A stochastic model for the analysis of a two-machine flexible manufacturing cell

A stochastic model for the analysis of a two-machine flexible

manufacturing cell

Majid M. Aldaihani *, Mehmet Savsar

Department of Industrial and Management Systems Engineering, College of Engineering and Petroleum, Kuwait University,

P.O. Box 5969 Safat 13060 Kuwait

Received 4 October 2004; received in revised form 12 September 2005; accepted 29 September 2005

Abstract

This paper presents a stochastic model to determine the performance of a flexible manufacturing cell (FMC) under variable

operational conditions, including random machining times, random loading and unloading times, and random pallet transfer times.

The FMC under study consists of two machines, pallet handling system, and a loading/unloading robot. After delivering the blanks

by the pallet to the cell, the robot loads the first machine followed by the second. Unloading of a part starts with the machine that

finishes its part first, followed by the next machine. When the machining of all parts on the pallet is completed, the handling system

moves the pallet with finished parts out and brings in a new pallet with blanks. A model with these characteristics turns out to be a

Markov chain with a transition matrix of size 5nC3, where n is the number of parts on the pallet. In this paper, we present exact

numerical solutions and economic analysis to evaluate FMC systems, to determine optimal pallet capacity and robot speed that

minimize total FMC cost per unit of production.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Flexible manufacturing cells; Stochastic modeling; Production planning

1. Introduction

The demand for customized products has been increasing continuously in the recent years and a great deal of

attention has been given to the automation of manufacturing systems. In order to meet increased demand for

customized products and to reduce production lot sizes, the industry has adapted new techniques and production

concepts by introducing flexibility into production machines so that variety of products can be manufactured on the

same equipment. Past studies, including Chan and Bedworth (1990); Sohal, Fitzpatrick, and Power (2000); Kim, Park,

and Leachman (2001), and Lashkari, Balakrishnan, and Dutta (2002) indicated that flexible manufacturing cells

(FMCs) present a feasible approach for automating the job shop process, since they require lower investment, less

risk, and also satisfy many of the benefits gained through flexible manufacturing systems (FMSs). While FMSs are

very expensive and generally require investments in millions of dollars, FMCs are less costly, smaller and less

complex systems. Therefore, for smaller companies with restricted capital resources, a gradual integration is initiated

with limited investment in a small FMC, which facilitates subsequent integration into a larger system, an FMS. An

FMC consists of a robot, one or more flexible machines including inspection, and an external material handling

Computers & Industrial Engineering 49 (2005) 600–610

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0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cie.2005.09.002

* Corresponding author. Tel.: C965 4987257; fax: C965 4816731.

E-mail address: [email protected] (M.M. Aldaihani).

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M.M. Aldaihani, M. Savsar / Computers & Industrial Engineering 49 (2005) 600–610 601

system such as an automated pallet for moving blanks and finished parts into and out of the cell. The robot is utilized

for internal material handling which includes machine loading and unloading. The FMC is capable of doing different

operations on a variety of parts, which usually form a part family with selection by a group technology approach. The

cell performance depends on several operational and system characteristics, which includes part scheduling, robot,

machine and pallet characteristics.

Most of the researches related to operational characteristics of FMC are directed to the scheduling aspects.

Scheduling algorithms are used to determine the sequence of parts, which are continuously introduced to the cell.

Chan and Bedworth (1990); Hitomi and Yoshimura (1986); Seidmann (1987); Hutchinson, Leog, Snade, and Ward

(1991), and Agnetis, Alfieri, and Nicosia (2003) have developed models for static and dynamic scheduling in FMC.

However, system characteristics, such as configuration, design, and operation of an FMC, have significant effect on its

performance. Machining rate, pallet capacity, robot speed and pallet speed are important system characteristics

affecting FMC performance. Several models have been developed for FMS and FMC in relation to the effects of

different parameters on system performance. Buzacott and Yao (1986); Henneke and Choi (1990); Sabuncuoglu and

Hommertzheim (1989); Savsar and Cogun (1993), and Cogun and Savsar (1998) have presented stochastic and

simulation models for evaluating the performance of FMC and FMS with respect to system configuration

and component speeds, such as machining rate, robot and pallet speeds. Sohal et al. (2000); Kim et al. (2001), and

Lashkari et al. (2002) presented systematic approaches for the study and control of FMCs as well as allocation of

pallets in FMCs. Savsar (2000); Koulamas (1992) have looked into the reliability and maintenance aspects and

presented stochastic models for the FMC, which operate under stochastic environment with tool failure and

replacement consideration. They developed Markov models to study the effects of tool failures on system

performance.

In this study, we present a stochastic model to determine the performance of an FMC under variable operational

conditions, including random processing times, random machine loading and unloading times, and random pallet

transfer times. Stochastic modeling is necessary due to different operational requirements of a variety of parts

dynamically scheduled to enter into the cell. In particular, utilization rates of the machine tools, the robot and the

pallet handling system are formulated and determined under different operational parameters. The model should help

the decision makers to select the best pallet capacity and robot speed, which minimize FMC cost per unit of

production.

2. Operation of the cell

The FMC system considered in this study is illustrated in Fig. 1. An automated pallet handling system delivers n

blanks consisting of different parts into the cell. The robot reaches to the pallet, grips a blank, moves to the first

machine, and loads the blank. While the first machine starts operation on the part, the robot reaches the pallet, grips

the second part, moves to the second machine, and loads it to the machine. Next, robot reaches to the machine which

finishes its operation first, unloads the finished part and loads a new part. The loading/unloading operation continues

in this way with the preference given to the machine which finishes its operation first. After the machining operations

of all parts on the pallet are completed, the pallet with n finished parts moves out and a new pallet with n blanks are

delivered into the cell automatically. Due to the introduction of different parts into FMC and the characteristics of the

system operation, processing times as well as the loading/unloading times are random, which present a complication

Machine 1

Machine 2 Robot

Pallet In/out

Fig. 1. A flexible manufacturing cell with a robot, two machines, and a pallet.

M.M. Aldaihani, M. Savsar / Computers & Industrial Engineering 49 (2005) 600–610602

in studying and modeling the cell performance. If there was no randomness in system parameters, the problem could

be analyzed by a man-machine assignment chart for non-identical machines, and by a symbolic formulation for

identical machines. The reader is referred to Francis, McGinnis, and White (1998) for more information on classical

man-machine assignment problem and the related analysis.

3. Stochastic modeling of cell operations

In order to analyze the FMC with stochastic operation parameters, the following model is developed. Processing

times on the machine, robot loading and unloading times, pallet transfer times, and machine operation times are all

assumed as random quantities that follow exponential distribution. In order to model FMC operation, the following

notations are introduced:

Sijkl state of the FMC in steady state

Pijkl steady state probability that the system will be in state Sijkl

i number of unprocessed items on the pallet

j state of the production machine 1 (jZ0 if the M/C is idle; jZ1 if the machine is operating on a part; and jZ2 if the machine is waiting for the robot)

k state of the production machine 2 (kZ0 if the M/C is idle; kZ1 if the machine is operating on a part; and

kZ2 if the machine is waiting for the robot)

l state of the robot (lZ0 if the robot is idle; lZ1 if the robot is loading/unloading machine 1; lZ2 if the robot

is loading/unloading machine 2)

im loading rate of the robot for machine m (mZ1,2) (parts/unit time)

um unloading rate of the robot for machine m (mZ1,2) (parts/unit time)

zm combined loading/unloading rate of the robot for machine m (mZ1,2)

w pallet transfer rate (pallets/unit time)

nm machining rate (or production rate) of machine m (mZ1,2) (parts/unit time)

n pallet capacity (number of parts/pallet)

Qc production output rate of the cell in terms of parts/unit time

It should be noted that the robot transportation times are assumed to be included in the loading and unloading

times. Using the state probability definitions and the above notation, the probability transition diagram of the FMC

operation, with stochastic operations of the machine tool and the robot, is shown in Fig. 2. Using the fact that the net

flow rate at each state is equal to the difference between the rates of flow in and flow out, the following system of

difference equations are constructed for the stochastic FMC to compute the steady state probabilities.

wP0;000Kl1Pn;001 Z 0

l1Pn;001 Kðv1 C l2ÞPnK1;102 Z 0

v1PnK1;102Kl2PnK1;202 Z 0

l2PnK1;102 Kðv1 Cv2ÞPnK2;110 Z 0

v1PnK2;110 C l2PnK1;202Kðv1 Cz1ÞPnK2;011 Z 0

v2PnK2;110Kðv1 Cz2ÞPnK2;102 Z 0

v2PnK2;011Kz1PnK2;021 Z 0

z1PnK2;011 Cz2PnK2;102 Kðv1 Cv2ÞPnK3;110 Z 0

v1PnK2;102Kz2PnK2;202 Z 0

v2PnK3;110 Cz1PnK2;021 Kðv1 Cz2ÞPnK3;102 Z 0

v1PnK3;110 Cz2PnK2;202 Kðv2 Cz1ÞPnK3;011 Z 0

«

v2P0;110 Cz1P1;021Kðv1 Cu2ÞP0;102 Z 0

v1P0;110 Cz2P1;202Kðv1 Cu1ÞP0;011 Z 0

v1P0;102 Ku2P0;202 Z 0

u2P0;102 Kv1P0;100 Z 0

u1P0;011 Kv2P0;010 Z 0

v2P0;011 Ku1P0;021 Z 0

v1P0;102 Ku2P0;202Ku1P0;001 Z 0

v2P0;010 Cu1P0;021Ku2P0;002 Z 0

u1P0;001 Cu2P0;002KwP0;000 Z 0

(1)

The resulting set of difference equations must be solved in order to obtain the state probabilities. There are 5nC3

equations and equal number of unknowns. It is difficult to obtain a closed form solution for these equations for a

general n. However, exact numerical solutions can be obtained for fixed values of system parameter n. For example,

for nZ4, number of system states, as well as number of equations is 5(4)C3Z23. In order to determine numerical

solutions, for 23 state probabilities represented by the vector, P, the set of equations given by PTZ0 must be solved

for P, where T is the probability transition rate matrix.

Since matrix T is known to be singular for Markov Chains, we must add the normalizing condition given below:

Xn

iZ0

X2

jZ0

X2

kZ0

X2

lZ0

Pijkl Z 1 (2)

Exact numerical solutions can be obtained for each state probability. In the following section, first we present a

case example to illustrate the model for the case of nZ4. Then we analyze FMC performance measures by varying the

pallet capacity, n, and the robot speed. Finally a cost model is developed and utilized to optimize total system cost

with respect to various cost elements, the pallet capacity and the robot speeds.


4. Case problems and results

Several case problems have been considered with specific cell parameters in order to illustrate the application of

the model. For the first case, the following are the assumed mean values for various cell parameters. It should be noted

that the mean is the inverse of the rate in each case:

Operation time per part Z1/nmZ2 time units, mZ1,2

Robot loading time for the first part Z1/imZ0.25 time units, mZ1,2

Robot loading/unloading time for subsequent parts Z1/zmZ0.5 time units, mZ1,2

Robot unloading time for the last part Z1/um Z0.25 time units, mZ1,2

Pallet transfer time Z1/uZ1 time units per pallet

Pallet capacity, nZ4 units.

As mentioned in the previous section, for nZ4 the system can be in any one of 23 states at any time. The steady

state probability of each state represents the fraction of time that the system is in that state. The probability transition

matrix, T is constructed for nZ4 and PTZ0 is solved for the steady state probability vector P. The solution vector P is

used to determine various system performance measures.

System performance is measured by FMC production rate as well as machine, robot, and pallet utilization rates.

Table 1 summarizes system performance measures and the system states, which are used to determine these measures.

Production output rate of the cell, Qc, is defined as the number of parts completed by the cell per unit time. The cell

will produce one unit when only one of the machines operates and it will produce two units when both machines

operate. Therefore, to determine cell output rate, percent of time that only one machine is operating and the percent of

time that both machines are operating must be determined separately if the machines have different machining rates.

l1

Sn-2,1,1,0

Sn-1,1,0,2

Sn-1,2,0,2

Sn,0,0,1

Sn-2,0,1,1 Sn-2,1,0,2

S0,0,0,0

Sn-2,2,0,2Sn-3,1,1,0Sn-2,0,2,1

Sn-3,0,1,1Sn-3,1,0,2

Sn-3,0,2,1Sn-4,1,1,0Sn-3,2,0,2

Sn-4,1,0,2Sn-4,0,1,1

S1,2,0,2S0,1,1,0S1,0,2,1

S0,0,1,1S0,1,0,2

S0,0,2,1S0,0,1,0S0,1,0,0

S0,0,0,2S0,0,0,1

S0,2,0,2

l2v1

v1v2l2

z2v2 z1 v1

z1 v2 v1 z2

z1 v2z2v1

z1v2v1z2

z1 v2 v1 z2

v1 u2 u1 v2

u2 v2v1 u1

u1 u2

w

Fig. 2. Probability transition flow diagram for the flexible cell operation.


Table 1

States of the system in which each component is busy, idle, or waiting

System condition States of the system in which the condition occurs and total percent of time the system is in these states for

the case nZ4 units per pallet

Pallet Busy P0000

M/C1 Busy P3102CP2110CP2102CP1110CP1102CP0110CP0102CP0100

M/C1 Waits P3202CP2202CP1202CP0202

M/C1 Idle P0000CP4001CP2011CP2022CP1011CP1021CP0011CP0010CP0021CP0001CP0002

M/C2 Busy P2110CP2011CP1110CP1011CP0110CP0011CP0010

M/C2 Waits P2021CP1021CP0021

M/C2 Idle P0000CP4001CP3102CP3202CP2102CP2202CP1102CP1202CP0102CP0202CP0100CP0001CP0002

Robot Busy P4001CP3102CP3202CP2011CP2102CP2021CP2202CP1102CP1011CP1202CP1021CP0011CP0102C

P0202CP0021CP0001CP0002

Robot Idle P0000CP2110CP1110CP0110CP0100CP0010


Percent of time M/C 1 is busy onlyZLmc1ZP3102CP2102CP1102CP0102CP0100

Percent of time M/C 2 is busy onlyZLmc2ZP2011CP1011CP0011CP0010

Percent of time both machines are busyZLmc12ZP2110CP1110CP0110

FMC production rate is given by QcZn1(Lmc1)Cn2(Lmc2)C(n1Cn2)(Lmc12)

Based on this formulation above, the cell production rate is found to be 0.5505 parts per unit time.

For the case example with a pallet capacity of nZ4, two additional cases are solved for different cell parameters and

the results are summarized in two graphs as given in Figs. 3 and 4. For the first case, v1Zv2Z1.0 and for the second case,

v1Zv2Z2.0 parts/unit time; for both cases l1Zl2Z4.0; z1Zz2Z2.0; u1Zu2Z4.0 parts/unit time; and w changing

between 1 and 10 parts/unit time. The first graph compares the production output rate with respect to the pallet transfer

rate for the two machining rates. The second graph compares the utilization of the machines and the robot with respect to

pallet transfer rates under both cases. The dotted lines represent the first case (v1Zv2Z1.0) while the solid lines

represent the second case (v1Zv2Z2.0). While the production rate is very high in the second case, machine utilizations

are lower than the first case due to higher machine operation rates. Other comparisons can be seen in the graphs.

The stochastic model presented above can be utilized to analyze the FMC performance under several other

parametric conditions. The effects of pallet capacity, n, on various cell performance measures can be analyzed. A case

example was selected to investigate this aspect of FMC cell operation. Fig. 5 shows the effects of pallet capacity on

the FMC production rate under different robot speeds for a cell with machining rates v1Zv2Z1.0 parts/unit time,

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1 2 4 7 9 10

Pallet Transfer Rate/Min. (w)

Pro

duct

ion

Rat

e/M

in. (

Q)

V1=V2=1.0 V1=V2=2.0

3 65 8

Fig. 3. Production rates (dotted lines is the first case).

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

1 6 10

Pallet Transfer Rate/Min. (w)

Util

izat

ion

Rat

es

Robot M/C1 M/C2 Robot M/C1 M/C2

2 3 4 5 7 8 9

Fig. 4. Utilizations of components (dotted lines are the first case).


robot loading rate in the range l1Zl2Z2–14 parts/unit time with increments of 2 units, combined robot

loading/unloading rate in the range z1Zz2Z1–7 parts/unit time with increments of 1 unit, and robot unloading rates

within the range u1Zu2Z2–14 parts/unit time with increments of 2 units. The pallet transfer rate was set to 1

pallet/unit time. As seen in Fig. 5, the FMC production rate significantly increases with increasing pallet capacity for

any robot speed. The robot speeds are denoted by RSi, where i changes from 1 to 7, 1 corresponding to the lowest

speed and 7 corresponding to the highest speed as specified in the speed range. The graph clearly illustrates how the

increasing robot speed affects FMC production rate at various pallet capacities.

Figs. 6–9 illustrate the effects of robot speeds on machine 1, machine 2, robot, and pallet utilizations respectively

under different pallet capacities. Utilizations of the fist three FMC components (machine 1, machine2, and robot)

increase with increasing pallet capacity and the robot speeds. However, the utilization of the pallet decreases with

increasing the pallet capacity and decreasing the robot speed. The same notation is used for the robot speeds as in

Fig. 5. As it is seen in these figures, pallet capacity, as well as robot speeds, have significant effects on machine, robot,

and pallet utilizations, in addition to the FMC production rate.

Production Rate (n,RS)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 1 5 7 8 9 10 11

Pallet Capacity

Pro

duct

ion

Rat

e

RS1 RS2 RS3 RS4 RS5 RS6 RS7

2 3 4 6

Fig. 5. Effects of robot speeds on FMC production rate.

Utilization of Machine 1

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 3 10 11

n

Util

izat

ion

RS1 RS2 RS3 RS4 RS 5 RS6 RS7

1 2 4 5 6 7 8 9

Fig. 6. Effects of robot speeds on machine 1 utilization.


5. Modeling and optimization of FMC system costs

FMC performance can be further investigated and optimized with the use of a cost model by incorporating various

cost elements in the analysis. For a man-machine assignment problem, with m machines assigned to one operator,

the following cost model has been suggested by Francis, McGinnis, and White (1998).

TCðmÞ Z ðC1 CmC2ÞT=m (3)

where

TC(m)Z Cost per unit of production if m machines are served by one operator

C1ZOperator or robot cost per unit time

C2ZMachine cost per unit time (assuming identical machines)

mZNumber of machines assigned to the operator

TZCycle time during which m parts are produced

Utilizatio n of Machine 2

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

n

Util

izat

ion


0 3 10 111 2 4 5 6 7 8 9

Fig. 7. Effects of robot speeds on machine 2.

Robot Utilization (n, RS)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

n

Util

izat

ion


0 3 10 111 2 4 5 6 7 8 9

Fig. 8. Effects of robot speeds on robot utilization.


We have changed the FMC cost model above, which is based on a constant cycle time, to the following form for

the stochastic case with variable cycle time, incorporating pallet costs in addition to machine and robot costs.

TCðmÞ Z ðC1 CC2 CCr CCpÞ=Qc (4)

where

C1ZCost of machine 1 per unit time (including capital recovery and other costs)

C2ZCost of machine 2 per unit time (including capital recovery and other costs)

CrZCost of robot per unit time

CpZCost of pallet per unit time

QcZCell production rate per unit time, which is given by the following:

Pallet Utilization (n, RS)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n

Util

izat

ion


0 3 10 111 2 4 5 6 7 8 9

Fig. 9. Effects of robot speeds on pallet utilization.

Optimal Pallet Capacity

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

n

cost

3 10 111 2 4 5 6 7 8 9

Fig. 10. Optimum pallet capacity, which results in minimum total FMC cost.


Qc Z r1n1 Cr2n2

where, r1 and r2 are percent utilizations of machine 1 and machine 2, respectively, n1 and n2 are production rates of

machine 1 and machine 2 per unit time, respectively.

Eq. (4) gives the total system operation cost per unit of production for the stochastic FMC system. It

includes the operation costs of the machines, the robot, and the pallet. For the case example presented in the

previous section, all costs were normalized with respect to machine costs, which were set to C1ZC2Z1.0 cost

per unit/unit time for both machines. Since the robot cost depends on the speed of the robot itself, this cost was

set as a function of the speed z as follows: Robot cost, CrZ0.1C0.1z. Thus, this cost includes a fixed portion,

which is about 10% of the machine cost, and a variable portion which increases with respect to the speed.

Similarly, the pallet cost was set as a function of pallet capacity n as follows: Pallet cost, CpZ0.1C0.08n.

Based on these normalized costs, in which fixed robot and fixed pallet operation costs are 10% of machine

operation costs, optimum pallet capacity was found to be nZ6 units at the lowest level of robot speeds as

shown in Fig. 10. The corresponding total system operation cost was 4.17.

Table 2 shows the optimum pallet capacity under all robot speeds specified. Among all the speeds and pallet

capacities, the optimum corresponds to a pallet capacity of nZ9 units and to a robot speed of RS5, which was l1Zl2Z10 parts/unit time, z1Zz2Z5 parts/unit time, and u1Zu2Z10 parts/unit time.

Table 2

Minimum total system cost with varying robot speeds and pallet capacity

N RS1 RS2 RS3 RS4 RS5 RS6 RS7

2 4.920 4.096 3.927 3.910 3.952 4.023 4.110

3 4.444 3.549 3.344 3.297 3.313 3.358 3.418

4 4.257 3.309 3.080 3.017 3.018 3.049 3.096

5 4.185 3.192 2.945 2.870 2.860 2.882 2.922

6 4.170 3.136 2.873 2.789 2.772 2.787 2.821

7 4.188 3.115 2.839 2.746 2.723 2.733 2.762

8 4.227 3.116 2.827 2.727 2.699 2.705 2.729

9 4.278 3.132 2.830 2.724 2.690 2.693 2.714

10 4.341 3.157 2.844 2.731 2.694 2.693 2.712

n* 6 7 8 9 9 10 10


6. Conclusions

Flexible manufacturing cells are gaining wide acceptance in today’s dynamic manufacturing environment. Thus, it is

essential to be able to analyze these systems in detail either before implementation or during their operations. While

modeling and analysis of traditional machines and production systems have been a subject of extensive research over the

past several years, FMC systems have not received the same amount of attention. The stochastic models and the

procedure for numerical solutions obtained in this paper could be used to analyze the productivity of a FMC under

different machine, robot, and pallet operational characteristics. While exact numerical solutions can be used to investigate

system performance and system behavior under different operational conditions, it is still desirable, and the work is in

progress, to obtain closed form solutions for state probabilities, which would result in simple formulas for various system

performance measures as functions of operational parameters of system components and the pallet capacity n.

The model was utilized to analyze FMC system performance measures with respect to pallet transfer rates, robot

speeds, and pallet capacities. Amount of increase in FMC production rate and the utilizations of its components with

respect to pallet and robot characteristics are determined using the model presented. A cost model was developed to

optimize system performance with respect to various cost elements under different conditions. In particular, minimum

FMC operation cost corresponding to specific pallet capacity and robot speeds was also determined. Design engineers

and operation managers can benefit from these analysis either during the FMC design phase for the appropriate

selection of its components or during its operation phase.

The stochastic model of the described FMC can be extended in many directions. A new problem emerges if the

number of machines in the FMC is increased to three or more machines instead of two. It will be attractive if the

model of this problem turns out to be a Markov chain with an exact solution. Another related problem arises when we

assume that each machine is served by a dedicated robot. Although the robot utilization would to be less in this case, it

may be interesting to witness the effect on the production rate and machines’ utilization and compare them to our

model. A further extension is to study and analyze an FMC with unreliable machines and/or robot.

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Documents

A stochastic model for the analysis of a two-machine flexible manufacturing cell