Multi-product budget-constrained acquistion and pricing with uncertain demand and supplier quantity discounts

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1 Unversty of Wndsor Scholarshp at UWndsor Mechancal, Automotve & Materals Engneerng Publcatons Department of Mechancal, Automotve & Materals Engneerng Mult-product budget-constraned acquston and prcng wth uncertan demand and suppler quantty dscounts Janma Sh Guoqng Zhang Unversty of Wndsor Follow ths and addtonal works at: Part of the Busness Admnstraton, Management, and Operatons Commons, Busness Intellgence Commons, Operatons and Supply Chan Management Commons, and the Operatons Research, Systems Engneerng and Industral Engneerng Commons Recommended Ctaton Sh, Janma and Zhang, Guoqng. (2010). Mult-product budget-constraned acquston and prcng wth uncertan demand and suppler quantty dscounts. Internatonal Journal of Producton Economcs, 128 (1), Ths Artcle s brought to you for free and open access by the Department of Mechancal, Automotve & Materals Engneerng at Scholarshp at UWndsor. It has been accepted for ncluson n Mechancal, Automotve & Materals Engneerng Publcatons by an authorzed admnstrator of Scholarshp at UWndsor. For more nformaton, please contact scholarshp@uwndsor.ca.

2 Mult-product budget-constraned acquston and prcng wth uncertan demand and suppler quantty dscounts Janma Sh a,b, Guoqng Zhang a* a Department of Industral and Manufacturng Systems Engneerng, Unversty of Wndsor, 401 Sunset Avenue, Wndsor, Ontaro, Canada N9B 3P4 b School of Informaton System and Management, Natonal Unversty of Defense Technology, Changsha, Hunan, Chna *Correspondng author. Emal: gzhang@uwndsor.ca Abstract: We consder the jont acquston and prcng problem where the retaler sells multple products wth uncertan demands and the supplers provde all unt quantty dscounts. The problem s to determne the optmal acquston quanttes and sellng prces so as to maxmze the retaler s expected proft, subject to a budget constrant. Ths s the frst extenson to consder suppler dscounts n the constraned mult-product newsvendor prcng problem. We establsh a mxed nteger nonlnear programmng (MINLP) model to formulate the problem, and develop a Lagrangan based soluton approach. Computatonal results for the test problems nvolvng up to thousand products are reported, whch show that the Lagrangan based approach can obtan hgh qualty solutons n a very short tme. Keywords: Newsvendor, prcng, dscount, acquston, uncertan demand; 1

3 1. Introducton Due to demand uncertanty, the matchng of supply and demand s a constant challenge faced by a retaler. Product acquston and prcng are used as two levers n the retaler s up-stream and down-stream to better match supply and demand. A retaler can use prcng to manage demand and ncrease the revenue, and optmze acquston quantty or nventory level to reduce the msmatch and cost by explotng economes of scale. How to ntegrate both prcng and acquston decsons under uncertan demand s a challengng problem. The stuaton becomes more complcated when supplers provde quantty dscounts: the retaler can procure products at a lower unt prce f the acquston quantty s over a certan value - the threshold; however, snce the demand s uncertan, the retaler s overstockng rsk wll ncrease. Through settng a sutable prce, the retaler can reduce overstockng rsk and ncrease revenue. Thus coordnatng the acquston decson and prcng wth uncertan demands becomes more practcal and challengng when supplers offer quantty dscounts. Motvated by the observaton, ths research nvestgates the jont acquston and prcng problem wth uncertan demand and suppler dscounts. The problem s to determne the optmal orderng quanttes and sellng prces smultaneously so as to maxmze the retaler s expected proft. The problem s an extenson of the newsvendor problem. The newsvendor problem s a classcal model that s used to optmze the orderng quantty under uncertan demand. Due to ts practcal and theoretcal mportance, the newsvendor problem has been wdely studed. Khouja (1999) presented a comprehensve revew and classfed the extensons of the newsvendor problem nto eleven categores. Among those extensons are mult-product 2

4 acquston, newsvendor prcng, and suppler dscounts. Extensons to mult-product nvolve two or more products, usually wth resource constrants. The constraned mult-product newsvendor model was frst proposed by Hadley and Whtn (1963). Snce orderng multple products under budget or other constrants s common, the constraned mult-product newsvendor problem s wdely studed n the last two decades. Representatve work n ths area ncludes that by Lau and Lau (1995 and 1996), Erlebacher (2000), Abdel-Malek and Montanar (2005a, 2005b), and Nederhoff (2007). Incorporatng prcng decson nto the newsvendor problem was frst presented by Whtn (1955), where sellng prce and stockng quantty are determned smultaneously. Then t was extensvely studed by Petruzz and Dada (1999), Webster and Weng (2008), and Chen and Bell (2009). Another mportant extenson of the newsvendor problem s to take nto account the suppler dscount, whch s a common polcy for supplers to promote ther products. The notable work ncludes those of Pantumsncha and Knowles (1991), Khouja (1996), Ln and Kroll (1997), and Zhang (2010). So far, the three extensons to mult-product, prcng, and suppler dscounts have been wdely studed separately. To the best of our knowledge, t s the frst nvestgaton n the lterature that studes these three ssues n one ntegrated model. As dscussed before, through the ntegrated model, the coordnaton of the up-stream and down-stream s decsons makes the problem more practcal and challengng. Our objectve s to develop the optmal acquston and sellng polcy for the retaler, who faces uncertan demand and suppler dscounts. Snce supplers provde quantty dscounts, the product costs are pecewse lnear. We develop a Mxed Integer Nonlnear Programmng (MINLP) model to formulate the problem, and present a Lagrangan based soluton approach, whch s very effcent for large scale nstances. An outlne of ths paper s as follows. Secton 2 provdes a bref lterature revew of the 3

5 related research. Secton 3 presents the MINLP model for the problem. A Lagrangan based soluton approach s developed n Secton 4, and numercal examples and computatonal results are presented n Secton 5. We fnally conclude the paper n Secton Related Research There are numerous works that address the newsvendor problem and varous extensons. Here we manly revew the related studes on the extensons to mult-product, quantty dscount, and newsvendor prcng. For a more comprehensve revew, the reader s referred to Khouja (1999). A bref comparson of the features of the revewed papers wth the proposed model n ths research s llustrated n Table 1. [Insert Table 1 here] The mult-product acquston under uncertan demand s usually modeled as the mult-product newsvendor problem. A budget or other resource constrants are always assocated wth the problem otherwse t can be treated as a sngle product newsvendor problem. The Hadley and Whtn (1963) frst presented a formulaton for the constraned mult-product newsvendor problem and developed a soluton method for the problem. Then Lau and Lau (1995, 1996) presented a formulaton and a soluton procedure for the mult-product constraned newsvendor problem, whch can effcently solve large scale problems nvolvng 1000 products. Abdel-Malek and Montanar (2005a, 2005b) nvestgated the soluton spaces for the mult-product newsvendor problem wth one and two constrants respectvely. Abdel-Malek and Areeratchakul (2007) developed a quadratc programmng model for the mult-product newsvendor problem wth sde constrants, whch can be solved by famlar lnear programmng software packages such as Excel Solver and Lngo. Nederhoff (2007) presented an approxmaton method for the mult-product mult-constrant 4

6 newsvendor problem by approxmatng the objectve functon wth the pecewse lnear nterpolates. Zhang et al. (2009) presented a bnary soluton algorthm for the mult-product newsvendor problem wth budget constrant. More artcles on the mult-product newsvendor problem are ncluded n Table 1. Quantty dscount s a common and effectve polcy for supplers to promote ther products. Quantty dscount s based on the quantty of an tem purchased - promotng the buyer to order large quanttes of a gven tem. Pantumsncha and Knowles (1991) formulated a sngle-perod nventory problem wth the consderaton of standard contaner sze dscounts. Khouja (1995) formulated a newsvendor problem n whch multple dscounts are used to sell excess nventory, whle Khouja and Mehrez (1996) studed the mult-product constraned newsvendor problem under progressve multple dscounts. Khouja (1996) studed the newsvendor problem that consders both multple dscounts used by retalers to sell excess nventory and all-unts quantty dscounts offered by the supplers. However, the model does not consder any resource constrant. Ln and Kroll (1997) nvestgated the sngle-tem newsvendor problem wth quantty dscount and dual performance measure consderaton. The soluton approaches for the all unt quantty dscount and ncremental dscount are developed. Zhang (2010) ntroduced suppler dscounts to the constraned newsvendor problem, and presented a mxed nteger nonlnear programmng model. A Lagrangan heurstc s developed to solve the problem. However, the problem does not consder prcng decson. By ncorporatng prcng nto the newsvendor problem, Whtn (1955) frst nvestgated the optmzaton problem of determnng the stockng quantty and sellng prce smultaneously under uncertan demand envronment. Petruzz and Data (1999) presented a comprehensve revew and some meanngful extensons for the newsvendor prcng problem. Parlar and Weng (2006) studed the effects of coordnatng prcng and producton decsons 5

7 on the mprovement of a frm's poston n a prce-compettve envronment and found that by coordnatng ther prcng and producton decsons, the frm can ncrease ther proftablty, especally when condtons are unfavorable. Karakul (2008) studed the jont prcng and procurement of fashon products n the exstence of clearance markets. Serel (2008) studed a sngle-perod nventory and prcng problem where the rsky supply s consdered. Webster and Weng (2008) nvestgated a jont orderng and prcng problem for a manufacturng and dstrbuton supply chan for fashon products. Chen (2009) addressed the smultaneous determnaton of prce and nventory replenshment when customers return product to the frm. Pan et al. (2009) constructed a two-perod model to determne the prcng and orderng problem for a domnant retaler wth uncertan demand n a declnng prce envronment. As shown n Table 1, the three extensons to the newsvendor problem have been wdely nvestgated separately. There are a couple of artcles that combne two of those three extensons n the newsvendor problem but none on all those three. The model presented n ths paper enrches the newsvendor problem by consderng prcng, quantty dscount, and multple products smultaneously. The propertes of the newsvendor prcng problem wth suppler quantty dscounts are studed and a soluton approach s developed based on Lagrangan method. 3. Model formulaton In ths secton, the mult-product acquston and prcng problem s formulated as a MINLP model, whch s developed based on the followng assumptons. The retaler sells multple products. It s assumed that the demand for each product s ndependent. We also assume that the demand s prce-senstve and stochastc: the relatonshp between demand and prce s D p, u D p u ~, where D p a b p (where the range a 0 and b 0) s the expected demand and A,B wth mean and standard devaton. In order to assure that the 6 u s the stochastc term defned on

8 demand s nonnegatve for some range of p, A should not be less than a. Ths assumpton for the demand has been appled wdely n revenue management and operatons research lterature (Petruzz and Dada 1999; and Pan et al. 2009). We assume that the retaler has a budget constrant and supplers provde all-unt quantty dscounts. For the detals on all-unt quantty dscounts, the reader s referred to Khouja (1996) and Zhang (2010). 3.1 Notaton The followng notatons are used to formulate the problem: Indces: = 1,..., I: ndex of products k = the number of prce dscounts for product offered by supplers j = 1,..., k: ndex of prce segments for product offered by supplers. Parameters: cj = the unt acquston prce of product after dscount on dscount segment j L d j = the lower bound on the quantty of product on dscount segment j U d j = the upper bound on the quantty of product on dscount segment j BG = the avalable budget for the retaler g = the estmated understockng cost (the loss of goodwll) of one unt of product s = the estmated overstockng cost of one unt of product D a b p, the prce-dependent expected demand functon for product p p,u D p u, the prce-dependent stochastc demand functon for product D u = stochastc term defned on the range [A, B] wth mean, n ths paper A a, B and f, 0 F = pdf and cdf of the dstrbuton of u. 7

9 We defne the followng decson varables: p = the retal prce of product q j = the acquston quantty of product at dscount segment j (explaned n the model descrpton) yj = 1 f the retaler buys product at dscount segment j; otherwse 0. Followng Petruzz and Dada (1999), we also defne: k z q j D (p ) q j j1 k j1 a b p. k Where q j s the acquston quantty of product. The ntroducton of the decson j1 varable z facltates the modelng and analyss of the problem: there s overstockng cost f z s larger than u; otherwse understockng cost occurs. From the defnton, we can see that the lower bound on z s a (where both acquston quantty and the prce are set to zero), whch equals the lower bound A on u. The upper bound on z can be nfnte. Thus the varable z has the range [A, B]. Actually, t s common to apply the range of u to varable z (Petruzz and Dada 1999; Pan et al. 2009). 3.2 Model The model for the jont acquston and prcng problem can be formulated as: Max I 1 z A B p D p u s z u f u du p D p z g u z f u I k 1 j1 c j q j z du (1) subject to I k c j q j B G, (2) 1 j1 8

10 q j d j L y j, j, (3) q j d U j y j, j, (4) k y j 1, (5) j1 D k p z q j, (6) j1 A z B,, (7) p 0, q j 0, y j 0,1,, j. (8) The objectve functon s to maxmze the retaler s expected proft: the frst term of represents the expected revenue mnus the overstockng cost when the orderng quanttes are above the actual demand levels; the second term represents the expected revenue mnus the understockng costs when the orderng quanttes are lower than the actual demands; the revenue s evaluated based on sellng quantty, whch s equal to D(p) +u when overstockng, or D(p) + z for understockng; the thrd term s the total acquston cost. Constrant (2) s the budget lmtaton. Constrants (3)-(5) are the quantty dscount constrants: (3) and (4) ensure the amount purchased from the suppler at the prce level postons n the correspondng dscount nterval. Constrants (5) ensure only one dscount level s eventually appled, whch mples that for each product only one of qj, j = 1,..., k, could be non-zero. Constrants (6) gve the relatonshp between acquston quantty and determnstc demand, as the defnton on z. Constrants (7) gve the bounds on z. Constrants (8) are nonnegatve and ntegral constrants. The formulae gven by (1) to (8) s a MINLP model. It s hard to obtan the exact optmal soluton to the problem, especally for large scale nstances. In the next secton we propose a Lagrangan based approach to solve the problem. 9

11 4. Soluton Approach The Lagrangan based approach conssts of the followng three phases: frst, we construct the Lagrangan relaxaton problem by relaxng the budget constrant (2); second, the Lagrangan relaxaton problem s solved by bsecton algorthm. The soluton obtaned may volate the budget constrant (2). Thus, n the last phase, a feasblty algorthm s developed to construct a feasble soluton. Detals of each phase are presented n the followng. 4.1 Lagrangan Relaxaton By ntroducng a Lagrange multpler λ, we relax the budget constrant (2) and construct the followng Lagrangan relaxaton problem: LR Max 1 I I k z A 1 j1 B p D p u s z u f u du p D p z g u z f u c j q j B G I k 1 j1 c jqj z du (9) subject to (3)-(8). Then the relaxed problem can be decomposed nto I sngle-product subproblems: Subproblem LRp : LRp Max z p A B D p u s z u f u du p D p z g u z f u k 1 j1 c j q j z du (10) subject to (3)-(8). Substtutng (10) nto (9), the relaxed problem can be wrtten as Max LR I 1 LRp B G 10

12 subject to (3)-(8) Propertes for sngle product newsvendor prcng problem In subproblem LRp, constrant (5) ensures that only one dscount segment can be selected. It mples that the soluton to subproblem LRp must locate n one nterval of k dscount levels. Thus, we can solve subproblem LRp by solvng the k subproblems and each of them s assocated wth one prce level. Then the best soluton of the k subproblems s the optmal soluton of subproblem LRp. Actually, as Proposton 2 shows later on, t s unnecessary to solve all subproblems n many stuatons. For prce level j, we have the followng subproblem LRp j : LRp j Max c j z A B p D p u s z u f u du p D p z g u z f u 1 D p z z du (11) subject to D D L p z d (12) j U p z d (13) j A z B, (7) p 0. (8) We frst ntroduce two Lemmas for the objectve functon of LRp j, whch present the soluton approach for the problem constrants (12) and (13). LRp j wthout consderng the dscount nterval Lemma 1. For a fxed z, the optmal sellng prce to maxmze the objectve functon LRp j s determned unquely as a functon of z : 11

13 where Substtutng p j j j p z p 0 z, 2b z B u z f udu and z j p 0 a b c j 1. 2b Lemma 1 has been ntroduced by Petruzz and Dada (1999). p j p j nto these condtons can be descrbed as follows. z LRp j, the optmzaton becomes a maxmzaton over j the sngle varable z : Maxmze LRp j z, p z. Petruzz and Dada (1999) present the z suffcent condtons for the unmodalty of the objectve functon LRp j. For our problem, Lemma 2: If b c c 2g A 0 a and the j j F s a dstrbuton functon z satsfyng the condton 2r z 2 dr z /dz 0 for A z B, where r z f z 1 F z and there s an unque s the hazard rate, then functon z n the regon j A,B that satsfes j LRp j z, p z s unmodal n z, j dlrp j z, p z dz 0. All the followng propostons and algorthms are developed based on the assumpton that the condtons n Lemma 2 are satsfed. Lemmas 1 and 2 provde a way to fnd the optmum n the regon A,B for functon j LRp, but mostly not for the subproblem LRp j snce the soluton satsfes constrant (7) but may volate constrant (12) or (13). We call the soluton realzable f t satsfes constrants (12) and (13). The followng propostons are presented for the stuaton when the soluton s unrealzable. We proceed to analyze the stuaton when constrant (12) s volated. As ndcated n Lemma 2, functon LRp j s unmodal n z. Thus, f constrant (12) s volated, then the optmal soluton to subproblem LRp j s obtaned at the dscount break pont, that s, D p z d L j. It follows that p a L z d j b. We defne p jl a L z d j z b and 12

14 substtute t nto functon LRp j. Then the optmzaton of functon LRp j n D z d j L p becomes a maxmzaton over the sngle varable z : Maxmze z Let and denote the optmal soluton of functon LRp j n p jl z jl We have the followng proposton. Proposton 1: When the orderng quantty s set to LRp j z, p jl z. D z d j L. p d L j, the soluton to maxmze functon LRp j s unque, and jl p a jl z L d j b where z jl s the unque soluton n the A,B that satsfes dz 0. dlrp j z, p jl z regon The proof of proposton 1 s provded n the Appendx. Proposton 1 provdes the way to solve subproblem LRp j when the soluton to functon LRp j volates constrant (12). Smlarly, we can solve subproblem LRp j for the stuaton that the soluton to functon LRp j volates constrant (13). We omt the parallel analyss snce the stuaton does not happen n our algorthm, whch s dscussed n the next secton. The followng propostons gve the relatonshps among the solutons of the subproblems LRp j, j 1,...,k,.e., wth dfferent prce levels. Proposton 2: Let LRp j denote the maxmum value of functon If the condtons for Lemma 2 are satsfed, then we have Proposton 3: The optmal solutons j 1,...,k, satsfy z j z j 1, and j z and LRp, j1 j* D p z j* j 1* j D p z 1*. LRp j. LRp j, for p to maxmze functon j j 1,...,k. LRp j, for Proofs of Propostons 2 and 3 are provded n the Appendx. Proposton 2 mples that we do not need to solve the subproblems wth hgher prce levels f the soluton to functon LRp j at a prce level s realzable. Proposton 3 shows that the optmal order quantty for lower prce level s larger than that for hgher prce level. 13

15 4.1.2 Soluton algorthm for subproblem LRp Based on Lemmas 1-2 and Propostons 1-3, an algorthm for subproblem LRp, called as Algorthm A, s developed. The man steps of Algorthm A are descrbed below and a flow chart s presented n Fgure 1. Let z and p denote the optmal solutons for subproblem LRp. Algorthm A: Step 0: Intalzaton Intalze and set j k. Step 1: Calculate the optmal j z and j p for maxmzng functon LRp j : Step 2: j z s obtaned by Lemma 2 and j p s obtaned by Lemma 1. If d L j j D p z j d U j, j S j and go to Step 4; Otherwse go to Step 3. Step 3: Calculate the optmal p jl and z jl for maxmzng functon LRp j wth D z d j L p Set accordng to Proposton 1. j j 1 and go to Step 1. Step 4:, j S j S. j j j j j j jl jl z p arg max LRp p, z, LRp p, z [Insert Fgure 1 here]. The algorthm starts wth the lowest prce. Step 1 calculates the optmal soluton for functon LRp j. Step 2 checks f the optmal orderng quantty s realzable,.e., 14

16 d L j D j j U p z d j. If so, accordng to Propostons 2 and 3 we don t need to calculate the optmal solutons to LRp j for j j S, snce ther optmal values are less than that of LRp S j. Otherwse, by Proposton 2 we know that the optmal orderng quantty at ths dscount segment s the dscount break pont. Thus Step 3 calculates the optmal p jl and z jl for maxmzng functon LRp j wth Step 1 to calculate the optmal soluton to the next dscount segment. Step 4 evaluates and compares wth all the optmal solutons of subproblem D z d j L. Then the algorthm goes back to p LRp j, for LRp S j j j S. Then the best soluton s the optmum of subproblem LRp. Algorthm A s smlar to the procedure followed to solve the sngle tem newsvendor problem wth quantty dscount (Ln and Kroll 1997), but we ncorporate the prcng decson nto the problem. As ndcated before, the algorthm does not need to consder the stuaton that the soluton to functon LRpk volates constrant (13): Usually, the upper bound on the dscount segment wth the lowest prce s a large value up to nfnte, thus the optmal orderng quantty for functon LRp s ether wthn prce range or less than k L d k. If the soluton s n the dscount range, the algorthm stops; otherwse go to the second lowest prce. Accordng to Proposton 3, we know the soluton to functon LRp j must satsfy constrant (13). 4.2 Solvng the Lagrangan Dual Problem For a gven value of λ, the Lagrangan relaxaton problem provdes an upper bound to the orgnal problem. Lagrangan Dual Problem s to fnd the optmal Lagrangan multpler that mnmzes the upper bound Bsecton algorthm We frst set λ=0 and solve subproblems LRp, for =1,,I, by Algorthm A. If 15

17 I k 1 j1 c j q j B G, t ndcates that the capacty constrant (2) s non-operatve, and the optmal solutons wth λ=0 are optmal to the orgnal problem. Otherwse, we need to solve the Lagrangan dual problem to fnd the optmal Lagrange multpler to mnmze the upper bound. To solve the dual problem, the bsecton teraton algorthm s ntroduced as follows: Step 0: Set 1 0 and 2 max ( max s explaned n the next secton). Step 1: Let Algorthm A, and get ther optmal solutons Step 2: Calculate ( 1 2 )/2, solve all the subproblems B error. z and p. LRp for 1,..., I, by Step 3: If absb error 1 or Step 4: If B error 0, then set abs 1 2 2, then Stop. 1 ; else set 2. Go to Step 1. I k Where, we defne B error c j q j B G. 1 and 2 are parameters for stop crtera. In 1 j1 our case, 1 1 and Our computatonal experments show that: for small scale problems such as nvolvng less than 20 products, the algorthm stops wth condton absb error 1 ; for large scale problems such as nvolvng hundreds of products, the algorthm stops wth condton abs It mples that the bsecton algorthm can obtan the optmal solutons for small scale problems, but for large scale problems, the dual soluton from the bsecton algorthm may volate budget constrant (2). A feasblty algorthm s needed to construct a feasble soluton when the budget constrant s volated Observatons In each teraton of the bsecton algorthm, Algorthm A s repeatedly employed to solve subproblems LRp for 1,..., followng nonlnear equatons should be solved: I. From Algorthm A we can see that a number of the 16

18 j dlrp j z, p z dz 0, j j S,...,k, (a) dlrp j z, p jl z dz 0, j j S,...,k. (b) Equaton (a) s used to fnd the optmal soluton to maxmze functon LRp j, whle equaton (b) s used to get the optmal soluton to maxmze functon LRp j n D z d j L, that s, the optmal soluton at dscount break pont j. It s tme-consumng to p solve these nonlnear equatons. The followng property of equatons (b) can used to reduce ther solvng tmes. Observaton 1: The solutons of equatons (b) are ndependent of the Lagrange multpler. Snce dlrp j z, p d j L dz z d L j a 2z b g s F z 1 z uf udu d L j b A b g, the equatons (b) are ndependent of the Lagrange multpler. Observaton 1 mples that the optmal sellng prce and the optmal value of z at the prce break pont are constant and they don t vary wth the change of. We only need to solve equaton (b) at most one tme for each product at each dscount break pont. Observaton 2: There s an upper bound on the Lagrange multpler. In terms of condtons n Lemma 2, we should keep a b c j c j 2g A 0 for all, j, n order to make sure that every equaton (a) has an unque root, that s, a A 2g b b c j 1,, j. Let max mn a A 2g b b c j 1, j, whch gves the upper bound on. As ponted by Lau and Lau (1995, 1996), when the budget capacty s too small, some orderng quanttes durng the process of the bsecton algorthm may be negatve, whch are 17

19 nfeasble. Thus, n each teraton, the orderng quanttes for all products, q D p z for =1,,I, should be checked and the negatve ones are set to zero. 4.3 Feasblty Algorthm The soluton obtaned by the bsecton algorthm may volate the budget constrant. We have I k defned that B error c j q j B G. If B 0, then the soluton s nfeasble. Whle 1 j1 er r or B error0 mples the soluton s feasble but the budget s not suffcently utlzed. Hence, we develop a feasblty procedure to ether adjust the dual soluton to be feasble or mprove the soluton. The feasblty algorthm s descrbed as follows. Step 0: Sort the products n the descendng order n terms of unt acquston cost. Step 1: If B 0, decrease the acquston quanttes of the products n the order untl er r or the total budget reaches ts balance. Step 2: If B error0, ncrease the acquston quanttes for the products n the reverse order untl the budget s fully utlzed. Step 3: Recalculate the optmal sellng prces for the adjusted products by Proposton 1. The basc dea of the feasblty algorthm s straghtforward. Each product has an upper bound on the acquston quantty, whch s the optmal order quantty wthout the budget lmtaton. The acquston quantty for each product adjusted n Step 2 cannot be more than the upper bound. For most of the cases, the balance s reached by adjustng only one product s acquston quantty snce the Lagrangan dual solutons are very near to the optmal soluton. Fgure 2 shows the complete structure of the Lagrangan based approach, whch manly conssts of bsecton algorthm and feasblty algorthm. [Insert Fgure 2 here] 18

20 5. Numercal Example and Computatonal Results The proposed approach s tested on the randomly produced examples. The algorthms are mplemented wth Matlab. The computatonal experences for all the examples are conducted on the IBM T60 laptop wth Wndows XP (Intel Core 2 Duo CPU, 1GB of RAM). 5.1 Numercal example Ths secton presents a numercal example to llustrate our procedures. We nvestgated the supply and random prce-dependent demand of fashon clothes such as prntable garment at a retal store n Chna. Ths example s desgned based on the nvestgaton data from the acquston and prcng process of sweatshrts wth fve styles. The avalable budget (B G) for the sweatshrts s $ 125,000 and the random part ( u ) of the prce-dependent demand s assumed to follow the normal dstrbuton wth a mean of zero. The suppler offers three dscount segments: less than 2000, from 2000 to 4000, and over Other parameters for the example are shown n Table 2. [Insert Table 2 here] We apply the proposed Lagrangan approach to the nstance. We frst solve the Lagrangan dual problem wth the bsecton algorthm. In the dual soluton, the budget requred , whch s slghtly less than the avalable budget of 125,000. Thus the dual soluton s feasble. The upper bound obtaned by the dual soluton s 184,725.75, whle the proft of the feasble soluton s $184, The relatve gap between the feasble soluton and the bound, defned as (upper bound-lower bound)/lower bound, s 1.67E-07, whch s very small. Thus the feasble soluton s very close to the optmal soluton. Snce the budget requred almost reaches the lmt and the gap s so small, t s unnecessary to employ the feasblty algorthm to adjust the dual soluton for ths nstance. The optmal order quanttes and sellng prces for the example are presented n Table 3. [Insert Table 3 here] 19

21 5.2 Manageral analyss and comparsons We use the above example to nvestgate the relatonshps between the solutons and crtcal parameters to gan some nsghts to the jont orderng and prcng problem. The relatonshp between the expected proft and the budget capacty We observe the change of the expected proft by varyng the budget capacty from 115,000 to 175,000 whle the other parameters are fxed. The relatonshp between the proft and the budget capacty s shown n Fgure 3. It can be seen that the total expected proft ncreases when the budget capacty ncreases, but t keeps unchanged beyond about 145,000. It ndcates that the proft can be ncreased by addng the avalable budget, but does not ncrease any more after a certan pont, snce the addtonal budget s not fully utlzed due to the lmtaton on demands. [Insert Fgure 3 here] The acquston polcy versus the standard devaton of a product s demand To observe mpacts of demand uncertanty on the acquston polcy, Fgure 4 llustrates how the acquston quanttes of the products change when the standard devaton of product 1 s demand vares from 200 to 2,000. It can be seen that the orderng quanttes for products 4 and 5 keep unchanged, whle the orderng quantty for product 1 ncreases and the orderng quanttes for products 2 and 3 decrease, when the standard devaton ncreases from 200 to 800 and from 1400 to It mples that the retaler would shft the budget from the products wth lower rsk to the products wth hgher rsk. It concdes wth the result of classcal newsvendor problem when the order quantty s above the demand average. But when the standard devaton vares from 800 to 1,400, the orderng quanttes for all the products keep unchanged. Ths s because the orderng quantty for product 1 s at the dscount break pont 4,000. It shows that the acquston polcy s less senstve to the uncertanty of demand when the orderng quantty s at the dscount break pont. 20

22 [Insert Fgure 4 here] The prcng polcy versus the standard devaton of a product s demand We also observe how demand uncertanty affects the prcng polcy by varyng the standard devaton of product 1 s demand from 200 to 2,000. From Fgure 5, t s nterestng to see that the prce of product 1 decreases when the standard devaton vares form 200 to 800 and from 1,400 to 2,000 respectvely, and ncreases when the standard devaton vares from 800 to 1,400. Note from Fgure 4 that the order quantty of product 1 ncreases when the standard devaton vares form 200 to 800 and from 1,400 to 2,000. Thus retaler reduces the sellng prce to nduce demand ncrease. When the standard devaton vares from 800 to 1,400 the product 1 keeps the orderng quantty unchanged at 4000, whch s the dscount break pont. In the stuaton the retaler reduces the understockng rsk through ncreasng the sellng prce. [Insert Fgure 5 here] Comparson of the solutons of dscount case wth that of non-dscount case Snce the problem wthout dscount s a specal case of the problem wth dscounts, the non-dscount case can also be solved by the Lagrangan based soluton approach, and the solutons are presented n Table 3. The optmal proft s $ 175,191.11, whch s a lttle less than that of the dscount case. Comparng the solutons for the two cases, we can see that the orderng quanttes n the dscount case are more than that of the non-dscount case, whle the sellng prces n the dscount case are less than that of the non-dscount case. It ndcates that the suppler quantty dscounts can promote the retaler orders more products, and the retaler can ncrease the proft by offerng lower sellng prces to customers. Therefore both supplers and retalers can beneft from the suppler dscount polcy. 5.3 Performances of the Soluton Approach In order to further test the performance of the soluton approach, thrty test problems 21

23 wth the szes of 20, 200 and 1000 products are randomly produced based on our nvestgaton data, and then solved by the proposed soluton approach. Table 4 llustrates the runnng tmes and soluton gaps for the problems. Accordng to Table 4, the maxmal relatve gap for the small sze examples s 9.59E-04 whle the average gap s 2.21E-04, and the computatonal tmes for all the small examples are wthn 2 seconds. For the mddle sze examples, the maxmal and average gaps are 2.00E-04 and 2.01E-05 respectvely, and the computatonal tmes are less than 16 seconds. For the large sze examples, the maxmal and average gaps are 3.86E-05 and 6.20E-06 respectvely, whch are less than that of small and mddle sze problems. The runnng tme for the problem wth 1000 products s no more than 80 seconds. It s concluded that our soluton approach can present very good solutons for all scale examples n a short computatonal tme. [Insert Table 4 here] 6 Conclusons Ths paper nvestgates the mult-product acquston and prcng problem when uncertan demands and suppler quantty dscounts are present. We llustrate that the problem s an extenson of the newsvendor prcng problem. It s the frst work to combne suppler dscounts wth the constraned mult-product newsvendor prcng problem. The combnaton makes the problem more practcal and challengng. Through the proposed MINLP model and soluton approach, ths research provdes the retaler an effectve way to use both acquston and prcng decsons as levers to better match demand and supply, and ncrease the proft under the crcumstance. We analyze the propertes of the newsvendor prcng problem wth suppler quantty dscounts. Based on the propertes, we develop a Lagrangan based soluton approach. The bsecton algorthm s appled to solve the Lagrangan dual problem to obtan an upper bound. Thrty numercal examples are randomly produced for testng the approach. The 22

24 computatonal results show that the proposed soluton approach can provde hgh qualty solutons n terms of the relatve gaps, n a very short tme. Through a numercal example, we also compare the solutons of two cases wth and wthout the dscounts. It s found that the dscount schemes have mportant mpacts on the newsvendor s acquston and prcng polces: order more from supplers and offer lower prces to customers, whch beneft supplers and also ncrease newsvendor s proft. In ths paper, we only consder the retaler s budget constrant, whle n practce the retaler may have multple resource constrants. Thus, a natural extenson of our study s to consder the problem wth multple constrants. In ths scenaro, bsecton algorthm s unsutable, and the subgradent algorthm can be employed to solve such more complcated problem. Another extenson to our problem s to take nto account other dscount schemes, such as ncremental quantty dscount, volume dscount, and bundle dscount, whch are also common n current busness practce. Then new models and soluton approaches need to be nvestgated. Acknowledgment Ths research s partally supported by NSERC Dscovery grant, CFI fundng, and Chna State Scholarshp Fund. We would lke to thank the anonymous revewers for ther comments and suggestons that helped mprove ths paper. Appendx Proof of Proposton 1. LRp j z, p jl z B z z A L a z d j d L j z u s z u f u du b L a z d j b d j L g u z f u du c j 1 d L j. The frst and second dervatves of LRp z, p jl z are as follows: 23

25 dlrp j z, p jl z dz d L j a 2z b g s F z 1 z uf udu d L j b A b g, and d 2 LRp z, p jl z 2 F 2 z a z d L j g s f dz b b 2 F z a z d L j g s f z 0. b b z Thus the frst dervatve dlrp j z, p jl z dz s a monotoncally decreasng functon of z. Snce d L j B, dlrp j B, p jl B dz 2d L j a 2B g s 0, b and dlrp j A, p jl A dz d L j s 0. b Therefore there s a unque dlrp j z, p jl z dz 0. Proof of proposton 2: Let z j and p j z jl n the regon denote the optmal solutons to maxmze functon A,B that satsfes LRp j, for j 1,...,k. Then we have LRp j = LRp j z j j, p. Snce LRp j z j j, p LRp j z j1 j1, p and LRp j z j1 j1, p LRp, j1 z j1 j1 j1 j1, p c, j1 c j Dp z 0, we have Let LRp j z j, p j Proof of proposton 3: LRp, j1 z j1 j1, p. 24

26 for R j z dlrp j j z, p dz j 1,...,k. z 1 c j g a b c j 1 g s z 1 F 2b 2b 1 F c j 2 z g a g s z 1 F z, 2b 2b z As know that Snce R j z j z j 1 and p j j R j z 0, for R j Furthermore z j 1 are the optmal solutons to j 1,...,k. j 1 R, j 1 z 1 F z 2 j 1 R, j 1 z 0. R j j 1 LRp j, for A a b c j c j 2g A 0. 2b c j c, j 1 0, j 1,...,k, from lemma 2 we Therefore the unque root z j j z 1. Let q j j D p z j, Substtute p j j p Functon j z nto z j j 1,...,k. q j a b c j1 2 q j for R j belongs to range z q j, and we obtan z j 2 z a b c j 1 z 2 2 z j. j 1* A,z, that s, z monotonously ncreases n z, for j 1,...,k, as dq j z 1 F z dz 2 z j j z 1 j j, thus j j 1 z q z q. 0. b, cj c j j j j j j j1 j1 j1 q z q z 0, therefore z q z 2 q. j j* j1 j1* j j j1 j1 We can get q z q z, that s, p z D p z D. 25

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