Production Planning for Perishable Products with Partial Postponement Strategy

Transcription

1 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 Productio Plaig for Perishable Products with Partial Postpoemet Strategy A. Shabradi Departmet of Idustrial Egieerig, Faculty of Egieerig, Tarbiat Moallem Uiversity, Karaj, Ira M. Mohammadi Departmet of Idustrial Egieerig, Faculty of Egieerig, Tarbiat Moallem Uiversity, Karaj, Ira ABSTRACT This work addressed the tactical productio plaig for perishable products with partial postpoemet strategy i which the cost is miimized subject to a set of costraits such as labor workig time ad machie time. For a specific fial cofiguratio some portio of demad has less demad fluctuatio ad we ca use make-to-stock strategy to respod demad. But for other portio of demad which has high degree of ucertaity, we use cocept of postpoemet ad it will be beeficial. We compute the level of postpoemet capacity ad fiished goods ivetory by developig a mixed iteger oliear programmig model. We have aalyzed the behavior of the model to make maagerial isight uder differet scearios of fill rates ad fiished-goods ivetory costs. Our fidig shows the fiished goods ivetory ad fill rates ca decrease the expected total cost ad icreases postpoemet capacity. Geeral Terms Productio Keywords Productio maagemet, mass customizatio, partial postpoemet, perishable 1. INTRODUCTION I this age of globalizatio, rapid techological advacemet, icreased ecoomic ucertaity ad demadig customers lead to short product cycle ad shorter lead times which require quick respose to highly customized customers eeds, better product ad service quality. Postpoemet strategy is cosidered a effective way to respodig evirometal ucertaity ad achievig mass customizatio. With today s market characteristics ucertai ature should be viewed as a opportuity ad ot a problem. I geeral, postpoemet is a strategy which is used to determie the efficiet maer to make products ad services available to ed user. Postpoemet, also kow as ed of lie cofiguratio, late customizatio or delayed product differetiatio, is ow defied as a orgaizatioal cocept whereby chagig, timig ad/or placig some of the processes i the supply chai (such as purchasig, desigig, maufacturig, productio, packagig, labelig ad marketig) would take place after some key iformatio about the customer s specific eeds or requiremets is revealed. The goal of postpoemet is to supply desirable products to customers at a relatively low cost i a resposive way. I this paper we develop a geeral tactical productio plaig model for perishable products that have sigificat setup costs. Perishable products regarded products that caot be stored for a log time because they rot or ca o loger be used. Aother type of perishable products are computers ad mobile phoes which sale volumes drop dramatically ad their value decreases quickly with the lapse of time due to costat improvemet of techology. Seasoal products like high fashio apparel, Christmas gifts ad caledars are aother type of these products. These are sold oly below full price after a day or a seaso. I order to deal with the productio plaig uder limited resources for perishable products we adopt partial postpoemet strategy. Postpoemet i productio refers to geeric itermediate products beig maufacture i early stage ad accordig to differetiatio optio fial activity such as packagig; fial assembly is postpoed to later util customer order iformatio received. Partial postpoemet meas to stock products both i geeric form ad customized form so that customers demad satisfied either from customized goods ivetory or geeric goods ivetory. Partial postpoemet is a special form of form postpoemet. It has bee proved it s ot beeficial from ay amout of postpoemet from service levelivetory trade off perspective [1]. Partial postpoemet uses effectively some portio of the demad for product which has less demad fluctuatio ad is kow with certaity. It is idicate less expesive make-to-stock sequece of operatio would be used to produce this portio of fiished-goods ivetory. This work maily motivated by dairy productio compaies that products differet flavored dairy productio ad depedet products demad such as yoghurt. This paper orgaized as follow. I sectio related literatures are itroduced. After that problem descriptio ad model formulatio is preseted. I sectio 4 we provide umerical illustratio. I sectio 5 sesitive aalyses of fill rates ad fiished-goods holdig cost are preseted. Sectio 5 cotais a summary, coclusio ad futures works.. LITERATURE I geeral, cost models for aalyzig postpoemet strategies ca be classified ito four categories. They are descriptive models, determiistic models, stochastic models ad heuristic models. Accordig to Hiller ad Lieberma [] stochastic models is defied as a ivetory model where demad i ay period is radom. May authors describe various approaches to hadle ucertaity i their model. This trai starts with ladmark work of Zi ad Bowersox [3] by usig a ormative cost model ad discrimiate aalysis. They cocluded product value is the most importat variables that justify postpoemet. Iyer et al., [4] developed a two-stage capacity plaig problem 8

2 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 model uder demad postpoemet. The mai idea behid the model is that by preemptig stockouts through demad postpoemet we reduce the total cost. Ad this is achievable by postpoig a fractio of the demad i a period to permit capacity to be procured to fulfill demads later. Their approach is to formulate a two-stage stochastic dyamic program with updatig of demad iformatio. Recet stochastic postpoemet Bish et al., [5] study the advatages of two perishable idivisible ad flexibility resources structure uder demad forecast ucertaity ad demad variability i a firm that to satisfy two demad types. The mai goal was to delay resource allocatio decisio util more iformatio o customer demad is achieved Al-Salim ad Choobieh [6] propose two oliear biary optimizatio models for determiig the optimal stage for differetiatig multi-product productio system. Their two developed models maximize the expected value of profit ad the value of optios to postpoe product differetiatio respectively. Their results show the profit model recommeds a later product differetiatio tha the optio model. If coefficiet of variatio of product is low demad correlatio is ot importat ad egative correlatio amog product demad is better tha positive correlatio. Recet work, Sog ad Kusiak [7] preset a geeral framework for fidig optimal modules i a delayed product differetiatio sceario. The mai goal was to miimize the mea umber of assembly operatio ad pre-assembly cost. Aupidi ad Jiag [8] presets a duopoly model where firms decisios o capacity, productio, ad price uder demad ucertaity are made i a three-stage decisio makig framework. I their model the firm always decides about capacity before ad price after the demad realizatio, while there is a differece i the timig of the productio decisio. Their study show that the optimal capacity of the flexible resource is icreasig i demad variability ad that the value of flexibility is maily sigificat if the demad levels are highly variable. Grama ad Saders [9] modeled the tradeoff betwee postpoemet capacity ad forecast accuracy. Their fudig shows postpoemet capacity ad forecast accuracy resultig i reduced ivetory levels while maitaiig a costat service level. Their study uses ivetory reductio as the oly beefit of improvig forecast accuracy ad icreasig postpoemet capacity. The value of a product is a factor sets limits o the beefit that ca be achieved by icreasig postpoemet capacity ad improvig forecast accuracy. Bish et al., [1] i their paper study the effect of two mai driver product, substitutio ad operatioal postpoemet, o the firms optimized capacity decisio i a moopolistic settig. They cosider price ad quatity (PQ) ad price oly as a two type of postpoemet strategy. They formulate both postpoemet strategies as two-stage stochastic programmig problems Tog [11] presets a two-stage supply chai model with a sigle ew product maufacturer ad a sigle retailer. The maufacturer orgaizes his productio after takig orders from the retailer. He show for products featurig a expoetial demad with Gamma prior, if the demad forecast improves over time ad the maufacturer sets the wholesale price, the expected order size icreases whe the order is placed later. Recetly partial postpoemet has become a icreasigly attractive research subject. Swamiatha ad Tayur [1] aalyze a fial assembly process with productio capacity costrait where ivetory is stored i the itermediate form called vailla boxes. This model compares performace of both assemble to order (where compoets are stocked ad products assembled from the compoets after demad is realized) ad make-tostock (where ivetory is carried i fiished form oly) semifiished assembly process. Their approach allows for multiple poits of differetiatio. They develop a stochastic iteger program to determie the optimal types of vailla boxes as well as their ivetory levels which miimize the expected holdig ad pealty costs i sigle ad multi-period settigs. A sigleperiod, multi-product, capacitated ivetory model cosidered by Grama ad Magazie [13] where ivetory ca be stored i a itermediate form. To satisfy the demad subjected to a capacity costrait the fiished goods are used first ad the the semi-fiished product. They used a ivetory- service level approach without costs to determie the levels of fiished goods ivetory for a specified level of capacity. Their approach miimizes total ivetory subject to meetig a specified service level rather tha cosiderig shortage costs. Chopra ad Midl [14] described a strategy of producig the amout of products that is very likely to sell usig the lower cost productio. He called that tailored postpoemet. Silver ad Mier [15] developed a partial postpoemet strategy for a fast food idustry usig a ewsvedor-style approach. They help a pizza shop determie the quatities of each type of fiished ad ufiished pizzas. Grama [1] developed a oliear programmig icludig a sigle-period, two product, order-upcost model to aid i settig the levels of fiished goods ivetory ad postpoemet capacity. He developed previous work of Grama ad Magazie [13] by icorporatig a decisio cost model to settig the levels of ivetory. He foud out the expected total cost decreases ad postpoemet capacity icreases as the value of the postpoemet commo item ad the cost of postpoemet decrease also the cost of packagig becomes greater tha the assembly cost. Recetly i productio plaig literature usig postpoemet strategy we ca refer to Leug ad Ng [16] which defie a twostage stochastic model for determiig the productio loadig plaig for perishable products i particularly holiday-themed toys. The mai deferece betwee this work ad our paper is that we use a cotiuous demad distributio while leug ad Ng use a demad uder four differet scearios i each period. They foud out with postpoemet strategy a savig about 7% i total cost (icludig operatioal cost, ivetory cost, ad hirig cost ad lay-off cost) is made i compariso with opostpoemet strategy. I aother work they also [17] use a preemptive goal programmig model to deal with the aggregate productio plaig problem for that firm. Demad for differet period is cosidered Determiistic ad time-sesitive. Three major objectives with target values are optimized hierarchically. Their fidig idicate i order to meet the dramatically icrease i demad some semi-fiished products recommeded to be produced i the earlier plaig horizo. Previous work by Grama [1] provides a basis for our itegratio partial postpoemet strategy ito tactical productio plaig while we have limited resources ad sigificat setup cost. 3. PROBLEM DESCRIPTION AND MODEL DEVELOPMENT I this study the tactical productio plaig problem for perishable products is ivestigated. We developed sigle period, two product order up-to partial postpoemet uder limited 9

3 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 resources. I every period we determie how may fiished products (direct productio), geeric products (master productio) ad trasfer product (fial activity) should be produce at the begiig of the period so that use better utilizatio of limited resources. Productio starts from raw material to produce geeric products. After geeric products productio some portio of geeric product is ot postpoed ad immediately fial activity is carried out ad stored as a fiished goods ivetory. Remai of geeric products held as semifiished goods to satisfy demad by combiatio with fiished goods ivetory. Demad i excess of the o-postpoed ivetory is met if possible through doig fial activity of the geeric ivetory. Without loss of geerality ad better uderstadig of the model we assume oe uit of geeric item makes oe uit of fiished product. 3.1 Problem formulatio Notatio: Parameters: C SF,i setup cost for productio of direct fiished product i from raw material C SS,i setup cost for productio of geeric product i C ST,i setup cost for productio of fiished product i from geeric product C W labour cost C RF,i regular-time uit productio cost to produce oe uit of direct fiished product i C RS,i regular-time uit productio cost to produce oe uit of commo item from raw material C RT,i regular-time uit productio cost to produce oe uit of fiished product i from semi-fiished products C OF,i overtime uit productio cost to produce oe uit of fiished product i from raw materials C OS,i overtime uit productio cost to produce oe uit of commo item from raw materials C OT,i overtime uit productio cost to produce oe uit of fiished product i from commo item H p ivetory holdig cost for oe uit of commo item H pp the cost of postpoemet materials ad activities for oe uit of the commo item H F,i ivetory holdig cost for oe uit of fiished product i. λ W fractio of regular workforce available for over-time δ regular workig hours of labor i each period γ i the target umber of stockouts for product i λ M fractio of regular machie capacity available for over-time use M t maximum regular time machie capacity M a big umber a F,i ma hours required to produce oe uit of fiished product i from raw materials a S,i ma hours required to produce oe uit of commo item product i from raw materials a T,i ma hours required to produce oe uit of fiished product i from semi-fiished products b F,i machiig time required to produce oe uit of fiished product i from raw materials b S,i machiig time required to produce oe uit of semifiished product i from raw materials b T,i machiig time required to produce oe uit of fiished product i from commo item products Ucertai parameters D i radom variable represetig the demad for product i Decisio variables cotiuous variable W Number of workers Z RF,i amout direct product i durig regular time Z OF,i amout direct product i durig over-time Cap R the amout of postpoemet capacity produces i regular time Cap O the amout of postpoemet capacity produces i over-time P R,i the amout of postpoemet capacity used to meet demad for product i i regular time P O,i the amout of postpoemet capacity used to meet demad for product i i over-time Biary variables Y F,i = 1 if Z RF,i otherwise Y S,i = 1 if Z RS,i otherwise Y T,i = 1 if Z RT,i otherwise Expected value fuctio E TC PP the expected total cost of a partial postpoemet strategy (if CapR + CapO>) E TC Npp the expected total cost of a partial postpoemet strategy (if CapR + CapO=) E SO i the expected umber of stock out of product i i terms of geeric products. 3. Compoet of objective fuctio 3..1 Fial activity cost The fial activity such as packagig, labelig ad fial assembly occurs after the productio of geeric item. There are cost C RT,i ad C OT,i associated with regular ad over-time productio cost to carry out fial activity for product i. Now we compute this cost accordig to the differet regio of Figure 1. if demad for product oe occurrig i regios {1}, {}, {3} or {13} ad demad for product two occurrig i regios {1}, {4}, {1} or {11} are met usig Z RF,1 + Z OF,1 ad Z RF, + Z OF, respectively. So, the expected total cost to occur fial activity for both fiished goods ivetories is give by: C RF,i. Z RF,i + C OF,i. Z OF,i If demad for product oe occur i regio {4}-{1} i additio to i Z RF,it + i Z OF,it the postpoed ivetory will also trasformed to the fiished goods to met demad. The expected cost for both items will be: C RT,i. E P RT,i + C OT,i. E P OF,it 1 3

4 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 The expected umber of uits of the geeric item to carry out fial activity for product i is equals to: E P RT,i = E P OT,i = K K E P RT,i K E P RT,i K i = 1, 3 i = 1, 4 The expressio ad limits itegratio for each regio of iterest ca be foud i Appedix A. Figure 1: Graphical depictio of ivetory levels uder partial postpoemet strategy a 1 = + Cap O a = + Cap R a 3 = + Cap R + Cap O b 1 = + Cap O b = + Cap R b = + Cap R + Cap O 3.. Holdig cost of fiished goods ivetory If demad for product oe occurs i regios {1}, {}, {3} or {13} ad demad for product two occurs i regios {1}, {4}, {1} or {11} it is possible to have some fiished goods ivetory. The total expected cost for period t is give by E H F,i Z RF,i + Z OF,i x i + The expected quatity of fiished goods left over of product oe at the ed of period is E Z RF,1 + Z OF,1 x 1 + {1,,3,15} = Z RF,1 + Z OF,1 x 1 f t x 1, x dx 1 dx 6 Where Z RF,i + Z OF,i x i + = max, Z RF,i + Z OF,i x i The expressio for f x 1, x was developed usig Morgester s copula. Copula provides a method of costructig multivariate distributios from kow margial distributios. 5 The demad for product two will be solely from fiished-goods ivetory if it occurs i regios {1}, {4}, {1} or {11}. Expressio for product two are similar to product oe Holdig cost of postpoed ivetory At the begiig of the period a ivetory of postpoed items exist. It s give by H P. Cap R + Cap O E P RT,i + P OT,i 3..4 Shortage cost Demad i differet regios are met by combiatio of Z RF,it, Z OF,it, P RT,i,t ad P OT,i,t. The expected umber of stockouts for each product i is the sum of the expected umber of stockouts for each regio i which stockouts ca occur: E SO i = K E SO i K i = 1, 8 The detail expressio for the expected umber of stockouts for product oe i regio k ca be foud i Appedix B. Objective fuctio: The objective fuctio of the mixed-iteger oliear program icludes six terms. The first ad the secod term are setup cost ad labour cost respectively. The third term is productio cost of direct fiished products cost associated with regular ad over-time. The fourth term is productio cost of geeric product cost associated with regular ad over-time. Fial term is the expected total holdig cost of fiished-goods ivetory. mi E TC PP = mi E TC NONPP = Costraits: C SF,i Y F,i + C SS,i Y S,i + C ST,i Y T,i + C RF,i Z RF,i + C OF,i Z OF,i + C RS,i + H pp + H p Cap R + C OS,i + H pp + H p Cap O T + E C RT,i H p P RT,i t=1 + C OT,i H p P OT,i + E H F Z RF,i + Z OF,i D i + T C SF,i Y F,i + C SS,i Y S,i + C ST,i Y T,i T 7 + C W W t=1 + C W W + C RF,i Z RF,i + C OF,i Z OF,i t=1 + E H F Z RF,i + Z OF,i D i + E SO i γ i i = 1,

5 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 Z RF,i + Z OF,i + Cap R + Cap O a i i = 1, 1 Z RF,i + Z OF,i + Cap R + Cap O b i i = 1, 13 P RT,i + P OT,i Cap R + Cap O 14 Cap R + Cap O K 15 a F,i Z RF,i + a Si Cap R + a T,i P R,i a F,i Z OF,i + a Si Z OS,i + a T,i P O,i b F,i Z RF,i + b Si Z RS,i + b T,i P R,i b F,i Z OF,i + b S,i Z OS,i + b T,i P O,i δw 16 λ t W δw 17 M 18 λ M M 19 Z RF,i + Z OF,i MY F,i i = 1, Cap R + Cap O MY S,i i = 1, 1 P R,i + P O,i MY T,i i = 1, Z RF,i, Z OF,i i = 1, 3 Cap R, Cap O, W, K 4 P R,i, P O,i i = 1, 5 Y F,i ε, 1 i = 1, 6 Y S,i ε, 1 i = 1, 7 Y T,i ε, 1 i = 1, 8 The objective fuctios miimize the expected total cost with ad without postpoemet. Costrait (11) isures that the target fill rates for each product are met. Costrait (1) ad costrait (13) are lower ad upper boud of demad. Costrait (14) isures that the amout of the postpoed ivetory used by both products is less tha or equal to the postpoed ivetory. Costrait (15) is used to creatig the opostpoemet or some specific amout of postpoed capacity. Costraits (16) ad (17) limit the labour workig hours durig regular time ad overtime respectively. Similarly, costraits (18) ad (19) limit the machiig time durig regular time ad overtime respectively. Costraits () () esure that setup costs will be icurred whe the correspodig productio activities started. Costraits (3)-(5) are the o-egative costraits. Costraits (6)-(8) are used for the setup idicatio of the product activity. 4. COMPUTATIONAL RESULTS I this sectio to validate our proposed model we use a umerical example. The iput parameters are give i table 1. to table 3. We assume the demad of two products are idetical ad uiform distributed i 48,15. Products demad are idepedet expect i the discussio of correlated demad. We use differet measuremet, umber of stockouts for customer service level. The expected umber of stockouts for product i is defied as bellow: E SO i γ i 9 That γ i is give by: γ i = 1 β i. μ i 3 That β i is fill rate for product i. the fill rates are set to β 1 = β =.975. Table 1: Productio cost Product Cost Direct productio Regular time Over-time Geeric Productio Regular time Over-time Trasfer productio Regular time Over-time Table : Operatig ad cost data Product Direct Geeric Trasformed Set up cost Labor time Machie time Table 3: Machie ad workforce level Maximum workforce level 1 Maximum machie capacity 16 Fractio of workforce available for over-time.3 Fractio of machie capacity available for over-time.4 The optimal solutio for the base case uder postpoemet strategy is = = 818, P 1,R = 35, P,R = 366, P 1,O = 31 ad P,O =. Table 4. ad table 5. preset a compariso betwee two strategy (PP ad NONPP). As we see applyig postpoemet strategy lead to reductio i fiished goods ivetory ad direct productio cost ad savig about 6% is made. Table 4: Variables value (Base Case) Without PP With PP Product1 Product Product1 Product ZRF ZOF 7 PRT POT 31 YF YS 1 1 YT 1 1 CapR 7 CapO 3

6 Postpoemet Capacity Percet Cost reducctio compared to idepedet-nopostpomet Table 5: Optimal Solutio of Base Case (Demad Correlatio=+.33) Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 Labor cost Setup cost D.productio cost G.& T. productio cost F.G. ivetory cost PP.cost Total cost With PP Without PP SENSITIVE ANALYSIS I this sectio we study the effect of fill rate o postpoemet capacity ad percet reductio i expected total cost compared to idepedet demad ad opostpoemet case. Although Morgester s Bivarate Uiform distributio limit the correlatio coefficiet to a rage of -1/3 to +1/3. It s eable us to gai isight ito the impact of demad correlatio o expected total cost ad postpoemet capacity. 5.1 Fill rate The various levels of fill rates betwee 9% ad 1% are cosidered to examie the behavior of differet objective fuctio with differet demad correlatio. These variatios are show i Figure. I this graph percet cost reductio compared to idepedet products demad ad opostpoemet case are depicted. Accordig to Figure. to Figure 5. we ca coclude icreasig the fill rates results i higher postpoemet capacity level, higher fiished goods ivetory ad icreasig percet reductio i the expected total cost of partial postpoemet compared to idepedet products demad ad opostpoemet case. But i a very high level of fill rate the level of postpoemet capacity decreases while the fiished goods ivetory cotiue to icrease i a more aggressive mode. We ca see relatioship betwee postpoemet capacity ad fiished goods ivetory i Figure. to Figure 5. The fill rate level which percet reductio i expected total cost begis to decrease is differet for various demad correlatio. The lower demad correlatio results to higher fill rate levels that percet reductio i expected total cost icreases. i.e. for -.33 correlatio coefficiet case percet reductio i expected total cost icreases util to 99.87% fill rate level. For the higher level of fill rate beeficiary of partial postpoemet decreases i a very speed ad make-to-stock strategy is better. I other side for +.33 demad correlatio case the percet reductio fill rate level is lower tha 99.98% ad equal to 99.68%. Total Cost Reductio as Fuctio of Fill Rate 3% 5% % 15% 1% 5% % 9% 93% 95% 98% 1% Fill rate Figure : Effect of fill rate o the expected total cost for selected values of the coefficiet of demad Postpoemet Capacity as a Fuctio of Fill rate % 9% 93% 95% 96% 98% 99% Fill Rate Figure 3: Effect of fill rate o the postpoemet capacity for selected values of the correlatio of demad 33

7 F.Givetory Cost with PP Strategy Postpoemet Capacity Thousads Percet Cost reducctio compared to idepedet-nopostpomet Total expected cost Thousads Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 Total Expected Cost as a fuctio of Fill rate correlatio shows that the percet reductio of the total expected cost for egatively-correlated demads is greater tha for positively-correlated demad. But for a postpoemet capacity as holdig cost icreases the postpoemet capacity icrease i differet patter for differet demad correlatio. Other costraits i our model affect the level of postpoemet for differet holdig cost. These affects are illustrated i Figure % 9% 94% 96% 98% 1% Fill Rate Figure 4: Effect of fill rate o the expected total cost for selected values of the coefficiet of demad Fiished-Goods ivetory Cost as a fuctio of Fill Rate Percet reductio i total expected cost as a fuctio of fiished goods holdig cost 6% 5% 4% 3% % 1% % -5% 5% 145% 38% Variatio of Holdig Cost of Fiished goods % 9% 93% 95% 96% 98% 99% Fill Rate Figure 6: Total expected cost as a fuctio of fiished goods holdig cost % 17% 133% 5% 367% 483% Figure 5: Effect of fill rate o the fiished-goods ivetory cost for selected values of the correlatio of demad 5. Fiished goods ivetory cost We ow examie the impact of various value of fiished goods ivetory cost o expected total cost ad postpoemet capacity i various demad correlatio. The fiished-goods holdig cost for base case is 6. Graph i Figure 6. showed from -5% of holdig cost of base case to +5% from base case. As holdig cost icreases thus the postpoemet capacity (CapR+CapO) icreases. The effects of differet demad correlatio o total expected cost are the same as the oes used i previous sectio. Compariso of the graphs at each level of the demad Figure 7: Percet reductio i total expected cost as a fuctio of fiished goods holdig cost 6. SUMMARY AND FUTURE WORKS I this study we developed a mixed iteger oliear programmig model for sigle period, two products productio plaig usig a partial postpoemet strategy i which demads of products are depedet ad uiformly-distributed to HF

8 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 settig the level of fiished goods ivetory ad postpoemet capacity. We derived expressio for the expected umber of stockouts ad expected operatioal cost. The computatioal results show for dealig ucertaity partial postpoemet is a good strategy. We foud out that expected total cost decrease ad postpoemet icreases as (1) the value of fiished goods ivetory cost icreases () product demads became more egatively correlated. The expected total cost icreases ad capacity decreases as the fill rate approach 1%. There is still room for improvemet. First real data from compaies ca be used to validate our proposed model ad to aalysis its sesitivity to chages i productio plaig strategies. Secod we developed the model for two products. However we are facig more tha two products i real world. Third we developed a sigle period, short productio plaig. It s valuable to develop a model for itermediate ad log term productio plaig. 7. APPENDIX.A Expected of postpoemet capacity used by product 1 I below expressio we assume: z i = Z RF,i + Z OF,i i = 1, A. 1 Regios {1,, 3, 15} If demad for product oe occurs i regios {1}, {}, {3}, or {15} of Fig., the oe of the postpoed items will be used to meet demad. E P RT,1 {1,,3,15} = A. Regio {7} E P RT,1 7 +Cap R + +Cap R +Cap O x = P RT,1 f x 1, x dx 1 dx A Cap R x +Cap R + +Cap R +Cap O x E P OT,1 {7} = x 1 + +Cap R x Cap R f x 1, x dx 1 dx A. 9 Regio {8} E P RT,1 {8} +Cap R = P 1R f x 1, x dx dx 1 A. 1 +Cap R Cap O E P OT,1 {8} + +Cap R x 1 +Cap R = x 1 +Cap R Cap O + +Cap R x 1 Cap R f x 1, x dx dx 1 A. 11 Regio {9} + +Cap R +Cap O x E P OT,1 9 = x 1 +Cap R Cap R f x 1, x dx 1 dx A. 1 E P OT,1 {1,,3,15} = A. 3 Regio {4} +Cap R E P RT,1 {4} = x 1 f t x 1, x dx 1 dx A. 4 E P OT,1 {4} = A. 4 Regio {5} +Cap R + +Cap R x E P RT,1 {5} = x 1 z1 f x 1, x dx 1 dx A. 5 E P OT,1 {5} = A. 6 Regio {6} E P RT,1 {9} = P 1R f x 1, x dx 1 dx A. 13 Regio {1} E P OT, Cap R +Cap O x +Cap R +Cap R +Cap O = x 1 +Cap R Cap R f x 1, x dx 1 dx A. 14 E P RT,1 {1} +Cap R +Cap O = P 1R f x 1, x dx 1 dx A. 15 +Cap R Regio {11} E P RT,1 {6} = A. 7 +Cap R +Cap O + +Cap R +Cap O x E P OT,1 {6} = x 1 +Cap R f t x 1, x dx 1 dx E P OT,1 {11} = Cap O f x 1, x dx 1 dx +Cap R +Cap O A

9 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 E P RT,1 {11} = Cap R f x 1, x dx 1 dx +Cap R +Cap O A. 17 E SO 1 = x 1 P RT,1 P OT,1 f x 1, x dx 1 dx B. 3 Regio {1, 13, 14} +Cap R + +Cap R x E SO 1 5 = x 1 P RT,1 f x 1, x dx 1 dx B. 3 E P RT,1 {1} = P RT,1 f x 1, x dx 1 dx +Cap R + +Cap R x P RT,1,t,t f x 1, x dx 1 dx +Cap R + +Cap R +Cap O x P RT,1 f x 1, x dx 1 dx +Cap R + +Cap R x P RT,1 f x 1, x dx dx 1 +Cap R Cap O + +Cap R x 1 + +Cap R +Cap O x P RT,1,t +Cap R f x 1, x dx 1 dx A. 18 E P OT,1 {1} = P 1O f x 1, x dx 1 dx +Cap R +Cap O + +Cap R +Cap O x P OT,1 +Cap R f x 1, x dx 1 dx +Cap R + +Cap R +Cap O x P OT,1 + +Cap R x f x 1, x dx 1 dx +Cap R P OT,1 f x 1, x dx dx 1 +Cap R Cap O + +Cap R x 1 + +Cap R +Cap O x P OT,1 +Cap R f x 1, x dx 1 dx A APPENDIX B. Expected Stock out of Product oe: E SO 1 11 = +Cap R +Cap O x 1 Cap R Cap O f x 1, x dx 1 dx B. 1 E SO 1 1,13,14 = E SO 1 E SO E SO E SO 1 7,8,9 B. +Cap R +Cap O + +Cap R +Cap O x E SO 1 6 = x 1 +Cap R P OT,1 f x 1, x dx 1 dx B. 4 +Cap R + +Cap R +Cap O x E SO 1 {7,8,9} = x 1 + +Cap R x Cap R P OT,1 f x 1, x dx 1 dx B REFERENCES [1] Grama, GA. 1. A partial-postpoemet decisio cost model. Europea Joural of Operatioal Research, 1(1):34-44 [] Hillier, FS. ad Lieberma, GJ. 1. Itroductio to Operatios Research, 7th Editio, McGraw-Hill, Sigapore. [3] Zi, W. ad Bowersox, D.J Plaig physical distributio with the priciple of postpoemet. Joural of Busiess Logistics, 9(): [4] Iyer, VA, Deshpade, V. Wu, Z. 3. A postpoemet model for demad maagemet. Maagemet Sciece, 49(8): [5] Bish, EK. Li, KY. Hog, S-J. 8. Allocatio of flexible ad idivisible resources with decisio postpoemet ad demad learig. Europea Joural of Operatioal Research 187:49 44 [6] Al-Salim, B. Choobieh, F. 9. Redesig of productio flow lies to postpoe product differetiatio Iteratioal Joural of Productio Research 47(): [7] Sog, M. Kusiak, A. 1. Miig Pareto-optimal modules for delayed product differetiatio. Europea Joural of Operatioal Research, 1: [8] Aupidi, R. Jiag, L. 8 Capacity ivestmet uder postpoemet strategies, market competitio ad demad ucertaity. maagemet sciece 54(11): [9] Grama GA, Saders NR (9) Modellig the tradeoff betwee postpoemet capacity ad forecast accuracy. Productio Plaig ad Cotrol (3):6 15 [1] Bish, EK. Suwadechochai, R. 1. optimal capacity for substitutable products uder operatioal postpoemet. Europea Joural of Operatioal Research 7:

10 Iteratioal Joural of Computer Applicatios ( ) Volume 31 No.5, October 11 [11] Tog, C. 1. Order postpoemet i a supply chai i the presece of expoetial demad with gamma prior. Operatios Research Letters, [1] Swamiatha, JM. ad Tayur, SH Maagig Broader Product Lies Through Delayed Product Differetiatio Usig Vailla Boxes, Maagemet Sciece, 44(1):S161 S17. [13] Grama, G.A. ad Magazie, M.J.. A umerical aalysis of capacitated postpoemet. Productio ad Operatios Maagemet 11 (3), [14] Chopra, S. ad Meidl P. 3. Safety ivetory. I: Supply chai maagemet: Strategy, plaig, ad operatio. d ed. Upper Saddle River, NJ: Pearso Pretice Hall, 34. [15] Silver, EA. ad Mier, S. 5. A repleishmet decisio ivolvig partial postpoemet. OR Spectrum, 7:1 19. [16] Leug, SC. ad Ng, WL. 7a. A goal programmig model for productio plaig of perishable products with postpoemet. Computers ad Idustrial Egieerig, 53: [17] Leug, SC. Ng, WL. 7b. A stochastic programmig model for productio plaig of perishable products with postpoemet. Productio Plaig ad Cotrol, 18(3):19. 37