Capacitated Lot-Sizing and Scheduling with Sequence ...ieomsociety.org/ieom_2016/pdfs/532.pdf · Capacitated Lot-Sizing and Scheduling with Sequence-Dependent setups in Petrochemical

Capacitated Lot-Sizing and Scheduling with Sequence-Dependent setups in Petrochemical Plants

Eng. Sari Abdullah Dr. Abdulrahim Shamayleh

Dr. Malick Ndiaye Industrial Engineering Department

American University of Sharjah Sharjah, UAE

[email protected] [email protected]

[email protected]

Abstract — In this work we developed a mathematical model to study the problem of capacitated lot sizing. The problem under study is concerned with multi-period planning and scheduling for a company that runs multiple plants to produce multiple grades of product. The company owns multiple warehouses and the raw material needed for production is provided by different suppliers. There is a setup cost for switching the production between the different product grades and the setup cost is sequence dependent. The problem under study is in the petrochemical manufacturing field. When different grades are produced in a petrochemical facility, a non-prime material is formed, this material depends on the production sequence of the differ grades. The model will answer questions related to supplier selection for raw material, production plan for each plant, inventory levels of the different grades at each plant, and warehouse selection to satisfy demand.

Keywords — capacitated lot-sizing; sequence-dependent change-overs; mixed-integer linear programming; multi-grade polypropylene; petrochemical production

I. INTRODUCTION

The global market is becoming more competitive than before. Companies started to be aware that production planning is considered as an important activity that effects the organization performance. There are three levels of planning: Strategic Planning (Long Term), Tactical Planning (Medium Term) and Operational Planning (Short Term). Production planning involves many important decisions such as lot-sizing and scheduling which are the most important and challenging. Lot-sizing is deciding about the production orders or lots to satisfy the customers demand while considering the minimum cost. Lot-sizing problems vary in their complexity depending on the industry application that impacts the features taken in the model [1].

Lot-sizing problems can differ in terms of specifications and features, Hacer et al. [2] reviewed the literature between 2000 and 2010 and summarized the lot-sizing problem according to number of manufacturing levels, capacity constraints, setup time and cost, planning horizon, demand type , and inventory shortages.

There are two categories of production systems, the single level system where raw materials are transferred to a final product with a single operation process and the multiple level systems where raw materials need more than one operation in order to be transferred to a final product. The demand of the single level products can come directly from customers or from the forecasts. This type of demand is called Independent. On the other hand, the demand of the multiple level products is called dependent because there is a relation between the items to get the final product.

As for the capacity, it can be identified as the resources availability or the production capability. If there are no limits on the resources, the lot-sizing problem is called uncapacitated problem where in this case the complexity will be less than if it is a capacitated problem.

Setup time/cost occurs when a production changeover happen to produce different products. The structure of setup is divided into two types, simple and complex. The setup time and cost can be considered simple if it doesn’t depend on the sequences of the products decisions. On the other hand, the complex setup structure consists of three types. The first type is setup carryover that involves continuation with the production of the same product in the next period without additional setup time or cost. The second type is family or major setup that happens when other products from a different product family is produced at the same period. The third type is sequence-dependent setup where production sequences influence the production sequence.

Planning horizon is the time interval that the production master plan schedule into the future. The planning horizon can be one of three types, infinite, finite and rolling.

1835© IEOM Society International

Proceedings - International Conference on Industrial Engineering and Operations Management, Kuala Lumpur, Malaysia, March 8-10, 2016

Proceedings of the 2016 International ConfKuala Lumpur, Malaysia, March 8-10, 2016

Infinite planning horizon is used usually when the demand of the products is dynamiperiods where the demand is known, the firperiod.

The demand can be divided into two tyconfidence. Deterministic demand can be sttype the demand is known based on some pr

Inventory shortage or backlogging is a fe

The lot sizing problem can be classified sizing problem (ELSP), Discrete lot-sizing problem (PLSP), General lot-sizing problem

Economic lot-sizing problem (ELSP) ibacklogging is not allowed, knowing the cohow many units to order while minimizingproblem (DLSP) is considered a small bucktime horizon is finite and divided into disdeterministic but dynamic. The objective is t

Capacitated lot-sizing problem (CLSP) The CLSP includes the sequence-dependentlot-sizing problem (PLSP) is concerned wiproducts can be produced and the sequence tthe DLSP and CLSP, where it is a mixed of small micro periods with a variable length, periods scheduling all together are considere

The Petrochemical Market is becoming pressure on the petrochemical producers besurvive and compete in the global market, Cand efficiency of the way of managing anEnterprise-Resource-planning systems, supProgramming [8].

SABIC is a manufacturing company manufacturers polymers, fertilizers, and meproducts. When it comes to production plaffiliate the monthly production plan. The pinventory level of products in the current peperiod. The department start matching the

Figure 1process map at SABIC af

ference on Industrial Engineering and Operations Manag6

when the demand of the products is constant, while finiteic. In the rolling planning horizon the planning is preparerst production decision is made then the horizon will be u

ypes, a deterministic demand where the value of the dematic or dynamic. The other type of demand is the stochast

robabilities.

eature where the demand of the current period can be satis

into five categories according Karimi et al. [3]. The cateproblem (DLSP), Capacitated lot-sizing problem (CLSP

m (GLSP).

s for single-item and deterministic demand with a finist function and the holding cost at each period, the object

g the holding and ordering cost over the planning horizoket problem where one product can be produced at most

screte periods that are used for planning production. Deto minimize the costs of holding and changeover costs [5].

is a large bucket problem, where several products can bet setup times and costs, in addition to the carry-over setupth multiple products produced to satisfy a dynamic demtime and cost is consider [6]. General lot-sizing problem (small-bucket and big-bucket. Each time unit in the planninthe sequence can be obtained by assigning each lot to a

ed as a macro period [7].

II. PROBLEM DESCRIPTION competitive recently. In the last years the World Trade Oecause of the incremental in the intensity of the environCompanies in Saudi Arabia started looking for methods tond producing. Achieving such goals can be done by sevupply-chain management and Optimization tools such

that is located in Saudi Arabia, it is a chemical ietals. SABIC owns a lot of joint-ventures with other companning, the headquarters planning department is respon

plan depends on three factors that should be taken into coeriod, production capacity in the next period and the expee supply and the demand for each affiliate after takin

previous factors. When the produbusiness department modify the plato consider the uncertainty in the mfinalized, each affiliate is requestamount of different products. The dare made by the production engproducts are produced through schedule of the lot is planned in atransition cost. This transition cowhen a setup happens to produce considered expensive due to the nat

The traditional process of pproducts pass through three procethese processes.

affiliate

gement

e planning horizon is used d for a certain number of updated towards the next

mand is known with high tic or probabilistic, in this

fied by the next period.

gories are: Economic lot-P), Proportional lot-sizing

ite planning horizon and tive becomes determining on [4]. Discrete lot-sizing t in a certain period. The emand of the products is .

e produced per time unit. p option [1]. Proportional

mand. It is the most. Two GLSP) is hybrid between ng horizon is divided into micro period. The micro

Organization increased the nmental laws. In order to o maximize effectiveness veral ways, for example,

as Linear and Integer

industrial company that panies to produce certain

nsible for providing each nsideration: the expected ected demand in the next

ng into consideration the uction plan is ready, the an by a certain percentage market. After the plan is ed to produce a certain

disaggregation of the plan ineers in each affiliate, lots, the size and the

a way that minimize the ost or setup cost occurs

a different product, it is ture of the process.

producing petrochemical ess, figure 1 summarize


Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-10, 2016

In stage one the raw materials arrive and enter a reactor to be processed, this reactor has a known capacity, many conditions must be calibrated such as temperature, pressure, and feed rates of raw materials in order to produce a certain grade. Changing these conditions can result with much different type of products, where these products may differ in physical and chemical properties. This makes a market for these different grades with different usages. Producing different grades need a changeover period in the reactor, in this period an amount of transitional material is produced. This material does not conform to the quality of any grade, which is why the selling price of it is lower than the prime grades. The amount of this material depends on the sequence of the production. Stage two some additives will be added to the grades to which can differ the specs of the same grade. Stage three the output of stage two will be packed into different packaging types.

The objective of this research is to develop a Mixed-Integer Linear Program (MILP) that integrates the planning process of supplier selection, lot-sizing and sequencing decisions and inventory levels in addition to the logistics functions of transportation and warehousing to minimize the overall affiliates cost which will increase the efficiency of the whole supply-chain of SABIC. In the objective function the costs of raw materials, production cost, transition costs, inventory holding costs and transportation costs will be minimized over the decision horizon.

III. LITRETURE REVIEW As mention previously, the capacitated lot sizing problem can have different configurations with different level of

complexity. In this research, the papers that were reviewed consider mainly a single level, dynamic, capacitated lot sizing with a difference in machine configuration, setup cost and time dependence and the solution algorithm.

A. CLSP with Single Machine Configuration Menezes and Clark [7] developed a model a single machine configuration with a sequence-dependent cost and time in

addition to a carry-over setup. The real contribution was in considering that the setup costs natural doesn’t follow the so-called triangular inequality. The triangular inequality means if there are a three products that needed to be processed, the cost and time required to setup the machine for the first product to the second then the third is always less than the sum of time required when the setting up considering an in-between product. This can’t be the case in many industries such as (pharmaceutical, chemical, food, etc). James and Almada-Lobo [9] propose two models that differs in the machine configuration, where the first consider a single machine capacitated lot-sizing with sequence dependent time and cost, and the second consider a parallel machine capacitated lot-sizing with sequence dependent time and cost. Both of the models is considered NP-Hard, therefore a new iterative MIP-based neighborhood search heuristics is developed.

Kwak and Jeong [10] considered a model that deals with a single machine capacitated lot-sizing with sequence dependent time and cost. A special form of setup cost is considered where the sequence-dependent setup time requires more when a larger product is produced. A two level hierarchical approach is used to solve the model. Mirabi [11] modeled a single machine capacitated lot-sizing problem with a sequence-dependent cost only with considering the time. Then a hybrid simulated annealing is used to solve the Hard-NP.

Shim [12] considered a single machine capacitated lot-sizing problem with sequence-dependent cost and time, in addition to the carry-over setup. Then a two-stage heuristic is suggested to solve the model. Almada-Lobo and James [13] addressed a problem were commercial solvers fail to solve Large-sized NP-Hard problems. The multi-product capacitated lot sizing and scheduling problem with single machine configuration and sequence-dependent setup times and costs was considered to be solved using a tabu search and a variable neighbourhood search meta-heuristic, then a set of computational experiments were performed to show the effectiveness of the approach.

B. CLSP with Parallel Machines Configuration Recently Xiao et al. [14] examined the parallel-machine configuration with sequence-dependent setup times, time

windows, machine eligibility and preference in addition to the backlogging option. These properties is common in the semiconductor manufacturing industry. The model is solved using the hybrid Lagrangian-simulated annealing-based heuristic algorithm.

Fiorotto & Araujo [15] considered a capacitated lot sizing problem for a single level of manufacturing, multiple products can be produced, Independent setup time that it’s not affected by the sequence of the lot-sizes and a configuration of parallel-machine. Then the model was solved using a Langrangian Heuristics with a reformation of the model as a shortest path problem. The results of the Heuristics were compared with high-performance Mixed Integer Program software.

C. CLSP with Flow Shop Configuration Mohammadi et al. [16] present a capacitated lot-sizing model with a sequence-dependent cost and time in a pure flow

shop configuration. Solving the CLSP is equivalent to solving the multiple dependent TSPs. Since TSP is NP-Hard, CLSP is NP-Hard as well.



Five MIP-based Heuristics were proposed, the first three is considered in the large instances of problems and the rest two consider the flow shop problem. Babaei et al. [17] propose a model that consider a flow shop configuration with sequence-dependent costs, setup carry-over and backlogging where this option has been considered recently in the literature. This model is solved using a genetic algorithm.

Ramezanian & Mehrabad [18] developed a multi-period capacitated lot-sizing in a flow shop configuration. The novelty of the work was in proposing an efficient mathematical model that makes the solution complexity easier. Sequence-dependent setups in terms of time and cost were introduced in the model, in addition to the carry-over option. The model was solved using Heuristic algorithms based on rolling horizon and the performance was evaluated through the trial of variant scale of the problem. Mohammadi & Ghomi [19] consider a capacitated lot-sizing problem in a flow shop configuration with sequence-dependent setups. The contribution was in developing a genetic algorithm-based heuristic that solve this NP-Hard problem. The algorithms combine genetic algorithms with rolling horizon approach. The evaluation of the effectiveness of the method was by comparing the final results of this method by others heuristics. The results were superior in compare with the others.

Mohammadi et al. [20] propose a new algorithmic approach to solve a capacitated lot-sizing problem with sequence-dependent setups in a flow shop configuration. The approach was held using two solution algorithms, the rolling horizon approach and a specific heuristics.

D. CLSP with Distinctive Properties Almada-Lobo et al. [21] address a problem that occur in the glass container industry, the model that was developed

consider a multiple machine configuration with a sequence-dependent cost and time in addition to a carry-over setup. The raw materials that are used in the manufacturing process are the same, in other words, all products share the same resource. Since CLSP is considered NP-Hard, the solution method that are used is a Lagrangian decomposition based heuristic. Nascimento et al. [22] considered a problem where multi plants produce multi products with the possibility of transfers between the plants, in other words, plant A can produce and send to plant B. capacitated lot-sizing and scheduling was considered in the plants however with sequence independent setups. Greedy Randomized Adaptive Search Procedure heuristic as well as a path-relinking intensification procedure were developed to solve this NP-Hard problem. In addition this heuristic was applied on parallel machine configuration. The results were confirmed by statistical tests.

Zhang [23] studied a capacitated lot sizing problem with an option of outsourcing, production capacity was considered as constant on the other hand the outsourcing capacity was considered uncapacitated. A dynamic programming-based algorithm was developed to solve the model while considering a polynomial time. Almedera et al [24] recognized an import dimension that classical capacitated lot-sizing problems don’t consider which is the Lead time. The multi-level CLSP models usually don’t mimic the correct resource requirements or the precedence relations, therefore two models were presented one that consider batch production with lead times and the other consider the allowance of lot-streaming. Results were compared with current models and cost savings of 30–40 percent were observed.

Tempelmeier & Hilger [25] considered a situation where lot-sizing is required but with a present of a stochastic dynamic demand. Multi-items with constant capacity and backorders are considered, the model was developed using linear programming, and then it was solved using a fix-and-optimize heuristics. Kantas et al. [26] formulated a model for capacitated lot-sizing that takes into consideration CO2 emissions and water excessive. The model minimize the cost of producing ethanol production while taking into consideration the different sources of biomass that are considered as the raw materials while considering the different impact of each source on the CO2 emissions level.

Lu et al. [27] integrated the classical capacitated lot-sizing model with preventive maintenance consideration. Many plants requires the system reliability to be above a certain value where this can impact the production in many ways, the reliability constraints were linearized to have a mixed-integer linear program. Three stages heuristics were developed to solve to model that includes Lagrangian-based heuristic to solve the CLSP. A numerical experiments were performed to support the efficiency of the heuristics. Xiao et al. [28] examined a capacitated lot-sizing problem with sequence-dependent setup times in a parallel machine configuration. The contribution was in adding the machine eligibility and preference constraints. These constraints frequently happens in the semiconductor manufacturing industry. A mixed integer program was developed and solved by fix-and-optimize algorithms to get the solutions.

IV. MODEL The following MILP model was developed to determine the optimal solution in terms of: supplier selection for raw

materials, selection, quantity, sequence and inventory level of each grade in each plant, and warehouse selection to satisfy the customer demands. The assumptions, Indices, Parameters, Decision Variables, Objective Function and Constraints are presented below.



A. Assumptions

Many assumptions were done while formulating the model, the following points illustrate these assumptions.

• Each supplier has a limited capacity.

• Production is made in several affiliates.

• Each affiliate has a limited capacity.

• Each affiliate may produce all the grades.

• Each grade has different profit per unit.

• Each grade has different demand.

• All grades use the same type of raw material.

• All grades has the same usage of raw material.

• Demand must be satisfied unless unfeasible.

• Inventory is allowed in the affiliates.

• Inventory is not allowed in the warehouses.

• All grades need to be transported to the warehouse then to the customer.

• The production can start from any grade each time unit.

• No carry-over setup is allowed.

B. Indices

C. Parameters

= supplier number, g = 1, ... , = affiliate number, = 1, ... , = grade number, = 1, ... , = warehouse number, = 1, ... , = customer number,

g Gi i Ij j Je e Ek = 1, ... , = month number, = 1, ... ,

k Kt t T

= number of customers = number of affiliates = number of suppliers = number of warehouses

= supply capacity at supplier

= production capacity at affiliate = Cost of producing grade

g

i

ijt

mnltS g

A iC j at affiliate at month

= Cost of nonprime grade that is produced at month ijht

i t

N t



D. Decision Variables

E. Objective Function Minimize the total cost that is generated from producing the prime and non-prime grades while taking into consideration

the holding cost in each affiliate for each grade, transportation costs from the suppliers to the affiliates, from the affiliates to warehouses and from the warehouses to the customers.

The above objective function is optimized subject to the following constraints.

F. Constraints 1) Supplier Capacity Constraint

in transition between grades and at affiliate = amount of nonprime grade that is produced at month

in transition between grades and at affiliate = holding cost of grade at affi

ijht

ijt

j h iO t

j h iH j liate at month

= demand of grade for customer at month

1 = cost of shipping one unit from supplier to affiliate at month

2 = cost of shipping one unit of grade from affili

kjt

git

ijet

i t

D j k t

C g i t

C j ate to warehouse at month

3 = cost of shipping one unit of grade from warehouse to customer at month

M = a large numberejkt

i e t

C j k t

= amount of grade produced in affiliate at month

= inventory level of grade stored in affiliate at month

1, if grade is produced in affiliate at month = 0 other wise

ijt

ijt

ijt

i

x j i t

I j i t

j i ty

f

⎧⎨⎩1, if higher grade than grade is produced in affiliate at month =

0 other wise1, if transition is made between grades and in affiliate at month =

0 other wise = amount

jt

ijht

git

j i t

j h i tz

sa

⎧⎨⎩⎧⎨⎩

of raw material shipped from supplier to affiliate at month

= amount of grade shipped from affiliate to warehouse at month

= amount of grade shipped from warehouse to cusijet

ejkt

g i t

aw j i e t

wc j e tomer at month k t

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

* + * * + *

+ 1 * + 2 * + 3 *

I J T I J H T I J T

ijt ijt ijht ijht ijht ijt ijti j t i j h t i j t

G I T I J E T E J K T

git git ijet ijet ejkt ejktg i t i j e t e j k t

Min C x N o z H I

C sa C aw C wc

= = = = = = = = = =

= = = = = = = = = = =

= ∑∑∑ ∑∑∑∑ ∑∑∑

∑∑∑ ∑∑∑∑ ∑∑∑∑

1

, g 1,..., = 1,...,I

git gi

sa S G t T=

≤ =∑



The Total amount purchased from each supplier g at time t cannot exceed its availability.

2) Affiliate Capacity Constraint

The total amount produced per time unit of prime and non-prime grades in each plant cannot exceed the plant’s capacity. 3) Customer Demand Constraint

The total amount transported from each warehouse should be equal to the demand for each customer. 4) Warehouse Inventory Constraint

The amount of grades that are transported from the affiliates should be transported to the customers without keeping any inventory at the warehouse.

5) Transportation Constraints

The amount of grades that are transported from the affiliates should be transported to the customers without keeping any inventory at the warehouse.

The production amount for each grade in addition to the previous period inventory amount should equal to the transported amount to the warehouses and the inventory amount of the current period.

6) Sequenceing Constraints Transitions can be made from grade j to a higher grade j in each plant i.

To ensure Yij = 1, if grade j is produced in affiliate i at time unit t.

1 1 1

* , 1,..., = 1,...,J J H

ijt ijht ijht ij j h

x o z A i I t T= = =

+ ≤ =∑ ∑∑

1

, 1,..., 1,..., = 1,...,E

ejkt jkte

wc D j J k K t T=

= = =∑

1 1

- 0, 1,..., 1,..., = 1,...,I K

ijet ejkti k

aw wc j J e E t T= =

= = =∑ ∑

1 1

, 1,..., = 1,...,G J

git ijtg j

sa x i I t T= =

= =∑ ∑

11

+ , 1,..., 1,..., = 1,...,E

ijt ijt ijet ijte

x I aw I i I j J t T−=

+ = = =∑

,

,

1,..., 1,..., = 1,...,

1,..., 1,..., = 1,...,ijt ijt

ijt ijt

x My i I j J t T

y x i I j J t T

≤ = =

≤ = =



To ensure fij = 1 if higher grade than grade j is produced in affiliate i at time unit t.

To insure only one of the higher grades h is chosen as the immediate successor of grade j.

V. CONCLUSION

In this work, a capacitated lot-sizing with sequence-dependent costs mathematical model was developed to determine the optimal solution of selecting suppliers for raw materials procuring and production planning and warehouse selection to satisfy the customer demand. This model was applied in the field of petrochemical production plants while considering the nature of switch-overs in that industry. It was mentioned in the literature review that the CLSP is hard-NP, this gives a motivation to develop a heuristics to solve a large-scale problems where the commercial software become inefficient in terms of competing time when the scale of the problem increase.

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1

1

, 1,..., 1,..., = 1,...,

, 1,..., 1,..., = 1,...,

J

iht ijth j

J

iht ijth j

y Mf i I j J t T

y f i I j J t T

≥ +

≥ +

≤ = =

≥ = =

∑

∑

0.5*( ),

1

1

1,..., 1,..., 1,..., = 1,...,

1, 1,..., 1,..., = 1,...,

1, 1,..., 1,..., = 1,...,

ijt ihtijht y y

J

ijht ijt ijth j

J

ijhth j

z i I j J h j J t T

z y f i I j J t T

z i I h j J t T

≤ +

≥ +

≥ +

= = = +

≥ + − = =

≤ = = +

∑

∑



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Capacitated Lot-Sizing and Scheduling with Sequence ...ieomsociety.org/ieom_2016/pdfs/532.pdf · Capacitated Lot-Sizing and Scheduling with Sequence-Dependent setups in Petrochemical