Engineering King Saud University and Mr. Mohammed. ALthebyani .... executed, aircraft waiting for landing clearance at an airport, class scheduling at school ...
King Saud University College of Engineering Department of Industrial Engineering
An Investigation of Due-Date Setting Rules at Stochastic Production Environment: A Case Study By: Eng. Talal Alsulaiman Supervised By: Dr. Ibrahim M. Alharkan Date: April 2008
Submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department with College of Engineering King Saud University
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We hereby approve the Master of Science Thesis report entitled:
An Investigation of Due-Date Setting Rules at Stochastic Production Environment: A Case Study Prepared By: Eng. Talal Alsulaiman COMMITTEE MEMBERS SUBERVSOR:
Dr. Ibrahim M. AL-Harkan
Signature: _______________________
EXAMINERS:
Dr. Anis Gharbi
Dr. Salah Addin Bendak
Signature: _______________________
Signature:_______________________
April 2008
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© Copyright by TALAL ALSULAIMAN 2008 All Rights Reserved.
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In the memory of my Grand Father Hamed Alsulaiman
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Acknowledgment “Thanks to Allah for all his uncountable blessings”
In this occasion I would like to thank Dr. Abdullrahmna Alahmari Dean of Industrial Engineering King Saud University and Mr. Mohammed ALthebyani Division Head - Samba Financial Group for granting me this opportunity to complete my Masters in Industrial Engineering Sciences. Also, I would also thank to Eng. Abdulaziz Alsaleh Production Emg. Advance Electronic Company for allowing me to conducted my research studies in his prestigious organization. My Sincere thanks to greater professors, Dr.Abdulaziz Altameemi, Dr. Salah Addin Bendak, Dr. Abdelghani Bouras, Dr. Mohammed Afwazan and Dr. Monsser Hareqa, from which I have learned a lot and got invaluable knowledge during my two years of studies in King Saud University. I would also like thanks Mr. Ibrahim Alessa f. And Dr. Khalid ALessa and for their valuable feedback and suggestions for completing my thesis. I am also greatful to my Family, my friend and Office colleagues Mr.Abdullah Al Nasseer, Mr. Imran Kazim and Mr. Abdulrahman Albosaili. for their kind support and encouragement. Finally, I would like to express my sincere gratitude and deep appreciation to the person who honored me to be one of his students. Dr. Ibrahim Alharkan Thank you for every thing.
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ABSTRACT This study investigates a dynamic approach that assigns due-dates to jobs based on workshop information. Flow shop structure thirteen machines was modeled and simulated using the Awisem package. Twenty four experiments were designed to investigate the combinations between due date assignment methods and due date factors or allowance level. The investigated due dates in this study were Authenticated Rule of the Shop, Total Work Contents Rule (TWK), Number of Operations Rule (NOP), and Processing Plus Waiting Time Rule (PPW). Theses rules were investigated under allowance level of one, five ,and ten. These combinations were examined a military radio product. This product is called Radio Panther where the investigations were performed on the full Radio Panther and on a major part of it that is called Transceiver. The performance measure of the experiments was the absolute lateness of not meeting the due date. The absolute lateness is concerned with both earliness and tardiness. The simulation results were test for the significant using Analysis of Variance and ranked by using Tukey’s test. The results suggest that assigning due date based on TWK with allowance level equal one will minimize the absolute lateness for both Radio Panther and Transceiver.
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TABLE OF CONTENTS Chapter1:
Introduction
1
Definitions Levels of Scheduling Classes and Categories of scheduling problems Machine Environments Processing Constrains Criteria and Objective Functions of Scheduling Solution Methodologies for Scheduling Problems Purpose of the Study Scheduling Environment Definition Design of Experiments Simulation Modeling and Analysis Test of Hypothesis Analysis of Variance Tukey’s Test
1 2 3 4 5 6 9 11 11 13 14 15 19 20 22
Literature Review
23
3.1. 3.2. 3.3. 3.4.
Brief History of Scheduling Theory Dispatching Rules Due-Date Setting Rules Related Previous Studies on Scheduling Problems
23 25 28 31
Chapter4:
Case Study: Advanced Electronic Company
37
Company’s Profile AEC’s Products and services Production Shops Parts of Radio Panther Process Plan of Radio Panther
37 40 42 47 48
Systems Modeling and Implementation
53
5.1. 5.2.
Designing the Experiments Simulation Model
53 56
5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6
AweSim Language Selecting Input Probability Distributions Statement of the Problem and Assumptions Translation of Simulation Network Verification and Validation Simulation Results
56 67 72 74 79 80
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.
Chapter2: 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
Chapter3:
4.1. 4.2. 4.3. 4.4. 4.5.
Chapter5:
Research Methodology
IX
5.3
Testing Stated Hypotheses
81
5.3.1 5.3.2
Constructed Analysis of Variance (ANOVA) Table Ranking the Effecting Factors
82 86
Chapter6:
Conclusions and Recommendations
90
6.1. 6.2. 6.3. 6.4.
Summary Conclusion Contributions Recommendations for Further Researches
90 92 96 96
Appendix A: Appendix B: Appendix C: Appendix D: Appendix E:
References Fitted Distributions for Collected Data Simulation Network Echo Report Simulation Output ANOVA Model and Tucky’s Test Output
98 101 105 106 119 144
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LIST OF FIGURES Figure #
Figure Title
Page #
Figure 2.1
Flow Chart of the Research Methodology
12
Figure 2.2
General Model of process or system
14
Figure 2.3 Figure 2.4
Flow Charts of Simulation Model Critical Regions for One-sided and Two-sided Testes of the Hypothesis
16 19
Figure 4.1
Basic Organization Chart of AEC
39
Figure 4.2
Floor Shop Layout of AEC
43
Figure 4.3
Parts Family Tree of Radio Panther
49
Figure 4.4
Precedent Constraints of each Part of Radio Panther
50
Figure 4.5
Process Plan of Radio Panther
52
Figure 5.1
Composition of AweSim
59
Figure 5.2
Process Flow Chart
78
Figure 5.3
Model Comparison for Radio Panther
81
Figure 5.4
Model Comparison for Transceiver
81
Figure 5.5
Interaction between DDSR and DDF
86
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Figure A.1
Fitted Probability Mass Functions for Size of Order of Panther Radio and Transfer
102
Figure A.2
P-P Plots for the Order Size of Panther Radio and Transfer
104
Figure E.1
Normal Probability Plot
144
Figure E.2.
Residual Plot
145
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LIST OF TABLES Figure #
Figure Title
Page #
Table 1.1
Criteria Associated with each of the Three Measures
7
Table 2.1
Possible Results of any Hypothesis Test
20
Table 2.2
ANOVA Table for one factor A
21
Table 3.1
Versions of Dispatching Rules
27
Table 4.1
Shop of Panther Work Cell
45
Table 5.1
Designed Experiments
56
Table 5.2
The functional of cv and τ
69
Table 5.3
Order size Distribution of Radio Panther
71
Table 5.4 Table 5.5
Operations Processing Time Simulation Results on Overall Mean of Absolute lateness
71 80
Table 5.6
Observations of the Experiments
83
Table 5.7
ANOVA Outcomes
84
Table 5.8
Due Date Setting Rules Ranking
87
Table 5.9
Due Date Factor Ranking
87
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Table 5.10
Due Date Setting Rules and Due Date Factor Ranking for Radio Panther
88
Table 511
Due Date Setting Rules and Due Date Factor Ranking for Transceiver
89
Table A.1
Historical Data for the Order Size of Panther Radio and Transceiver
101
Table A.2
Descriptive Statistic and Goodness of Fit for Fitted Probability Mass Function for Size of Order of Panther Radio and Transceiver
103
Table A.3
Mass Functions for Order Size of Radio Panther and Transceiver
103
Table E.1
ANOVA Table by SPSS
146
Table E.2
Tukey’s Test for Due Date Setting Rules
147
Table E.3
Tukey’s Test for Allowance Level
147
Table E.4
Tukey’s Test for Due Date Setting Rules and Allowance Level
148
Chapter
1
Introduction Sequencing and scheduling is one of the most important activities in production planning and control. The effective scheduling is important for any organization in terms of meeting customer expectations, reducing the holding cost or the penalty cost of not meeting the target date. Pinedo [1] further discussed the importance of the sequencing and scheduling problem: …Sequencing and scheduling are forms of decision-making, which play a crucial role in manufacturing as well as in service industries. In the current competitive environment, effective sequencing and scheduling has become a necessity for survival in the marketplace. Companies have to meet shipping dates committed to the customers, as failure to do so may result in a significant loss of good will. They also have to schedule activities in such a way as to use the resources available in an efficient manner. The sequencing and scheduling problem may occur in different industries. It may occur in manufacturing area by which parts waiting for processing in a manufacturing plant, computer operating system by which the central processing unit CPU devotes to the different programs that have to be executed, aircraft waiting for landing clearance at an airport, class scheduling at school, and waiting customers for some explicit service.
1.1. Definitions It is important to distinguish between terms “scheduling” and “sequencing”. The definition of sequencing is common among researchers. According to Alharkan [2,3] sequencing can be defined as the order in which the jobs (tasks) are processed through the machines (resources). However, the researchers did not agree on one definition of scheduling but all of the definitions are in the way of allocate the means to perform a collection of jobs overtime as a process of decision making that aims to optimize one or several objective functions.
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Pinedo [1] defined scheduling as: … Scheduling deals with the allocation of scarce resources to tasks over time. It is a decision-making process with goal of optimizing one or more objectives. Baker [4] defined scheduling as: ...Scheduling is the allocation of resources over time to perform a collection of tasks. Scheduling is a decision-making function: it is the process of determining a schedule. Scheduling is a body of theory: it is a collection of principles, models, techniques, and logical conclusions that provide insight into the scheduling function. Morton and Pentico [5] defined scheduling as follows: ...Scheduling is the process of organizing, choosing, and timing resource usage to carry out all the activities necessary to produce the desired outputs at the desired times, while satisfying a large number of time and relationship constraints among the activities and the resources. Jobs that have to be performed are the activities such as tasks, delivery, transportation, machining, milling, grinding, painting, sanding…etc. Means are the resources that utilized to complete the job. Examples of means are machines, men, operators, power…etc. However, the generated schedule has to meet two conditions. First condition is that all technological constraints are met which means that schedule is feasible. Second condition is that all objective functions are optimized, so that the schedule is sufficient.
1.2. Levels of Scheduling According to Morton and Pentico [5] sequencing and scheduling are caught up at several levels of the decision-making process. These levels are as follows: 1) Long-Range Planning: the types of jobs or projects at this level include location sizing, designing, and expansion of plants, warehouses, departments and transfer lines. Usually, the horizon time for this level is between 2 to 5 years. The problems that laminate the scheduling models
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are grouping, aggregation, and desegregation and the forecasting issues amongst the long horizon. 2) Middle-Range Planning: at this level, production smoothing deals with resources reconfiguration over time to provide balanced capacity facing seasonal demand. The horizon time for this level is between 1 to 2 years. Even though, the issue of grouping, aggregation, and desegregation is dominate at this level but forecasting is still an issue. 3) Short Range Planning: This level includes Material Requirements planning (MRP) and due-date setting. Its usual horizon time is between 3 to 6 months. 4) Predictive scheduling: This level includes shop routing, assembly line balancing and process batch sizing. Its usual horizon time is between 2 to 6 weeks. Sequencing, timing and routing are dominated at this level although forecasting and Management information System (MIS) need more focusing. 5) Reactive Scheduling or Control: which is performed every day or every three days. A few examples are: hot jobs, down machines, and late material. Level four is the concern of this research, and therefore, sequencing and scheduling methodologies for only this level will be discussed. Specifically, environments, general assumptions, categories, criteria, decision-making goals, and solution methods for the sequencing and scheduling problems will be explained.
1.3. Classes and Categories of scheduling problems According to Pinedo [1] there are three classes in scheduling problems, which are sequencing, scheduling, and scheduling policy. Sequencing usually corresponds to a permutation of the n jobs or the orders in which jobs are to be processed on a given machine. Scheduling usually refers to the allocation of jobs within a more complicated setting of machines. A policy of scheduling prescribes an appropriate action for any one of the states the system may be in.
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The concept of scheduling policy is often used in stochastic setting. In deterministic models, usually only sequences or schedules are of importance. Moreover, scheduling problems may be divided into four categories. According to Alharkan [3] the categories of scheduling problems are: 1) Deterministic: is the category where all components of the system such as the state of the arrival of the jobs to the shop, due-dates of jobs, ordering, processing time are known and determined in advance. 2) Static: is the same as deterministic problems expect that the nature of the job arrival is different. The set of jobs over time does not change and it is available before hand. 3) Dynamic: The difference between dynamic and static is that in dynamic problems the set of jobs change over time and jobs arrive at different time. 4) Stochastic: occurs when at least one component of the system is having a stochastic behavior and it is undetermined in advanced.
1.4. Machine Environments According to Pinedo [1] scheduling problems may be described in three fields. The triplet is α β γ , where α is the machine environments and contains a single entry, β filed provides details of processing characteristic and constraints and may contain no entry at all or multiple entries, γ filed describes the objective function and usually contains one entry. There are several machine environments; the following are some of most known environments: 1) Single Machine (1): one machine and n jobs to be processed. 2) Identical Machines in Parallel ( Pm ): m identical machines in parallel along with n jobs that requires a single operation. 3) Different Speed of Parallel Machines ( Qm ): is same as Pm except that the speed of machines are different.
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4) Unrelated Machines in Parallel ( RM ): is same as Qm expect the speed of the machine is differed based on the job nature. 5) Flow Shop ( Fm ): The jobs have to proceed over a series m machines following the same rout. 6) Flexible Flow Shop ( FFc ): is a combination between parallel machines and flow shop environments. 7) Job Shop ( J m ): in a job shop with m machines, each job has its own predetermined rout to follow. 8) Flexible Job Shop ( FJ c ): is a combination between parallel machines and job shop environments. 9) Assembly Job Shop ( AJ m ): a job shop with jobs that have at least two component items and at least one assembly operation. 10) Hybrid Job Shop ( HJ M ): the precedence ordering of the operations of some jobs are the same. 11) Hybrid Assembly Job Shop: is a combination between assembly and hybrid job shops. 12) Open Shop ( Om ): where there are no restrictions of jobs routing through machines. In other word, there is no specific pattern for the jobs. 13)Closed Shop ( Cm ): it is a job shop; however, all production orders are generated because of inventory replenishment decisions. In other words, the production is not affected by the customer order.
1.5. Processing Constraints The shop may contain specific constraints that should be taken into consideration. These constraints may be defined in the general description of scheduling problem. They may be introduced in the β filed where no or multiple constrains are present. Depending on the nature of environment, these
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constraints are differed. According to Pinedo [1], some useful constraints are as following: 1) Release Dates ( rj ): meaning that the jobs may not start before it is ready. It may also called ready time. 2) Preemption ( prmp ): it implies that it is allowed to replace the job that on the machine by another job at any point in time. 3) Precedence Constraints ( prec ): it implies that one or more jobs are depending on other jobs, which means that it or they cannot be started unless the previous jobs are completed. 4) Breakdowns ( brkdwn ): it implies that machines are not continuously available. 5) Permutation ( prmu ): it implies that the order in which jobs go through the first machine is maintained through the system. It often appears in flow shop. 6) Blocking ( block ): Blocking is a phenomena that may occur in flow shop. If a flow shop has a limited buffer in between two successive machines, then in case of puffer is over the upstream machine then, it is not allowed to release a completed job. 7) No-wait( nwt ): it another phenomena that may occur in flow shops. it is an opposite of blocking, which means that the jobs are not allowed to wait between two successive machines. For additional constraints that have been used when solving sequencing and scheduling problems, it is suggested here to refer to Pinedo [1], Alharkan [2,3], Backer [4], Morton and Pentico [5], and Conway, Maxwell and Miller [6].
1.6. Criteria and Objective Functions of Scheduling One of scheduling model features is the diversified objective function. According to Rinnoy [7] and French [8], the performance of any sequence of jobs can be measured through completion time criteria, on due date criteria,
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and inventory and on machine utilization criteria. These criteria represent the objective functions that are defined in the third γ filed of the general description of scheduling problems. Table 1.1 shows the performance measurement criteria that was introduced by Rinnoy [7] and French [8]. Table 1.1 Criteria Associated with each of the Three Measures Criteria based on completion times
Average completion time C .
Completion time of job i ; Ci
The total weighted completion time
Flow time of job i;
n ∑w C i=1 i i
The total weighted waiting time .
Fi = Ci -ri
n m ∑ w ∑ W i = 1 i j = 1 ij
.
Maximum completion time (the schedule time, total production time, or makespan) ; max
Cmax = 1,...,n {Ci}.
The total flow time;
n ∑F i=1 i
The total weighted flow time ;
F.
Average flow time ;
Waiting time of job I; Wi =
.
Maximum flow time;
m F − ∑ Pij i j=1 .
The total completion time =
n ∑C i=1 i
The total waiting
n ∑w F i=1 i i
.
Fmax.
n m ∑ ∑ W ij time i = 1 j = 1
.
Average waiting time W . .
Criteria based on inventory and machine utilization Average number of jobs waiting for machines Nw .
Average number of unfinished jobs N u .
Average number of jobs completed N c .
Average number of jobs actually being processed N p .
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Table 1.1 Criteria Associated with each of the Three Measures (Continued) Average number of machines idle I .
Maximum machine idle time I max .
n m
Average utilization U = ∑ ∑ Pij / m. C max i =1 j=1
Criteria based on due-dates n
The total lateness; ∑ L i . i=1
Lateness of job i; Li = Ci - di.
The total weighted lateness:
n ∑w L i =1 i i
Average lateness; L . .
max
Maximum lateness;
Lmax = 1,...,n {Li}.
Earliness of job I;
Ei = 1,...,n {0, -Li}
max
The total tardiness ; Average tardiness;
n ∑T i=1 i
.
Tardiness of job I; Ti = max {0, Li} 1,...,n
Maximum Earliness; Emax = max {Ei} 1,...,n
The total weighted tardiness; Maximum tardiness;
T.
∑α wE
Absolute Lateness; Li
i
i
n ∑w T i=1 i i
.
Tmax = max {Ti} 1,...,n
+ β iTi
n
Number of jobs tardy NT = ∑ δ( Ti ) , δ(Ti) = 1 if Ti > 0 and δ(Ti) = 0 if Ti ≤ 0. i=1
These criteria might help to make the right decision. According to Backer [4], the goals of decision-making are efficient utilization of machines, rapid response to demand and chose conformance to prescribed deadlines. The efficient utilization of machines might be achieved by associating the criteria of minimizing maximum completion time, average number of machine ideal or by maximizing average number of job completed or average utilization of the machine. The rapid response to demand might be achieved by minimizing summation of jobs completion time, flow time, lateness, summation of jobs to machines waiting time, or by minimizing the average completion time, flow time, lateness and waiting time. The close conformance to prescribed deadlines might be achieved by minimizing the maximum lateness, tardiness,
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summation of jobs’ tardiness, average of tardiness, or summation of waiting time multiplied by tardiness of each job.
1.7. Solution Methodologies for Scheduling Problems Several methods have been developed to solve and model sequencing and scheduling problems that belong to any of the four categories (deterministic, static, dynamic, and stochastic). Alharkan [2] and Morton and Pentico [5] have summarized several approaches to solve scheduling problems. Following is the summarization: Mathematical Program Approach proved its validity over the years. Mathematical program chooses an objective function to optimize, formalize the resources and constrains and then solve the problem by one of mathematical program techniques such as branch and bound. Balas and Gomomory developed a modern integer program in 1960. Later on, Srinivasan and others used dynamic programming for sequencing problems in 1971. The disadvantage of both integer and dynamic programs is the exponential growth in problem size. For this reason various heuristic methods such as neighborhood search begun to be used for approximate solution. Computer Simulation approach is one of the oldest approaches to measure the performance of developed schedules. The advantage of using simulation lies in its experimental flexibility. Simulation is a stochastic technique meaning that it is based on the use of random numbers and probability density function to investigate problems. However, the disadvantage of the simulation approach is that the obtained results are not even approximately optimal. It is more of performance measurement technique than optimization technique. Recently, Meta-heuristic approaches are utilized to decipher scheduling problems. The Meta-heuristic consists of advanced algorithms such as Genetic Algorithms, Simulated Annealing, and Tabu Search. Genetic Algorithms is a search process that is simulating the natural evolutionary theory by which the best solutions are merged to produce new generation of solutions (children) that have a mixed features of the existing solutions (parents). Simulated Annealing is an extension of neighborhood search. It was originated in material science. It is applied as a local search tool to improve the objective function of scheduling problems. Tabu Search is also a neighborhood search
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that has a list of recent search positions. Tabu comes from the fact that these positions may not be repeated while in the active list and the search of improvement is continued in anticipation of finding a local optimum. In addition, Approaches such as Expert Systems, Neural Network, Artificial Intelligence (AI), and Decision Support Systems (DSS) are recently utilized to find solutions of scheduling problems.
Chapter
2
Research Methodology In this chapter, research methodology is shown. First, purpose of the study is introduced. Then, definition of existing environment is represented. After that, mathematical notifications of the research topic for design of experiments, simulation analysis, analysis of variance (ANOVA) and Tukey’s test concepts are illustrated.
2.1. Purpose of the Study The aim of this thesis is to test a method of re-assigning job due-dates at a particular industrial application that has a flexible flow shop environment. Evaluating and comparing variety of due-date setting rules can accomplish this. The selected production environment is having a stochastic behavior and since then simulating the existing system will make the evaluation. In order to utilize the proposed approach a case study will be introduced. The case study will be on a product of the Advanced Electronic Company (AEC). In brief, the case study is implemented on two products of that produced by AEC. The case study is having, in scheduling terminology, 13 machines and 42 operations for the first product and 10 machines and 31 operations for the second product The uncertainty of the selected case study is coming from the order size and the processing time of the operations. However, the process plan of the products are almost the same except some specific details that will be explained later in Chapter 4 and 5. Moreover, the case study is having production capacity constraint and some precedence constraints. The details of the precedence constraints are shown in Chapter 4 and 5. In- addition to that, the objective of re-assigning the due dates is to reduce the absolute lateness in order to complete the production of an order at the agreed target date regard less the earliness or tardiness. The
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goal of selecting the absolute lateness as an objective function is to minimize the holding and penalty costs. The flow of the research methodology is as following. First, experiments will be designed. Second, simulation model will be developed. Third hypotheses will be stated. Fourth testing the hypotheses will be performed by utilizing analysis of variance (ANOVA). Fifth ranking the experiments will be completed by utilizing Tuky’s test. Following is the flow chart of the research methodology.
Design the experiments
Develop a simulation model in order to represents the system to measure the performance of each experiment on the stated objective function
Test the hypotheses to measure the significant between the experiments by using Analysis of Variance and Tukey’s test
Figure 2.1 Flow Chart of the Research Methodology The research study attempts to answer the following questions: 1- Is there a significant difference between due dates setting rules? 2- What is the effect of allowance level k ? 3- Does the due date setting rules and due date factor interact? 4- What is the best combination between due date setting rules and allowance level k ?
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This study designed and developed required components to implement a simulation analysis that investigates and measures the effectiveness of assignments of due dates continuously. However, the methodology of research is distributed among chapters of this thesis as follows. First a literature review will be introduced. The literature review contains a brief history of scheduling theory, survey of dispatching and due date setting rules and preview of pervious related studies. Second, an introduction of the case study including full description of system components and shop layout will be presented. Then, implementation of investigation procedure will be employed. The implementation includes designing of the experiments, determination of system of constrained and assumptions, developing simulation models, and analyzing of variance (ANOVA) and tuck’s test. Finally, recommendations and further future studies will be discussed.
2.2. Scheduling Environment Definition As pre-described in chapter 1 and according to Pinedo [1] scheduling problem may be described in three fields. The triplet is α β γ , where α is the machine environment and contains a single entry, β field provides details of processing characteristic and constraints and may contain no entry at all or multiple entries, γ field describes the objective function and usually contains one entry. Thus, the case study may be presented as following: F13 / prec / Li
Where F13 means that the machine environment follows flexible flow shop environment where the jobs have to precede partially over a series of 13 machines following the same rout. Precedence Constraints ( prec ) means that one or more jobs are depending on other jobs, which implies that it or they cannot be started unless the previous jobs have been completed. Li Is the objective function and it means the absolute lateness. Absolute lateness is the difference between completions time of job ( i ) and the due date of that job. It can be presented mathematically as following: Li = Ci − di
(2.1)
Where Li is the absolute lateness, Ci is the completion time of job i , and di is the due date of job i . Thus, the objective function of the problem is to
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minimize the absolute lateness to be as closest as possible to the due date in order to limit tardiness and earliness.
2.3. Design of Experiments The objective of the research project is to test the significance between several proposed due date setting rules and dispatching rules in order to find best combination that minimizes absolute lateness. Design of experiments concept is an ideal perception to do so. Formally and according to Montgomery [9] experiments can be defined as a test or series of tests in which purposeful changes are made to the input variables of process or system, so that we may observe and identify the reasons for change that may be observed in the output response. The experimentation plays an important role in product activities, which consist of new product design and formulation, manufacturing process development, and process improvement. Design of experiments is a statistical technique that measures the effect of input factors on the response or output factors. However, the general model of process or systems is presented in Figure 2.2.
Effecting Factors … Inputs
Output Process
… Effecting Factors
Figure 2.2 General Model of process or system In our scheduling problem, there are multiple factors and it has been stated by Montgomery [9] that the best approach to dealing with several factors is to conduct a factorial experiment. The advantage of using factorial experiments is that the interaction between the factors can be easily detected.
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The total of factorial experiments is relying on the number of the factors and their levels or treatments. It can be presented mathematically as following: nk
Where k and n are the numbers of the factors and levels respectively.
2.4 Simulation Modeling and Analysis A system can be defined as a collection of entities that react together to provide an explicit conclusion. The system can be a facility or a process. In order to evaluate any type of systems, we often represent it in model format. Analytical solution can evaluate such system, if it is simple enough, by using mathematics concepts such as algebra, calculus, and statistics and probability theories. On the other hand, if the system intends to be large and complex, numerical methods can be utilized for evaluation. Simulation is the methodology to represent the entire system on the computer in order to evaluate it numerically. It is one most used operation research techniques. It has been ranked as one of the three most important techniques in operation research beside mathematical programming and statistics, as per Lane, Mansour, and Hapell survey [10]. Moreover, Grupta has surveyed 1924 journals of Interface and initiated that simulation was second only to mathematical program among 13 different techniques. There are various applications of simulation. It can be applied in manufacturing systems, communication networks, transportation systems and computer systems. In addition, it can be utilized in the process of reengineering process, services organizations, inventory systems as we as economic and financial structure. Simulation model can be developed through stages that are explicated in Figure 2.3. Simulation can be defined as stochastic technique, meaning it is based on the use of random numbers and probability density function to investigate problems [11]. The MCS method has been used in many fields such as, economics, nuclear physics, traffic…etc. Evidently, the way they are applied varies widely from field to field, and there are dozens of subsets of simulation experiments even within any specific filed. However, strictly speaking, to call
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something a simulation experiment, all you need to do is to use random numbers to examine the problem. Simply, in simulation experiments, the random selection process is repeated many times to create multiple scenarios. Each time a value is randomly selected, it forms one possible scenario and gives a solution to the problem. Together, these scenarios give a range of possible solutions, some of which are more probable and some less probable. When repeated for many scenarios [10,000 or more], the average solution will give an approximate answer to the problem. Accuracy of this answer can be improved by simulating more scenarios. In fact, the accuracy of simulation experiment is proportional to the square root of the number of scenarios used. Random numbers can be generated through developed algorithms that can generate a set of numbers from uniform distribution on the interval of 0 to 1. Problem formulation and data collection
Developing the suggested simulation model
Determining the model’s parameters and constructing the computational model in the computer
Verification of the model
No Is the model verified? Yes Validation of the model
No Is the model Valid
Yes Representation of the output in terms of tables & graphs
Figure 2.3 Flow Charts of Simulation Model
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Algorithms Linear Congruential Generators (LCG) [10], Mixed Congruential Generators, Multiplicative Congruential Generators, and Multiple Recursive Generators are widely used to generate numbers that are Independent and Identically Distributed IID and have less pseudo randomness. Many random number generators use LCG that was introduced by Lehmer in 1951 [10]. Even though with few drawbacks LCG has proved over the years that it is a sophisticated generator of random numbers. The algorithm of LCG is as follows: 1- Find the seed for each individual by using: Z i = (aZ i −1 + C )(mod m)
(2.2)
Where zi is the seed for iteration i , m is the modulus, a is the multiplier and C is the increment. 2- Divide (aZi −1 + C ) over m and let zi be the reminder where 0 ≤ zi ≤ m − 1 . 3- Obtain the value of the uniform random number by dividing the seed over the modulus: Ui =
zi m
(2.3)
After that, linking the generated random number to generated random variates can draw simulation conclusion. A simulation that has any random aspect at all must involve sampling or generating random variables from the desired distribution. Nevertheless, the basic ingredient needed for every method of generating random variates from any distribution or random process is a sources of IID U (0, 1) random variates. For this reason it is essential that a statistical reliable U (0, 1) random number generator be available. No need to mention that most of the simulation packages have a convenient random number generator and without availability of acceptable random number generator, it is impossible to generate random variables correctly from any distribution. However, there are many techniques for generating random variates and the particular algorithm used, must be dependent on the distribution from
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which we wish to generate. However, in this section we show the algorithm of generating random variates from normal distribution. For further reading, refer to Simulation Modeling And Analysis for Law and Kelton [10]. The algorithm that presented by Polar [10] for generating random variates for normal distribution can be summarized as following: 1- The expected variable x ' can be obtained by: x ' = μ + σx
(2.4)
Where μ & σ are the mean and standard deviations of the observation respectively and x is the random variable. 2- In order to find x ' , we need to refer to the generated random number U1 and U 2 as IID that follow (0, 1) and then let: V1 = 2U i − 1
∀i = 1,2
(2.5)
and W = V1 + V22 2
(2.6)
3- If W > 1 go to step 2, otherwise go to step 4.
4- Let − 2 ln w w
y=
(2.7)
then, x1 = V1 y
(2.8)
x2 = V2 y
(2.9)
'
'
5- Substitute the values of x1' & x2' in equation (2.4) to estimate the expected variable x ' .
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6- Go to step 2 and repeat the algorithm for a large number of iterations.
2.5. Test of Hypothesis Setting up and testing hypothesis are essential parts of statistical inference. A statistical hypothesis test is a statement either about the parameter of a probability distribution or the parameter of a model. The hypothesis is often about the parameter of population like expected value and variance. Usually, the testes of hypothesis are represented by the null hypothesis ( H 0 ) and alternative hypothesis ( H1 ) where the drawn conclusion is often reject or fail to reject the null hypothesis. An example of the null hypothesis test is hypothesis of two sample means [9] and it is represented as H 0 : μ1 = μ 2 H1 : μ1 ≠ μ 2
The above example is containing two-sided alternative hypothesis. However, if the alternative hypothesis is having greater than or less than signs (>, 1 1
