Optimal ordering quantities for substitutable deteriorating items under joint replenishment with cost of substitution

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1 J Ind Eng Int (7) 3:38 39 DOI.7/s z ORIGINAL RESEARCH Optimal ordering quantities for substitutable deteriorating items under joint replenisment wit cost of substitution Vinod Kumar Misra Received: October 6 / Accepted: 7 February 7 / Publised online: 4 Marc 7 Ó Te Autor(s) 7. Tis article is publised wit open access at Springerlink.com Abstract In tis paper we develop an inventory model, to determine te optimal ordering quantities, for a set of two substitutable deteriorating items. In tis inventory model te inventory level of bot items depleted due to demands and deterioration and wen an item is out of stock, its demands are partially fulfilled by te oter item and all unsatisfied demand is lost. Eac substituted item incurs a cost of substitution and te demands and deterioration is considered to be deterministic and constant. Items are order jointly in eac ordering cycle, to take te advantages of joint replenisment. Te problem is formulated and a solution procedure is developed to determine te optimal ordering quantities tat minimize te total inventory cost. We provide an extensive numerical and sensitivity analysis to illustrate te effect of different parameter on te model. Te key observation on te basis of numerical analysis, tere is substantial improvement in te optimal total cost of te inventory model wit substitution over witout substitution. Keywords Inventory control Substitutable items Cost of Substitution Deterioration Optimal ordering quantity Joint replenisment & Vinod Kumar Misra vkmisra5@gmail.com Department of Computer Science and Engineering, B T Kumaon Institute of Tecnology, Dwaraat, Almora, Uttarakand 63653, India Introduction As we know tat at any retails or supermarket te occurrence of temporary stock-outs is a very common penomenon in te categories of frequently purcased items and it is also very common to see at any retails or supermarket, customers wo willing to purcase certain items will be willing to purcase te substitute items, if tey faced te situation of te stock-outs. A survey report of Anupindi et al. (998) also observed te same penomenon, in wic e found tat 8 88% of buyer would be willing to buy te substitute items if te desired items are out of stock. Te substitutable items in wic sufficient deterioration can take place during te normal storage period of te units and consequently tis loss must be taken into account wen analyzing te inventory system of substitutable items, i.e. te effect of deterioration plays a vital role in te decision of ordering quantity of substitutable deteriorating items. Wen substitution will take place an additional cost is incurred, known as substitution cost. Suc substitution costs may arise due to a variety of reasons: te cost of te reworking required on an item to make it substitutable for te oter, loss of a customer s goodwill due to substitution, etc. Deterioration of pysical goods in stock is a very realistic feature and tere is a big need to consider it in te inventory modelling of substitutable items. Tang and Yin (7) categorizes te substitution as stock-out-based substitution, price-based substitution and assortment-based substitution. Recently, Kim and Bell () categorizes te substitution as symmetrical substitution and asymmetrical substitution. As tey define, te definition of tese categories are: stock-out based substitution corresponds to a situation in wic a customer may purcase anoter product as a substitute, wen te preferred product is out of stock, price-based

2 38 J Ind Eng Int (7) 3:38 39 substitution corresponds to a situation in wic a retailer uses different pricing to make certain products substitutable, assortment-based substitution occurs wen products wit similar attributes are substitutable wile symmetrical substitution occurs wen all of te unfulfilled demands of one items are completely fulfilled by te demands of te substitute items and asymmetrical substitution occur wen partial fraction of unfulfilled demands are added to te demands of te substitutable item. Based on te categories defined as above, tis paper lies in te category of asymmetrical stock-out-based substitution. In te recent years, little bit attention as been given in te researc for te stock-out-based substitution witin te EOQ setting under deterministic demand and to te best of our knowledge no one consider te concept of deterioration for te substitutable items wit deterministic demand and joint replenisment. Recently, Salame et al. (4) developed te joint replenisment policy for substitution by considering te deterministic demand, closely related to tis paper, but tey ave not considered te concept of deterioration wic is more realistic to determine te accurate optimal ordering quantity in te current era of competitive business strategies. Large numbers of te literature are available in inventory modelling of substitutable and deteriorating item separately. Tus, in subsequent paragrap, first, we discuss about recent and previous advancement in te inventory modelling of deteriorating items ten inventory modelling of substitutable items. Te journey of inventory modelling was started in second decade of nineteent century wen Harris (95) developed te first inventory model and tis model was generalized by Wilson (934) by deriving te formula to obtain te economic order quantity (EOQ). Te inventory of deteriorating items was first studied by Witin (957) in wic e considered te fasion goods as deteriorating items. Furter tere are several researcer wo gave different inventory model of deteriorating item under different realistic situations, te reader may refer te review paper on inventory of deteriorating items of Raafat (99), Goyal and Giri (3), Li et al. () and Bakker et al. () and Kanlarzade et al. (4) for detailed review of te literature of inventory of deteriorating items. Recently, Taleizade (4a, b) gave economic order quantity model for deteriorating and evaporating items wit consecutive and advanced payment, respectively. Te first inventory model of substitutable item was studied by McGillivray and Silver (978) by considering tat all of te substitutable items ave te same unit variable cost and sortage penalty. Parlar and Goyal (984) developed te similar model for optimal ordering decisions for stocastic demands. Pasternack and Drezner (99) numerically proved tat if te products are not substitutable ten te associated optimal order quantities can be larger or smaller. In tis sequence, Drezner et al. (995) developed an EOQ model wit substitution for two substitutable products and compare te results wit no substitution. Ernst and Kouvelis (999) suggested an efficient numerical searc algoritm for te optimal stocking levels for tree partially substitutable products. Gurnani and Drezner () extended te model of Drezner et al. (995) for multiple products. Misra and Ragunatan (4) gave new explanation for wy retailers migt be interested in vendor-managed inventory and sowed tat vendor-managed inventory intensifies te competition between two manufacturers of competing brands. Under joint replenisment policy (JRP), Porras and Dekker (8) provided a complete analysis and presented a new inventory model over JRP wen a correction is made for te empty replenisment and Hong and Kim (9) gave a genetic algoritm for JRP and devised an unbiased estimator to find out te exact cost. In continuation of tis, Sculz and Tela () teoretically sowed tat te JRP wit constant demands may ave no polynomial-time algoritm. Taleizade et al. (5) gave Joint optimization of price, replenisment frequency, replenisment cycle and production rate in vendor-managed inventory system wit deteriorating items. Recently, Krommyda et al. (5), Salame et al. (4), Rasouli and NakaiKamalabadi (4), and Gercack and Grosfeld (999) developed inventory model for two substitutable item wit deterministic demand, constant olding cost and fixed ordering cost but no one considered te effect of deterioration in inventory decision of substitutable items. Zao et al. (4) analyzed te pricing decision for two substitutable product wit price-dependent probabilistic demand wit fixed ordering cost and constant olding cost wile Ye (4), Huang and Ke (4), Li et al. (3), Li and You (), Hsie (), and Xue and Song (7) developed te inventory policy for multiple substitutable item wit stocastic demand, fixed ordering cost and constant olding cost. Tis paper makes te model of Krommyda et al. (5), Salame et al. (4), and Rasouli and NakaiKamalabadi (4) more realistic and applicable by taking into account te effect of deterioration and cost of substitution on inventory of te substitutable items. Demand is considered as a constant function for bot mutually substitutable items. If one of te items is out of stock ten its demand will be fulfilled by te second item and if any demand is not met by substitutable item it will be completely lost. Bot te products are order jointly and replenisment cycle is same for bot te items. Te rest of te paper is organized as follows: In te next section, we describe te assumptions and notations used in te entire article, Formulation and solution Section gives

3 J Ind Eng Int (7) 3: te detail of matematical formulation and solution procedure of te model, extensive numerical analysis and convexity sown grapically in Numerical and sensitivity analysis Section and article ends wit summary and conclusions of te article. Assumptions and notations For te matematical formulation of te inventory model, te following assumptions and notations are used. Assumptions (a) (b) (c) (d) Notations Te two items are ordered jointly in every ordering cycle. Te demand rates and deterioration rates are known and constant for bot items. Te procurement lead time is zero and replenisment rates for bot items are infinite. Wen an item is completely depleted and it subsequently becomes out of stock and tere is on-and inventory of te second item available, ten te second item wile supplying its own demands substitutes te demands of te first item during stock-out period. Tis substitution need not be te full substitution. It can be limited to a fraction (known as te substitution rate) of te total demand of te first item during stock-out period. Te remaining un-substituted demand of te first item is lost. Notation is grouped into parameters of te model, intermediate variables, and derived functions. Parameters D, D Demand rate of item and item Deterioration rate a, a Substitution rate of item by item Q ; Q Optimal ordering quantities of item and item, respectively A, A Fixed ordering cost per order of item and item i Rate of olding cost of item and item C, C Item cost per unit of item and item C S Unit substitution cost for item if substituted by item C S Unit substitution cost for item if substituted by item p, p Lost sale cost per unit of item and item, respectively Intermediate variables p Portion of time wen substitution occur z Inventory level of item wen oter item is out of stock t Time wen level of inventory of substituted item completely depleted t Time wen level of inventory of substitute item completely depleted in case of no substitution Derived functions I ðtþ Inventory level of item wen item depletes before item at time t, t t I ðtþ Inventory level of item wen item depletes before item at time t, t t I3 ðtþ Inventory level of item wen item depletes before item and substitution take place at time t, t t þ p I ðtþ Inventory level of item wen item depletes before item at time t, t t I ðtþ Inventory level of item wen item depletes before item at time t, t t I3 ðtþ Inventory level of item wen item depletes before item and substitution take place at time t, t t þ p TC (Q, Q ) Total cost wit substitution in per ordering cycle of item TC (Q, Q ) Total cost wit substitution in per ordering cycle of item TC (Q, Q ) Total cost wit substitution in per ordering cycle of bot item TC (Q, Q ) Total cost per unit time wit substitution in per ordering cycle of bot item wen item deplete before item TC (Q, Q ) Total cost per unit time wit substitution in per ordering cycle of bot item wen item deplete before item TC WS (Q, Q ) Total cost per unit time witout substitution Formulation and solution As stated before, we consider an inventory system for two mutually substitutable deteriorating items under te assumptions as mentioned in Assumptions and notations Section. At te beginning of te replenisment cycle, te retailer orders Q and Q units of item and item, respectively, wose consumption rates are D and D. Te inventory level of bot items gradually depletes due to deterioration and consumption rate. Tere are tree possibilities cases, Case Item depletes before item, i.e. if at time t te inventory of item is out of stock, as depicted in Fig., ten item partially substitutes te item wit substitution rate a and a portion of unmet demand for item is assumed to be lost wit te rate of ( - a ).

4 384 J Ind Eng Int (7) 3:38 39 Fig. Grapical representation of first scenario of inventory model wen t \ t Ordering Quantity Q ( ) Q ( ) 3 ( ) z t t p Sortage Time Case Item depletes before item, i.e. if at time t te inventory of item is out of stock, as depicted in Fig., ten item partially substitutes te item wit substitution rate a. A portion of unmet demand for item is assumed to be lost wit te rate of ( - a ). Case 3 Te inventory level of bot items becomes zero simultaneously and no substitution will take place, as depicted in Fig. 3. Case (Fig. ) Item is substituted by item (t \ t ). As depicted in Fig., te inventory level of bot item is governed by te following differential equations di ðþ t þ I ðþ¼ D t ; t t dt ðþ wit boundary condition I ðþ ¼Q and I ðtþ ¼: di ðþ t þ I ðþ¼ D t ; t t dt wit boundary condition I ðþ ¼Q and I ðt Þ¼I3 ðt Þ: ðþ di ðþ t 3 þ I ðþ¼ ðd t 3 þ a D Þ; t t t þ p dt wit boundary condition I3 ðt Þ¼I ðt Þ and I3 ðt þ pþ ¼: ð3þ I I Te solutions of Eqs. () (3) are ðþ¼ t D e ðt tþ ; t t ð4þ ð ðþ¼ t Q þ D Þ e t D ; t t ð5þ Fig. Grapical representation of second scenario of inventory model wen t [ t Ordering Quan ty Q ( ) Q ( ) 3 ( ) t t z p Sortage Time

5 J Ind Eng Int (7) 3: Q Q D D Time t t t Fig. 3 Grapical representation of tird scenario under joint replenisment I ðþ¼ t D þ a D e ðtþp tþ ; t 3 t t þ p ð6þ Te cost components per cycle consist of (a) costs related to item (b) costs related to item (c) lost sale costs and (d) substitution costs. (a) (b) (c) Te total cost associated wit item per ordering cycle consists of fixed ordering cost, purcase cost and olding cost, and can be expressed as TCðQ ; Q Þ¼ A þ C Q þ ic Q D ln Q þ D : D ð7þ Te total cost associated wit item per ordering cycle consists of fixed ordering cost, purcase cost and olding cost, and expressed as TCðQ ; Q Þ A þ C Q þ ic Q D ln Q þ D D ¼ B ic ðd a þ D Þ ln D ða Q þ D a þ Q þ D Þ A : ðd a þ D ÞðQ þ D Þ ð8þ Te Lost sale cost is incurred due to demand for te item, wic can be expressed as Lost sale cost ¼ p D ð a Þ ln D ða Q þ D a þ Q þ D Þ : ðd a þ D ÞðQ þ D Þ ð9þ (d) Te substitution cost is incurred according to te number of units of item substituted by item at te rate of C S per unit substituted and can be expressed as Substitution cost ¼ C SD a ln D ða Q þ D a þ Q þ D Þ : ðd a þ D ÞðQ þ D Þ ðþ Tus, te total cost per ordering cycle TC (Q, Q ), from Eqs. (7) () is given as TCðQ ; Q Þ¼ A þ A þ C Q þ C Q þ ic Q D ln Q 3 þ D D þ ic Q D ln Q þ D D ic ðd a þ D Þ ln D ða Q þ D a þ Q þ D Þ : ðd a þ D ÞðQ þ D Þ þ p D ð a Þ ln D ða Q þ D a þ Q þ D Þ ðd a þ D ÞðQ þ D Þ 6 þ C SD a ln D 7 4 ða Q þ D a þ Q þ D Þ 5 ðd a þ D ÞðQ þ D Þ ðþ Finally, for case (wen t \ t ), TC (Q, Q ), te average total cost per unit time (say a year) is obtained by multiplying te total cost per ordering cycle by te average. number of cycles per year is given as ln a Q þd a þq þd D a þd and

6 386 J Ind Eng Int (7) 3:38 39 TCðQ ; Q Þ¼ ln aqþdaþqþd D a þd A þ A þ C Q þ C Q þ ic Q D ln Q þ D D þ ic Q D ln Q þ D D ic ðd a þ D Þ ln D ða Q þ D a þ Q þ D Þ ðd a þ D ÞðQ þ D Þ þ p D ð a Þ ln D ða Q þ D a þ Q þ D Þ ðd a þ D ÞðQ þ D Þ B þ C SD a ln D ða Q þ D a þ Q þ D Þ A ðd a þ D ÞðQ þ D Þ ðþ Case (Fig. ) Item is substituted by item (t \ t ). Following an approac analogous to case, for case (wen t [ t ), TC (Q, Q ), te average total cost per unit time (say a year) is TCðQ ; Q Þ¼ ln aqþdaþqþd D a þd A þ A þ C Q þ C Q þ ic Q D ln Q þ D D þ ic Q D ln Q þ D D ic ðd a þ D Þ ln D ða Q þ D a þ Q þ D Þ ðd a þ D ÞðQ þ D Þ þ p D ð a Þ ln D ða Q þ D a þ Q þ D Þ ðd a þ D ÞðQ þ D Þ B þ C SD a ln D ða Q þ D a þ Q þ D Þ A ðd a þ D ÞðQ þ D Þ Case 3 (Fig. 3) No Substitution. ð3þ Figure 3 illustrates te inventory levels for te case of no substitution. Under a joint replenisment policy, te inventories of bot items deplete to zero simultaneously, i.e. Q /D = Q /D. Te average total cost per unit time for an inventory system witout substitution under joint replenisment, TC WS (Q, Q ), consists only of setup costs, purcase costs and olding costs and is given as TC WS ðq ; Q Þ¼ ln QþD D A þ A þ C Q þ C Q þ ic Q D ln Q þ D D B þ ic Q D ln Q þ D A : D ð4þ Solution procedure To determine te optimal ordering quantities from total cost function. In te next section we will grapically sow tat te total cost function is a strictly convex function and using tis property of total cost function, next we provide an algoritm to obtain te optimal ordering quantities. Algoritm to obtain te optimal ordering quantities. Step Initialize all te parameters of te model. Step Solve te constraint optimization problem P : ðq ; Q Þ tat min TCðQ ; Q Þ subject to Q Q Q ;Q D D and P : ðq ; Q Þ tat min TCðQ ; Q Þ subject to Q Q Q ;Q D D Step 3 Obtain te optimum solution by ðq ; Q Þ = Min ðp ; P Þ. Step 4 Stop. Numerical and sensitivity analysis In tis section, we provide a numerical example to illustrate te proposed model using Maple matematical modelling package. In Numerical analysis, we use te parameter as defined in Table unless oterwise mentioned. According te algoritm as defined above, we solve te constraint optimization problem as defined in step- by maple matematical software. Te outputs of step- are P : Q = 6.8, Q = 9.34, TCðQ ; Q Þ =.79 and P: Q = 99.45, Q = 9.34, TCðQ ; Q Þ = Wen comparing te total cost of bot constraint optimization problems in step-3 of te algoritm ten first optimization problems (P ) lead to te optimal solution. Tus, te optimal ordering quantities in tis case is ðq ; Q Þ = (6.8, 9.34) and te optimal total cost is.79. Te optimal total cost of inventory model wit no substitution under same assumptions using Eq. (4) wit traditional calculus metod te outputs are Q = 78.7, Q = 44.67, TCðQ ; Q Þ = Comparing te output (optimal total cost) of bot inventory model ten difference in optimal total cost and percentage improvement are 96.9 and 4.59, respectively. To examine te nature of average total cost function, we plot te total cost function wit varying order quantity of item and item. Te results are sown in Figs. 4, 5 and 6.

7 J Ind Eng Int (7) 3: As Figs. 4, 5 and 6 sow tat te average total cost function is strictly convex function. Tus, average total cost function (Eq. ) always leads to te unique optimal solution. Next, we carry out a sensitivity analysis of te optimal total cost and optimal ordering quantities according to given values of different parameter of te model. Te percentage improvements in te optimal total cost according to te values of various parameters are also presented. Te numerical results are given in Table. Now, we investigate te decrease in total cost wen substitution is possible and compared to te case witout substitution wit respect to various parameters of te system. Te results are sown in Figs. 7, 8 and 9. Wile te extent and relative rate may vary, general nature of te percentage improvement in TC over TC WS follows te intuitive reasoning. Te findings are presented in Table 3. As we know tat te main objective of an organization, dealing te inventory control, is to provide te rigt products to te rigt place, at te optimal price and time, and in te good quality wit optimal quantity. Wit te elp of tis inventory model, wic provides a detailed analysis of substitutable deteriorating items wit numerical examples, manager can increase te firm s capability and teir performance to deal te inventory of teir organizations. Summary and conclusions In tis paper, we presented an inventory model for two substitutable deteriorating items under joint replenisment in eac replenisment cycle. If at any moment of time, te inventory level of one item is out of stock ten second item will partially substitute te first item and vice versa. A Table Initial parameters used for numerical analysis Parameter Item- Item- Consumption rate (D, D ) 5 Deterioration rates ().. Substitution rates (a, a )..4 Setup costs (A, A ) 3 3 Rate of olding cost (i) Cost per unit (C, C ) 3 3 Lost sale costs (p, p ) 6 4 Substitution costs (C S, C S ) Fig. 5 Total cost (TC) versus Q at fixed Q Fig. 4 Total cost (TC) versus Q at fixed Q Fig. 6 Total cost (TC) versus ordering quantity of item and item

8 388 J Ind Eng Int (7) 3:38 39 Table Sensitivity analysis optimal total cost and optimal ordering quantities Parameter Value of parameter Optimal total cost and optimal ordering quantity wit substitution Optimal total cost and optimal ordering quantity witout substitution % Improvement in optimal total cost Q Q TC Q WO Q WO TC WS C /C / / / / / i p A = A C S a portion of unmet demand for bot items is assumed to be lost and eac substituted item incurs a cost of substitution. For te situation as mentioned above, we matematically formulated an inventory model and developed a solution procedure to obtain te optimal ordering quantities. Te numerical analysis sowed tat as rate of olding cost, setup cost and deterioration rate increases, te percentage improvement in optimal total cost wit substitution over witout substitution also increases and if we increase te sortage cost or cost of substitution, te percentage improvement decreases in total optimal cost wit substitution over witout substitution wile percentage improvement becomes constant wen substitution rate of item becomes equal or more from substitution rate of item.

9 J Ind Eng Int (7) 3: TC TCWS % improvement TC TCWS % Improvement Rate of Holding Cost (i) Percentage Cange Deteriora on Rate (θ) Percentage Cange Fig. 7 % improvement in TC over TCWS wit ratios of item cost (C /C ) and rate of olding Cost (i) 5 TC TCWS % Improvement 5. 3 TC TCWS % Improvement Percentage Cange Percentage Cange Sortage Cost (π ) Setup Cost (A =A ). Fig. 8 % improvement in TC over TC WS wit variation in sortage cost (p ) and setup cost (A = A ) TC TCWS % Improvement Cost of Subs tu on (C S ) Percentage Cange TC TCWS % Improvement Subs tu on Rate (α ) Percentage Cange Fig. 9 % improvement in TC over TC WS wit cost of substitution (C S ) and substitution rate (a )

10 39 J Ind Eng Int (7) 3:38 39 Table 3 Improvement in total optimal cost wit substitution over total cost witout substitution Parameters Variation Optimal total cost witout substitution TC WS Optimal total cost TC Percentage improvement C /C Increases Increases Increases Increases I Increases Increases Increases Increases p Increases Constant Increases Decreases A Increases Increases Increases Increases C S Increases Constant Increases Decreases a Increases Constant Increases Decreases Increases Increases Increases Increases In addition, as tis paper only considered two deteriorated items, joint replenisment and same replenisment cycle, wile future researces can focus on more tan two items, different replenisment cycle for different items, multiple supplier and retailers, trade credit mecanism, supplier retailer cooperation, etc. Acknowledgement I would like to tanks Dr. Kripa Sanker, Professor, Department of Industrial and Management Engineering (IME), Indian Institute of Tecnology, Kanpur, Uttar Prades for teir valuable suggestion and support and KIT-IITK for giving me an opportunity under visiting researcer program to carried out tis researc. Open Access Tis article is distributed under te terms of te Creative Commons Attribution 4. International License (ttp://crea tivecommons.org/licenses/by/4./), wic permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to te original autor(s) and te source, provide a link to te Creative Commons license, and indicate if canges were made. References Anupindi R, Dada M, Gupta S (998) Estimation of consumer demand wit stock out based substitution: an application to vending macine products. Market Sci 7:46 43 Bakker M, Riezebos J, Teunter RH () Review of inventory systems wit deterioration since. Eur J Oper Res :75 84 Drezner Z, Gurnani H, Pasternack BA (995) EOQ model wit substitution between products. J Oper Res Soc 46: Ernst R, Kouvelis P (999) Te effects of selling packaged goods on inventory decisions. Manag Sci 45(8):4 55 Gercak Y, Grosfeld-Nir A (999) Lot-sizing for substitutable, production-to-order parts wit random functionality yields. Int J Flex Manuf Syst : Goyal SK, Giri BC (3) Recent trends in modeling of deteriorating inventory. Eur J Oper Res 34(): 6 Gurnani H, Drezner Z () Deterministic ierarcical substitution inventory models. J Oper Res Soc 5:9 33 Harris FW (95) Operations and Costs. A. W. Saw Company, Cicago, pp Hong S, Kim Y (9) A genetic algoritm for joint replenisment based on te exact inventory cost. Comput Oper Res 36():67 75 Hsie Y-J () Demand switcing criteria for multiple products: an inventory cost analysis. Omega 39:3 37 Huang H, Ke H (4) Pricing decision problem for substitutable products based on uncertainty teory. J Intell Manuf. doi:.7/s Kanlarzade N, Yegane BY, Kamalabadi IN, Farugi H (4) Inventory control wit deteriorating items: a state-of-te-art literature review. Int J Ind Eng Comput 5:79 98 Kim S-W, Bell PC () Optimal pricing and production decisions in te presence of symmetrical and asymmetrical substitution. Omega 39(5): Krommyda IP, Skouri K, Konstantaras I (5) Optimal ordering quantities for substitutable products wit stock-dependent demand. Appl Mat Model 39():47 64 Li H, You T () Capacity commitment and pricing for substitutable products under competition. J Syst Sci Syst Eng (4): Li R, Hongjie L, Mawinney JR () A review on deteriorating inventory study. J Serv Sci Manag 3:7 9 Li X, Nukala S, Moebbi S (3) Game teory metodology for optimizing retailers pricing and self-space allocation decisions on competing substitutable products. Int J Adv Manuf Tecnol 68: McGillivray AR, Silver EA (978) Some concepts for inventory control under substitutable demand. Inf Syst Oper Res 6:47 63 Misra BK, Ragunatan S (4) Retailer- vs. vendor-managed inventory and brand competition. Manag Sci 5(4): Parlar M, Goyal S (984) Optimal ordering decisions for two substitutable products wit stocastic demands. Opsearc : 5 Pasternack B, Drezner Z (99) Optimal inventory policies for substitutable commodities wit stocastic demand. Nav Res Logist 38: 4 Porras E, Dekker R (8) Generalized solutions for te joint replenisment problem wit correction factor. Int J Prod Econ 3(): Raafat F (99) Survey of literature on continuously deteriorating inventory model. J Oper Res Soc 4():7 37 Rasouli N, NakaiKamalabadi I (4) Joint pricing and inventory control for seasonal and substitutable goods mentioning te symmetrical and asymmetrical substitution. Int J Eng (IJE) Trans C Asp 7(9): Salame MK, Yassine AA, Madda B, Gaddar L (4) Joint replenisment model wit substitution. Appl Mat Model 38(4): Sculz A, Tela C () Approximation algoritms and ardness results for te joint replenisment problem wit constant demands. Lecture notes computer science, vol 694, pp

11 J Ind Eng Int (7) 3: Taleizade AA (4a) An economic order quantity model wit consecutive payments for deteriorating items. Appl Mat Model 38: Taleizade AA (4b) An economic order quantity model wit partial backordering and advance payments for an evaporating item. Int J Prod Econ 55:85 93 Taleizade AA, Noori-daryan M, Cárdenas-Barrón LE (5) Joint optimization of price, replenisment frequency, replenisment cycle and production rate in vendor managed inventory system wit deteriorating items. Int J Prod Econ 59:85 95 Tang CS, Yin R (7) Joint ordering and pricing strategies for managing substitutable products. Prod Oper Manag 6:38 53 Witin TM (957) Te teory of inventory management, nd edn. Princeton University Press, Princeton Wilson RH (934) A scientific routine for stock control. Harv Bus Rev 3:6 8 Xue Z, Song J (7) Demand management and inventory control for substitutable products. Working paper, Te Fuqua Scool of Business, Duke University, Duram Ye T (4) Inventory management wit simultaneously orizontal and vertical substitution. Int J Prod Econ 56:6 34 Zao J, Wei J, Li Y (4) Pricing decisions for substitutable products in a two-ecelon supply cain wit firms different cannel powers. Int J Prod Econ 53:43 5