.

The model's control module acts as the cell's scheduler. Control though is not an activity as it does not take time, it acts as such by controlling the flow of entities within the model. Its calling time and its output are shown at table 4.7.

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Table 4.6: Entities, queues, activities and attributes of FMS2 simulation model

ENTITY Q U E U E S ACTIVITIES ATTRIBUTES

Bar Bar Store

L oad station

M agazine

B ar issue

Bar transport

Bar feed

bar type bar sizeQ uantity o f bars D estinationC om ponent's rough lengthR em aining length o f barBatch s iz e in number o f com ponents

N um ber o f bars in ltn

Request requestsource

w ait to b e executed

B ar issuebar type bar sizeQ uantity o f bars D estinationC om ponent's rough lengthBatch s iz e in number o f com ponents

N um ber o f bars in ltn

Lathe S et up Idle

W ait to be allocated

W ait for tool change

C on flict

Lathe set up

P rocessin gBar feedT est p iece manufacture

T ool change

Breakdown

Repair

lathe type com ponent bin capacity inspection quantity bar sizebar feeder change num ber o f tool changes order numberbar quantity remained to b e issuedlathe status for bar deliveriescurrent tooling com pon ent processing tim e

continued in the next page

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ENTITY Q U E U ES ACTIVITIES ATTRIBUTES

T est p iece test p iece source

T est p iece manufacture com ponent type

w ait for inspection Inspection lathe

G auge idle gauge Inspection

Crane 1 crane 1 idle Bar issue

Crane 2 crane 2 idle Bar transport

Operator free Lathe set upT ool change Inspection

U nload bins

B in em pty bin queue

C onvey fu ll b in com ponent type

storage o f full bins

Em ty bin transfer m ax. bin capacity

local bin queue

Break w ait to flag breakdown breakdown

Order cell workload 'control' com ponent type qty o f com ponents

lathe workload m ax bin capacity inspection qty com ponent's rough length lathe typebar type bar qty bar sizeprocessing tim e tool package

continued next page

ACTIVITY ST A R T C O N D ITIO N S E N D A CTIO NS DURATIO N

Bar issue 1 there are bars in the store2 there is a request for bars3 crane 1 is free4 load station is em pty

1 p lace bars in load station2 set crane 1 free3 p lace request in its source

for recycling

3 m inutes (estimate average)

Bar transport 1 there are bars in load station2 crane 2 is free

1 place bars in destination m agazine2 set crane 2 free

0.33 * 5 1 L-7 1 + 2 (minutes)

Bar feed 1 lathe is idle2 there is bar in the m agazine3 batch quantity has been producedC s Drem aining length o f bar is less than the com ponent's rough length

1. place bar back in store (recycling)2. IF batch is com pleted TH EN put lathe in 'wait for allocation' and call the control m odule ELSE put lathe in id le queue

2 m inutes (estimate average)

Lathe set up 1. there is a lathe w aiting for set up (the lathe is linked with a bar request)

2. there is an operator free

1. set lathe idle2. set operator free3. activate request for bars i.e

put request in 'wait to be executed'

20 + 3 x no. o f tool chgs + bar feeder change x 30+ part processing tim e (proof run)

Table 4.7: Activities,their start conditions, end actions and duration in FMS 2

continued next page

ACTIVITY ST A R T CO N D ITIO N S E N D A CTIO NS D URATIO N

P rocessing 1. lathe is idle2. there is bar in the magazine3. rem aining bar length is bigger than the com ponent's rough length4 . batch not com pleted

IF tool change is required TH EN put lathe in 'wait for tool change' queue EL SE put lathe in idle queue

com ponent's processing time

T ool change 1. there is a lathe w aiting for tool change2. there is a free operator

1. put lathe in idle queue2. set operator free

2% o f batch tim e length' for batches o f 3 to 6 hrs.5% o f the batch length for batches bigger than 6 hrs.

T est p iece manufacture

1. lathe is idle2. test p iece source is not em pty3. inspection quantity has been reached

1. put test p iece in 'wait for inspection '2. put lathe in idle queue

com ponent's p rocessing time

Inspection 1. gauge is free2. there are test p ieces waiting for

inspection3. there is a free operator

1. put test p iece in source for recycling2. set gauge free3. set operator free

5 m inutes ( estim ate average)

C onvey bin is fullC ° D

batch is com pletedC s D

inspection quantity is reached

put bin in storage 0 .2 x ( 6 + 5 x L)

The model provides for counters to measure the elapsed times referred to different entities as well as a facility to collect information on tool holder utilisation.

ACTIVITY ST A R T C O N D ITIO N S E N D ACTIO NS D U R A TIO N

Tranfer o f em pty bins in local queue

1. there is on e bin left in local bin queue2. there are bins in em pty bin queue

put bin in local bin queue 0 .2 x 5 x L

U nload bins 1. there is a free operator2. the storage space is fu ll

1. set operator free2. put bins in em pty bin queue

no. o f bins in storage x 0 .25

Repair1. there is a flag for breakdown2. lathe is idle

1. rem ove breakdown flag2. put lathe in id le queue3. schedule next breakdown

duration drawn from uniform distribution betw een 0 .5 to 24 hrs.

'control' a batch is com pleted 1. lathes are put in queue 'wait for set up'2 . lathes are linked with bar request

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4.3.4 System performance under the anticipated workload

Model statusThe model at the first stage of its development did not incorporate the activities of1) bin unloading,2) tool change3) breakdown and repairAdditional assumption Inspection does not involve the presence of an operator.Option Lathes are held during the inspection of the test piece, waiting for the inspection results before they start producing again. This policy will be followed in reality until confidence in the system has been built up.

Input dataThe input data file consists of orders for 162 components which have been selected to be manufactured in the ce ll. Twenty-six components require the GE65 type of lathe, one hundred and sixteen components require the GS30 lathe type and the remaining components (20) can be manufactured by either type (see figure 4.8).

Initial part allocation to lathe types

9.71% 12.62%q parts allocated to GE65s gH parts allocated to GS30s □ alternative allocation

Figure 4.8 : Part allocation to lathe type

Processing times are estimated since components were not proved at the time on the newly acquired lathes. For the same reason, tool packages were not known. Random tool packages are used instead( 24 number are drawn randomly from 200 numbers representing the 200 different tool assemblies; NAG library routines are used). The order quantity is equal to the projected four weeks demand. Initially production in four weeks

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batches is investigated.

CASE 1 ~ with option. Results are summarised in table 4.8.

Utilisation of equipment is measured from when the system is fully loaded until the time the last order is loaded in the system. Thus, utilisation figures are not distorted by the run's transients which will not be present in the real situation.

PRO DUCTIO N TIM E 38308.3 m in. or 79 .8 shifts (3 shifts/day) or 26.67 days (6 days /week)

UTILISATION IN % (measured until 36870 .8 m in .)

la th e ty p e p r o c e s s in g se t up in s p e c t io n b a r issu e te s t p ie c e id le

G E65s 52 .75 3.7 4 .0 0.4 1.75 37.1

G S30s 88 .65 4.5 5 .0 0 .6 1.2 0

GAUGE C R A N E 1 C R A N E 2

24 .77 .586.63

d e la y s in in sp e c t io n 3.63d e la y s in b a r d e liv er ie s d u e to b u sy cra n es o.5

Table 4.8 : Performance results with the initial workload

Discussion1. The current workload exceeds the system's capacity for four weeks production requirements even when 3 shifts per day for a 6 day week is assumed (P.T.=26.67 days). Note that breakdowns, tool change or operator unavailability is not incorporated in the model at this stage (6 operators are assumed to serve the system) in which case a greater production time should be expected. However, under the normal operating strategy, lathes will not have to wait for the inspection results as they do now, thus spending unproductively 4.28% of their time on average.

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B production time□ GE65 utilisation□ GS30 utilisation

Figure 4.9 : System performance with lathes working during inspection

The improvement which could be achieved in this case (CASE 2) is shown in the bar chart of figure 4.9, where the production time and the average utilisation of the GE65s and GS30s are depicted for both cases. Production time has been reduced by 5.66% . This reduction is more than the average time lathes spent waiting for the inspection results which is 4.7%. The additional gains may be attributed to the gains of the test piece delays in inspection (0.6 % per lathe on average). The system however, remains overloaded since the production time is still 25.16 days for 24 days production requirements.2. Changeover time for four-week batches is on average 4.1%. This is expected to increase for smaller batches3. The time lathes spend for bar issuing is very small. The two cranes cope well with the bar delivery requests. Delays in bar deliveries are only 0.5%. Both cranes have low utilisation.4. The gauge utilisation is 24.7 % and delays in test piece inspection are 3.63%. These delays reflect to a delay of 0.6% in average for each lathe.5. The two GE65s are underutilised as compared to GS30s, the reason being the lack of orders requiring this type of lathe for its manufacture. Figure 4.8 shows the unbalance of workload between the two lathe types.When all alternative type allocations are designated to GE65s ( CASE 3 ) the situation becomes slightly better balanced as figure4.10 shows and the improvements achieved may be seen in the bar chart of figure 4.11.

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□ parts alienated to GE65s B parts allocated to GS30s

Figure 4.10 : Allocation of parts on lathe type without alternative allocation

100

(O>s<d"Oc©Ecoo=3X)oQ.

o'_ccocdto©"Occd

80 -

60 -

40 -

20 -

01

experimental cases

■ production time □ GE65 utilisation E3 GS30 utilisation

Figure 4.11 : System performance for increased GE65 workload

Utilisation of GE65 lathes has been increased but they are still underutilised. A more radical action is therefore required. The production time is 33.5 days which is a surprising result This situation arises because time lengthy batches were allocated late in the schedule, occupying lathes for long time after the last order was loaded into the system. Indeed, analytical results showed that the time the last order was loaded is at21.6 days, almost 11 days before all orders are completed. Figure 4.12 shows the

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relationship between the production time and the last order loading time for the three cases investigated so far.

Figure 4.12 : Makespan and last order loading time

The interesting question which arises from the above observation is whether the result of case 3 is likely to arise often and whether the cell scheduler takes into account the batch length for order allocation. The next section is set to give an answer to these questions and investigate the effect of different scheduling order allocation policies.

4.3.5 System performance under different order allocation policies.

First, an analysis of the order lead time distibution is carried out for case 3 (figure4.13). It demonstrates the highly dispersed lead times which characterise the orders. As a result, the situation which has arisen above (case 3, figure 4.11) is very likely to occur. To eliminate this possibility the order allocation policy should take into account the batch time length.

Alternative allocation policies examined are:

POLICY A

1. Select the order for which a bar feeder change is not required.(This is the case where the component to be produced requires bars of the same size as the component whose batch has previously been completed on the lathe considered for allocation.)

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1.1 If there is more than one order without a bar feeder change needed, choose the order which requires the smallest number of tool assembly changes.

1.1.1 If there are more than one orders with the same minimum number of tool assembly changes choose an order randomly.

2. If there is not an order available for which a bar feeder change is not needed, select the order which will require the smallest number of tool changes

2.1. If more than one orders exist with the same number of tool assembly changes needed select an order randomly.

This scheduling policy is the original allocation policy which was devised by the company with the aim to reduce the changeover time between orders.

Figure 4.13 : Lead time histogram

POLICY B

1. Select the order for which a bar feeder change is not needed.1.1 If there is more than one order with a bar feeder change not needed select the order with the maximum batch length.

1.1.1 In case more than one orders in 1.1 have the same batch length select an order in random

2. If there is not an order available for which a bar feeder change is not needed, select the order with the greater batch length.

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2.1 In case more than one orders in 2 have the same batch length select an order in random

POLICY C

1. Select the order with the maximum batch length.1.1 If there are more than orders with the same batch length select one in random

The system performance for case 3 under the three order allocation policies is summarised in figure 4.14. Figure 4.15 shows the system performance for case 2 under policy A and policy B.

■ production time □ last order in H lathe set up

time

Figure 4.14 : System performance under different allocation policies, case 3

Both policies B and C reduce the time distance between the production time and the last order loading time . Policy B gives the best results. Production time is reduced to a minimum of 26.10 days and the mean set up time for lathes is at the lowest level at 4.4%. Policy C fails to be as good as policy B due to the high mean set up time for lathes (5.36%) which has a negative effect on the production time.This result indicates the importance of the bar feeder change criterion in order allocation. On the contrary, the number of tool assembly changes as a criterion for order allocation does not have a big effect

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policy A policy Border allocation policy

■ production time □ last order in @ lathe set up

time

Figure 4.15 : System performance under different allocation policies, case 2

Results in figure 4.15 also show the relative unimportance of the number of tool changes as a criterion for order allocation since policy A gives only a slightly increased mean set up time for lathes (from 4.46% for policy A to the 4.58% for policy B).

4.3.6 System performance under different manning levels

Six operators were so far employed in the system to ensure operator availability. This number is now gradually reduced in order to investigate the system's performance under a constraint operator availability. Experiments on different number of operators refer to case 4.

CASE 4 uses 1) the more balanced allocation for GE65s ( figure 4.10)2) order allocation policy : B3) the option of lathes working during inspection

Results are shown graphically in figure 4.16.

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-o- production timedelays in lathe set up operator utilis.

Figure 4.16 : System performance with varying manning levels

Results show that the system could be served by two operators without a severeincrease of the production time and with mean operator utilisation at 27%. However, at this stage of the model activities like tool change and bin unloading are not included in the model and operators are not involved in inspection. These activities will increase the operators' workload. In addition, operator workload is expected to increase for small batch production because of the more frequent changeovers which are expected to occur.

4.3.7 System performance under the updated workload

Model statusConsultation with the company on the previously presented matters resulted in the following decisions:1. Order allocation policy B is to be employed.2. Lathes will keep producing while inspection takes place.3. The activity of bin unloading is incorporated.4. An operator is involved in inspection to allow a better investigation of the operators utilisation.5. Three operators are to be employed.

Input dataThe input data file has been updated. The workload is not decreased because of the desired versatility and the unknown patterns of demand. The system must be able and

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ready to produce any of the components according to actual demand (production quantities used in the simulation runs are only projected from the past aggregate demand)

The new input data file comprises 173 components:31 components require the GE65 lathe type 122 components require the GS30 type of lathe 20 components may be processed by both lathe types.

Figure 4.17 shows graphically the workload for each type of lathe.

El parts allocated to GE65s IH parts allocated to GS30s □ parts allocated to both types

70.0%

Figure 4.17 : Part allocation with the updated workload

Alternative lathe type processing is kept to ensure system flexibility in order allocation (load balancing). The low GE65's utilisation is to be increased by undertakingsubcontracting work.

Results under the new data file are given in table 4.9.

Under the updated workload GE65s have a utilisation of 50% approximately and GS30s a utilisation of 92%. The production time is 35 days. The three operators have a mean utilisation of 20 % and the greatest part of this time is spent for lathe set up.

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PRODUCTION TIME 50316.5 min. or 104.8 shifts (3 shifts/day) or 34.9 days (6 days /week)

UTILISATION IN % (measured until 5 0256 .0 m in .)

la th e ty p e p r o c e s s in g se t up b ar issu e te s t p iece id le

G E 65s 49.8 5.25 0.3 1.65 42.8

G S30s 91.85 6.37 0.45 1.27 0

m ea n se t u p in s p e c t io n u n lo a d b in s to ta l

o p er a to r

u t i l i s a t i o n11.63 6.83 0 .66 19.13

GAUGE 20.59CRANE 1 5 .8CRANE 2 5 .09

d e la y s in in sp e c tio n 2.11%d e la y s in b ar d e liv er ie s d u e to b u sy cra n es 0-27 %

1 2 9 %d e la y s m la th e s e t u p d u e to b u sy op erator : *

Table 4.9 : System performance under the updated workload

4.3.8 Effect of lathe breakdown on system performance

Lathe breakdown was incorporated in the model in order to investigate its effect on the system performance. Breakdown may not occur while a lathe is in the course of manufacturing a component due to the limitations of the simulation model. Instead, if the lathe is working, a breakdown is flagged through entity break and breakdown takes place as soon as the lathe finishes processing the component. This limitation is not serious because the average processing time is in the range of 2 minutes which is very small as compared with the mean time between failures (in the range of 5000 minutes). Breakdown data about the system do not exist because the cell was in the course of being installed at the time of the simulation exercise. It is expected, however, that the mean lathe downtime will be at maximum 12% and the repair time can be anything between half an hour and twenty-four hours. Random number generating routines were used

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from the NAG library to simulate breakdowns. The time between failures was drawn randomly from a normal distribution and the repair time from a uniform distribution between 0.5 to 24 hours. Three cases were examined with increasing lathe down time using the following normal distributions :

CASE 1 N ( 14000,4000)CASE 2 N ( 10000,4000 )CASE 3 N ( 5000,2000 )

The increase in production time with increasing lathe down time is shown graphically in figure 4.18.

®Ecoo3"Oo

Figure 4.18 : Makespan versus mean lathe downtime

The percentage increase in production time due to lathe breakdown is directly related to the mean lathe downtime. The reason for this direct relationship is that each component is completed in one lathe so that breakdown of a lathe does not affect the work of the rest. Therefore, lathe downtime is reflected directly to increased production time.

4.3.9 System performance under production in small batches

The aim of the company is to produce components in one week batches. The bar store however, deals with full lengths of bars and it was therefore decided to define as a lower bound on the batch size the number of components which can be produced by one bar.

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This bound may be ignored in case there is an order which requires the same bar material and size and is waiting to be processed.Then, the minimum batch is one component. This latter possibility is not incorporated into the model. In order to compensate for this model inefficiency, the extreme case, according to which there is always an order which requires the same bar, is considered. It is then enough to produce components in the batch sizes required without consideration for a minimum batch of one bar. In reality results will lie between the values given for the two cases of minimum batch size of 1 bar and of the case where no minimum exists.

Careful examination of results was carried out to examine the effect of the minimum batch of one bar on the batch sizes. Figure 4.19 shows the percentage of orders for which the number of parts produced by 1 bar corresponds to production requirements of one and more weeks.

£©■ Eoo©o>©a8.

60

50

40

30

20

10

01 2 3 4 5

p r o d u c t io n r e q u i r e m e n ts in n u m b e r o f w e e k s

J--------------------

t------ 1------ r

Figure 4.19 : Number of parts produced from one bar

Therefore for a total of 83 components out of 173 production quantity out of one bar covers the production requirements of more than one week.

The system performance for the case of production in four week batches and one week batches with minimum batches of one bar, is shown in figure 4.20.

100

85

-oc<tiwItTX)

60 -

-o- production time GS30 utiiis. GE65 utiiis.

-o- operator utiiis, delays in set up

0 1 2 3 4 5batch size in weeks

Figure 4.20 : System performance for batches of different size

The production time shows an increase from 34.9 days to 41.73 days. Lathe utilisation has decreased for the GS30s from 91.9 % to 81.2 % in average, whereas GE65s stay at the same utilisation level of 49.6% in average. Operator utilisation shows an increase from 19.2 % in average to 34.7 % caused by the increase in the time the operator spends for setting up due to more frequent changeovers. Delays in setting up due to busy operators are doubled from 1.29 % to 2.69% but they are still very low.

Figure 4.21 shows the system performance for producing in one week batches with and without the constraint of 1 bar on the minimum batch. Changeovers are more frequent when producing in batches of one week without minimum batches of one bar because of the lack of the batch size spread existing in the opposite case (figure 4.18). The effect of more frequent changeovers results in an increased production time, operator utilisation and set up delays, and decreased lathe utilisation when compared with the case of existing of a minimum batch.

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coojw3T3

</>S'■ o

80

60

40

20

0

100

min batch of 1 beer

■ production time Q GS30 utilis.E! GE65 utilis.□ operator utilis. G delays in set up

Figure 4.21 : System performance with batch sizes of one week’s required production with a lower bound on batch size equal to the number of parts produced bv one bar

4.3.10 System performance under tool changing due to tool wear

Tool life records are held on the computer for each component machined in the cell. The tool life for each tool used to machine the component is based on the number of components the tool can produce. A second record of tool life is held against each lathe. For each tool assembly loaded to a lathe the remaining tool life is held as a percentage. As it is desirable to keep tool changes to a minimum, when a tool is due for a change or when tool offset limit is reached, the computer will scan tool life information relating to all other tools loaded on the same lathe. If any tool changes are due within a predetermined time, they will be changed at the same time.

Detailed representation of the above policy in the model would mean a duplication of the cell's real time software which would result in an unnecessarily complicated model. In order to avoid duplication, tool changing is approximated in the model as follows:1) A tool change is to be incorporated in the changeover time for batches which do not last for more than 3 hours.2) Batches whose duration is greater than 3 hours need a tool change. The tool change occurrence and duration is given in Table 4.10.

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B A T C HD U R A T IO N

TOOLC H A N G E

O C C U R E N C ETO O L CH A N G E D U R A T IO N

3 to 6 hours 3 hrs 2 % o f the batch length

greater than 6 hrs 3 hrs 5 % o f the batch length

Table 4.10 : Data on tool change occurrence and duration

Results on the effect of tool changing are summarised in the table 4.11. Two cases are examined. The first, is for production in 4 week batches and the second for production in 1 week batches with minimum batches of one bar.

BATCH SIZEPRODUCTION

TIME (days)M EA N ]UTILIS

G S30s

LATHEATION

GE65s

AVERAGE TIM E SPENT IN TOOL CHANGE(%) (G S30s only)

D ELAYS IN TOOL

CHANGE (%)

4 week batches

36 .4 88.12 47 .4 3 .5 4 (4.2) 0.45 (4 oper)

1 week batches m inimum batch

o f 1 bar43.1 78.48 48 .4 2.74 (3.17) 2.77 (3 oper)

Table 4.11 : System performance with tool changing

Comparison of the results given above with the results of the simulation runs without tool changing show that (figure 4.22, figure 4.23):1) Production time increases by 4.2% in the first case and by 3.27 % in the second. The need for tool changing and its duration is greater for the case of production in 4 week batches (longer batches). The increase is directly related to the average time GS30 lathes spend in tool changing due to tool wear. GE65 lathes are underloaded and therefore they do not affect production time. Lathe utilisation (in processing) is decreased since some of the previously productive time is spent in tool changing.

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production time GS30 utilis. GE65 utilis.

Figure 4.22 : System performance with and without tool changing for batch sizes of four weeks required production

Figure 4.23 : System performance with and without tool changing for batch sizes of one week required production

2) An operator is involved in tool changing. For the first case, 3 operators are unable to cope with the increased operator needs in the cell. The reason for the increased number of operators in this case, is the simultaneous occurrence of requests for operator service which are all impossible to complete. The result is a jam of full bins at the output storage which even when tolerated until an operator is freed, result in the starving of lathes from bins and then production stops. If unloading is ignored, then the sys tern could be

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served by 3 operators but delays in setup and other activities involving operators are expected to be severe. Delays in set up and tool changing for the second case are 2.8%.

4.3.11 Tool holder utilisation

A data file consisting of 42 components for which tool packages were available is used to collect information on tool holder utilisation. The model was run for 4 weeks production requirements in 4 weeks batches and for production in 1 week batches with minimum batches of one bar. Tool change due to tool wear is not considered at this stage.

Results on the tool holder utilisation are given in table 4.12 for the first case.

TOOL HOLDER USE in %(production time: 17.31 days)

Tool N U M B ER O F C O F >IESholder 0 1 2 3 4 5 6 7 8 9

100 49.2 13.1 3.2 19.3 14.8101 4.5 4.1 3.2 0.7 16.1 35.2 35.3 0.5102 0 0 0 0 0 7.6 22.8 13.7 15.5 1.2103 0 0 0.2 2.5 65.3 1.8 22.0 7.7104 21.2 11.5 35.2 30.8 0.2 0.6107 0 13.1 31.0 40.3 15.1110 99.4 0 0.2121 0 13.1 31.0 40.3 15.1122 40.4 31.9 27.3132 91.3 8.2135 0 0 7.8 0.7 1.6 7.3 15.5 9.3 4.0 16.9136 91.3 8.2138 74.1 18.3 7.2139 35.2 64.3141 0 9.7 11.2 18.2 26.7 9.9 1.0 10.0 4.4 8.6144 91.5 8.0145 76.0 23.6152 18.2 30.6 40.2 10.5153 91.5 8.0

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155 49.6 50.0161 61.2 38.3168 0 0 0 0 68.0 1.8 29.7169 48.1 38.9 10.8 1.8176 0 28.0 42.0 29.6181 0 0 14.2 10.1 45.6 29.7199 15.7 6.7 51.3 25.9999 35.2 64.3

number of copiesType 10 11 12 13 14 15 16 17

102 4.3 1.1 0 15.2 0 10.1135 0 4.7 2.3 0 15.0 7.2 6.7 0.5

Table 4.12 : Tool holder utilisation

Results on the tool holder utilisation showed an unbalanced use of tool holders. For example, tool holder type 135 needed an availability of 17 copies for 0.5%. Also tool holder type 102 requires 22.8% of the time 6 copies, 15.5% 8 copies and 15.2% 13 copies. Tool holders should be used in a balanced way in order to reduce the number of copies which should be available and at the same time, not to delay production. This can only be achieved by devising a tool holder scheduling policy. Experimentation with different order dispatching rules to achieve balancing of the tool holder utilisation were

attempted but did not give any substantially better results. Further investigation is needed in this direction.

4.3.12 Conclusions

1. FMS2 machining equipment is highly utilised(tables 4.8 and 4.9). The comparative low utilisation of GE65s with the workload examined indicates that the production of parts requiring this type of lathe could possibly be satisfied with one GE65.2. For 4 weeks production requirements, production in 1 week batches rather than 4 week batches increases production time by about a week ( table 4.10 and figures 4.20 and 4.21).3. Tool changing due to tool wear increases production time for about a day to a day and a half (figures 4.22 and 4.23).4. Lathe breakdown has an almost linear effect on production time which increases by

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15% for 12% down time (figure 4.11).5. Three operators are able to cope with system requirements with utilisation staying at 20 to 35% (table 4.9, figure 4.20).The system could be run with two operators if bin unloading is accomodated in their free time so that no jam is created at the unloading station.6. Scheduling of FMS2 refers merely in the sequencing of orders for entering the system. Experiments showed that occurence of bar feeder change and the batch length are important in keeping changeover time and production time down (figures 4.13 and4.14). Results of the tool holder usage indicate the necessity of developing a scheduling policy which will take into account the tool holder requirements of each order with the aim to balance their use.

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CHAPTER 5

A PERIODIC PRODUCTION POLICY FOR FMS SCHEDULING

5.1 Introduction

A new production policy for FMS has been developed to tackle the problem of FMS scheduling.The objective of the FMS scheduler is to satisfy a known, time varying demand for a family of parts that is dictated by the Master Production Plan. Generally, a production target is specified, to be attained over a certain time horizon. At the operational level, however, this objective often translates to "minimise the makespan or the total completion time of the production target". Another major objective is to minimise work-in-process (WIP) which is one of the main cost drivers of FMSs.

Most researchers have so far treated work-in-process as a secondary objective or ignored it and as a result, parts are introduced into the system faster than they can be processed. They are then stored in buffers or in the transportation system while waiting for machines to be free, resulting in an undesirably large work-in-process. Kinemia & Gershwin [14] suggest a hierarchical production scheduling policy which controls WIP levels by respecting the system's capacity constraints. The system's capacity depends on the operational states of machines.

Combinatorial approaches to FMS scheduling have to deal with the massive computational effort imposed by problems of even moderate size. Most of them adopt simplified heuristic techniques to cut down computational requirements [69].

The multi-objective FMS scheduling problem is approached here by introducing a "periodic production policy". Its basic idea is to schedule a sample of the production mix and repeat this schedule at regular intervals until production requirements for the planning horizon are met. Periodic production in sets of parts, in the context of flexible production, allows fast response to changes in market requirements. The mere existence of these changes would not justify long term production schedules. Moreover, quick response to market changes means that safety stocks and therefore inventory costs may be kept at very low levels.

The total number of parts in the planning horizon is so large that we can consider the system to be in an optimal state of operation when it produces parts in the prescribed ratios at a steady maximum rate averaged over suitable time intervals [38]. The additional

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time saving possible by using schedules which minimise the makespan for the total set of parts in the planning horizon will usually be very small.

The idea of periodic scheduling was first introduced by Hitz for flexible flow shops. The sample produced periodically is called the minimal part set Its size is determined mainly by external factors like for example, when the various parts are required for assembly into a larger unit [38,39]. Then, all parts needed to assemble a number of products in an assembly period should be manufactured in a corresponding machining period to reduce inventory volumes [70]. Wittrock [71], defines the minimal part set as the smallest possible set of parts meeting the production requirements. I f , for example, the mix of the production requirements in the planning period is 3000 parts of type A, 2000 of type B and 1000 parts of type C, the minimal part set is 3 parts of type A, 2 of B and 1 of C .

The period for flexible flowshops is equal to the maximum machine work load [38,71]. Periodic release strategies for the general FMS case have been studied by a number of researchers [40, 41, 42]. These works are devoted to the analysis of the system's dynamic behaviour under periodic releasing as well as the definition of procedures to search for a 'good' periodic releasing strategy. They showed that FMSs for which the number of parts in the system is fixed by the number of available fixtures for each part type, have a periodic steady state [42]. Periodic releasing of parts to an FMS with FCFS discipline used at each machine buffer results also in a periodic steady state [40]. These works assume known part routings. Last, a study of periodic releasing of parts to an FMS without considering the detailed operations on the machines and where each part type is characterised by a given flow time has been made [41 ].The period is chosen for each part type so that it is greater or equal to the given flow time and is large enough to allow the most loaded machine of the system to perform all the work associated with the minimal part set. In a later work, detailed scheduling concentrated on strategies which take into account flow time constraints.

The approach presented here determines the size and period of the minimal part set so as to minimise the production cost as measured by the cost associated with the WIP and the cost of the system's idle time. The period is less or equal to the makespan of the optimal schedule of the minimal part set. The approach can be conceptually separated into two subproblems.i) the first is called the sizing problem and it refers to the choice of the size of the minimal part set.ii) the second, called the 'scheduling' problem, is to determine the best schedule for the minimal part set whose size was found in (i).

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Scheduling the parts belonging to the minimal part set is again a difficult optimisation problem and algorithms for its solution are discussed in detail in the subsequent chapter. For this chapter it is enough to assume that given a minimal part set, an algorithm may be employed to compute the best or a good schedule, in terms of the makespan. It is worth noting, however, that since we are referring only to a small sample of the parts that are to be produced in the planning horizon, the scheduling problem is of reduced size as compared to the problem of scheduling all the parts.

The approach can be summarised as follows:A mathematical model for the production cost per part in a minimal part set is developed. Since the cost is associated with the schedule which is the outcome of the 'scheduling subproblem', a simple iterative procedure is introduced by which the best minimal part set size and schedule is produced. The dynamic behaviour of the system under the periodic production policy is discussed and an improved approach is then examined where MPSs are allowed to overlap to the extent that can be achieved without conflict in machine scheduling. The result is a period time which is less than the MPS's makespan.

Application of the policy for FMSs with failure prone machines is also described. Hildebrant & Suri [13] and Kinemia & Gershwin [14] have developed on-line hierarchical control algorithms for FMS with unreliable workstations. In Hildebrant's approach the top level of the hierarchy chooses the part routing for each failure condition of the system by means of a non-linear optimisation algorithm. Dynamic feedback control policies are adopted by Kinemia & Gershwin. The top level of the control hierarchy chooses the part mix as a function of the failure state of the machines and the current production levels. In the approach presented here, each time the machine state changes, the first level of control calculates a new MPS for periodic production under the new system status and the production ratios required. Numerical examples to illustrate the suggested policy are presented in chapter 7 after the introduction of scheduling algorithms in chapter 6

5. 2 Problem statement

Definitions- Notations

Minimal Part Set fMPS)The periodic production of a set of parts leads to the notion of a " Minimal Part Set".

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Definition,i A Minimal Part Set is a set of integers P = (nj, n2» n j , n j ) such that:

Jn: = r: Xn;J J i=l

where rj : the production ratio of part type j J : total number of part types

Size of the Minimal Part Set fMPSlDefinition : the number of parts comprised in the MPS represent its size which is denoted by N.

JN = 2 n ji=l

Period of scheduleDefinition: the elapsed time between the time the first part of the MPS starts being processed and the time the last part of the MPS is completed, i.e the MPS's makespan, is called the period of the schedule and is denoted by Tpr.

Frequency of scheduleDefinition: the number of times the minimal part set (MPS) must be repeated in order to satisfy the production requirements is called the production frequency.

The problem to be solved for the periodic production scheduling policy can be expressed as follows:

Given a description of an FMS system ( number of machines M, buffer capacities , travel times between machines) and a desired production ( number of part types J, production ratios required rj, and operation sequences and processing times for each part type) determine :

the size and the schedule with minimum makespan for a set of parts which is to be produced periodically so that:

the production cost per part is minimised subject to the constraint that parts are to be produced in the prescribed ratios.

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5.3 A mathematical model for the production cost per part when producing in Minimal Part Sets

Since our objective is of a scheduling nature, the production cost per part refers to the average scheduling cost; it is therefore equal to the total cost of the MPS's schedule divided by the number of parts in the set. The total cost of a schedule is a complex combination of capital costs, inventory costs, idle-time costs and costs due to late delivery. RinnooyKan [72] and Gupta and Dudek [73] discuss the form of total scheduling costs. We concentrate in this analysis on the inventory and idle time costs as the main cost elements although the analysis may as well be applied to a more complex cost formula. Lateness costs are ignored on the assumption that parts produced by the system are delivered before their required delivered time. The analysis will remain general; any practical situation will exhibit its own characteristic features which must explicitly be taken into account.

a) With respect to idle time costs, the machine with the highest utilisation, i.e the minimum idle time, can be taken to be a measure of the system's idle time. This is used to calculate the idle time costs. The choice of this cost rather than the machine idle costs is justified if we consider that, assuming that there is a machine such that its idle time is zero, the system performance cannot be further improved by rescheduling or by an increase in the MPS size; the only possible improvement could arise from a different product mix or machine mix. Hence, the idle time cost is taken to be equal to:

iq min {Tpr- £ 4 } m eM p ie Im

rather than I K j( Tpr - X t | ) meM i s lm

where Im : the set of operations scheduled on machine m M : the set of machines tj : processing time of operation i

and Kj represents the opportunity cost of business forgone per unit idle time which is the prime costs of idle resources. Capital costs and fixed costs associated with the machines can be ignored since we are dealing with an operating decision rather than an investment decision.

if we denote by T ^ the system's idle time, then the idle time cost is equal to Kj T ^

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b) Inventory costs include the work-in process inventory cost whichis t*icur?<iffom the time the part enters the system to its completion, and the finished inventory cost which «s incured during the period starting at the part’s completion and ending with the part's delivery. The inventory cost is calculated here for the two extreme cases of:1. The case where the production in MPSs is viewed as equivalent to the traditional batch production i.e raw material for the production of all parts in the MPS is presented at the system's input buffer at a certain time and parts are exiting the system after all parts in the set have been completed, and2. The case where parts are issued at the system's input buffer as soon as they are required to start their first operation and exiting the system as soon as they are completed, irrespective of the periodic production in MPS.Practical cases lie between those extremes and relevant examples are studied at chapter 7. The inventory cost may be expressed mathematically for these two cases as follows:For the first case the inventory cost is equal to:

JTpr J n jC j

where cj is the inventory cost per time unit for the part of type j

To simplify the formula, we take the average in-process inventory cost rate which we denote by C j, i.e

JQ= (X n: C: ) / N

j=l J J

In the case where Cj = constant for every j, Cj = Cj

Therefore inventory cost for the first case is equal to:N Cj Tpr

and for the second case is equal to:NCj [ (X F„ ) /N ]

pe P ^where Fp is the flow time for part pand Cj is the inventory cost per time unit per part

The expression in parenthesis is the mean flow time for the parts in the MPS and let us

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denote it by F. Therefore the inventory cost for the second case is equal to N Cj F

For the first case, the size of the MPS is a measure of the inventory level whereas for the second case, the inventory level is merely determined by the system's local and central buffer capacity. A large inventory helps to ensure that parts are always available for processing so that machine utilisation is kept high. However, high inventory levels mean that buffer storages should be of large capacity and material handling systems must be devoted to storage. In addition, working capital is tied up in the parts and is not recovered until the processing of the parts is complete and the inventory is sold. A low inventory level means that machines might starve, in which case valuable capital is under-utilised. The best inventory level should strive a balance between the inventory cost and the idle time cost

The total cost C, of producing all parts in the MPS of size N, comprises the cost of inventory and the cost of the system's idle time. Mathematically it can be expressed as follows:

C = N Cj Tpr + Kj Tj^ for set deliveries or C = N Cj F + Kj Tj^ for part deliveries

and the cost per part is:

K1 Tidc/n = q ( t +-------- )F q n

orK1 Ti clC/N = Cj ( F + -----------)q n

Let q denote the ratio of the cost rate for idle resources to the inventory costrate i.e : Cr = Kj / Cjand assume that Cj =1 with no loss of generality.

Then, the production cost per part of the MPS is given by :

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C /N = Tpr + Cr (Tid /N ) (5.1)or

C /N = F + q . (Tid /N )

We want to minimise (5.1) with respect to N. The value of N which minimises (5.1), the production cost per part, gives the best MPS size. If N is known, then nj's can bedetermined from the equadon nj = rj N and therefore the Minimal Part Set is fullydescribed. Once determined, the MPS is produced periodically until the production requirements for the planning horizon are met

5.4 Nature of the objective function

The makespan Tpn the mean flow time F and the system's idle time Tjd correspondingto a MPS are interrelated and depend on the value of N (i.e the size of the MPS), and its schedule. Three numerical examples are employed here to illustrate the tight relationship between N and T^d , and N and Tpr and F. Three types of parts in equal productionratios are assumed to be produced by a system comprising five machines. Routings and operation times are created randomly. The schedule associated with each part set is produced by the algorithm introduced in the next chapter. Figure 5.1 shows the relationship of the percentage system idle time Tjd to the MPS size N.

system idle tim e vs MPS's size50

example 1 example 2 example 3

0 10 20 30number of parts In the MPS

Figure 5.1: Relationship of the percentage system idle time to the MPS size

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Results show that idle time is reduced in an exponential way with increasing number of parts, reaching a minimum value for some N, something one would intuitively expect when the system becomes overloaded [74]. In example 2, zero idle time is achieved for sizes bigger or equal to twelve.

Figure 5.2 shows how the makespan and the mean flow time of parts in the MPS is related to the size of the MPS.

H3- makespan 1 mean flow 1

-B- makespan 2 mean flow 2 makespan 3

-o* mean flow 3

Figure 5. 2 : Relationship of the MPS's makespan and mean part flow time to its size

There is a close almost linear relationship between Tpr and N. Increasing values of Nresult in bigger completion times. The relation of N to the mean flow time F appears to be linear for the range of small sizes but with a smaller slope compared to the slope of makespan. It is expected, however, to become constant on average for some big size in a steady state (e.g example 3 in figure 5.2) because the system is then overloaded. Results of other studies on the relationship between the mean flow time and the number of parts in the system portrayed also a linear relation [74].

Idle time costs and inventory costs added together give the total cost whose general shape for example 2 is given in figures 5.3 and 5.4 for the two extreme cases. The cost per part is displayed in the figures for different Cr values. The cost ratio Cj. is animportant parameter, its value depending on the particular application, in determining the location of the minimum i.e the optimal size for the MPS. The higher the cost ratio the

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bigger the size of the optimal set.

cost per part for varying cost ratio

-o- costratio=5 costratio=10

-»■ costratio=30 costratio=50

Figure 5.3: Cost per part for varying cost ratio with set deliveries

cost per part for varying cost ratio

■Q* cost ratio=5 cost ratio=10

-®- cost ratio=30 -©- cost ratio=50

Figure 5.4: Cost per part for varying cost ratio with part deliveries

The slope of the cost function in sizes bigger than the optimum is greater in the case of set deliveries (figure 5.3) as compared to the case of individual part deliveries (figure5.4). As a result, production in sets of optimal size when deliveries are in sets, is important, since deviation from the optimal size in both directions ( towards smaller or

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bigger set sizes ) would mean substantial cost penalties. On the other hand, the individual part deliveries case should involve set sizes which enable the system to be utilised at its highest, since idle time costs are important to minimise, but cost penalties for producing in sets greater than the optimal are not severe.

This is so because increased set sizes leave the mean flow time and consequently the inventory cost almost unaffected since, by definition, parts are not introduced in the system before they are ready to start their first operation according to the set's schedule and they leave the system as soon as they are completed. Nevertheless, a small set size requires less memory and computational time in the search for a 'good' schedule, and unless externally imposed reasons require production in greater set sizes, the optimal size is recommended.

For low cost ratios, in the case of part deliveries, the cost may become very flat over an extensive range of size values rendering the optimisation of the set size cost ineffective (figure 5.4). However, low cost ratios imply a closeness in values for the cost rate for idle time and the cost rate for inventory. Consequently and due to the high capital equipment involved in a FMS, a very high value for the inventory cost rate is implied. The inventory cost rate is related to the part's value and production of very valuable components is not a frequent case in practice.

Figure 5.5 : Cost per part for set and part deliveries

The case of part deliveries is equivalent to zero finished inventory, a situation which is

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most unlikely to occur in practice; the extra cost for the finished inventory which would occur in practice, would push the cost function towards the case of figure 5.3. Figure5.5 shows the cost per part for example 2 with cost ratio equal to 30 for the two extreme delivery cases. The area between the two curves is the area where the cost per part for a real case will lie.

5.5 Discrete nature of the optimising parameter

The number of parts in the MPS, N , is the unknown variable of equation (5.1) which we wish to define. This is a discrete variable with a fixed step. It is integer since it refers to the number of parts in the MPS but it does not take all integer values since parts in the MPS should satisfy production ratios. It takes values from the set:

{ AN, 2AN, 3A N ,..... }

where AN is the smallest MPS.

If Tpr , F and T ^ could be expressed analytically in terms of N, equation (5.1) could be solved mathematically and the minimum N found. However, in this approach, N values are mapped to Tpr and T ^ values via an algorithm which is here the algorithm introduced in the next chapter, to find the best or a good schedule for the N parts with which Tpr , F and T ^ are associated. Thus, a simple iterative procedure is introduced next for the determination of the best N .

5.6 Solution procedure

The method suggested here solves the problem using an iterative procedure. Starting from an initial value of N, an algorithm is used to produce a schedule for the N parts . The corresponding values of Tpr and Tj^ are then fed into equation (5.1) to calculate theproduction cost per part corresponding to the current N value. The optimal N can be found in a straight forward way i.e by trying increasing values of N from its set of permissible values starting from the smallest value until the minimum production cost per part is found, or with the use of a simple algorithm. The procedure is formally described below. It uses a simple algorithm to reduce the number of iterations.

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1. For N = N0 = AN and N = N = k AN where k is sufficiently largei) use an algorithm to find

a schedule S(NQ) so that Tpr (N0) is minimised ( or (NQ)) and a schedule S(Nj) so that Tpr (N j) is minimised (or Tj^(N j))

2. Given the ratio of cost rates Cj. calculateC(N0) = Tpr (N0) + C, ( Tid(N0) / N0)C(Nj) = Tpr(N!) + C,. ( Tid(N[) / N0 )

3. Calculate the new N' from the equation:NoCCN^ + N! C(N0)N'= ------------------------- —

C(N0) + CCNj)

If N' & k AN for some ke N thenfind keN such that /N - kAN/ is minimum and set N'= kAN

end if

If C(N') > min { C(N0) , C ^ ) }set N0 = N'and = kAN > N'

set N0 = N’ and Nj = N'Nj = N t N0 = N0

endif

4. Go to step 2 until C(N') - C(N0) < 8

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5.7 Dynamic behaviour of the system under the periodic production policy

The case of set deliveriesParts are released to the system's input buffer in sets of size and content prespecified by the MPS periodically. Output is then periodic in the steady state with a period equal to the releasing period and steady state starts with the first set periodically released.That is, the v-th set is released at time (v-l)Tpr and exits periodically every Tpr time units.The case of individual part deliveriesParts are released to the system's input buffer in a sequence and times determined by the schedule of the MPS. The sequence is repeated periodically with period Tpr Part p of the v-th MPS will be released in the input buffer of the system at time

(v ~ l) Tpr + Tp

where Tp is a constant lower than TpP characteristic of part p and which can be found from the MPS's schedule.The succession of parts in the MPS ordered with the times Tp will be named the input or loading sequence:

(Pl» Tj ) , (p2, )> ••••’ (PN» W

with Ti - x2 < Tpr

For example, for the MPS's schedule shown in figure 5.6 , the input sequence of parts is: ( 1, 0.0 ) , ( 2, 0.0 ) , ( 3, 2.0 )

M1 1 3 2

M2 2 3 1

M3 2 1 30 1 2 3 4 5 6

Figure 5.6: An example MPS schedule

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Let us denote by ep the time that part p of a MPS is completed. Then Fp = ep - is theflow time of part p. For the example of figure5.6^ 7.0, F2= 5.5 and F3 = 4.0It is obvious that a loading period lower than the part's flow time will induce an unlimited increase on the amount of work waiting to be performed at the shop's input. To avoid this happening, we must have a releasing period greater or equal to each part's flow time. Then, there exists a periodic steady state output with a period equal to the releasing period. This steady state starts with the first part periodically released [41]. x

The loading period in this approach is equal to the makespan of the MPS which clearly satisfies the above requirement. Thus, the system's steady state output is again periodic with the same period starting with the first part periodically released.Part p of the v-th MPS will exit the system at time

(v- l ) Tpr + epwhere ep < TprThe succession of parts in the MPS ordered with the times ep will be named the outputsequence. This is in general not the same as the input sequence. For the example of figure 5.6, the output sequence is: ( 2, 5.5 ), ( 3, 6.0 ), ( 1, 7.0 )

5.8 Improved periodic releasing for FMS

Figure 5.7 shows the schedule of the production of two successive MPS for the example of figure 5.6.

M1 1 3 2 1 3 2

M2 2 3 1 2 3 1

M3 2 1 3 2 1 30 1 r> 3 4 5 6 7 8 9 10 11 12 13

Figure 5.7 : Schedule of two successive MPSs

It is clear that the two schedules could overlap without creating a machine conflict as it is shown in figure 5.8.

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M1 1 3 2 1 3 2

M2 2 3 1 2 3 1

M3 2 1 3 2 1 30 1 2 3 4 5 6 7 8 9 10 11 12

Figure 5. 8 : Improved schedule of two successive MPSs

Thus unnecessary idle time could be avoided by improving our periodic production as in figure 5.8. The improved period is now less than the MPS's makespan and is defined formally as follows:Tper = Tpr - min { xm + ( Tpr - em ) }

where xm is the starting time for machine m in the MPS's scheduleand em is the finishing time for machine m in the MPS's schedule

The production period becomes equal to the MPS's makespan when the minimum time allowance in the MPS's schedule is zero.

The releasing time for each part is now as described in the previous section but with the improved period i.e is equal to : ( v -1 ) Tper + XpSimilarly the the time a part is exiting the system is equal to : ( v -1 ) Tper + Fp

5.9 Periodic production in an FMS with failure prone machines

Flexible manufacturing systems, like all manufacturing systems are subject to random disturbances in the form of machine failures and repairs, material unavailability, 'hot' items or batches and other phenomena. These disruptive effects complicate an already difficult optimisation problem. In this section, the application of the periodic production policy in a system with failure prone machines, is examined.

Naturally, a MPS whose size and schedule are best for the case of full machine availability can not in general be maintained during a breakdown. What is then needed, is

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to calculate a new MPS based on the new system status and the current production ratios required.

The production target is specified for each part type j as Qj parts of type j which have tobe produced in the planning horizon (PH). The total number of all parts to be produced in (PH) is denoted by Q.

That is, Q = S Qi]e J J

instantaneous production ratios required, are calculated from production requirements as follows:

qj=Qj /Q

Let us denote by Wj(v) the cumulative production of part type j by period v. The cumulative production must equal Qj in the end of the production. In order to attain this objective in a fully reliable system it is enough to set the actual production ratios rj , for which a MPS is determined, equal to qj.

Let us now examine the case with failure prone machines.AssumptionThe scheduling problem of the transition period between the occurrence of a change in the machine operational state (i.e a machine breakdown) and the beginning of production of the new minimal part set is ignored. The implied assumption is that in practice, parts belonging to the set which d re being produced when the breakdown occurred are completed using alternative machines or scrapped.

Machine state- DefinitionDenote by the operational state of machine m. Then am is defined as :

am =01

if machine m is down at period v if machine m is up at period v

If there are several identical class machines am is equal to the ratio of the operational machines to the total number of identical machines.

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In general, the machine state is defined b y : a = ( aj, ^ ..., aj^)

Feasibility of production ratiosA production ratio is feasible if it satisfies the constraints :

JSum* ri ^ am Vm e M j= l

and rj > 0 Vj

where umj “if part type j does not use machine m if part type j uses machine m

Thus, if am = 0 for some m, all rj which use machine m are set to zero.

As machines fail or get repaired i.e as the machine state changes, the set of feasible production ratios change. Production of those parts requiring the broken down machines falls behind. When those machines are repaired, the actual production ratios should compensate for the production loss. The objective is to compute the production ratios so that cumulative production equals the required production Qj in the end of the productionhorizon. This is achieved by computing the actual production ratios rj dynamically, each time there is a change in the machine state.

Suppose that there is a change of the machine state at the end of the v-th schedule period for the production of the v-th MPS. The new machine state is a. Then:1. Calculate the current qj required as follows:

Qj = Qj - w j(v)Q'= iQ 'jj s / Jqj = Q j / Q '

2. The actual production ratios are then found as follows:n = min { q : . ajn }

meMV j

am e awhere

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For example, in case ^ = 0 for some machine m which part type j should visit, then rj = 0. On the other hand, if there is full availability of the machines required for part type j, then rj = qj , in which case the actual production ratio equals the currently required production ratio.

When the actual production ratios are determined, then the best MPS and its schedule will be recomputed and implemented until the next change of the machine state.

Ill

CHAPTER 6

THE FMS PART SCHEDULING PROBLEM

6.1 Introduction

The complete FMS scheduling problem might be a difficult problem to solve because of its complexity. Many researchers decomposed it into subproblems which should be solved more easily. They mainly use a two stage procedure:

1) machine loading2) part sequencing

The machine loading problem refers to the allocation of operations and tools to machines. It is stated formally according to Stecke [33] as follows:

Allocate the operations and required tools of the selected part set among the machines (or machine groups) subject to technological and capacity constraints. Machine groups refer to machines with identical tooling so that they can be used interchangeably.

The machine loading problem in FMSs has been formulated and solved by a relatively large number of authors, one of the most complex non-linear integer programming formulations being presented by Stecke [33]. A number of linear integer programming formulations are discussed by Kusiak [75]. The non-linearity in the formulation of Stecke was due to the design of the tools and tool magazines. A common practice in machine loading is to balance the workload of the workstations. However, it has been shown that this is not always the optimal [31].

Machine loading simplifies the part sequencing problem since the sequence of machine visits is established before the parts are dispatched into the system. However, the flexibility in determining the sequence of machine visits in real time as the part makes its way through the system is lo st. Thus, the optimal solution for these two problems is not necessarily optimal for the complete FMS scheduling problem. The dynamic determination of machine visits is equivalent to the use of alternate operations.

An alternate operation could be used if one machine is temporarily overloaded while another is idle. Then, alternate operations may be used to offload bottleneck machines to speed up the flow of parts and balance utilisation, even though a time penalty may incur. Time penalties usually incur for two reasons :1) due to tooling assigned to the machine center. For example, a two inch milling cutter might be used instead of a four inch cutter, but a longer processing time will be required.

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2) in case a system is provided with a tool transport network so that tooling is not fixed, a machine may be used for an alternate operation after the tool needed is transported to it. Then,the incurred time penalty refers to the tool transporting time.

Pooling of machines into groups (assigning identical tooling) improves the system's performance [54, 31], but can only be implemented in case the required tools can be accommodated by tool changers. In other cases, improvement of performance requires the use of alternate operations.

The importance of alternate operations has been indicated in studies made for the conventional job shop [76] and for FMS [77]. However, none of these studies investigates the case of dynamic tooling assignments for systems provided with tool transport. Iv/a.ta [69], devised a heuristic dispatching algorithm for the dynamic selection of machines, tools and transport devices .

The part sequencing problem resulting from the loading problem refers to the problem of finding the sequence of parts on each machine when the sequence of machines that each part should visit, is fixed. It is the same as the classical job-shop problem. In some cases, this problem might be easy to solve, e.g when there is only one operation to be performed on each part or each part can be processed on one to three machines [34]; in this case the part sequencing problem is decomposed into scheduling subproblems with one to three machines respectively. Scheduling problems with one or two machines and the special case of scheduling with three machines can be very efficiently solved. Algorithms of complexity nlogn are available for solving the single machine scheduling problem [78,79] When this is not the case, simulation has been mainly used to study the loading problem in conjunction with various dispatching rules for part sequencing[80].

In this chapter, a general model for the static job scheduling problem is presented together with a review of all the main schedule generation and search procedures. These procedures are presented in an algorithmic form which makes them easily translatable to code-programs and allows direct comparisons to be made. The relative strengths and weaknesses of each procedure are discussed in detail. A new algorithm is then proposed which allows for full flexibility. Machine loading and part sequencing are studied in conjunction. The algorithm is based on a simulation model for FMSs. Simulation is considered to be an appropriate tool for the composite problem of the complete FMS scheduling whereas a mathematical model would be very complex. The proposed algorithm is original in that it combines simulation techniques with search procedures to find a solution for the complete FMS scheduling problem.

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Simulation studies have investigated different aspects of the FMS scheduling problem like the effectiveness of dispatching rules, the effect of different arrival rates on workload balance, the control of release of parts in conjunction with the dispatching rules to achieve a balance on the workload [80] and others. However, none of them used simulation to formally develop an algorithm for the solution of the complete FMS scheduling problem.

The FMS scheduling problem is analysed here in a deterministic context. Processing and travel times are assumed known. The problem can be expressed as follows :

< Produce in a time efficient manner a set of N parts which belong to a specific range of part types with prescribed constant ratios. Each part type requires a prescribed sequence of operations. Any number of operations are allowed to a given part type and each machine is able to perform one or more operations. >

This problem is identical to the classical static deterministic job-shop scheduling problem although the most common formulation of the classical job-shop specifies that each part has M operations, one on each machine. However, the most general case presented here is conceptually no more difficult. The problem is static, because the number of parts and their ready time are known and fixed in advance. It is deterministic, because processing times and all other parameters are known and fixed. The performance measure is the total production time or makespan. This is equivalent to the mean machine idle time and the mean number of parts actually being processed [79].

The closeness of the mathematical theory of scheduling for FMSs and job shops leads us to present the job-shop theory first and then it is easy to follow the development of this theory in the FMS context. Thus, the first few sections of this chapter give a brief survey of the conventional job-shop problem.

6.2 A general model for the static job shop scheduling problem

The principal assumptions underlying the job shop model are the following:

1. No machine may process more than one operation at a time.2. Each operation once started must be performed to completion.3. No part may be processed by more than one machine at one time.4. A known, finite time is required to perform each operation, and each operation must be completed before any operation which it must precede can

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begin.5. Processing times of operations are independent of the order in which the operations are performed.6. Machines are continuously available.7. All parts are ready to start processing before the period under consideration begins.8. The time required to transport parts between machines is negligible.9. operations are processed as compactly as possible subject to technological constraints.(i.e. no superfluous idle time exists)

The model can be stated as follows:1. A set of machines is given and designated: M = { 1, 2 ,.., m ,.., M }

2. A collection of parts to be scheduled is given and designated:P = { 1 ,2 , . . , p , .., P }

3. Each part consists of several operations.i denotes the operationIp is the set of operations of part pIp is the number of operations of part p/ is the set of all operations and,I the total number of operations

An operation is usually designated by the triplet (p, i, m) to denote the fact that operation i of part p requires machine m.The numbering of operations is usually sequential for a given part to indicate the linear operation sequence.

tjp : the time required for operation i of part pTpr : the makespan or the time the last operation is completed

4. The machine sequence matrix is givenwith dimension P x max IDpeP por in case at least one part visits all machines once, the dimension is : PxM

5.It is desired to find :The part sequence matrix which gives the sequence of parts on machines which is optimal according to a measure of performance.

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with dimension M x max Im where Im is the no. of operations me A/ allocated to machine mor in case at least one machine is visited by all parts once, the dimension is MxP

Gantt chart representation of a scheduleA useful notation related to this model is the Gantt chart which is a graphical description of the job shop problem. The Gantt chart consists of a collection of blocks, each of which is identified by a part-operation-machine triplet. The length of the block is equal to the processing time of the associated operation. If the operation blocks are placed as compactly as possible on the Gantt chart in some arbitrary fashion as in figure 6.1a , the chart describes the workload for each machine but is unlikely to represent a valid schedule. A schedule is a feasible resolution of the resource constraints when no two operations ever occupy the same machine simultaneously and in addition no two operations of the same part are processed simultaneously by different machines. A feasible schedule is shown in figure 6.1 b . Data for this example are given in tabular form in table 6.1. For simplicity of representation the number on each block refers to the part number only.

M l

M 2

M 3

1 2 3

1 2 30 10 12 142 4 6 8

tim e ax isFigure 6.1a : Graphical representation of machine workload

16

Figure 6.1ft : Gantt chart of a feasible schedule

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Tpr = 15 for this schedule

Part operation sequence and duration(operation type. : operation time)

1 2 :2 3 :4 1 :32 2 :3 1 :3 3 :13 1 :4 2 :2 3 :4

Table 6.1 : Routing and processing times for the example of figure 6.1

A Gantt chart can also be given in a matrix form as follows:Assume that time is measured in discrete units. Then a Gantt chart is defined as follows:

G = { Gm>t} m = 1, 2, .., M

t = At, 2 A t , T prp (the part index) if operation (p, i, m) is processed on

machine m at time t 0 otherwisewhere 'm,t

An example of such a representation is given in table 6.2 for the schedule of figure 6.1 b

Ml 3 3 3 3 0 0 0 0 0 2 2 2 1 1 1 M2 1 1 0 0 3 3 2 2 2 0 0 0 0 0 0 M3 0 0 1 1 1 1 3 3 3 3 0 0 2 0 0

Table 6.2 : Tabular representation of a Gantt chart

Most researchers are working with models which are somewhat less complex than the one just described by making various simplifying assumptions to arrive at an amenable analytic treatment of the problem. One direction of simplification is to reduce the number of machines considered and/or to assume that all jobs have the same routing which is the case of a flow-shop.

There are two major approaches to deal with the job-shop scheduling problem: Combinatorial approaches and mathematical programming.

Mathematical programming deals with constraint optimisation problems. The scheduling

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problem can be translated into a mixed integer programme. Integer programming formulations of scheduling problems are discussed in [36, 78, 72, 79]. However, integer programs are not easier to solve than the original problem. Methods of solving integer programs are reviewed in Garfinkel & Nemhauser [90]. Those methods can be classified as either implicit enumeration (branch and bound) or cutting plane. Both require much computation. Moreover, they are based on properties of integer programs in general and pay no regard to particular properties of the problem being solved. As a result they tend to take longer to find a solution than those designed specifically for a particular class of problems.Thus, most texts on scheduling suggest a direct approach rather than an indirect via integer programming. [78,79]

Next a review of the combinatorial approaches is given.

6.3 Review of combinatorial approaches to job shop scheduling

6.3.1 Problem description and solution subsets

Nature of the problem : Operation scheduling for processing P parts on M machines is a combinatorial problem in nature since there are P! different part sequences possible for each machine, and therefore (P !)^ different schedules amongst which an optimal solution exists according to a certain measure of performance. The actual number of schedules is usually smaller than (P !)^ because of the sequence or technological constraints. Nevertheless, the significance of the size of this set is that it discourages the full enumeration of solutions. The question arises, whether, there is an efficient algorithm that could give the optimal solution for this set.

N-P completeness : It has been shown [91], that for P>2 and/or M>2 the generaljob-shop problem is a N-P complete problem. The implication is that this problem can not be solved by a polynomial bounded algorithm but only by some form of enumeration(tree search). Clearly, it would be helpful to be able to ignore many of these schedules so that to reduce the search for an optimum to more manageable sets.

Subsets of the solution set: A number of 'good' subsets of the solution set exists and are reviewed next. Most of the combinatorial algorithms developed, search for an optimum in one of these subsets. Subsets of the solution set by decreasing size are:

i) the set of semi-active schedulesii) the set of active schedulesiii) the set of non-delay schedules

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Set of semi-active schedules

Definition : A schedule is called semi-active when no operation can start earlier in time without altering the operation sequence on any machine.

Adjusting the start time of an operation while preserving the operation sequence is equivalent to moving the operation block to the left on the Gantt chart. This type of adjustment is called a local left shift. An example of a semi active schedule is given in figure 6.1 b . In principle, one could insert an arbitrary amount of idle time at any machine between adjacent pairs of operations. It is clear, however, that once the operation sequence on machines is specified, this kind of idle time can not be helpful for any regular measure of performance.

The set of semi-active schedules dominates the set of all schedules, which means that it is sufficient to consider only semi-active schedules to optimise any regular measure of performance.The exact number of semi-active schedules is difficult to determine but if the sequences on each machine were entirely independent there would be (P !)^ semi-active schedules. However, technological considerations render some of these combinations of sequences infeasible.

Set of active schedules

Definition : A schedule is called active when there is no operation which could start earlier without delaying any of the other operations.

Figure 6.2 : Gantt chart of active schedule resulting bv altering the semi-active schedule of figure 6.1^.

Tpr =13 for this schedule

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In the example of Figure 6.1 b, operation (1, 3,1) could be shifted to the left and beyond operation (2 , 2 , 1) already scheduled on machine 1. The resulting schedule is shown in Figure 6.2 . This is an active schedule since no other left shift of this kind can be made. This type of adjustment is called a global left-shift. Therefore, the set of active schedules can be defined as the set of schedules in which no global left-shift can be made and is clearly a subset of the semi-active schedules.

The number of active schedules still tends to be large but it is a dominant set. A set is dominant when it contains the optimum and this has been proved for the set of active schedules[92]. It has been shown [92], that to generate active schedules, one must consider at each step only those schedulable operations for which the earliest possible starting time is less than or equal to the minimum of the earliest completion time.

Set of non-delay schedules

Definition : A schedule is called a non-delay schedule where no machine is kept idle at a time when it could begin processing some operation.

For example, in Figure 6Ab , machine 2 at time 2 remains idle when it could start processing operation (2, 1, 2). In figure 6.3, the schedule is active and non-delay.

M l

M 2

M 3

3 2 1

1 2 3

1 3 20 2 4 6 8 10 12 14 16

tim e ax isFigure 6.3 Non-delav schedule of the example given in table 6.1

Tpr = 12 for this schedule

All non-delay schedules are active since no left shifting is possible. On the other hand, many active schedules may not be non-delay. This means that the number of non-delay schedules can be significantly less than the number of active schedules. However, there is no guarantee that the non-delay subset will contain the optimum. Nevertheless, the best non-delay schedule can usually be expected to provide a very good solution if not an

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optimum. [78]. Non-delay schedules are generated if at each step we consider only the schedulable operations with earliest starting time equal to the minimum of the starting times.

6.3.2 Schedule generation procedures

General notation and methodologySchedule generation procedures are fundamental to a variety of solution techniques for the job-shop problem.Most of them treat operations in an order consistent with the problem’s precedence relations. In other words no operation is considered until all its predecessors have been scheduled.Schedulable operation : An operation (p, i, m) is said to be schedulable regardless of the actual time at which the next scheduling decision is required, once all its predecessors are scheduled.Partial schedule : A partial schedule is made up by those operations which have already been assigned a starting time.Given a partial schedule^ unique corresponding set of schedulable operations may be constructed.

In general, we can assume that a partial schedule consists of a vector: pd = ( sl» s2» sd ) for d = 0, 1, 2,..., I

Each s^ is a member of a finite linearly ordered set i.e

sd: the set of schedulable operations corresponding to the partial schedule P ^ i-

An exhaustive search must consider the elements :S i xS2 x .... x Sj

as potential solutions. A way to organise such an exhaustive search is backtracking.

Exhaustive search procedure bv backtrackingBacktracking is a general technique used in a wide range of search problems including scheduling, game playing, parsing e t c It works by continually trying to extend a partial solution. At each stage of the search, if an extension of the partial solution is not possible we backtrack to a shorter partial solution and try again.Initially, we start with the null vector as our partial solution. We then calculate Sj and

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according to the type of schedule that we want to generate/ we find those members of S j

which are candidates for s^. All candidates for s^ belong to a set which is designated by

A j . Clearly, S j z> A . We choose s^ e A j so that we now have the partial solution ? i = ( s^). In general, A^ comprises candidates for the extension of the partialsolution (sj, S2, s ^ . j ) to (sj, S2, s ^ , s^). If A^ = 0 , which means that there are not any member candidates for s^, we backtrack and make a new choice for s^.j and so on.

It is helpful to picture the process in terms of the depth- first traversal.The subset (S j x ^2 x ...X ) for d = 1, 2, .. ,1 can be represented by asearch tree as follows: The root of the tree ( the Oth level) is the null vector. Its sons are the choices for s and in general, the nodes at the dth level are the choices for s^ giventhe choices made for s^, S2,.., s^_ j as indicated by the ancestors of those nodes.

start

choices for s i

choices for s2 given s i

choices for s3 given s l& s2

choices for s4 given s i , s2, ands3

Figure 6.4 : The search tree of partial solutions

The tree and the backtracking procedure is shown in figure 6.4. Backtrack traverses the nodes as indicated by the dashed lines. The backtracking algorithm for exhaustive search of schedules is described formally as follows:

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Denote b y :

Algorithm 1

Let

while d>0 DO

a - : the earliest time at which operation i e could be started

<j>i: the earliest time at which operation i e could becompleted

: backtracking for exhaustive search of schedules

d = 1Po = ()Ad =Sl

while Arf * 0 <

DO

the null partial schedulethe set of operations with no predecessors9 sidestep- Choose s^ g Arf

~Ad =Ad - l sd>-Sd = Srf { s^ }- = ( s^, S2,.., s^)If is a solution then record itelse advance- d = d + 1- create S^ by adding the direct successor of operation s^ to S^_j

- Determine <j)* =min { } for the generationieS o f ac tive schedules

or

-Determine

G * = m in { G j } fo r the generation ie Stf o f n o n -d e la y sched

m : the m ach in e on w h ic h (J) o r G * c o u ld be realised

-Find Arf whose elements require machine m & for which Gj < <j) or Gj = G (5^ )

backtrack - d = d - 1

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Branch and bound procedure

Branch and bound procedure is a well -known variation on back track to curtail search by branch pruning; i.e by removing subtrees from the search tree. Curtailing is based on the assumption that each solution has an associated performance measure and the optimal solution is to be found. In order for the branch and bound to be applicable ,the performance measure must be well defined for the partial solutions and in this case is called the bound.For all partial solutions and all extensions we must have :

bound ( sj, S2, < bound ( sj, S2,.., s ^ , sd)The bounding procedure involves the calculation of bounds for the partial solutions of the current level; A complete solution is also obtained with an associated performancevalue © . It is called trial solution and it may be obtained in the course of the tree search by pursuing the tree directly to the bottom as rapidly as possible. The boundingprocedure will be to compare the lower bounds at each node with the value of 0 for the current trial schedule. Then, a partial solution can be discarded if its bound is greater or equal to the performance value of the trial solution; i.e :

if bound (P^) > © then is discarded where © : performance measure of the trial solution.It is clear that if bound(P^) > © , it is not possible to improve upon the trial solution by exploring that branch further and hence we eliminate that node and all nodes beyond it inthe branch. If boundCP^) < 0 the node can not be eliminated and the branch must beexplored beyond i t If at the final node we find that the schedule has a value 0 less than that of the trial schedule then this schedule becomes the new trial schedule. Eventually, all nodes will be explored or eliminated and the trial solution that remains must be optimal.

The performance measure which has received substantial attention in job-shop research is the makespan. Basic work in this area was carried out by Brooks and White [93]. Their approach in the calculation of bounds recognises the complementary roles played by jobs and machines. They suggest the use of a job based bound and a machine based bound. The calculation is based on a given partial schedule P^ and an associated set ofschedulable operations .

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Analytically, a job based bound is : bj = max { Gj + U j} ieSda machine based bound is : b2 = ma x ( f m + Ym )

IDEMwhere U j: the unscheduled processing time for the job corresponding to operation i where fm : the latest completion time of an operation on machine m and Ym : the unscheduled processing time that will require machine m

The final bound suggested is B = max(b^,b2)

The machine based bound can be improved slightly if fm is replaced by the earliest time at which some unscheduled operation could begin on machine m . If this time is denoted by f m then

b 2 = m a x ( f m + w m }me m

The determination of f m requires examination of not only the operations in set butpotentially several of their successors as well. Nevertheless, the improvement in the lower bound may be worth the effort.

Branch and bound is easily incorporated into the general back track algorithm. The formal description of the algorithm is given next.

Algorithm 2 : Generalised branch and bound

Let d = 1P o - oAd = S i

min Tpr = ©o bound = 0

the null partial schedulethe set of operations with no predecessorsthe minimum makespanthe lower bound

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SidgStSB

whiled>0do4

-choose s Arf -Ad =Ad " ( sd )

-s d = sd - 1 sd )" Pd = (®1* s2>-' sd)-bound=bound(P(j)

while

-if is a solution and bound< minTpr theni) save as the schedule with the minimal Tpr so farii) put minTpj. = bound

< else advanceAd * 0 & d = d + 1bound<minTprdo

create 5^ by adding the direct successor of operation s^ to

- Determine <j>* = min { } for the generationie S d of active schedulesa* = min { G j} for the generation i g Sd of non-delay sched.

- Determine m*: the machine on which <j>*o r G *c o u ld be realised

backtrack - d = d -1

-Find A a whose elements require machine m* and for which < <p or Gj = a ( 5^ □ Ad )

bound = boundCP^)Lageweg reviewed implicit enumeration techniques for job-shop scheduling and came to the conclusion that only very small problems can be solved optimally within reasonable time [94]. For large problems even controlled enumeration applied to the non- delay tree may be impractical. In such cases, it is necessary to consider heuristic methods.

H euristic methods

For large problems, the vast computational effort required for enumerating the schedule

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tree, may render such an approach infeasible, even when the enumeration is curtailed by a branch and bound scheme. At the other extreme, a suboptimal approach that generates only one complete schedule might involve a light computational effort even in large problems. Heuristic methods can be used to find an approximate solution and they lie somewhere between these two extremes.

1. Priority dispatching rules

The schedule generation procedures may create several branches in the tree of partial schedules. In the case of active schedules, a branch is created for each operation with<Ti«t> and in the case of non-delay schedules for each operation with = a . In aheuristic procedure, one full schedule is only generated and therefore one branch is only created by using a priority rule. A generalised heuristic algorithm can be formally described as follows:

Algorithm 3 : Generalised heuristic dispatching

Let d = 1P q = () the null partial schedule

Ad = S 1

DO

- V s^ g Arf calculate a priority index according to a specific rule- Find s^ with the smallest index- P ^ = ( s ^ , S 2 , . . , S j j )

- If P^ is a solution STOPadvance- d = d + 1

i - Create by adding the direct successor of operation s^ to S ^

- Determine <J)* =min {<j): } for active dispatchingieSd

or a* =min { a; } for nondelay dispatching

-Determine m*: the machine on which <{>* or c*can be realised- Find Arf a subset of S^ whose elements require machine m

sjc *and for which a- < (|> or C |< o

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The two heuristic algorithms devised in this way are for active and non-delay dispatching. Many priority rules have been examined in the literature [95, 96, 97]. In makespan problems there was no priority rule that dominated the others. The most successful have been the MWRR ( most work remaining) producing the best makespan and SPT (shortest processing time) which has produced best schedules in a few test problems. The most significant result of the makespan study was that non-delay dispatching was a better basis for heuristic schedule generation [97]. For example, in 20 test problems containing 10 jobs and 4 machines, active dispatching produced a better schedule than non-delay dispatching only twice.

2. Heuristic branch and bound

A rather different approach to the generation of a single schedule with a dispatching procedure is based on the branch and bound solution. It is identical to the priority dispatching in that the algorithm is terminated after the first feasible solution is obtained.

Algorithm 4 : Heuristic branch and bound

Let d = 1P q = () the null partial schedule

Arf = Sj set of operations with no predecessors

DO

-V s^ g Arf create = ( s j , s ^ ), and calculate boundCP^)- Find min { boundCP^) } & Set P^ = argmin { boundfP^) }- I f Pd is solution STOPadvance - d = d + 1

< - Create by adding the direct successor of operation s^to S^_j

- Determine (f)* = min { <j): } for active dispatchingieSd

or a* = min { O; } for non- delay dispatchingi e S d

- Determine m* on which or a* can be realised -Find Arf a subset of S^ whose elements require m* and for which dj < <|>* or dj = a*

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This is called branch and bound without backtracking. Selection of the branch is achieved by considering the lower bounds of the partial schedules rather than a priority index. At each level, branching is done from the node with the minimum lower bound.

Sampling procedures

Sampling procedures examine a number of schedules in the population of active or non-delay schedules and select the best amongst them. This is done by resolving the conflict between the schedulable operations randomly rather than based on a priority dispatching algorithm. When it is repeated several times a collection of different schedules is generated. However, it is difficult to draw quantitative conclusions about the best schedule in the sample. Work by Giffler,Thompson & Van ness [98], compared the best schedule obtained with sampling to the schedule obtained with the SPT rule with the makespan as criterion but theiewas no clear preference for one approach or the other. Bakhru and Rao [99], used sampling to examine the difference between active and non-delay population. They found that non-delay population is preferable for sampling.

Probabilistic dispatching

This method combines the philosophies of priority dispatching rules and random sampling. It resolves conflicts of schedulable operations by random selections that are biased towards a reliable priority rule. The choice is made, first by ranking the operations by the given priority rule and then by assigning selection probability aj tooperation i using a random mechanism ; i.e the first operation is assigned the largest aj and so on.

Results of research work on probabilistic dispatching show that probabilistic dispatching is better than random sampling. However, probabilistic dispatching like random sampling suffers from the lack of quantitative knowledge about the best schedule in the sample and the related implications for selecting a sample size. Therefore it is really difficult to know whether the improvement of probabilistic dispatching over the priority dispatching is worth the added computational burden. Conway, Maxwell & Miller conclude for instance that probabilistic dispatching will provide only diminishing returns in larger problems [36].

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6.4 A new simulation-based algorithm for the generation of non-delay schedules for a FMS (SHA)

Many algorithms have been reported in the literature for the generation of active and non-delay schedules for job-shops and they basically follow the lines of algorithm 1. [92, 100]. For the FMS case, an algorithm considering the set of machines and the processing times of operations, is too simple to result in realistic schedules for a system as complex as the flexible manufacturing system. Some effort to devise a more realistic algorithm was made by Iwata , who incorporated set up times and travel times in his branch and bound algorithm [101] and in his heuristic dispatching algorithm [102]. Later on, he also incorporated selection of tools and transport devices [69]. However, travel times are not the only complication of flexible manufacturing. Tooling considerations, routine maintenance activities are only some of the factors which influence the operation of an FMS.

In this section, an algorithm using a simulated model of an FMS is suggested. The algorithm generates all feasible non-delay schedules for the FMS. The set of non-delay schedules was chosen because of its more manageable size and because it can be easily implemented. Furthermore, studies for job shop indicated that the set of non-delay schedules is a better basis for heuristic schedule generation.The simulation model used, is the one described in chapter 3. The use of a simulated model guarantees the generation of realistic schedules thus alleviating the disadvantages of the classical algorithms. This algorithm may subsequently used as the basis on which operational research techniques can be implemented to produce the best or a 'good' non-delay schedule.

The simulation model described in chapter 3, has some features which facilitate the development of the schedule generating algorithm. Scheduling decisions are taken in the control module. The control module is called when a scheduling decision is required. This is the time a machine is released from its previous activity and is ready to work on a new part or the time a part becomes available for processing and there is an idle machine.We call this time, the allocation time a. Decisions at the allocation time concern the choice of a part from those available, which can be processed by the released or idle machine. All available parts which can be processed by the released or idle machine for their next operation belong to a set called the choice set of this machine and which is designated by Cm to denote the fact that its members refer to machine m. Choice sets are formed in the control module and then a decision is taken based on a priority rule. Choice sets can be thought as corresponding to the A^ sets of the classical algorithm

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since it is only their members which can be considered as candidates for extending thepartial schedule at a stage. The allocation time a can be thought as corresponding to the minimum of the earliest starting time of the schedulable operations.

The formal description of the simulation based algorithm for the generation of non delay schedules is as follows:

Denote by C j71

Md

Pdm

Pd

Kd

the choice set of machine m at stage d

the set of idle machines at time athe part that belongs to the choice set ofmachine m at stage darray of parts selected, for eachmachine in Mdthe cartesian product of the choice sets at stage d, i.e Kd = C ^ x . ^ C j 71

Algorithm 5 : A simulation based, backtracking search of non-delay schedules for FMS

Let d = 1the null partial schedule the allocation timethe set of machines associated with a It comprises all parts whose first operation can be performed on m

Po = ( )a = 0Md =M C / > :

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whiled > 0 i while do Kd * 0

do

sidestep- choose p^111 e C J71 V me Md- Pd = ( Pd1- ’ Pdm )

Kd =Kd -P d■ ^d = ( Ph P2’”» Pd )- if is a solution then

record it else advance- Simulate the system until the next allocation time a- d = d + 1- the control module generates Md and C j71 V me Md

backtrackd = d -1

^ d ~ ^ d ~ P d

If we compare the classical schedule generation algorithm and the simulation based algorithm we note the following:

Comparison of the classical approach with the simulation based algorithm.

In the classical approach, computation of the earliest starting time o for an operation i involves the computation of the maximum of the earliest finishing time of the predecessor of this operation and the earliest time the machine required by operation i finishes processing another operation. In practice, however, the time a machine becomes available for processing a new operation or the time an operation finishes, is a function of many factors which are not taken into account in the classical case. For example,i) Tool wear might enforce a tool change on a machine. The presence of an operator might be necessary for the tool change, or the tool needed, might not be currently duplicated on the machine's magazine and a new tool might have to be brought in. It is then inevitable for the machine to wait and thus to delay the time it will become ready to process a new part.ii) there are also other activities involving a machine, like routine maintenance or house keeping tasks which add on the complexity of the calculation of the time this machine will be ready to work on an operation.

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It is clear that this difficulty is easily overcome when using a simulated model. The time a machine becomes ready to process a new part is a distinct event in the simulation model and is called the allocation time.

The allocation time is in general different to the earliest starting time of the operation allocated to a machine at this time in that:1) the part involved has to be transported to the machine in question. Travel time depends on the distance between the part's current location and its destination, and the speed of the transport device. If the part's location is the same to its destination, then allocation time coincides with the earliest starting time. Otherwise, travel time and waiting due to busy transport devices delay the start of the operation. Travelling is no more than an activity concerning the system and its entities and is simulated in the model. Simulation keeps track of the location of the different parts in the system and the transport devices and it can therefore cope with complications like this with no additional effort.2) In systems provided with tool transport networks a tool might have to be transported to the machine to enable the processing of the allocated operation to take place (for example, in the case of the use of alternate operations). Thus, the operations starting time is delayed by the tool transport time.In general, the relationship between the allocation time a and the starting time a* is :<Tj = a + max { (part transport time + delays in part transport network) ,

( tool transport + delays in tool transport network) }Delays refer to transport devices being busy or broken down.

A choice set is also different in general to the subset A of schedulable operations. The set of schedulable operations S comprises all operations with predecessor operations completed. Those operations which require machine m and belong to this set can be considered for allocation to machine m . In practice, however, a part may not be processed by machine m for its next operation although it is theoretically possible or it may not be available at the time a . For example :i) machine m* might require a tool which is not currently available ( for example it may at the moment being used by another machine ).ii) in some cases inspection is applied only to a sample of the parts of the same type produced rather than all of them. Then, a part might be bound to an inspection facility and it can therefore not be available although its next operation may be processed by machine m .All these possibilities may be covered in the simulation based algorithm. The choice set

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for machine m can only contain those parts which are available for processing by this machine.

In summary, the simulation model can accommodate any degree of complexity that a particular FMS has to show. Thus, resulting schedules are much more realistic than those created by computational algorithms. Furthermore, use of a simulated model facilitates the collection of different categories of information for each of the generated schedules so that the best nondelay schedule can be found according to any regular measure of performance.

Computational time depends on the size of the problem. One simulation run i.e generation of one schedule corresponds to a small fraction of a second and it is mainly dependent on the level of detail of the simulation model and the degree of complication the particular FMS which has been simulated presents ( for example , the different activities involved during the operation of the system ). However, if the number of non-delay schedules is very big, then computational time may reach the non-economical level. The in-built flexibility on operation allocation apparently increases the number of non-delay schedules examined as compared with the conventional job-shop case. However,due to the nature of the allocation time and of the choice sets, the number of levels and the number of alternatives at each stage in the tree of non delay schedules, are less than they would be in the classical case because the added complications of the FMS operation render some of the alternatives infeasible. The smaller number of feasible schedules counteracts the increasing tendency caused by the in-built flexibility.

Memory requirements are relatively small. They are limited to the storage of the generated choice sets in a depth first visit. Also, in case the generated choice set is of size one, no alternative or branching exists and therefore the choice set may not be saved. The depth first traversal backtracking method used, is in general a memory efficient method for exhaustive enumeration since not all nodes have to be saved.

6.5 Heuristic approaches for FMS scheduling

If the size of the problem is big enough to render the enumeration of the non-delay schedules uneconomical cost or time wise, we can resort to a heuristic method so as to compromise the quality of the answer with the quantity of computation.

Branch and bound curtailing of the tree search is not a good alternative due to the inefficiency of the lower bounds proposed so far for job-shop scheduling. Branch and

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bound finds the best solution but to be effective is necessary to use bounds which are good enough to prune large parts of the tree in early stages. The bounds suggested by Brooks & White [93] are very weak particularly for the simulation based algorithm. The computation of bounds is based on processing times and it is therefore unlikely to satisfy the curtailing condition bound>0 particularly in early stages since, for the calculation of 0 , which is the makespan of the trial solution, many time consuming activities were involved apart from operations.

A simulation model can be regarded as a heuristic dispatching method for the generation of one full non-delay schedule. In an algorithmic form, it could be given as follows:

Algorithm 6: A Simulation based, priority dispatching method for FMSscheduling

Let d = 1Po = ( ) the null partial schedulea = 0 the allocation time

choice set of machine m , V me M^

(a) 1. Vpê C j71 calculate a priority indexVm e M j do according to a priority rule

DO

2. find p^m with the smallest index

< Pd = ( Pd1- Pd2’-P dm > where ms Md

(b) pd = ( Pl> P 2 - P d >if is a solution STOP

advance(c) Simulate the system until the next allocation time a

Put d = d + 1the control module generates C j 71 & VmeM ^

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Priority dispatching for the generation of a non-delay schedule is similar to the dynamic dispatching which is frequently used in practice for the dynamic job shop scheduling. Due to the inconclusive and sometimes conflicting evidence about the best dispatching rule( as discussed in the heuristic dispatching for job shops) production of one schedule using a priority rule is the least one could do to improve on the schedule of an FMS.

6.6 A new simulation based heuristic algorithm for FMS scheduling

The algorithm suggested in this section for the derivation of a good schedule for FMS belongs to the category of heuristic dispatching. The priority rule which is used toallocate a part p e Cm at each decision stage is a looking ahead rule. Looking ahead is realised by making use of the simulation model. The priority index concerns the performance measure (makespan) associated with the looking ahead simulation run and it is a prediction of the effect of the alternative decision on the overall performance of the schedule. The predicted makespan is not in general the best, one could get, given the partial schedule . This is so, because the resulting makespan corresponds to aschedule which is generated by extending the partial schedule P^ solving conflicts inrandom until the generation of a full schedule. An obvious variation of this algorithm is to use a good priority rule rather than a random choice for the extension of the partial schedules in the prediction section. It is helpful to picture the algorithm in terms of the search tree of the non delay schedules.

start

Figure 6.5 : The search tee of the simulation based heuristic dispatching algorithmusing a 1 look ahead1 rule

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The prediction run for a partial schedule is equivalent to a branch visit starting from the node representing down to a terminal node. The tree and the algorithm procedureis shown in figure 6.5. The algorithm procedure is indicated by the dashed lines. The node with the minimum predicted makespan is encircled. Makespan values are noted next to nodes. The prediction run, given a partial schedule P^, is equivalent to the visitof a branch, starting from the node representing the partial schedule P^ down to a terminal node of the subtree.

The algorithm can be described formally as follows :

Algorithm 7 : A simulation based, heuristic dispatching method using alooking ahead rule

Let d = 1Po = ()a =0

Md =M

c<r ■■

the null partial schedule the allocation timethe set of machines associated with a the choice set of machine me Md

1)

while K d * 0 , DO

create alternative partial schedule for stage d -choose p^111 e C f 1 V m g Md

-Pd = (Pd1- ’Pdm)

'K d = K d -P d

Pd = ( Pl> P2’-’ Pd)-if is a solution then

save makespan( P ^ )elsepredict its makespan with a look ahead simulation run

d' = d

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-Simulate the system until the next allocation time a' d' = d' + 1M g : the set of machines associated with a 'Cd^ : the choice set of machine m , me Af^

choose p ^ e C^t71 VmeA/^Pd' = (P d ^ -’Pd'™)Pd' = (P l-.P d ’Pd')If Pjj* is a full schedule then

save makespan( P^-) and exit endif

2) Find the minimum makespan ( P^«) and save P^ as the partial solution with the best predicted performance so far.3) If P^ is a solution STOP4) Simulate the system until the next allocation time a.5) d = d + 1

The control module generates : Md : the set of machines associated with aC j71 : the choice set of machine me Md

6) Go to step 1

6.6.1 Evaluation of the simulation based heuristic algorithm (SHA)

Ideally one would like to know the probabilistic behaviour of a heuristic algorithm for all the possible instances and forms of a problem. For the general case of heuristic algorithms, it is not possible to derive this information with mathematical analysis. What , can be done, is to look at a number of sizes and data structures of the problem and

derive a probabilistic empirical profile for its performance.

The FMS scheduling problem structure depends on a number of parameters:1) the size as an indicator of complexity2) the variance of processing times amongst different parts in the set of

DO

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schedulable parts3) the sequence constraints

The heuristic algorithm described above has been evaluated on the basis of solution values obtained over a number of problems of different sizes and different variance of processing times, where precedence relations are selected at random, using the NAG library routine G05EHF. Table 6.3 gives the sizes and variances examined.

Problem sizeN x M 6 x 4 10x4 20 x 5 20 x 10

Operationtime

distributionuniformUR[1,36]

normal N [20,10]

normal N [ 20, 5 ]

Variance high medium low

Table 6.3 : Size of problems and distributions of operation times

Travel times and other activities present in the real FMS operation were ignored in this stage in order to make the algorithm comparable to algorithms developed for the job shop case. Five samples were examined for each combination of problem size and variance of processing time.

Each problem was solved using two variations of the SHA as well as using the priority dispatching simulation based algorithm ( algorithm 6 ) for comparison purposes. The two SHA variations are created by using two different rules in the extension of the alternative partial schedules at each stage in order to predict their makespan ( namely the FCFS and the SPT rule ). Thus 120 problems were solved in all. The smallest problem size was also solved using the full enumeration of the non-delay schedules and the best non delay schedule found.

A feasible solution obtained with a heuristic is certainly appropriate only for the particular problem under consideration and does not allow any comparison with other similar problems. To overcome this difficulty a normalisation of the solution is applied. For the small problems (6x4), the algorithm solution efficiency is calculated as the percentage ratio of the best non delay schedule value to the value found by the algorithm. For bigger size problems, where finding the best non delay schedule becomes

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computationally unacceptable, a percentage of the solution value deviation from an absolute lower bound is used and is designated by D^. has the advantage of being based on the known feasible solution and not on the unknown optimal.

The percentage solution deviation from the lower bound is given by the following formula:

S(A)-B!Du = ------------------x 100S(A)where S (A) : is the solution value obtained using the algorithm A

Bj : the theoretical lower bound to the overall problem solution

Bj may be calculated as follows:

M NBj = max [ max X tpm , max X tpm ] p m=l F m p= lF

where tpm is the processing time of part p on machine m.Table 6.4 gives the average number of decision stages and nodes explored by the algorithm SHA and the computer time required for each problem size (this is an average over 30 samples).

A summary of the results from the experimental study of the simulation based heuristic algorithm are presented in tables 6.5 and 6.6. Table 6.5 gives the average solution efficiency for the 6x4 problem and the average percentage solution deviation from Dj forall problem sizes. Worst case values observed are given in table 6.6 represented for the 6x4 problem by the minimum value of solution efficiency and for all problem sizes by the maximum value of the solution deviation.

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Problem average number of CPU time in secsize decision stages nodes FTN5 compiler6x4 6 16.9 0.428

best ND 1845.53 45. 940

10x4 18.86 73.96 2.9

20x5 61.1 842.70 100.820x10 82.23 650.23 210.51

Table 6.4 : Computational requirements of the SHA algorithm

problem

sizeUR [1, FCFS

36]

SPT

N [ 2

FCFS0, 10]

SPT

N [ 2

FCFS

0, 5]

SPT

6x4 98.57 100 97.62 96.78 95.30 94.28

problem

size

UR [1,

FCFS

36]

SPT

N [ 2

FCFS

so. 10]SPT

N [ 2

FCFS

0, 5]

SPT

6 x4 13.9 12.67 9.84 10.4 15.8 16.61

10x4 5.06 6.30 3.49 5.18 5.23 7.33

20x5 0.0 1.15 0.56 2.99 1.49 3.03

20x10 12.63 13.66 11.01 13.10 14.39 16.10

Table 6.5 : Average percentage efficiency of the SHA solutions compared with the best non delay schedule and average percentage deviation of solution compared with makespan lower bound.

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problem UR [1, 36] N [ 20, 10] N [ 20, 5]

size FCFS SPT FCFS SPT FCFS SPT

6x4 92.85 100 93.29 91.45 90.18 91.71

problem

size

UR [1,

FCFS

36]

SPT

N [ 2

FCFS

0, 10] SPT

N [ 2

FCFS

0, 5]

SPT

6 x4 23.16 23.16 18.71 23.61 22.28 24.86

10x4 11.56 10.82 10.24 13.6 9.60 15.35

20x5 0.0 4.31 2.80 10.02 7.45 10.02

20x10 15.88 19.34 15.70 17.85 17.72 17.85

Table 6.6 : Worst case values of solution efficiency and solution deviation.

The solution values obtained by using the SHA ( algorithm 7 ) were then compared to those obtained using the priority dispatching algorithm (algorithm 3) with the SPT used as the dispatching rule.

problem size U R [1 ,36] N [20,10] N[20,5]

6x4 8.90 8.01 10 .95

1 0 x4 14 .95 13 .56 10 .92

2 0 x 5 7.55 13 .29 13 .59

2 0 x 1 0 14 .00 12 .67 15 .47

Table 6.7 : Average percentage improvement of makespan value produced by the simulation based heuristic algorithm compared with the value produced by using thesimulation based priority dispatching with shortest processing time rule.

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Table 6.7 shows the average percentage improvement achieved for the different problem sizes and processing time distributions. It is worth noting that the average 10% improvement refers to the schedule of the minimal part set which in absolute terms corresponds in many hours saved or in considerable increase of production in the planning horizon.

6.6.2 Discussion

The comparison of the full enumeration and heuristic results for the 6x4 problem (table6.5) shows that SHA achieves a high level of solution efficiency with the maximum deviation being only 5.7% from the best non delay schedule. If we compare these results with the same ones as analysed with the deviation from the lower bound, it can be seen that the comparison with the lower bound makespan undervalues the heuristic algorithm. The worst result (N(20, 5) distribution and SPT rule) suggests that deviations from the lower bound is 16.6% whereas the solution efficiency shows that is only 5.7%. In general, the solution deviation values are small in problems with large number of parts in relation to machines (e.g 20 x 5 ). This is not unexpected as for a large number of jobs, there are always some schedulable jobs in the choice set and machine idleness may be avoided. Therefore, the makespan found by the algorithm is much nearer to the lower bound. Where the reverse is true (e.g 10x4, 20x10) idleness is introduced due to precedence constraints and the comparative shortage of jobs in the choice set. This is higher for bigger sized problems (table 65).

In almost all cases, (the only exception being the case of high variance data at the 6 x 4 problem) the use of the FCFS rule for the look ahead simulation run gives a better performance than the SPT rule (table 6.5 and table 6.6). The value of variance is also significant with the solution deviation increasing as the variance decreases. The solution efficiency is also better for high variances. Both size and variance have some impact on the worst values. It is worth noting that the worst case error is 23%.

Comparison of the solution values obtained by the SHA algorithm to those obtained using the simple dispatching simulation with SPT used as the dispatching rule, shows that in all cases the SHA gives significant improvements. ^ b I e G.Tj

The cost of obtaining a solution is not high. Computer time increases with the size of the problem but it still remains in acceptable levels even for big problems. In present terms a ten machine FMS is fairly large. For a given number of parts, CPU time appears to be approximately linear with the number of machines whereas for a given number of

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machines CPU time displays a disproportionate increase with respect to the number of parts. Also, when the ratio of the number of parts to the number of machines is high, the proportion of decision stages with respect to the number of nodes examined is small. The comparison of the full enumeration of the 6x4 problem with the heuristic solution illustrates that even with small problems the full procedure is too computationally demanding for practical use.

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CHAPTER 7

COMPUTATIONAL EXAMPLESFOR THE PERIODIC PRODUCTION SCHEDULING POLICY FOR FMS

7.1 Introduction

This chapter reviews and discusses the methods proposed in this thesis for improved production control of flexible manufacturing systems. The large number of parts in the planning horizon, the frequent changes in the production requirements and the changes of system status due to the machine failures led to a new definition for the optimal state of operation in a FMS (Chapter 4). "The FMS is operating optimally if it produces parts in the prescribed ratios at steady maximum rate averaged over suitable time intervals. An optimal scheduling policy should also be able to deal with machine failures and demand changes." A periodic production policy was devised with the aim of achieving this optimal and in this chapter it is applied step by step on two example FMSs. Though not real they cover a multitude of aspects of FMS that are likely to be encountered in practice. The first is a system for the manufacture of parts of varying complexity as measured by the number of operations. Each operation may be performed by more than one machine apart from inspection which is performed last. The second is basically a flexible flow shop in the sense that parts of different types have to go through the same sequence of operations. Each kind of operation is undertaken by a group of identical machines.As the policy unfolds, its merits and limitations are discussed.

7.2 Example 1

A shop consisting of five machines and two transport devices is required to produce eight different types of parts in equal production ratios. Parts require up to nine operations in order to be completed. Their operation sequence and durations is given (table 7.1) but machine routing is not prespecified. Machines are able to perform different kinds of operations (up to three, table 7.2) and they are subject to failure. Transport times are distributed normally with a mean of half a minute.Operations may be performed alternatively by up to two machines. Parts may visit a machine more than once.

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Part type Operation sequence and duration ( operation type. : time)

1 1: 39 2 : 22 10 : 82 3 :2 2 1:27 6 :1 4 4 :3 10:113 1 : 38 7 : 23 2 : 17 8 : 14 9 : 21 7 :3 10 : 64 5 : 37 3 : 35 1: 16 10 : 95 1 :8 2 :13 3 :4 4 :2 6 5 :31 6 : 17 7 :16 8 : 39 10: 206 5 :13 9 :19 8 :8 7 :6 6 :2 2 10: 87 1:15 8 : 15 2 :9 9 :2 2 3 : 27 10: 148 4 :1 2 5 :1 3 6 :8 7 :3 10:11

Table 7.1: Operation sequence and durations of example 1

Machine Operations

1 1 2 32 4 5 63 7 8 94 3 5 75 10

Table 7.2: Machine operations in example 1

The periodic production policy presented in chapter 4 requires the determination of the best size and schedule of the minimal part set which is to be produced periodically. If a change then occurs, in production requirements or in a machine’s operational state, the size and schedule of the MPS has to be recomputed.

STEP 1: Determination of the size and schedule of the Minimal Part Set

In this example, there are eight parts to be produced in equal production ratios. Therefore, the smallest set which meets this production ratio is of size eight and it will consist of one part of each type. Bigger sets will have sizes which will be multiples of eight ; for example, size sixteen consists of two parts from each type. The heuristic simulation based algorithm described at chapter 5, was used to derive the schedule of the minimal part set for sizes eight and sixteen. The makespan, the system idle time and the average flow time per part for the two sizes of the set, size 8 and size 16, are given in

146

table 7.3. The time to produce 16 parts is less than the time to produce two sets of eight. However, the choice of the right size of set will be based on the production cost per part (chapter 4). The values of makespan, system idle time and average part flow time are used to calculate the cost per part for different values of the cost ratio for the two cases of both set and part deliveries (figure 7.1 and 7.2) using the formulas ( 5.1). The cost ratio is the ratio of the cost of idle resources to the inventory cost, it depends on the company's accounting methods and it is particular to each application.

production in MPSs improved periodic productionMPS size Makespan system idle time Period system idle time

average flow time

8 244 40.5 208 4.4 150.69

16 442 34.0 418 10 212.5

Table 7.3 : Schedule results for varying set sizes.

set deliveriescost per part for varying cost ratio

-o n=8n=16

Figure 7.1 : Cost per part for varying cost ratios for set deliveries

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part deliveries

mps size -o- n=8

n=16

Figure 7.2 : Cost per part for varying cost ratios for part deliveries

In both the above cases, set size sixteen becomes economical for cost ratios greater than twenty for the case of part deliveries and for ratios greater than sixty five for the case of set deliveries. Before the final choice on the set size is made we must consider the improved periodic production.

The production period can be less than the time to complete all parts in the MPS (chapter 4 ). If a machine has finished its work on a set, then instead of staying idle until all parts of the set are completed, it can start work on the subsequent set as long as this does not interfere with the schedule. The finishing and starting times of machines are shown in figure 7.3 and 7.4 for the two set sizes respectively. The improved periodic production is given in the same figures. It is realised if we think of the second set's schedule sliding to the left until it reaches the first set's schedule. Numerical results given in table 7.3 above show that the improvement achieved when production in sets of 8 parts is much greater than when producing in sets of 16 parts . Note that double the system idle time when producing in improved periodic sets of 8 is less than the idle time when producing insets of 16parts.

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starting and finishing times of machines for MPS size=8

©coaE

Tpr=244 min.

initial and final □ idle time £2 MPS schedule

improved periodic production for MPS size=8

Tperiod = 208 min.

inter set idle time first set

second set

0 50 100 150 200 250 300 350 400 450 500time

Figure 7.3 : Improved periodic production in sets of eight

Thus, the cost per part in producing in sets of eight will always be less than the cost per part for producing in sets of sixteen:

set of 8 parts: C/N = 208 +Cj. 0.55 for set deliveriesset of 16 parts: C/N = 418 +Cj. 0.625 for set deliveries

149

starting and finishing times of machines for MPS size=16

©

o©E

time

Tpr=442min

initial and final □ idle time ESI MPS schedule

improved periodic production for MPS slze=16

©co©E

0 200 400 600 800 1000

Tperiod=418 min

inter set □ idle time E3 first set

second set

time

Figure 7.4 : Improved periodic production in sets of sixteen

Therefore, production should be in sets of eight. In case of set deliveries, a set of eight parts will be released at the system's input buffer every 208 minutes and the sets of completed parts will exit the system periodically with the same period of 208 minutes. If parts are released and exit the system individually, the loading and output sequence and times are given in table 7.4. These sequences were extracted from the schedule of the MPS.

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Loading sequence Output sequencePart type input time Part type output time

5 0.0 8 76.08 0.0 4 157.04 0.5 6 174.53 8.5 3 189.56 25.5 7 204.02 38.0 2 215.57 60.5 1 224.01 147.0 5 244.5

Table 7.4 : Loading and output sequences and times for periodic production in sets of eight parts

The machine utilisations for the improved periodic production are given in table 7.5.

Machine Utilisation

1 97.92 76.03 81.64 69.15 41.7

Table 7.5 : Machine utilisation for periodic production in sets of 8 parts

STEP2 : Periodic production’s dynamic behaviour

Desired production : 40 parts of eight types in equal production ratios.

If the system status is unchanged production of 40 parts is equivalent to the production of 5 consecutive sets of 8 parts. Scheduling using two methods was compared; periodic scheduling in sets of eight and the SPT dispatching rule. SPT rule performs well for problems of makespan minimisation. The makespan and the average flow time achieved by both methods are given in table 7.6.

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Makespan Average flow time

Periodicproduction 1076 min. 150.6 min

SPT 1171 min. 229.3 min.

Table 7.6 : Performance of the SPT rule and the periodic scheduling policy for the desired production

Periodic production gives the best results with respect to both measures of performance. Moreover, it guarantees that each set of eight parts will comprise a part of each type which is very useful if say the eight parts will be subsequently assembled to form a product. The SPT rule and most of the other priority rules lacks the ability to control the production in such a way. For example, the content of the parts completed around the middle of the production time ( at 600 minutes ) with the SPT rule is as given in table 7.7, which shows that production is far from the required production ratios.

Part type 1 2 3 4 5 6 7 8

Part quantity 0 5 0 1 2 5 5 5

Table 7.7 : Content of parts completed in the middle of the schedule when using the SPT rule

In general, when using the SPT rule for scheduling, the average flow time increases with increasing desired production requirements, whereas in periodic production average flow is constant and thus unaffected by the production target (figure 7.5).

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Figure 7.5 : Average part flow time with increasing production target for periodic production and production with the SPT rule

STEP 3 : Periodic production under machine failures

Desired production : 80 parts of 8 types in equal production ratios.

This is equivalent to 10 sets of eight parts if the production status does not change. Here, however, we will assume that a change in the operational state will take place.

Change of machine operational state : machine 3 breaks down after having completedthe sixth set of eight parts.

That is, 48 parts have been produced until that time , 6 of each type.

The breakdown affects the production ratio because some of the part types, specifically part type 3, 5, 6 and 7 can not be produced at all.. Thus, the new production ratios will be as follows:New production ratios: part type 1 2 3 4 5 6 7 8

ratio 1 1 0 1 0 0 0 1

Now, the system comprises 4 machines and there are 4 different types of parts which can be produced by it. Therefore, the smallest size for the MPS is 4 consisting of one part from types 1, 2, 4 and 8. Set sizes four and eight are examined and the results are

153

given below:

production in MPSs improved periodic productionMPS size Makespan system idle time Period system idle time

average flow time

4 118.5 14.5 106 2.0 94.75

8 220.0 13 210.5 3.0 114.2

Table 7.8 : Schedule results for varying set sizes with one machine broken down

The cost per part in both cases will be given by:set of 4 parts set of 8 partsset deliveries : C/N = 106 + q 2/4 C/N = 210.5 + q 3/8part deliveries : C/N =* 94.75 + q 2/4 C/N = 114.2 + q 3/8The cost per part is less in the case of producing in sets of 4 unless q has a value greaterthan 150 for the case of part deliveries and a much greater value for the case of set deliveries. We will assume that this is not the case here and thus set size 4 is preferred. The finishing and starting times of machines for the production of two consecutive sets is given in figure 7.6 .

o

o(0E

inter set L-J idle time0 f i r s t s e t

0 s e c o n d s e t

Figure 7.6 : Periodic production in sets of 4 with machine 3 broken down

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The machine utilisation during this period is given below:

machine_____ 1 2 3 4 5utilisation 98.1 82.0 0 56.6 36.8

Let us assume that machine 3 is repaired within the next 3 1/2 hours from the time of breakdown so that the production of all types of parts may be resumed after the second set in the previous operational state has been produced. The production ratios of the different part types has however now been altered since production of types 3, 5, 6, 7 has fallen behind. The new production ratios required are given below:

part type 1 2 3 4 5 6 7 8ratio 1 1 2 1 2 2 2 1

So, the smallest size of MPS is 12. This comprises one part from type 1, 2 ,4 and 8 and two parts from type 3, 5, 6 and 7. The set sizes of 12 and 24 are examined and the results are given below:

set size makespan system idle time periodimproved system idle time averageflow

12 414 77 377.5 40.5 199.224 737 76 701.5 40.5 265.5

Table 7.9: Schedule results for varying set sizes after the repair of the broken down machine.

The cost per part was calculated for both part and set deliveries for different values of cost ratio and is given in figure 7.7 .

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set deliveries

part deliveries

-t> N=12 N=24

N=12N=24

Figure 7.7: Cost per part for varying cost ratio for set and part deliveries.

It is worth noting that in this case the small difference in the average flow between the two set sizes renders production in sets of 24 parts economical for a comparatively small value of cost ratio (approximately 40) whereas if production is delivered in sets set size 24 becomes economical for a much higher value of cost ratio (approximately 200).

We assume that parts are delivered rather than sets and that the cost ratio is high enough to render set size 24 economical. Therefore, after machine 3 is repaired a set of 24 parts is produced and the desired production is reached. The starting and finishing time of

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machines are given in figure 7.8.

Machine utilisations are given below:machine 1 2 3 4_____5utilisation 94.4 71.6 92.1 56.0 38.4

The production of the 80 parts in the presence of machine's 3 downtime is shown in figure 7.9 from the fifth set onwards.

time

inter set n idle time

E2 fifth set S sixth set Q seventh set § eight h set E3 ninth set

E3 broken down

Figure 7.9 : Dynamic periodic production in the presence of machine failure

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The total completion time for all 80 parts is 2196 minutes or 37 hours approximately. Figure 7.9 shows how production is changing with operational state changes (here, this is the breakdown and repair of machine 3) from sets of 8 parts to sets of 4 and last to the set of 24 parts of the remaining production. During production, average part flow switches to known values (determined by the MPS's schedule) when the operational state changes until the next change in status. Machine utilisation stays in satisfactory levels and total production is achieved by satisfying production ratios all along the production time as far as it is feasible.The periodic production in the presence of failures does not, however, look explicitly at the production schedule during the transient time between the machine's failure and the start of the new schedule under the current operational state. The implied assumption is that if a machine breaks down in the course of producing a set of parts, unfinished parts are either rerouted or are included in the remaining production quantity for subsequent consideration in the calculation of production ratios.

7.3 Example 2

A shop comprising five machines is intended to produce ten different kinds of parts. Machines are grouped into three groups, each machine in a group being able to perform the same number of operations at the same time and cost, i.e machines in a group are equivalent Each part type has to pass subsequently from the first to the last group before being completed. The first four types of parts are much bigger than the rest. The system may be regarded as a flexible flow shop. Input data for this system is given in table 7,10.

Part type Operation durations

1 39 22 182 22 27 143 38 23 174 37 35 165 8 13 46 13 19 87 15 15 98 12 13 89 8 7 410 7 14 5

Group Machines

1 1 , 22 3 , 43 5

Table 7.10 : Input data for example 2

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STEP 1: Determination of the size and schedule of the MPS

In this example, there are 10 different types of parts to be produced in equal production ratios and therefore the smallest size of the MPS is 10 and it will consist of one part from each type. Bigger sets will have sizes which will be multiples of lO.The heuristic simulation based algorithm described at chapter 5, was used to derive the schedule of the minimal part set for sizes 10,20 and 30.The makespan, the system idle time, the average flow time per part and the corresponding values of the period and the system's idle time for the improved periodic schedule for these set sizes are given in table 7.11.

set size makespan system idle time Periodimproved system idle time

averageflow

10 154 51 127.5 24.5 57.9

20 260 54 228.0 22.5 64.9

30 367 52.5 345.0 30.0 74.05

Table 7.11: Schedule results for varying set sizes

Cr=10 Cr=50

-a- Cr=100

Figure 7.10 : Cost per part as a function of the set size for the set delivery policy

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The choice of the best set size is based on the cost per part calculated for each of the set sizes and the kind of delivery policy. Figures 7.10 and 7.11 plot the cost per part against the set size for different cost ratios for the two delivery policies respectively. The best set size in each case is encircled.

-£> Cr=10 Cr=50 Cr=100

Figure 7.11 : Cost per part as a function of the set size for the part delivery policy

It is assumed that parts are introduced and delivered individually and that the cost ratio is less or equal to fifty. Therefore the best size is that of twenty parts.

oco(0E

-- 1--,-- --- 1--«--1--«-- 1--'--0 100 200 300 400 500

time

initial and final □ idle time E3 first set

E3 second set

Figure 7.12: Periodic production in sets of 20 parts

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The finishing and starting times of machines in the periodic production for two consecutive sets is given in figure 7.12 above.

The machine utilisations under this policy is :

Machine_____ 1____2________ 3______4______£Utilisation 87.3 84.6 77.2 87.7 90.35

All machines are highly utilised with machine 5 being the most heavily loaded.

STEP 2 : Periodic production's dynamic behaviour

Desired production : 80 parts of ten types in equal production ratios.

This is equivalent to the production of 4 sets of size 20 if the demand and machine status does not change. Periodic scheduling in sets of 20 and the SPT dispatching rule were used for the production of the 80 parts. The makespan and the average part flow time achieved by both methods are given in table 7.12 .

makespan average flow timePeriodicproduction 944 64.5

SPT 923.5 94.49

Table 7.12 : Performance of the SPT rule and the periodic policy for the desired production

Results show that periodic production gives a much smaller average flow time than SPT rule but the makespan achieved by SPT rule is slightly better. In general, as it was also shown in the previous example, use of the SPT rule for scheduling means that the average flow time is increasing with the total production quantity whereas in periodic production it is constant (figure 7.13).

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-o- SPTperiodic

Figure 7.13: Average part flow rime for the periodic production policy and the SPT rule

The makespan is also increasing but at a rate slightly smaller than the one in periodic production which increases proportionally with the total number of parts (figure 7.14).

-o- SPT-*• periodic

Figure 7.14 : Production makespan against production target for the periodic production policy and the SPT rule

This means that for some production target the SPT rule completes all parts in less time. However, the full control over the production of parts in the prescribed ratios at a steady

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rate and average flow and consequently the full control over the cost of production compensates fully for the marginal makespan gains which may be achieved if the schedule for producing all parts in the planning horizon would be considered

STEP 3 : Periodic production under machine failure

The flow shop nature of the system means that if all machines in a group are broken down, the production will have to stop until one machine of the group is at least repaired. We will therefore examine the situation by assuming that only one machine of a group is broken down and thus production does not cease.

Desired production : 80 parts of 10 types in equal production ratios

Machine operational state : machine 2 breaks down after completion of the second set of20 parts.

When machine 2 breaks down, 40 parts of each type are completed. Production ratios remain the same.The smallest size of the MPS is again 10. Sets comprising 10 and 20 parts are scheduled under the new operational state using the simulation based heuristic algorithm. Results are shown in table 7.13.

set size makespan system idle time period improved system idle time

averageflow

10 222 23.5 203.5 5 53.9

20 422 25 408 11 53.9

Table 7.13 : Schedule results for varying set sizes with one machine broken down

The cost per part in both cases is given by:

set of 20 parts C/N=408+Cr 11/20CyN=53.9+Cr 11/20

set of 10 partsset deliveries : C/N=203.5+ Cj. 5/10 part deliveries : C/N=53.9+Cr 5/10

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The cost per part is minimum when producing in sets of 10 for any values of Cr and forboth delivery policies. Thus production is continued under the new operational state and the finishing and starting times of machines for two consecutive sets is given in figure 7.15.

oco(0E

time

inter set □ idle time

E3 first set 0 second set

Figure 7.15 ; Periodic production in sets of 10 with machine 2 broken down.

The machine utilisation during this period is given below:

machine 1_____ 2 3______4_____£utilisation 97.0 0 52.9 39.2 50.5

Note that the effect of the breakdown is to shift the bottleneck to machine 1 with the result that the rest of machines are moderately utilised.

We assume that at the end of the production of two sets with machine 2 broken down, machine 2 is repaired and production may be resumed. The actual production is then 60 parts with the prescribed production ratios. Twenty more parts are left to be produced and since production ratios remain the same, the best size and schedule for the MPS is as calculated at the beginning for full machine availability. Thus, a set of twenty parts is produced and the desired production is reached. Figure 7.16 shows the five sets (starting and finishing times of machines per set) produced by the system.

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a>coCOE

time

inter set □ idle time

0 first set 0 second set 0 third set m fourth set E2 fifth set

s broken down

Figure 7.16 : Dynamic periodic production in the presence of machine failure

The final makespan is 1076.5 minutes or approximately 18 hours as compared to 944 minutes for full machine availability (table 7.12).

7.4 Conclusion

The periodic production policy is designed for the particular production needs of FMSs with the aim to realise the potentials offered by such systems. As the examples of this chapter show, the policy's dynamic behaviour in the FMS stochastic environment (due to machine's failures and/or demand changes) is very good; steady state is reached with the first part periodically produced. Part flow time is constant for unchanged operating and demand state. Furthermore, average part flow is known in advance together with the periodic loading and output sequences and times of parts which means that WIP costs are completely controllable.The periodic output of parts in the prescribed ratios result in minimised lead times across the shop floor and thus a just in time approach to inventory control may be realised. Just in time inventory control is, ideally, equivalent to the inventory elimination and it has been identified as one of the most important potential benefits from the use of FMS. This benefit can however be realised only if the right production policy is adopted. Periodic production decreases the size of the scheduling problem. A schedule of only a sample of parts is required, that of the MPS, rather than a schedule for the total production. Thus, the scheduling problem does not have to be decomposed to subproblems, for example the routing problem and the part flow problem, as in other approaches in order to be simplified. With the help of the simulation based heuristic algorithm, a good realistic schedule can be found. The computational

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requirements for the calculation of the MPS's schedule are small taking into account the time scale to which such a schedule refers (time between changes in machine and demand state). It can therefore be used on line in a practical system. The scheduler will be called each time the operational or demand state changes and the controller will then implement the resulting schedule periodically until the next change.

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CHAPTER 8

MACHINING PARAMETER OPTIMISATION IN FMSs

8.1 The role of machining parameters in FMS scheduling

In the traditional approach to scheduling, determination of the optimal values of the machining parameters ( cutting speed and feed rate) was assumed to be uninfluenced by the production planners, who would use some other parameters such as overtime or the number of workers to meet production requirements and dead lines. Thus, optimisation of the regular time criteria ( for eg. mean flow time, total completion time etc) were adequate scheduling objectives from an economic point of view[72]. However, classical hiring and firing policies are not applicable in an FMS because of the high degree of automation. Machining parameters should therefore be incorporated in the scheduling model as decision variables to influence production capacity. Production capacity can be varied by changing cutting speed and feed rate and these in turn influence the production costs. The cost of tool wear which typically increases with the cutting speed and feed rate also affects production cost.

Incorporation of the machining parameters in the scheduling model influences the objective function and changes the problem's constraints. A typical formulation of the scheduling problem including machining parameters may be found in Kusiak [34] and in appendix. The production function involved in such a formulation is strongly non-linear [103]. Thus, an already complex problem is made even more complicated.

Machining parameters manipulation in an FMS context has been used in conjunction with scheduling for correcting deviations from a precalculated schedule [11, 104, 105]. These deviations are likely to occur due to the normative calculation which is performed in the factories to obtain the operations' time values or the stochastic nature of the operation times when adaptively controlled machine tools are used in the FMS. The uncertainty on operation times suggests that our predictions on the future course of the system can not be accurate and perturbations like overloading of certain machines, idle time or delayed start of certain jobs are inevitable to occur in practice. These perturbations however may be eliminated by the manipulation of the machining parameters.

Machining parameters' manipulation was also used for reducing production costs. The inevitable machine idle time which occurs due to the scheduling constraints even for

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realistic work in process levels is exploited. Reduction in the tool cost component of the total costs is achieved by reducing machining rates where the time span available to process a job is longer than required for maximum production rate conditions [105]. This approach does not however derive the production cost function for the complete schedule. Machining parameters are selected from values which correspond to the efficiency range of each individual operation. Efficiency range values are bracketed by the machining parameter values for minimum machining cost and the values for maximum productivity for each operation taken individually.

The method presented here decomposes the general FMS scheduling problem into two levels. The top level solves the scheduling problem with fixed machining parameters and the lower minimises production cost under the schedule constraints where machining parameters are the problem's decision variables; i.e the solution of the top level problem (scheduling) imposes a constraint on the lower level problem (cost minimisation).

The scheduling problem was discussed in the previous chapters. In this chapter we concentrate on the formulation and solution of the problem presented at the second level, i.e the production cost minimisation under schedule constraints. This problem is formulated as a non-linear programming optimisation problem with linear constraints. The production cost is expressed as a function of the starting and finishing times of operations which are the problems decision variables. These variables are subject to technological constraints, minimum and maximum duration constraints and last, schedule constraints imposed by the predetermined schedule. The resulting minimum cost schedule has the same makespan as the initial one but minimum cost. A standard NAG library routine is used to solve the problem. It uses a sequential quadratic programming algorithm in which the search direction is the solution of a quadratic programming problem.

Machining economics for single stage manufacturing in the FMS context are first presented followed by the problem formulation and solution. Computational results from a representative example are then given and subsequently discussed.

8.2 Single stage machining economics for FMSs

Significant attention has been paid to the optimisation of elementary operations in order to determine the optimal machining conditions to be utilised on the machine tools. Machining parameters include the cutting speed, the feed rate and the depth of cut. For single stage manufacturing, where a part is processed on a single machine tool, basic

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mathematical models have been developed based upon three evaluation criteria, namely: the maximum production rate, the minimum cost and the maximum profit rate criterion. One of these criteria is employed according to the manufacturing objective and optimal machining conditions (usually machining speed and feed rate) are determined using unconstrained or constrained optimisation [107]. Multi-stage manufacturing, when a part is completed after being processed in a series of machine tools and workstations, has also been investigated by many researchers [107, 108, 109] and optimal machining speeds at different stages determined.

In the study of the traditional single stage machining economics, unit production cost comprises the following elements:

1. material cost2. set-up cost3. machining cost4. tool replacement cost

Costs which relate to machining time, tool changing time and tool usage are fundamentally affected by decisions on machining parameters,whereas material cost and set up cost are not influenced by such decisions.

Set-up costs are high in traditional machining shops and they mainly influence the determination of the most economical batch size. In the case of FMSs, the highly sophisticated machine tools allowed a considerable reduction in set up times and their associated cost, so that production in small batches even of one part became economically feasible. For the purpose of this analysis it is therefore enough to consider only the cost of machining and the cost of tool replacement.

In the formulation presented here, the following notation is used:

cj production cost of operation iR production ratet machining timeT the tool lifeTc the time to change a tool in the case of high speed steel tool or

index an insert in the case of cemented carbide tool Km the machining cost in £ per unit timeKt the average cost per regrind or per edge

L

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D the diameter of the barL the cutting lengthS feed per revolutionV cutting speedVc the cutting speed for minimum costVp the cutting speed for maximum production rateNp number of parts per tool change

It is assumed that the Taylor's simple life equation is valid, i.e:

v r 1/k = c 1

where T : the tool lifek, Ci : constants

k is normally negative; it takes values in the the range of -2.5 to -5

The tool life constants depend on the kind of cutting tool used, the work material, the machine tool, the presence of cutting fluids and other machining conditions.

Cutting parameters are related to machining time through the equation:

7tDL t= ------SV(8. 1)

The cost of machining is equal to : Kmt

The cost of tool replacement includes the cost of the tool Kt and the cost of tool changing KmTc . The cost of tool replacement per operation is therefore given by the equation:

KmTc+KtNp

The number of parts per tool change Np is, in turn, equal to:

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170

Np = T / t

or n _ = crk (——)‘k r^1F jtDL

Therefore the tool replacement cost is equal to :

K m T c + K t ^ D L v k , k + l

crk (— )"k t‘s

The total operation cost is the sum of the machining cost and the tool replacement cost

Ci =Kmy K t

crk (■ ^ 2 L ) -k tk+i + K „ tkm (8.2)

The cost c can also be expressed in terms of the machining parameters S, V :

ci “ ^m s crk( 7 t P L ) y - k - l (8.3)

The cutting speed for minimum cost is obtained if we differentiate (8.3) with respect to V and set it equal to zero:

Vc = Ci (-Km

(KmTc-fKt)(-k-l).)- 1/k (8.4)

The production time for an operation is the sum of the machining time and the tool changing tim e:

1/R = t + Tc / Np or 1/R = t + (T c / T ) t

1/R = t +

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or 1/R = ( fP t ) (V - l + - 5 - V-k"1 ) S C f k (8.5)

The cutting speed for maximum production rate can be found from equation (8.5), if we differentiate with respect to V and set it equal to zero.

The cutting speed for maximum production rate is given by:

Vp = C! (------------)‘1/k (8.6)v Tc(-k-l)

If we compare the cutting speed for minimum cost with the cutting speed for maximum production rate it is clear that:

V p>V c

The minimum cost and the maximum rate cost can also be expressed analytically as functions of the tool life constants and the machining feed rate :

Ci(Vc) = (K^Tp+K,)-1 [(-k-l)-1 +SC (-k-l)k+1/k'

] Km +1^ (8.6a)

TrDI KrrT'cÎ^t C i(V p ) = — [ mSC! (Tc(-k-l))fc+U/k + Km(Tc(-k-l))-1(k ] (8.6b)

The cost and the production rate for an operation, described by equations (8.3) and (8.5) have been graphed against the cutting speed for the cases of high speed steel tool and the case of cemented carbide tool in figures 8.1 and 8.2 .

The data used for the derivation of these curves are summarised in table 8.1.

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machining t o o l t y p econstants cemented carbide high speed steel

Km (£/min) 9 9Kt (£) 2.5 10Tc (min) 1 2S (mm/rev) 0.5 0.5k -2.5 -5Cl 300 80D (mm) 50 50L (mm) 200 200

Table 8.1: Typical values for machining constants(source:B.Mills & A.H Redford. Machinabilitv of Engineering Material. Applied Science Publishers)

Figure 8.1 : Operation cost and production rate for high speed steel tool operation

L

173

o+*EcJ20 31a.

250

Figure 8.2 : Operation cost and production rate for cemented carbide tool operation

The range [Vc , Vp ] is called the high efficiency range. Traditionally, cutting speed values are chosen from this range. The difference of C|(Vp) from Cj(Vc) is a measure of the potential cost savings which can incur if the machining time is increased from its value for Vp to its value of Vc .

Using the same data and the equations developed above, we obtain the results summarised in table 8.2.

cemented carbide tool high speed toolmin. cost max rate min cost max rate

V 231.3 255.1 48.31 52.78t 0.2716 0.2463 1.300 1.190T 1.916 1.500 12.453 8.00Np 7.054 6.089 9.579 6.720R 2.419 2.436 0.663 0.672Ci 4.074 4.105 14.625 14.880

Table 8.2 : Single stage optimal performance results

The following observations can be made :

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1. The speed corresponding to the maximum production rate Vp , constitutes an upper bound on the range of economically feasible speed values since for speeds higher than Vp , the production rate is decreasing and the cost is increasing.2. The rate of change for both cost and production rate with the cutting speed, is much greater for high speed steel tools than for cemented carbide tools.3. The high efficiency range is very small in both cases.Values of Vc and Vp are close. Inboth cases examined above (table 8.2), Vp is about 9.5% higher than Vc . Using equations (8.4) and (8.6) we can actually calculate the fractional difference between Vp and Vc in terms of the tool parameters:

Vp-Vcv c

= ( 1 + Kt___L_ )-l/kTc Km

-l

Usually, this is in the range 0.05 - 0.15 (or 5% - 15%) for the FMS case.4.The cost curve is quite flat in the region of the Vc , Vp values, even for the case ofhigh speed steel tools. In the examples above, Cj(Vp) is only about 1% higher than ci(Vc). The fractional difference between the cost at maximum rate and the minimum cost can be calculated from equations (8.6a) and (8.6b):

Ci(Vp)-Ci(Vc) Kt Kt- i - l — _ = ( i ---------— ) ( i + _ L _ ) i /k _iC i(V c) kKmTc KmTc

This is usually in the range 0.005 - 0.020 (or 0.5% to 2%). Therefore, there are not considerable savings to be attained if cutting speeds are allowed to take values from what is the efficiency range of each individual operation. Such an approach is followed by McCartney & Hinds [106] who achieve high cost savings only because the function , which describes the relationship between cost savings derived for an operation and the extra time used, was chosen rather than calculated. It can be therefore concluded that slowing down the machine tool by reducing the cutting speed from Vp to Vc is unlikely to result in substantial cost savings.

However, as it will be made clear at the following section, there might be scope for reducing the cutting speed beyond the Vc value if the operation in question, is one amongst others which should be performed on a set of parts on a number of

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workstations. This is because system economics include the cost of idle time which has not been considered in the above analysis. Idle time and its associated cost can be reduced by extending machining times. At the same time, use of low speeds will reduce tool costs. Thus, optimal machining speeds when system economics are considered may be lowerfejthe machining speeds resulting from the optimisation of each individual operation. Cutting speeds however can be reduced until a minimum value which depends on the surface finish requirements for the particular operation.

8.3 Machining parameter optimisation under schedule constraints

The policy presented here for the machining parameter optimisation in the context of FMSs comprises two levels.At the higher level, a schedule is constructed for a set of operations.Aim:Minimisation of the schedule's makespan.The makespan is the time period from the start of the first operation to the finish of the last Machining times used for the schedule construction:The operation times utilised in the schedule construction are minimised. They correspond to cutting speeds for maximum production rate. These cutting speed values constitute an upper level on the range of economically feasible speed values.The use of such machining times for the schedule construction is reasonable since at this level the aim is to minimise the schedule's makespan.Set of operations to scheduleThe set of operations whose schedule is constructed at the initial level, belong to a set of parts which is to be produced periodically by the FMS. This set is called the minimal part set and it was introduced at Chapter 5. Parts in the minimal part set confine to production requirements as regards the part mix ratio.The size of the minimal part set is the outcome of an optimisation problem which strikes a balance between the cost of holding jobs and the cost of idle resources ( Chapter 5).Algorithm used for the schedule constructionThe schedule is constructed in this work with the help of the SHA algorithm introduced in Chapter 6. This is a heuristic algorithm which searches the set of non delay schedules. It does not guarantee the optimal but it finds a schedule which is on average 10% away from an absolute lower bound in terms of the makespan.Any algorithm which minimises the makespan could be used at this level, to construct a schedule for the set of operations.

At the lower level the cost of production associated with the schedule constructed at the

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higher level, is minimised under the schedule constraints.

AimMinimisation of production cost of a schedule holding the makespan constant. The resulting schedule has the same makespan to the initial schedule but minimum cost Production cost of scheduleThis cost comprises the cost of each individual operation and the cost of idle time which inevitably occurs in a schedule. The extent of idle time depends on the part mix, the routine and the schedule effectiveness. It is minimum for the optimal schedule. A mathematical model for this cost is developed in the next section.Solution methodThe problem is formulated as a non linear optimisation problem with linear constraints. A standard NAG library routine is used to solve the problem.

8.3.1 Problem formulation

Production cost of a schedule - mathematical model

The following notation is used:Pm

m

TVciîdCT4

part index , p e Pmachine index, m e M

operation index, i e Iset of operations needed to complete part pset of operations scheduled on machine m the production time (makespan) for set I operation cost cost of idle timetotal production cost of schedule time required for operation i

The production cost Gp is equal to the sum of the individual operation costs and the idle time cost:

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C j= X + (8.7)i e l

The cost of idle time is given by :Cid’= Kj Tj

where:Kl : the cost of lost production due to idle resources per unit time

and Ti : the sum of all machines idle time

but, Ti = 2* (Tpr t | ) msM y is Im

or Ti = M Tp r- S t iis I

where M is the cardinality of set M ; i.e the system comprises M machine tools

The cost of idle time is therefore equal to :

qd = K1MT_r -K 1 S q (8.8)y i 6 /

If we substitute the cost of idle time and the cost of operation q from equations (8.8) and (8.2) into equation (8.7), the total production cost for a schedule becomes:

G r= XKmT,

is / C f kc+ K t t.k + i

" s i

+ ^ *ii s /

+ K, M Tpr - K[ I t jis /

Let v > TC+Kt x / DiLi vC fk Si

-k

Kj is a constant for each operation and is dependent only on the tool used and the

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machining to be done. Then, the total cost of production is:

Cx = I Kj tik+1 is /

i g /

+ K1M Tpr (8.9)

If the feed rate Sj is set to the maximum determined from the feed and/or roughness constraints when the motor power is large enough, the machining times t| are the problem’s decision variables. From those, cutting speeds Vj can be determined from equation (8.1).

Machining times can be expressed in terms of starting and finishing times of operations:

li = x2i - x2i-l

where X2j :the finishing time of operation iand x2i-l : starting dme of operation i

The total production cost, expressed as a function of the starting and finishing times of operations is given by :

CT = I , K i (*2i - x2M )k+1

, ^ / x2 i_x2i-1 i e l

+ K{ M T pr (8.10)

The total cost of production given by equation (8.10), is the objective function of the machining parameter optimisation problem under schedule constraints. Before we proceed to the constraint formulation, let us discuss the form of the objective function

[

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and its implications.

1. The objective function is non linear. It is the sum of a non-linear, a linear and a constant element The non-linearity is due to the tool costing element2. The constant cost element depends on the makespan Tpr . It is minimum forminimum makespan and for the purpose of optimisation under schedule constraints need not be considered.3. The contribution of the linear cost element to the total cost depends on the values of the costing factors Km and K j. Those values depend heavily on the accounting system of the company. Three cases can be distinguished :

i) Km = K[ In that case the total cost is equal to the tooling cost and the constantelement which decreases monotonically as k is negative and between -2.5 and-5 with increasing machining times. Therefore, the optimal solution will lie on a vertex of the feasible region which is determined by the constraints of the decision variables.ii) Km< Kj The objective function is again monotonous with rate of decreasegreater than the rate of the previous case. The optimal solution will lie in a vertex of the feasible region as before.iii) Km>K| The objective function has in this case a minimum which might lieinside the feasible region. In practice, th is means that machining times in the optimal solution might not have been extended to their bounds. Constraints are then satisfied as strict inequalities.

It is difficult to comment on the values of the costing parameters for a practical situation as they depend heavily on the company and the particular application. However, the shape of the objective function, and therefore the location of its minimum, is mainly determined by the tool costing element irrespective of the values of the costing parameters. The increase of Kj relative to Km contributes negatively to the linear costelement and positively to the constant element; consequently, the total contribution in the cost is negligible.

Constraint formulation

Total production cost is to be minimised without altering any measures of schedule performance. This means that the schedule resulting from the cost minimisation should be of the same makespan Tpr as the initial schedule constructed with minimum

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machining times. The notion of critical operation is introduced to help deriving those constraints.

Definition : Critical operations are all the operations in the schedule which directly influence our objective to keep the makespan Tpr constant.

Critical operations can be found as follows:i) A critical status is assigned automatically to those operations which end at time Tpr.ii) These operations are then examined to identify which preceding operations have dictated when they could start.iii) Those operations are then made critical and the procedure goes again to step ii, until the beginning of the makespan i.e until time zero.Referring to figure 8.3, let a be the operation representing the final operation to be performed at the schedule. Then three possibilities can occur.i) the previous job P on the same machine directly paces aii) the previous operation on the same part a-1 directly paces a .iii) both p and a-1 directly pace a .

i)a-1

ii)a-1

p a

iii)a-1

Figure 8.3 : Critical operations

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If the makespan Tpr is to be maintained , then the starting and finishing times of critical operations should be fixed, i.e:

x2i = fi |and x2 i- l= bi V i s / C (8.11)where Ic : the set of critical operations. Clearly : I 3 Ic

b j , fj : the start and finish time of operation i in the initial schedule

Decision variables are also subject to the following constraints:1. Schedule constraints: They enforce the sequence of jobs on machines. They guarantee that the finishing time of an operation is less or equal to the starting time of the next operation on the same machine :

x2i "x2 i'-l - 0-0 where i, \ e Im , V m e M (8.12)

If i (or 1 ) is a critical operation ,the finishing time X2j of operation i (or starting time x2i"_i°f operation i') is fixed and therefore (8.12) is a lower (or upper) bound for variable X2i"_j (orx2j).

2. Makespan constraints: The makespan is an upper bound for the finishing time of all the operations which are final on each machine.

x2im- Tpr V m e M (8.13)

where im denotes the last operation scheduled on machine m The total number of schedule and makespan constraints is I.

3. Technological constraints : They guarantee that the finishing time of an operation is less or equal to the starting time of the next operation for the same part.

x2i ' x2 i'-l - 0-0 where i, \ e Ip V p e P If i or Y is a critical operation, constraint (8.14) is a bound for X2j'_i or X2i*

(8.14)

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The total number of technological constraints is I-P.

4. Minimum duration constraints: They guarantee that cutting speeds do not take values higher than the cutting speed for maximum production rate Vp .This is desirable sincefor higher speed values production rate decreases and production cost increases. Therefore, machining times should always be greater or equal to the machining times for maximum production rate.

x2 i" x2i-l - di V i e / (8.15)

where dj is the machining time of operation i for maximum production rate cutting speed. There are I minimum duration constraints.

5. Maximum duration constraints: They guarantee that cutting speeds do not take values lower than the ones necessary to meet the surface finish requirements for the particular operation. There is therefore a maximum machining time permissible for the desirable surface finish.

x2 i- x2i-l - hi V i e / (8.16)

where hj is the maximum permissible machining time for operation i. There are I maximum duration constraints.

6. Semi-positiveness constraints: They ensure that the starting and finishing times of operations will not take negative values.

x2i >0.0

x2i-l -0 -0 Vi 6 / (8.17)There are 21 semi-positiveness constraints but they can be reduced to min(M,P) as follows: Zero is a lower bound for the starting time of all first operations; all remaining semi-positiveness constraints are satisfied by the minimum duration, the technological and the schedule constraints.

The formulation of the cost minimisation problem under schedule constraints is next

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illustrated with the help of a simple example.

ExampleTwo parts are to be produced by a system of three machine tools. The machining sequences and the machining times for maximum production rate are given below. We also assume that a schedule is devised and the resulting job sequences are given.

Part Operation sequence and duration( operation T y p e ': time)

1 1:3 3 :4 2 :42 2 :4 1 :4 3 :2

Machine Part sequences1 1 22 2 13 1 2

A Gantt chart of the schedule is given in Figure 8.4.

Figure 8.4 : Gantt chart for the operations of an example

To simplify the notation, we name the operations as they appear on the Gantt chart (number on the top of the boxes) instead of the twilet (p,m).Critical operations are shadowed. Note that all operations of part 1 are critical.

Therefore we have:Critical equalities : xj = 0.0 xg = 11.0

X2 = 3.0 xg = 3.0 X7 = 7.0 xjq = 7.0

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Schedule constraints:

Makespan constraints:

mXVI<NX

= > 3.0 < x3

x6 - x7 = > x6 < 7.0xio< ;xn = > 7.0 <; x u

X 4 < 1 1 . 0

x12< 11.0( schedule and makespan constraints =5 instead of 1=6)

Technological constraints : ( 2 instead of I-P=4)x6 S x 3x4 S x n

Minimum duration constraints: (3 instead of 6=1)X4 - X3 > 4.0x6 ' x5 - 4 0 x12" X11 - 2,0

Semi- positiveness constraints : ( 1 instead of 2=min {M, P })X5 > 0.0

Comment : Positiveness of X3 and xjj is satisfied from schedule constraints. Positiveness of X4, x^, and x 1 2 is satisfied from the minimum duration constraints.

8.3.2 Problem solution

The problem of cost minimisation under schedule constraints was formulated as a non-linear constrained optimisation problem subject to linear constraints and simple bounds.

For the solution of this problem a standard NAG library routine is used, namely E04VCF. This routine is designed to solve the non-linear programming problem-the minimisation of a smooth non-linear function, subject to a set of constraints on the

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variables. The routine uses a sequential quadratic programming algorithm, in which the search direction is the solution of a quadratic programming problem. The problem has to be stated in the following form:

minimise FUN(x)XG Rn

subject to : I < [ ] < ua Lx

where FUN(x) : is a smooth non-linear function a l : is a constant matrix I, u : are the lower and upper bound arrays

Upper and lower bounds are specified for all the constraints. This form allows full generality in specifying all types of constraints; for example, equality constraints can be defined by setting 1 = u. However, in case the problem is sparse and large, the MINOS/AUGMENTED program could be used since this routine treats all matrices as dense.

A program was developed to transform the initially given data on the schedule which is desired to be minimised, to a form which is acceptable by the NAG library routine. Simple algorithms were used to form the necessary constraint and bound matrices. Specifically:Input of original problem : The input of the original problem for the case of MxP when all parts visit all machines once, consists of three matrices which describe the part routing, the sequence of parts on each machine and last the matrix giving the scheduled starting and finishing times of operations on each machine.

1. The machine sequence matrix. It gives the sequence of machines that a part should visit in order to be completed.with dimension P x max Ipeor when all parts visit all machines once, the dimension is : PxM2. The part sequence matrix. It gives the sequence of parts on machines.

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with dimension M x max ImmeM

or when all parts visit all machines once, the dimension is : M xP

3.Schedule matrix. It gives the starting and finishing times of operations which correspond to the parts sequenced as in the sequence matrix.with dimension : M x2 max Im

m eM

or when all parts visit all machines once, the dimension is : M x 2P

An initial solution is provided for the optimisation to start and this is given by the values of the schedule matrix which are the starting and finishing times of operations for maximum production rate. The makespan of the initial schedule is found by the program by finding the maximum of the finishing times. Critical values are found with the help of a simple algorithm operating on the schedule matrix. Values at critical positions are replaced by zeros so that the outcome matrix has 0.0 values on the positions of critical operations. Creation of bound arrays and of the constraint matrix is performed in the program with the help of a simple algorithm which operates on the schedule matrix with the zero values on critical times. This algorithm works as it is described next:The schedule matrix is scanned and when a zero value is encountered, an equality constraint is created for this variable (since it is critical) and a lower or upper bound is defined for the immediate neighbour variable at the same row, if it is not a zero ( a lower bound if the non-zero position is on the left of the currently examined zero valued variable or an upper bound if it is on the right; note that it is impossible to have a zero value between two non-zero values on the the same row because a critical operation has both starting and finishing times set to zero). When a non-zero value is encountered, then minimum and maximum duration constraints are created involving the two variables of the starting and finishing times of the operation which corresponds to the non-zero value, and schedule and makespan constraints are created if the next operation on the same row is not critical or there is none (i.e the operation examined is the last on this machine).

For the derivation of the technological constraints,the matrix of the operation sequence for each part in terms of the starting and finishing times of operations is created by transforming the schedule matrix. Then, technological constraints are created for the non-zero positions involving the starting and finishing times of the non-critical

187

o p c ' c c t t i c r v ^ ^ <_■ £ “t k e , ,vi € , L g K b o u . v o p e .K ’c i^ i c r v i is C ' f i t i C Q . i j I o v J e . ' r <1y \c I u j^ jp c Y t c u v \ i

redefined if there are greater or smaller than the already existing bounds.

8.4 Results and discussion

The cost minimisation routine was applied in a job shop-like FMS which consists of five machining centres and is intended to process four different types of parts. Input data is given in table 8.3.

Part type Operation sequence and duration (operation ± y p e : time)1 1 :6 2 :4 3 :3 4 :8 5 :22 3 :3 1 :7 2 :5 4 : 10 5 :23 2 :4 4 :9 1 :6 3 :3 5 :34 1 :5 2 :4 3 :5 5 :2 4 :8

Table 8.3 : Operation sequence and durations

The workload chosen for the cell does not result in a balanced loading on machines (figure 8.5). Machine loading varies gready from bottleneck (machine 4) to little used (machine 5). It is recognised that potential savings from a cell depend upon the loading level variations on machines and a more balanced workload would result in less savings. The above workload distribution was however selected in order to facilitate the investigation of the effectiveness of the cost minimisation under schedule constraints with respect to the WIP level and machine utilisation levels.

machine

Figure 8.5 : Machine workload

> a r e

188

The heuristic simulation based algorithm was used to calculate realistic schedules for increasing set sizes (increasing WIP levels), with machining times corresponding to the maximum production rates. These schedules were then fed as inputs to the cost minimisation routine to produce the minimum cost schedules. The percentage cost reduction derived for each periodic schedule is given in figure 8.6. For the calculation of the schedule cost we se t: Dj = 50 mm and Sj = 0.5 mm/rev for every i. The cuttinglength Lj is set proportional to the machining time for maximum production rate and Km is set equal to Kj.

Figure 8.6 : Total cell percentage savings for increasing MPS sizes

Cost savings initially rise sharply with the WIP and then fall off. This can be explained in terms of two opposing trends that affect scheduling. First, there is an increase because there are simply more parts being processed by the system and therefore greater opportunities for savings arise. As the system approaches saturation, there is a reduction in idle time occurrence, thus constraining any extension of operation duration.

189

As can be seen from this example the percentage cost reduction ranges from 13 to 20% of the initial cost for each periodic schedule. The net cost savings for the whole of the production horizon are therefore a considerable amount of money. However, as discussed earlier, the level of potential cost savings depends upon the particular work mix. If the cell has a more balanced workload, savings would be less. For the case of one machine bottleneck as in the example investigated here, there is a varying degree of potential savings with each machining centre. There is, for example, very little opportunity for the bottleneck machine to offer tool cost savings due to the lack of idle time. This is shown in figure 8.7 where the % cost savings of each machine is plotted.

-o machine 1 machine 2

-o- machine 3 machine 5

Figure 8.7 : Percentage machine cost savings

190

The bottleneck machine (number 4) has zero savings whereas the remaining machines still maintain an effective savings performance. The underlying trend is an inverse relationship between machine utilisation and machine cost savings. Figure 8.8 shows this trend for this example.

0>o>c0)oo©JCo(0E

a MPS = 4 ♦ MPS = 8 □ MPS =12 A MPS = 16

Figure 8.8 : Machine cost savings dependence on workload

The NAG library routine used for the solution of the cost minimisation problem performed very well giving an optimal solution in every case. Computational requirements increase almost proportionally with problem size but memory space requirements grow rapidly with the problem size. Table 8.4 shows the workspace arrays needed for the cases examined above and it is clear that this might cause a problem for computers with no virtual memory.

MPS size workspace required by the NAG routine4 IW ( 146 ) W (5154)8 IW (316) W (20863)12 IW (477 ) W (45830)16 IW (638 ) W (80477)

Table 8.4 : Memory requirements

It is worth noting that memory requirements would be even greater if critical operations were not used in the problem solution. For example, problem size 4 needs arrays IW

191

(171) and W ( 6329) with no calculation of the critical operations and this increase in memory requirements is growing almost exponentially by size. The use of critical operations reduces the problem size both directly by cutting down on the number of constraints and bounds and indirectly by transforming some inequality constraints into simple bounds. Specifically:1) The minimum and maximum duration constraints for the critical operations are transformed into simple lower and upper bounds (or equality constraints) for the variables corresponding to the starting and finishing times of those operations.2) Schedule and technological constraints which involve a critical operation are transformed into simple bounds.3) Schedule and technological constraints involving two critical operations omitted since they are implied by the equality constraints of the critical operations.4) Semi-positiveness and makespan constraints involving a critical operation are not considered as they are implied in the equality constraints.As a result the size of the constraint matrix is reduced.

A more effective approach would be to employ the cost optimisation on a selective basis i.e to exclude operations which are performed by bottleneck or highly utilised machines as these machines do not offer opportunities for tool cost savings due to the lack of idle time. On the other hand, a routine which deals with sparse problems and has therefore less memory requirements could be used instead, as the constraint matrix is rather sparse in this case since only two variables are involved in each of the inequality constraints.

The problem of machining parameter optimisation can be formulated as a large optimisation problem where schedule and machining parameter issues are tackled together rather than separately in a two level approach. Appendix presents a simple formulation of the general optimisation problem as a non-linear mixed integer programming problem under constraints. A review of the available codes which can solve this kind of problem can be found in [111]. Such a formulation however has serious memory and computational time requirements which is a serious disadvantage. Table 8.5 below compares the number of constraints for the present formulation when critical operations are not considered and the general formulation given in appendix. It is reminded that the present formulation results in a problem of less size when critical operations are taken into account.

192

constraint type present formulation general formulationschedule & makespan I I ( P - l )

technological I - P I - Pminimum & maximum duration I I

semi-positiveness 21 reduced to min { M, P ) 21 reduced to Ptotal number of constraints 31 - P - min{M,P} I ( P + 1)

number of variables 21 I ( P + 3 ) / 2

Table 8.5 : Number of variables and constraints for the present and the general formulation of cost minimisation

For a problem containing 4 parts (i.e P=4 ) and 3 machines (i.e M=3 => 1=12 ), the formulation under schedule constraints requires 24 variables and 29 constraints which are further reduced by considering the critical operations whereas the general formulation requires 60 constraints and 42 variables (refer to example in section 8.3.1). This gap is widening with size and therefore the computational demands for big size problems may be severe.

193

CHAPTER 9CONCLUSIONS AND RECOMMENDATIONS

Conclusions drawn from the various aspects of the theoretical and computational investigation on the production scheduling policy for FMSs, and possible extensions of the research work, may be summarised as follows:

1) Review of the previous work indicates that some sort of hierarchical policy is necessary; there is no one FMS model from those used for studying scheduling and control problems which successfully addresses the complete FMS scheduling problem; queueing models are restricted as they tend to appear in aggregate levels, simulation models are mostly used in a trial and error process in an attempt to find an optimal solution and scheduling approaches have unacceptable computational requirements.

2) The activity centered methodology for simulating FMSs leads to easy construction of FMS models. Complicated linking is avoided, the flow of entities is decided by the control module which comprises all decisions concerning the FMS operation and thus, control and scheduling issues are easy to study.

The given FMS modelling method could be enhanced with the development of a program generator where activity diagrams would be given and linking would be done automatically. Graphic facilities could also be added.

3) The periodic production policy is a method by which:a) steady state is periodic in sets of parts which satisfy the required production ratios;b) production ratios are calculated based on the current operational and demand state so that long term production target is satisfied;c) WIP is completely controllable as part flow time is constant during production under the same operational state;d) lead times and production cost are minimised;e) fast response to a change in the operational or demand state is achieved;f) the part scheduling problem is decreased in size andg) the policy can be implemented on line in practical systems since the calculation of the MPS schedule is required only each time the operational or demand state changes.

In the periodic production scheduling policy for FMS, it would be desirable to eliminate the machine idle time within the minimal part set schedule in order to fully utilise machinery. A natural extension of the present study would therefore be to look at the

194

possibility of scheduling parts of subsequent sets when no parts of the present set are available for machining while preserving periodic output of parts at the required ratios. The transition period between two steady states of an FMS which occurs when the system's operational state changes could also be looked at in detail to gain insight on the system's behaviour during that time.

4) The part scheduling problem in a minimal part set is solved without decomposition in routing and part flow problems; thus, operational flexibility is preserved. The simulation based heuristic algorithm guarantees the derivation of a realistic schedule. The optimal solution is not guaranteed but the resulting makespan deviates in average 10% from the lower makespan bound and it is better from the makespan derived by the use of the SPT rule. Computational requirements increase with the size of the system and the minimal part set but they remain in acceptable level for realistic system sizes.

It would be desirable to examine alternative ways for future performance estimates needed in the operation of the simulation heuristic algorithm. For example, a study could be made to establish the best priority rule which should be used, or a method could be suggested by which a weighted number of estimates are used for each node.

5) Machining parameter optimisation under schedule constraints minimises production costs by both saving in tooling costs and reducing machine idle time. Potential cost savings depend however on the loading level variations on machines. A more balanced workload results in less savings; therefore, machining parameter optimisation with its memory and computational requirements is not justified for this case.

Investigations on the computational performance of the presently existing codes for integer programming problems or even the development of a new code which would take advantage of the special characteristics of the general FMS optimisation problem may also lead to a computationally acceptable solution which would give the schedule of the minimal part set and the machining speeds for minimum cost production.

195

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204

APPENDIX

General FMS optimisation problem

A general approach does not preassume the existence of a schedule as in the two level approach (chapter 8). The objective function of the cost minimisation problem is of the same form as equation (8.10) but the makespan Tpr is not constant anymore but it is rather a function of the machining times and the unknown schedule.

The makespan Tpr in this case can be thought as the maximum operation finishing time:

Tpr = max { x2 i ) i e l

The objective function for the general case is therefore given by :

CT = X K j (x2 i ' x2i-l)lC+1 i e l

S Cx2i -x2i.i) i e l

{ x2i } (A.l)

Technological constraints, minimum and maximum duration constraints and semi-positiveness constraints are identical to those developed for the previous problem. Critical operation equalities and makespan constraints are not valid since a schedule is not predefined. However, schedule constraints must be employed to assure that no two operations are processed simultaneously by the same machine.

Suppose for example that part p precedes part q on machine m which means that operation i(p,m) is completed before operation i(q,m). Then it is necessary to have :

x2i(q,m)-l - x2i(p,m)

+ (Km -K !)

+ Kj M max

On the other hand, if part q precedes part p on machine m, then it is necessary to have:

205

x2i(p,m)-l - x2i(q,m)

These are called disjunctive constraints because one or the other must hold.In order to accomodate these constraints in the formulation, we define an indicator variable Ypqm as follows :

Ypqm = * if p precedes q on machine m = 0 otherwise

Then, the constraints become:

x2i(q,m )-l" x2i(p,m) + H (1~ Ypqm) - 0*0

x2i(p,m)-l' x2i(q,m) + ^ ( Ypqm) - 0*0 (A.2)

where H represents a very large positive number.

The general FMS optimisation problem is therefore to minimise total cost given by equation (A.l) under the constaints given by equations (8.14), (8.15), (8.16), (8.17) and (A.2).

Type of problemThe problem, is a non-linear mixed integer programming optimisation problem under constraints. It is a mixed integer problem due to the introduction of the integer variables ypqm in addition to the continuous x variables.

Number of variables and number of constraintsThere are 21 x variables, and I( P-l)/2 Ypqm variables since yqpm need not be definedif Ypqm is in the formulation.The total number of variables is therefore : I ( P+3)/2

There are I minimum and maximum duration constraints , 21 semi positiveness constraints which can be reduced to P, I-P technological constraints and I(P-l) schedule constraintsThe total number of constraints is therefore: I(P+1)

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PRODUCTION SCHEDULING POLICY FOR FLEXIBLE … · Last, in the third part (Chapter 8) the effect of machining parameters on production scheduling for FMS is investigated. The cost