AN EXTENDED NEWSVENDOR MODEL FOR SOLVING CAPACITY CONSTRAINT PROBLEMS IN A MULTI-ITEM, MULTI-PERIOD ENVIROMENT

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1 Production Systems nd Informtion Engineering Volume V (2009), pp AN EXTENDED NEWSVENDOR MODEL FOR SOLVING CAPACITY CONSTRAINT PROBLEMS IN A MULTI-ITEM, MULTI-PERIOD ENVIROMENT PETER MILEFF University of Miskolc, Hungry Deprtment of Informtion Technology mileff@iit.uni-miskolc.hu KAROLY NEHEZ University of Miskolc, Hungry Deprtment of Informtion Engineering nehez@it.iit.uni-miskolc.hu [Received: Jn 0, 2009] Abstrct. In the lst few yers, the effective mngement of the inventory control problems becme ever more criticl problem of the supplier compnies. In this pper, on the bsis of demnds of mjor Hungrin mss production compny, n extension of n nlyticl inventory control model is presented considering the condition of globl cpcity constrint. This model ws elborted in our previous pper [6] nd models the one customer - one supplier reltion. Our im is to determine n optiml holding-production policy of the supplier, which mkes cost-optiml stockpiling policy possible for n optionl long production time horizon. We show, tht building on our former results, the globl cpcity constrint stisfying policy cn be determined by the help of new heuristic method. Keywords: stockpiling policy, extended newsvendor model, globl cpcity constrint. Introduction In the lst 5 yers, the business environment of compnies in the field of mss production hs been ltered. The demnd rte for mss products hs remined t high level but numerous new requirements hve ppered on the mrket. Chnges in the business environment influence engineering nd logistic reltions between compnies nd suppliers. The former, simple buying-selling (so-clled cool ) reltion hs become much wrmer. This mens tht coopertive nd collbortive methods nd ctivities hve become the min object in SCM development. Reltions of the mrketing orgniztions, end-product mnufcturers nd supplier compnies cn be very complicted nd vrious in prctice. This

2 4 P. MILEFF, K. NEHEZ motivtes wide exmintion of the vilble models nd further investigtion of effective decision supporting nd plnning methods. In the literture we cn meet the wide scle of stockpiling models [6]. In the lter we del with one of the most known stochstic method, so-clled newsvendor model. The model is certinly mong the most importnt models in the field of opertions mngement. It is pplied in wide vriety of stockpiling problems. In this pper in complince with this, n extended newsvendor model [7] will be exmined on the bsis of former results in multi-product nd cpcity constrint cse. The properties of our extended newsvendor model tht the inventory control problem of n optionl length production horizon cn be solved nlyticlly opened new opportunities to model multi-product cpcity constrint problems. Cpcity constrint problem ppers lmost ll lrger or smller commercil firms. These firms produce often severl products in complince of customer demnds. In cse of dynmiclly nd stochsticlly chnging demnds, often the cpcity problem cn be ppered. The question is lwys the sme: how much should be produced? Of course the question is very simple, but the solution is NP-hrd. The model conditions re fully identicl with the conditions presented in [5][6] publictions. The greter prt of the models in the literture solves the problem pplying the tool of the dynmic progrmming or some kind of serching method (soft-computing). Becuse of the lrge serching spce, these solutions require extremely long computtion time in cse of mny product number nd long production time horizon. During our reserch we investigted the cpcity constrint problem from one product one period to more products nd more periods. This pper presents only some of these methods... An Extended Newsvendor Model The clssicl newsvendor model cnnot be pplied properly to solve the tsks ppering t the customized mss production. The reson of this is the high vlue of the setup cost nd it cnnot hndle multi-period problems, where customer demnd cn vry stochsticly. During our reserch we hd ben developed new inventory controll method, which gives the optiml solution for the problem in n nlitic wy, nd ssures efficient stockpiling for the supplier. Summrizing the min chrcteristic of the model the objective function cn be formulted s follows:

3 K he q + pe SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS n( q23... n) = cf + cv[ q23... n I] + he[ q23... n D] + he[ q23... n D D2] + + [ D D... D ] + pe[ D q ] + pe[ ( D + D ) q ] n + [ [ n ] ], + [( D + D D ) q ] + pe D D + [ D q ] 2 2 n n n n n n (.) where the individul prmeters re the following: cf fixed cost. This cost lwys exists when the production of series is strted. [Ft / production] c v vrible cost. This cost type expresses the production cost of one product. [Ft / product] p penlty cost (or bck order cost). If there is less rw mteril in the inventory thn needed to stisfy the demnds, this is the penlty cost of the unstisfied orders. [Ft / product] h inventory nd stock holding cost. [Ft / product] D this mens the demnd from the receiver for the product, which is n optionl probbility vrible. [number / period] E[D] expected vlue of the D stochstic vrible. q the product quntity in the inventory. The decision of the inventory control policy concerns the product quntity in the inventory fter the product decision. This prmeter includes the initil inventory s well. If nothing is produced, then this quntity is equl to the initil quntity, i.e. concerning the existing inventory. I initil inventory level. We ssume tht the supplier possesses I products in the inventory t the beginning of the demnd of the delivery period. The new method is robust nd dequtly elegnt (detiled in pper [6][7]), becuse the solution is independent from the type of the distributed function: p cv hf ( q23... n) hf2( q23... n) hf23( q23... n)... hf23... n ( q23... n) F n( q n*) =, (.2) h+ p where () F represents the joint distribution function in complince with the number of periods drwn together. The q 23 n * - which stisfies the eqution - expresses how mny finished goods should be in the inventory t the time when customer demnd ppers with regrd to n periods. Of course the criticl inventory level mentioned first by Herbert Scrf [] for one-period production cn be pplied in the cse of joint production for n numbers of production cycles too. However this is not discussed in this pper.

4 6 P. MILEFF, K. NEHEZ 2. The One Product, Multi-Period Model Introducing the globl cpcity constrint for the multi-period extended newsvendor model we consider the chrcteristic of the model, the period bsed policy. Accordingly two type of optimiztion methods cn be differed: the service level bsed policy nd cost bsed policy. Of course these policies re in contrst with ech other. Only one of them cn be prevel in the stockpiling. 2.. Cost Bsed Policy This pproch models the type of supplier, which is directly in reltion to the mrket. In cse of this type of policy, the min objective is to minimize the costs. It should be decided tht how mny bck-orders cn be occurred during the time horizon. So the penlty cost is determined by the supplier itself [7]. Becuse in cse of cost bsed policy keeping the service level is not the min objective, the solution t the optimiztion of the production costs cn be the reduction of the number of setups or tking the risk of the penlty. The reduction of the number of jointly produced periods is not definitely the best solution, the miniml cost cse. It is possible tht tking the penlty cost is cheper for the supplier. Theorem : if the sum of the penlty cost ppering t producing the quntity ccordingly to the cpcity contrint nd the holding cost of quntity of the truncted period storing from the beginning of the time horizon, is less thn the cost of prepering new setup, then tking the risk of the penlty is the proper policy. We prove this theorem with the following explntion. Denote the cost vlue of the occurred bck-orders with vrible b. The next eqution helps to decide wht policy should be pplied. b+ if b+ i= i= h ( C q h ( C q ) c ) > c f f, then tking the risk,, then reducing the periods., (2.) where h is cummultive holding cost per products nd q < C is the quntity in complince with the number of jointly produced periods. Vrible mens the period number, which q vlue is even smller thn the C cpcity constrint. Then i= ( C q ) h ( C q ) = h (2.2)

5 SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS 7 represents the holding cost, which ppers s difference of the quntity of the cpcity constrint nd the quntity of the reduced jointly produced periods. The mening of the formul: In the cse, when the vlue of b+ h C q ) is ( cheper thn new setup cost ( c ), the cost will be miniml, when the supplier f chooses to tke the penlty nd produces the quntity of the cpcity constrint. Otherwise reducing the number of the jointly produced periods is good policy. Figure shows the method of the cost bsed policy properly. Figure. Applying cpcity constrint in cse of cost bse policy 2.. Service Level bsed Policy The min objective when choosing this policy is to ssure the predeterminded Service Level continully. This level is determined in complince with the objectivel of the compny. Applying cpcity constrint mens tht the number of unstisfied orders should be less thn the strict service level. Unkeeping this importnt rule tkes the risk of the reltionship of the customer nd the supplier. The reduction of the jointly produced periods gives the solution of this problem. If the optiml qntity clculted with the extended newsvendor model exceeds the vlue of the cpcity constrint, then the predetermined service level cn be kept only the wy, tht we reduce the number jointly produces periods until the quntity ccording to the reduced period stisfies the cpcity condition. The reson of this solution the per unit cost vrition curve detiled in pper [7]. 3. The Multi-Product, Multi-Period Model Solving cpcity constrint problems in cse of this model is the most complicted. In the literture the ABC method is pplied widely to solve the problem. In this cse, conclusions cn be mde bsed on the prepred Preto digrm bout the significnce distribution of the elements of product set [4]. But the method

6 8 P. MILEFF, K. NEHEZ does not nswer properly the questions ppering t the clcultion of the optiml stockpiling quntities. In the following we present new heuristic method to solve multi-product, multiperiod nd service level bsed cpcity constrint optimiztion problems. We ssume globl cpcity constrint, which mens tht products shre in one common production cpcity nd ssume tht the decisions of the inventory control policy regrd to long time horizon. In this presented solution we prefer Service Level bsed policy. The objective is to determine the reduced number of jointly produced periods per product in the wy tht the sum of the totl mounts stisfies the cpcity constrint condition. The min ide behind our new heuristic solution is the specific property of the perunit cost of the products. Figure 2 shows the vrition of the per-unit cost of product in function of jointly produced periods. Per-Unit Cost q* opt Period Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 Time horizon Figure 2. Per-unit cost vrition in cse of seven periods length time horizon Ech product hs similr per-unit curve [7]. If the sum of the quntities of n number of products is greter thn the vlue of cpcity constrint, then the solution should be modified. If n i= i i* opt i u ( q I ) C, the solution is optiml. (3.) j i* In the eqution i (i=,,n) mens the number of products, q opt j is the optiml quntity of the product i in cse of opt j number of jointly produced periods nd u i represents the cpcity usge of the product. I i denotes the initil inventory of product i. The question is, which setup number of product should be modified to hve the solution miniml cost? Strt from exmintion of the per-unit cost curve. The next figure shows the vrition of the per-unit cost for 4 periods bsed on the modifiction of Figure 2.

7 SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS 9 Figure 3. Increse of per-unit cost in function of jointly produced weeks From Figure 3 it is esy to see, if the optiml number (4) of jointly produced periods is decresed to three periods, the vlue of the per-unit cost will be certinly incresed. Bsed on this observtion: Theorem 2: the cpcity constrint suitble solution is optiml, when the sum of the per-unit cost increses risen from reducing the number of jointly produced periods t the products, is miniml. Denote FKV i the sum of performed per-unit cost vritions t product i. Then: n i= FKV min. (3.2) i Theorem 2 helps finding the optiml solution, but n essentil serching method is necessry, which cn clculte the sum of per-unit cost vritions fst in multiproduct, multi-period environment. In the following we present new, nd suitble lgorithm. 3. The Algorithm nd the Other Prts of the Method The bsic ide behind the lgorithm is the existence of the optiml solution per product without the cpcity constrint condition. The objective of the method is to move on the serching spce long the miniml per-unit cost vritions, becuse we lredy mentioned before tht the optiml solution hs miniml sum of per-unit vrition costs. The lgorithm cn be divided three min prts: () strting the method nd checking the cpcity constrint condition (2) selecting optimum possible modifictions (3) choosing the merge combintion to get better solution. During the solution serch these steps re continully repeted util the optiml supplier policy will be found in complince the cpcity constrint condition.

8 0 P. MILEFF, K. NEHEZ 3.. Strt nd Cpcity constrint condition test The first step of the method is to determine the optiml number of jointly produced periods bsed on the introduced per-unit cost model [8]. This opertion is performed only once during the running, t the strt. After tht it should be investigted tht is there ny products which production cn be shift forwrd. This cn be performed by compring the quntities in the inventory nd the optiml quntities ccording to the jointly produced periods. Regrding to the first period: v * ( q I i ) 0 c, j =,2,..., m, i =,2,..., n (3.3) i j If product cn be found where this eqution is fulfilled during the clcultion of the per-unit costs, then the production of this product cn be shifted forwrd long the time. After this, these products do not tke prt in the further steps. The next step is the evlution of the following cpcity constrint condition. n j= u i q i opt j Li I i C. (3.4) If the condition is stisfied, then the solution is optiml. The eqution hs new member L i which modifies the optiml number of jointly produced periods. L i represents the solution vector, which will store the reduced jointly produced periods of the products fter the itertion steps of the lgorithm. At the beginning of the itertion, this is zero vector. If the eqution is not stisfied next steps follows Choosing the optimum possible modifictions If the solution t the first step or in previous itertion do not stisfies the cpcity constrint, then the modifiction of the solution is required. At the second step of the lgorithm those products will be chosen, which cn be suitble t the determintion of the optiml solution. Choosing the optimum possible modifiction mens lwys the product, which per-unit cost vrition is miniml. To determine this product the following steps must be performed: the formerly clculted optiml number of jointly produced periods is reduced virtully with one period opt L +. j i ccordingly to the ctul modifictions (L i ). This mens t product i: ( ) Before nd fter the reduction, the per-unit cost nd then the per-unit cost vrition cn be clculted. Only one product hving the miniml per-unit cost vrition will be chosen during this itertion step. If product i hve been chosen, then the i. element of the solution vector will be incresed: L i = L i +. The following figure shows this reduction method properly.

9 SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS Figure 4. Reduction of the jointly produced periods in function of the solution vector The product with the mximum per-unit cost vrition will lso be chosen if merging ws performed in the previous itertion. This choice constitutes the bse of the lst step of the lgorithm, which forbids the conformtion of itertion loops (detiled t step 3) Choosing the merge combintion to get better solution Selecting the miniml per-setup cost is not enough to find the best solution. There cn be cses, when the sum of per-setup cost vrition of two products cn be substitute for n nother product per-setup cost vrition for better solution. Figure 5 presents the per-setup cost vrition of three products nd the possibility of substitution. Figure 5. Comprison nd substitution of per-setup cost increses

10 2 P. MILEFF, K. NEHEZ Explntion of figure 5: becuse of the originl computed solution does not fit to the cpcity constrint condition, the number of jointly produced periods is reduced by help of the lgorithm. According to the first step of the lgorithm, the product hving the miniml vrition vlue of per-unit cost will be chosen. Let it be the first product. Suppose tht, the solution through reduction of jointly produced periods from six to five still does not fit the cpcity constrint. In cse of one product, the substitution phse cnnot be explined, so the lgorithm runs on. In the second step, gin product will be chosen, which hve the miniml vrition vlue of per-unit cost. Let it be the second product now. It is not sure tht the reduction of jointly produced periods of the two chosen product is the optiml solution. Therefore we should exmine tht the sum of vrition increses of per-unit cost, gined from the reduction of the jointly produced periods of the two chosen products, cn be substituted for smller vrition of per-unit cost. Figure 5 shows properly tht the vrition of per-unit cost of product three is smller thn the vritions sum ppered t product one nd two. This mens tht the vritions sum ppered t product one nd two cn be substitute for reduction of jointly produced periods t product three. Exmine the cse, if there re four products. In the next figure, the per-unit cost vritions of product three nd four cn be seen in cse of one period decreses of its jointly produced periods. Figure 6. Substitution phse in cse of more thn three products In cse of more thn three products, question rises: which product should be chosen for substitution. In figure 6 cn be seen tht both of the vritions of perunit cost t product three nd four re smller thn the vritions sum ppered t the first two products. In this cse the product should be chosen, where the vrition flls the frthest from the vritions sum. The min reson of this is the following: if the result fter the substitution does not fit the cpcity constrint, the lgorithm in the next step

11 SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS 3 chooses gin product, which hs the miniml per-unit cost vrition. In this exmple, the first product will be this gin. Suppose tht the solution is optiml now. If product four is chosen, then the sum of per-unit cost vritions is certin smller s if product three would be chosen for substitution. During the substitution process we use the product with mximum per-unit cost vrition vlue found t the second step. This vlue nd product constitutes the bse of reference in the investigtion of the per-unit cost vritions. The possibility of merging will be exmined ccording to this vlue, becuse it cnnot be the chosen product. If the substitution performed successfully it is necessry to prepre the next itertion. The first step is to modify the solution vector. The vlue in the vector belonging to the selected product should set to zero. This ensures tht the lgorithm cn move on the serch spce long the miniml per-unit cost vritions. After this process the next itertion comes until the solution does not fit the cpcity constrint condition. Clcultions in prcticl show clerly tht finding the optiml solution do not need lot of itertion step. In cse of product, the optiml number of jointly produced periods is bout 7-8 periods. The nlytic solution of the extended newsvendor model ssures high performnce clcultion in n optionl multi-product, multiperiod environment in cse of long time horizon. 4. Conclusion In this pper we extended the previously elborted nd modified newsvendor model [6][7] with the condition of globl cpcity constrint. Bsed on the periodic chrcteristic of the model, two problem groups were differed nd presented. For the most complex, multi-period, multi-product cse new heuristic method ws elborted. This model mkes possible the determintion of cost-optiml stockpiling policy pplying cpcity constrint in cse of optionl number product nd optionl long production time horizon. Becuse of the individul pproch of the new model, it cn be used effectively in prctice in reltion to other models in the literture. Acknowledgements The reserch nd development summrized in this pper hs been crried out by the Production Informtion Engineering nd Reserch Tem (PIERT) estblished t the Deprtment of Informtion Engineering. The reserch is supported by the Hungrin Acdemy of Sciences nd the Hungrin Government with the NKFP VITAL Grnt. The finncil support of the reserch by the forementioned sources is grtefully cknowledged. Specil thnks to Ferenc Erdélyi for his vluble comments nd review works.

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13 SOLVING MULTI-PERIOD, MULTI-ITEM CAPACITY PROBLEMS 5 [5] MILNE, A.: The mthemticl theory of inventory nd production: The Stnford Studies fter 36 yers, In Workshop, August 994, Lke Blton. ISIR, Budpest, 996, [6] VON NEUMANN, J. AND MORGENSTERN, O.: Theory of Gmes nd Economic Behvior, Princeton University Press, 944. [7] DVORETZKY, A., KIEFER, J., WOLFOWITZ, J.: On the optiml chrcter of the (s; S) policy in inventory theory, 953, Econometric 2: