SCHEDULING ROTARY INJECTION MOLDING MACHINE. A thesis presented to. the faculty of. the Fritz J. and Dolores H Russ

Transcription

1 SCHEDULING ROTARY INJECTION MOLDING MACHINE A thesis presented to the faculty of the Fritz J. and Dolores H Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Shravan B. Urs November 2005

2 This thesis entitled SCHEDULING ROTARY INJECTION MOLDING MACHINE by Shravan B. Urs has been approved for the Department of Industrial and Manufacturing Systems Engineering and the Fritz J. and Dolores H. Russ College of Engineering and Technology by Gürsel A. Süer Professor of Industrial and Manufacturing Systems Engineering Dennis Irwin Dean, Fritz J. and Dolores H. Russ College of Engineering and Technology

3 URS, SHRAVAN B. M. S. November Industrial and Manufacturing Systems Engineering Scheduling Rotary Injection Molding Machine (118 pp.) Director of Thesis: Gürsel A. Süer This thesis focuses on scheduling a rotary injection molding machine at a shoe manufacturing plant. A group of three scheduling heuristics were developed by applying a basic machine scheduling methodology. The three heuristics are: HP-2 (Left/Right Rule), HP-3 (Min Ct Rule) and HP-4 (Left/Right with Min Ct Rule). The objective of these heuristic procedures is to minimize the makespan in the rotary machine scheduling. The heuristics are encoded using Matlab, which is a software application used in engineering and scientific applications. The software application is used to test and compare the performance of the above heuristics with respect to makespan. After performing extensive experimentation using varied sets of data, it is observed that, HP-3 (Min Ct Rule) consistently performed better than the rest and is thus suggested as the heuristic of choice. Approved: Gürsel A. Süer Professor of Industrial and Manufacturing Systems Engineering

4 Dedication I would like to dedicate this thesis to my advisor Dr. Gürsel A. Süer, who has been a constant source of help and support all through my years at Ohio University. Further, I would like to dedicate my thesis to my family and friends for their love and support.

5 Table of Contents 5 Abstract...3 Dedication List of Tables..9 List of Figures...12 CHAPTER 1: INTRODUCTION Scheduling Functions Machine Scheduling Classification of Scheduling Problems with respect to Shop Configuration Classification of Scheduling Problems with respect to Job Arrival Classification of Scheduling Problems with respect to Operation Times Scheduling Solution Procedures Rotary Injection Molding Machine Performance Measures Computer Program - Matlab Objective Justification Sections of the Thesis CHAPTER 2: LITERATURE REVIEW CHAPTER 3: PROBLEM DESCRIPTION A Cell Group Product Family Formation Demand vs. Size Molds Shoe Sizes Injection Time, Cycle Time, and Set-up Time The Rotary Injection Molding Machine Problem Loading of the Rotary Machine (One Complete Turn)... 32

6 6 CHAPTER 4: METHODOLOGY Heuristic Procedure 1 (From Paper Süer [1]) Phase 1 Initial Assignment Phase 2 Interval Identification Phase 3 Scheduling Example Problem Rotary Machine Loading Heuristic Procedure 2 (Left/Right Rule) Phase 1 Initial Assignment Phase 2 Interval Identification Heuristic Procedure 3 (Minimum Ct Rule) Phase 1 Initial Assignment Phase 2 Interval Identification Phase 3 - Scheduling Heuristic Procedure 4 (Minimum Ct Rule with L/R Division) Phase 1 Initial Assignment Phase 2 Interval Identification Heuristics Applied to Multi Products Heuristic Procedure 1 (From Paper Süer [1]) Heuristic Procedure 2 (Left/Right Rule) Heuristic Procedure - 3 (Minimum Ct Rule) Heuristic Procedure - 4 (Minimum Ct Rule with L/R Division) Heuristics Applied to Multi Families Computer Software Software Validation CHAPTER 5: EXPERIMENTATION AND RESULTS Uniform Demands Small Problem Category Medium Problem Category Large Problem Category... 75

7 7 5.2 Demand in Equal Increments Equal Low Increments Small Problem Category Moderate Problem Category Large Problem Category Equal Moderate Increments Small Problem Category Medium Problem Category Large Problem Category Equal High Increments Small Problem Category Medium Problem Category Large Problem Category Increasing Increments Incremental High Demand Incremental Medium Demand Incremental Low Demand High Volume vs. Low Volume Products High Volume Products Low Volume Products High & Low Volume Products Combined Asymmetrical Demand Incremental High with Zero Demand for Larger Sizes Incremental High with Zero Demand for Large & Small Sizes Incremental High with Mixed Zero Demand Comparison with Subramanian s [2] Procedure Cycle Time Variation Increased Number of Pairs of Positions... 98

8 8 CHAPTER 6: CONCLUSIONS AND FUTURE WORK Conclusions Future Work Mold Availability Genetic Algorithms Increased Shoe Sizes Different Injection Times Incorporate Lasting & Finishing Alternate Approach for HP-2 and HP Alternate Approach for HP BIBLIOGRAPHY APPENDIX A APPENDIX B APPENDIX C

9 List of Tables 9 Table 3-1: Shoes sizes & parameters Table 4-1: Data for shoe sizes Table 4-2: Initialization phase Table 4-3: Intervals Table 4-4: Intervals Table 4-5: Left schedule Table 4-6: Right schedule Table 4-7: Final schedule for phase Table 4-8: Intervals Table 4-9: Intervals Table 4-10: Intervals Table 4-11: Intervals Table 4-12: Intervals Table 4-13: Intervals Table 4-14: Three product lines Table 4-15: Left hand part Table 4-16: Right hand part Table 4-17: Combined schedule Table 4-18: Phase -1 result Table 4-19: Multi family data Table 5-1: Uniform demands Table 5-2: Small problem category Table 5-3: Medium problem category Table 5-4: Large problem category Table 5-5: Equal low variance Table 5-6: Small problem category Table 5-7: Medium problem category Table 5-8: Large problem category Table 5-9: Equal moderate increments... 79

10 10 Table 5-10: Small problem category Table 5-11: Medium problem category Table 5-12: Large problem category Table 5-13: Equal high increments Table 5-14: Small problem category Table 5-15: Medium problem category Table 5-16: Large problem category Table 5-17: Incremental high increments Table 5-18: Results - incremental high increments Table 5-19: Data incremental medium increments Table 5-20: Results incremental medium increments Table 5-21: Data incremental low increments Table 5-22: Results incremental low increments Table 5-23: Data high volume Table 5-24: Results high volume Table 5-25: Data low volume products Table 5-26: Results low volume products Table 5-27: Results high & low Table 5-28: Larger sizes zero demands Table 5-29: Results zero demands for large sizes Table 5-30: Zero demands for large and small sizes Table 5-31: Results zero demands for large & small sizes Table 5-32: Mixed zero demands Table 5-33: Results for mixed zero demands Table 5-34: Comparison of procedures Table 5-35: Cycle time variation Table 5-36: Variation in pairs of positions Table 6-1: Analysis section Table 6-2: Analysis section Table 6-3: Analysis section

11 11 Table 6-4: Analysis section Table 6-5: Analysis section

12 List of Figures 12 Figure 1-1: Production System Figure 1-2: Sequential Loading Figure 3-1: A Cell Group Figure 3-2: Demand vs. Size Figure 3-3: Sequential Loading Figure 3-4: Sequential Loading Figure 3-5: Sequential Loading Figure 3-6: Sequential Loading Figure 3-7: Sequential Loading Figure 3-8: Sequential Loading Figure 3-9: Sequential Loading Figure 3-10: Sequential Loading Figure 4-1: Preliminary Phase Figure 4-2: Final Gantt Chart Figure 4-3: Sequential Loading Figure 4-4: Sequential Loading Figure 4-5: Sequential Loading Figure 4-6: Sequential Loading Figure 4-7: Sequential Loading Figure 4-8: Sequential Loading Figure 4-9: Sequential Loading Figure 4-10: Sequential Loading Figure 4-11: Sequential Loading Figure 4-12: Preliminary Gantt Chart Figure 4-13: Final Schedule Figure 4-14: Ct Matrix Figure 4-15: Clustered Matrix 8 8 ½ Figure 4-16: Clustered Matrix 8 8 ½ Figure 4-17: Clustered Matrix 8 8 ½ ½... 48

13 13 Figure 4-18: Clustered Matrix 8 8 ½ ½ Figure 4-19: Clustered Matrix 8 8 ½ ½ ½ Figure 4-20: Clustered Matrix 8 8 ½ ½ ½ Figure 4-21: Initial Gantt Chart Figure 4-22: Final Schedule Figure 4-23: Left Hand Matrix Figure 4-24: Clustered Matrix 8 ½ Figure 4-25: Clustered Matrix 8 ½ Figure 4-26: Right Hand Matrix Containing Larger Sizes Figure 4-27: Clustered Matrix 9 ½ Figure 4-28: Clustered Matrix 9 ½ ½ Figure 4-29: Initial Gantt Chart Figure 4-30: Final Schedule Figure 4-31: Initial Assignment Figure 4-32: Initial Assignment Figure 4-33: Ct Matrix Figure 4-34: Clustered Matrix 9 9 ½ Figure 4-35: Clustered Matrix 9 9 ½ Figure 4-36: Clustered Matrix 9 9 ½ Figure 4-37: Clustered Matrix 9 9 ½ ½ Figure 4-38: Clustered Matrix 9 9 ½ ½ Figure 4-39: Clustered Matrix 9 9 ½ ½ ½ Figure 4-40: Gantt Chart for Initial Assignment Figure 4-41: P3 Left Figure 4-42: P3 Left Figure 4-43: P3 Left Figure 4-44: Initial Assignment Figure 4-45: Family Gantt Chart Figure 5-1: Equal Increments Figure 5-2: Increasing Increments... 84

14 CHAPTER 1: INTRODUCTION 14 In this chapter, the different scheduling concepts are defined and introduced. The various types of scheduling problems are categorized. Next, different types of solution procedures are discussed. Performance measures that are employed in scheduling problems are discussed. A brief description of the rotary injection molding machine is given. The software application that is used to implement the algorithms is described in brief. Later, the objective of the thesis is discussed and justification is made. Finally, chapters of the thesis are summarized. 1.1 Scheduling Functions Machine scheduling is a very important aspect of a manufacturing firm. Scheduling is a decision making process that involves the allocation of resources to activities over time. The process of scheduling requires both sequencing and resource allocation decisions. The function of scheduling is to determine the order of products to be manufactured, and to determine their start times and completion times. Production activities are started when the customer places the order and is completed when the product is completed and shipped. Production control has to plan and control the production activities. The function of production control system is to satisfy the three aspects of demand, namely, quality, cost and timing. Scheduling is the activity that takes care of the timing. There is a planned hierarchy that exists in the decision making process in production. Baker [12], Pinedo [13] and Chao et al. [14] have illustrated the close interaction among production planning, master scheduling, capacity planning, material planning and scheduling. Production planning is the monthly production order of product families. Master production scheduling is the weekly production schedule of individual products. Capacity planning includes determining number of machines needed, manpower requirements, number of plants, number of shifts, etc. Rough-cut capacity planning involves identifying weekly capacity requirements. Material planning involves

15 15 planning of purchased and manufactured materials in synchronous with usage and production plans. The manufacturing system generates the demand by interacting with customers and schedules raw materials from suppliers. The scheduling part comes into picture once the demand has been set and availability of resources has been determined. In order to create real time schedules, there is a constant communication between the scheduling function and the shop floor to consider availability of machines, manpower and raw materials. Figure 1-1 illustrates the relationship chain between the various components of the production system. Resource Planning Production Planning Demand Management Rough-cut Capacity Planning Master Production Schedule Detailed Capacity Planning Detailed Material Planning Material & Capacity Plans Shop Floor Systems Purchasing Figure 1-1: Production System

16 1.2 Machine Scheduling 16 Machine scheduling is categorized based on shop configuration, job arrival and operation times Classification of Scheduling Problems with respect to Shop Configuration Machine scheduling problems can be classified in the following categories based on shop configuration: 1. Single machine: A single machine scheduling problem can be described as the optimal utilization of a single machine in a manufacturing facility to produce multiple products. 2. Parallel machine: In this type of configuration there are multiple machines that are similar (identical, uniform or unrelated) and each job requires one machine to be completed. 3. Flowshop: Flow shop can be described as a series of machines that are usually dissimilar where the output of a particular machine becomes the input for the successive machine. All jobs have unidirectional flow. 4. Jobshop: Job shop can be described as a shop floor consisting of a number of machines, where each machine is utilized to perform a specific function. In this configuration jobs follow different machine sequences. 5. Assembly Lines: Assembly lines can be described as a series of stations that performs functions in an order that is required to produce one single product at a time. 6. Flexible Manufacturing Systems: This type of a shop configuration consists of highly automated machine centers. Machines are capable of running different products and perform multiple operations. 7. Cellular Manufacturing: Here, machines are grouped together as cells in order to perform a specific group of functions usually in order to manufacture a certain family of products. The scheduling problem associated with this thesis can be considered as a special case of the Parallel Machine Scheduling problem. Even though the rotary injection molding machine is a single machine, it behaves like a parallel machine. Therefore, scheduling the rotary injection machine is more of a parallel machine scheduling problem.

17 1.2.2 Classification of Scheduling Problems with respect to Job Arrival 17 Scheduling problems can also be classified based on job arrival: 1. Static: A scheduling problem is said to be static if the number of jobs and their ready times are known and remain constant. 2. Dynamic: If the arrival of jobs is random, the problems are said to be dynamic. In this thesis, jobs will be assumed to be known in advance; therefore it is a static scheduling problem Classification of Scheduling Problems with respect to Operation Times Scheduling problems are classified as deterministic or probabilistic based on processing and set-up times. 1. Deterministic: Scheduling problems are said to be deterministic if the processing times and set-up times are known and remain constant. 2. Probabilistic: If either the processing times or the set-up times are uncertain, the problem is said to be probabilistic. The rotary injection molding machine problem is deterministic in nature since, the processing times and set-up times are known and remain constant Scheduling Solution Procedures The solution procedures can be classified as: 1. Optimizing: The optimizing procedures can be described as those that guarantee an optimal solution. Optimizing procedures are only viable in solving small scale problems. 2. Heuristics: Heuristic procedures can be described as those that produce optimal or a near optimal solution. However, they do not guarantee optimal solutions. They are developed for a specific type of problems. 3. Meta-Heuristics: These procedures can be applied to a broader scope of problems. They include Genetic Algorithms, Tabu Search and Simulated Annealing among others. In this thesis, heuristic solution procedures are proposed to solve the problem in question.

18 1.3 Rotary Injection Molding Machine 18 This is a real life problem associated with Timberland Inc., a shoe is manufacturing plant located in Puerto Rico. In the company s shop floor, there are six manufacturing cells that contain one rotary injection molding machine per cell. This thesis focuses on scheduling various product families assigned to a single rotary injection molding machine. A single rotary injection molding machine produces shoes of different types. There are (n) pairs of positions available on the rotary injection molding machine. Shoes/Jobs are sequentially loaded on to the (n) pairs with the help of the rotational capability of the machine (refer to Figure 1-2). Thus, several shoes/jobs are being produced at any given point of time. This aspect of the rotary injection molding machine makes it to behave like a parallel machine. The scheduling of this machine thus resembles a parallel machine scheduling problem. In a parallel machine scheduling problem, each machine is a separate entity. The machines are independent of each other. In a rotary machine, the positions are dependent on each other. The restrictions in a rotary machine are that only similar products can be produced at a time. Also, if the machine stops, production at all positions come to a stop. 0 = Operator position Load size X (left & right) on positions 1& 2 belonging to Pair Figure 1-2: Sequential Loading

19 1.4 Performance Measures 19 Performance measures are parameters which are used to measure the performance of the manufacturing system. Some of the performance measures utilized in the scheduling literature are: 1. Makespan: can be defined as the maximum of completion times. 2. Tardiness: Tardiness can be defined as the amount of time the job is late. 3. Flowtime: Flowtime is defined as the amount of time the product stays in the system before completion. 4. Waiting Time: Waiting time is defined as the amount of time the product has to wait in between operations on two different machines. The performance measure of interest in this thesis is Makespan. The main aim is to determine an optimal sequence of loading that will result in minimum makespan. Makespan was chosen as the measure of performance in this thesis because efficient machine utilization is extremely critical and lower makespan results in better utilization of the rotary machine. The makespan for the production of one particular family of products will affect the completion times of the next series of families to be produced. 1.5 Computer Program - Matlab Matlab is an engineering tool and analysis software from Mathworks Inc. It is built using the Java programming language. It contains many essential built-in functions that make writing programs and executing algorithms very efficient. It utilizes matrix based computations for scientific and engineering use. The most important aspect of Matlab that made writing the code efficient and less cumbersome is the relative use of the concept of pointers. Since the problem at hand consists of data that needs to be referenced and cross references in many steps of the algorithm, writing the code in languages such as C, would prove to be more cumbersome and time consuming. Matlab s in-built functions handle the referencing of variables in an efficient and easy manner. 1.6 Objective The objective of this thesis is to develop heuristic procedures in order to minimize the makespan in scheduling the Rotary Injection Molding Machine and compare their

20 performance. A software program is developed using Matlab to accomplish this task. The procedures have been compared by using various data sets. Extensive experimentation has been conducted in order to determine trends in the behavior of procedures under different conditions Justification This particular rotary injection molding machine problem is found to be unique in the sense that, this is a single machine scheduling problem that behaves similar to a parallel machine scheduling problem. It is found that there hasn t been any significant research work done in the past but for two cases. A paper by Süer [1] is the first work done in order to solve such a scheduling problem. Subramanian [2] proposed a scheduling procedure for the rotary injection molding machine. The procedure assumes no mold restrictions for all shoe sizes. The procedure attempts to schedule a particular shoe size in every position of the rotary machine. Once the demand has been met, the next shoe size is scheduled in every position. This method seems to be impractical in a real life scenario, since it would require too many molds of each type to be available in the system. This thesis is an extension to the work done by Süer [1]. These new heuristic procedures that are a part of this thesis focus on producing better results in terms of minimizing the makespan. 1.8 Sections of the Thesis This thesis starts with a basic introduction to machine scheduling. The important aspect that makes the Rotary Injection Molding machine problem, different from conventional single machine and parallel machine scheduling problems are briefly described. Chapter 1, defines and states the performance measure used in the proposed methodologies. The objective of this thesis is discussed. A brief description of the software program is discussed. The main purpose of the software program is to aid as an essential tool to conduct experimentations efficiently and consistently. Chapter 2 provides a review of previous literature related to machine scheduling and minimizing makespan. Chapter 3 presents the problem statement and describes the basic functioning

21 of the rotary injection molding machine. Chapter 4 presents the methodologies which have been used in order to determine possible solutions to the problem. Chapter 5 presents the results of the experimentation and provides an analysis of the results. In Chapter 6, the conclusions and future work that can be done to improve the proposed procedures is discussed. 21

22 CHAPTER 2: LITERATURE REVIEW 22 This chapter presents a summary of research done on topics relevant to parallel machine scheduling and minimizing makespan on parallel machines. Considerable amount of work has been done in scheduling parallel and identical machines with respect to minimizing makespan. The scheduling problem pertaining to this thesis is a unique one. Previous research work on this type of a scheduling problem has been attempted by Süer et al. [1] and Subramanian [2]. Min and Cheng et al. [3] attempted to minimize the makespan in an identical parallel machine problem. In order to solve a large scale identical machine scheduling problem, the authors used a Genetic Algorithm based on machine code in order to minimize makespan. The authors used several scale numerical examples to prove that genetic algorithm works very efficiently for large scale identical parallel machine scheduling problem for minimizing the makespan. The authors also suggest that the quality of the solutions is better than those of heuristic procedures and simulated annealing. Haouari and Gharbi [4] considered minimizing the makespan on identical machines subject to release dates and delivery dates. The authors have developed branch and bound algorithms to solve this problem. The authors developed a preprocessing algorithm to accelerate the convergence of the proposed algorithms. The search tree in the branch and bound algorithm is also reduced using polynomial selection algorithm. Zhang, Li and Li [5] analyzed a problem of solving n jobs on m identical parallel batch machines. The authors state that each job in the batch has a release time and processing time. The authors attempted to schedule jobs in batches that result in having the same starting and completion times. Thus, the processing time in a batch is determined by the largest processing time of any job in the batch. Cheng and Kovalyov [6] have studied a problem of scheduling n independent jobs on m unrelated parallel machines. A particular job has a processing time and a deadline. The authors grouped the jobs in batches that contain continuous scheduled jobs. The completion time for the batch depends on the completion time of the last job in the

23 23 batch. The authors consider a constant setup time before the first job of the batch has been processed. As an example the authors considered the production of different part types by several machining centers working in parallel in a machine shop. All part types in a single batch share the same completion times. Shin and Kim et al. [7] have presented a restricted Tabu search algorithm that schedules jobs on parallel machines in order to minimize maximum lateness of jobs. The jobs are said to have release times, due dates and sequence dependant setup times. The authors claim that the restricted search algorithm reduces search efforts and produces efficient final schedules. The authors also claim that the proposed algorithms produces better solutions than simulated annealing and heuristic procedures such as Rolling Horizon Procedure. Lin and Liao et al. [8] have considered a two uniform parallel machine problem in order to minimize makespan. The authors have transformed the problem into a special problem that that considers work load instead of completion times. An optimal algorithm is developed for he transformed problem. The authors claim that this algorithm can generate optimal solutions for large sized problems in a short amount of time. Dhaenens-Flipo and Dupont et al. [9] have analyzed a batch processing machine that can simultaneously process several jobs forming a batch. The authors have considered the problem of scheduling jobs with non-identical capacity requirements, on a single batch processing machine to minimize makespan. The authors have used a branch and bound technique to solve this problem. Each job is said to have a specific capacity requirement and the batch size is determined by the capacity that the machine can handle. Kempf, Uzsoy and Wang et al. [10] have studied the problem of minimizing total completion time and makespan on a single batch processing machine with job families and secondary resource constraints. The authors discuss that there are n simultaneously available jobs on a single batch machine. Each job is said to require one position and job processing times are deterministic. The jobs have been divided into families such that all jobs of a family have the same processing time. Once the batch processing begins, it can not be interrupted to add or remove jobs until the processing time for the batch has been completed.

24 24 Ruiz-Torres and Gupta [11] have considered an identical parallel machine problem of minimizing makespan subject to minimum flow time. The authors have proposed an algorithm where an equivalent makespan problem is created with n jobs. A list of jobs is created by combining two sub-lists, A and B. The job assignment to the two sub-lists is iterative. The jobs are scheduled using LPT or SPT rule. Süer [1] developed a heuristic procedure in order to minimize the make-span in the rotary injection-molding problem. This particular procedure is discussed in brief. The procedure is a 3-phase heuristic procedure. The first phase is the Initial Assignment phase. The Largest Number of Turns rule is used which is very similar to the Largest Processing Time (LPT) rule. Since the cycle times are not yet known, jobs or shoe sizes are sorted based on largest number of turns required to complete the injection process. The initial assignment is done by the minimum load rule. The second phase is the Interval Identification phase. The injection times and cycle times are a function of the size of the shoe. Hence the cycle time used to set the machine depends on what other sizes have been assigned to the machine at the same time. The largest of the injection times for the sizes will become the cycle time for the machine and thus the processing time is computed. The first sizes assigned to each position are considered and set S is formed. The minimum of total turns on all positions in set S is determined and the first interval is thus found. Similarly the subsequent intervals are determined by including the next shoe size assigned to the set. In the third phase the completion times are computed for all the intervals by adding the setup times. After computing the completion times, the final schedule is determined. Lai et al. [15] have considered a general scheduling problem with the objective of minimizing makespan under uncertain scheduling environments. The authors have assumed the processing time of an operation to be a function of a known probability distribution function. The scheduling environment is assumed to be uncertain. The processing time is only known as an upper and lower bound of the function. The authors

25 25 have presented an approach based on an improved stability analysis of an optimal makespan schedule. Goemans [16] has considered a scheduling problem with the objective of minimizing makespan on three dedicated parallel machines. Three machines are considered with a set of n jobs. The author has assumed that there is no idle time preceding a job on all machines. The processing time is thus the total completion times irrespective of the schedule of jobs. The approximation algorithm presented chooses the best schedule out of eighteen schedules. An approximation algorithm is proposed that optimizes the objective function of minimizing the makespan time between jobs. Mokotoff [17] has considered a deterministic scheduling problem of minimizing the makespan on identical parallel machines. The author has analyzed the problem which is said to have a polyhedral structure. The author has also identified inequalities in for fixed values of the makespan. The author has used linear programming relaxations that are iteratively solved until an integer solution is obtained. The author has used several constructive rules in the computation of the upper bound values. The algorithm uses a preprocessing phase in which the problem is reformulated and simplified. The author has shown through several computations that the proposed algorithm gives an optimal makespan in the majority of the tested cases. Chang et al [18] consider the objective function of minimizing the makespan for machines that can process multiple jobs with non identical processing times at the same time as a batch. The objective function comprises of two parts namely assigning batches and scheduling them on parallel processing machines. An approach similar to the simulated annealing process of metallurgy has been considered. Significant results were obtained for multiple batches of problem adding complexity. The problem can also be expanded to include other important job attributes, making the model more practical. Subramanian [2] attempted to solve the scheduling problem as a part of the cell scheduling problem. The author assumes unlimited availability of the molds for every size of the shoe. An algorithm was developed wherein a particular shoe size of a certain demand is manufactured continuously until the demand is met. The next shoe size is assigned to all positions of the rotary machine and continuously produced until its

26 26 demand is met. This method is not feasible in an actual industry setting because mold availability is always limited. Caffrey et al. [20] attempted to reduce the makespan in a five machine scheduling problem. The authors decided to limit the size of the problem to a five job, five machine scheduling problem. The authors state that the problem tends to grow exponentially as the number of jobs or number of machines increase. Every job is said to have is own distinct processing time. This was done with the intention of accommodating all possible sequences in a reasonable computation time. Lin et al. [21] attempted to schedule an identical parallel machine with an objective to minimize the makespan as well as the flowtime. The lower bound calculation and the job replacement rule were employed in order to increase computational efficiency. The problem can be considered as n jobs on m identical machines. Each job is said to have its own fixed processing time. The authors contend that even though the algorithm has an exponential time complexity, it is quite efficient in finding the optimal solution. Further they argue that heuristic procedures can only be used to solve scheduling problems of a small scale. Genetic algorithms are thus suitable for large scale scheduling problems because of characteristics such as near optimization, high speed and easy realization. Also, the authors suggest that the quality of the solution to the scheduling problem is better than LPT or simulated annealing. Baker et al [22] have considered a scheduling problem wherein a single processor was utilized by two or more customers. The authors have examined the results of minimizing an aggregate scheduling objective function in which jobs belonging to various customers are evaluated based on their respective criteria. The three scheduling criteria examined are minimizing makespan, minimizing maximum lateness and minimizing total weighted completion times. The authors have demonstrated that the problem tends to become NP hard while determining a minimizing a set of all criteria considered simultaneously. Gupta et al. [23] have considered solving a scheduling problem on two identical parallel machines in order to minimize makespan and total flowtime. A multifit algorithm is described to find a solution in polynomial computational times. The algorithm has two

27 phases. The first phase generates an initial solution for the upper bound. The second phase searches for an optimal solution. The authors have compared the results of the multifit algorithm with some known heuristics and shown that the multifit algorithm provides an efficient solution to the scheduling problem. 27

28 CHAPTER 3: PROBLEM DESCRIPTION 28 Timberland Inc. is a shoe manufacturing plant located in Puerto Rico. The problem outlined in this chapter describes the specific scheduling problem experienced in the Timberland plant. 3.1 A Cell Group The Timberland plant consists mainly of six connected cell groups (Figure 3-1). Each cell group consists of a Lasting Cell, a Rotary Machine Cell and a Finishing/Packaging Cell. The Lasting Cell prepares the shoes for injection molding. The Rotary Machine Cell consists of six pairs of stations that, at a time process one pair of shoes each. The Finishing/Packaging Cell removes excess material from the sole, performs finishing operations and packs the shoes for delivery. Each cell group functions independently from the others. Lasting Cell Rotary M/C Finishing/ Packaging Cell Figure 3-1: A Cell Group It is assumed that cell loading has already been done, thus the order of products to be processed in each cell group has been determined. This thesis does not consider product or family sequencing decisions on Lasting and Finishing Cells. The procedures discussed in this thesis determine family sequence and product sequence in each family that go into scheduling the Rotary Injection Molding Machine.

29 3.2 Product Family Formation 29 Product families are determined based on the color and material of the shoes. The colors available are black, honey and nicotine. The two types of materials used are PVC and TPR. An example of different product families is shown below. Family 1 consists of Product lines P1, P2 & P3 P1: This line consists of men s formal shoes of color Black and material PVC. P2: This line consists of men s semi-formal shoes of color Black and material PVC. P3: This line consists of men s casual shoes of color Black and material PVC. Family 2 consists of Product lines P4 & P5 P4: This line consists of women s formal shoes of color Nicotine and material TPR. P5: This line consists of women s casual shoes of color Nicotine and material TPR. Family 3 consists of Product lines P6, P7 & P8 P6: This line consists of men s hiking shoes of color Black and material TPR. P7: This line consists of men s dress shoes of color Black and material TPR. P8: This line consists of men s boots of color Black and material TPR. 3.3 Demand vs. Size The demand of the shoe is related to the size. It is observed that the larger and smaller sizes have lesser demand compared to the medium size shoes. The curve for Demand vs. Size follows a normal distribution as shown in Figure 3-2. Demand vs. Size Demand Volume Series1 Shoe Sizes Figure 3-2: Demand vs. Size

30 3.4 Molds 30 Every shoe size has its own distinct mold. The pigment to manufacture the shoe has to be injected into the particular mold. The size of the mold depends on the shoe size. Thus, larger shoe sizes require larger molds. 3.5 Shoe Sizes Shoe sizes can vary from smaller sizes to larger sizes. The scope of this thesis limits the sizes of shoes from size 71/2 through size 11. Also, it is assumed that the mold sizes and injection times for similar sizes of shoes for women and men remain constant for that particular size of the shoe. Table 3-1 shows the demand (Di), upper sole availability (Ui), outer sole availability (Oi) and Injection times (Iti) for shoe sizes 71/2 through 11. Sizes 71/2 8 81/2 9 91/ /2 11 Di Ui Oi Iti Table 3-1: Shoes sizes & parameters 3.6 Injection Time, Cycle Time, and Set-up Time Injection time is defined as the time required for injecting the pigment inside the mold. The injection time is a function of the size of the shoe. Larger shoe sizes require larger injection times, since the injection time is a function of the mold size. The cycle time is set to the maximum of injection times. When a mold change occurs due to a change in shoe size, set-up times have to be introduced. The new molds have to be incorporated into the rotary machine. Each family change comes with change in colors and/or change in materials. Therefore the rotary machine has to be cleaned so as to not contaminate the next set of shoes that needs to be produced. The set-up times include

31 31 mold changing time, tank cleaning time and any other time necessary to have a functional mold for the next shoe size. It is assumed for the purpose of all calculations in this thesis that the setup time takes 15 minutes. 0 = Operator position Figure 3-3: Sequential Loading Consider the rotary machine with six positions (refer to Figure 3-3). Positions 1 & 2 contain sizes 71/2 (left & right) with injection time of 16 seconds, positions 3 & 4 contain size 9 (left & right) with injection time of 21 seconds, positions 5 & 6 contain size 81/2 (left & right) with injection time of 19 seconds. Thus, the cycle time is calculated as the maximum of the injection times of sizes 71/2, 9 & 81/2, that is, cycle time is 21 seconds. 3.7 The Rotary Injection Molding Machine Problem A description of the existing problem is given below: The rotary injection-molding machine has (n) pairs of positions available. Therefore, at any given time, there are (n) pairs of shoes/jobs loaded on the machine. The shoes are loaded with the help of the rotational capability of the molding machine. The activities that go into loading the machine are given as follows: 1. Remove previously injected shoe 2. Place it on the racks 3. Get new upper from the racks 4. Place it on the mold 5. Inject bottom and rotate the table

32 32 The machine rotates every cycle time. The machine is made to stop when there is a mold change due to the size change. The aim is to produce all sizes as early as possible i.e. to minimize make-span. 3.8 Loading of the Rotary Machine (One Complete Turn) The sequential loading of the rotary machine is illustrated with the help of Figures 3-4 through This is the description of one complete cycle. 1. Time now = 0 0 = Operator position Load size 8 ½ (left) on position 1. Inject the material into the mold Figure 3-4: Sequential Loading Time now = 20 (seconds) 0 = Operator position Now, the machine is rotated anti-clockwise to get position 2 to operator position. Load size 8 ½ (right) on position 2. Inject the material into the mold. 5 Figure 3-5: Sequential Loading -2

33 3. Time now = 40 (seconds) 0 = Operator position Now, the machine is rotated anti-clockwise to get the position 3 to the operator position and size 9 (left) is loaded on to position 3. The molds in positions 1 and 2 are cooling off Figure 3-6: Sequential Loading Time now = 60 (seconds) 0 = Operator position Now, the machine is rotated anticlockwise to bring position 4 to operator position. Size 9 (right) is loaded on position 4. The previous sizes are now cooling off Figure 3-7: Sequential Loading Time now = 80 (seconds) 0 = Operator position Now, the machine is rotated anti-clockwise to bring position 5 to the operator position. Size 9 ½ (left) is loaded to position Figure 3-8: Sequential Loading -5

34 34 6. Time now = 100 (seconds) 0 = Operator position Now, the machine is rotated anti-clockwise to bring position 6 to the operator position. Size 9 ½ (right) is loaded to position Figure 3-9: Sequential Loading Time now = 120 (seconds) O = Operator position Now, the machine is rotated anti-clockwise back to the initial position. Position 1 is in the operator position. Remove the shoe size 8 ½ and inject material into same mold to make next shoe Figure 3-10: Sequential Loading -7

35 CHAPTER 4: METHODOLOGY 35 This chapter discusses the heuristic procedures that have been employed to solve the scheduling of the rotary injection molding machine. The first procedure, Heuristic procedure 1 (HP-1), is taken from Süer et al. [1]. Heuristic procedures - 2, 3 & 4 have been developed in order to find a better solution compared to HP Heuristic Procedure 1 (From Paper Süer [1]) The procedure is applied to a single product line consisting of shoe sizes 71/2 through 11. It is assumed that a single mold is available for each size. A three phase heuristic procedure is developed Phase 1 Initial Assignment The production quantity Ti is determined by considering the minimum of demand (Di), outsole availability (Oi), and upper availability (Ui). The jobs are sorted based on largest number of turns. Sizes are assigned to different positions based on minimum load. The procedure sorts jobs based on largest number of turns. The number of turns cannot be converted into processing times yet. The data is given below in Table 4-1 taken from Süer [1]. Sizes 71/2 8 81/2 9 91/ /2 11 Di Ui Oi Ti Iti Table 4-1: Data for shoe sizes

36 36 The steps of phase 1 are: 1. Determine the number of turns of each size, Ti = min (di, ui, oi). 2. Sort sizes based on largest number of turns T [1] T [2] T [3] where T [i] is the number of turns for the product in the ith order. 3. Choose the next size from the list and assign it to the position with minimum number of turns. 4. Continue until all sizes are assigned, and then go to Phase Phase 2 Interval Identification In this phase, the intervals are identified for different cycle times. The injection times and cycle times are a function of the size of the shoe. Hence, the cycle time used to set the machine depends on what other sizes have been assigned to the machine at the same time. The largest of the injection times for the sizes will become the cycle time for the machine and thus the processing time is computed. The steps for phase 2 are: 5. Consider the first sizes assigned to each position and form set S. 6. Determine the minimum of total turns on all positions in set S. Lmin = min (Li), where Li is the load on position i. 7. The first interval is determined as (0, Lmin). 8. Eliminate the size with minimum total turns from the set. 9. Include the next size assigned in the same position in the set. 10. Similarly determine the rest of the intervals Phase 3 Scheduling 11. Consider all sizes in the 1st interval and choose the max of injection times as the cycle time (Ct). 12. Determine the processing time of that interval (Pt = TT*Ct*2*n), where n=number of pairs of positions, where TT = number of turns in that interval. 13. Repeat the procedure for all intervals and compute all processing times. 14. Add setup times and complete the schedule.

37 4.1.4 Example Problem 37 The heuristic procedure described in Sections is illustrated with an example problem. Consider the data given in Table 4-1 of section Phase 1: Table 4-2 gives the steps of the initialization phase. The procedure followed is stated in Section The Gantt chart for the preliminary phase is shown in Figure 4-1. Step 81/2-9-91/ /2-101/2-11 Size Position Total turns 1 81/ / / / Table 4-2: Initialization phase Size 81/2 9 91/ Turns 8 71/ / Figure 4-1: Preliminary Phase

38 38 Phase 2: In phase-2, intervals are identified by determining the minimum of total turns on all positions in set S. The intervals are shown in Tables 4-3 and 4-4. Step Total turns (TT) 190,180, ,180, ,280, ,310,315 Min TT Interval TT Sizes 81/2,9,91/2 81/2,9,8 81/2,8,10 8,10,71/2 It (Injection times) 19,21,22 19,21,18 19,18,24 16,18,24 Ct Completion times Table 4-3: Intervals 1-4 Step Total turns (TT) 310,315, ,415, , Min TT Interval TT Sizes 10,71/2,101/2 71/2,101/2,11 71/2,101/2 11 It (Injection times) 24,16,25 16,25,27 16,25 27 Ct Completion times Table 4-4: Intervals 5-8 Min TT = Minimum of Total Turns. Ct = Cycle Times (Maximum of Injection Times).

39 39 Phase 3: In this phase, completion times are computed for all the intervals by adding the setup times. After computing the completion times, the final schedule is determined. In the first set in the table, we consider sizes 8 ½, 9, 9 ½. The first interval is calculated as follows: Consider the minimum of the total turns, i.e Consider the maximum of the 3 injection times, i.e. 22 (seconds). The number of positions is equal to 2*3 = 6. The completion time for the first interval is 140*22*6 / 60 = 308 (minutes). Similarly the rest of the intervals are calculated: Interval minutes Interval = 407 minutes Interval = 446 minutes Interval = 667 minutes Interval = 767 minutes Interval = minutes Interval = minutes Interval = 1116 minutes Final schedule is shown with completion times in Figure 4-2. HP-1 gives a make-span of 1116 minutes. Size 91/2 81/ Time (minutes) / / Figure 4-2: Final Gantt Chart 4.2 Rotary Machine Loading The sequential loading of the rotary machine for the example problem discussed in Section is shown in Figures 4-3 through 4-11.

40 40 1. Time, t = 0 Load size 8 ½ on positions 1 and Similarly load size 9 to position 3 and 4, and size 9 ½ to positions 5 and 6 by rotating the rotary machine. Figure 4-3: Sequential Loading 2. Time, t = 308 (minutes) Now, the machine has completed 140 pairs of sizes 8 ½, 9, and 9 ½. Remove size 9 ½ and load size 8 on to positions 5 and Load size 8 on positions 5 and 6. Figure 4-4: Sequential Loading 3. Time, t = 407 (minutes) Now, the machine has completed 180 pairs of sizes 8 ½, and 9, and 40 pairs of size 8. Remove size 9 and load size 10 on to positions 3 and Load size 10 on positions 3 and 4. Figure 4-5: Sequential Loading

41 4. Time, t = 446 (minutes) Now, the machine has completed 190 pairs of size 8 ½, 50 pairs of size 8, and 10 pairs of size 10. Remove size 8 ½ from positions 1 and Load size 7 ½ on positions 1 and 2. Figure 4-6: Sequential Loading 5. Time, t = 667 (minutes) Now, the machine has completed 80 pairs of size 8, 40 pairs of size 10 and 30 pairs of size 7 ½. Remove size 8 from positions 5 and Load size 10 ½ on positions 5 and 6. Figure 4-7: Sequential Loading 5. Time, t = 767 (minutes). 1 Now, the machine has completed 30 pairs of size 10 ½, 70 pairs of size 10, and 60 pairs of size 7 ½. Remove size 10 from positions 3 and Load size 11 on positions 3 and 4. Figure 4-8: Sequential Loading