J Ind Eng Int (205) :53 542 DOI 0.007/s40092-05-09-5 OIGINAL ESEACH A two-level discount model fo coodinating a decentalized supply chain consideing stochastic pice-sensitive demand Jafa Heydai Yousef Noouzinasab eceived: 28 May 205 / Accepted: 4 July 205 / Published online: 24 July 205 The Autho(s) 205. This aticle is published with open access at Spingelin.com Abstact In this pape, a discount model is poposed to coodinate picing and odeing decisions in a two-echelon supply chain (SC). Demand is stochastic and pice sensitive while lead times ae fixed. Decentalized decision maing whee downsteam decides on selling pice and ode size is investigated. Then, joint picing and odeing decisions ae extacted whee both membes act as a single entity aim to maximize whole SC pofit. Finally, a coodination mechanism based on quantity discount is poposed to coodinate both picing and odeing decisions simultaneously. The poposed two-level discount policy can be chaacteized fom two aspects: () maeting viewpoint: a etail pice discount to incease the demand, and (2) opeations management viewpoint: a wholesale pice discount to induce the etaile to adjust its ode quantity and selling pice jointly. esults of numeical expeiments demonstate that the poposed policy is suitable to coodinate SC and impove the pofitability of SC as well as all SC membes in compaison with decentalized decision maing. Keywods Two-level discount Supply chain coodination Stochastic pice-sensitive demand Multi-echelon inventoy systems & Jafa Heydai j.heydaiut.ac.i School of Industial Engineeing, College of Engineeing, Univesity of Tehan, Tehan, Ian Intoduction Unde taditional decision maing, each supply chain (SC) membe maes decisions based on its own inteests. Since SC membes ae affected by each othe, it is necessay to find mechanisms that impove the pefomance of all SC paties beyond those in the taditional decision maing. Among all SC decisions, the impotance of eplenishment and maeting decisions is undeniable. To ensue the satisfying custome demand without delay and scimping the uncetain events impact, it is essential to eep the inventoy in a ational level. eplenishment policies consist of two categoies () decision on ode quantity (lot size) o poduction ate and (2) decision on eode point. The fist one is one of this pape s concens. On the othe hand, in the supply chain management, optimal picing stategy impoves the pofitability of system significantly (Gallego and Van yzin 994). Picing stategies play an impotant ole when custome demand is pice sensitive and also when poduction/distibution decisions can be complemented with picing stategies in manufactuing envionments (Simchi-Levi et al. 204). Picing stategies has a geat impact on etail and manufactuing industies that use the intenet and Diect-to- Custome models such as Dell Computes and Amazon.- com (Chan et al. 2004). Optimal decision maing on picing stategy equies nowing the custome demand at a specific pice. Maet demand in addition to the pice can depend on othe vaiables such as band name, quality of poduct o delivey length, but hee we focus on a model the demand of which is sensitive only to the poduct pice. Integation of individual decisions on odeing and picing policies thoughout the SC can impove the pefomance of SC in a same way that evenue management has enhanced the efficiency of the ental ca companies,
532 J Ind Eng Int (205) :53 542 hotels and ailine (Chan et al. 2004). In paticula, this may be moe applicable fo e-commece opeating systems, since pice can be changed easily and data fo demand ae available eadily (Chan et al. 2004). This pape aims to coodinate odeing and picing stategies thoughout the SC simultaneously using an intensive discount scheme. The poposed coodination mechanism impoves SC pofitability fom two aspects: () inceasing evenue fom the customes by selling moe and (2) educing cost of mateial flow between SC membes. The liteatue includes some wos that theoetically studied the impotance of integation SC odeing and picing policies to mitigate the taditional decision-maing disadvantages, but the coodination of these decisions simultaneously with a finite poduction ate and lost sales inventoy system is often neglected. We efe the eades to Gloc (202) fo a compehensive liteatue eview on integated odeing policies and to Chen and Simchi-Levi (202) fo a compehensive eview on integated inventoy planning policies and picing stategies. Stating fom the evaluation of individual decision maing on odeing and picing policies in a one-etaile and one-supplie system, this pape pesents and evaluates a two-level discount scheme to coodinate SC membes decisions simultaneously to maximize whole SC pofit as well as all membes pofitability. The esults show a damatic impovement in the SC and its membes pefomance when the poposed incentive plan is placed. The est of this pape is oganized as follows. Liteatue eview pesents a liteatue eview on the supply chain coodination which involved odeing, picing and pice discount in inventoy systems. In Mathematical modeling, a desciption of the poposed model is povided. The optimization models in decentalized and centalized modes ae pesented, the poposed coodination plan is explained and a method based on Li and Liu s wo (Li and Liu 2006) is developed to divide the inceased pofit aised fom joint decision maing between both sides of SC. Numeical expeiments pesents numeical expeiments to demonstate the capability of poposed scheme and finally Conclusions concludes the pape. Liteatue eview This wo is involved with two main categoies in the liteatue: () the supply chain coodination that examines how to align SC membes decisions to impove thei pofitability and subsequently the pofitability of the whole SC and (2) the odeing policies, picing stategies and pice discount in inventoy systems. If SC membes coodinate thei decision based on the whole SC optimal objective, the pefomance of SC will be impoved (Cachon 2003). A compehensive liteatue eview of supply chain coodination though contacts has been povided by Cachon (2003). Among vaious SC coodination mechanisms, buybac policies (Yao et al. 20; Ai et al. 202), evenue-shaing mechanisms (Palsule-Desai 203), sales ebate contacts (Saha 203), quantity-flexibility models (Kaaaya and Baal 203) and quantity discount contacts ae the most common ones. uantity discount contacts ae moe common in pactice among afoementioned contacts. Unde quantity discount contacts, the supplie impoves his pefomance though moe sales of poducts and eduction of his opeational costs; on the othe hand, the etaile benefits fom a discount in wholesale pice (Sadian and Yoon 994; Munson and osenblatt 998; Lin 2008, 203; Chung et al. 204; Taleizadeh and Pentico 204). Monahan (984) demonstated that the quantity discount offeed by a vendo is able to induce the buye to incease his ode quantity. ecent eseaches, by consideing this fact, have shown that this incentive tool could be used to coodinate SC decisions. Weng (995) investigated a supplie buye system faced with the deteministic demand. It has been shown that a quantity discount policy egadless of its fom (incemental o all-unit discount policy) is an effective plan to induce the buye to align its decision vaiables with the joint decision-maing plan. i et al. (2004) investigated a supplie etaile SC faced with the demand disuption duing the planning hoizon. They showed that a wholesale quantity discount policy is able to evise the poduction plan and coodinate the SC. Li and Liu (2006) designed a wholesale quantity discount scheme unde demand uncetainty to induce the etaile to incease its ode quantity in a one-supplie one-etaile supply chain. They showed thee is feasible solution that the supplie and etaile accept to paticipate in the joint decision maing. Futhe, the authos designed a method to divide the inceased pofit aised fom joint decision maing between both sides of SC. Xie et al. (200) studied an ealy ode commitment (EOC)-based discount to coodinate a SC consisting of one manufactue and multiple independent etailes. The authos demonstated that the wholesale pice discount policy can encouage the etailes to paticipate in EOC that esult in lowe SC costs. Sinha and Samah (200) developed a multiple picing schedule to influence odeing polices of a goup of heteogeneous buyes to paticipate in coodination plan povided by a supplie. Thei study showed that by inceasing the numbe of picing schedules, the benefits of coodination plan would incease. Wang et al. (20) developed an all-unit discount coodination scheme, fo a one-supplie multiple-buye SC. To educe waehousing costs, the supplie should encouage the buyes to synchonize thei odes with the supplie s
J Ind Eng Int (205) :53 542 533 eplenishments policies; in thei model, encouagement is planned by poviding an all-unit pice discount. Du et al. (203) studied a hybid cedit-wholesale pice discount scheme to induce the SC membes consisting of one supplie and one buye to mae thei decisions in a way to impove the entie SC pofitability in addition to thei own pofit. Peng and Zhou (203) investigated efficiency of a quantity discount scheme to coodinate a fashion supply chain faced with stochastic demand and uncetain yields. The esults of thei study showed that the negative effects of uncetain yields and stochastic demand can be educed by implementing the poposed policy. Zhang et al. (204b) poposed a quantity discount scheme to coodinate and impove the efficiency of an integated poduction-inventoy system. They assumed that the poduct has a fixed lifetime and poduction ate is finite. A coodination plan was developed in which a vendo by poposing an all-unit quantity discount induces the buye to puchase lage lot size in an integated inventoy system involving defective items (Lin and Lin 204). Zhang et al. (204a) investigated a manufactue etaile supply chain whee the manufactue is is avese and delives lowe amount of the etaile s ode quantity when the etaile has a delay in payments. In this situation, they poposed a modified quantity discount in which the manufactue induces the etaile to incease its ode quantity while having an advanced payment. Heydai (204) poposed a time-based tempoay pice discount coodination plan fo a two-echelon SC, in which the selle ties to induce the buye to globally optimize safety stoc. Yin et al. (204) developed a game theoetic model to help a manufactue fo supplie selection and having a longtem elationship using a quantity discount coodination scheme. Yang et al. (204) discussed about thee plans: quantity discount, cedit peiod and centalized SC in a two-echelon system consisting of one manufactue and one etaile whee demand depends on stoc level of the etaile. The authos showed that the cedit peiod contact is pefeed to induce the etaile to incease its ode quantity when the manufactue inteest ate is less than the etaile inteest ate. In that model, iespective of inteest ates, centalized decision maing esult in equal o highe SC pofit than two othe ones. A pice discount policy was developed fo a fim with an opaque selling stategy to encouage flexible customes to postpone thei demand (Wu and Wu 205). esults showed that using this intensive scheme, the fim educes shotages and capacity wastes. Saha and Goyal (205) analytically investigated thee SC coodination plans namely wholesale pice contact, cost shaing contact and joint ebate contact to coodinate one-manufactue one-etaile SC faced with a pice stoc-sensitive demand. esults showed that the pefeences of paties between thee afoementioned plans ae not always aligned. Futhemoe, they found out that the etaile with a highe bagaining powe pefes the wholesale pice contact. In addition to discount contacts as an intensive scheme to integate system s decisions, we eview some ecent papes in the field of picing, eplenishment policies and coodinating SC with pice-sensitive demand. Chen and Bell (20) investigated a single manufactue single etaile system whee custome demand is pice sensitive and the system expeiences custome etuns. They poposed an incentive contact that includes two buybac pices: () fo unsold poducts and (2) fo etuns of custome. Taleizadeh and Nooi-dayan (204) investigated a decentalized supply chain consisting of one supplie, one poduce, and some etailes by consideing pice-sensitive demand. They aim to educe system-wide cost by coodinating SC decisions that includes: supplie and poduce pice and shipment numbes eceived by the supplie and the poduce. Lin and Wu (204) designed an integated system opeations policy unde uncetain and pice-sensitive demand that simultaneously detemines the pice of poduct and opeational levels of pocuement, poduction, and distibution. Taleizadeh et al. (205a) developed an economic poduction and inventoy model fo a system consisting of one distibuto, one manufactue, and one etaile. They detemined the distibuto ode quantity and its selling pice and also the manufactue and the etaile selling pice with the aim of maximizing all system membes pofit. In anothe wo, Taleizadeh et al. (205c) investigated a poblem of joint detemination of eplenishment lot size, selling pice, and shipment numbes with assuming ewo on defective items fo an economic poduction quantity (EP) model. Taleizadeh et al. (205b) developed a vendo managed inventoy (VMI) model fo a two-echelon system including one supplie and seveal noncompeting etailes facing a pice-sensitive demand. They poposed an inventoy model to jointly optimize the eplenishment fequency of aw mateial, the poduction ate, the eplenishment cycle of the poduct, and the etail pice. They assumed diffeent deteioation ates fo the aw mateial and the finished poduct. Extending the pevious studies, this pape consides a two-echelon supply chain consisting of one-supplie and one-etaile faced with a stochastic pice-sensitive demand. In this model, coodination of picing and odeing decisions simultaneously with consideing a finite supplie poduction ate and lost sales inventoy system, is investigated. To achieve it, a two-level pice discount scheme is poposed: () discount in selling pice to induce customes to buy moe, and (2) a wholesale pice discount to induce the etaile to adjust its ode quantity and selling pice based on joint decision maing. In the poposed model, the
534 J Ind Eng Int (205) :53 542 SC opeates in a competitive maet whee demand will be lost if it is not met immediately. In paticula, in the poposed model, the etaile has the authoity of deciding on ode quantity and selling pice but both of these decision vaiables have a substantial impact on the supplie pofitability. The etaile maes decisions on both the ode quantity and selling pice in ode to maximize its own pofit. On the othe hand, in a picesensitive maet, the etaile should povide a selling pice to maximize its evenue fom the customes. Optimizing ode quantity and selling pice by the etaile maximizes its pofit egadless of othe SC membes pofit. On the othe hand, optimizing ode quantity and etail pice fom the viewpoint of whole SC may educe the etaile pofit. In this study by modeling of SC membes cost stuctue, a two-level discount scheme is poposed to coodinate SC membes decisions to maximize whole SC pofit as well as all membes pofitability. The poposed two-level discount consists of two pats: () a etail pice discount initiated by the etaile to induce customes to buy moe and (2) a wholesale quantity discount initiated by the supplie to convince the etaile to globally optimize etail pice and also the size of odes to educe SC opeational costs. Decision models ae developed in thee cases decentalized, centalized, and coodinated stuctues. Unde the decentalized decision stuctue, the etaile decides on both ode quantity and etail pice based on its own cost stuctue. Unde centalized stuctue, it is assumed that thee is a single decision mae who aims to maximize whole SC pofit. Finally, a coodination mechanism is poposed to induce both membes to decide based on whole SC viewpoint while both membes pofitability is guaanteed. A two-level discount model is poposed as incentive scheme to shae benefit of coodinated decision maing between SC membes faily. Mathematical modeling The consideed SC consists of one supplie and one etaile. The supplie manufactues the poduct. The etaile faces with a stochastic pice-sensitive demand with nomal distibution. Lead time is deteministic and constant. Demand will be lost if the custome s needs ae not met immediately. The etaile uses a (, ) continuous eview system. Based on demand infomation and its cost stuctue, the etaile decides on the selling pice to the customes and ode quantity fom the supplie. Afte etaile s odeing, the supplie detemines the quantity of poduction. It is assumed that the supplie has enough and finite capacity to meet the etaile s odes. In addition, expected custome s demand is consideed as a linea function of Fig. Supplie etaile customes inteactions unde decentalized model Fig. 2 Supplie etaile customes inteactions unde coodinated model etail pice, while the standad deviation of demand is fixed and it is not a function of etail pice. Figues and 2 illustate the investigated poblem unde decentalized and coodinated stuctues, espectively. The following notations ae applied in the mathematical models: p E[D(p)] D S h p L h s w S s c n etaile s unit selling pice (decision vaiable) Expected custome s demand ate pe yea at selling pice p Standad deviation of custome s demand etaile s ode quantity (decision vaiable) etaile s odeing costs pe ode etaile s unit holding cost pe yea etaile s unit shotage cost etaile s eplenishment lead time Safety stoc facto Supplie s unit holding cost pe yea Supplie s wholesale pice befoe applying discount Supplie s fixed cost that is incued with each handling the etaile s ode Supplie s unit poduction cost Supplie s lot size multiplie (decision vaiable) Supplie s annual poduction capacity Decentalized decision maing Unde decentalized decision maing, each membe ties to maximize its own pofit function individually egadless of othe membe. The expected annual custome demand is a linea function of etail pice given by E[D(p)] = a - bp, whee a is maet size and b is the pice-elasticity coefficient of demand. Let P (, p) denote the etaile s expected annual pofit function, then it can be fomulated as:
J Ind Eng Int (205) :53 542 535 a bp P ð; pþ ¼ ðp wþða bpþ ð Þ S h 2 þ ð þ Þ a bp ð Þ ðp þ p wþ DL ðþ whee the fist tem denotes etaile s expected annual goss pofit. The second and thid tems denote expected annual odeing cost and annual holding cost, espectively. The last tem denotes expected annual lost sales penalty and oppotunity costs. DL is standad deviation of custome s demand duing lead time: pffiffiffi DL ¼ ð2þ Since demand comes fom nomal pobability distibution, then expected shotages will be DL (Silve et al. 998, pp. 722 723): ¼ Z ðz Þp ffiffiffiffiffi expðz 2 =2Þ dz ð3þ 2p Accoding to Eq. (), the etaile decides on and p to maximize its own pofit function. Poposition The etaile pofit function is concave with espect to and p simultaneously when: 2b 2ða bpþððp þ p wþ þs Þ 3 [ ðbðp 2p þ wþþaþ bs 2 2 Poof See Appendix. Let and p denote the optimal values of etaile decision vaiables that maximize its pofit function unde the decentalized model. Using fist-ode condition, we have: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ða bp Þh ððp þ p wþ þs Þ ð4þ h p ¼ D ð L p wþþ w þ S 2 Gu ðþ þ a ð5þ 2b Since the values of and p ae ciculaly depending on each othe, then an iteative pocedue can be used to calculate optimal values as follows: Step Set p = 0 (minimum feasible value fo p ) Step 2 Calculate using (4) Step 3 Calculate p using (5) based on obtained Step 4 epeat second and thid steps to convege Step 5 The obtained values of and p ae optimum. Chec optimality condition of Poposition to ensue global optimization. Now let P s (n) be the supplie s expected annual pofit function, then it can be fomulated as: P s ðnþ ¼ðw cþða bp Þ ða bp Þ n S s 2 h s n ðn2þða bp Þ ð6þ Poposition 2 The supplie pofit function is concave with espect to n. Poof n s See Appendix. is the supplie lot size multiplie so that it maximizes its pofit function unde the decentalized model. Let n s be a value of n that maximizes the supplie pofit eleasing the constaint that n is an intege, it can be calculated as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n s ¼ 2S s Gu ðþ a bp h s bp þ a þ DL a bp ð7þ Since n must be an intege vaiable, eithe the smallest following intege o lagest pevious intege of n s whicheve esults in lage value of P s (n) will be optimum value of n fom the supplie pespective (i.e., n s ). Centalized decision maing Unde centalized decision maing, a cental decision mae aims to maximize the whole SC pofit. In this situation, sale and eplenishment policies ae detemined fom viewpoint of the entie SC. Let P sc (, p, n) be the expected annual pofit function of SC that is the sum of the supplie and etaile annual expected pofit: P sc ð; p; nþ ¼P ð; pþþp s ðnþ ¼ðpcÞða bpþ ða bpþ S þ DL p þ Ss n h 2 þ ð þ Þ 2 h s n ðn 2Þða bpþ ð8þ Obsevation The SC expected annual pofit function is concave with espect to, p and n unde some cicumstances (fo details see Appendix ). Let sc, p sc and n sc denote the values of SC decision vaiables that maximize SC pofit function eleasing the constaint that n is an intege. Then, we have:
536 J Ind Eng Int (205) :53 542 h 3 sc ¼ 3 ð h s n sc abp sc 2 a bp sc whee, h 3 Þðn sc 2Þ S s n sc þ S þ p sc c þ p þ h DL Although the above pocedue finds optimal decisions fo the whole SC, the found solution does not ceate moe pofitability than decentalized model fo each SC membe. To guaantee paticipation of both SC membes in the joint decision-maing pocess, it is necessay to designate an incentive scheme to encouage SC membes though inceasing thei pofit. 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ffiffi 3 08 a bp sc S 2 8 a bp 3 S s s sc n sc þ S þ p sc c þ p 3 DL u n t sc ð h ¼ h s n s abp scþðn sc 2Þ þ h B þ 26 a C bp sc DL S s A n sc 4 h s n sc a 2 bp sc ð nsc 2Þ þ h ð9þ! p sc ¼ a 2 b þ c þ S s n sc h s scð n sc 2Þ ð þ p þs Þ sc 2 sc ð0þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n sc ¼ 2S s sc þ a bp sc sc h s bp sc þ a sc þ DL a bp sc ðþ Since the values of sc, p sc and n sc ae ciculaly depending on each othe, then an iteative pocedue can be applied to calculate optimum values of decision vaiables as follows: Step Set p sc = 0 (minimum feasible value fo p sc ) Step 2 Set n sc = (minimum feasible value fo n sc ) Step 3 Calculate sc using Eq. (9) Step 4 Calculate p sc using Eq. (0) Step 5 Calculate n sc using Eq. () Step 6 epeat thid, fouth and fifth steps to convege values of sc, p sc and n sc Step 7 Calculate SC pofit function at the smallest following intege and lagest pevious intege of n sc ; whicheve esults in lage value of P sc (, p, n) is selected as n sc Step 8 Obtained values of sc, p sc and n sc ae optimum. Chec optimality condition of Obsevation to ensue global optimum solution. Coodination mechanism and incentive scheme It is clea that substituting the optimal values of whole SC decision vaiables fo values of decision vaiables that ae made individually by membes, SC pofit will be impoved. Nevetheless, it cannot guaantee impovement of all SC membes pofit. To encouage the etaile and supplie to ente joint decision maing, thei expected annual pofit must be impoved in compaison with decentalized mode. Both decision vaiables and p ae unde the authoity of the etaile, while decision vaiable n has no effect on the etaile pofit function. In the decentalized model, the etaile decides on both decision vaiables based on its own pofitability; theefoe, changing decisions fom decentalized mode cause pofit loss fo the etaile. In this section, we popose a mechanism that equies the supplie to educe wholesale pice to encouage the etaile to change its decisions on and p with espect to the whole SC optimal decisions. Conside that the etaile is equied to change its ode size fom to sc and also its selling pice fom p to p sc. Let K be a coefficient that the etaile is equied to apply on its ode size and d be a coefficient to apply on its selling pice, then we have: K ¼ sc ð2þ
J Ind Eng Int (205) :53 542 537 d ¼ p sc p ð3þ On the othe hand, the supplie is equied to apply coefficient d on its wholesale pice w to ceate a discounted wholesale pice. Substituting K fo and d p fo p and d w fo w in the etaile pofit function (Eq. ()), we get the impoved pofit of the etaile in migating fom decentalized model towad joint decision maing as: P d ; K ; d p P ; p ¼ d p d w a bd p a bd p K K S h 2 þ ð þ Þ a bd p p þ d p d w DL P ; p K ð4þ The etaile accepts an ageement that maes Eq. (4) geate than zeo, thus we should have: d wabd p DL K abd p K ðpþd p Þ DL d p abd p þs K h 2 þ ð þ Þ P ;p ð5þ In fact, the maximum value of coefficient d that can encouage the etaile to paticipate is: d max ¼ wabd p DL K 0 a bd p ðp þ d K p Þ DL d p a bd p þ S B K C h 2 þ ð þ Þ P ; A p ð6þ By the simila pocedue, the minimum acceptable value of coefficient d that guaantees moe pofitability fo the supplie can be calculated as: Let IP sc be the inceased pofit of whole SC afte joint decision maing: IP sc ¼ P sc sc ; p sc ; n sc P ; p þ Ps n s ð8þ If the supplie implements d min, the whole IP sc will be assigned to the etaile, while implementing d max assigns all pofits to the supplie. Assume that a (0 \ a \ ) faction of IP sc is assigned to the etaile and subsequently - a faction is assigned to the supplie whee a is bagaining powe of the etaile with espect to the supplie. Let IP be the inceased pofit of the etaile afte implementation of quantity discount policy. Based on the above-mentioned analysis, we get: IP ðd ; d ; KÞ ¼ a IP sc ¼ a IP d min ; d ; K þ ð aþip d max ð9þ ; d ; K Simplifying Eq. (9), we can calculate d based on bagaining powe a as: d ¼ a d min þ ð aþd max ð20þ Nomally, the etaile attends d min while the supplie wants to implement d max. Using Eq. (20) accoding to bagaining powe a, the value of d will be specified. Numeical expeiments Using a set of test poblems, the pefomance of the poposed model is demonstated. Table shows the thee investigated test poblems. The esults of unning the poposed model on the test poblems show that the poposed intensive scheme is able to coodinate SC. Coodination scheme impoves pofitability of both SC membes as well as whole SC pofit. Table 2 compaes the values of decision vaiables and pofit functions obtained fom the decentalized, centalized, and coodinated decision-maing models. Note that since vaiable n should tae an intege value, then obtained d min ¼ wabd p DL K 0 2 Kh s n sc a bd p n DL sc 2 K B a bd p C S s =n sc þ Ps n A þ c w s K K ð7þ
538 J Ind Eng Int (205) :53 542 Table Data fo the investigated test poblems Test poblem Test poblem 2 Test poblem 3 w 200 205 20 a 3000 4000 9000 b 0 35 h 40 32 35 h s 35 32 37 S 8000 6000 4000 S s 9000 6500 6550 p 4 2 3 c 50 60 85 L 4 9 D 40 30 50 4500 5500 0,000 0.95 0.95 0.95 Table 2 The esults of decentalized, centalized, and coodinated decision-maing models Test poblem Test poblem 2 Test poblem 3 Decentalized mode 4.94 584.80 390.24 p 259.92 289.54 238.78 n s n s.43.06.26 P 4204.99 4792.39 307.65 P s 045.50 25863.66 4760.76 P sc 4656.49 73056.47 7778.40 E[D(p)] 400.80 85.06 642.70 Centalized mode sc 849.46 970.76 256 p sc 239.45 27.33 229.79 n sc n sc 0.70 0.73 0.49 P -2364.49 4953.02-7955.6 P s 267.02 33982.37 665.27 P sc 9306.53 75935.39 8696. E[D(p)] 605.62 05.39 956.93 Coodinated mode K 2.062.6600 3.234 d 0.922 0.937 0.9624 d max 0.9453 0.9746 0.9452 d min 0.9065 0.9522 0.9406 d 0.9259 0.9634 0.9429 P 653.64 49500.74 3476.50 P s 2774.89 26434.65 529.6 P sc 9306.53 75935.39 8696. decimal values fo this vaiable ae infeasible. As discussed befoe, the smallest following and lagest pevious intege of n may be optimal. As shown in Table 2, in the centalized model, the etaile should incease its ode Fig. 3 The effect of b on the etaile pofit quantity and educe the selling pice compaed to decentalized model. The esults show the expected pofit of SC in centalized model is inceased in compaison with decentalized model. In centalized model, the expected pofit of supplie is inceased but the etaile loses money. Thus, nomally the etaile efuses to change its mind about its decision vaiables and p. A mechanism that inceases both SC membes pofit should be designated. The maximum and minimum values of wholesale discount coefficient d ae obtained so that both the etaile and supplie benefit fom inceased pofit of SC. Wholesale discount coefficient d can be detemined based on bagaining powe a using Eq. (20). In Table 2, wholesale discount coefficient d is detemined supposing equal bagaining powe fo SC membes (i.e., a = 0.5). With ageement of SC membes to paticipate in coodination plan, the etaile should apply ates K and d on its ode quantity and selling pice, espectively. On the othe hand, the supplie should apply ate d on its wholesale pice. Table 2 shows that the etaile and supplie have a moe pofit beyond those in the decentalized decision maing. Thus, the poposed incentive scheme is applicable. In addition, the poposed model inceases SC pofit same as centalized decision maing in all investigated test poblems, theefoe the poposed model is able to achieve channel coodination. To investigate the impact of two significant paametes b and L on the pofitability of SC and its membes, a set of sensitivity analyses is conducted. The equied data fo sensitivity analyses ae taen fom test poblem. Figues 3 and 4 illustate changing of the etaile and the supplie pofit functions in the decentalized, centalized, and coodinated models ove inceasing b. As expected, both membes of SC have a lowe pofit when b has a lage value that subsequently esults in lowe SC pofit. In addition, both membes pofitability unde coodinated model is geate than decentalized model which implies that the coodination model is applicable fom both
J Ind Eng Int (205) :53 542 539 Fig. 4 The effect of b on the supplie pofit Fig. 7 The effect of b on the SC pofit Fig. 5 The effect of L on the etaile pofit Fig. 8 The effect of L on the SC pofit Figues 7 and 8 illustate changes on whole SC pofit by inceasing b and L, espectively. As expected, by inceasing b and L, SC pofitability deceases. Nevetheless, the poposed coodination model mitigates negative effects of inceasing b and L on SC pofitability with espect to decentalized decision-maing model. Conclusions Fig. 6 The effect of L on the supplie pofit membes pespective. Based on Figs. 3 and 4, thee is a theshold of b beyond that the business is disadvantageous; the poposed coodination model is able to enhance this theshold fo both membes with espect to decentalized decision-maing model. Figues 5 and 6 illustate the effect of lead time L on the etaile and supplie pofitability unde the decentalized, centalized, and coodinated models. As the Figues show, the pofitability of both SC membes deceases with inceasing L that is the diect esult of moe lost sales by polonging the lead times. Howeve, pofitability of both membes unde coodinated decision maing is always geate than decentalized model. This pape has investigated a two-level pice discount as a coodination plan which is chaacteized by two diffeent aspects: () Maeting management: discount in etail pice to incease the demand, and (2) Opeations management: discount in wholesale pice to induce the etaile to adjust its ode quantity and etail pice based on joint decision maing. In the poposed model, SC faces with a stochastic pice-sensitive demand. In the decentalized decision-maing model, the etaile decides on the ode quantity and selling pice based on its own expected pofit function. Howeve, centalized modeling of SC decisions evealed that thee is a bette solution fom the whole SC viewpoint. Ou numeical expeiment esults show that the pofitability of the entie SC and supplie impoves in the centalized mode, but the etaile loses money. In this situation, by applying a discount on wholesale pice, the inceased pofit
540 J Ind Eng Int (205) :53 542 would be divided between SC membes so that the supplie and etaile impove thei pofit. Uppe bound and lowe bound fo the wholesale pice discount fom both the etaile and supplie pespectives ae extacted. Numeical expeiments and sensitivity analyses show that the poposed model has the ability of achieving channel coodination. The poposed model can be extended to conside othe demand functions. Also, in addition to sensitivity of the demand to pice, othe paametes such as lead time length and poduct quality that can affect the demand can be consideed. Anothe altenative fo futue study is about consideing the impact of selling pice on demand standad deviation. Open Access This aticle is distibuted unde the tems of the Ceative Commons Attibution 4.0 Intenational License (http://cea tivecommons.og/licenses/by/4.0/), which pemits unesticted use, distibution, and epoduction in any medium, povided you give appopiate cedit to the oiginal autho(s) and the souce, povide a lin to the Ceative Commons license, and indicate if changes wee made. Appendix Poof of Poposition To pove concavity of the etaile pofit function with espect to and p, the Hessian matix of the etaile s expected annual pofit function should be calculated. If the Hessian matix is negative definite, the poposition will be poved. We have: HP ð ð; pþþ ¼ o2 P ð; pþ=o 2 o 2 P ð; pþ=oop o 2 P ð; pþ=opo o 2 P ð; pþ=op 2 main diagonal is also negative. Theefoe, the equisite fo Hessian matix to be negative definite is satisfied. The fist pincipal mino of the above Hessian matix is the same as the fist element of the main diagonal that has a negative value. The second pinciple mino is positive when: 2 a bp 2b ð Þ ðp þ p wþ ð þs Þ 3 b p 2p þ w [ ð ð ÞþaÞ bs 2 Satisfying the above condition esults in Hessian matix of the etaile s expected pofit function to be negative definite. With espect to ational values of paametes, this condition would be satisfied. Poof of Poposition 2 It is enough to show that the second-ode deivative of the supplie s expected annual pofit function with espect to n is negative: o 2 P s ðnþ 2ða bpþ S s on 2 ¼ n 3 As mentioned, we have a pofitable business when Gu() DL \ then above tem is negative. Details of obsevation To show concavity of SC pofit function with espect to, p, and n the Hessian matix of the SC expected annual pofit function should be calculated. If the Hessian matix is negative definite, the obsevation will be poved. We have: 2 3 o 2 P sc ð; p; nþ=o 2 o 2 P sc ð; p; nþ=oop o 2 P sc ð; p; nþ=oon HP ð sc ð; p; nþþ ¼ 4 o 2 P sc ð; p; nþ=opo o 2 P sc ð; p; nþ=op 2 o 2 P sc ð; p; nþ=opon 5 o 2 P sc ð; p; nþ=ono o 2 P sc ð; p; nþ=onop o 2 P sc ð; p; nþ=on 2 whee o 2 P ð; pþ 2 a bp o 2 ¼ ð Þ ð ðp þ p wþ þs Þ 3 o 2 P ð; pþ op 2 ¼2b o 2 P ð; pþ ¼ o2 P ð; pþ ¼ ð b ð p 2p þ w ÞþaÞ bs oop opo 2 Since p [ w, then the fist element of the main diagonal is negative. On the othe hand, we would have a pofitable business if Gu() DL \ then the second element of the whee: o 2 P sc ð; p; nþ 2 a bp o 2 ¼ ð Þ 3 ððp c þ pþ DL þs Þ 2 a bp þ ð Þ ð 3 Þ n 4 S s o 2 P sc ð; p; nþ op 2 ¼2b o 2 P sc ð; p; nþ 2 a bp on 2 ¼ ð Þ ð ÞS s n 3 2
J Ind Eng Int (205) :53 542 54 o 2 P sc ð; p; nþ ¼ o2 P sc ð; p; nþ oop opo a bp bpc ¼ ð ð ÞÞ bs ð þ p DL Þ 2 þ b ð 2 ÞS s n 3 2 h sbn ð 2Þ o 2 P sc ð; p; nþ ¼ o2 P sc ð; p; nþ oon ono a bp ¼ ð Þ ð 2 ÞS s n 2 3 2 h a bp sp ð Þ o 2 P sc ð; p; nþ ¼ o2 P sc ð; p; nþ ¼ bð ÞS s opon onop n 2 2 bh 2 ð DL Þ 2 The fist element of the main diagonal is negative unde below condition. Howeve, with espect to ational values of the model paametes, this condition would be satisfied. 3 DL S s \ ðp c þ pþ DL þs ð2þ n Accoding to the peceding explanation, the second and thid elements of main diagonal ae negative. Theefoe, the equisite fo Hessian matix to be negative definite is satisfied. The fist pincipal mino of the above Hessian matix is the same as the fist element of the main diagonal that has a negative value unde condition 2. The second pinciple mino is positive when: 2 a bp 2b ð Þ 3 ððp c þ pþ DL þs Þ 2 a bp ð Þ 3 ð Þ n 4 S s a bp bpc [ ð ð ÞÞ bs ð þ p DL Þ 2 þ b ð 2 ÞS s n 3 2 2 h sbn ð 2Þ ð22þ And the thid pinciple mino is negative when: 0 2 a bp 4b ð Þ 2 a bp 3 ððp c þ pþ DL þs Þþ ð Þ 3 ð Þ n 4 S s B a bp ð DL Þ ð Þ ð C ÞS s A n 3 3 þ 2b ða bpþ ð 2DL ÞS s n 2 3 2 h a bp sp ð Þ 2 0 2 a bp ð Þ 2 a bp 3 ððp c þ pþ DL þs Þþ ð Þ 3 ð Þ n 4 S s þ B b ð DL ÞS s n 2 2 þ bh s ð DL Þ 2 C A 2 0 ða bp bp ð cþþ DL bs ð þ p DL Þ 2 þ b ð 2 ÞS s n 3 2 2 h sbn ð 2Þ þ B 2 a bp ð Þ ð C ÞS s A n 3 0 2 ða bp bp ð cþþ DL bs ð þ p DL Þ 2 2 þ b ð 2 ÞS s n 3 2 h sbn ð 2Þ 0 ða bpþð2 DL ÞS s \ n 2 3 b ð DL ÞS s B 2 h a bp sp ð Þ C A n 2 2 þ bh s ð DL Þ B 2 C A ð23þ
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