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A NETWORK APPROACH TO A CLASS OF MULTIFACILITY, MULTIPRODUCT PRODUCTION SCHEDULING PROBLEMS By ROBERT CALEB DORSEY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTSS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1973 TO LOU ACKNOWLEDGMENTS I wish to acknowledge the considerable contributions to this work by Dr. H. Donald Ratliff and Dr. Thom J. Hodgson, who acted respectively as the Chairman and CoChairman of my Doctoral Committee. Their energy, insight and clarity of expression have contributed greatly to both the form and content of this dissertation. I thank the remainder of my committee, Dr. Richard L. Francis, Dr. Kerry E. Kilpatrick and Dr. Richard A. Elnicki, for their efforts which contributed to the completion of this work. I would also like to acknowledge Dr. Robert N. Braswell, Dr. Richard S. Leavenworth, Dr. Eginhard Muth, Dr. B. D. Sivazlian and Dr. M. E. Thomas for their assistance and support throughout my stay at the University of Florida. Finally, I thank my wife, Lou, and my parents, Caleb and Dot Dorsey, for their encouragement throughout the course of this work. This dissertation was supported in part under grants DAAROD3112470G92A and DAAROD3112471G183 and ONR contract number N0001468A01730013. iii TABLE OF CONTENTS Page ACKNO WLEDGMENTS . ..... ...... iii LIST OF FIGURES . . vi ABSTRACT . . . vii CHAPTER 1 INTRODUCTION . . .. .1 An Overview. . ... ..... 1 Literature Review . . 2 Practical Motivations . . 5 A General Problem Description . 7 Organization of the Dissertation . 9 2 A NETWORK APPROACH TO THE PRODUCTIONINVENTORY SCHEDULING PROBLEM WITHOUT BACKORDERING .. 10 Introduction . . 10 Problem Development and Formulation . 11 A Solution Procedure . ... .16 An Example . .. 20 A Warehouse Space Limitation . ... 21 Extensions of the Model . .. 22 Conclusions . . 25 3 A NETWORK APPROACH TO THE PRODUCTIONINVENTORY SCHEDULING PROBLEM WITH BACKORDERING .. 28 Introduction . . 28 A Mixed Integer Programming Formulation .. 29 Reformulation as an All Integer Program .. 34 Formulation as a MinimalCost Flow Problem 37 Conclusions . . .. 43 4 A MULTIPLE FACILITY, MULTIPLE PRODUCT PRODUCTION SCHEDULING PROBLEM WITH OVERTIME . .. 45 Introduction . .. .. 45 The Overtime Production Scheduling Problem .. 47 A Special Case of the Overtime Scheduling Problem 54 Conclusions . . 61 TABLE OF CONTENTS (Continued) CHAPTER Page 5 AN EFFICIENT INTEGER PROGRAMMIING ALGORITHM FOR A MULTIPLE BATCH PRODUCTION SCHEDULING PROBLEM 62 Introduction . . ... 62 An Integer Program . . ... 64 An Algorithm for Solving the Integer Program .. 66 An Example . . ... 67 Other Properties of the Integer Program ... 69 The Production Scheduling Problem . .. 70 Computational Experience . ... 81 Conclusions . . ... 81 6 SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH ...... 82 APPENDIX PROOF OF THEOREM 5.1 . .. 85 BIBLIOGRAPHY . . ... 87 BIOGRAPHICAL SKETCH . . ... 91 LIST OF FIGURES Figure Page 2.1 Product Inventory Change during a Production Period 13 2.2 Network Example of Problem P2.1 . ... 17 2.3 Network Example with Facility and Period Dependent Costs . .... 26 3.1 Network Example of Problem P3.1 . ... 33 3.2 Network Example of Problem P3.3 . ... 42 4.1 Network Example of Problem P4.2 .. 53 5.1 Geometric Interpretation of the Algorithmic Example 68 5.2 Network Example of Constraint Set P5.6 .. 80 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A NETWORK APPROACH TO A CLASS OF MULTIFACILITY, MULTIPRODUCT PRODUCTION SCHEDULING PROBLEMS By Robert Caleb Dorsey March, 1973 Chairman: Dr. H. Donald Ratliff CoChairman: Dr. Thom J. Hodgson Major Department: Industrial and Systems Engineering A class of multiple facility, multiple product, production inventory scheduling problems is considered over a finite planning horizon. The horizon consists of discrete production periods during each of which at most one product can be assigned to a facility. Product demands are constant over a period but not necessarily identi cal in all periods. Each product has a minimum batch size which can be made, and only integer multiples of this basic batch size are allowed. The problem objective is to determine an assignment of products to facil ities which minimizes the sum of any production, inventory, backordering and facility usage charges which occur over the horizon. Four differ ent scenarios for this class of problems are considered. In a straightforward manner, each of the scenarios which are considered is formulated as an integer program. It is then shown that each of the problems can be reformulated as a network flow problem which can be solved using any of the very efficient algorithms for finding minimalcost flows in single commodity networks. In addition, for some important special cases, single pass procedures are developed which are more efficient than the standard flow algorithms. In the fourth sce nario the horizon length is also considered to be a variable in the optimization. In this scenario a specially structured integer program ming formulation is identified, and an extremely efficient algorithm is developed for determining its solution. viii CHAPTER 1 INTRODUCTION An Overview Inventory management and production control represent two major components of the manufacturing process. Traditionally, research in productioninventory systems has been divided into two groups based on the assumptions regarding the interaction of these components. In one, the assumption is made that the two components are independent, and optimization can be achieved by optimizing the control of each component separately. In the other, inventory management and production control are not assumed to be independent, but rather the components interact and optimization must be achieved by considering the two components jointly. The research in this dissertation falls in the second group of thought and deals more specifically with the analysis of a multi facility, multiproduct, integrated productioninventory system. Realistic examples of the multiple facility, multiple product system include textile spinning machines, metal stamping operations, steel rolling facilities, glass molding lines, container manufacturing machines and packaging lines. Contributions within this dissertation include original formu lations of the multifacility, multiproduct problem as integer programs and network flow problems which offer considerable insightsinto the operation of the productioninventory system. These insights allow us to enrich the problem generously to include many other realistic aspects of the productioninventory system heretofore not solvable with present techniques. In addition, efficient algorithms are developed for solving different types of associated problems which in many cases are single pass procedures which can be easily carried out by hand. Literature Review Problems of the type considered in this dissertation generally have been modeled as onefacility, deterministic problems where produc tion rates and demand rates for each of the products take on known con stant values. Only one product can be produced on the facility at a time, and a setup is necessary each time a different product is to be produced. Stockouts are not permitted, and backordering to meet demands may or may not be allowed. Inventory carrying charges for each of the products are assessed in direct proportion to the amount of product on hand. The problem is to determine a schedule for allocating the produc tion time of the facility to each of the products so that the costsasso ciated with setting up, producing, backordering and carrying inventory are minimized over a given planning horizon. One general difference in models dealing with this problem is in the definition of the planning horizon. In one approach the length of the planning horizon is assumed to be infinite, and in the other the length of the horizon is assumed to be fixed and finite. In the finite horizon case, the optimization problem involves the determination of a feasible assignment of products to the facilities which minimizes costs. In the infinite horizon approach, the existence of a finite, but variable length, production cycle is assumed where this production cycle repeats itself infinitely many times throughout the horizon. Hence, the infinite horizon problem involves the joint determination of the optimal production cycle and the optimal assignment of products to the facility during this production cycle. A wellknown approach to the infinite horizon form of the problem is to assume that each product is set up and produced exactly once dur ing each production cycle where the amount produced for each product at each setup is proportional to the length of the production cycle. In this case the object of the optimization reduces simply to the determi nation of the optimal length of the production cycle. This approach is typically referred to as a model with a pure rotation cycle. An optimal macroplan for this model has been determined by Magee [12] and by Rodgers [14]. The more general approach to the infinite horizon problem is not to require a pure rotation cycle. Rather each product may be pro duced on an individual production cycle where the only restriction is that feasibility must be maintained with respect to the facility. At present, there is no general method for determining an optimal macro plan for this model. However, upper and lower bounds can be determined on the cost of the optimal solution. A lower bound is determined by solving the onefacility, oneproduct problem with a pure rotation cycle for each of the products and summing their respective costs. If, in fact, the optimal production quantities determined by solving each of these problems can be scheduled on the facility without causing scheduling conflicts, then this schedule is an optimal macroplan for the multiple cycles problem. In like manner, an upper bound on the optimal cost is determined by solving the onefacility, multiple product problem with a pure rotation cycle and noting the associated cost. Researchers who have investigated this problem include Bomberger [1], Elmaghraby [3,4], Haessler [9], Hodgson [10], Madigan [11], Magee [12], Maxwell [13], and Stankard and Gupta [15]. Of primary interest in this dissertation is the finite horizon approach to the problem. In this approach the horizon is assumed to consist of discrete production periods (shifts, for example) during which at most one product can be assigned to the facility. During any period product demands are assumed to be constant but not necessarily the same in all periods. The productioninventory scheduling problem was initially modeled as a finite horizon problem by Wagner and Whitin [16]. They considered a onefacility problem for the case where backordering is not allowed and production amounts in any period can vary. They developed a dynamic programming procedure for solving the problem and were able to realistically enrich the problem to include period depend ent product demands, inventory costs and setup costs. Zangwill [17] extended the results of Wagner and Whitin to include backordering and showed that the problem could be formulated and solved as a minimalcost flow problem. In addition, he generalized the cost function to include concave inventory and backordering costs. In order to bring the finite horizon approach to the problem more closely in parallel to the real environment, Elmaghraby and Mallik [2 ] considered the problem in a slightly different context. They assumed that the production amounts of each product at each assignment of the product to the facility are fixed and constant. They then developed a dynamic programming recursion for solving the problem for the case where demand rates and production costs are assumed to be con stant and identical in all periods. For the batch type production sys tem where product inventories are updated only at the end of each pro duction period, their results apply to a special case of the problem considered in Chapter 3 of this dissertation. Practical Motivations While the assumptions in the previous models provide a basis for analyzing industrial scheduling problems, it is frequently true that the characteristics of the real system are much more complex than these models allow. For example, many production operations consist of numer ous facilities (M). When customer demands are received which require processing in this operation, the system controller must decide, in addition to what, when and how much of the product to produce, which facility or facilities should be used. Underlying in the previous models is the assumption that facility selection is a deterministic problem which is not part of the decision making, and hence system optimization can be achieved by solving M onefacility problems. However, this is not always the case in the real environment as system controllers frequently interchange products between facilities in order to meet demand requirements and minimize costs. Another consideration is initial product inventories. In the infinite horizon approaches, the assumption is made that the initial product inventories are "in phase with" the optimal macroplan of allo cating the production time of the facility to each of the products. In the industrial environment this is not necessarily true, and indus trial schedulers in using the infinite horizon approach may still be left with a subhorizon of unknown time duration during which the facil ity must be scheduled "out of phase" with the optimal macroplan. In the industrial environment, it is rare that decision making is based on a single plan of operation as many of the previously described models assume. For example, an industrial scheduler fre quently may vary production run lengths, schedule overtime, schedule alternate facilities, purchase product from outside sources, etc., in order to effectively control the system. Another consideration which is frequently applicable to many production operations is a fluctuating demand rate. The fluctuations of the demand rate can arise in numerous ways at least some of which can be predicted in advance by good forecasting. Many industries have seasonal demands yearinandyearout where the demand might be repre sentable by a cyclic function. Some industries have a record of steady, continued growth which can be represented possibly by a ramp or exponen tial growth function. Other industries have consistent sales through out the year with large increases around the holiday season (impulse function). Temporary plant shutdowns for maintenance purposes or vacations also occur in which case the demand rate for the supplier to these plants may suffer a sudden drop (negative impulse function). New plants are being bL. and new uses are being found for different products either of nich might correspond to a step function in the demand rate forecast. As these examples indicate, the real environment is often much more complex than the assumptions of existing models, and the develop ment of new insights is necessary if optimal system control is to be achieved. This dissertation deals specifically with some of the prob lems associated with the operating productioninventory system. The objective of the research is to provide original insights into the achievement of optimal control in this complex system as well as to provide efficient procedures, based on these insights, for solving related problems in the real environment. A General Problem Description An industrial process made up of M facilities in parallel is considered where there are N different products to be produced over a finite, but possibly variable length, planning horizon. The horizon is made up of production periods (shifts, for example) which have equal time duration. During any period at most one product can be scheduled on each facility. When any product i is assigned to a facility for one period, an integer multiple of p. units of product is made. Production 1 completed is added to each product's inventory, and nonnegative demands (dik units for product i in period k) are satisfied from this inventory. iK. Backordering may or may not be allowed (depending on the application), and setups are assumed to occur between production periods. Initially, each product i is assumed to have a nonnegative beginning inventory level denoted by Ii0 (units of product i), and when applicable, a non negative initial backorder level denoted by Bi. (units of product i). During the horizon the system controller must schedule the products to the facilities in such a manner as to meet all product demands, and at the same time build up each product's ending inventory level to a desired nonnegative amount denoted by I.i. The problem objective is to iH determine such a production schedule which minimizes the sum of any pro duction, inventory, backordering and facility usage charges over the horizon. As an example of this problem, consider a simplified version of a twoproduct, onefacility problem over a threeperiod horizon where at most p. units of each product i can be produced by the facility dur ing any period. During any period, then, the system controller has three options. He can schedule either product one or product two to the facility, or he can leave the facility idle. Hence in this problem 3 there are 3 or 27 possible solutions to the scheduling problem. Depend ing on the imposed constraints, all, some or none of these schedules may be feasible solutions to the system controller's problem. What the system controller must attempt to do is determine a feasible schedule which minimizes costs. In general for a Nproduct, Mfacility, Hperiod problem of this type, there will be (N+1) possible schedules to eval uate. Hence, it is clear that something significantly better than total enumeration is needed for the system controller to be able to solve realistic kinds of problems. One of the goals of this dissertation is to provide an efficient procedure for solving this problem. Organization of the Dissertation In Chapter 2 a fixed horizon problem with production and inven tory costs is addressed. A mathematical formulation of this problem is developed which allows the problem to be solved using any of the algo rithms for finding minimalcost flows in single commodity networks. However, for an important special case, a single pass algorithm is given which is even more efficient than the standard flow algorithms. In Chapter 3 a fixed horizon problem with production, inventory, and backordering charges is considered. In a straightforward manner, this problem is formulated as a linear, mixed integer program. It is shown, however, that the problem can be reformulated as an all integer program which can be solved as a minimalcost flow problem. In Chapter 4 a fixed horizon problem is considered where the system controller has an overtime option available for use in schedul ing the system. This problem is formulated as an integer program, and a solution procedure is developed for solving the problem which involves solving a minimalcost flow problem. In Chapter 5 the problem of bringing initial product inventories "in phase with" a desired infinite horizon cycle plan is considered. The problem is formulated as an integer program which has special struc ture, and a very efficient algorithm is developed for solving the problem. A summary of the results of this research and suggestions for future research are given in Chapter 6. CHAPTER 2 A NETWORK APPROACH TO THE PRODUCTIONINVENTORY SCHEDULING PROBLEM WITHOUT BACKORDERING Introduction The productioninventory scheduling problem is considered over a fixed planning horizon made up of H production periods. Each of the M facilities is considered to be identical in that product i can be produced by any facility at a fixed production rate of p. units per period and a cost of c. dollars per period. Demands for each product i 1 occur continuously during each period k according to a fixed demand rate of dik units per period. Inventory carrying charges are assessed on product inventories. During the horizon the system controller must schedule the products to the facilities in such a manner as to meet all demands, without backorders, and at the same time build up each product i's inventory level to a desired minimum ending amount denoted by I. . The objective is to determine a production schedule which minimizes the sum of production and inventory carrying charges over the horizon. This problem is formulated as a network flow problem for which efficient solution procedures are known. However, for an important special case a single pass algorithm is given which is more efficient than the standard flow algorithms. In addition, the general model can be enriched generously to include numerous other realistic aspects of the production system without severely complicating the solution proce dure. These generalizations include such things as product and facil ity dependent production costs, facility usage costs, period dependent facility availability, period dependent inventory carrying costs, inven tory constraints, facility and product assignment restrictions, and horizon dependent facility allocation costs. For presentation purposes, only basic assumptions are used in the development of the problem with the more detailed aspects of the model being identified in the "Extensions of the Model" section. Many of these enrichments will no doubt become apparent in the development. Problem Development and Formulation For the problem being considered, backordering is not allowed and product inventory changes during any period are assumed to occur linearly (i.e., during any period the production and demand rates are constant). Thus stockouts on product i will occur during period k if and only if at least one of the ending period inventories, I or i,kl Iik, is negative. Hence, requiring that ending period inventories be nonnegative for all products and periods is a necessary and sufficient condition for preventing stockouts. Define wik as the minimum number of times (integer) product i must be produced during the first k periods in order to have a non negative inventory level at the end of period k. Then j=1 ik 1 ( k = 1,2,...,H1 j=1 where (a) denotes the smallest integer greater than or equal to a. In addition to being nonnegative, product i's inventory level at the end of the horizon must be greater than or equal to I. Hence, define H H WiH = max [0, ((iH + E di iO)/Pi> i = 1,2,...,N, j=1 and let W be an N by H matrix with components [w i]. Define x.k as the number of facilities scheduled to produce product i during period k where X is defined as an N by H matrix with component [x ik. The inventory level for product i at the end of per iod k can then be written as k k I p x. + I d k = 1,2,...,H. ik 1 10 ij j=1 j=1 Figure 2.1 gives an example illustration of the change in inventory of product i during a period in which the amount produced on product i exceeds the amount demanded. Obviously, the linear change in inventory of product i during period k is a function of the number of facilities scheduled to produce product i, and the slope of the line can take on M+1 possible values (i.e., xik can be 0, 1,2,...,M1 or M) in period k. The average amount of inventory on hand for product i during period k in any of these cases is given by kI ki (I ()(p x + 2Z x..+ 21. d 2 E d. i,k1 k 2 Piik j ij 2i0 ik2 j ij j = ,2,j=o k = 1,2,...,H where xio and di are defined to be zero. 10 i0 TIME (IN PERIODS) Figure 2.1 Product Inventory Change during a Production Period. I ,kI 0 ik 0 Denoting by 0. the inventory carrying charge per unit of product 1 i carried in inventory for one period, where 0i > 0, the total inventory cost over the horizon is given by N H k1 N H k1 ( i2)0iPixik+2 E x. i+ Z ()(2 d 2E d i 2k 2 1 d i. i=1 k1 j=0 i=l k=l j=O 1 Letting i = ()0iPi N H k1 K= 2 ()0.2ii dik 2 E d. . i=1 k=l j=0 and noting that H k1 H S2 Z x.. = E (2H2k)x , k=l j=0 1 k=l k the total cost of carrying inventory and producing the N products over the horizon H can be written as N H N H S 2 (2H2k+l)$.xik + K + E c. E xi (2.1) ik 1 ik i=l k=l i=1 k=l where K is a constant. The system controller's objective is to minimize (2.1) subject to the constraints that (a) only M facilities are available for produc tion during each period, (b) demands must be met without allowing back orders, and (c) at most one product can be scheduled on each facility each period where a facility, when assigned to product i, is assumed to produce p. units of product i. A mathematical formulation of this problem is as follows problem is as follows: N H N H Minimize E E (2H2k+l) ii + K+ E c. E x i=1 k=l i ik i=1 I k=1 ik subject to N [P2.1] E xik < M k = 1,2,...,H i=l k k x w i = 1,2,...,N j= ij ik k = 1,2,...,H x 0, integer 1,...,N ik k = 1,2,...,H It is easily shown that problem P2.1 has a feasible solution if and only N if E Wik < AMk for all k = 1,2,....,H. i=1 Note in problem P2.1 that all costs are positive and that any H feasible solution (X) to the problem must have E xik wH for all k=l i = 1,2,...,N. Hence, it follows that any optimal solution will have each product i produced exactly wiH times during the horizon. This N H implies that the term, c. Z Xik is a constant in any optimal solu i=l1 k=l tion to problem P2.1. Knowing that constants do not affect the optimi zation, problem P2.1 can be reformulated as follows: N H Minimize E Z (2H2k+l) i.x i=l k=l 1 subject to N [P2.2] Z x M k = 1,2,...,H k2 ik . i=l k E x 2 w i = 1,2,...,N j=lij ik k = 1,2,..,H X i. 0, integer 1,2,...,N ik k = 1,2,...,H . This problem can be viewed as a flow problem which can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. For specific details, see Ford and Fulkerson [ 5]. An example network for a Mmachine, twoproduct, threeperiod problem is shown in Figure 2.2. Note in this network that the number of machines does not affect the number of nodes and arcs. Rather for an Nproduct, Hperiod problem, there are H(N+1)+2 nodes and H(2N+1) arcs. In this network the variable xik represents the flow from node k to node ik along the arc (k,ik). Efficient algorithms are available for solving this type of problem (e.g., Fulkerson [6 ]). However, because of its special structure, a more direct solution procedure can be derived. A Solution Procedure The problem formulation calls for each product i to be assigned at least wiH times during the horizon H, and since all costs are posi tive, each product will be assigned exactly wiH times in any optimal solution. For all periods the increase in cost for assigning product i to a facility in period k' as opposed to a facility in period k where k < k : H is 2(kkl)i Hence, from the standpoint of product i, it is economically more advantageous to assign product i in period k rather than period k'. Suppose that the products are numbered so that N N1" 2 .21* If there is an optimal solution to problem P2.2, then there is an optimal solution (X) to problem P2.2 such that Costs Upper Bounds M Lower Bounds "w11 W 2 I W2 1 w12 w22 w23 Figure 2.2 Network Example of Problem P2.1. if p > q and H H if Xpk < (wpH x where E x. 0 (2.2) j=k+l j=H+1 i then x = 0. qk To see that this result is true, assume that an optimal solution (X*) to problem P2.2 has H * x < (w w ) E x pk pH p,kl pj j=k+l and x > 0. Let k be the latest period such that k < k and x > 0. qk pk Such a period must exist if the solution is feasible. By making a pairwise swap, a feasible solution (X) can be constructed where x.. + 1 for (i=p,j=k) and (i=q,j=k') 1J x.. x. 1 for (i=p,jk') and (i=q,j=k) 13 x.. otherwise. This solution has objective function value z = z + 2(kk')q 2(kk')4 Sz since > . *p q H This process can be continued until either xk = (w w ) E x pk pH pk1 j pJ j=k+l or x = 0. In either case an optimal solution with respect to p and q has been constructed. Similarly, by starting with k = H and then letting k = Hl, k = H2,...,k = 2, an optimal solution can be con structed such that condition (2.2) is satisfied for all products and periods. It should also be noted that once the products are numbered there is at most one feasible solution which satisfies condition (2.2). Hence, if a solution can be constructed which satisfies condition (2.2) for all products and periods, the solution is optimal. An algorithm based on the above result can now be given for solving problem P2.2. In the algorithm each planning period is con sidered once where period H is the first period assigned. During each period, the priority in which assignments of products to facilities are made is based on the costs i.. The larger the cost, the greater the product's priority. Considering each product in the order of its pri ority, the number of facility assignments made on product i in period k is based on (a) the availability of facilities after other products with a higher cost have been assigned and (b) the maximum desirable num ber of assignments of product i in period k. Formally, the algorithm is as follows where M denotes the availability of facilities at intermedi ate stages of the procedure and p and t indicate respectively the cur rent product and period under consideration. Step 0. Order the product indices so that N *N1 "'* 2 1 Define wi0 =0 for all i = 1,2,...,N, and set t = H. Step 1. Set M' = M and p = N. Step 2. Determine: H x = min (M', (wH w ) Z x p t pH ,t j=t+ M = M x pt Step 3. Set p = p1. If p 1, go to Step 2. Otherwise, go to Step 4. Step 4. Set t = t1. If t 1, go to Step 1. Otherwise, go to Step 5. H Step 5. Terminate. If xi = wiH for all i = 1,2,...,N, k=l k then an optimal solution (X) to problem P2 has been determined. Otherwise, the problem does not have a feasible solution. The algorithm is a one pass procedure which terminates after exactly HN iterations since each product is considered once during each period. At termination, the constructed sequence is deemed to be feasible or infeasible depending on whether or not wiH facility assign ments have been made for each product. Obviously, an infeasible solu tion can be made feasible only if the number of facilities is increased or if the no stockout restriction is relaxed. Optimality of a con structed, feasible solution follows from the result given in (2.2). An Example As an example of the solution procedure, consider a twoproduct, threeperiod problem where M= 102 312 W = = (10,20) 3 6 8 The product indices are already ordered so that 2 1. Thus the solution procedure goes as follows: a. Assign product 2 twice to period 3 since min {10,2} is 2. b. Assign product 1 eight times to period 3 since min (8,93 is 8. c. Assign product 2 three times to period 2 since min (10,3] is 3. d. Assign product 1 twice to period 2 since min {7,23 is 2. e. Assign product 2 three times to period 1 since min (10,33 is 3. f. Assign product 1 twice to period 1 since min [7,23 is 2. 2 2 8 g. Stop. An optimal solution is X = with objective 3 3 2 function value z = 760. A Warehouse Space Limitation In many production environments, there exists a space limitation on the maximum amount of material which can be carried in inventory at any given time. In the problem considered in this chapter, a warehousing restriction of this type can easily be incorporated into the model for the special case where one machineperiod of production of product i takes up the same amount of space as one machineperiod of production of any other product. For this special case, let Sk denote the maximum number of machineperiods of production which can be carried in inven tory during period k where S1 is assumed sufficient to store the initial product inventories. For the inventories to be within limits, then, it must be the case that the inventory on hand at the end of each period k is less than or equal to Sk and Sk+.l Equivalently, this implies that N k N k N E Z x. min {S Sk + E Z (d. /p.) Z (o /P.) (2.3) i=1 j= 1 k+l i=l j=1 i= for all k = 1,2,...,H. Obviously, (2.3) can never be satisfied if the righthand side is strictly negative. Hence, we are only interested in pursuing the problem for the case where the righthand side is nonnegative for all k = 1,2,...,H. Also, xik must be integer valued for all i = 1,2,...,N and k = 1,2,...,H. Therefore, we can define Uk as the largest integer less than or equal to the righthand side of (2.3). The inventory con straint can then be included in problem P2.2 by annexing the constraint N k E Ex. Uk. i=1 j= 1 The revised problem is still a network problem which can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. In fact, this warehousing limitation can easily be incorporated into the previously given solution procedure by setting X = 0 in step 0 and in step 1 redefining N r M = min (M, min (U E x.)}. r=t,...,H i=l a=t Extensions of the Model There are a number of other realistic aspects which can be easily incorporated into the model. With the exception of (a) below, the previous algorithm is not satisfactory for determining an optimal solution when any or all of these additional enrichments are included in the model. However, an optimal solution can easily be determined for these cases by a minimumcost flow algorithm. The enrichments include: (a) The number of facilities available in each period may vary. This assumption is easily included in the previously given solution procedure by substituting Mt for M in step 1 where Mt denotes the number of facilities available in period t. (b) In addition to the cost of producing each product for one machineperiod, there can be a cost (convex) incurred on the number of facilities in use at any given time. This can be introduced into the network interpretation of the problem by using the standard trick'of replacing each arc (s,k) by M different arcs ((s,k) ; I = 1,2,...,M) each having an upper bound of one on the flow through that arc. Any flow on arc (s,k)2 would incur a cost A the difference in cost for using I facilities during period k rather than 11 facilities. Any cost for not using any facilities is simply a constant which does not affect the optimization. The convex cost assumption implies that AM M1 2 1 which insures that any flow in period k will be placed on arc (s,k)2 before arc (s,k)+1 unless equality holds in which case it does not matter. (c) There can be a technological restriction on the number of facilities (m.) which can be used to produce product i simultaneously. This assumption can be introduced into the network interpretation of the problem by placing upper bounds of m. on all arcs (k,ik). 1 (d) There can be physical limitations (L.) on the maximum number of units of product i which can be carried in inventory at any given time so long as k L. r max [PiWik +iO di i = 1,2,...,N . k=0,1,...,H j=0 This assumption can be introduced into the network interpretation of the problem by placing upper bounds uik on the arcs (ik;i,k+l) where r k Uik = L(Li I1. + Zd )/p k = 1,2,...,1 j= k = 1,2,...,H and [a] is defined as the largest integer less than or equal to a. (e) Period dependent production costs (c ) can be included in ik the network interpretation of the problem by adding cik to the cost (2H2k+l)*i already on the arcs (k,ik). (f) In the formulation of the problem P2.2, it has been assumed that M facilities have been allocated to the production process for use throughout the entire horizon H. Another realistic consideration is a problem where there is competition for these facilities on a long term basis (i.e., the value of M becomes a variable in the optimization). This competition might be denoted by a nonnegative charge f(M) where logically f(M) would be a monotonically nondecreasing function of M. When this charge is included, a procedure for problem optimization is to solve problem P2.2 for the maximum possible value of M. From the optimal solution to this problem, determine the maximum number of facility assignments (r) in any period. Then an optimal solution to the allocation problem occurs for that value of M (M = 0,1,...,r) which min imizes z(M) + f(M) where z(M) is the corresponding optimal solution to problem P2.2. This requires that problem P2.2 be solved up to r times. (g) The cost of producing product i for one machineperiod can be dependent on the facility as well as the period in which the assignment is made. For this case the problem formulation becomes N H M N H M Minimize Z Z (2H2k+l) i. x. + E E c. CiXik i=l k=l 1=1 i=1 k=l a=1 subject to N M [P2.3] Z E xik < M k = 1,2,...,H i=1 =1 k M S x w i = 1,2,...,N j=l E=1 ij ik k = 1,2,...,H i = 1,2,...,N xilk = O or 1 1 = 1,2,...,M k = 1,2,...,H where ci k is the cost of producing product i on facility I in period k and xilk denotes the number of times product i is assigned to facil ity I in period k. An example network interpretation of this problem is given in Figure 2.3. In this example the variable x. k represents flow from node mek to node nik along the arc (m k,nik). (h) Realistically, it may be technologically impossible to produce product i on facility a. This restriction can be included in the model by omitting arcs (mik,nik) in all periods k = 1,2,...,H. Conclusions A multifacility, productioninventory scheduling problem has been considered which can be solved as a minimumcost flow problem. However, because of the nature of the objective function, an even simpler solution procedure has been developed which, even for large problems, can easily be carried out by hand. In addition, the model can be generalized to allow for the inclusion of other realistic Costs m I 5 I + C V2 +C211 Upper Bounds S4V5 22 + c221 m "1 + C 112 1(2 3v 2 + 212 12 \ 22( 3T V I+C122_ "22 3' 2 + C222 \ ('13 +c C113 n 1 32 + c213 13 I 2 I + C 12 n223 Y /2 + C223 Figure 2.3 Network Example with Facility and Period Dependent Costs. Lower Bounds Iw w21 "12 W22 assumptions. Efficient minimumcost flow algorithms are available for solving the general problem when any or all of these enrichments are included. In this model it has been assumed that inventories change linearly from the end of one period to the end of the following period as a result of constant production and demand rates. However, the results (with the exception of the warehouse space limitation) are equally valid for any system in which inventory changes from the end of one period to the end of the following period do not drop below the minimum of the beginning and ending period inventories. This includes the inventory assumptions of a batch type production system in which inventories are updated only at the end of each production period. The only revision in the model would be in the assessment of inventory carrying charges. CHAPTER 3 A NETWORK APPROACH TO THE PRODUCTIONINVENTORY SCHEDULING PROBLEM WITH BACKORDERING Introduction In Chapter 2 the multiple facility, multiple product production inventory scheduling problem without backordering was considered. In this chapter we address the more general problem where backordering to meet demands is allowed. Specifically, the cost of producing each product is dependent on the facility on which the product is produced as well as the period during which the assignment is made. Production completed during period k is added to each product's inventory at the end of the period, and outstanding backorders (B. units) and non negative demands (di units) for product i are satisfied from this inventory. Period and product dependent inventory carrying charges and backordering costs respectively are assessed on material remaining in inventory and outstanding demands at the end of each period. During the horizon the system controller must schedule the products to the facilities in such a manner as to meet all product demands on a first come, firstserved basis, and at the same time build up each product's inventory to a desired nonnegative ending amount (iH units). The prob iH lem objective is to determine a production schedule which minimizes the sum of production, backordering and inventory carrying charges over the horizon. In succeeding sections we formulate this multiproduct, multi facility production scheduling problem as a linear, mixed integer pro gram. Once formulated, this problem can be given a network flow inter pretation which looks very similar to the network interpretation given by Zangwill [18] for his onefacility problem. However, in contrast to Zangwill's problem, this mixed integer programming formulation can be solved as a minimumcost flow problem only for a very special case. We show, however, that this problem can be reformulated so that the general case can also be solved using any of the algorithms for finding minimal cost flows in single commodity networks. This reformulation, in addition to being a model for a very general multifacility, multi product problem can be enriched to include numerous realistic aspects of the production system without severely complicating the.solution procedure. A Mixed Integer Programming Formulation Initially, we will assume that there is a nonnegative number of backorders on the books for each product (i.e., B. o 0) and that initial product inventories for each product i are zero (i.e., I. =0). iO This assumption causes no loss in generality since the firstcome, firstserved rule implies that the first I units of demand for each product i will be satisfied from the initial product inventories irre spective of any decisions made in the problem optimization. Hence, we can assume that any initial inventories have been used to adjust the amount of initial backorders and demands appropriately so that initial inventories can be assumed to be zero. The formulation of this production scheduling problem is further facilitated by redefining demands in terms of cumulative machineperiods of production. This is done by defining wik as the number of machine periods of production of product i required to satisfy all demands in periods one through k. Hence, wk = E d.. + B. /p k = 1,2,...,H1 ik j=1 13 k = 1,2,...,H and H iiH idik + 0 + iH/Pi Sin E \d ik + B + IiH3. A 1,2,...,N. k= Define xijk as the number of times product i is assigned to facility j in period k, and denote by c.. the cost of each of these ijk assignments where c.ij 0. Define Iik as the number of machineperiods of production of product i in inventory at the end of period k, and define Bik as the amount (in machineperiods of production) of unsatis fied demand for product i at the end of period k (i.e., the total amount of product i on backorder at the end of period k). Now let 0ik be the cost incurred per machineperiod of production of product i in inven tory at the end of period k, and let bik be the cost incurred per machineperiod of production of unsatisfied demand for product i at the end of period k where 0i > 0 and bi > 0. The system controller's ik Ik problem is to minimize the sum of production, inventory and backorder ing costs over the planning horizon subject to the constraints that (a) all demands must eventually be satisfied, (b) only one product can be assigned to a facility each period where a facility, when assigned to produce product i, is assumed to produce exactly p. units of the product, and (c) the inventory level for each product i at the end of the horizon must be built up to a minimum ending amount of iH units. A mathematical formulation of this problem is as follows: A mathematical formulation of this problem is as follows: N M H N H1 Minimize E Z 2 c.. x. + E E 0 I i=1 j=1 k=l i=l k=l N H1 ik + E bik Bik ik k k= i=l k=l subject to M k SE x. Iik + B = wi ^ t iji ik ik ik j=1 1=1 M H Z E xi. i w j=1 k=ljk N Sx. < 1 1ijk i i=l k I B 0 ik Bik x. i 0, integer ijk i = 1,2,...,N k = 1,2,...,H1 i = 1,2,...,N j = 1,2,...,M k = 1,2,...,H i = 1,2,...,N k = 1,2,...,H1 i = 1,2,...,N j = 1,2,...,M k = 1,2,...,H Problem P3.1 is a linear, mixed integer program which can be given a network flow interpretation. To construct the network, define a node m. corresponding to each facility j and period k, and define jk a node nik corresponding to each product i and period k. From each node mjk construct a forward arc (m ,ni ) to each node nk for all jk jk ik ik i = 1,2,...,N. The cost per unit of flow on each of these arcs is c.ik. 1Jk Next connect each node nik to node ni,k+l by two parallel forward arcs (nik ,n ) and one reverse arc (n ,n. ). The flow through one of ik ,k+l i,k+l ik the forward arcs must be exactly Wik. Hence, upper and lower bounds of Wik are placed on the flow through that arc. Each unit of flow on the [P3.1] (3.1) other forward arc incurs an inventory charge (0ik), and each unit of flow onthe reverse arc incurs a backordering charge (b i). Finally, ik define a source node s and a sink node t where a forward arc (s,m. ) Ik from node s to node m. is constructed for all j = 1,2,...,M and Jk k = 1,2,...,H and a forward arc (n ,t) from node niH to node t is con iH iH structed for all i = 1,2,...,N. Since only one product can be assigned to a facility each period, a capacity of one is placed on the flow through each of the arcs (s,m k), and since wiH machineperiods of product i must be produced during the horizon, a lower bound of wiH is placed on the flow through the arc (n ,t). Then xik denotes the flow iH ijk from node mjk to node nik along the arc (mk ,nik), Iik and wik corre jk ik jk ik ik ik spond to the amount of flow from node nik to node n. along each of ik 1,k+l the forward arcs (n ni,k+l ), and Bik denotes the flow from node ni, ithe forward arcs kk ik 1 i,k+l to node nik along the reverse arc (ni,k+l,nik). An example network for a twoproduct, onefacility, threeperiod problem is given in Figure 3.1. In this problem if the w 's are all integer, then problem P3.1 can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. For specific details see Ford and Fulkerson [ 5]. However, because of the way in which the w. 's have been defined (i.e., in cumulative machineperiods of production), it is unlikely that these will all be integers. In this case, using a minimal cost flow algorithm will not necessarily provide an optimal solution to the problem which has all xijk integer. In the remainder of this chap ter, we will show how problem P3.1 can be reformulated so that it can be solved using any of the algorithms for finding minimalcost flows in single commodity networks when the ik's are not all integer. Production Costs Upper and Backordering Lower Bounds Costs 1 Figure 3.1 Network Example of Problem P3.1. Reformulation as an All Integer Program An optimal solution to problem P3.1 has the property that inventory carrying charges and backordering costs will not occur simul taneously for any product i and period k (i.e., I. B. = 0). This ik ik result follows from the fact that I.k and Bik occur in only one con ik ik strain, and the cost coefficients associated with each variable are strictly positive. Consider an optimal solution to problem P3.1 having B. > 0 Ik for some product i and period k. Then, from the above result I = 0 ik and from constraint (3.1) M k 2 x. + B = k (3.2) j=l 1 1J k ik Now, breaking up each term into its integer part and its fractional part where [r] denotes the largest integer less than or equal to r and f(r) denotes the quantity r [r] (i.e., for nonnegative r the fractional part of r is denoted by f(r) where 0 f(r) < 1), equation (3.2) becomes M k 2 E x.j + [Bik] [ ] = f( ik) f(Bik) (3.3) j=1 2 j1 ik .k The lefthand side of equation (3.3) is made up of integer terms only, which implies that f(w ) f(Bi ) must be an integer. This, coupled with the fact that 0 : f( ik) < 1 and 0 : f(B i) < 1, implies ik ik further that f(wik) f(B ik) = 0. Hence, an optimal solution to problem ik Ik P3.1 having B > 0 has ik f(Bk) = f(wi) ( ik ik (3.4) In similar fashion consider next an optimal solution to problem P3.1 having Ik > 0 for some product i and period k. Then B = 0 and Ik ik from constraint (3.1) M k S E i I =w ik (3.5) j=1l = i ik ik Breaking up each term into its integer part and its fractional part, equation (3.5) becomes M k S xij [Iik] ik = f(wik) + f(Iik). (3.6) j=1 1 k Ik As in the backordering case, the lefthand side of equation (3.6) is made up of integer terms only which implies that f(w ) + f(I ) must be an integer. This, coupled with the fact that (a) f(w. ) equals ik zero when wik is an integer and is strictly positive when wik is not an integer, (b) 0 f( ik) < 1, and (c) 0 f(Ii ) < 1, implies further ik ik that f(w ik) + f(ik) equals one if wik is not an integer and equals zero otherwise. Hence, an optimal solution to problem P3.1 having Ik > 0 Ik has 1i f(w ) if wik integer Ik ik f(ik) = (3.7) 0, otherwise. Finally, by defining I as a 01 variable which takes on the ik value one if Iik > 0 and wik integer and zero otherwise, relationships (3.4) and (3.7) become f(B i) = f(wik) 1i Iik f(I ) = [I f(wi))i . ik ik ik (3.8) Sufficient relationships have now been developed so that a new formulation of the production scheduling problem can be given. To I / formulate this problem, let I and Bk correspond to [I. ] and [Bk.], ik ik ik Ik respectively, and for notational convenience define aik = f(w ik) and wik = [w ]. Now replacing Ik and Bi in problem P3.1 by their integer ik ik Ik ik and fractional parts and then substituting the expressions in (3.8) for f(B ik) and f(Iik ), we obtain the following formulation: ik ik M H N H11 E Zc.. x. + 2 E {0 Ii + jl kCijk ijk+ i=l ik ik j=1 k=l i=l k=I N H1 + E bik k + a ik i=1 k=l subject to M k Sx. I I + B =w = =1 I ik ik ik ik j=1 X=1 M H E E x. w j=1 k=l jk x < 1 1 ijk  i=l if Ii > 0 and w ik integer /I ik k ik Otherwise O otherwise X..jk I B k 0, integer 1jk ik' ik i = 1,2,...,N k = 1,2,...,H1 i = 1,2,...,N j = 1,2,...,M k = 1,2,...,H i = 1,2,...,N k = 1,2,...,H1 j = 1,2,...,M k = 1,2,... ,H N Min E i=l [P3.2] (3.9) (3.10) (3.11) ((1 ia i k0 a b iklI Problem P3.2 is an all integer program. It follows from our previously derived results that solving problem P3.2 is equivalent to solving problem P3.1. We now proceed to show that problem P3.2 can be converted into a minimalcost flow problem. Formulation as a MinimalCost Flow Problem Note in the objective function of problem P3.2 that the term N H1 E E aik b is simply a constant, and since constants do not affect i=l k=l the optimization, this term can be omitted. Note also that any feasible solution to problem P3.2 must have all x integer. Hence w in con ijk iH straint (3.10) can be replaced by the smallest integer greater than or equal to wH (i.e., (w )) without any effect on the optimization. In iH Finally, we will show that constraint (3.11) can be replaced by I =0 o r e (3.12) ik ik k =1,2,...,H1 where eik is a constant defined to be one if wik integer and is zero otherwise. To see that (3.11) can be replaced by (3.12), let P3.2' denote problem P3.2 with (3.12) used in place of (3.11). From equation (3.5), I.k > 0 if and only if ik M k E x.ij > w.k wi' (3.13) j=1 1_ ik ik and from constraint (3.2) and the integrality requirements of xijk Iik = O if and only if ik ik M k Z Z xi. < w w. < (3.14) j=l 1= 1 ik ik Together, (3.13) and (3.14) imply that constraint (3.11) is equivalent to M k 1 ,if E xi wik >0 and w i integer j=1 2=1 i i = 1,2,... ,N ik k = 1,2,...,H1. S, otherwise (3.15) Note next that problem P3.2' is a relaxed version of problem P3.2. Hence, an optimal solution to problem P3.2 is also an optimal solution to problem P3.2 if it is feasible to problem P3.2. From (3.15) this implies that an optimal solution to problem P3.2 is also an optimal solution to problem P3.2 if in this solution M k (a) Ii = 1 implies E x.. w >0 and w. / integer ik j 1 iji ik ik j=1 =1i and M k (b) I = 0 implies E Z x.. w <0 and/or w. = integer. ik j= .1 ij ik ik Case (a): Obviously, wik cannot be integer valued since this would imply that I/ would be restricted to the value zero. Hence, to prove this ik M k case we assume that Ii = 1, w i integer, and E E xij wik<0 for ik ik l I.J ik j=1 =1 some product i and period k in an optimal solution to problem P3.2 . Then in order for constraint (3.9) to be satisfied, it must be the case that B. 1. This, however, implies that there exists a feasible ik solution to problem P3.2' having I" and Bi each reduced in value by ik ik one yielding an objective function value improvement given by Az = (la )0ik+ a b b ikik ik ik ik < 0 since (1a k), bik, 0k > 0. Hence a contradiction since we could not have started with an optimal solution. Case (b): Obviously, (b) holds if Wik= integer. Hence to prove this case M k we assume that I integer, and E x. >0 for some ik ik j=a 1 i ik product i and period k in an optimal solution to problem P3.2 Then in order for constraint (3.9) to be satisfied, it must be the case that Iik 1. This, however, implies that there exists a feasible solution to problem P3.2' having Ik increased in value by one and I' decreased ik ik in value by one yielding an objective function value improvement given by Az = ik + (1 aikik aikbik ik ik ik < 0 since aik, bik' 0ik > 0. Hence a contradiction since we could not have started with an optimal solution. Therefore, cases (a) and (b) do hold at optimality in problem P3.2 which implies that solving problem P3.2' is equivalent to solving problem P3.2. Making the specified changes in the objective function and constraint (3.10) in problem P3.2 and replacing constraint (3.11) by (3.12), problem P3.2 becomes as follows: N M H N Hl Min E E E c x + E E~li Ik ijk ijk ik i=l j=1 k=1 i=l k=1 N Hi + (l(aik) ikaikb. JI k + E bikBik i=l k=l subject to M k E x E' + i = 1,2,. ..,N j=1 2 1ij ik ik ikik k = 1,2,...,H1 M H [P3.3] E E xj (w ) i = 1,2,...,N j=1 k=l ik Nj E x < 1 j = 1,2,...,M Sijk k = 1,2,...,H [ < e i = 1,2,... ,N ik ik k = 1,2,...,H1 i = 1,2,...,N ik ik Ik Bi k 0, integer j = 1,2,...,M ijk' kk = 1,2,...,H . The multiple facility, multiple product productioninventory scheduling problem with backordering is now in the format of problems which can be solved using any of the algorithms for finding minimal cost flows in single commodity networks. Efficient algorithms are available for solving this type of problem (e.g., Fulkerson [ 6]). The network structure corresponding to problem P3.3 is similar to that of problem P3.1. The primary difference is in the construction of a second arc (nik,n i,k+) from each node nik to node ni,k+. Flow through each of these arcs is limited by the capacity e. and a cost, Ik (1a ik ik aikbik, is incurred for each unit of flow placed on the arc (nik ,n k+l). The actual amount of flow placed on this arc is denoted by the variable I An Nproduct, Mfacility, Hperiod problem ik of this type will have H(M+N)+2 nodes and H(M+MN+4N)3N arcs. An example network is shown in Figure 3.2 for a onefacility, twoproduct, three period problem where for notational convenience sik is defined to be ((1aik)ik aikbik). In addition to being a model for a very general multifacility, multiproduct productionscheduling problem, there are a number of other realistic enrichments to the problem which can be included in the model without severely complicating the solution procedure. These are dis cussed below: (a) There can be a technological restriction on the number of facilities (qik) which can be used to produce product i simultaneously in period k. This generalization can be introduced into problem P3.3 by adding the constraint x < q i = 1,2,...,N Mx j=1 ijk ik k = 1,2,...,H. (b) In addition to the cost of producing each product for one machineperiod, there can be a cost incurred on the number of facil ities in use at any given time. Let rk denote the cost per facility in use during period k where rk 2 0. Then this enrichment is included in the problem formulation by adding the following term to the objec tive function, H N M Srk Z xjk k=l i=l j=l Production Costs Upper and Lower Bounds Figure 3.2 Network Example of Problemc P3.3. Upper Bounds However, this cost function does not have to be linear. Rather the costs can be convex. The insertion of a convex cost term is a stan dard trick in networks and is discussed in detail in Ford and Fulkerson [ 5]. (c) The number of facilities available in each period may vary. This generalization is incorporated in the model by omitting those nodes mk (and the associated arcs) in period k corresponding to any jk facility j which is unavailable. (d) Upper bounds can be placed on the values of I' and Bk imply ik ik ing physical limitations on the amount of product i which can be carried in inventory or on backorder during period k+1. (e) Realistically, it may be technologically impossible to produce product i on facility j. This restriction can be included in the model by omitting arcs (mk ,nik) in all periods k = 1,2,...,H. Conclusions A multiple facility, multiple product productioninventory scheduling problem with backordering has been considered. In a straightforward fashion this problem has been formulated as a linear, mixed integer program which can be given a network flow interpretation. However, this formulation can be solved as a minimalcost flow problem only for a very special case. In this paper it has been shown that the general problem can be reformulated so that it can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. This reformulation, in addition to being a model for a very 44 general multifacility, multiproduct problem, can be enriched to include numerous other realistic aspects of the production system without severely complicating the solution procedure. CHAPTER 4 A MULTIPLE FACILITY, MULTIPLE PRODUCT PRODUCTION SCHEDULING PROBLEM WITH OVERTIME Introduction In the real environment, many production systems face fluctuating demand patterns which often tax the productive capability of the system to meet these demands. As a consequence, many schedulers, or system con trollers, use overtime to increase the productive capacity of the system. When this option is available, the cost of the overtime must be taken into account, and production costs become a function of the amount of overtime used as well as the straight time production amounts. In this chapter the problem of scheduling lots to facilities is considered for the case where an overtime option is available to the system controller. Specifically, an industrial process made up of M facilities in parallel is considered where there are N different products to be pro duced over a finite planning horizon. During any period k, M facilities are available for production where Mk < M. All of these facilities are considered to be identical in that p. units per period of product i can be produced by any available facility. During any given period at most one product can be scheduled on each facility. Corresponding to each of these assignments is a cost, denoted by c.k which includes the cost of using facility j in period k to produce product i as well as the of using facility j in period k to produce product i as well as the associated costs of producing p. units of product i. At the end of each production period, the system controller has the additional option of scheduling an overtime shift for producing product i on facility j when product i is scheduled on straight time on facility j. The incre mental cost per period for this overtime production is denoted by gi where gi > 0. (Alternatively, this problem can be thought of in a batch type production process where the scheduler must decide whether to pro duce zero, one, or two batches of product each time the product is assigned to a facility for one period. It can also be viewed in the context of a continuous production process where the scheduler has the capability of doubling the production rate of each of the facilities.) Production completed on straight time and overtime during period k is added to each product's inventory at the end of the period, and nonneg ative demands (dik units) are satisfied from this inventory. During the horizon the system controller must schedule the products to the facilities in such a manner as to meet all product demands, without suffering backorders or stockouts, and at the same time build up each product's ending inventory level to a minimum nonnegative amount denoted by IiH. The problem objective is to determine such a production sched ule which minimizes the sum of straight time and overtime production costs over the horizon. In succeeding sections, the overtime production scheduling problem is formulated as a linear, integer program. This formulation can be given a network flow interpretation, but in general the problem cannot be solved using any of the minimalcost flow algorithms. How ever, it is shown that this problem can be solved by solving a relaxed version of the problem. This relaxed problem has the property that it can be solved using any of the very efficient algorithms for finding minimalcost flows in single commodity networks. Finally, a single pass algorithm is developed for solving a special case of the overtime problem. The Overtime Production Scheduling Problem In this problem product inventories are updated only at the end of each production period, and product demands must be satisfied from these endofperiod inventories without allowing backordering or stock outs. Hence, requiring that the net ending period inventories be non negative for all products and periods is a necessary and sufficient con dition for all demands to be met without backorders or stockouts. To facilitate the formulation of this problem, define wik as the minimum number of machineperiods (integer) during which product i must be produced in the first k periods in order to have a nonnegative inventory level at the end of period k. Then, k ik = max [0, (( Z d. )/Pi k 1,2,...,H j=1 where (a) denotes the smallest integer greater than or equal to a. In addition to being nonnegative, product i's inventory level at the end of the horizon must be greater than or equal to IiH. Hence, define 1H H w =max (O, ((iH + E d.. i )/p i = 1,2,...,N j=l and let W be an N by H matrix with components [w ik. ik Define yijk as the number of times facility j is scheduled to produce product i on straight time during period k, and denote by c.i 1jk the cost of each of these assignments where cijk > 0. Similarly, define xijk as the number of times facility j is scheduled to produce product i on overtime during period k, and denote g. as the incremental cost (product dependent only) of producing product i for any overtime machineperiod where g. > 0. Then the system controller's problem is to minimize the sum of straight time and overtime production costs sub ject to the constraints that (a) only M facilities are available for production in period k, (b) demands must be satisfied without allowing backorders or stockouts, (c) product i can be scheduled on overtime on facility j during period k only if it has already been scheduled on straight time on facility j during period k, and (d) only one product can be scheduled on a facility each period where a facility, when assigned to produce product i, is assumed to produce exactly p. units of the product each machineperiod. A mathematical formulation of this problem is as follows: N H k Minimize E E c yijk + g.x. i=1 k=l j=l k subject to k Mk Z Z[Y + x ] 2 w i(4.1) =1 j=1 ij ij ik k = 1,2,... ,H N < k = 1,2,... ,H [P4.1] y 1 (4.2) i= ijk j = 1,2,...,Mk i = 1,2,... ,N yjk xjk k 1,2,...,H (4.3) j = 1,2,... ,Mk i = 1,2,...,N Yijk' Xijk O, integer k = 1,2,...,H (4.4) j = 1,2,...,Mk Problem P4.1 is a linear, integer program which can be given a network flow interpretation. However, the problem cannot in general be solved using any of the minimalcost flow algorithms. We will show, though, that a relaxed version of the problem can be formulated and that this relaxed problem, when solved, will provide an optimal solu tion to problem P4.1. This relaxed problem will have the property that it can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. The formulation of this problem is based, in part, on the following result: Lemma 4.1 A necessary and sufficient condition for problem P4.1 to have a feasible solution is that there exists a set of y ijk's satisfying constraints (4.2) and (4.4) and having k M =1 j=l ijI (wik/2) (4.5) for all i = 1,2,...,N and k = 1,2,...,H. Proof of Lemma 4.1 A. Necessary Condition: Assume that there exists a feasible solution (Y*,X*) to problem P4.1 having k MI E Y j (wik /2) (4.6) 1=1 j=1 ij2 ik for at least one product i and period k. Then (4.6) and constraint (4.3) together imply that k MI E E (y* + x <2 (W /2) 1 < wi .c=i ij2 + Xij^2 ik1 1 ^ik L=1 j=1 ; ]k k A contradiction exists since constraint (4.1) cannot be satisfied unless (4.5) holds. B. Sufficient Condition: Set xijk = jk for all i = 1,2,...,N; k = 1,2,...,H; and j = 1,2,...,Mk. Then, by construction, constraints (4.1), (4.3) and (4.4) are satisfied. Hence, (Y,X) is a feasible solution to problem P4.1. Q.E.D. Since condition (4.5) must be satisfied by all feasible solu tions to problem P4.1, it can be included in the problem as a redundant constraint. To formulate a relaxed version of problem P4.1, then, we add constraint (4.5) to the problem, omit constraint (4.3), and make H Mk the variable transformation z. = E i x The relaxed problem S k=l j=l can be formulated as follows: N H A N Minimize E Z E c.ij Yijk + E g.z i=l k=l j=l i=l g subject to k M M i 1 j= i ,2,..., N [P4.2] yik < 1 k = 1,2,"... (4.8) i=1 jk j = 1,2,...,M k M i1l i = 1,2,...,N (4.9) = j=1 Yij w Wik/2) k = 1,2,... ,H i = 1,2,... ,N yijk zi 0, integer k = 1,2,...,H j = 1,2,... ,M Since (a) problem P4.2 is a relaxed version of problem P4.1, and (b) problem P4.2 includes, as constraints, the necessary and sufficient conditions specified by Lemma 4.1, it follows that problem P4.1 has a feasible solution if and only if problem P4.2 has a feasible solution. Let (Y,Z) be an optimal solution to problem P4.2. Then (Y ,X ) is an optimal solution to problem P4.1 where, by construction, o Yijk = ijk k j i 1,2,...,N yj if E E y ir z k = 1,2,...,H (4.10) 1=1 r=l j = 1,2,..., 0 o xijk 0 otherwise. To show that (Y ,Xo) is a feasible solution to problem P4.1, note first that constraints (4.2), (4.3) and (4.4) are satisfied by construction since (Y,Z) is a feasible solution to problem P4.2. To show that constraint (4.1) is satisfied, we note from (4.10) that Y = Y and that either k aM k (a) E E x = E y ij (4.11) a=1 j=1 1 =1 j=l1 or k MI o (b) E E x = z (4.12) =1l j=l j1 i for all i = 1,2,...,N and k = 1,2,...,H. Case (a): After substituting Y for Y, constraint (4.9) and equation (4.11) together imply that k M SZ yo( + xo. 2(wik/2) > wi. 11 j=1 i k Hence constraint (4.1) is satisfied for this case. Case (b): Constraint (4.7) and equation (4.12), together with the fact that Yo = Y implies that k M Z Z ( +y + x . 1=1 j=1 ij ij ik Hence, constraint (4.1) is also satisfied for this case, and (Y ,X ) is a feasible solution to problem P4.1. In minimization problems, it is well known that the optimal objective function value for a relaxed version of a problem represents a lower bound on the value of the objective function of any feasible solution to the unrelaxed problem. Therefore, since (a) (Y ,X ) is a feasible solution to problem P4.1, and (b) the objective function value of the solution (Y ,X ) is equal to the objective function value of the optimal solution (Y,Z), it follows that (Y ,X ) is an optimal solution to problem P4.1. Problem P4.2 can be viewed as a flow problem which can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. An example network for a oneproduct, threeperiod problem with two facilities available in each period is shown in Figure 4.1. In this network the variable y.ij represents the flow from ijk c 0 0(0 tn0 00 node mj to node n along the arc (mjk,ni ), and the variable zi rep jk ik jk ik 1 resents the flow from node s to node oil along the arc (s,o ). Efficient algorithms are available for solving this type of problem (e.g., Fulkerson [6]). A Special Case of the Overtime Scheduling Problem For the special case where the production costs are not facility and period dependent (i.e., c.. = c. for all k = 1,2,...,H and ijk i j = 1,2,...,Mk), a single pass procedure can be developed for solving the overtime production scheduling problem. This procedure, even for large problems, can easily be carried out by hand. Before describing the procedure, however, note first that facility discrimination is no "k longer necessary, and we can let Yik = 2y Problem P4.1, then, j=1 reduces to the following formulation: N H N Minimize E E c.y + Z g.z. ik i.i i=l k=l i=l subject to k z + y w = 1,2,... (4.13) i ai ik k = 1,2,...H [P4.3] N ik y Mk k = 1,2,...,H (4.14) i=l k I iik k = 1,2,...,H i = 1,2,...,N ik' zi 0, integer k = 1,2,...,H The procedure to be given for solving problem P4.3 involves two phases. Phase I determines a set of yik' s (if they exist) which satisfy the necessary and sufficient conditions for a feasible solution to problem P4.3 to exist as stated in Lemma 4.1. Phase II, then, builds onto this Phase I solution in order to determine an optimal solution to problem P4.3. Problem P4.3 has a feasible solution if and only if there exists a set of ik 's which satisfy constraints (4.14) and (4.15) for all i = 1,2,...,N and k = 1,2,...,H. The set of yik's is determined by omitting constraint (4.13) and solving the relaxed problem. In terms of the network given in Figure 4.1, this is equivalent to omitting those nodes oik for all i = 1,2,...,N and k = 1,2,...,H and all arcs leading into and out of these nodes. To solve this relaxed problem, a backward procedure is used which is essentially equivalent to a flow algorithm. The procedure is as follows where R denotes the availability of facil ities at intermediate stages of the procedure and p and t indicate respectively the current product and period under consideration. Phase I Step 0. Define (w i/2) = 0 for all i = 1,2,...,N and set t = H. Step 1. Set R = Mt and p = N. Step 2. Determine: H pt = min {R, (w pH2) (Wp t_/2) Z y p j=t+l H where Z y 0 j=H+1 R = R pt Step 3. Set p = p1. If p 1,.go to Step 2. Otherwise, go to Step 4. Step 4. Set t = t1. If t 1, go to Step 1. Otherwise, go to Step 5. H Step 5. Terminate. If yik = (w iH/2) for all i = 1,2,...,N, k=l then an optimal solution has been determined for the relaxed problem. Otherwise, problem P4.3 does not have a feasible solution. If, in Phase I, an optimal solution to the relaxed problem is determined, then we proceed to Phase II, since from Lemma 4.1 a feasible solution to problem P4.3 is known to exist. Otherwise, we stop, since problem P4.3 does not have a feasible solution. Before proceeding, however, note that any feasible solution to the relaxed problem having each product i assigned exactly (w i/2) times during the horizon is an optimal solution to the relaxed problem. Hence, the relaxed problem may have many alternate optimal solutions. Using the backward proce dure approach to determine one of these optimal solutions has the fol lowing important advantage, however, in determining an optimal solution. The Phase I procedure at termination for each period k has the property that either (a) all MA facilities are assigned, or (b) the only product assignments which can be reassigned to period k (without violating feasibility or assigning a product more than (wiH 2) times) are those which are already assigned to a period r where r k. Example 1 As an example of the Phase I procedure for solving the relaxed version of problem P4.3, consider a twoproduct, threeperiod problem where c = (6,3) W M = (8,8,10) An optimal solution is det a. Assign product b. Assign product min (9,93 = 9. c. Assign product min (3,8) = 3. d. Assign product e. Assign product min (3,8] = 3. f. Assign product 4 6 24 (w/2) S 6 12 14] ermined as follows: 12 7 . two once to period 3 since min (1,10) = 1. one nine times to period three since two three times to period two since one once to period two three times to two since min (1,53 = 1. period one since one twice to period one since min (2,5] = 2. 2 1 9 g. Stop. An optimal solution is Y = 3 3 1 Assuming Phase I yields a feasible solution to the relaxed problem, we now proceed to Phase II. In this phase an optimal solution to problem P4.3 is determined by building on to the Phase I solution. This is equivalent to assuming that the part of an optimal solution which has been determined in Phase I is fixed (i.e., let Y' denote the solution obtained in Phase I) and now all that remains is the determin ation of the remaining portion of the solution (i.e., denote the remain ing portion of the solution by (Y',Z)). The Phase II problem to be solved is formulated as follows: N H N Minimize E E [cik + ciYk + E g.z i=l k=l i=l subject to k k S// i i = 1,2,...,N zi + Y i ik Yi k = 1,2,...,H 2=1 i =I [P4.4] N N yik Mk ik k = 1,2,...,H i=1 1=1 a i = 1,2,...,N Yik' z. 0, integer i 1,2,...,H ik 1 k = 1,2,...,H A network interpretation of problem P4.4 is obtained from Figure 4.1 by omitting all arcs (nik,ni,k+l) for all i = 1,2,...,N and k = 1,2,...,H1, and all arcs (n. ,t) for all i = 1,2,...,N. Also N redefine MA = h E yik for all k = 1,2,...,H. Now note from this i=l network interpretation that for all products having ci g gi that it is at least as advantageous to assign overtime as it is to assign straight time. Since this can be done without causing facility conflicts, all demands for these products are satisfied by overtime assignments. For those products having ci < gi, it is more advantageous to assign straight time than it is overtime. However, there are a limited number of avail able facilities left. Hence, it is more advantageous to assign a product i having (c.g.) < (c.g.) to a vacant facility than it is a product j. To determine an optimal solution to problem P4.4, we again use a back ward procedure because of the useful property which it allowed in the Phase I procedure. In the procedure, sets S and S are defined where S contains those products for which an optimal form is known (i.e., ini tially those products having c. > g.), and S contains those products 1 1 for which an optimal solution form is not known. The procedure then considers one product in S at a time(i.e., the first product considered is that product in S having the minimum value of c.g.). Once the form of an optimal solution is determined for this product, then it is trans ferred from set S to S. This process continues until S is an empty set. In the procedure we build on to the yik's directly rather than using the Y" and Y notation. Phase II Step 0. (a) Order the product indices so that C11 c2g2 ... CN gN. (b) Define the sets S = [ilcgi < 03 and S = ilc.gi a 03. I i 1 i H (c) Set z. = W Z yik for all i = 1,2,...,N. k=1 Step 1. If S Z 0, go to Step 2. Otherwise, stop. An optimal solution is (Y,Z). Step 2. Set p = max i and t = H. ViES Step 3. (a) Determine N t1 a = min z p, Mt Ey i z p + E y w 3 i=1 &=0 where y and w are defined to be zero for all i = 1,2,...,N. (b) Set pt = pt + a and z. = z. a. 1 1 Step 4. If t = 1 or zi = 0, go to Step 5. Otherwise, set t = t1 and return to Step 3. Step 5. Set S = (Sp3 and S = [S+p], and go to Step 1. Since overtime periods are assigned without facility conflict, it is clear that the only way there could exist a better optimal solu tion to problem P4.3 than that obtained by Phases I and II is if there exists an optimal solution (Y*,Z) having H H H H E k= E y + 1 and E = k y 1 ik ik jk ik k=l k=l k=l kk= for at least two products i and j where c.g. < c.g.. However such a solution cannot exist since (a) in Phase II product i was assigned to all periods before product j, and (b) in Phase I the backward proce dure has the property that there does not exist a feasible interchange of these two product assignments. Example 2 As an example of the Phase II procedure, reconsider the problem given in Example 1 where, in addition, g = (5,9). An optimal solution is determined as follows: a. The products are already indexed so that clg 1 c2g2. Hence, since c g1 = 1 and c2g2 = 6, we define S = '13 and S = (2]. b. Set y23 = 1 + 0 = 1 since min (7,0,1} = 0. c. Set y22 = 3 + 4 = 7 since min [7,4,4] = 4. d. Set y21 = 3 + 3 = 6 since min [4,3,3j = 3. e. Stop. An optimal solution is Y = and Z = (12,0). 16 7 1] Conclusions A multiple facility, multiple product overtime scheduling problem has been considered. A linear, integer programming formulation was given for the problem which in this case can be solved by solving a relaxed version of the problem. This relaxed problem was shown to have the property that it can be solved using any of the algorithms for finding minimalcost flows in single commodity networks. However, for an important special case a single pass algorithm was developed which is more efficient than the standard flow algorithms. CHAPTER 5 AN EFFICIENT INTEGER PROGRAMMING ALGORITHM FOR A MULTIPLE BATCH PRODUCTION SCHEDULING PROBLEM Introduction In this chapter a multiple batch production process is considered where the horizon length is a variable in the optimization. During any production period, each of the M facilities has the capability of pro ducing up to m. batches of product. A learning effect is experienced 1 in the production of these batches and is exhibited in the production costs. The cost (c ) of the Ith batch produced on a facility during any period is no less than the cost (ci, +) of the 4+1st batch produced for all I = 1,2,...,m.1. A batch of product i contains p. units, and 1 only whole batches are produced. Production completed during period k is added to each product's inventory at the end of the period, and non negative demands (dik units) for product i are satisfied from this inventory. It is assumed here that under normal operating conditions the scheduler, or system controller, for this process has a production plan for scheduling the N products to the M facilities. This production plan may be a schedule based on the wellknown infinite horizon economic manu facturing quantity which was discussed in Chapter 1. In these formula tions of the problem all demands must be met without allowing backorder ing, and the problem objective is to determine a repetitive production cycle for allocating the production time of a facility to each of the N products so that costs are minimized. In most treatments the addi tional assumption is made that initial product inventory levels are "in phase with" the infinite horizon cycle plan meaning that the product inventory levels are all sufficiently large for an infinite horizon cycle plan to begin immediately without suffering stockouts. When ini tial product inventories are considered in the problem, there may exist a subhorizon of unknown time duration (h) during which the facilities must be scheduled "out of phase with" the infinite horizon cycle plan in order to avoid stockouts or backorders. In this chapter an extremely efficient procedure is developed for bringing a system into phase with a given production pattern such as an infinite horizon cycle plan. Let iH denote the minimal inventory level in units for each iH product i which, when taken collectively for all products, represents an acceptable condition or "state of the system" for the system con troller's desired production pattern to begin. Realistically, there is usually more than one such state which can be used since the entry point into a repetitive production cycle is ordinarily not unique. (However, the number of possible entry states is finite since time is discrete.) In this chapter we develop a solution procedure for a single entry state and leave to the reader the straightforward reapplication of the solution procedure for other entry states. The scheduler's problem then is to determine an assignment of products to facilities and the corresponding production amounts at each of these assignments over a variable horizon h subject to the constraints that (a) all demands are satisfied without allowing backordering, and (b) the nonnegative, initial product inventory level I (units of product i) for all products is converted to a desired nonnegative, ending inventory level liH. The objective is to minimize the sum of production costs and a general cost function f(h) which is a monotonically nondecreasing func tion of the horizon length h. In succeeding sections, we identify a specially structured integer program and present an extremely efficient algorithm for its solution. After identifying special properties of this general problem, the multiple facility, multiple product production scheduling problem is formulated as an integer program and shown to have the same special structure. An Integer Program Noteworthy contributions have been made to the present stateof theart of solving discrete optimization problems (i.e., for a survey of many of these, see Geoffrion and Marsten [7 ]), but to date the research has not provided an efficient procedure for solving the general problem. Rather, researchers have been most successful in (a) identify ing problem structures which insure integer termination of the corre sponding relaxed (noninteger) optimization problem, and (b) determin ing solution techniques which are applicable to specific classes of discrete problems. In this section a specially structured integer programming formulation is identified for which a very efficient algorithm will be presented for determining its solution. The problem formulation is given below where (a) J is the index set of integers, (b) all functions 1 are real valued, (c) gk (*) exists for all k = 1,2,...,, and (d) for any vectors e = (e ,e ,... ,e) and x = (x ,x 2...,x ) having e. > 0 1 n 2' n i and x. 2 L. for all i = 1,2,...,n, f(x) < f(x + e) and 1 1 gk (hk (x) gk hk(x + e)3. Minimize z = f(x) subject to [P5.1] k(x ) hk(x) W O k = 1,2,...,m 3k k k EJ5 xi > L., integer i = 1,2,...,n (5.2) 1 1 A result which provides the basis for an algorithm for solving problem P5.1 is the following: Lemma 5.1 If lower bounds, Li, are known for all variables x i = 1,2,...,n, in problem P5.1, then for the variable x. in any constraint k of (5.1), 1 3k L. = max {L., ( is also a lower bound on the variable x in problem P5.1. (Note that Jk (a) implies the smallest integer greater than or equal to a.) Proof of Lemma 5.1 Assume that there exists an x. < L. in a feasible solution to 3k Jk problem P5.1. By assumption, L. is a known lower bound on the value 3k of x in any feasible solution to the problem. Hence it must be the case that x. 2 L. which implies from (5.3) that 3k 3k 1 L. = (g hk(L))> (5.4) k But (5.4) is just constraint k of (5.1) rewritten as an equality with 1 each variable x. set at its known lower bound. Since gk h k(L) < 1 * g [h (L + e)) for any e 2 0, the only way a variable x. < L. can be 3k 3k feasible to problem P5.1 is if some x. = L. is decreased in value. Hence, 1 1 a contradiction exists since L is a lower bound on the value of x. in 1 1 any feasible solution to problem P5.1. Q.E.D. An Algorithm for Solving the Integer Program As is, the problem formulation has an interesting structure. The objective function is a nondecreasing function for all increasing values of x.. Each of the constraints in (5.1) has one variable which can be expressed as a nondecreasing function for all increasing values of x.. A feasible solution to the problem requires that all variables be integer valued and bounded from below. Using these properties and assuming that a feasible solution to the problem is known to exist, an algorithm for solving the problem can be described as follows: Step 0. Set x = L. Step 1. If any constraint in (5.1) is not satisfied, go to Step 2. Otherwise, stop. An optimal solution is (x). Step 2. Pick any unsatisfied constraint in (5.1), redefine 1 x. = (g kh (x)3) and return to Step 1. and return to Step 1. At each iteration of the procedure, the algorithm either terminates or the value of a variable in an unsatisfied constraint is increased by at least one. Therefore, since (a) a feasible solution is known to exist, and (b) from Lemma 5.1 each variable is always reset at a value equivalent to a known lower bound on that variable, the algorithm must terminate in a finite number of iterations. Furthermore, since the value of the objective function in problem P5.1 is nondecreas ing for an increasing value of any x., optimality of the solution at termination also follows from Lemma 5.1. An Example Consider the following example which is in the format of problem P5.1. Minimize z= 0.5x + 2x subject to 2x1 0.5x2 2 10  x + 4x 2 8 xI 0.1x2 2 1 x x2 2 0, integer (5.5) (5.6) The problem is illustrated in Figure 5.1 where an optimal solution is obtained by the iterative scheme in the following manner. a. Set x = 0. b. Constraint (5.5) is not satisfied so redefine x1 = (<10 + 0.5(0)5/2) = 5. c. Constraint (5.4) is not satisfied so redefine x2 = ((8 + 1(5)/4) = 4. I 0.1x2= I 2x 0.Ix2 =10 Feasible Integer Region 20 15 10 5 20 Geometric Interpretation of the Algorithmic Example. xi +4x2= 8 Figure 5.1 d. Constraint (5.5) is not satisfied so redefine x1 = ([10 + 0.5(4))/2) = 6. e. Stop. All constraints are satisfied. An optimal solution is x = (6,4) with objective function value z = 0.5(6) + 2(4) = 11. Other Properties of the Integer Program In addition to the ease of solution, problem P5.1 has a number of additional properties which are of interest. Property 1 Consider any two problems having identical constraint sets of the form required in problem P5.1, but different objective functions. The same optimal algorithmic solution solves both problems irrespective of the objective function f(x) so long as each satisfies the condition f(x) : f(x + e) for all x and any e 0. Property 2 Any feasible solution (x) to a problem P where problem P includes the constraint set of problem P5.1, has x L x* where x* is the optimal algorithmic solution to problem P5.1. Property 3 The algorithmic solution to problem P5.1 is the unique optimal solution to the problem if f(x) < f(x + e) for all x and all e 2 0 where at least one component of e is strictly positive. Property 4 The corresponding continuous portion of problem P5.1 (omitting the integer requirements on x in constraint (5.2)3 has an optimal solu O O tion (x ) which has all x. integer when a feasible solution exists if 1 1 gk [(hk(x)] is integer valued for all x = integer and k = 1,2,...,m. Note finally that in the development i the algorithmic proce dure, termination was insured since a feasil ,. solution to problem P5.1 is known to exist. If a feasible solution is not known to exist, then an additional stopping rule must be added to the procedL.e. This can be done in a straightforward manner when necessary (e.g., place upper bounds on each of the decision variables and terminate the solution pro cedure when any one of these bounds is exceeded). The Production Scheduling Problem The system controller's problem is to determine an assignment of N products to M facilities and the corresponding production amounts at each of these assignments over a variable horizon h. This is to be done in such a manner as to [a] satisfy all demands (di units for ik product i in period k) without allowing backordering, [b] convert the nonnegative, initial inventory level (I ) for all products i into 10 a desired nonnegative, minimum ending inventory level (IiH), and [c] minimize costs. The costs include production charges (i.e., c.i denotes *th the incremental cost of producing an th batch of product i during any machineperiod) and a general cost function f(h) which is a monotonically nondecreasing function of the horizon length h. A batch of product i contains exactly pi units of the product and only whole batches can be produced. Only one product can be assigned to a facility each period, and up to m. batches of that product i can be produced during any 1 machineperiod. By assumption, the initial and required ending period inventory levels are nonnegative which implies that backordering, if it occurs, must occur at the end of at least one of the intermediate periods (k = 1,2,...,hl). Define wik as the minimum number (integer) of batches of product i which must be produced during the first k periods in order to prevent backorders from occurring at the end of period k. Then, k wik = max [0, ( d _ i )/Pi k = 1,2,...,h1. kkl 0 k = 1,2,...,h1. j=1 Likewise, define h w = max [0, ((Ih + d.. I. )/Pi i = 1,2,...,N. j=l 1 Now by defining xilk as the number of facilities scheduled to produce exactly I batches of product i during period k, a mathematical formulation of this problem can be given as follows: N h mi Minimize f(h) + Z E E E c. x.i i=l k=l 2=1 r=l subject to k mi Z E2 x. > i wN (5.7) j=l =1 j ik k = 1,2,...,h [P5.2] N m. S 1 x. i = 1,2,... ,N h, x. i 0, integer I = 1,2,... ,mi k 1,2,...,h k = 1,2,...,h . Problem P5.2 is a nonlinear, integer program since the index h is a variable in the optimization. Rather than solve problem P5.2 directly, a relaxed version of the problem will be formulated and solved. This relaxed problem can be put into the format of problem P5.1 and solved very efficiently by the previously given algorithm. Once this relaxed problem has been solved, a simple procedure for solv ing problem P5.2 will be given. The relaxed problem is formulated by (a) omitting those con straints in (5.7) corresponding to k = 1,2,...,h1, (b) relaxing the k constraints given in (5.8) to the single constraint given in (5.10) h below, and (c) making the variable transformation yil = Z xi. . k=l The problem, then, is as follows: N i Minimize f(h) + E E c. y. i=l &=1 r=l subject to m. 1 [P5.3] E ly Wih i = 1,2,... ,N (5.9) N mi h (1/M) E E yi (5.10) i=1 1=1 h, y 0, integer 1,2,...,N ia = 1,2,...,mi. Theorem 5.1 A sufficient condition for problem P5.3 to have a feasible solution is that N 2 (d ik/m.p.) < M (5.11) i=l ik k 1 i=l for all k = 1,2,...,h. Relation (5.11) is a facility capacity condition which is common in continuous infinite horizon productioninventory problems. For example, see Hadley and Whitin [8 ]. A proof for this multiple facility, multiple batch discrete problem is given in the Appendix where the proof given is along the lines of that given by Elmaghraby and Mallik [2 ] for the case m. = 1 and M = 1. In this paper it is 1 assumed that condition (5.11) is satisfied. Note that problem P5.3 is in the format of problem P5.1 for the special case where m. equals one for all i = 1,2,...,N. However, in general, it is not in the correct format. Define the variable v. as the number of machineperiods having at least I batches of product i produced. Then, for all i = 1,2,...,N, vi vi,V +l 2 = 1,2,...,m.1 1 Yi= v. A = mi . 1 M1 Making this variable transformation in problem P5.3, we obtain the following equivalent program. N mi Minimize f(h) + S ci vi. i=l i=1 subject to mi [P5.4] v v = 12,...,N 1 i,^l u = 1,2,...,m.1 1 N h & (1/M) Z vi i=l h, v 0, integer = 1,2,..., S= 1,2,...,m. . 1 Now making the additional transformation, t. = v. 1i ir r=l 2 = 1,2,..,m, problem P5.4 can be reformulated equivalently as follows: m.l N 1 Minimize f(h) + [c.i t. Z (c. i ci )ti 2 u i,m. i,m 1,+1 subject to t. 1,m. 1 Wih 2til t2 0 2t ti,2 ti 0 ti ti,1 0 t21i,21 i = 1,2,...,N i = 1,2,...,N i = 1,2,...,N L = 3,4,...,m. i = 1,2,...,N S= 2,3,...,m. i N h > (1/M) 2 t. i=l h, t. 0, integer 12 = 1,2,...,N = 1,2,...,m.. Note that for the case m. = 2, constraints (5.12) should be omitted. The constraint set of problem P5.5 is now in the format of problem P5.1 (i.e., each constraint has one variable which can be written as a nondecreasing function of other nondecreasing variables and a feasible solution is known to exist). Hence, an optimal solution to problem P5.5 can be obtained by the algorithmic procedure since c c c ... > c. 0 il i2 i,m. 1 implies that the objective function is a nondecreasing function of any increasing variable. [P5.5] (5.12) Upon solving problem P5.5 by the algorithmic procedure, the optimal solution (h,T) can be transformed back into an optimal solution (h,Y) to problem P5.3. This solution has the property that m. batches 1 of each product i will be produced each time product i is assigned for one machineperiod with the possible exception of one assignment. m.1 i Hence, Z yi, equals 0 or 1 for all products i = 1,2,...,N. a=l Lemma 5.2 The optimal algorithmic solution (h,T) to problem P5.3 has m.1 E y i= 0 or 1 (5.13) =1 for all products i = 1,2,...,N. Proof of Lemma 5.2 From Property 1, the optimal algorithmic solution to problem P5.3 is the same irrespective of whether the objective function value is strictly increasing or just nondecreasing in each variable. Hence, in problem P5.3 it can be assumed that c. >c > ..> c il i2 m. 1 for all products i = 1,2,...,N. Assume that the optimal solution to problem P5.3 does not satisfy (5.13). Then an optimal solution (h*,Y*) mi1 * must have E yi. > 2 which implies that either 2=1 m.1 (a) Z lYi = < mi or m.l (b) 'E y = 9 > m. acilyi Case (a): There exists a feasible solution (h,Y) having h = h* and 1 if = e if M. YiI = if I= m 0 otherwise with objective function value z: z (Ci ci ) < z since (ci c ) > 0. Hence, there is a contradiction since (h*,Y*) cannot be an optimal solution. Case (b): There exists a feasible solution (h,Y) having h = h* and (1 if = [6/m]m. and i > 0 Yi i + [/m if = m. S, otherwise with objective function value z z (c c. ) il i,m. < z since (c.i ) > 0 . il ,m. Hence, there is a contradiction since (h*,Y*) cannot be an optimal solution. Q.E.D. The optimal algorithmic solution (h ,Y ) to problem P5.3 implies that the shortest horizon over which all product inventories can be 0 built up simultaneously to an amount IiH is h production periods. mi During this horizon each product should be assigned Z y times where all but at most one of these assignments should include the production m.l i o of m. batches of product i. If Z yi. equals one, then the other m.i 1 assignment of product i should include the production of E 0yi i iA batches. Clearly, since problem P5.3 is a relaxed version of problem P5.2, (h ,Yo) is also an optimal solution to problem P5.2 if it is feasible to problem P5.2. Without loss of generality, it can be assumed that the first 0 imi assignments of product i during the horizon h have m. batches m. of product produced where E y assignments are made. Let z denote 21i as ik i1 the number of these machineperiod assignments which are made in period k. Then, obviously, (h ,Y ) can be converted into a feasible solution to problem P5.2 if and only if there exists a feasible solution to the following constraint set: ho mi h 1i Z k = i = 1,2,...,N k=l k =1 i Z z (w / m /M i = 1,2,... ,N l ij ik k = 1,2,...,ho1 j=1 [P5.6] N Sz < M k = 1,2,...,h ik i=l i = 1,2,...,N z 0, integer ik k = 1,2,...,ho. m i Since 2 yi (w iho/mi) for all i = 1,2,...,N, it follows L=1 directly that there exists a feasible solution to constraint set P5.6 if and only if N 2 (w /m) 5 Mk i=1 for all k = 1,2,...,h This, coupled with the following result and the fact that any feasible solution to problem P5.2 must have h ho (Property 2), implies that problem P5.2 has a feasible solution if and only if (h ,Y ) can be converted into a feasible solution to problem P5.2. Lemma 5.3 Problem P5.2 has a feasible solution if and only if N S(w i/m.) Mk (5.14) i=l for all k = 1,2,...,h. Proof of Lemma 5.3 The "if" part follows since the algorithmic solution (h ,Y ) has already been shown to be a feasible solution when (5.14) is satis fied. The "only if" part follows by summing over all i in constraint (5.7), assuming m. batches are produced at each facility assignment, 1 and summing over all k in constraint (5.8). From (5.7), then, any feasible solution must satisfy m. m. N k i i Z E E x. > i(w /m.) (5.15) 1 ik I i=1 j=l =1 i=l for all k = 1,2,...,h and from (5.8) any feasible solution must satisfy N k mi E E E x.. Mk (5.16) i=l j=1 Y1 for all k = 1,2,...,h. Together, relations (5.15) and (5.16) along with the integrality requirement on xilk imply that (5.14) must be satisfied by any feasible solution to problem P5.2. Q.E.D. A procedure for solving problem P5.2 is to solve the relaxed version of the problem, and then convert this solution to a feasible solution to problem P5.2 by finding a feasible solution to constraint set P5.6. Any feasible solution determined by the procedure represents an optimal solution to problem P5.2 since the optimal objective function value of a relaxed version of a problem is a lower bound on the optimal objective function value of the general problem. In addition, if a feasible solution cannot be determined to constraint set P5.6, then it is known that problem P5.2 does not have a feasible solution. Constraint set P5.6 describes a network for which a twoproduct, Mfacility, threeperiod example is shown in Figure 5.2. From the net work it can be seen that a simple procedure for determining a solution to this constraint set is to assign each product i (w i/mi) times to vacant facilities in period one. Then assign each product i (wi2/mi)  (w i/m.) times to vacant facilities in periods one and two, and in gen eral assign each product i (w ik/m) 'w.i,kl/mi) additional times to ik i i,k1 i vacant facilities in periods one through k for all remaining periods k = 3,4,...,h The solution obtained is optimal unless at some stage in the assignment procedure there are insufficient facilities available to complete all assignments in which case problem P5.2 does not have a feasible solution. Lower Bounds Figure 5.1 Network Example of Constraint Set P5.6. Computational Experience The algorithmic procedure was programmed in Fortran H and used to solve the production scheduling problem. An IBM 370/165 computer was used, and 200 different problems were considered where parameters were generated randomly. Problems having as many as 20,000 variables and 20,000 constraints were generated. All problems were solved, and the computation times which were obtained were very nearly a linear function of the number of constraints. The average solution time (as 4 measured in terms of constraints in the problem) was 5.8 X 104 4 seconds per constraint with a range of approximately 3 104 seconds per constraint about the mean. The computer program completed approximately 100,000 iterations of the algorithm each second. Conclusions A specially structured integer program has been identified, and an extremely efficient algorithm for determining its solution has been developed. The specially structured program is a generalization of a multifacility, multiproduct production scheduling problem which has been presented. The scheduling problem arises, realistically, in industrial situations where schedulers use repetitive cycle plans such as the wellknown economic manufacturing quantity models for controlling production systems. CHAPTER 6 SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH In this dissertation a class of multiple facility, multiple product, production inventory scheduling problems has been considered. Specifically, four different scenarios for this class have been addressed. In a straightforward manner, each of these was formulated as an integer program. It was then shown that each of the problems could be reformu lated as a network flow problem which can be solved using any of the very efficient algorithms for minimalcost flows in single commodity networks. In addition, for some important special cases, single pass procedures were developed which are more efficient than the standard flow algorithms. In the fourth scenario, a specially structured integer programming formulation was identified, and an extremely efficient algo rithm was developed for determining its solution. The scenarios which have been considered have included very general and realistic assumptions. The originality of the formulations for this class of problems, in addition to allowing us to determine optimal solutions efficiently, has offered considerable insight into the operation of the production inventory system. These insights have allowed us to enrich the problems generously to include many other real istic aspects of the production system heretofore not solvable with present techniques. As should be the case, however, the research for this disserta tion has been openended in character. Each question answered has allowed new questions to be posed. Some of the logical and important extensions which it would seem desirable to pursue in future research in this area include: 1. It has been assumed throughout the dissertation that setups occur between production periods, whereas in many production systems, setups actually occur during production periods. In these situations the productive capacity of the system is depleted when setups occur, and an important extension of the results in this research would be to obtain equivalent results for the case where setups are assumed to occur during the production period. 2. The facility usage costs which have been used do not take into account the product assigned to the facility during the previous period. Realistically, however, the usage costs often should only be assessed when a different product is produced on the facility in each of the two periods. Solving the productioninventory scheduling prob lems for this more general case would be a considerable contribution, but the problem appears to be quite difficult. 3. Each time a product is assigned to a facility for one period, it has been assumed that only an integer multiple of the basic batch size for each product can be produced. An important generalization of the results in this research could be obtained by determining equiva lent results for the case where partial batches are allowed. 84 4. Product demand rates, though not necessarily identical in all periods, have been assumed to be deterministic. Considerable addi tional insight into the multifacility, multiproduct production inventory system could be achieved by studying similar problems having stochastic demand rates. APPENDIX PROOF OF THEOREM 5.1 Define N h = TT (m.p.) i=l 1 1 Z (dik/m.p.) =1 Yi = 0 if I = m., otherwise otherwise i = 1,2,... ,N. Then, from (Al), Iy = h S(di /p ) k=l = ih (ill  i )/Pi i0O for all i = 1,2,...,N, and N mi h N E E yi = Z (d ik/mii) i=1 i k=1 i=1 h < Z M, since from (5.11) k=l N Z (d ikmpPi) < M. i=l k 1 1 Relationship (A2) implies that N mi Mh E i y, = s > 1. i=l A=1 (Al) (A2) Thus, during the horizon h, all product demands are satisfied exactly, and there are s additional unassigned periods available. Hence, by assigning each product i, pi additional times during the horizon h where 0 5 .i max (0 (i io )/Pi) and N Z = s, i=l at least one product must become discretely closer to satisfying con straint (5.9). If all constraints are now satisfied, then a feasible solution has been determined. Otherwise, redefine (ill i = (iH i im and h+1, h+2, ... = 1,2,..., and repeat the procedure. A feasible solution must be obtained in a finite number of iterations since at each iteration, all demands are satisfied and at least one product is assigned at least one additional time. Q.E.D. BIBLIOGRAPHY References Cited [1] Bomberger, Earl E., "A Dynamic Programming Approach to a Lot Size Scheduling Problem," Management Science, 12, pp. 778784, 1966. [2] Elmaghraby, Salah E. and Arup K. Mallik, "The Scheduling of a MultiProduct Facility," OR Report Number 76, North Carolina State University, April, 1972. [3] Elmaghraby, Salah E., "Some Recent Developments in Aggregate Pro duction Planning," Paper presented at the Joint ORSATIMSAIIE Meeting, 1972. [4] Elmaghraby, Salah E., "The Machine Sequencing Problem Review and Extensions," Naval Research Logistics Quarterly, 15, pp. 205232, 1968. [5] Ford, L. R. and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, New Jersey, 1962. [6] Fulkerson, D. R., "An OutofKilter Method for MinimalCost Flow Problems," J. Soc. Indust. Appl. Math., 9, pp. 1827, 1961. [7] Geoffrion, A. M. and R. E. Marsten, "Integer Programming Algo rithms: A Framework and Stateofthe Art Survey," Management Science, 18, pp. 465491, 1972. [8] Hadley, G. and T. M. Whitin, Analysis of Inventory Systems, PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1963. [9] Haessler, R. W., "A Note on Scheduling a MultiProduct Single Machine System for an Infinite Planning Period," Management Science, 18, pp. B240B241, 1971. [10] Hodgson, Thorn J., "Addendum to Stankard and Gupta's Note on Lot Size Scheduling," Management Science, 16, pp. 514517, 1970. [11] Madigan, J. G., "Scheduling a MultiProduct SingleMachine System for an Infinite Planning Period," Management Science, 15, pp. 713 719, 1968. [12] Magee, J. F. Production Planning and Inventory Control, McGraw Hill, New York, New York, 1958. [13] Maxwell, William L., "The Scheduling of Economic Lot Sizes, Naval Research Logistics Quarterly, 11, pp. 89124, 1964. [14] Rodgers, Jack, "A Computational Approach to the Economic Lot Scheduling Problem," Management Science, 4, pp. 264291, 1958. [15] Stankard, Martin F. and Shiv K. Gupta, "A Note on Bomberger's Approach to Lot Size Scheduling: Heuristic Proposed," Management Science, 15, pp. 449452, 1969. [16] Wagner, Harvey M. and T. M. Whitin, "Dynamic Version of the Econom ic Lot Size Model," Management Science, 5, pp. 8996, 1959. [17] Zangwill, Willard I., "A Deterministic MultiPeriod Production Scheduling Model with Backlogging," Management Science, 13, pp. 105119, 1966. [18] Zangwill, Willard I., "A Backlogging Model and a MultiEchelon Model of a Dynamic Economic Lot Size Production System A Net work Approach," Management Science, 15, pp. 507527, 1969. Additional References Baker, Kenneth R., Control Policies for an Integrated Production and Inventory Systems, Ph.D. Thesis, Cornell University, 1969. Bowman, E. H., "Production Scheduling by the Transportation Method of Linear Programming," Operations Research, 4, pp. 100103, 1956. Brown, Robert G., Decision Rules for Inventory Management, Holt, Rinehart and Winston, Inc., New York, New York, 1967. Buffa, Elwood S., ProductionInventory Systems, Richard D. Irwin, Inc., Homewood, Illinois, 1968. Carroll, Donald C., Heuristic Sequencing of Single and Multiple Jobs, Ph.D. Thesis, Massachusetts Institute of Technology, 1965. Conway, R. W., W. L. Maxwell and L. W. Miller, Theory of Schedul ing, AddisonWesley Publishing Company, Reading, Massachusetts, 1967. Dooley, J. E. and Ben Bernholtz, "A ManyProduct SingleMachine Recursive Scheduling Procedure," Canadian Operational Research Society Journal, 7, pp. 165176, 1969. Eilon, Samuel, Elements of Production Planning and Control, John Wiley and Sons, Inc., New York, New York, 1962. Elmaghraby, Salah E., Arup K. Mallik and Henry L. W. Nuttle, "The Scheduling of Lots on a Single Facility," AIIE Transactions, II, pp. 203213, 1970. Hanssman, Fred, Operations Research in Production and Inventory Control, John Wiley and Sons, Inc., New, New York, 1962. Hodgson, Thom J. "An Analytic Model of a TwoProduct, One Machine, ProductionInventory System," Management Science, 19, pp. 391405, 1972. Hodgson, Thom J., SequenceLot Size Policies in a Production Inventory System, Ph.D. Thesis, University of Michigan, 1970. Holt, C., F. Modigliani, J. Muth and H. Simon, Planning Production Inventories and Work Force, PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1960. Kanodia, Lalit, Heuristic Sequencing of Jobs and Batch Size Deter mination in the Dynamic Closed Job Shop, Ph.D. Thesis, Massachusetts Institute of Technology, 1967. Keachie, E. C. and Robert J. Fontana, "Effects of Learning on Optimal Lot Size," Management Science, 13, pp. B102B108, 1966. Kortanek, K. O. and A. L. Soyster, "On the Status of Some Multi Product MultiPeriod Production Scheduling Models," Management Science, 17, pp. B560B561, 1971. Kunreuther, Howard, "ProductionPlanning Algorithms for the InventoryOvertime Tradeoff," Operations Research, 19, pp. 1717 1719, 1971. Love, Stephen F., "A Facilities in Series Inventory Model with Nested Schedules," Management Science, 18, pp. 327338, 1972. Manne, Alan S., "Programming of Economic Lot Sizes," Management Science, 4, pp. 115135, 1958. Parsons, J A., "Multiproduct Lotsize Determination when Certain Restrictions Are Active," Journal of Industrial Engineering, 17, pp. 360365, 1966. Salveson, M. E., "A Mathematical Theory of Production and Schedul ing," Journal of Industrial Engineering, 4, pp. 36, 1953. Taha, Hamby A. and Ronald W. Skeith, "The Economic Lot Sizes in Multistage Production Systems," AIIE Transactions, II, pp. 157 165, 1970. Veinott, Jr., Arthur F., "The Status of Mathematical Inventory Theory," Management Science, 12, pp. 745777, 1966. Veinott, Jr., Arthur F., "Optimal Policy for a MultiProduct, Dynamic, Nonstationary Inventory Problem," Management Y.ionce, 12, pp. 206222, 1965. Veinott, Jr., Arthur F., "Minimum ConcaveCost Solution of Leontief Substitution Models of MultiFacility Inventory Systems," Operations Research, 17, pp. 262291, 1969. Zangwill, Willard I., "Minimum Concave Cost Flows in Certain Networks," Management Science, 14, pp. 429450, 1968. Zangwill, Willard I., "A Deterministic Multiproduct, Multifacility Production and Inventory Model," Operations Research, 14, pp. 486 507, 1966. BIOGRAPHICAL SKETCH Robert Caleb Dorsey was born May 5, 1943, in Memphis, Tennessee. He received his elementary and secondary education in the Memphis school system, graduating from White Station High School in June, 1961. He attended Georgia Institute of Technology from 1961 to 1964 and received a Bachelor's degree with a major in Industrial Engineering in August, 1964. After graduating from Georgia Tech, he worked for Tennessee Eastman Company in Kingsport, Tennessee, from September, 1964, through March, 1970. While working with Tennessee Eastman Company, Robert Dorsey pursued his studies through a night program at the University of Tennessee and was awarded a Master of Science degree with a major in Industrial Engineering in December, 1967. In March, 1970, Robert Dorsey took an educational leave of absence from Tennessee Eastman in order to enter the graduate program in the Industrial and Systems Engineering Department at the University of Florida. He received a Master of Engineering degree from the Uni versity of Florida in June, 1971. Robert Dorsey is a registered Professional Engineer and is a member of Alpha Tau Omega fraternity, Alpha Pi Mu honorary engineering society, and the American Institute of Industrial Engineers, The Insti tute of Management Science, and Operations Research Society of America professional societies. He is married to the former Lou Gilliam of Kingsport, Tennessee, and is the father of one son, Trent. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 11. Donald Ratliff ,lairnian Assistant Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the decree of Doctor of Philosophy. Thom J. Hod son, CoChairman Assistant Pr fessor of Industrial and Systems Engin ering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard L. Francis Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it co.nior',Ls to acceptable standards of scholarly presentation and is fully ad'q!u.te, in .c:ope and quality, as a dissertation for the degree of )DoCr tor of Pil 'o'hy. KerF. E. Ki trick As. istant i ofcssor of Industrial and Systen's engineering g 