A SINGLE MACHINE SEQUENCING PROBLEM WITH MULTIPLE CRITERIA. by AVINASH G. TILAK, B. Tech. A THESIS IN INDUSTRIAL ENGINEERING

Transcription

1 A SINGLE MACHINE SEQUENCING PROBLEM WITH MULTIPLE CRITERIA by AVINASH G. TILAK, B. Tech. A THESIS IN INDUSTRIAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN INDUSTRIAL ENGINEERING Approved Accepted August, 97

2 ACKNOWLEDGEMENTS I wish to express my appreciation to Dr. Shrikant S. Panwalkar for his direction and guidance through all phases of this research. I would also like to thank the other members of my committee, Dr. Richard A. Dudek, Dr. Milton L. Smith and Dr. James E. Archer for the suggestions given especially in the final phase of this thesis. My special thanks are extended to Anheuser-Busch, Inc. for providing the data needed for this research. Finally, I wish to thank the National Science Foundation which provided a partial financial support of this research under Grant Number GK-9 A#l in the form of a research assistantship and the large amount of computation time necessary for this research.

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES ii V I. INTRODUCTION. Definition and Classification of the Scheduling Problem. Use of Multiple Criteria in Scheduling Research. Rational Behind the Approach Taken in This Research 7. Purpose of This Research 0. Outline of Succeeding Chapters... II. PROBLEM DESCRIPTION. Definition of Various Terms and the Description of the Problem.... Analysis of Available Data 0. Literature Survey 0 III. EFFECT OF MULTIPLE CRITERIA ON TOTAL COST.... Introduction. Scheduling Rule to Minimize Changeover Cost 9. Scheduling Rule to Minimize the Penalty Cost. Description of the Sequential Evaluation Methods. simulation Results t

4 IV. RESULTS OF THE STUDY OF DUE DATE ASSIGNMENT METHODS. Introduction. Development of Due Date Assignment Rule A and Simulation Results Development of the Due Date Assignment Rule B 0. Simulation Results of the Due Date Assignment Rule B V. CONCLUSIONS AND RECOMMENDATIONS 7. Conclusions 7. Recommendations 7 REFERENCES APPENDIX IV

5 LIST OF TABLES Table Page.. PROBABILITY DISTRIBUTION OF HAVING n JOBS IN AN ORDER.. PROBABILITY DISTRIBUTION OF DUE DATES.. PROBABILITY DISTRIBUTION OF COMBINATIONS..... CHANGEOVER COST MATRIX FOR LINE.. JOB LISTING FOR THE SAMPLE PROBLEM.. SCHEDULE FOR THE SAMPLE PROBLEM BY METHOD C-P 7.. RESULTS FOR THE SAMPLE PROBLEM BY METHOD P-C, AFTER APPLICATION OF THE PRIMARY CRITERION.. SCHEDULE FOR THE SAMPLE PROBLEM BY METHOD P-C 9.. RESULTS OF SIMULATION WITH RANDOM DUE DATES..... DUE DATE DISTRIBUTION FOR THE SIMULATION RUNS WITH RANDOM DUE DATES.. IMPORTANT RESULTS FROM TABLE.... RESULTS OF SIMULATION WITH DUE DATE ASSIGNMENT RULE A 9.. DUE DATE DISTRIBUTION BY RULE A 0.. DUE DATE DISTRIBUTION OF JOBS FOR RULE B DUE DATE ASSIGNMENT RULE B., RESULTS OF SIMULATION WITH DUE DATE ASSIGNMENT RULE B 7

6 .. DUE DATE DISTRIBUTION BY RULE B 7.. IMPORTANT RESULTS FROM TABLE.. 7 VI

7 CHAPTER I INTRODUCTION. Definition and Classification of the Scheduling Problem "The sequencing problem is the problem of defining order (rank, priority and the like) over a set of jobs (tasks, items, commodities and the like) as they proceed from one machine (processor, facility, operation and so on) to another or over the same processor. Thus the sequencing problem involves the determination of the relative position of job j to all other jobs" []. The problem arises due to the limitation on the availability of the resources to do the given tasks. The sequencing problem is of common day to day occurrence. Examples of sequencing can be found in schools (sequencing of classes to classrooms), hospitals (sequencing of patients to test facilities), airports (sequencing of airplanes to runways), transportation (sequencing of cargo to a number of trucks). However, the analytical research in the area of sequencing has been reported only since 9 []. Generally, researchers divide the scheduling problem into a number of categories, depending on the problem

8 characteristics. For example, scheduling may be distinguished by one or more of the following: ) a single processor versus more than one processor. ) a unique flow pattern for each job versus identical flow pattern for all jobs. ) a fixed, finite number of jobs to be processed on various machines versus jobs arriving at the shop in a continuous fashion. ) deterministic information relative to jobs and machines versus one or more of the events involved in a probabilistic manner. Elmaghraby [] classifies the scheduling problem as shown in Figure... Within Elmaghraby's framework of classification, the situation considered in this research is the one, in which a number of jobs are to be scheduled on a single machine and in which some of the jobs have to meet different but specified due dates. At a first glance, the one machine sequencing problem appears to be too trivial to merit any attention, but the study of the single processor problem is important for at least two reasons. fl ) Many complete operational systems can be \\ modeled as single processor systems. Examples are paint manufacturing, canning and

9 Operational System Static Shop Dynamic Shop Single Processor Multiple Processor Jobs with Due Dates Jobs without Due Dates In Series Hybrid System Fig.... Classification of the scheduling problem. bottling processes, and data processing systems. ) There is always the hope that the study of the single processor situation may shed light on the more complex multi-processor situation. Several different criteria have been used by researchers for the evaluation of the one machine sequencing situation. Some of these criteria are as follows: Minimize ) maximum tardiness ) weighted sum of completion times ) weighted sum of tardiness I I ); total cost of tardiness ) total penalty for 'related' jobs

10 ) total setup time (or cost) 7) total cost of processing for single state-variable systems ) number of changeovers 9) cost of changeovers Research in the one machine sequencing situation has been quite extensive. Methods have been developed to generate optimum sequences with each of the above criteria under simplifying assumptions. However, no one has attempted to devise a sequencing system which combines the above criteria, operationally, to establish a total 'payoff function []. This research considers the effect of applying multiple criteria, sequentially, to a single machine sequencing problem.. Use of Multiple Criteria in Scheduling Research In 99 and 97, the Sequencing Research Group at Texas Tech University conducted a survey of industrial scheduling problems. Their findings have been reported in reference, One of the conclusions they reached is stated as follows:,,, the schedulers are faced with satisfying a multiple number of criteria. Meeting due dates ranked as the most prevalent criterion to be satisfied, while other criteria such as minimizing setup time/cost, inprocess inventory and makespan were considered about equal in importance with the latter being utilized the least.

11 Various approaches have been proposed to tackle the problem of multiple criteria. Some of these approaches are as follows [] :. Composite evaluation approach. Various weights are attached to each single measure, such that the combination of various measures results in an overall or grand criterion. For example, total cost may be considered as a combination of job waiting cost, penalty cost, machine idle cost, setup/changeover cost with equal weights attached to each measure,. Sequential evaluation approach. The evaluation proceeds in such a manner that each measure is treated individually, by selecting a primary criterion, secondary criterion and so on. Each criterion is optimized, one at a time, within the framework of the higher level criteria already evaluated and ignoring the lower level criteria to be evaluated subsequently.. Constrained evaluation approach. This approach is similar to the sequential evaluation approach, except that, secondary, tertiary etc. criteria are treated as constraints on the solution space. From the above discussion it can be seen that these represent evaluation approaches rather than specific solution procedures. The solutions are obtained basically by intution, experience, enumeration and/or some heuristic

12 methods. The first step towards the development of solution procedure(s), which take into account the application of multiple criteria, is to study the relationship between various criteria. If a 'cost' can be assigned to each of the factors involved, the objective can be represented by the minimization of the total cost of production. The total cost of production as utilized in a scheduling criterion is not necessarily the cost used for accounting purposes, since costs such as delay/penalty cost rarely assume a dollar value in actual accounting figures, other costs can be considered as constant, etc. In the case of a single processor, the total cost can be composed of the following components.. Processing cost. This cost is the cost of resources required for actual processing. This cost can assumed to be constant, i.e. independent of the sequence.. Setup/Changeover cost. This cost will contribute to the total cost in one machine case.. Penalty cost. This cost occurs if a job cannot be completed by its due date. This cost will be a significant part of the total cost in the one machine case, although it is difficult to measure or assign. Other costs such as inprocess inventory cost do not play a significant role in the one machine case, because the only inventory will be in the form of raw materials before

13 processing and finished goods after completion of jobs. Cost such as machine idle cost can be assumed to be zero, as this cost is not a function of the scheduling decisions. Thus in a single processor case, the components of the total cost, defined above, that can be generally controlled (i.e. that are dependent on a sequence) are penalty cost and changeover cost. It is clear that if an attempt is made to minimize the changeover cost, due dates for a number of jobs may not be met, thus increasing the penalty cost. On the other hand, if scheduling is done within the due dates, then the changeover cost will usually increase.. Rational Behind the Approach Taken in This Research After S. M. Johnson's success with the n-job, - machine problem [9], the trend in the sequencing research was towards the development of algorithms which optimized some criteria of performance. However, these algorithms have not found much application in practice. The reason for this is that the problem size that can be handled economically by these algorithms is quite small compared to the majority of the industrial scheduling problems. This fact promoted the development of heuristics. The heuristic methods do not guarantee the optimum solutions, but do 'obtain solutions close to the optimum. The use of heuristic methods permits attacking larger

14 problems. Many heuristic methods have been widely applied in industry because of their simplicity and their ability to produce acceptable solutions. It appears, however, that there are instances where available heuristics which are appropriate for industrial applications are not utilized by practitioners. This may be due to a nximber of factors; among these are: () practitioners may be unaware of research developments on the heuristics, and () the heuristics have not been tested sufficiently under realistic conditions for practitioners to regard the heuristics as reliable and valuable. A survey conducted by the Sequencing Research Group at Texas Tech University revealed [] that while most industries consider due date related criteria to be of prime importance they would also like to consider other criteria related to setup, inprocess inventory etc. However, very little research has involved multiple criteria for optimization []. One of the factors contributing to the lack of research in this area is that multiple criteria make it very difficult to solve problems analytically. The approach generally taken in sequencing research has been to formulate a general problem under some simplifying assumptions; solve the problem analytically or by I t heuristic methods;- and then attempt to apply the solution procedure to a particular problem. But as mentioned before.

15 this method has not met with much success. It is possible to attack the problem of sequencing in another manner using the following approach: ) Start with a typical problem from an industry. ) Study the relationship between the components of the total cost. If possible, find an optimal scheduling procedure for the problem under consideration. ) Relax some of the constraints that do not affect the solution procedure and seek a wider application of the solution procedure. ) If possible, relax some additional constraints and study the sensitivity of the solution procedure to the changes in the constraints. Develop new solution methods and study the improvements. ) Apply the above four steps to a number of industrial problems and attempt to identify patterns developing out of these several studies such that solution method(s) can be applied to different scheduling problems. With these primary findings, the Texas Tech Sequencing Research Group selected an industrial problem and decided to apply the research approach of generalization after the complete analysis of a specific case [], This research

16 0 considers the analysis of one such special case.. Purpose of This Research This research considers the application of sequencing research to an industrial problem. Although ideally it is desirable to use the steps described in the previous section for approaching such a problem, certain modifications have been made. The purpose of this research can be described as follows. This research studies the relationship between the components of the total cost for a specific problem involving the one machine case. Specifically, its concern is with the interacting effects of penalty cost and changeover cost. A sequential approach explained earlier is used in this study. The data is generated by using sample data from an industrial setting. The problem that is studied in this research is from Anheuser-Busch, Inc., St. Louis, Missouri, The reasons for the selection of this problem along with the description of the problem are discussed in Chapter II. To calculate the penalty cost, it is necessary to assign the due dates to the jobs. In the sequencing literature, researchers have generally assigned due dates at random. It is felt that there is a need for using an approach different from the random due date assignment. Although this research initially uses the random assignment

17 technique the feasibility of assigning the due dates based on the job and the shop characteristics of the problem under consideration will be studied. These characteristics will be explained in Chapter IV. An attempt will be made to develop a procedure for assignment of due dates; and the relationship between the components of the total cost will be studied.. Outline of Succeeding Chapters The problem under consideration is described in detail in Chapter II. The analysis of the available data is also presented. The results of this analysis are used for simulating the research problem. Chapter II also contains a survey of literature pertinent to the problem under consideration. Chapter III includes the discussion of the procedures used to minimize the changeover cost and the penalty cost. The results of the simulation used to study the relationship between the above two cost components are presented in the final section of Chapter III, Chapter IV contains the feasibility study of due date assignment based on the job and the shop characteristics. Chapter V summerizes the conclusions of this research and presents recommendations for future rjesearch.

18 CHAPTER II PROBLEM DESCRIPTION. Definition of Various Terms and the Description of the Problem To describe the problem it will be necessary to define some of the terms used in the description that follows. Order: An order consists of different items requested by a customer. Job: Each item on a customer list constitutes a job. Changeover/Setup cost matrix: This matrix gives the cost of processing the job j after job i on a given machine. This cost is a fxmction of time required for the changeover and material required for the changeover. In case of a single processor there will be only one changeover/setup cost matrix. P-Time matrix: This matrix gives the processing time of job i on machine m. There will be only one column in the matrix since a one machine problem is considered. The processing time will not include the changeover/setup time. Production line or Line: Production line is a group of machines ccbnnected by a continuous conveyor system, such that for all practical purpose, these machines

19 can be considered as one machine. Due date: Due date is the deadline by which an order and/ or job must be completed to avoid penalty. Scheduling period/horizon: Scheduling period is the length of time in the future for which scheduling is to be done. The problem to be described is an actual problem from industry, namely Anheuser-Busch, Inc., St. Louis, Missouri, The reasons for selecting this particular problem were the availability of the data, the adequacy of the information system and cooperation on the part of the company. "Anheuser-Busch, Inc., the largest brewer in the world, produces four different brands of malt beverages. The Company has its plants located in nine cities throughout the United States; its largest brewery and the corporate headquarters are located in St, Louis, Missouri. The description of the problem given below represents the beer canning operations of the St, Louis plant which is representative of the canning production system within the Company, Four brands of malt beverages Budweiser, Michelob, Busch and Budweiser Malt Liquor are brewed and packaged. In addition to brand, alcohol content for one brand may assume different levels to meet certain requirements of the states where beer is sold. Four categories of containers are used in packaging the beer: these are returnable

20 bottles, non-returnable bottles, cans and kegs. Several sizes of cans and bottles are used, and these cans or bottles must conform to different state requirements; also on cans the lid may have to have a printed tax stamp or statement as specified by the state. An example of a complete specification of a product is as follows: Budweiser with per cent alcohol in oz. can with Oklahoma tax lid and tab top opener is packaged in packs with cans to the case. When all possibilities are considered, the St, Louis plant has about 00 combinations of brand, alcohol content, container size, package type and tax requirements just for beer packaged in cans and bottles. On a typical can line there are four operations involved: filling and sealing cans, pasteurizing, packing and palletizing.... The palletized product is delivered to the loading area and/or warehouse from which it is shipped to its destination by means of trucks and railcars. There is a limited amount of storage space in the loading area which permits the storage of only hours production. Certain items which have infrequent demands and/or low volume production cannot be stored in large quantity because of the company quality policies regarding length of storing period for packaged product, even though larger lot sizes would reduce! production costs. However, for low volume products there is a limited amount produced for

21 inventory; the policies maintain a high quality product delivered to the customer. For this discussion, a single job is defined as the batch of beer using the same packaging, labels, etc., i.e. which does not require any change in the setup of the operating line. Further, two identical orders, meeting the above requirement from two customers will be treated as two distinct jobs, but no setup change is required between these jobs. At present the scheduling of the production line is performed in the following manner. A typical order is received by the Inventory Programming (IP) Department approximately three weeks prior to production and shipment. A brand-packaging-labeling code is assigned to each item in the order, and the personnel of the IP section decide on the day and production time block within which the viz. p.m.- a,m,, a.m.-7 a.m., 7 a.m.- a.m., a.m.- p.m., p.m.-7 p,m, and 7 p.m.- p.m. The production time block is determined by matching with the truck schedulers program of desired and assigned loading periods for distributor owned or contract haulers trucks. Orders transported by common carrier are assigned a production time block based on times when common carrier trucks can be obtained by the Company, Rail shipments are assigned production time blocks last, since railcar loading can take

22 place any time during the day. This information is key punched for computer input and from this data the 'Brewery Order Account' is produced which includes the week's orders in total barrels by brand and package. As indicated earlier this occurs approximately three weeks prior to production. The collection and dissemination of information continues from this point. Orders are filed in a dated Daily File. A two-week ahead schedule specifying the number of shifts of various units which will be required is compiled, and this is translated to a daily requirement in terms of the number of shift units needed. Orders are time stamped with designated loading time period and confirmed to the wholesaler about ten days to two weeks prior to production. The daily requirement information generated by the IP Department is then sent to the Computer and Scheduling Departments. At this point it should be noted that production information is collected periodically through each shift relative to units of production, time of completion, units loaded, etc. Based on this information and the daily requirements projection, several reports are sent to the Scheduling Department. These include a -day projection, inventory on hand,jand production reports. The Scheduling Department determines the daily production time block

23 7 detailed schedule one day prior to production based upon order requirements and inventory on hand under the criterion of minimizing setup times and costs. In the event that certain order requirements cannot be met, the schedulers meet with IP personnel to make modifications for the next day's schedule. Analysis of the system of order processing, scheduling and production data covering a period of weeks, permitted the research team to make the following observations in sequencing/scheduling terms. Once the product mix for a given day is determined, the daily schedules are determined under the criterion of minimum setup time since the criterion for (due date) the -hour scheduling period during which the product must arrive at the loading dock has already been met. Thus, dual critera are involved in this problem, but it is possible to partition the problem and handle each criterion separately. The product mix for a given shift is governed by the truck arrival schedule; thus an improvement in overall performance might be achieved if sequencing/scheduling involved a joint determination of daily production schedule and arrival schedule. The bottling/canning rates for each line are known. Thus, for all practical purposes the problem is one of deterministic processing times" [],

24 Within the general framework of the problem as described above, this research effort was defined. For the purpose of this study, however, the company is considered to be a hypothetical one for the following reasons: ) By considering a hypothetical company the actual product codings and processing time data can be kept confidential. ) The actual problem data was used to develop distributions from which new hypothetical order and jobs were generated for the simulation runs. The description of the problem parameters as established for this research is given below. It is based on a seven day data set of orders for production. This data set was utilized to establish product mix, number of products and probability distributions of the parameters involved. The hypothetical data generated was forced to display characteristics similar to the data set provided by the Anheuser-Busch, Inc. A company produces six different products. These products are packed in containers of various sizes. In all, fifteen different container sizes are used. The containers are marked with different codes for identification purpose. These markings are based on the type of product and perhaps the geographical location of the customer to

25 9 whom the container is to be shipped. Twenty different markings are possible. A job may be represented by a combination of a given product in some container with a particular marking on the container, for a specific customer. There are 00 combinations of product, container and marking possible. However, the total number of combinations produced is slightly below 00. This is due to the fact that some of the combinations are infeasible while some others, though feasible, did not appear in the seven day data set. The production shop has production lines ( machines). A particular container size is assumed to be manufactured on only a specified line. The present scheduling procedure is as follows. The orders are received by the sales department about to 0 days in advance. The scheduling period (the day of production) is decided by considering the availability of the raw materials and other related factors. After deciding on the scheduling period, the due date (i.e. the time of the day by which the order should be ready) is determined by discussion with the truck loading department. The criteria used are the availability of the truck loading docks, the customer preference, etc. If an order is to be shipped by railcar no due date is assigned to the order. Due date information is then passed on to the customer, so that the customer can send his truck at the proper time. Scheduling

26 up to this point is done about ten days prior to the actual 0 production. The final sequencing of jobs is done on the day before the actual production. At this point, the criterion used in sequencing the jobs is to minimize the changeover cost within the due dates. This is clearly the sequential evaluation approach. The primary criterion is meeting the due dates and the secondary criterion is minimization of changeovers. This type of problem can be found in many types of industries, such as canning and bottling industry, paint manufacturing, dairy products, and business forms printing industry, to name a few.. Analysis of Available Data Data for a seven day period was provided by the company. It was assumed that this data represents a typical production period. This data was analyzed in great detail to obtain the empirical distributions which were used in problem simulations. The company receives N orders per day, where N is distributed normally with a mean of 0 and standard deviation of. eleven jobs. Each order can consist of between one to The probability of having n jobs in an order is given in Table,., The scheduling horizon is one day. A full day is divided into six equal parts of four hours each. An order can have a "due date" of through

27 TABLE,. PROBABILITY DISTRIBUTION OF HAVING n JOBS IN AN ORDER Number of Jobs Cummulative in an Order Probability Probability.70.70,, , depending on the time by which the order should be ready on the given day. Some orders were assigned an arbitrary due date of which implies that these orders do not have a fixed time by which the order should be ready on the given day. These orders were scheduled for production any time during the day under consideration. The probability distribution of due d^tes is given in Table... * There are a total of product-container-marking combinations. The probability that a given job consists

28 TABLE.. PROBABILITY DISTRIBUTION OF DUE DATES Cummulative Due Date Probability Probability.0, ,000 of one of these combinations is given in Table... The average processing time (in seconds) for these combinations is also given in the same table. It was found that the actual processing time of a given job varied from as little as one tenth of the average value to as high as twice the average value. If two consecutive jobs processed on a machine involve either a product change or a marking change or both, there is a cost involved in the changeover. This cost can be due either to the loss of product or the loss of time or both. The cost of a marking change involves only the lost time. While the cost of product change has two components, loss of product and loss of time. When the product change and

29 Serial Number TABLE. PROBABILITY DISTRIBUTION PR- l i CNT- MK OF COMBINATIONS Mean PTime ProbedDility , ,

30 TABLE.. Continued Serial Number PR- «I li CNT MK Mean PTime Probability ,

31 TABLE,. Continued Serial Number PR-( ; ::NT- MK Mean PTime Probability , ,

32 TABLE.. Continued Serial Number PR-CNT-MK Mean PTime Probability ( ,0.000,

33 7 TABLE.. Continued Serial Number PR-(! :NT- MK Mean PTime Probability ,00090, ,0009,

34 TABLE.. ( Continued Serial Number PR-CNT-MK Mean PTime Probability ; ,007, , ,

35 TABLE.. Continued 9 Serial Number PR-CNT-MK Mean PTime Probability the marking change are carried out simultaneously the cost becomes equal to [ (loss of product) x (cost factor) + [max (time to change product, time to change marking) x (cost factor)], The time required to change the product is the same as the time required to change the marking, the changeover time utilized for this research was set at minutes per changeover. This means that the cost of changing the product as well as the marking is composed of the cost due to loss of product and the cost due to loss of time in changing the ( product. Thus the'cost of changing the product as well as the marking is same as the cost of changing the product

36 0 alone. For the problem under consideration these costs are as follows: Cost of marking change Cost of product change = unit = units Cost of product and marking change = units The numerical cost figures used above are hypothetical values for these costs. If the above three costs are x, y, z units respectively, then the only relationship between these three costs, necessary for this research is (x < y = z). These costs result in a specially structured changeover matrix. For example Table.. gives changeover cost matrix for production line number. At this point no quantitative units are assigned for penalty cost as it is very difficult to quantify the penalty cost. The penalty cost can vary drastically from one problem to another or even from one customer to another for the problem described previously.. Literature Survey The specific case studied in this research is that of sequencing on parallel (not identical) lines, with sequence dependent setup costs and a penalty associated with late orders. For this reason, the literature survey is confined to one machine research considering either penalty cost or setup cost or both. There are many articles [, 0,,,, ]

37 CM CM w < in CHANGEOVER COST MATRIX FOR LINE Final State Next Job -> Present Job - o n ro CO ro CO Initial.. State o CO CO rh r-i H O -- o CO ro rh H o rh -- o ro ro H o H H -- o ro ro o r-i rh rh --0 o rh O ro ro ro ro -- o o ih ro ro ro ro --

38 published about the one machine situation, where the jobs have the due dates and the penalty costs associated with them. McNaughton [0] discusses scheduling of several one stage jobs on several machines which are capable of processing the jobs with varying degree of efficiency. Each job has a due date, and the penalty associated with each job is a function of tardiness. The objective is to minimize the total penalty cost. Setup time is included in the processing time and is independent of the sequence. McNaughton has proved some theorems regarding necessary and sufficient conditions for the optimality of a sequence; but he has not presented any scheduling procedure. Root [] considers the problem of scheduling n jobs with due dates and loss functions on m parallel and identical machines in a single stage production. The optimal schedule minimizes the cost of tardiness. A common deadline t is asstimed for all the jobs. All the n jobs are o available for processing at an arbitrary time zero. All the jobs must be processed exactly once on one of the m machines. Each machine begins operation at time zero and operates continuously without idle time, on one job at a time, lontil all jobs assigned to that machine are completed. Associated with each job is processing time SL^., for the job i on machine j'. This time includes setup and tear down time. Also associated with a job is a loss function f^.

39 such that if job i is x units late, the loss for the job i is f^(x). An algoritlim is developed which schedules the n jobs to minimize Z f,(x). i=l ^ Elmaghraby [] has developed an algorithm which solves the problem of minimizing the total cost of tardiness, for the one machine case, as measured by I C.(d.,T.) where i=l ^ ^ ^ d^ is the due date of job i, T. is the completion time of job i, and C. is any non-decreasing penalty function. The proof of optimality is included for the case, when C.'s are linear. Moore [] considers the problem of minimizing the number of late jobs on a single processor. The procedure is then extended to solve the problem in which each job is associated with a continuous monotonic non-decreasing deferral cost function. The objective is to produce a schedule where the maximum deferral cost incurred is minimal. Two algorithms which accomplish the above two objectives respectively (namely minimizing the number of late jobs and minimizing the maximum deferral cost) are presented; and the proofs of optimality are also included. Shrinivasan [] has developed an algorithm which solves the problem of sequencing n jobs on one machine, where processing times are sequence independent and include setup times. All the jobs have due dates. The objective

40 is to minimize total tardiness. References [,, 7, ] consider the effect of the sequence on the setup cost. The objective is to sequence n jobs on a single production facility, so that some aspect of changeover cost is minimized. Gavett [7] discusses the performance of three scheduling rules for the minimization of setup cost on a single facility. The rules are ) Next best rule ) Next best rule with variable origin ) Next best rule after column deductions The procedure of the next best rule is to always select that unassigned job which has the least setup cost relative to the job which has just been completed. The first job is selected arbitrarily. For the second rule, all possible jobs are tried in the starting position and then the next best rule is applied. Hence the second rule gives n sequences, out of which the best one is chosen. The third rule consists of applying the next best rule after subtracting the minimum value in each column of the setup cost matrix from all other values in that column. Gavett concluded that the nd and rd rules worked a little better than st rule. Presby and Wolfson [] developed a solution procedure for the problem of minimizing setup cost on a single

41 processor. The procedure guarantees the optimum solution. However, as the procedure is essentially curtailed enumeration, solution of problems with more than 0 jobs become economically infeasible. Elmaghraby [] discusses sequencing of n jobs on a facility, where a subset of the jobs are 'related' to each other in such a manner that regardless of which job is completed first, the utility of that job is hampered until all the other jobs in the same subset are also completed. There is a cost associated with this loss of utility. This cost is given by C.. = a.. T. - T., where jobs i and j are ij ij J related to each other and T., T. are their respective com- ^ J pletion times, a.. is the cost of waiting per unit time. i<j The total cost is given by a.. T. -T.. Elmaghraby, - -^ has presented an algorithm to sequence the n jobs on a single machine so as to minimize the total cost. Buzacott and Dutta [] discuss the problem of scheduling n jobs on a single facility which has various settings. They define a state to be a unique combination of these settings. For example, the single facility might be a computer, the different settings might be one or two or three disks, 00 K or 0 K or 0 K memory partition, one or two tape drives etc.; and one combination of these settings might be disks, one tape drive with 0 K memory partition

42 There is a choice of states in which to process a job; and the cost of processing depends on the state. In addition, there is also a sequence dependent changeover cost between the states. The problem is to schedule the jobs and pick an optimum setting for each job, so as to minimize the overall operating costs. A dynamic programming model is developed for obtaining an optimal solution to the problem. Glassey [] solves the case where a finite horizon delivery schedule for a number of products is given. All the products are processed by the same processor. The problem is to schedule the production on a single machine, to meet all the due dates and then to minimize the number of product changes. Thus the changeover matrix has all diagonal elements zero and all the remaining elements are one. The solution procedure is to define a state space such that the shortest distance through the network corresponds to the minimum number of product changes. This solution procedure always results in an optimal sequence. From the above discussion it can be seen that all these articles deal with only a single criterion, either the penalty cost or the setup cost. The article by Glassey considers multiple criteria indirectly. The primary objective of Glassey is to meet all the due dates (infinite penalty) and then to minimize the number of product changes. Although none of the above articles consider the

43 problem of multiple criteria, these articles were of considerable help in gaining significant insight into the 7 problem of multiple criteria. Specifically, the algorithms presented by Gavett and Moore were useful in developing the scheduling rules for minimization of the setup cost and the penalty cost for the sequential evaluation approach used in the succeeding chapters.

44 CHAPTER III EFFECT OF MULTIPLE CRITERIA ON TOTAL COST. Introduction One objective of this research is to study the relationship between the penalty cost and the changeover cost for a given configuration of the problem. The approach used for this purpose is the sequential evaluation approach As explained earlier it is difficult to assign a penalty cost to different jobs or orders. It was therefore assumed that all the jobs are equally important. This means that minimizing the number of late jobs can be used as an objective criterion equivalent to the objective of minimizing the penalty cost. Two methods were used to solve the given problem. In one method, called C-P, the primary criterion is minimization of the changeover cost and the secondary criterion is minimization of penalty cost. In the second method, called P-C, the primary criterion is minimization of penalty cost and the secondary criterion is minimization of changeover cost. The problem was then simulated using the distributions developed earlier and using the following procedure: ) Generate the total number of orders to be produced in one scheduling period (i.e. one

45 9 day). ) Generate the number of jobs in each order and assign product, container, marking codes and processing time to each job. ) Assign a due date to all the jobs in every order. ) Schedule the jobs by method C-P and by method P-C (described later). Calculate the number of changeovers and average job lateness and compare the results. ) Repeat steps through to reduce the random effects. As explained earlier, the sequential evaluation approach involves scheduling of jobs first by the primary criterion alone. After this, the jobs are rescheduled using the secondary criterion but within the framework of the primary criterion. At this point it will be helpful to discuss the scheduling rules that are used in this research when minimizing the changeover cost is the primary criterion and when minimizing the penalty cost (number of late jobs) is the primary criterion.. Scheduling Rule to Minimize Changeover Cost The special structure of the changeover cost matrix (Table..) was helpful in developing a scheduling rule for minimizing the changeover cost. It may be noticed that

46 0 all the diagonal elements of the changeover cost matrix are zero, all the elements in the small squares around the diagonal have a value of unit when the present job and the next job have the same product code. The remaining elements are equal to units. The scheduling rule is as follows: ) Given that a job X is in sequence position i, the best candidate for the sequence position (i+) is any other job with the same product code and the same marking code (and of course the same container code) as job X. ) If no such job is found, select any job with the same product code as job X. The marking code does not matter. ) If no such job is found, then any job is a candidate for the next sequence position. It can be seen that this is essentially an application of the next best rule of Gavett [7] adopted for this special configuration. It can be easily proved that the above procedure will result in an optimum solution, which minimizes the total changeover cost. The proof is given below. Let there be n jobs to be processed on a given production line. Let C denote the minimum number of combinan tions (of product code, container code, marking code) covering all these n jobs. It is clear that at least C setups

47 will be required before all the jobs can be completed, assuming that initially the line was not setup for any job. Any sequence which produces all the jobs with the same combination together and then starts with a job having a different combination will have exactly C setups. As menn tioned in Chapter II, the cost of a product change is greater then the cost of a marking change, while the cost of both a product and marking change is equal to the cost of a product change alone. This implies that if a sequence has only C changeovers, and if it also minimizes the numn ber of product changes, then the sequence will be an optimum sequence. If all the jobs with the same product are grouped together, then the resulting sequence will have the minimum number of product changes. To be precise, the number of product changes will be equal to the number of different products used by all the jobs; and this is an absolute lower limit. Thus a sequence generated by steps, and gives one optimum sequence, which minimizes the cost of changeovers.. Scheduling Rule to Minimize the Penalty Cost (Number of Late Jobs) Characteristics of the scheduling rule used to minimize the penalty cost will now be discussed. An order is late if any job in that order is late. Thus if the number of late jobs is minimized, the chance of an order being

48 late is also decreased, thus reducing the penalty. Moore [] has presented an algorithm to minimize the ntmiber of late jobs. Woolsey [7] describes this algorithm as follows: "Assume that you have a batch of N jobs with known due dates, (d^), and known or estimated processing times, (Pi).. Order the jobs from left to right in order of increasing due dates. That is, the job with the earliest due date is first and the job with latest due date is last. This is the current sequence.. Using the current sequence, find the first late job. If one is found, GO TO Step. If no job is late, STOP. Sequence is optimal.. Look at the subsequence up to and including the late job. Find the job in this subsequence with the largest processing time and reject it. Consider the resulting sequence as the current sequence. GO TO Step " [7]. It can be seen that initially this algorithm gives priority to due dates. If a job does not meet the due date, the sequence is rearranged giving priority to the shorter processing times. If any of the rearranged jobs still does

49 not meet the due date, the job under consideration is rejected, and the algorithm goes back to step. The algorithm terminates when the jobs are sequenced according to their due dates followed by a set of jobs that have been rejected. This concept has been modified as follows: ) Arrange all the jobs with due dates in ascending order of due dates. ) If two jobs have the same due date, arrange these two jobs according to the ascending order of processing time. ) After sequencing all the jobs with due dates, start sequencing the jobs without due dates as follows: Arrange these jobs in ascending order of processing time. Go on inserting these jobs in the original sequence without violating any of the due dates. The jobs that are left over are added at the end of the sequence.. Description of the Sequential Evaluation Methods After determining the scheduling rules for minimizing the changeover cost alone and for minimizing the penalty cost alone, the following procedures were used for the sequential evaluation method C-P and the method P-C.

50 Method C-P. Given that a job X is in sequence position i, determine the job that is a best candidate for the sequence position (i+) using the scheduling rule to minimize changeover cost (Section.).. If two or more jobs tie for the sequence position (i+), break the tie using the scheduling rule to minimize penalty cost (Section.).. If two or more jobs remain tied for the sequence position (i+), select any one of these jobs. Method P-C The procedure is similar to that in method C-P, except that the scheduling rule to minimize penalty cost is applied first and the ties are resolved by the application of the scheduling rule to minimize changeover cost. The schedule obtained by the P-C method described above can be improved substantially using a simple technique. This is done as follows. If there are two jobs with the same product, container and marking codes in two consecutive due date periods and are separated by at least one job with a different code, then the above two jobs are moved within their respective due date periods to avoid a changeover. (i.e. one job is scheduled at the end of the period and the other job is scheduled at the beginning of the next period). If there are several pairs of jobs satisfying the above requirement,

51 only one pair can be picked, this decision is arbitrary. Before describing the results of the simulation, a sample problem is scheduled by the methods C-P and P-C. The data for the sample problem is given in Table... Job Number TABLE.... JOB LISTING FOR THE Due Date SAMPLE PROBLEM Pr-i Cnt--Mk PTime ( 000

52 The sequencing procedure using the method C-P proceeds as follows: Sequence position (S.P.) : by the changeover rule. all the jobs are candidates Select jobs 9 and 99 by the due date rule as step. Select job 9 by the due date rule as step. S.P. : job 0 and 70 are candidates by the changeover rule. Select job 0 by the due date rule. SP. : job 70 is the only candidate. Proceeding in this manner, the final sequence by method C-P is presented in Table... The procedure for method P-C is presented by means of two tables. Table.. gives the results of arranging the jobs according to the due date criterion (primary criterion) and is self-explanatory. The secondary criterion (changeover criterion) is applied to the sequence in Table.. to obtain the final sequence as listed in Table.... Simulation Results For the purpose of this simulation, job due dates were assigned randomly to match with the distribution of due dates developed from the sample data. The production shop was simulated for ten scheduling periods (i.e. ten days) using different random seed numbers for each day. orders were scheduled by both methods C-P and P-C. The A computer program was written for the purpose of scheduling and

53 7 TABLE... SCHEDULE FOR THE SAMPLE PROBLEM BY METHOD C-P Job Numbe sr Due Date Pr-( :n-mk PTime is documented in the Appendix. This program was used to produce a partial application of the scheduling rules and the remaining scheduling was done manually as follows. In the method C-P, the computer program grouped together all the jobs on a given line by product and marking codes. In each group the jobs were arranged in increasing order of due date