A three-layer supply chain integrated production-inventory model under permissible delay in payments in uncertain environments

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1 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 RESEARCH Open Acce A three-layer upply chain integrated production-inventory model under permiible delay in payment in uncertain environment Dipak Kumar Jana 1*, Kalipada Maity and apan Kumar Roy 3 *Correpondence: dipakjana@gmail.com 1 Department of Engineering Science, Haldia Intitute of echnology, Haldia Purba Midna Pur 71657, Wet Bengal, India Full lit of author information i available at the end of the article Abtract In thi paper, an integrated production-inventory model i preented for a upplier, manufacturer, and retailer upply chain under conditionally permiible delay in payment in uncertain environment. he upplier produce the item at a certain rate, which i a deciion variable, and purchae the item to the manufacturer. he manufacturer ha alo purchaed and produced the item in a finite rate. he manufacturer ell the product to the retailer and alo give the delay in payment to the retailer. he retailer purchae the item from the manufacture to ell it to the cutomer. Ideal cot of upplier, manufacturer, and retailer have been taken into account. An integrated model ha been developed and olved analytically in crip and uncertain environment, and finally, correponding individual profit are calculated numerically and graphically. Keyword: Production inventory ytem; Uncertain variable; hree-layer upplier chain; Delay in payment Introduction Supply chain management ha taken a very important and critical role for any company in the increaing globalization and competition in the market. A upply chain model SCM i a network of upplier, producer, ditributor, and cutomer which ynchronize a erie of interrelated buine proce in order to have 1 optimal procurement of raw material from nature, tranportation of raw material into a warehoue, 3 production of good in the production center, and 4 ditribution of thee finihed good to retailer for ale to the cutomer. With a recent paradigm hift to the upply chain SC, the ultimate ucce of a firm may depend on it ability to link upply chain member eamlely. One of the earliet effort to create an integrated SCM ha been developed by Oliver and Webber 1, Cohen and Baghanan, and Cachon and Zipkin 3. hey developed a production, ditribution, and inventory PDI planning ytem that integrated three upply chain egment compriing upply, torage/location, and cutomer demand planning. he core of the PDI ytem wa a network model and diagram that increaed the deciion 13 Jana et al.; licenee Springer. hi i an Open Acce article ditributed under the term of the Creative Common Attribution Licene which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited.

2 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page of 17 maker inight into upply chain connectivity. he model however wa confined to a ingle-period and ingle-objective problem. Viwanathan and Piplani 4 were concerned an integrated inventory model through common replenihment in the SC. Khouja 5 wa the firt to conider a three-tage upply chain with one or more firm at each tage. Agarwal et al. 6 have developed a dynamic balancing of inventory model in upply chain management. Rau et al. 7 developed an integrated SCM of a deteriorating item with hortage. Lee 8 added a new dimenion to the ingle vendor-ingle buyer problem by etting the number of raw material hipment received by the vendor per cycle to be a deciion variable. Ben-Daya et al. 9 have developed an integrated productioninventory model with raw material replenihment conideration in a three-layer upply chain. Sana 1 ha integrated a production-inventory model of imperfect quality product in a three-layer upply chain. Recently, Pal et al. 11 have developed a three-layer upply chain model with production-inventory model for reworkable item. All of the abovementioned SCM are conidered with contant, known demand and production rate in a crip environment. Different type of uncertainty uch a fuzzine, randomne, and roughne are common factor in SCM. In many cae, it i found that ome inventory parameter involve fuzzy uncertainty. For example, inventory-related cot uch a holding cot and etup cot, demand, and elling price depend on everal factor uch a bank interet, tock amount, and market ituation which are uncertain in a fuzzy ene. o be more pecific, inventory holding cot i ometime repreented by a fuzzy number, and it depend on the torage amount which may be imprecie and range within an interval due to everal factor uch a carcity of torage pace, market fluctuation, human etimation, and/or thought proce. he following paper have been developed in thee environment. Wang and Shu 1 developed a fuzzy deciion methodology that provide an alternative framework to handle SC uncertaintie and to determine SC inventory trategie, while there i a lack of certainty in data or even a lack of available hitorical data. Fuzzy et theory i ued to model SC uncertainty. A fuzzy SC model baed on poibility theory i developed to evaluate SC performance. Baed on the propoed fuzzy SC model, a genetic algorithm approach i developed to determine the order-up-to level of tock-keeping unit in the SC to minimize SC inventory cot ubject to the retriction of fulfilling the target fill rate of the finihed product. he propoed model allow deciion maker to expre their rik attitude and to analyze the trade-off between cutomer ervice level and inventory invetment in the SC, o that better SC inventory trategie can be made. Da et al. 13 have preented a joint performance of an SC with two warehoue facilitie in a fuzzy environment. A realitic two-warehoue and multi-collection-productioninventory model with contant/tock-dependent demand, defective production ytem, and fuzzy budget contraint ha been formulated and olved in an SC context. Later Chen et al. 14 developed a multi-criteria fuzzy optimization for locating warehoue and ditribution center in a upply chain network. Peidro et al. 15 developed a fuzzy linear programming model for tactical upply chain planning in a multi-echelon, multi-product, multi-level, multi-period upply chain network in a fuzzy environment. In thi approach, the demand, proce, and upply uncertaintie are jointly conidered. he aim i to centralize multi-node deciion

3 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 3 of 17 imultaneouly to achieve the bet ue of the available reource along the time horizon o that cutomer demand are met at a minimum cot. hi propoal i teted uing data from a real automobile SC. he fuzzy model provide the deciion maker with alternative deciion plan with different degree of atifaction. Chu 16 developed the upply chain flexibility that ha become increaingly important. hi tudy thu build a group deciion-making tructure model of flexibility in upply chain management development. Recently, Jana et al. 17 have developed a fuzzy imulation via contractive mapping genetic algorithm approach to an imprecie production-inventory model under volume flexibility. hi tudy preent a framework to evaluate the upply chain flexibility that comprie two part: 1 an evaluation hierarchy with flexibility dimenion and related metric and an evaluation cheme that ue a three-tage proce to evaluate the upply chain flexibility. hi tudy then propoe an algorithm to determine the degree of upply chain flexibility uing a fuzzy linguitic approach. Evaluation of the degree of upply chain flexibility can identify the need to improve upply chain flexibility and identify pecific dimenion of upply chain flexibility a the bet direction for improvement. he reult of thi tudy are more objective and unbiaed for two reaon. Firt, the reult are generated by group deciion-making with interactive conenu analyi. Second, the fuzzy linguitic approach ued in thi tudy ha more advantage to preerve no lo of information than other method. Additionally, thi tudy preent an example uing a cae tudy to illutrate the availability of the propoed method and compare it with other method. Kritianto et al. 18 developed an adaptive fuzzy control application to upport a vendor-managed inventory VMI. hi paper alo guide the management in allocating inventory by coordinating with upplier and buyer to enure minimum inventory level acro a upply chain. Adaptive fuzzy VMI control i the main contribution of thi paper. However, the uncertainty theory wa developed by Liu 19, and it can be ued to handle ubjective imprecie quantity. Much reearch work ha been done on the development of the uncertainty theory and related theoretical work. You proved ome convergence theorem of uncertain equence. Liu 1 ha defined uncertain proce and Liu ha dicued uncertain theory. In thi paper, we developed for the firt time a three-layer upply chain model under delay in payment in an uncertain environment. In the traditional economic order quantity EOQ model, it often aumed that the retailer mut pay off a oon a the item are received. In fact, the upplier offer the retailer a delay period, known a trade credit period, in paying for the purchaing cot, which i a very common buine practice. Supplier often offer trade credit a a marketing trategy to increae ale and reduce on-hand tock level. Once a trade credit ha been offered, the amount of the tied up retailer capital in tock i reduced, and that lead to a reduction in the retailer holding cot of finance. In addition, during the trade credit period, the retailer can accumulate revenue by elling item and by earning interet. A a matter of fact, retailer, epecially of mall buinee which tend to have a limited number of financing opportunitie, rely on trade credit a a ource of hort-term fund. In thi reearch field, Goyal 3 wa the firt to etablih an EOQ model with a contant demand rate under the condition of permiible delay in payment. Khanra, Ghoh, and Chaudhuri 4 have developed an EOQ model for a deteriorating item with time-dependent quadratic demand under permiible delay in payment. Alo,

4 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 4 of 17 Maihami and Abadi 5 have etablihed joint control of inventory and it pricing for non-intantaneouly deteriorating item under permiible delay in payment and partial backlogging. he propoed model conider a three-layer upply chain involving the upplier, manufacturer, and retailer who are reponible in tranforming the raw material into finihed product and making them available to atify the cutomer demand time. Inventory and production deciion are made at the upplier, manufacturer, and retailer level in uncertain environment. he problem i to coordinate production and inventory deciion acro the upply chain o that the total profit of the chain i maximized. Neceary knowledge about uncertain variable o better decribe ubjective imprecie quantity, Liu in 19 propoed an uncertain meaure and further developed an uncertainty theory which i an axiomatic ytem of normality, monotonicity, elf-duality, countable ubadditivity and product meaure. Definition 1. Let Ɣ be a non-empty et and L be a σ algebra over Ɣ. Eachelement L i called an event. A et function M{ } i called an uncertain meaure if it atifie the following four axiom of Liu 19: Axiom 1. Normality M{ } =1 Axiom. Monotonicity M{ }+M{ C }=1, for any event Axiom 3. Countable ubadditivity For every countable equence of event 1,,..., we have { } } M i M { i. i=1 i=1 Definition. Liu 19 he uncertainty ditribution : R, 1 of an uncertain variable ξ i defined by t = M{ξ t}. Definition 3. Liu 19 Let ξ be an uncertain variable. hen the expected value of ξ i defined by Eξ = M{ξ r}dr M{ξ r}dr, provided that at leat one of the two integral i finite. heorem 1. Liu 6 Let ξ be an uncertain variable with uncertainty ditribution. If the expected value exit, then Eξ = 1 1 αdα. Lemma 1. Let ξ La, b, c be a zigzag uncertain variable. hen it invere uncertainty ditribution 1 α = 1 1 αa + b + αc, and it can be expreed a 1 1 a + b + c Eξ = 1 αa + b + αc dα =. 1 4

5 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 5 of 17 heorem. Liu 6 Let ξ 1, ξ,..., ξ n be independent uncertain variable with uncertainty ditribution φ 1, φ,..., φ n, repectively. If f i a trictly increaing function, then ξ = f ξ 1, ξ,..., ξ n i an uncertain variable with invere uncertainty ditribution 1 α = f 1 1 α, 1 α,..., 1 n α. heorem 3. Liu 6 Let ξ 1 and ξ be independent uncertain variable with finite expected value. hen for any real number a 1 and a, we have Ea 1 ξ + a η = a 1 Eξ +a Eη. Aumption and notation Aumption he following aumption are conidered to develop the model: i ii iii iv v vi vii Model are developed for ingle item product. Lead time i negligible. Joint effect of upplier, manufacturer, retailer i conider in a upply chain management. Supplier produced the item with contant rate unit per unit time, which i a deciion variable. otal production rate of manufacturer i equal to the demand rate of manufacturer. he manufacturer give the opportunity to the retailer conditionally permiible delay in payment. Idle cot of upplier, manufacturer and retailer are alo aumed. Notation he following notation are conidered to develop the model: p = production rate for the upplier, which i a deciion variable. = demand rate or production rate for the manufacturer. D r = contant demand rate for the retailer. = contant demand rate of cutomer. C = purchae cot of unit item for upplier. C m = elling price of unit item for upplier which i alo the purchae cot for manufacturer. C r = elling price of unit item for manufacturer which i alo the purchae cot for retailer. C r1 = elling price for retailer. t = production time for uppler. = cycle length for the upplier. r = time duration where order i upplied by the manufacturer, by retailer cycle length. = lat cycle length of the retailer. = total time for the integrated model. h = holding cot per unit per unit time for upplier. h m = holding cot per unit per unit time for manufacturer. h r = holding cot per unit per unit time for retailer.

6 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 6 of 17 A = ordering cot for upplier. A m = ordering cot for manufacturer. h r = ordering cot for retailer. id, ĩd = idle cot per unit time for upplier in crip and uncertain environment, repectively. id m, ĩd m = idle cot per unit time for manufacturer in crip and uncertain environment, repectively. id r, ĩd r = idle cot per unit time for retailer in crip and uncertain environment, repectively. n =numberofcycleforretailer. r = number of cycle where manufacturer top production. M = retailer trade credit period offered by the manufacturer to the retailer in year, which i the fraction of the year. I p = interet payable to the manufacturer by the retailer. I e, Ĩ e = interet earned by the retailer in crip and uncertain environment, repectively. AP = average total profit for the integrated model. Pm =optimumvalueofp m for integrated model. AP = optimum value of average total profit for the integrated model. Model decription and diagrammatic repreentation he integrated inventory model Figure 1 tart when t = andtockizero.atthat time, the upplier tart their production with the rate p unit per unit time and purchae at the rate unit per unit time to the manufacturer. When t = t, upplier top their production, and at t =, the inventory level of upplier become zero. he total time of the integrated model i, o the idle time for upplier i. Similarly, the Figure 1 Inventory level for the integrated model.

7 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 7 of 17 manufacturer tart their production at the ame time t = with the production rate unit per unit time and purchae thi production D r unit to the retailer in the time gap R, which i the bulk pattern. At time t = = R, manufacturer top their production, and at t = n + 1 R n = D r, the tock of manufacturer i zero. hu, idle period for the manufacturer i n + 1 r. Retailer tart elling thi product to the cutomer at time t = r and end elling at = n+1 r + nd r. he idle period for retailer i r. Mathematical formulation of the model Formulation of upplier individual average profit Differential equation for the upplier in Figure in, igivenby dq p, t t =, t t, dt, t with boundary condition q t = andt =,. Solving the differential equation with the boundary condition, we have P P m t, t t q t = P m t, t < t 3, < t H = Holding cot of upplier. t = h p tdt + = h p t p t. t tdt he total idle cot = id R+P t 1Dc 1,purchaecot= c, elling price = c m, and ordering cot = A. APS = Average profit for upplier. = 1 revenue from ale-purchae cot-holding cot-idle cot-ordering cot. = 1 p c m c p t h t 1 p t id R + p t 1 A. 4 Figure Inventory level of upplier.

8 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 8 of 17 Formulation of manufacturer individual average profit Inventory level of manufacturer in Figure 3 in,igivenby t, t R t id R, i R < t i + 1 R i = 1,,..., r 1 t rd R, r R < t q m t = rd R, < t r + 1 R id R, i R < t i + 1 R i = r + 1, r +,..., n 1 nd R, n R < t n + 1 R, n + 1 R t 5 with boundary condition q m = andq m i R + = q m i R D R. H m = Holding cot for manufacturer. R r 1 i+1r = h m tdt + t id R dt + t rd R dt 1 i R r R r+1r n 1 i+1r + rd R dt + id R dt + r+1 i R n R = h m n R n + n r R D R p t. n+1r nd R dt he total idle cot = id m pm m nd R,purchaecot= c m, elling price = c r, and ordering cot = A m. Cae 1. When M R I em = I pr = Amount of interet earned by the manufacturer in,fromretailer. = Amount of interet paid by the retailer to the manufacturer in,. R = c r I p n D R tdt + nd R tdt M = nc ri p R D R + M MD R + cr I p + nd R M + M M nd R Figure 3 Inventory level of manufacturer.

9 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 9 of 17 APM 1 = Average profit of manufacturer. = 1 revenue from ale-purchae cot-holding cot-idle cot + earned interet-ordering cot. = 1 c r c m h m n R n + n r R D R p t pm m nd R id m + nc ri p R D R + M MD R pm nd R + c r I p + nd R M + M A m 6 Cae. When M R I em = I pr = Amount of interet earned by the manufacturer in, fromretailer. = Amount of interet paid by the retailer to the manufacturer in,. R = c r I p n D R tdt + nd R tdt M = nc ri p R D R + M MD R + cr I p + nd R M + M APR = Average profit of retailer. = 1 revenue from ale-purchae cot-holding cot M nd R + earned interet-payable interet-idle cot-ordering cot. = 1 c r1 c r h r p m n R n + 1 R D R + nc r 1 i e M + c r 1 i e nd R M nc ri p R D R + M MD R idr R A r Formulation of retailer individual average profit Inventory level of retailer in Figure 4 in, igivenby { t, i R t i + 1 R q r t = nd r t, n + 1 R t 7 8 with boundary condition q r n + 1 R = andq r =. H r = Holding cot of retailer. R = nh r D R tdt + = h r p m n R n + 1 R D R nd R tdt.

10 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 1 of 17 Figure 4 Inventory level of retailer. he total idle cot = id r R,purchaecot= c r, elling price = c r1,and ordering cot = A r. Cae 1. When M R Interet earned by the retailer for n + 1 cycle i given by I er = Amount of interet earned by the retailer from the bank in n+1 cycle. M = n + 1c r1 i e M t dt = n + 1c r 1 i e M, I pr = Amount of interet paid by the retailer to the manufacturer in,. R = c r I p n D R tdt + nd R tdt M R D R + M MD R pm nd R = nc r I p + c r I p + nd R M + M, M APR 1 = Average profit for retailer. = 1 revenue from ale-purchae cot-holding cot + earned interet-payable interet-idle cot-ordering cot. = 1 c r1 c r h r p m np D m R n + 1 R D R c + n + 1c r 1 i e M nc ri p R D R + M MD R idr R A r + nd R M + M pm nd R c r I p 9 Cae. When M R Interet earned by the retailer for n + 1 cycle i given by

11 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 11 of 17 I re = Retailer earned interet. = c r1 i e n M = nc r 1 i e M M t dt + + c r 1 i e nd R M t dt + M nd R Interet payable by the retailer for the firt n cycle i given by I rp = Retailer payable interet. R = c r I p n D R tdt = nc ri p M R D R + M MD R APR = Average profit for retailer. 1 = 1 revenue from ale-purchae cot-holding cot + earned interet-payable interet-idle cot-ordering cot = 1 c r1 c r h r p m np D m R n + 1 R D R c + nc r 1 i e M + c r 1 i e nd R M nc ri p R D R + M MD R idr R A r 11 Crip environment Cae 1. M R AP 1 = otal average profit for integrated model. = APS + APM 1 + APR 1 = D c p t + D R c m c h + c r c m h m = pm m nd R id m A m + c r1 c r h r p m + n + 1c r 1 I e M id r R A r Ap + D t Bp t + E, R where P m = P t, A = h m h p t n R n + n r id R + p t 1 1 R D R p t n R n + 1 R D R A 1 h r 13 B = c r1 c + nh r h m R id m + id 14

12 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 1 of 17 and E = n + n r n + 1 h m + h r R D R + n + 1c r 1 I e M + nid m id id r R + id A + A m + A r 15 d dp AP 1 = d dp p = D R B ± t A 4t AE BD R t A AP 1 <, if Ap + B < 4p t + Bt + D R herefore, AP 1 i concave if Ap + B < 4p t + Bt + D R. 18 Cae. M R AP = otal average profit for integrated model. = APS + APM 1 + APR 1 p t = c m c h p t + D R + c r c m h m n R n + n r pm m nd R id m = A m + c r1 c r h r p m id R + p t 1 1 R D R p t n R n + 1 R D R A + nc r 1 I e M + c r 1 I e nd R M id r R A r Ap + D t Bp t + F R 19 where A and B are given in 16 and 17, repectively and n + n r n + 1 F = h m + h r R D R + n + 1c r 1 I e M + nid m id id r R + id A + A m + A r d dp AP = p = D R B ± t A 4t AF BD R t A 1 d dp AP 1 <, if Ap + B < 4p t + Bt + D R. herefore, AP 1 i concave if

13 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 13 of 17 able 1 Input data of different parameter for Cae 1 and Cae Parameter c c m c r c r1 h h m n r ρ id id m id r I e A A m A r m i p D C D R Cae Cae Ap + B < 4p t + Bt + D R. 3 Propoed inventory model in uncertain environment Let u conider ĩd, ĩd r, ĩd m, and Ĩ re a zigzag uncertain variable where id = Lid 1,id,id 3, id r = Lid r1,id m,id r3, id m = Lid m1,id m,id m3, and I re = LI re1, I re, I re3. hen, the objective i reduce to the following: ForCae1M R c m c h p t 1 ĩd R + p t 1 A P 1 = p t + D R A + c r c m h m n R n + n r R D R p t A m + c r1 c r h r p m np D m R n + 1 R D R c ĩd pm m nd R m + n + 1c r 1 I re M ĩd r R A r. ForCae M R p t A P = c m c h p t + D R p ĩd 1 R + p t 1 m + c r c m h m n R n + n r R D R p t ĩd pm m nd R m A m + c r1 c r h r p m np D m R n + 1 R D R c { ncr1 M + + c } r 1 nd R M I re ĩd r R A r. 4 A 5 able Optimum reult for objective function and other parameter Parameter Cae 1 Cae AP 1, , p APS APM APR

14 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 14 of 17 able 3 Optimum value change due to parametric change Parameter Parametric P AP name value Cae 1 Cae Cae 1 Cae Cae 1 Cae c , , , 5.47 c m , , , 5.5 c r , , , 5.43 h , , , 5.43 h m , , , h r , , , he equivalent criodel Uing Lemma 1 and applying heorem, the expected total average profit i given by the following: ForCae1M R p t EA P 1 = c m c h p t + D R 1 Eĩd R + p t 1 A + c r c m h m n R n + n r R D R p t A m + c r1 c r h r p m np D m R n + 1 R D R c Eĩd pm m nd R m + n + 1c r 1 E I re M Eĩd r R A r. 6 ForCae M R able 4 Input data of different zigzag parameter for Cae 1 and Cae Parameter in uncertain environment id Cae 1 L.8,1.,1.4 L1.5,.,.5 L1.4,,.3 L.4,.6,.8 Cae L1.4,1.8,.4 L,.3,.9 L1.4,.1,.5 L.6,.8,.1 id m id r Ĩ e

15 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 15 of 17 able 5 Optimal value of objective and deciion variable Parameter Cae 1 Cae AP 1, , 5.43 p APS APM APR p t EA P = c m c h p t + D R + c r c m h m n R n + n r Eĩd pm m nd R m A m + c r1 c r h r p m np D m R n + 1 R D R c { ncr1 M + Eĩd 1 R + p t R D R p t 1 + c } r 1 nd R M E I re Eĩd r R A r. A Numerical example Crip environment he input data of different parameter for Cae 1 and Cae are hown in able 1, and the expected optimum value of the total profit i given in able. 7 Senitivity analyi he major contribution of the upply chain i mainly in the incluion of the manufacturer. We conider product reworking of defective item which are reworked jut after regular production, with a different holding cot for good and defective item in the three-layer upply chain. An integrated production-inventory model i Figure 5 Average profit veru upply rate of upplier in Cae 1 and Cae.

16 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 16 of 17 preented for the upplier, manufacturer, and retailer upply chain under conditionally permiible delay in payment which ha developed in both crip and uncertain environment able 3. Uncertain environment Input data of different crip parameter for Cae 1 and Cae are given in able 1; the remaining uncertain parameter are depicted in able 4. Optimal value of objective and deciion variable are given in able 5 and graphically repreented in Figure 5. Achievement and concluion of the model In thi model, we developed a three-layer production-inventory upply chain model in an uncertain environment. Here, the upplier are alo the manufacturer; they collect the raw material ore and produce the raw material of the actual manufacturer. For example, in the petroleum indutry, upplier collect the ore and produce the naphthalene, which i the raw material of the manufacturer. hen manufacturer produce the uable product to ell to the retailer. In thi paper, we have developed a production-inventory upply chain model under an uncertain environment. he paper can be extended to imperfect production-inventory ytem. Deterioration can be allowed for produced item of the retailer and manufacturer. In the cae of the retailer, it might be intereting to conider the effect that only a percent of imperfect quality product could be reworked by manufacturer and that other crap item mut be eliminated immediately. In order to how the uncertaintie, the preent model could be extended, applying tochatic demand and production rate in each member of the upply chain. hee are ome topic of ongoing and future reearch, among other. Author detail 1 Department of Engineering Science, Haldia Intitute of echnology, Haldia Purba Midna Pur 71657, Wet Bengal, India. Department of Mathematic, Mugberia Gangadhar Mahavidyalaya, Mugberia Purba Medinipur 7145, Wet Bengal, India. 3 Department of Mathematic, Bengal Engineering and Science Univerity, Shibpur Howrah 71113, Wet Bengal, India. Received: 11 April 13 Accepted: 6 July 13 Publihed: 13 Augut 13 Reference 1. Oliver, RK, Webber, MD: Supply-chain management: logitic catche up with trategy reprint from Outlook 198. Cohen, MA, Baghanan, MP: he tabilizing effect of inventory in upply chain. Oper. Re. 46, Cachon, GP, Zipkin, PH: Competitive and cooperative inventory policie in a two-tage upply chain. Manag. Sci. 45, Viwanathan, S, Piplani, R: Coordinating upply chain inventorie through common replenihment epoch. Eur. J. Oper. Re. 19, Khouja, M: Optimizing inventory deciion in a multitage multi cutomer upply chain. ranportation Reearch. 39, Agarwal, V, Chao, X, Sehadri, S: Dynamic balancing of inventory in upply chain. Eur. J. Oper. Re. 159, Rau, H, Wu, MY, Wee, HM: Deteriorating item inventory model with hortage due to upplier in an integrated upply chain. Int. J. Syt. Sci. 35, Lee, W: A joint economic lot ize model for raw material ordering, manufacturing etup,and finihed good delivering. Int. J. Manag. Sci. 33, Ben-Daya, M, Seliaman, M: An integrated production inventory model with raw material replenihment conideration in a three layer upply chain. Int. J. Prod. Econ. 8, Sana, S: A production-inventory model of imperfect quality product in a three-layer upply chain. Deciion Support Syt. 5, Pal, B, Sana, SS, Chaudhuri, K: hree-layer upply chain - a production-inventory model for reworkable item. Appl. Math. Comput. 19, Wang, J, Shu, YF: Fuzzy deciion modelling for upply chain management. Fuzzy Set Syt. 15,

17 Jana et al. Journal of Uncertainty Analyi and Application 13, 1:6 Page 17 of Da, B, Maity, K, Maiti, M: A two warehoue upply-chain model under poibility / neceity / credibility meaure. Math. Comput. Model. 46, Chen, C, Yuan,, Lee, W: Multi-criteria fuzzy optimization for locating warehoue and ditribution center in a upply chain network. J. Chin. Int. Chem. Eng. 38, Peidro, D, Mula, J, Jimenez, M, Botella, MM: A fuzzy linear programming baed approach for tactical upply chain planning in an uncertainty environment. Eur. J. Oper. Re. 5, Chu, SJ: Interactive group deciion-making uing a fuzzy linguitic approach for evaluating the flexibility in a upply chain. Eur. J. Oper. Re. 13, Jana, DK, Maity, K, Da, B, Roy, K: A fuzzy imulation via contractive mapping genetic algorithm approach to an imprecie production inventory model under volume flexibility. J. Simulation. 7, Kritianto, Y, Helo, P, Jiao, J, Sandhu, M: Adaptive fuzzy vendor managed inventory control for mitigating the Bullwhip effect in upply chain. Eur. J. Oper. Re. 16, Liu, B: Uncertainty heory. nd edn. Springer, Berlin 7. You, C: Some convergence theorem of uncertain equence. Math. Comput. Model. 49, Liu, B: Fuzzy proce, hybrid proce and uncertain proce. J. Uncertain Syt., Liu, B: Some reearch problem in uncertainty theory. J. Uncertain Syt. 3, Goyal, SK: Economic order quantity under condition of permiible delay in payment. J. Oper. Re. Soc. 36, Khanra, S, Ghoh, SK, Chaudhuri, KS: An EOQ model for a deteriorating item with time dependent quadratic demand under permiible delay in payment. Appl. Math. Comput. 18, Maihami, R, Abadi, INK: Joint control of inventory and it pricing for non-intantaneouly deteriorating item under permiible delay in payment and partial backlogging. Math. Comput. Model. 55, Liu, B: Uncertainty heory: A Branch of Mathematic for Modeling Human Uncertainty. Springer, Berlin 1 doi:1.1186/ Cite thi article a: Jana et al.: A three-layer upply chain integrated production-inventory model under permiible delay in payment in uncertain environment. Journal of Uncertainty Analyi and Application 13 1:6. Submit your manucript to a journal and benefit from: 7 Convenient online ubmiion 7 Rigorou peer review 7 Immediate publication on acceptance 7 Open acce: article freely available online 7 High viibility within the field 7 Retaining the copyright to your article Submit your next manucript at 7 pringeropen.com