- Email: [email protected]
output lot sizing in single stage batch production systems under constant demand
Computers ind. Engng Vol. 19, Nos 1-4, pp. 37-41, 1990 Printed in Great Britain.All rights reserved
0360-8352/90 $3.00+0.00 Copyright © 1990PergamonPress pie
INPUT/OUTPUT LOT SIZING IN SINGLE STAGE BATCH
PRODUCTION SYSTEMS UNDER CONSTANT DEMAND Avijit Banerjee 1, Cheickna Sylla2 and Somkiat Eiamkanchanalai 1 1Drexel University, Philadelphia PA 19104
2New Jersey Institute of Technology Newark, NJ 07102
ABSTRACT This paper develops an integrated input/output lot sizing model under deterministic conditions for a single product manufactured within a batch production environment. Our analysis incorporates the effects of work-inprocess inventories resulting from multiple input items that are converted to output at finite rates in a single stage process and encompasses, simultaneously, the lot sizing decisions for the end product, as well as the externally procured input items. The resulting model represents a nonlinear mixed integer optimization problem, for which an efficient heuristic solution technique is suggested. INTRODUCTION Since the development of the economic order quantity (EOQ) concept by Harris [7] inventory lot sizing has been the focus of a considerable amount of research. An early modification of the EOQ model incorporated a finite production rate, resulting in the well-known classical production lot size (PLS) formula. The single stage PLS model has since been greatly embellished and has found applications under a wide variety of conditions (see [11] for a survey). Most of these applications, however, do not consider the effects of work-in-process (WIP) inventories. In recent years significant progress has been made regarding multistage lot sizing models, which do incorporate WIP effects [4], [5], [8], [10]. While some of such work has focused on developing optimum seeking algorithms with simplifying assumptions, much effort has gone into developing heuristics involving less restrictive assumptions [2], [6]. For instance, Billington et al. [3] and Moily [9], among others, have made substantial contributions to the capacitated multistage lot sizing problem. Nevertheless, most of the research cited above tends to ignore the effects of the gradual conversion of input to output at a finite rate within each production stage. In contrast, we explicitly consider the effects of WlP inventories that result from the conversion process, albeit in single stage production environments. In a recent study, Banerjee and Burton [ 1] examined the implications of such transformation processes in terms of WIP inventories and studied their influence on the lot sizing decision in single stage systems. In their analysis, the order quantity of an input item is allowed to be either a fraction or a multiple of the end item production lot size and a heuristic solution methodology, based on a search technique, is developed. In this paper, we assume for simplicity, that the procurement lot size for any input item can be a fraction of its total amount necessary to produce a batch of the product, lot-for-lot ordering representing a special case of this scenario. This assumption is consistent with the just-in-time (JIT) philosophy, which advocates relatively small batch quantities, for the purpose of inventory reduction. In other words, consistent with JIT principles, we rule out those situations where relatively high ordering costs may warrant ordering input items in substantially large lots and justify limiting our attention to receiving any input item in one or more lots during the course of a production cycle. Furthermore, we develop a new heuristic procedure for the simultaneous determination of input and output lot sizes, that turns out to be substantially more efficient than the search technique proposed by Banerjee and Burton [1]. NOTATION AND ASSUMPTIONS The following notational scheme is used here: D = demand rate for the end item in units per period; P = end item production rate in units per period; S = fixed setup cost for producing an end item batch; C ---the product's unit variable cost (S/unit); Aj = fixed ordering cost for the jth input item in $/order(j = 1, 2 .... n); vj = unit purchase price of input item j; r = the inventory holding cost rate in S/S/period; T = total inventory cycle time for the end item (periods); t = production time per cycle required by the end item in periods; TRC = total relevant cost; Q = production lot size for the end item in units; Qj = order quantity for input item j in units. Note that, of necessity, P > D Also, without loss of generality, we assume that the manufacture of each unit of the product requires one unit of input j. For instance, if 4 lbs. of a raw material and 3 units of a part are required to produce a unit of an end item, a "unit" of the material is specified to be 4 lbs. and 3 units of the part are defined as one "unit" of this input, in order to satisfy this assumption. As mentioned below, it is further assumed that each input is procured independently, in order to avoid the complexities associated with the joint ordering of multiple 37
38
Proceedings of the 12th Annual Conference on Computers & Industrial Engineering
input items. In addition to the above, the following major assumptions are made in this paper: (1) a product is manufactured autonomously in a single stage batch production system and stored in a single location; (2) the input items (e.g. materials, parts, etc.) are ordered independently of one another and the entire order quantity of each input is received at the same time; (3) the production and storage environment is deterministic, i.e., the demand and production rates, lead times, cost parameters, etc. are all known and constant through time; (4) stockout situations for the end item as well as the inputs are unacceptable and not permitted; (5) no quantity purchase price discounts are available for the input items procured from external sources; (6) the planning horizon is infinite; (7) there are no constraints on storage space and capital invested in inventories; (8) the planning horizon is infinite and cost minimization is the optimization objective. (9) the ordering costs, relative to the carrying costs, are sufficiently low to enable the receipt of one or more lots of any input item during the production of a batch of the end product; (10) the lot sizes are allowed to be non-integers. THE MODEL AND ITS SOLUTION The classical PLS model ignores the presence of WIP inventories and their influence on the lot sizing decision. If the effects of such inventories are taken into account, this model is no longer valid. Figure 1 shows the inventory time plots for the finished and input items, depicting the case of a single input ordered, as an example, three times during each production run. In general, if Q is the production batch size, the procurement lot size for input item j is Q/kj, where kj is restricted to a positive integer. Thus, the average inventory of input j is lj = (Q/2kj)(t)(D]Q) = (Q/2kj)(Q/P)(D/Q)= DQ]2Pkj. Incorporating the effects of WIP inventories, the total relevant cost per period for an end product and an input item j, considered together, can be expressed as TRC(Q,kj) = (D/Q)(S+Ajkj) + (Qr/2)[C(I-D/P)+(D/P)(vj/kj)],
(1)
where the first term represents the fixed setup and ordering costs and the second term shows the inventory holding costs per period pertaining to the two items. Extending the above cost function to the case of n input items and defining the vector k = (k 1, k 2 ....... kn), we obtain n n TRC(Q, k) -- (D/Q)(S+~Ajkj) + (Qr/2)[C(1-D/P)+(D/P)~vj/kjl. (2) j=l j=l It can be easily shown that the cost function (2) above is convex in terms of Q, as well as each kj. The minimization of the RHS of (2) represents a nonlinear, mixed integer optimization problem, if the kj values are indeed restricted to integers. Strictly speaking, it is not guaranteed that integer kj values will yield the minimum total relevant cost. However, if any kj is allowed to be a non-integer, the resulting average inventory expression and the cost function become almost intractably complex. Thus, in order to avoid such complications, we impose the integer restriction on each kj. This, nonetheless, renders the minimization of (2) a daunting task and forces us to seek a heuristic solution procedure for finding near-optimal values of Q and the kjs. Towards this end, we utilize the convexity properties of (2) mentioned above. Setting to zero the first partial derivative of this cost function with respect to Q at Q=Q*, we obtain
Q*(k)
V r[C(1-D/P) + (D/P)~vj/kj] j=l
(3)
Substituting Q = Q* from (3) into (2) results in n
n
TRC(Q*,_k) = D{[(2r/P)(S+~Ajkj) {C(P/D- 1)+~vj/kj }], (4) j=l j=l which represents the total relevant cost per period resulting from the optimal production lot size, Q*, for a given set of kj values. Removing the integrality requirement for the kj values for a moment and setting to zero the first partial derivative of (4) with respect to each kj at kj = kj*, yields the following first order conditions for the optimality of
Banerjee et al.: Input/Output Lot Sizing Model
39
Inventory Output t QI
-"Ix
I I
\
lb,.;/ : I t=Q/P
Inventory of Input j
I I I I I I I I
Time
I I I I I T-- Q/D__.._~
-1
Time Figure 1. Inventory time plots for output and input j (k .= 3)
J
TABLE1 Summary of Resul~ Iteration
k1
k2
k3
k4
k5
k6
TRC
0
3.70
1.91
2.22
2.20
1.72
1.36
6704.59
1
3 2
** **
** **
** **
** **
** **
6705.70* 6706.84
2
3 3
1 2
** **
** **
** **
** **
6729.67 6706.46*
3
3 3
2 2
2 3
** **
** **
** **
6705.27* 6726.25
4
3 3
2 2
2 2
2 3
** **
** **
6704.86* 6713.55
5
3 3
2 2
2 2
2 2
1 2
** **
6724.03 6710.39"
6
3 3
2 2
2 2
2 2
2 2
1 2
6714.16" 6734.98
* indicates the current value of TRCmin. ** indicates the current value ofkj is set at kj* that were obtained at iteration 0.
40
Proceedings of the 12th Annual Conference on Computers & Industrial Engineering
the kj values: n
vj(S + , jkj) _
j--1
kj* =
forj = 1, 2 ...... n.
(5)
~9/
AjtC(P/D-1) + Zvj/kj] j=l Based on the results obtained above, we now outline our heuristic algorithm for obtaining the appropriate values of Q and the kjs towards the minimization of the cost function (2). The steps of our procedure are: Steo 1: Using (5), compute kj* for all values o f j . Substitute these kj* values in (4) to compute the current minimum value of TRC, say TRCmi n. Set kj= kj* for all j. If all kj* values are integers, the optimal solution has been found. Go to step 3. Step 2: For each of the non-integer kj found in step l, examine the integers surrounding each kj. Substitute these values for the first non-integer kj into (4) and calculate TRC. Let the lower of these two TRCs be the current TRCmi n and reset the corresponding kj value to the integer yielding TRCmi n. Repeat this step for each of the remaining non-integer kj values sequentially, until all of them are reset to integers. Step 3: Substitute the kj s obtained from above into (3) to calculate Q*, then determine Qj = Q*/kj for all j. Note that in order to solve the equation set (5), any available simultaneous equation solving software on either a PC or a mainframe may be employed. In this paper we use such a software package, TK-Solver, on a Macintosh PC. A NUMERICAL EXAMPLE In order to illustrate the heuristic algorithm described above, suppose that the following data are available for one of the products, requiring 6 input items, manufactured in a single stage production system: S = $600/setup, C = $100/unit, P = 4000 units/year,D = 2000 units/year and r = $0.25/$/year. Input Item, j : 1 2 3 4 5 6 Aj (S/order) :
8
10
15
6
15
20
vj (S/unit)
15
5
10
4
6
5
:
The results obtained from applying steps 1 and 2 of our solution method are summarized in Table 1. In this table, the initial kj values are all non-integers. Thus a total of six iterations of our algorithm are needed to arrive at the respective integer kjs, as shown, yielding a total relevant cost of $6714.16 per year. Substitution of the initial non-integer kj*s into (4) leads to a reasonable lower bound, i.e. $6704.59 per year in this case. In comparison with this lower bound, our feasible solution, with integer kj values, results in a cost penalty of only 0.14%, indicating the efficacy of our heuristic. The final step of our procedure yields a production batch size of 438.48 and lot sizes of 146.16, 219.24, 219.24, 219.24, 219.24 and 438.48, respectively, for input items 1 through 6. Incidentally, the application of the procedure suggested by Banerjee and Burton [1] in this case results in the same solution. In order to gain some insights into the efficiency and effectiveness of our heuristic solution procedure, we randomly generated a set of 120 problems, varying the number of input items from 2 to 8. We solved each of these problems using our heuristic, as well as by the procedure suggested by Banerjee and Burton [1]. In the case of 87 of these problems, the solutions yielded by both the techniques were identical. For 24 problems, our heuristic yielded lower cost solutions, whereas, the latter method resulted in slightly better solutions in only 9 of the cases. The feasible solutions yielded by our technique result in cost penalties (in comparison to the aforementioned lower bounds) that were less than 1% in all cases. Furthermore, the CPU time required by our algorithm was between 30% to 40% less than that required by the Banerjee and Burton search procedure [1] for each of the 120 problems. In view of these observations it appears that our suggested heuristic is relatively fast and effective in finding at least near-optimal solutions. SUMMARY AND CONCLUSIONS In this paper we have analyzed the effects of WIP inventories on lot sizing decisions in single stage systems under deterministic conditions. An integrated model for the simultaneous determination of the production batch size for a product and the procurement lot sizes of the associated multiple input items was formulated, leading to a mixed integer, nonlinear optimization problem. In order to keep our model and its solution tractable, it was
Banerjee et
al.:
Input/Output Lot Sizing Model
41
assumed that an integer number of lots for each input item is procured during any production cycle, ruling out input lots that last over two or more such cycles. This assumption is not inconsistent with JIT concepts, that advocate small lot sizes through fixed cost reduction. Although optimum seeking techniques, such as branch and bound, may be employed for solving this problem, we suspect that such approaches may not be very efficient, particularly for relatively large problems. For this reason, we suggest a heuristic algorithm as a solution technique. Our algorithm is illustrated through a numerical example and we show that the solution thus obtained is as good as the one resulting from the application of a search based methodology suggested in an earlier paper. Indeed, our computational experience, although limited, indicates that the approach suggested here is relatively faster and, more often than not, tends to be superior in terms of total relevant cost reduction. For future directions of research in this area, we suggest extension of our analysis to more complex multistage systems, relaxing the integer requirements on the kj values. No doubt, such extensions may prove to be difficult to analyze. Nevertheless, it is hoped that the insights obtained from our work will be able to facilitate future research in this important field. REFERENCES
1. Banerjee, A. and Burton, J.S., "Lot Sizing and Work-in-Process Inventories in Single Stage Production Systems", in Gulledge, T. R. and Litteral (eds.), Cost Analysis Applications of Economics and Onerations Research, New york: Springer-Verlag, 1989, pp. 283 -297. 2. Biggs, J. R., Goodman, S. H. and Hardy, S. T., "Lot Sizing Rules in a Hierarchical Multi-Stage Inventory System", Production ~nd Inventory Management. First Quarter (1977),pp. 104 - 115. 3. Billington, P. J., McClain, J. O. and Thomas, L. J., "Heuristics for Multi-Level Lot-Sizing with a Bottleneck", Management Science, Vol. 32, No. 8 (1986), pp. 989 - 1006. 4. Crownston, W. B., Wagner, M. H. and Williams, J. F., "Economics Lot Size Determination in Multi-Stage Assembly Systems", Management Science, Vol. 19, No. 5 (1973), pp. 517 - 526. 5. De Bodt, M. A., Gelders, L. F. and Van Wassenhove, L. N., "Lot Sizing Under Dynamic Demand Conditions: A Review", En~neering Costs and Production Eeonomic~ (Netherlands~. Vol. 8, No. 3 (1984), pp. 165 - 187. 6. Graves, S. C. "Multi-Stage Lot Sizing: an Iterative Procedure", in Schwarz, L. B. (ed.), Multi-L~vel Production/InventorvControl Systems: Theory and Practice. New York: North Holland, 1981, pp. 95 - 109. 7. Harris, F., Ooerations and Costs, Chicago: A. W. Shaw, 1915. 8. Lambrecht, M. R., Vander Eecken, J. and Vanderveken, H.,"Review of Optimal and Heuristics Methods for a Class of Facilities in Series Dynamic Lot-Sizing Problems", in Schwarz, L. B. (ed.), Multi-Level Production/Inventory Control Systems: Theory and Practice, New York: North Holland, 1981, pp. 69 - 94. 9. Moily, J. P., "Optimal and Heuristic Procedures for Component Lot-Splitting in Multi-Stage Manufacturing Systems", Mana~,ement Science, Vol. 32, No. 1 (1986), pp. 113 - 125. 10.Steinberg, E. and Napier, H. A., "Optimal Multi-Level Lot Sizing for Requirements Planning Systems", Management Science, Vol. 26, No. 12 (1980), pp. 1258 - 1271. 11.Tinarelli, G. U., "Inventory Control: Models and Problems", European Journal of Onerational Research, Vol. 14 (1983), pp. 1 - 12.
0360-8352/90 $3.00+0.00 Copyright © 1990PergamonPress pie
INPUT/OUTPUT LOT SIZING IN SINGLE STAGE BATCH
PRODUCTION SYSTEMS UNDER CONSTANT DEMAND Avijit Banerjee 1, Cheickna Sylla2 and Somkiat Eiamkanchanalai 1 1Drexel University, Philadelphia PA 19104
2New Jersey Institute of Technology Newark, NJ 07102
ABSTRACT This paper develops an integrated input/output lot sizing model under deterministic conditions for a single product manufactured within a batch production environment. Our analysis incorporates the effects of work-inprocess inventories resulting from multiple input items that are converted to output at finite rates in a single stage process and encompasses, simultaneously, the lot sizing decisions for the end product, as well as the externally procured input items. The resulting model represents a nonlinear mixed integer optimization problem, for which an efficient heuristic solution technique is suggested. INTRODUCTION Since the development of the economic order quantity (EOQ) concept by Harris [7] inventory lot sizing has been the focus of a considerable amount of research. An early modification of the EOQ model incorporated a finite production rate, resulting in the well-known classical production lot size (PLS) formula. The single stage PLS model has since been greatly embellished and has found applications under a wide variety of conditions (see [11] for a survey). Most of these applications, however, do not consider the effects of work-in-process (WIP) inventories. In recent years significant progress has been made regarding multistage lot sizing models, which do incorporate WIP effects [4], [5], [8], [10]. While some of such work has focused on developing optimum seeking algorithms with simplifying assumptions, much effort has gone into developing heuristics involving less restrictive assumptions [2], [6]. For instance, Billington et al. [3] and Moily [9], among others, have made substantial contributions to the capacitated multistage lot sizing problem. Nevertheless, most of the research cited above tends to ignore the effects of the gradual conversion of input to output at a finite rate within each production stage. In contrast, we explicitly consider the effects of WlP inventories that result from the conversion process, albeit in single stage production environments. In a recent study, Banerjee and Burton [ 1] examined the implications of such transformation processes in terms of WIP inventories and studied their influence on the lot sizing decision in single stage systems. In their analysis, the order quantity of an input item is allowed to be either a fraction or a multiple of the end item production lot size and a heuristic solution methodology, based on a search technique, is developed. In this paper, we assume for simplicity, that the procurement lot size for any input item can be a fraction of its total amount necessary to produce a batch of the product, lot-for-lot ordering representing a special case of this scenario. This assumption is consistent with the just-in-time (JIT) philosophy, which advocates relatively small batch quantities, for the purpose of inventory reduction. In other words, consistent with JIT principles, we rule out those situations where relatively high ordering costs may warrant ordering input items in substantially large lots and justify limiting our attention to receiving any input item in one or more lots during the course of a production cycle. Furthermore, we develop a new heuristic procedure for the simultaneous determination of input and output lot sizes, that turns out to be substantially more efficient than the search technique proposed by Banerjee and Burton [1]. NOTATION AND ASSUMPTIONS The following notational scheme is used here: D = demand rate for the end item in units per period; P = end item production rate in units per period; S = fixed setup cost for producing an end item batch; C ---the product's unit variable cost (S/unit); Aj = fixed ordering cost for the jth input item in $/order(j = 1, 2 .... n); vj = unit purchase price of input item j; r = the inventory holding cost rate in S/S/period; T = total inventory cycle time for the end item (periods); t = production time per cycle required by the end item in periods; TRC = total relevant cost; Q = production lot size for the end item in units; Qj = order quantity for input item j in units. Note that, of necessity, P > D Also, without loss of generality, we assume that the manufacture of each unit of the product requires one unit of input j. For instance, if 4 lbs. of a raw material and 3 units of a part are required to produce a unit of an end item, a "unit" of the material is specified to be 4 lbs. and 3 units of the part are defined as one "unit" of this input, in order to satisfy this assumption. As mentioned below, it is further assumed that each input is procured independently, in order to avoid the complexities associated with the joint ordering of multiple 37
38
Proceedings of the 12th Annual Conference on Computers & Industrial Engineering
input items. In addition to the above, the following major assumptions are made in this paper: (1) a product is manufactured autonomously in a single stage batch production system and stored in a single location; (2) the input items (e.g. materials, parts, etc.) are ordered independently of one another and the entire order quantity of each input is received at the same time; (3) the production and storage environment is deterministic, i.e., the demand and production rates, lead times, cost parameters, etc. are all known and constant through time; (4) stockout situations for the end item as well as the inputs are unacceptable and not permitted; (5) no quantity purchase price discounts are available for the input items procured from external sources; (6) the planning horizon is infinite; (7) there are no constraints on storage space and capital invested in inventories; (8) the planning horizon is infinite and cost minimization is the optimization objective. (9) the ordering costs, relative to the carrying costs, are sufficiently low to enable the receipt of one or more lots of any input item during the production of a batch of the end product; (10) the lot sizes are allowed to be non-integers. THE MODEL AND ITS SOLUTION The classical PLS model ignores the presence of WIP inventories and their influence on the lot sizing decision. If the effects of such inventories are taken into account, this model is no longer valid. Figure 1 shows the inventory time plots for the finished and input items, depicting the case of a single input ordered, as an example, three times during each production run. In general, if Q is the production batch size, the procurement lot size for input item j is Q/kj, where kj is restricted to a positive integer. Thus, the average inventory of input j is lj = (Q/2kj)(t)(D]Q) = (Q/2kj)(Q/P)(D/Q)= DQ]2Pkj. Incorporating the effects of WIP inventories, the total relevant cost per period for an end product and an input item j, considered together, can be expressed as TRC(Q,kj) = (D/Q)(S+Ajkj) + (Qr/2)[C(I-D/P)+(D/P)(vj/kj)],
(1)
where the first term represents the fixed setup and ordering costs and the second term shows the inventory holding costs per period pertaining to the two items. Extending the above cost function to the case of n input items and defining the vector k = (k 1, k 2 ....... kn), we obtain n n TRC(Q, k) -- (D/Q)(S+~Ajkj) + (Qr/2)[C(1-D/P)+(D/P)~vj/kjl. (2) j=l j=l It can be easily shown that the cost function (2) above is convex in terms of Q, as well as each kj. The minimization of the RHS of (2) represents a nonlinear, mixed integer optimization problem, if the kj values are indeed restricted to integers. Strictly speaking, it is not guaranteed that integer kj values will yield the minimum total relevant cost. However, if any kj is allowed to be a non-integer, the resulting average inventory expression and the cost function become almost intractably complex. Thus, in order to avoid such complications, we impose the integer restriction on each kj. This, nonetheless, renders the minimization of (2) a daunting task and forces us to seek a heuristic solution procedure for finding near-optimal values of Q and the kjs. Towards this end, we utilize the convexity properties of (2) mentioned above. Setting to zero the first partial derivative of this cost function with respect to Q at Q=Q*, we obtain
Q*(k)
V r[C(1-D/P) + (D/P)~vj/kj] j=l
(3)
Substituting Q = Q* from (3) into (2) results in n
n
TRC(Q*,_k) = D{[(2r/P)(S+~Ajkj) {C(P/D- 1)+~vj/kj }], (4) j=l j=l which represents the total relevant cost per period resulting from the optimal production lot size, Q*, for a given set of kj values. Removing the integrality requirement for the kj values for a moment and setting to zero the first partial derivative of (4) with respect to each kj at kj = kj*, yields the following first order conditions for the optimality of
Banerjee et al.: Input/Output Lot Sizing Model
39
Inventory Output t QI
-"Ix
I I
\
lb,.;/ : I t=Q/P
Inventory of Input j
I I I I I I I I
Time
I I I I I T-- Q/D__.._~
-1
Time Figure 1. Inventory time plots for output and input j (k .= 3)
J
TABLE1 Summary of Resul~ Iteration
k1
k2
k3
k4
k5
k6
TRC
0
3.70
1.91
2.22
2.20
1.72
1.36
6704.59
1
3 2
** **
** **
** **
** **
** **
6705.70* 6706.84
2
3 3
1 2
** **
** **
** **
** **
6729.67 6706.46*
3
3 3
2 2
2 3
** **
** **
** **
6705.27* 6726.25
4
3 3
2 2
2 2
2 3
** **
** **
6704.86* 6713.55
5
3 3
2 2
2 2
2 2
1 2
** **
6724.03 6710.39"
6
3 3
2 2
2 2
2 2
2 2
1 2
6714.16" 6734.98
* indicates the current value of TRCmin. ** indicates the current value ofkj is set at kj* that were obtained at iteration 0.
40
Proceedings of the 12th Annual Conference on Computers & Industrial Engineering
the kj values: n
vj(S + , jkj) _
j--1
kj* =
forj = 1, 2 ...... n.
(5)
~9/
AjtC(P/D-1) + Zvj/kj] j=l Based on the results obtained above, we now outline our heuristic algorithm for obtaining the appropriate values of Q and the kjs towards the minimization of the cost function (2). The steps of our procedure are: Steo 1: Using (5), compute kj* for all values o f j . Substitute these kj* values in (4) to compute the current minimum value of TRC, say TRCmi n. Set kj= kj* for all j. If all kj* values are integers, the optimal solution has been found. Go to step 3. Step 2: For each of the non-integer kj found in step l, examine the integers surrounding each kj. Substitute these values for the first non-integer kj into (4) and calculate TRC. Let the lower of these two TRCs be the current TRCmi n and reset the corresponding kj value to the integer yielding TRCmi n. Repeat this step for each of the remaining non-integer kj values sequentially, until all of them are reset to integers. Step 3: Substitute the kj s obtained from above into (3) to calculate Q*, then determine Qj = Q*/kj for all j. Note that in order to solve the equation set (5), any available simultaneous equation solving software on either a PC or a mainframe may be employed. In this paper we use such a software package, TK-Solver, on a Macintosh PC. A NUMERICAL EXAMPLE In order to illustrate the heuristic algorithm described above, suppose that the following data are available for one of the products, requiring 6 input items, manufactured in a single stage production system: S = $600/setup, C = $100/unit, P = 4000 units/year,D = 2000 units/year and r = $0.25/$/year. Input Item, j : 1 2 3 4 5 6 Aj (S/order) :
8
10
15
6
15
20
vj (S/unit)
15
5
10
4
6
5
:
The results obtained from applying steps 1 and 2 of our solution method are summarized in Table 1. In this table, the initial kj values are all non-integers. Thus a total of six iterations of our algorithm are needed to arrive at the respective integer kjs, as shown, yielding a total relevant cost of $6714.16 per year. Substitution of the initial non-integer kj*s into (4) leads to a reasonable lower bound, i.e. $6704.59 per year in this case. In comparison with this lower bound, our feasible solution, with integer kj values, results in a cost penalty of only 0.14%, indicating the efficacy of our heuristic. The final step of our procedure yields a production batch size of 438.48 and lot sizes of 146.16, 219.24, 219.24, 219.24, 219.24 and 438.48, respectively, for input items 1 through 6. Incidentally, the application of the procedure suggested by Banerjee and Burton [1] in this case results in the same solution. In order to gain some insights into the efficiency and effectiveness of our heuristic solution procedure, we randomly generated a set of 120 problems, varying the number of input items from 2 to 8. We solved each of these problems using our heuristic, as well as by the procedure suggested by Banerjee and Burton [1]. In the case of 87 of these problems, the solutions yielded by both the techniques were identical. For 24 problems, our heuristic yielded lower cost solutions, whereas, the latter method resulted in slightly better solutions in only 9 of the cases. The feasible solutions yielded by our technique result in cost penalties (in comparison to the aforementioned lower bounds) that were less than 1% in all cases. Furthermore, the CPU time required by our algorithm was between 30% to 40% less than that required by the Banerjee and Burton search procedure [1] for each of the 120 problems. In view of these observations it appears that our suggested heuristic is relatively fast and effective in finding at least near-optimal solutions. SUMMARY AND CONCLUSIONS In this paper we have analyzed the effects of WIP inventories on lot sizing decisions in single stage systems under deterministic conditions. An integrated model for the simultaneous determination of the production batch size for a product and the procurement lot sizes of the associated multiple input items was formulated, leading to a mixed integer, nonlinear optimization problem. In order to keep our model and its solution tractable, it was
Banerjee et
al.:
Input/Output Lot Sizing Model
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assumed that an integer number of lots for each input item is procured during any production cycle, ruling out input lots that last over two or more such cycles. This assumption is not inconsistent with JIT concepts, that advocate small lot sizes through fixed cost reduction. Although optimum seeking techniques, such as branch and bound, may be employed for solving this problem, we suspect that such approaches may not be very efficient, particularly for relatively large problems. For this reason, we suggest a heuristic algorithm as a solution technique. Our algorithm is illustrated through a numerical example and we show that the solution thus obtained is as good as the one resulting from the application of a search based methodology suggested in an earlier paper. Indeed, our computational experience, although limited, indicates that the approach suggested here is relatively faster and, more often than not, tends to be superior in terms of total relevant cost reduction. For future directions of research in this area, we suggest extension of our analysis to more complex multistage systems, relaxing the integer requirements on the kj values. No doubt, such extensions may prove to be difficult to analyze. Nevertheless, it is hoped that the insights obtained from our work will be able to facilitate future research in this important field. REFERENCES
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