MIXED INTEGER PROGRAMMING (MIP) MODEL FOR PRODUCTION
SCHEDULING IN AN AUTOMOTIVE PART COMPANY
AISYAH NABILAH BINTI KAMARUDDIN
A project report submitted in partial fulfillment of the
requirements for the award of the degree of
Master of Engineering (Industrial Engineering)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
DECEMBER 2010
iii
Thanks to
My beloved parents
Kamaruddin bin Hashim and Roslina binti Nik Mohd. Amin
My supportive fiancée
Mohd.Khairul Rijal bin Ab.Rahman
iv
ACKNOWLEDGEMENT
First and foremost, I would like to express a lot of thank to Allah S.W.T for His
Blessing and giving me the strength to complete this project. I manage to complete
my Master Project without facing any major problems.
My sincere and heartfelt thanks to my project supervisor, Dr.Wong Kuan Yew, for
enlightening supervision and countless hours spent in sharing his insightful
understanding, profound knowledge and valuable experiences in order to make sure
my project is done successfully. As a supervisor, he has been a source of motivation
towards the completion of this project.
Thousands gratitude credited to Mr.Mohamad Ikbar bin Abdul Wahab for helping to
make this project success. I would also like to express a special thank to my family
members for their priceless support, encouragement, constant love, valuable advices,
and their understanding of me. Last but not least, thanks to all my friends, and to all
people who have involved directly and indirectly in making my research project
become flourishing and successful.
v
ABSTRACT
Scheduling is defined as the allocation of resources to task over time in such a
way that a predefined performance measure is optimized. Scheduling is a decision
making function that plays an important role in manufacturing industries. In this
research, a model of Mixed Integer Programming is proposed based on the
scheduling problems in an automotive part production company. This mathematical
method provides a theoretical framework for the formulation and solution of the
models which involve the translation of the real problem into a model. This model
targets to optimize the scheduling production in order to increase productivity. All
standard constraints encountered in the production scheduling are included in the
model (machine capacity, material balances, manpower restrictions and inventory
limitations). Machine setup time changeover and cost changeover for the different
product will influence the arrangement of scheduling process. The objective function
that is minimized considers all major sources of variable cost that depend on the
production schedule, such as changeover cost between products, inventory cost and
labor cost. The result from this study are presented and discussed based on the
optimal production schedule.
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ABSTRAK
Penjadualan ditakrifkan sebagai peruntukan sumber daya untuk tugas dari
masa ke masa sedemikian rupa sehingga saiz prestasi yang telah ditetapkan
dioptimumkan. Penjadualan adalah proses membuat keputusan yang memainkan
peranan penting dalam industri perkilangan. Dalam kajian ini, model „Mixed Integer
Programming‟ dibentuk berdasarkan kepada masalah penjadualan yang timbul di
bahagian pengeluaran automotif di syarikat. Kaedah matematik memberikan
kerangka teori untuk formulasi dan penyelesaian model yang diterjemahkan dari
masalah
nyata
mengoptimumkan
ke
dalam
bentuk
penjadualan
model.
Model
ini
mensasarkan
pengeluaran
dengan
tujuan
untuk
meningkatkan
produktiviti. Semua rintangan yang wujud dalam penjadualan pengeluaran akan
diserapkan ke dalam model (mesin kapasiti, keseimbangan bahan mentah, sekatan
tenaga kerja dan keterbatasan inventori). Pertukaran masa persediaan mesin dan
pertukaran kos untuk produk yang berbeza akan mempengaruhi tatacara proses
penjadualan. Fungsi objektif yang diminimalkan mempertimbangkan semua sumber
utama dari kos pembolehubah yang bergantung pada jadual pengeluaran, seperti kos
pertukaran di antara produk, kos persediaan dan kos tenaga kerja. Hasil dari kajian ini
disajikan dan dibahas mengikut penjadualan pengeluaran yang optimum.
vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
INTRODUCTION
1
1.1
Background of the Study
1
1.2
Problem Statement
2
1.3
Objectives
2
1.4
Scopes of the Study
3
1.5
Significance of the Study
3
1.6
Outline of the Report
4
1.7
Conclusion
7
SCHEDULING
8
2.1
Production Plannning
8
2.2
Scheduling
9
2.2.1
2.3
Flowshop Scheduling
Review of Previous Work in Scheduling
11
12
viii
2.4
Issues in Scheduling
16
2.5
Mixed Integer Programming
17
2.5.1
18
2.6
3
4
Conclusion
19
RESEARCH METHODOLOGY
20
3.1
Introduction
20
3.2
Research Methodology
20
3.3
Company I
23
3.3.1
24
Data Collection
3.4
AIMMS Modeling System Software
27
3.5
Conclusion
29
FORMULATION MODEL
30
4.1
Introduction
30
4.2
Problem Structure
30
4.3
Problem Definition
31
4.4
Model Formultion
34
4.4.1
Parameters
34
4.4.2
Decision Variables
35
4.4.3
Objective Function
37
4.4.4
Constraints
37
4.5
5
Mixed Integer Programming Algorithm
Conclusion
41
RESULTS AND DISCUSSION
42
5.1
Introduction
42
5.2
New Proposed Production Schedule
42
5.3
Conclusion
51
ix
6
CONCLUSION
52
6.1
Introduction
52
6.2
Conclusion
52
6.3
Limitations of Study
53
6.4
Future Research
54
REFERENCES
55
APPENDIX
60
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
1.1
Structure if the report
6
31
Product demand for a month period
24
3.2
Cycle time of each product in the production process
25
3.3
Changeover time from one product to another product
25
3.4
Changeover cost from one product to another product
26
3.5
Labor cost for a month period
26
4.1
Investigation of correctness of set (7)-(10)
42
5.1
The calculated production schedule
43
5.2
The flow of the whole process from one machine to another
46
machine with the quantity of products to be made
5.3
The flow of the whole process from one machine to another
47
machine with the quantity of products to be made
5.4
Daily production time including setup times
49
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
The overall methodology of the study
22
3.2
The process flow of the products in the company
23
4.1
Nomenclature contains indices, parameter and decision
36
variables used in the model
5.1
Percentage of total production days needed for each
product
48
CHAPTER 1
INTRODUCTION
1.1
Background of the Study
Scheduling can be broadly defined as the allocation of resources to tasks over
time in such a way that a predefined performance measure is optimized. From the
view point of production scheduling, the resources and tasks are commonly referred
to as machines and job and the commonly used performance measure is the
completion time of jobs [1]. In manufacturing, there are many types of scheduling
such as job shop, flow shop, sequencing, parallel and etc. Problems arise in
scheduling when the process of the production has many products to be produced.
Therefore, scheduling can be said as a strategy to determine what product will be
produced in a certain day based the certain limitations that need to be considered.
In order to have a good scheduling planning, a technique can be applied in
this problem is Mixed Integer Programming (MIP). Mixed integer programming is
known as a guiding quantitative decision in business, industrial engineering and to
the social and physical sciences. In World War II, this technique used to deal with
transportation, scheduling and allocations of resources under constraints like cost and
priority gave subject an impetus that carried it into the postwar era. This means that,
this technique can be used in order to help in solving scheduling problem. In this
study, MIP is used to model the real life situations in scheduling of automotive parts
production process.
2
1.2
Problem Statement
Scheduling is a complicated problem where there is a variety of products
needs to be scheduled in a certain period of time. In one day, production only can
produce a certain amount of product. This issue is further complicated where a
decision needs to be made in order to schedule the production process for all of the
products. This decision must take into consideration about the limitations that may
present during the scheduling process. Limitations that present in the scheduling
problem are machine capability, labor hour, changeover time for transition between
two products, change over cost for transition between two products and cycle time
for each product. Each type of product may have a different cycle time process where
it will restrict the amount of production due to the machine capacity. Besides, in one
day, there is possibility to have more than one product to be manufactured which
involve the changeover time and cost between those products. Therefore, a good
scheduling planning is needed to overcome all the restriction so that an optimal
production can be achieved.
1.3
Objective
In order to complete this project within the time frame and the scope given, several
objectives for this research have been identified and listed such as below:
a) To formulate a model based on Mixed Integer Programming (MIP)
method which can solve scheduling problem in an automotive part
company.
3
b) To determine the optimum schedule for the automotive part
production process.
1.4
Scope of the Study
In this research, the data used were taken from a company named company I
which is located at Pasir Gudang, Johor. This study focused only on the production
department where automotive component part is produced. The scope is narrow
down by only selecting a single production line as the subject in this study which is
capable to produce 8 different types of product. A complete data for a month
duration will be used in the study.
1.5
Significance of the Study
The significance of this study is to create a Mixed Integer Programming
model based on the company scheduling problem. This model will be used to solve
the scheduling problem in order to help the company to have a better planning in the
scheduling production process. Through this model, an effective scheduling planning
for the production process in a month period can be created as a guide to the
company so that an optimal production can be obtained.
4
1.6
Outline of the Report
The structure of the report can be summarized as shown in Table 1.1. The description
of each chapter in the report is as follow:
1) Chapter 1: Introduction
This chapter gives a broad idea about the whole study. The main topics
included in this chapter are background of scheduling and mixed integer
programming (MIP) technique, problem statement, objective, scope,
significance of study and the outline of the project report.
2) Chapter 2: Scheduling
This chapter summarized all the theories, information and previous
research that related to the scheduling problem and mixed integer
programming technique. Discussion based on the finding from journals,
books, internet, articles and etc are done in this chapter. Topics included
in this chapter are production planning, scheduling, mixed integer
programming (MIP) and previous research related to scheduling problem.
3) Chapter 3: Methodology
This chapter describes research methods used in this study. It gives
information on how the study are done and conducted through the whole
period in completing the study. Data will be collected from the
company‟s information during the case study. Once data have been
completed, an analysis of the data will be done by using AIMMS software
in order to produce optimal production schedule.
5
4) Chapter 4: Formulation Model
A mixed integer programming model formulation will be created and
discuss in this chapter. Information about the model is explained in details
here.
5) Chapter 5: Optimal Production Schedule
This chapter presents all the results of this study which was solved by
using AIMMS software. The solution of the problem will be proposed
once it satisfied all the restrictions in production process.
6) Chapter 6: Conclusion
This chapter summarizes the report. Recommendations for future works
will be suggested in this chapter also.
6
Table 1.1: Structure of the report
CHAPTER
CONTENTS
Chapter 1
This chapter examines the overall perspective for the
Introduction
research, such as:-
Chapter 2
Scheduling
-
Background of the study
-
Problem statement of the study
-
Objectives of the study
-
Scope of the study
-
Significant of the study
-
Outlines of the report
This chapter focuses on the following topics:
-
Production planning
-
Scheduling
-
Mixed integer programming
-
Previous research
Chapter 3
This chapter introduces the research methodology
Research
structure
Methodology
and research process which consists of:-
Research methodology
-
Data collected
Chapter 4
This chapter discuss about the data used in the process of
Formulation Model
created a MIP formulation model based on the company‟s
scheduling problem which consists of:
-
Problem structure
-
Problem definition
-
Model formulation
7
Chapter 5
Results
This chapter discuss about the data analyses, findings and
and the proposed solution. Topics included are:
Discussion
-
Result
-
Findings
-
Discussion
Chapter 6
This chapter is the last chapter of the thesis which is the
Conclusion
concluding and recommendation part.
1.7
-
Conclusion of the study
-
Recommendation for future works
Conclusion
As the conclusion, scheduling problem is a big issue to the manufacturer. A
good manufacturer will identify the problem of the company‟s scheduling planning
so that the problem can be solved by using certain suitable method. This study was
done in order to help in improving the company scheduling planning by using
application of the mathematical formulation. In the next chapter, further discussion
about the problem in scheduling and Mixed Integer Programming technique will be
discussed for more understanding on this research.
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CHAPTER 2
SCHEDULING
2.1
Production Planning
Production planning is a complex process that covers a wide range of
activities that insures the material, capacity and human resources are available when
needed. Production plan sets the overall level of output in broad terms in a
manufacturing industry. The plans includes the product families, financial plans,
customer demand, engineering capabilities, labor availability, inventory fluctuations,
supplier performance, and other considerations. Production planning is setting up in
the beginning of the production process where all the process flow, how it is done,
how much it cost, how much inventory it needs, how many labors its required, when
the production process should start and end, and etc [2]. This detail plans will guide
the whole production process from the start to the end.
Production planning is created based on analysis in a few factors which will
influence the production process. In developing production plans, production
planning play part to forecast of the future demand for both new and existing
products. The demand forecasting then will be translated into a production plan that
optimizes the company‟s objectives. In capacity planning section, this planning wills
analyzes how much of a product it must be able to produce with its capability in all
aspects from the starting process until the end of the process. Furthermore, in
location planning, it involves in analyzing facility sites in terms of proximity for raw
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materials, markets, availability of labor, and energy and transportation cost. For the
layout planning, the plan includes the designing of the facility so that customer‟s
needs are supplied for high-contract services and to enhance production efficiency.
In quality planning, which is the most important part, systems are developed to
ensure that products meet its company‟s quality product and also meet the customer
requirements [3].
There are several benefits of implementation the production planning in
company which are better response to customer orders as the result of improved
adherence to schedules, faster response to market changes, improved utilization of
facilities and labor, and reduced inventory levels [4]. Effective production planning
is essential in manufacturing industries and is the foundation on which the
manufacturing organization operates. A good process of generating a production
planning can provide a better view on the company‟s mission and will lead to the
way of success.
2.2
Scheduling
Scheduling can be broadly defined as the allocation of resources to task over
time in such a way that a predefined performance measure is optimized [5].
Scheduling is a decision making function that plays an important role in
manufacturing industries. Scheduling is applied in procurement and production, in
transportation and distribution, and in information processing and communication. In
manufacturing, scheduling function as to coordinated the flow of parts and products
through the system, and balances the workload on machines and personnel,
departments, and the entire plant. Scheduling is important in manufacturing
industries because a smooth and efficient schedule of production can help to optimal
the production of a high quality product and cut the production cost. Besides,
scheduling can help to produce product at the exact time required by the customer
demand [3, 6].
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Scheduling in a production process is a set of activities that are subject to
precedence constraints, where jobs that have to be completed before a given job is
allowed to start its processing are specified in details with its requirement. All
activities belong to a single project that has to be completed in a minimum time with
a specific amount of cost as the financial support. In manufacturing industries,
scheduling function typically uses mathematical optimization techniques to allocate
limited resources to the processing tasks [7]. It helps in the process of obtaining a
satisfactory or optimal schedule. Mathematical model can be describes number of
important characteristics. The first characteristic specifies the number of machines or
resources as well as their interrelationship with regard to the configuration which are
machine set up in series or in parallel. Study by Omar and Teo [8] can be relates with
this characteristic since their research considers a mathematical model, which is
Mixed Integer Programming model, to solve the problems of scheduling on a sets of
identical parallel machines, with distinct due dates, process time and early due date
restrictions.
The second characteristic is about a mathematical model that concerns of the
processing requirements and constraints. At this characteristic, it includes the setup
costs, setup times of the machine, and precedence constraints between various
activities. This characteristic is the main focus in the scheduling problems which
build the mathematical model to be solved. Various studies have use this
characteristic to build the mathematical model such as study by Blomer and Gunther
[9]. The last characteristic the last characteristic is in optimizes the objective which
may be a single objective or a composite of different objectives such as to minimize
or to maximize the production of product. Study by some researchers show the use of
this characteristic where the studies is about to minimize the total cost of makespan
and resource consumption function that is composed of release time reduction and
processing time reduction [10,11,12,13,14,15,16].
11
Scheduling have a significant on economic impact since in certain industries,
the viability of the company may depend on the effectiveness of its scheduling
systems. An effective scheduling system often allows an organization to conduct its
operation with minimum resources. The benefits of production scheduling are
process change-over reduction, inventory reduction, reduced scheduling effort,
increased production efficiency, labor load leveling, accurate delivery date quotes
and real time information.
2.2.1
Flowshop Scheduling
Flow shop scheduling is where a set of jobs are required to complete a
process to produce a product. There is more than one machine and each job must be
processed on each of the machines in order to complete process of production. The
number of operations for each job is equal with the number of machines. In this type
of scheduling, it fits with the type of product that has a set of flow process in order to
become a complete product. Commonly, this kind of scheduling is being applied to
products which are complicated to produce. In order to make the job easier, the
processes are separated by different process which will be done by different machine.
All the machines involve in the production have its own work and will do a specific
job only [17]. This is call as a line production. The goal of the flow shop scheduling
is deployment of an optimal schedule for N jobs on M machines.
Flow shop can raise issue in scheduling since it deals with a set of jobs where
for each job in a certain machine, the process will have cycle times which are
different with other process at other machine [18]. Total of the cycle time for each
product is known as the completion time processing. If the line production have
capability to produce many different types of product, some issues may present in the
scheduling. According to works before, the problem arises when to schedule the
different product which has different time setup. To schedule a flow shop production,
12
some limitations need to be considered such as product cycle time, machine
capability and labors [19,20]. A good planning in scheduling may help to optimal the
production and minimize the production cost.
2.3
Review of Previous Work in Scheduling
Various studies to solve scheduling problems in industries have been done by
many researchers throughout the whole world. These studies use various method and
technique in solving the problems. One of the techniques that have been used is
Mixed Integer Programming (MIP). This method is used to solve certain situation in
scheduling problem since it help industries in ways of cutting cost, optimality of the
production and build a systematic scheduling planning. Revision throughout the
previous study may help for a better understanding on how the previous researches
make use of the techniques and apply it into the scheduling problems.
Work by Doganis and Sarimveis [10] proposed a model based on Mixed
Integer Programming (MIP) where it is use in food industry for the yogurt
production. The purpose of this study is to minimize the costs that depend on the
production schedule and to optimally schedule the yogurt production operations for
18 different products on a single line production. In this problem, not every product
can be produce at the same time due to the production demand, due dates of orders,
sequencing of operations and available machine time and man hours. Therefore,
Mixed Integer Programming model are build based on the industry problem where
several aspects are taken into consideration which are the daily demand of each
product, the starting inventory, setup costs and times for transition between product,
the inventory holding cost, the production speed of each product, the labor cost for
three working shifts and the sequencing limitations. Objective of the model is to
decide and to calculate the product to be manufactured in each day and their
respective quantities, the machine time utilized by each product and the inventory
13
quantities of each product at the end of each day. Then, MIP optimization problem
was solved by using CPLEX algorithm which result to a complete production
schedule for the sequence of products that should be produced every day and the
respective quantities and the inventory levels at the end of each day.
Study by Harjunkoski et al, show how they apply Mixed Integer
Programming (MIP) in the paper industry. The application of MIP is use to minimize
cost for waste and the cutting time by producing a set of product paper reels from
larger raw paper reels. In solving a trim-loss problem, the variables are the lengths of
the product paper rolls which are its widths and lengths of raw paper rolls along with
the cutting patterns. The problem rises in this study is where some of the coefficients
and constraints are in the form equations of bilinear inequalities resulting to a nonconvex problem. Two-step optimization procedures are use in this optimization
problem in order to overcome the non-linearity which includes the MIP techniques.
First, generate all feasible cutting patterns which satisfying some of the constraints.
After the constraints are in linear problem, then second step is done by solving the
MIP problem with CPLEX program package. This optimization procedure has
success in saving time in cutting process approximately 10% and by minimizing the
production costs, the company has reduced the trim loss below 2% on average [11].
Giannelos and Georgiadis study focus on develop a new model based on
MILP techniques for scheduling the process of fast consumer goods manufacturing.
The new model is build to determine the equipment location, processing time and
rate, and the amounts of material being process in each task in order to optimize the
economic objective function. The study contribute a new simple formulation to solve
continues process scheduling problem in future [21]. In another study by Nystrom et
al, it also focuses on the optimization of continuously operated processes and
scheduling problem. The problems are dividing into two sections which are master
problem and primal problem. Master problem is using MIP techniques which solve
the production optimization problem including sequencing, allocation and durations
14
based on customer orders. Primal problem is using dynamic optimization (DO)
which solves part of production optimization problem in how to operate a process
[12].
Another study by Voudoris and Grossmann has proposed a method for the
integrated scheduling for a special class of multipurpose batch processes to maximize
the profitability of the process. The plant being studied is produce four types of
product which go through different processing stages and the manufacturing of the
products characterized through production routes. Since the processing stages are
different for each product, therefore, it will raise a problem in scheduling the
production process. The scheduling problems are separated in two phases where in
the first phase it is required that a given number of batches for every product have to
be produced in the minimum amount of time and in the second phase, a model of an
optimal design of a process considering the production schedule are created. This
study use three different approaches in solving the problem since it involve a
multipurpose plant. The main different between the approaches is the way with
which the scheduling sub problem is dealt with. A MIP model is build with
characteristic of fitting all the structural possibilities and final product inventories.
As the result, the MIP model can solve the problem by eliminating all the constraint
which can lead to scheduling delay [22].
Another study in food industry using application of the MIP method in
solving the scheduling problem is done by Simpson and Abakarov (2009). A canned
food plants are using batch processing as the mode of operations. The objective of
this research was to solve the problem of optimizing scheduling for the case where
given amounts of different canned food products, with specific requirements, would
be sterilized within a minimum plant operation time. In this case, there are 16
different products need to be sterilize which different types of product need a
different requirement of temperature and time in sterilization process. Canned food
plants with batch processing operations are unique in process line where the whole
process line operates continuously. Therefore, a good scheduling is needed to help in
maximizing the production and in the meanwhile, fit the capacity of autoclaves and
15
plant operation time. MIP techniques is chosen to model the problems where the
variables are the number of sterilization products, amount of each products, number
of sterilization vectors, set of possible sterilization vectors, set of sterilization time
and capacity of autoclaves. The MIP problem then solved by using two software
tools which are “lp_solve” version 5.5.0.10 and Online Web Service of COIN-OR
(Computational Infrastructure for Operations Research) : NEOS Server Version 5.0.
Result obtain from this study have shown that implementation of MIP techniques in
scheduling have been success in reducing the plant operation time approximately by
25% from 332.5 minutes to 249.05 minutes [13].
In year 1996, Lee and et al conduct a study of refinery short-term scheduling
of crude oil unloading with inventory management. This study consider a short-term
scheduling for crude oil inventory management problem which involves crude oil
unloading to storage tanks from crude vessels, crude oil transfer from storage tanks
to charging schedule for each crude distillation unit (CDU). A long-term refinery
planning is use as a basic to determine the crude supply plan and production rate.
Therefore, this study is done in order to minimizing cost involved in operation to
meet the resource supply and production planning. Operation cost in the crude
inventory management problem includes inventory cost, crude vessel harboring and
sea-waiting cost, and changeover cost for charged oil change in each CDU. Scope of
this study is to develop a rigorous MIP optimization model for short-term crude oil
unloading, tank inventory management, and CDU charging schedule. This study use
assumptions for modeling the problem where only one vessel docking station are use,
changeover times are neglected, the mixing time between oil feeding and charging
to charging tank is ignored and only specifics key components in crude or mixed oil
decide the property of each oil. Then the MIP model for the scheduling problem
proposed in this study are consist of minimizing the operating cost which subject to
vessel arrival and departure operation rules, material balance equations for the vessel,
material balance equations for the storage tank, material balance equations for
charging tank, material balance equations for component in the charging tank and
operating rules for crude oil charging. Then the MIP model formulation was solved
by using Linear Programming based branch and bound method [14].
16
2.4
Issues in Scheduling
In scheduling, there are many problems that contribute to the problem. If
there are multiple products that need to be produced, and at the same time, the
number of machines is limited, then the production needs to be schedule in order to
fulfill the demand for all the products. The problem will be easier if all the products
have the same flow of processes. Discussion about scheduling various products in a
single line production has been discussed in the previous research [10]. A schedule
planning need to be done in order to decide which product will be made on a certain
period of time so.
Another scheduling problem needs to be considered when involving in
production of multiple items is the sequence-dependent setup costs and setup times.
In this problem, decision maker must decide which items to be produce in which
periods, and must specify the exact production sequence and production quantities to
satisfy deterministic dynamic demand over multiple periods that span planning
horizon, in order to minimize the sum of setup and inventory holding costs
[23,24,25]. Therefore, in this case, several items can be produces in each time period
on the same machine, sequentially one after the other. Since the setup times and cost
are sequence dependent in the scheduling, it can affect the feasibility and cost.
Another restriction in scheduling may due to the cycle time and capacity of
the machine. Each different product has a different cycle time and will take a
different period of time in production to produce the end product. This problem is
further complicated by the machine capacity which only can work in a certain period
of time in a day [26,27]. Normally, any industry will have production operations not
more than 24 hours. Therefore, the quantities of product that can be produce in a day
are restricted by the machine capacity. A scheduling planning may help to arrange
the time scheduling so that in one day, the manufacturer can achieve optimal
production besides reducing the time wasted and cost of production.
17
These issues cannot be solving manually since it can be more complicated. In
fact, some of these issues cannot be solve since it may involve the mathematical
structure [28]. By previous research, there are proven that mixed integer
programming has been used widely in solving the scheduling problems in various
industries. Therefore, application of this technique should be further discuss in
solving the scheduling problem to help improve the scheduling system in industry.
2.5
Mixed Integer Programming
In year 1939, an introduction of the field in this techniques are introduced by
Leonid Kantorovich, George B.Dantzig and John von Neumann. Mixed integer
programming (MIP) is a mathematical modeling technique useful for guiding
quantitative decisions in business, industrial engineering, and to the extent of social
and physical sciences. Mixed integer programming is a powerful tool for planning
and control problems because of its modeling capability and the availability of good
solvers [29]. MIP problems involve the optimization of a linear objective function,
subject to linear equality and inequality constraints [30]. MIP deals with both
continuous and integer variables. It is well known that in the MIP problem, it is
necessary to restrict the decision variables of MIP models to be integers or binary
values [31].
MIP problems are much harder to be solving rather than pure linear
programming problems because the feasible region of MIP problems does not allow
the applications of the well known Simplex Algorithm. Even though it has lack in not
capable to use that algorithm, but MIP is a powerful technique which can be used in
solving a large data with large amounts of constraints [32 , 32]. One of the most used
of this technique is in solving the scheduling problem. Previous studies have shown
that this technique is successful in any type of scheduling problems
[34,35,36,37,38,39,40]. It shows that MIP can solve problems in scheduling by
18
proposed a better scheduling planning with objective function of optimization the
production.
2.5.1
Mixed Integer Programming Algorithm
1) Definition 1
A mixed integer programming is an optimization program involving continuos and
integer variables, and linear constraints [41]. Any MIP can be written as:
Where the set X is called the set of feasible solutions and is described by m linear
constraints, nonnegativity constraints on the x, y variables, and integrality restrictions
on the y variables. In matrix notation:
Where
Z(X) denotes the optimal objective value when the optimization is
performed over the feasible set X.
x and y denote, respectively, the n-dimensional (column) vector of
nonnegative continuous variables and the p-dimensional (column) vector of
nonnegative integer variables.
and
are the row vectors of objective coefficients.
is the (column) vector of right hand side coefficients of the m
constraints.
A and B are the matrices of constraints with real coefficients of dimensions
(m x n) and m x p), respectively.
19
2) Definition 2
A mixed binary linear program, or mixed 0-1 program, is a MIP (according to
definition 1) in which the integer variables y are further restricted to take binary
values. This means that the feasible set X of binary is defined by
By this definition, it is indicated the dimensions of all vectors for completeness.
2.6
Conclusion
MIP is a powerful technique which is suitable to help in scheduling a variety
of products as proven by the previous work done by the researchers. It can create a
model which can be used to plan the production of the product on what day to be
made and at the same time, help to reduce cost.
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
In this chapter, a complete explanation on how this research was done will be
explained. The main part is to define the problems that need to be focus on this study
which is solving the scheduling problem. A variety of techniques can be used to
solve scheduling problem, but the most commonly used is mixed integer
programming. Therefore, MIP is used as the technique in solving the scheduling
problem.
3.2
Research Methodology
The study was started by a visit to the company I. Through a detail
observation in production department, a problem in scheduling the production is
identified as the main problem for a group of product which known as M12. The
production department faces a problem in completing the production schedule as per
plan which leads to late shipment to the customers. A further discussion with the
production engineer about this problem in scheduling was made.
21
The next step is to gather all the information that involve in the production
process of products under M12 group. The data needed are monthly demand for each
product, machine speed for each product, setup time for transition between two
products, the lot size production and working hour in one day period. Information and
data related to production cost are also needed since the objective function of the
model is to minimize the cost. The cost that contributes to the production cost is labor
cost, setup cost between two products and inventory cost. All these data will be used
in solving the problem and further details about this will be discussed in the next
chapter.
The completed data will be used in the process of building the MIP model
formulation. The model will be constructed based on the limitations on the
production process which restricted the production schedule. Further discussion on
this limitations topic will be discussed in the next chapter. Once all the information
needed is completed, a MIP model consists of objective function, a set of constraints
and decision variables is created which fit with the production requirement.
Once the data collection is completed, then AIMMS software will be used to
solve the model. AIMMS software is known as an advanced development
environment for building optimization based operation research applications and
advanced planning systems. By using AIMMS, scheduling problem will be solved by
a new proposed optimal production schedule. The new schedule was constructed
based on the production limitations with the purposed of minimizing the production
cost. A discussion about the new proposed production schedule will be made in
Chapter 5.
22
Start
Define Problem
Define Objective
Company
Scheduling and
Mixed integer
programming
Define problem
in company
Literature Review
Data Collection
Construct MIP model :
1. Decision Variables
2. Objective function
3. Constraints
Solve model by using
AIMMS
Optimal
Stop
YE
sS
Figure 3.1 below show the overall methodology of this study
NO
23
3.3
Company
The study was conducted in Company I which is located in Pasir Gudang,
Johor. The company was established in year 1997 and the headquartered in
Singapore. The company operated through six main departments which are Human
Resource Department, Production Department, Quality Department, Engineering
Department, Visual Department and Planning Department. The main products
produce by this company are precision components for office automation, precision
components for hard disk drives, precision components for automotive, precision
rubber products and plating services. The scope in this study only focuses on the
Production Department and the production of automotive parts.
In automotive production department, there are various types of products
produced by the company. This study only focus on 8 products which have the same
process flows. Figure 3.2 below shows the process flow for the products:
Turning
Air Blow
Washing
Oven
Visual
Outgoing
and
Packaging
Figure 3.2 The process flow of the products in the company
24
3.3.1
Data Collection
All the data will be collected through visit to the company and by
interviewing the Process Engineer who work there. The data collected need to be
complete in order to have a good result in the end. Through the visit to the company,
the data collected are the demand of the products for a month period, cycle time for
each product, changeover cost between the products, changeover time between the
products, and the labor cost. All the data collected will be presented in the tables
below.
Table 3.1: Product demand for a month period.
PRODUCT
DEMAND
P1
4250
P2
1710
P3
3334
P4
3334
P5
1667
P6
200
P7
500
P8
2477
Table 3.1 shows the amount of demand from customer for a month period.
The highest demand is for Product 1, the second highest is the demand for Product 3
and Product 4 and followed by the rest of the product. The lowest demand is for the
Product 6. For all these product, earliness is possible but tardiness is not allow. The
company will distribute the product to the customer once the production process is
completed.
25
Table 3.2 : Cycle time of each product in the production process.
PRODUCT
MACHINE SPEED (IN SECOND)
P1
111
P2
126
P3
115
P4
135
P5
140
P6
147
P7
201
P8
204
Table 3.3: Changeover time from one product to another product (in hour).
FROM \ TO
P1
P2
P3
P4
P5
P6
P7
P8
„
P1
P2
P3
P4
P5
P6
P7
P8
0.350
0.283
0.383
0.390
0.333
0.383
0.350
0.283
0.390
0.416
0.390
0.333
0.390
0.350
0.370
0.400
0.390
0.383
0.350
0.383
0.301
0.390
0.370
0.400
0.283
0.416
0.383
0.400
26
Table 3.4: Changeover cost from one product to another product (in RM)
FROM \ TO
P1
P2
P1
P2
P3
P4
P5
P6
P7
P8
310
365
330
310
330
350
320
340
365
345
310
340
365
330
320
365
335
350
310
320
365
345
340
365
320
340
310
P3
P4
P5
P6
P7
350
P8
Table 3.5 : Labor cost for a month period ( for 1 shift)
POSITION
SALARY ( RM)
Operator
500
Technician
1000
27
As shown in Table 3.2, each product has different cycle time which restricts the total
quantity for each product to be produced in one day. Table 3.3 and Table 3.4 show
the changeover time and changeover cost between products when transition between
products takes place. The lowest changeover time and changeover cost between the
products will be put as the priority in choosing criteria if it is required to have more
than one products manufactured in same day. Table 3.5 shows the labor cost for a
month period for one shift. In production process, one technician and one operator
are needed to operate six machines.
3.4
AIMMS MODELING SYSTEM SOFTWARE
The AIMMS modeling system is especially designed to minimize the
mathematical complexity. The AIMMS modeling language incorporates indexing
facilities to guide in expressing and verifying complex mathematical models. This
study use AIMMS modeling system in order to solve the MIP model formulation
based on the production scheduling problem. A simple step will be explained on how
the MIP model formulation is solved by using this software.
STEP 1 : Declarations of model identifiers.
The first step is to set all the declaration sections that will used to store all the
model identifiers such as, Set Declaration, Parameter Declaration, Variables
Declaration, Constraints Declaration and etc.
STEP 2 : Entering sets and indices.
Next step is to declare each set used in the model under the Set Declaration
section. Then, declare all the indices as an attributes of the sets.
28
STEP 3 : Entering parameters and variables.
Next, declare the parameters and variables that are needed in the model. The
sets and their associated indices will be used to specify the index domain for
the parameters and variables.
STEP 4 : Entering constraints and the mathematical program.
As the step before, all the constraints need to be declared as what the model
required. Then, all the variables, constraints and objective function are
considered as part of the mathematical program.
STEP 5 : Entering set and parameter data.
In this step, all the data required in this study need to be entered under the
data page of each indexed parameter which is automatically filled with the
elements of corresponding sets.
STEP 6 : Computing the solution.
Once all the identifiers, their attributes and their data are completely entered
in the program, the next step is to build the procedures in order to be able to
instruct AIMMS to take action to solve the model.
29
3.5
Conclusion
This chapter explained about the methodology of the study. In order to solve
the model formulation of scheduling problem, data were collected in the production
department. All the data collected were presented in this chapter. The data collected
will be used along with the model formulation to produce an optimal production
schedule in chapter 5.
CHAPTER 4
FORMULATION OF MODEL
4.1
Introduction
In this chapter, a detail discussion is provided on how the production
conditions of Company I was transformed into the model formulation. All limitations
exist in the production process are taken into consideration while creating a suitable
MIP formulation model. The model consists of an objective function, a set of
inequalities of constraints and decision variables that represent all the limitations in
the production process in order to get an optimal production schedule.
4.2
Problem Structure
The purpose of this study is to optimally schedule the automotive parts production
which operated in a single production line. In this production, there are 8 different
products that were categorized under a same group called as M12. Products under
M12 group have similarity in the production process and have only slightly
difference in the aspects of changeover time and cost between each product. For all
these products, their production process has six process flows which start from
turning process, air blow process, washing process, oven process, visual inspection
31
and packaging. Each process needs a different cycle time to finish up its process. In
this production process, the turning process takes the longest cycle time compare to
other processes. Therefore, the turning process restricted the amount of product that
can be made for one day.
This study will produce an optimally schedule over a month horizon, with a
maximum of two 11-h shifts daily. Daily production time is operated for 22 hours
where the remaining of 2 hours is used for the machinery clearance. Each product has
different demand from customers with due dates within a month period. Production
must always be capable to produce the quantity requested by the demand and abide
with the due dates. Even though the due dates are at the end of each month, but the
company have policy to deliver the products that have been made on the same day of
the production of the products to the customer.
In production, no tardiness is allowed since it can disturb the scheduling
planning which can lead to additional cost. Earliness is possible but always restricted
by the inventory holding cost which gives a situation where a decision needs to be
made, either it is more beneficial to produce earlier or just in time. The model should
also take the available inventory at the beginning of the scheduling horizon into
consideration. The output should balance with the inventory at the beginning so that
the cost can be optimized.
4.3
Problem Definition
In this study, there are several factors that need to be taken into considerations
in building the model formulation. The factors are the monthly demand of each
product, cycle time for each product, the starting inventory, setup cost and setup time
if any transition between two products happens and the labor cost for the two working
32
shifts. The objective of the model is to decide and calculate the products to be
manufactured in each day and their respective quantities in a month period, the total
machine time utilized by each product and the inventory quantities of each product at
the end of the month. Beside all this factors, there are also constraints in the model
that should be met which are production demands, due dates of orders, sequencing of
operations and available machine time and man hours.
The basic characteristics of the proposed scheduling model are described
precisely as the following:
i.
Model formulation and time representation:
The model will use binary variables to indicate which products will be
made on a certain day. It is also used as a tool to decide whether
transition between two products will take place or not. The scheduling
horizon for this model is for a month period where the model will
decide what products will be made on each day.
ii.
Automotive components:
This study presented a problem of production scheduling in an
automotive parts company. All the restrictions presented in the
production process are taken into consideration in building the model
formulation. For instance, the priority for a certain product to be
manufactured first is determined by the durations of the completion
processing time of that product.
Changeover cost and changeover time between each product will be
the guide for the transition between products to take place or not in
order to minimize the cost. The company requirement for the
33
equipment clearance after each shifts end had specific the production
hours to only 22 hours a day. For the same reason, each production lot
must start and finish in the same day, although production in the next
day may start with the last product that is manufactured in the current
day.
iii.
Demand Satisfaction
Demand is the order requirement from the customer that needs to be
fulfilled before the due date of order. Demand in this study is
considered for a month. Early production is possible but tardiness is
not allowed.
iv.
Decision Variables
When solving the model, the decision variables will come out with
some possible combinations of their values in order to optimize the
objective function. Decision variables for this study are the quantity of
each product that will be produced in a day, the machine time utilized
by each product and inventory level of product k at the end of the day.
In that way, the optimization can decide which product should be
assign on a certain day and how many quantity it should produce in
order to reduce changeover cost between a variety of product.
v.
Objective Function
The objective function is directed in reducing the cost for setup cost,
inventory cost and labor cost.
34
4.4
Model Formulation
The model formulated is created based on the information provided by the
company. All information from the company has been transformed in the mixed
integer programming problem in order to solve the scheduling problem. This model
formulation was adapted from a study by Doganis and Sarimveis [10]. Further
explanation on the complete model will be discussed under this section.
4.4.1
Parameters
List of parameters used in this model formulation are:
Scheduling horizon
Number of the products
Demand of each products for a month period
Setup time and cost for possible transition between each product
Labor cost for each shift
Opening inventories at the end of the scheduling horizon.
35
4.4.2
Decision Variables
The solution from the scheduling model will provide the optimal values of the
decision variables and can be grouped into continuous variables and binary variables.
For each day in the duration of one month, the optimal values of the following
variables are obtained:
a) Continues variables
The amount of produced quantity for each product
The amount of machine utilization including the setup time.
b) Binary variables
Binary variables indicate whether the respective product is to be
produced or not in the particular day.
Binary variables indicate whether the transition between products
should take place or not.
36
NOMENCLATURE
Indices
i
days
k, l, m
products ,
Parameter
N
scheduling horizon days ,
i= 1,2,…..,N (based on days in
month)
P
number of products
Dk
total demand of product k for a month
Csetup(k,l)
changeover cost from product k to product l (RM)
tsetup(k,l)
changeover time from product k to product l (h)
cost11
labor cost for shift 1 (RM/h x 11 working hours)
cost22
labor costs for shift 2 (RM/h x 11 working hours)
inv(i,k)
inventory level of product k at the end of day i
openinv(k)
opening inventory level of product k at day i
u(k)
machine speed for product k (per hour)
M(k) ,
maximum and minimum production lots ( in one day, how
many product can be made :: depends on the cycle time for
each product / total time in a day (22 hours))
Decision Variables
prod(i,k)
quantity produced of product k on day i
Time(i)
total utilization of machine , including changeover times on
day i (h)
BIN(i,k)
production of product k on day i (1/0)
BINSETUP(i,k,m)
changeover from product k to product m on day i
(1/0)
Figure 4.1 : Nomenclature that contains indices, parameter and decision
variables used in the model.
37
4.4.3
Objective Function
Min
The objective function minimizes the total setup cost of transition between products (
if any possible transition happens on a particular day) , the inventory cost for each
product that will be produced in a month and the labors cost involved in the
production process. Raw material and utility costs are not considered in this case
because that cost does not depend on any particular scheduling problem.
4.4.4
Constraints
The following sets of equation represent the limitations in the production process that
must be satisfied. The parameters and variables used in this model are explained in
the nomenclature. The constraints in this model are as the following:
Constraint 1:
(1)
The total amount of product k that will be produced in that month must be
equal to the total demand of that product for that month.
38
Constraint 2:
(2)
This constraints define the range of the labor cost for one month period which
must be in a number greater than 0. The labor cost will be calculated based on
the pay for one labor in an hour cost. Then the cost for a labor per hour will be
multiplied by the total number of working hours in a day which is 11 hours.
Then, the sum of the cost for one day for a labor will be multiplied by the total
number of labors for both shifts. Finally, the total for the labors cost in one
day will be multiplied by the number of working days in a month in order to
get the actual cost for labor in one month period of time.
Constraint 3 and 4:
(3)
(4)
This equation shows relationship between continuous variables and binary
variables where M(k) and µ(k) indicate the maximum and the smallest lot size
allowed. Production of product k in day i is allowed (prod(i ,k) > 0) if and
only if the binary variable BIN(i ,k) takes the value of 1. Product k is not
manufactured in day i (prod(i ,k) = 0), if and only if the binary variable BIN(i
,k) takes the value of 0.
39
Constraint 5:
(5)
Constraint 5 is about the total material balance for each product throughout
the scheduling horizon. The summation of produced quantities of product k
throughout the production horizon (in a month) plus the initial inventory must
equal the sum of demand of that month plus the inventory of product k at the
end of the last day in the month. Opening inventory is set to zero for all 8
different products, which mean that the actual inventory levels at the
beginning of the time are assumed as minimum safety bounds that must be
satisfied at the end of day.
Constraint 6 :
(6)
Total machine time for each day Time (i) includes all the production time of
products k and the time used for setups. The first term in equation 6 summates
the production times for all the products k while the second term shows the
time used for setups between each products k. In one day, the production lot
cannot involve more than 22 hours, since all the equipment must be cleaned
up at the end of each shift. Given that two 11-h shifts are available
considering the 1-h cleaning time at the end of each shift.
40
Constraint 7, 8, 9 and 10:
(7)
(8)
(9)
(10)
Equations (7), (8),(9) and (10) relate to the setup constraints. These
constraints will take part where the variable BINSETUP( i,k,m) will have
value of 1 if and only if there is a changeover from product k to product m on
day i. In order to confirm the set of constraints (7)-(10) always produce
correct value of the binary variables, Table 4.1 is generated that will examine
all the different cases in this problem. For each case, there is only one possible
value of BINSETUP (i,k,m) which is becoming equal to 1, only if BIN(i,k) is
1 and BIN(i,m) is 1 and additionally all binary variables BIN(i,l) , l=k+1, …..
, m-1 are equal to 0.
41
Table 4.1 : Investigation of correctness of set (7)-(10)
BIN (i,k)
4.5
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
BIN(i,m)
0
0
0
0
1
1
1
1
BINSETUP(i,k,m)
0
0
0
0
0
0
1
0
Conclusion
This chapter provides information about how the Mixed Integer Programming
model formulation is created. Problem definition describes about all the factors that
need to be taken into consideration in creating the model. Further discussions on how
and why the factors were used in the model formulation will be explained. Then, a
whole complete model formulation was showed and discussed. As the conclusion,
this chapter discusses the main part in this study which is the mixed integer
programming model formulation which will be used in solving the scheduling
problem in the company. This model will be used to produce an optimal schedule
production
which
will
be
discussed
in
the
next
chapter.
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Introduction
This chapter presents the result of this study which was solved by using
AIMMS software. An optimal production schedule is proposed once it satisfied all
the restrictions in production process. The calculated production schedule shows the
quantity of products and on what day they will be made. Then, further details about
the result will be discussed in this chapter.
5.2
New Proposed Production Schedule
A MIP model based on the limitation in production scheduling has been
created in order to come out with a new proposed production schedule. There are
eight different products with different demand, where need to be decided what
product should be made on a certain day. Table 5.1 shows the calculated production
schedule with the respective quantity of each product.
43
Table 5.1 : The calculated production schedule
Product
\
Day
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
D16
D17
D18
D19
D20
D21
D22
D23
D24
D25
D26
D27
D28
D29
D30
D31
TOTAL
P1
P2
P3
P4
P5
P6
P7
P8
713
-------------- 394
--628
-------------388
----538
------586
------688
-----437
----200 --713
--------------388
-628
--------688
-----248
--375
------688
--------586
-----------388
--449
29
--- 106
-713
--------------388
----538
------586
--------341
--149
713
---------688
------------388
713
----------586
----248 400
---------586
-----------388
-54
133
-250
---4250 1710 3334 3334 1667 200 500 2477
TOTAL
PRODUCT
PRODUCE
IN 1 DAY
713
394
628
388
538
586
688
637
713
388
628
688
623
688
586
388
584
713
388
538
586
490
713
688
388
713
586
648
713
388
437
44
From the calculated production schedule, there are 25 days where only one
product will be produced on each day. Product 8 takes a total of six days to be
produced to meet the demand since it has the longest cycle time in the production
process. Therefore, the cycle time leads to the low quantity of Product 8 that can be
made on one day. Product 1 and Product 4 share the same total of five days for
production even though the cycle time for both products is not so long. Both of the
products have the highest demand among the other products.
Product 3 takes four whole days to finish up the high demand from the
customer. Product 2 and Product 5 required only two days each since the demand and
the cycle time of the production process are not so high. Since Product 6 and Product
7 have the lowest demand, its takes only one day each to produced the requested
quantity by customer. Within these 25 days, no transition between any products take
place which means that no changeover cost was added to the monthly production
cost.
Even though all the products have specific day for each for the production
process, but it still have a certain amount that required to be made on same day. A
transition between products takes place in order to finish up the amount of products to
meet the demand. From the Table 5.1, on Day 8, Day 13, Day 22 and Day 28, there
are two products were produced on the same day. Day 8 produce Product 1 and
Product 6 with the total of 637 products will be made on that day. In Day 13, a mixed
schedule production between Product 1 and Product 4 will take place while in Day
22, Product 5 and Product 8 will be made on the same day. Then, on Day 28, Product
1 and Product 2 production process take place on the same day.
For the rest of the other two days in the scheduling horizon, there will be three
products to be manufactured on the same day. The production takes place on Day 17
and Day 31. For the cases where two or three products are made on the same day, it
requires the transition setup between all those products. It means that to change from
one product to another product, the machine need to be changing its setup setting.
45
This process takes time before can be proceed to the production of the next schedule
products. Besides increasing the time in the total production time, this process also
increased the production cost in terms of changeover cost between those products.
The changeover cost and changeover time of the products can be referred to Table 3.3
and Table 3.4 in Chapter 3. The allocation of the production scheduling considers the
changeover cost and changeover time in order to get the lowest production cost.
As stated in Chapter 4, the total numbers of machine use in the production
process are six machines. The new proposed production schedule only consider for
one machine which is the machine for turning process. As discuss before, the turning
process takes the longest cycle time in the production process. The rest of the other
processes have lowest cycle time compared to the turning process. Therefore, this
production schedule use the turning process cycle time data in order to get the
maximum number of products to be made in one day for all the product since it
restricted the amount of production in one day. The rest of the process will continue
the processing job once it has been finish in the turning process with the same amount
of products.
For more understanding, Table 5.2 and Table 5.3 were created to show the flow of the
whole process flow in the production process.
31
Table 5.2 : The flow of the whole process from one machine to another machine with the quantity of products to be made.
PRODUCT
\
DAY
MACHINE 1
(TURNING)
MACHINE 2
(AIR BLOW)
MACHINE 3
(WASHING)
MACHINE 4
(OVEN)
MACHINE 5
(VISUAL
INSPECTION)
MACHINE 6
(PACKAGING)
D1
D2
D3
D4
D5
D6
D7
D8
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
P1 : 713
P7 : 394
P2: 628
P8 : 388
P5 : 538
P4 : 586
P3 : 688
P1 :
P6 :
437
200
P1 : 713
P8 : 388
P2: 628
P3 : 688
P1:
P4 :
248
375
P3 : 688
P4 : 586
P8 : 388
P3: P4: P7:
449 29 106
D9
D10
D11
D12
D13
D14
D15
D16
D17
32
Table 5.3 : The flow of the whole process from one machine to another machine with the quantity of products to be made.
PRODUCT
\
DAY
D18
D19
D20
D21
MACHINE 1
(TURNING)
MACHINE 2
(AIR BLOW)
MACHINE 3
(WASHING)
MACHINE 4
(OVEN)
P1 : 713
P8 : 388
P5 : 538
P4 : 586
P1 : 713
P8 : 388
P5 : 538
P4 : 586
P1 : 713
P8 : 388
P5 : 538
P4 : 586
P1 : 713
P8 : 388
P5 : 538
P4 : 586
D23
P8:
149
P1 : 713
P5:
P8:
341
149
P1 : 713
D24
P3 : 688
P3 : 688
P3 : 688
P3 : 688
P3 : 688
P3 : 688
D25
D26
P8 : 388
P1 : 713
P8 : 388
P1 : 713
P8 : 388
P1 : 713
P8 : 388
P1 : 713
P8 : 388
P1 : 713
P8 : 388
P1 : 713
D27
P4 : 586
P4 : 586
P4 : 586
P4 : 586
P4 : 586
P4 : 586
D28
P1:
P2:
248
400
P4 : 586
P8 : 388
P1:
P2:
248
400
P4 : 586
P8 : 388
D22
D29
D30
D31
P5:
341
P2:
54
P3:
133
P5:
250
P2:
54
P3:
133
P5:
250
P5:
P8:
341
149
P1 : 713
P1:
P2:
248
400
P4 : 586
P8 : 388
P2:
54
P3:
133
P5:
250
MACHINE 5
(VISUAL
INSPECTION)
P1 : 713
P8 : 388
P5 : 538
P4 : 586
P5:
P8:
341
149
P1 : 713
P1:
P2:
248
400
P4 : 586
P8 : 388
P2:
54
P3:
133
P5:
250
MACHINE 6
(PACKAGING)
P1 : 713
P8 : 388
P5 : 538
P4 : 586
P5:
P8:
341
149
P1 : 713
P1:
P2:
248
400
P4 : 586
P8 : 388
P2:
54
P3:
133
P5:
250
P5:
P8:
341
149
P1 : 713
P1:
P2:
248
400
P4 : 586
P8 : 388
P2:
54
P3:
133
P5:
250
48
Percentage of total production days for
each product
20%
21%
Product 1
Product 2
4%
9%
1%
Product 3
Product 4
10%
16%
19%
Product 5
Product 6
Product 7
Product 8
Figure 5.1 : Percentage of total production days needed for each product
In Figure 5.1 above, the chart represents the percentage for the number of days for
each product to complete the production process in the scheduling horizon period
which is within 31 days. The biggest percentage is for Product 8 with 21% of the
scheduling horizon period is used to finish the demand of the product. It is followed
by Product 1 with 20% , Product 4 with 19%, Product 3 with 16%, Product 5 with
10%, Product 2 with 9%, Product 7 with 4% and Product 6 with 1% only.
Table 5.4 : Daily production time including setup times ( in hour)
49
Product
\
Day
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
D16
D17
D18
D19
D20
D21
D22
D23
D24
D25
D26
D27
D28
D29
D30
D31
P1
P2
P3
P4
P5
P6
P7
P8
21.9
-------------- 21.9
--21.98
-------------- 21.98
----20.9
------21.97
------21.97
-----13.47
----8.16 --21.9
--------------- 21.98
-21.98
--------21.97
-----7.64
--14.06
------21.97
--------21.97
------------ 21.98
--14.34 1.08
--- 5.91
-21.9
--------------- 21.98
----20.9
------21.97
--------13.26 --8.44
21.9
---------21.97
------------- 21.98
21.9
----------21.97
----7.64 14.0
---------21.97
------------ 21.98
-1.89 4.24
-9.72
----
TOTAL
TIME
(INCLUDE
SETUP
TIME)
21.9
21.9
21.98
21.98
20.9
21.97
21.97
21.96
21.9
21.98
21.98
21.97
21.98
21.97
21.97
21.98
21.98
21.9
21.98
20.9
21.97
21.98
21.9
21.97
21.98
21.9
21.97
21.99
21.97
21.98
16.549
In one day production, the working hour is limited only to 22 hours, which
are divided into two working shifts. One working shift will have 11 hours period of
time for the operations hour. Another 2 hours balance was used for the clearance of
tools and machinery for the maintenance purpose. Therefore, the production process
in one day must not exceed the working hours.
50
Table 5.4 shows daily machine utilization time allocated to each product
including the time setup. As shown by the data in the table, all the production time
are under 22 hours, as restricted by the working hours. Products that need the most
times to finish the production as in proposed schedule are Product 2 and Product 8
with the total time of 21.98 hours (21 hour and 58 minutes). Product 3 and Product 4
has the second higher time with total of 21.97 hours while Product 1 and Product 7
has used total of 21.9 hours in one day production.
Even though the total time machine utilization for these products is same, but
the amount of product produces is difference, depending on its cycle time as
discussed before.
Product 2 takes 21.98 hours to produce for 628 amounts of
products while Product 8 use the same time period to produce only 388 products. For
21.97 hours period of time, the machine can produce 688 products for Product 3 and
586 products for Product 4. In 21.9 hours duration, the machine can produce 713 of
Product 1 while for Product 7, it only can produce for 394 products. For Product 5, it
takes 20.9 hours to produce the amount of 538 products. Product 6 needs the lowest
period of time to finish its demand which only for 8.16 hours due to the low number
of demands.
For the day that transition between products take place, the total time utilize
are including the time setup for the changeover. For the whole scheduling horizon,
the working time will be use totally without wasting even for a second. This can help
to increase productivity which can bring benefits to the company.
Observation through the discussion in this chapter shows that production is
accommodated towards the minimization of cost. The allocations of the production
schedule are made based on the lowest production cost. This is the main reason why
the proposed new production schedule has the most day in the scheduling horizon to
produce only one product in one day. This case did not require any changeover cost
for transition between products to take place. For the case when transition between
51
products is happen, the model allocates the transition to happen at the most lowest
changeover cost in order to achieve the goal.
The new proposed production schedule offer a better production cost for a
month period which is for RM 6115 only. This total cost is considered much lower
compare to the production cost the Company I need to spend before. Therefore, this
new propose production schedule plan offer a better scheduling planning which can
help the company I to gain more benefits and profits.
5.3
Conclusion
This chapter discussed the result from this study and the findings. The new
proposed production schedule offer a better scheduling planning with a lower
production cost. Beside, the new proposed production schedule offers an efficient
and optimal production of product with maximum utilization of the working hour‟s
time.
CHAPTER 6
CONCLUSION
6.1
Introduction
This chapter discusses the conclusions of this research. Besides that, there are
also discussion on the limitations of the research and a few recommendations for
further study in the same area which can be conducted in the future.
6.2
Conclusion
This study aimed to proposed a new production scheduling plan and
minimize the production cost for the company production process. The case study
focuses only on automotive component production department in Company I. It is a
company that supplies the automotive components to the automotive industry. This
company had faced a problem in the production scheduling where they cannot meet
the customer demand and the production process had exceeded the due dates.
53
In this study, a MIP model formulation was proposed to solve the scheduling
problem. The MIP model was created with an objective functions to satisfied a set of
constraints and decision variables. The model considered all the limitations that
present in the production process which may restrict the production of the products.
The limitations that affect the production process are the cycle time, the changeover
time, the changeover cost, working hour period, demand of the products and the due
dates. This mathematical model provides a theoretical framework for the formulation
and solution of those models where the translation from problem to model is
included.
By using AIMMS modeling system software, the optimal production
schedule was obtained which consist of the amount of products to be made on certain
day. Most of the days in the scheduling horizon will produce only one product in
one day, but the transition between products also take place on certain day which
only happen for six days. The allocation of the production scheduling was made with
the purpose to minimize the production cost. The result of this study indicates that
the new proposed production schedule can optimize the production process and at the
same time, reducing the production cost which give benefits to the company.
6.3
Limitations of Study
The main limitation in completing this study is due to the certain confidential
data that cannot be obtained. This is intended to protect the confidential information
of the company and secure them from the other competitors in the same industry.
However, the company I willing to give cooperation as long as this study was
conducted as an academic research and the information was keep closely confidential
from any other party outside the factory. Another factor that limits this study is the
time length and the travels distant to the company limited the visit. Again, the
56
company gives a good cooperation to help in completing this study by stayed
connected and data gathered through email.
6.4
Future Research
As the research is implemented in a short period of time, it still can be
improvised and modify to improve the findings and the results generated through this
research project. Future extensions of this research can be directed towards
scheduling planning for multiple parallel machines that is a set of machines, where
due to technical restrictions there are dependencies on the operations across the
machines. The scope of the study also can be widen up by doing a scheduling
planning for a bigger amount of products.
55
References
[1] Ozguven.C,Ozbakir.L.,Yavuz.Y. (2009). Mathematical Models for Job-Shop
Scheduling Problems with Routing and Process Plan Flexibility. Applied
Mathematical Modelling.In Press.
[2] Kanishka Bedi. (2007). Production and Operations Management. Second
editon. Oxford University Press. India.
[3] Heizer.J. and Render.B. Operations Management. 9th edition. Upper Saddle
River, N.J.Pearson Prentice Hall. 2008.
[4] Wazed.M.A, Ahmed.S. and Nukman.Y. (2009) .Commonality Indices in
Design and Production Planning. Journal of Advanced Manufacturing
Technology. Vol. 3(1). Pp 1-20.
[5] Yavuz.Y. et al. (2009). Mathematical Models for Job-Shop Scheduling
Problems with Routing and Process Plan Flexibility. Applied Mathematical
Modeling. Elsevier Inc. In Press.
[6] Pochet.Y. and Wolsey.L.A. Production Planning by Mixed Integer
Programming. United States of America. Springer .2006
[7] Yavuz.Y. et al. (2009). Mathematical Models for Job-Shop Scheduling
Problems with Routing and Process Plan Flexibility. Applied Mathematical
Modeling. Elsevier Inc. In Press.
[8] Omar.K.M and Teo.S.C. (2006). Minimizing the Sum of Earliness/Tardiness
in Identical Parallel Machines Schedule with Incompatible Job Families : An
Improved MIP Approach. Applied Mathematics and Computation. Vol.181. pp
1008-1017.
[9] Blomer.F and Gunther.H.O. (1998). Scheduling of a Multi Product Batch
Process in the Chemical Industry. Computers in Industry. Vol.36. pp 245-259
56
[10] Doganis. P and Sarimveis.H. (2006). Optimal Scheduling In A Yogurt
Production Line Based On Mixed Integer Linear Programming. Journal of Food
Engineering. 80:445-453.
[11] Harjunkoski.I , Isaksson.J and Westerland.T. 1998. Solving a Twodimensional Trim-Loss Problem with MILP. European Journal of Operational
Research. Vol.104. pp 572-581.
[12] Nystrom.R.H , Harjunkoski.I and Kroll.A . 2006. Production Optimization
for Continuously Operated Processes with Optimal Operation and Scheduling of
Multiple Units. Computers and Chemical Engineering. Vol.30 . pp 392- 406.
[13] R.Simpson and A.Abakarov. 2009. Optimal Scheduling of Canned Food
Plants Including Simultaneous Sterilization. Journal of Food Engineering.
Vol.90. pp 53-59.
[14] Lee.H, Pinto.J.M, Grossman.I.E and Park.S. (1996). Mixed-Integer Linear
Programming Model for Refinery Short-term Scheduling of Crude Oil Unloading
with Inventory Management.Industrial Engineering Chemistry. Vol.35. pp 16301641.
[15] Choi.B.C et al. (2007). Single Machine Scheduling Problem with
Controllable
Processing
Times
and
Resources
Dependent
Release
Times.European Journal of Operational Research. Vol 181. Pp.645-653.
[16] Mendez.C.A., Grossmann.I.E., Harjunkoski.I. and Kabor.P. (2006). A
Simultaneous Optimization Approach for Off-line Blending and Scheduling of
Oil-Refinery Operations. Computers and Chemical Engineering. Vol.30. pp 614634.
[17] T‟kindt.V., Jatinder.N.D.G, and Billau.J.C. (2003). Two Machine Flowshop
Scheduling with a Secondary Criterion. Computers & Operations Research. Vol
30.
Pp
505-526
57
[18] Tang.L. and Liu.P. (2009). Two Machine Flowshop Scheduling Problems
Involving
a
Batching
Machine
with
Transportation
or
Deterioration
Consideration. Applied Mathematical Modelling. Vol.33. pp 1187-1199.
[19] Martin.C.H. (2009).
Programing Approach
A Hybrid Genetic Algorithm/ Mathematical
to the Multi-Family Flowshop Scheduling Problem
with Lot Streaming. Omega. Vol.37. pp126-137.
[20] Khosla.I. (1995). The scheduling Where Multiple Machines Compete for a
Common Local Buffer. European Journal of Operational Research. Vol.84. pp
330-342.
[21] Giannelos.N.F and Georgiadis.M.C. 2003. Efficient Scheduling of Consumer
Goods Manufacturing Processes in the Continuous Time Domain. Computers &
Operations Research.Vol.30. pp 1367-1381.
[22] Voudouris.V.T and Grossmann.I.E. 1996. MILP Model for Scheduling and
Design of A Special Class of Multipurpose Batch Plants. Computer Chemical
Engineering. Vol.20.No.11. pp 1335-1360.
[23] Kovacs.A. , Brown.K. and Tarim.S.A.(2009). An efficient MIP model for the
Capacitated Lot Sizing Scheduling Problem with Sequence Dependent Setup. Int.
J.Production Economics. Vol.118. pp282-291.
[24] Adonyi.R., Biros.G.,Holczinger.T. and Friedler.F. (2008 ). Effective
Scheduling of Large Scale Paint Production System. Journal of Cleaner
Production. Vol.16. pp 225-232.
[25] Eren.T. and Guner.E. (2006). A Bicriteria Scheduling with Sequence
Dependent Setup Times. Applied Mathematics and Computation. Vol.179. pp
378-385.
[26] Tang.L., Liu.J., Rong.A. and Yang.Z. (2001). A Review of Planning and
Scheduling Systems and Methods for
Integrated Steel Production. European
Journal
of
Operational
Research.
Vol.133.
pp
1-20.
58
[27] Orcun.S., Altinel.I.K. and Hortacsu.O. (2001). General Continuous Time
Models for Production Planning and Scheduling of Batch Processing Plants:
Mixed Integer Linear Program Formulations and Computational Issues.
Computers and Chemical Engineering. Vol.25. pp 371-389.
[28] F.Werner. (1997). Parallel Machine Scheduling Problems with a Single
Server. Mathematical Computater Modelling. Vol. 26. Pp 1-11
[29] Earl.M.G. (2005). Iterative MILP Methods for Vehicle-Control Problems.
IEEE Transactions on Robotics. Vol.21.
[30]Grotschel.M. and Nemhauser.G.L. (2008). Georgre Dantzig’s Contributions
to Integer Programming.Discrete Optimization. Vol.5. pp 168-173.
[31] Jin.M., Luo.Y. and Eksioglu.S.D. (2008). Integration of Production
Sequencing and Outbound Logistics in the Automotive Industry. Int.J.Production
Economics. Vol.113. pp 766-774.
[32] Kim.s. et al. (2008). A Mixed Integer Programming Model for the
Multiperiod Scheduling Problem of Blending Zinc Bearing Materials to Minimize
Setups in the Hydrometalurrgical Process. Metallurgical and Materials
Transactions B.Vol.39B.pp 896-905.
[33] Omar.M.K. and Teo.S.C. (2006). Minimizing the Sum of Earliness/Tardiness
in Identical Parallel Machined Schedule with Incompatible Job Families : An
Improved MIP Approach.Applied Mathematics and Computation. Vol.181. pp
1008-1017.
[34] Chen.K. and Ji.P. (2007). A Mixed Integer Programming Model for
Advanced Planning and Scheduling (APS). European Journal of Operational
Research. Vol.181. pp515-522.
[35] Wolsey.L.A. (1997). MIP Modelling of Changeovers in Production
Planning and Scheduling Problems. European Journal of Operational Research.
Vol.99. pp 154-165.
[36] Rocha.P.L. et al. (2008). Exact Algorithms for A Scheduling Problem with
Unrelated Parallel Machines and Sequence and Machine Dependent Setup
Times. Computers & Operations Research. Vol.35. pp 1250-1264.
59
[37] Dastidar.S.G. and Nagi.R. (2005). Scheduling Injection Molding Operations
with Multiple Resource Constraints ans Sequence Dependent Setup Times And
Costs. Computers & Operational Research. Vol.32. pp 2987-3005.
[38] Roslof.J. et al. (2002). Solving a Large Scale Industrial Scheduling Problem
Using MILP Conbined with a Heuristic Procedure. European Journal of
Operational Research. Vol. 138. Pp 29-42.
[39] Mokotoff.E. (2004). An Exact Algorithm for the Identical Parallel Machine
Scheduling Problem. European Journal of Operational Research. Vol.152. pp
758-769.
[40] Pochet.Y. and Warichet.F. (2008). A Tighter Continuous Time Formulation
for the Cyclis Scheduling of a Mixed Plant. Computers and Chemical
Engineering. Vol.32. pp 2723-2744.
[41] Taha.H.A. Operations Research: An Introduction. 8th edition. Upper Saddle
River, N.J . prentice Hall .2007.
60
APPENDIX
FORMULATION MODEL
61
DATA : DEMAND FOR ONE MONTH
DATA : CHANGEOVER COST
62
DATA : CHANGEOVER TIME ( IN SECOND)
DATA : CYCLE TIME FOR EACH PRODUCT
63
CONSTRAINT : TOTAL PRODUCT
CONSTRAINT : MAXIMUM LOT
64
CONSTRAINT : MINIMUM LOT
CONSTRAINT : TOTAL MATERIAL BALANCE
65
CONSTRAINT : TOTAL MACHINE TIME
CONSTRAINT : SETUP 1
66
CONSTRAINT : SETUP 2
CONSTRAINT : SETUP 3
67
CONSTRAINT : SETUP 4
OBJECTIVE FUNCTION
68
DATA USED IN THE PROJECT
69
RESULT OBTAIN FROM AIMMS SOFTWARE