Research Article Strategic Wholesale Pricing and Commonality Strategy in a Supply Chain with Quality Segmentation

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1 indawi Publihing Corporation Matheatical Proble in Engineering Volue 015 Article ID page Reearch Article Strategic Wholeale Pricing and Coonality Strategy in a Supply Chain with Quality Segentation Tiantian Xu Tiaojun Xiao and Chen Tian School of Manageent Science and Engineering Nanjing Univerity Nanjing China Correpondence hould be addreed to Tiaojun Xiao; xiaotj@nju.edu.cn Received 6 Septeber 014; Accepted 19 March 015 Acadeic Editor: Ben T. Nohara Copyright 015 Tiantian Xu et al. Thi i an open acce article ditributed under the Creative Coon Attribution icene which perit unretricted ue ditribution and reproduction in any ediu provided the original work i properly cited. We develop two gae odel of a one-upplier and one-anufacturer upply chain to invetigate the upplier trategic wholeale pricing deciion and the anufacturer coonality trategy. The anufacturer ha three coonality trategie for the highend and low-end product: coon high-quality coponent coon low-quality coponent and dedicated coponent. We conider both wholeale price firt cenario and coonality trategy firt cenario. Under the wholeale price firt cenario we identify the range of each coonality trategy and find that (i) the coon low-quality coponent trategy i harful to the upplier; (ii) if the quality of low-quality coponent and the unit production cot of high-quality coponent are ufficiently low the upplier induce the coon high-quality coponent trategy by trategically decreaing the unit wholeale price of highquality coponent while if they are ufficiently high the upplier induce the dedicated coponent trategy by increaing the unit wholeale price of high-quality coponent and decreaing that of low-quality one. Under the coonality trategy firt cenario the coon low-quality coponent trategy ay exit. By coparing the two cenario we find that (i) if the unit production cot of low-quality coponent i ediu the equilibriu outcoe under both cenario are identical; (ii) there exit a firt-over advantage for the two player. 1. Introduction In reality conuer are heterogeneou following their valuation over product quality. To target different conuer anufacturer provide variant product with different quality level. For exaple enovo offer two ain erie IdeaPad and ThinkPad laptop and provide ix different type for each erie. When deigning product line one of the iportantprobleforanageriwhethertoueacooncoponent acro different product or not. In practice coon coponent ha been widely ued a a buine trategy; for exaple General Motor ue the ae engine and platfor acro exu and Cary [1]. In 013 Apple ued Cellular and Wirele Diplay eadphone and Connector a coon coponent within the new product iphone 5 and iphone 5c. Soe anufacturer purchae key coponent fro the uptrea upplier and then ue the to aeble different product with their own aterial and production kill. When purchaing key coponent the anufacturer ue dedicated coponent to differentiate the product or adopt a coon coponent for a high quality-price perforance. For exaple in coputer indutry Tohiba ue the ae type of Intel CPU to produce coputer uch a 40- AC05W1 and C40-Q-AT01W1 together with Tohiba own hard dik. owever when releaing a new product T40 enovo offer two verion with different Intel CPU: I3 and I5. Product line configuration of the anufacturer affect product ubtitutability a well a the price copetition between the product within a product line. Further it affect the uptrea upplier wholeale price deciion. In thi paper we will invetigate the trategic wholeale pricing deciion of the upplier. The extant literature about coon coponent conider the anufacturer coonality trategy [] where the upplier wholeale price behavior i ignored. owever the coonality trategy affect the order quantitie of key coponent a well a the profit of the upplier providing key coponent. When the anufacturer chooe a coonality trategy to axiize hi profit the upplier benefit ay be hared. Thu the upplier will trategically adjut her

2 Matheatical Proble in Engineering unit wholeale price to induce the anufacturer to chooe the coonality trategy that axiize the upplier profit. The anufacturer coonality trategy coplicate the upplier wholeale price deciion. Thi paper will invetigate how the upplier trategically ake the wholeale price deciion when the anufacturer decide whether to ue a coon coponent. To be pecific we develop two gae odel of a oneupplier and one-anufacturer upply chain to invetigate the upplier wholeale pricing trategy and the anufacturer coonality trategy where the anufacturer produce two product. Each product i aebled fro two key coponent where one i purchaed fro the upplier while theotheriproducedin-houe.theanufacturerdecide whethertobuytheaecoponent.iftheanufacturer chooe a coon coponent then the anufacturer decide to chooe the coon high-quality coponent or the coon low-quality one fro the upplier. We derive the deandfunctionfrotheconuer utilitywhichiinfluenced by the quality level and price of the product. We conider two cenario: wholeale price firt (WPF) cenario and coonality trategy firt (CSF) cenario. Under WPF cenario the upplier trategically offer a unit wholeale price to induce a coonality trategy. We identify the range of each coonality trategy and find that the coon low-quality coponent trategy i not an equilibriu becaue product differentiation can iprove the upplier profit uch that the upplier trategically offer a unit wholeale price to induce dedicated coponent trategy. Under CSF cenario the anufacturer chooe coonality trategy before the upplier offer the unit wholeale price. We give the range of uing a coonality trategy and find that the coon lowquality coponent trategy exit when the unit production cot of the low-quality coponent i ediu which i inconitent with that under WPF cenario; the upplier profit under WPF cenario i higher than under CSF cenario while the anufacturer profit under WPF cenario i lower.. iterature Review Thi paper i cloely related to the literature that exaine the iplication of coon coponent in product line deign proble. Whether to ue coon coponent acro product [ 3] and uing which kind of coon coponent [4] are two very iportant deciion. Berntein et al. [5] deontrate that the coonality deciion will not affect equilibriu deciion when the coponent i ued in the whole line while it can influence the optial deciion when the coponent i ued in part of line. Coon coponent ay benefit the low-end product while har the high-end one [6]. Uing coon coponent ay be accopanied with econoie of cale [7 8] and deign cot reduction effect [1 9]. Deai et al. [1] jut conider one kind of deign cot while eee and Swainathan[9] replenih their work with everal cot for. Subraanian et al. [10] tudy the ipact of reanufacturing on the coonality trategy. There are two review done by abro [11]andFixon[1]. The above odel focu on coonality iue fro the anufacturer perpective and tudy the trade-off between the cot reduction and cannibalization effect. owever we tudy the coonality trategy fro the upply chain perpective epecially the vertical interaction between the upplier and the anufacturer. Our paper i alo related to product line deign proble. Within a product line product can be vertically or horizontally different. There are odel that integrate both horizontal and vertical differentiation to olve the product line deign proble [1 8 13]. For the horizontal one oe reearcher how the optial variety level through aking the trade-off between arket deand and production cot [14 16]. The other conider the product line deciion in the upply chain etting to how the relationhip between variety and double arginalization effect [17 19]. The vertical differentiationodelipioneeredbymuaandroen[0] and extended by Moorthy [1]. After that variou product line deign proble of the anufacturer are conidered uch a facing a downtrea ditributor [] coon coponent [1 8] pecial developent intenive product with fixed cot [3] and different production cot [4]. Thee odel ainly aue that there are two exogenou arket egent. Market egentation can be influenced by oe arket tool uch a product poitioning [5] and operationrelated cot [6]. We invetigate when a coon coponent hould be ued in a upply chain where there are two arket egent. Thi paper i alo related to the trategic wholeale pricing deciion. Although the wholeale price i a very coon deciion in a upply chain odel trategic wholeale pricing deciion i rarely tudied. Strategic wholeale pricing ean that the upplier can trategically offer a unit wholeale price to induce the downtrea fir or the potential entrant upplier to chooe a trategy that i beneficial to the upplier. For exaple the upplier can trategically chooe the unit wholeale price to deter or allow the entry of the potential entrant upplier [7]. That i the uptrea upplier can induce the downtrea fir to chooe a trategy through adjuting the unit wholeale price. In thi paper we invetigate how the upplier offer a trategic wholeale pricing deciion to induce a coonality trategy. In uary thi paper contribute to the literature by conidering the trategic wholeale pricing and coonality deciion in a one-upplier and one-anufacturer upply chain. We invetigate how the upplier ake the wholeale price deciion when facing a downtrea anufacturer who ha three potential product line configuration trategie. We conider WPF cenario and CSF cenario. We exaine the effect of gae equence on equilibriu outcoe and profit by coparing the two cenario. 3. The Baic Model (Wholeale Price Firt (WPF) Scenario) Conider a upply chain coniting of one upplier (he) and one anufacturer (he) where the anufacturer purchae

3 Matheatical Proble in Engineering 3 one of key coponent denoted by A frotheupplier. The anufacturer produce the product with coponent A and hi own coponent B. Totargetdifferentconuer egent the anufacturer provide high-end product () and low-end product () We refer to product a egent i i =.FollowingDeaietal.[1] we aue that the product quality perceived by conuer can be expreed a q i i=whereq i i the product quality of egent i; q >q >0.Providingqualityq Ai for coponent A in egent i incur a arginal cot c Ai to the upplier; c A >c A.To focu on trategic factor we noralize the anufacturer arginal cot for coponent B in egent to zero. et the anufacturer arginal cot for coponent B in egent be c B >0. The anufacturer can chooe one or two type of coponent A to finih the production of the end product together with hi own coponent B. The anufacturer ha three coonality trategie: (a) coon low-quality coponent A ; (b) coon high-quality coponent A ;(c)two different quality level for coponent A to provide dedicated product (dedicated coponent). For exaple we ue upercript to repreent coonality trategy. The quality of the end product depend on it configuration profile.theproductqualitycanbeperceivedbyconuera q i =ω A q Ai +ω B q Bi i=whereω A and ω B are iportance weight that conuer endow to coponent A and B repectively. If the anufacturer chooe trategy the qualitie are q i =ω A q A +ω B q Bi while quality level can be expreed a q i =ω A q A +ω B q Bi if the anufacturer adopt trategy.sowehaveq =q q =q andq q =ω B (q B q B )=q q. We conider a arket in which conuer have heterogeneou valuation over quality. et the quality valuation of conuer be θ and the retail price of product i be p i. The utility function of the conuer buying product i i U(θ p i q i )=θq i p i. We noralize arket ize to one. Following Chayet et al. [5] and Yu[6] we aue that quality valuation θ i uniforly ditributed over the interval [0 1]. A conuer buy a product when the conuer can achieve the axial poitive utility. The conuer buy the high-end product only when θq p θq p and θq p 0 where θq p θq p i equivalent to θ θ 1 = (p p )/(q q ) and the conuer buy the low-end product when θq p θq p and θq p 0; explicitly θ (p /q θ 1 ).SiilartoYu[6] we can depict Figure 1. Defineθ = p /q (> 0). Froθ > θ 1 > θ it follow that θq p 0; that i the participation contraint of the conuer with θ>θ 1 i atified. The conuer with θ [θ 1 1]will buy the high-end product while the conuer with θ [θ θ 1 ) will buy the low-end product. Further the deand for the high-end product i D =1 θ 1 and that for the low-end one i D =θ 1 θ. 0<θ <θ 1 <1hould be atifiedtoenurebothproducthavepoitivedeand. Fro the definition of θ 1 and θ weobtainp =q θ 1 + q (θ θ 1 ) and p = θ q.thuwecanubtituteθ 1 and θ for p and p a the deciion variable. If the anufacturer chooe trategy (i.e. uing low- (high-) quality U(θ) U(θ) = θq p U(θ) = θq p 0 θ θ 1 1 p p Figure 1: Utility of conuer veru quality valuation. coponent A and B to produce low- (high-) quality product) the upplier profit i π (θ 1 θ w A w A ) =(1 θ 1 )(w A c A )+(θ 1 θ )(w A c A ) and the anufacturer profit i π (θ 1θ w A w A ) =(1 θ 1 )[q θ 1 +q (θ θ 1 ) w A c B ] +(θ 1 θ )(θ q w A ). If the anufacturer chooe trategy the upplier profit i π (θ 1 θ w A )=(1 θ )(w A c A ) (3) where 1 θ repreent the total purchaing quantity for uing coon low-quality coponent A. The anufacturer profit i π (θ 1θ w A ) =(1 θ 1 )[q θ 1 +q (θ θ 1 ) w A c B ] +(θ 1 θ )(q θ w A ) where the firt-ter i the anufacturer profit fro the high-end product while the econd-ter i the profit fro the low-end one. If the anufacturer chooe trategy the upplier profit i π (θ 1 θ w A )=(1 θ )(w A c A ) (5) and the anufacturer profit i π (θ 1θ w A ) =(1 θ 1 )[q θ 1 +q (θ θ 1 ) w A c B ] +(θ 1 θ )(q θ w A ). θ (1) () (4) (6)

4 4 Matheatical Proble in Engineering According to the tie equence of the unit wholeale price and coonality deciion we divide the dicuion into two cenario: WPF cenario and CSF cenario. Under WPF cenario the anufacturer chooe coonality trategy only when he oberve the unit wholeale price. Specificallythetieequenceofthegaeiafollow. (i) The upplier deterine the unit wholeale price of coponent A(w A w A ). (ii) Oberving the unit wholeale price the anufacturer firt chooe coonality trategy or and then decide the retail price. By uing backward induction technique we can olve the ubgae perfect Nah equilibriu (SPNE). 4. Equilibriu Analyi 4.1. Reaction of the Manufacturer. Given the unit wholeale price we can obtain the retail price reaction of the anufacturer (θ 1 and θ ). Furtherore we can obtain the deand and retail price of the high-end and the low-end product which are uarized in Propoition 1. Toenurethatthe deand for each product i poitive we aue that product differentiation i very large throughout thi paper: q q > c B. According to the definition of product quality we have q q >q q =q q. Propoition 1. (i) Under trategy the deand are D D (w Aw A )= 1 c B +(w A w A ) (q q ) (w Aw A )= c Bq and the retail price are p +w A q w A q q (w A) = (c B +q (q q ) +w A) p (w A )= (q +w A ) ; (ii) under trategy the deand function are (7) (8) (iii) under trategy the deand are D (w A )= and the retail price are p D (q =D c B q (w A) =p (w A) ) w A q p (w A )= (q +w A ). (11) (1) Propoition 1 iplie that if the anufacturer ue trategy the deand of the high-end product increae with product differentiation (the difference between the product quality level). When product differentiation increae oe conuer who buy the low-end product ay turn to buy the high-end one. The deand of the low-end product increae with it quality level while it decreae with the quality of high-end product. That i when the quality level of the low-end product increae the total deand of the two product increae while the deand of the high-end one decreae. owever when the quality of the high-end product increae the deand of the low-end one decreae becaue oe conuer are attracted to buy the high-end product. When the unit production cot of coponent B increae the deand of the high-end product decreae while the low-end one increae. If the anufacturer chooe a coon coponent the deand of the high-end product only relate to quality difference and the unit production cot of coponent B.Froq =q q we ee that the deand of the q high-end product under trategy i equal to that under trategy ;thatid =D. Fro Propoition 1 we know that the retail price of the high-end product increae with it own quality the unit production cot of coponent B and the unit wholeale price. The retail price of the high-end product under trategie and are identical becaue of q = q.iftheanufacturer ue trategy theretailpriceilowerduetothe lower quality. Siilarly we can know that the retail price of the low-end product under trategie and are identical which i lower than that under trategy. The deand of both high-end and low-end product hould be poitive. ea give the valid paraeter etting. and the retail price are D = 1 c B (q q ) D (w c A) = B (q q ) w A q p +w A) (w A) = (c B +q p (w A) =p (w A ); (9) (10) ea. If the unit wholeale price atify w A [(q q ) c B] <w A < c Bq q w A q c B <w A < c Bq q q q q then the anufacturer will provide two end product. (13) Under trategy or when the unit wholeale price increae the deand of the low-end one will decreae due

5 Matheatical Proble in Engineering 5 w A w A0 w A =w A +Δw I w A = (w A B 1 )/A 1 II w A0 III w A = q /q w A +q q q w A Figure : Coonality trategy veru the unit wholeale price. to a higher retail price. Fro ea weknowthatthe unit wholeale price hould be ufficiently low (w A < (c B q /(q q )) w A < c B q /(q q ))when the anufacturer ue coon coponent. By inerting the reaction function into () (4) and(6) we can obtain the profit function of the anufacturer: π (w Aw A ) π (w A) and π (w A). ea 3 decribe the effect of the unit wholeale price on the anufacturer profit. ea 3. (i) Under trategy the anufacturer profit i a convex and decreaing function of the unit wholeale price in each egent; (ii) under trategy or theanufacturer profitiaconvexanddecreaingfunctionoftheunitwholeale price. ea 3 iplie that within the feaible region given by ea the decreaing peed of the anufacturer profit in the unit wholeale price becoe fater. The anufacturer deterine the coonality trategy baed on the profit under different condition. Define F 1 = π (w Aw A ) π (w A) F = π (w Aw A ) π (w A) andf 3 = π (w A) π (w A). Theanufacturer will ue trategy if F 1 0 and F 0the anufacturerwouldliketouetrategy if F <0and F 3 0 and the anufacturer would like to ue trategy if F 1 <0and F 3 <0. Propoition 4 uarize the coonality trategy of the anufacturer given the unit wholeale price w A w A ; alo ee Figure. Propoition 4. If (w A w A ) Itheanufacturerue trategy ; if (w A w A ) II the anufacturer will ue trategy while the anufacturer ue trategy if (w A w A ) III where region I={(w A w A ) w A +Δw w A (w A B 1 ) A 1 } II = {(w A w A ) w A > ax {w A +Δw w A }} B 1 A 1 = q = q c B w A = q (q q q q + (q Δw =((q ) (q q w A +q q )(q q q q q q q )(q (q q ) )q q q q c B )(q q ) (q q c B ) (q (q q ) 1. q )(q q )) )q q (14) Propoition 4 not only anwer how the anufacturer react to the unit wholeale price but alo how guidance about how the upplier hould et her optial unit wholeale price. Figure how that threhold w A0 and w A0 are iportant value and the anufacturer ue trategy only when both unit wholeale price are lower than thee threhold. Fro ea 3 we know that a high unit wholeale price of the high-quality coponent A will lead to a low profit for the anufacturer under trategy while it doe not change the profit of the anufacturer uing trategy. So the upplier can et a ufficiently high unit wholeale price w A to induce the anufacturer to only purchae the lowquality coponent A. Siilarly when the upplier offer a ufficiently high unit wholeale price w A theanufacturer will ue trategy. When the unit wholeale price i high enough (w A >w A0 or w A >w A0 ) the anufacturer get a higher profit fro uing a coon coponent. Fro Proof of Propoition 4 we know that the lope of the line w A =(w A B 1 )/A 1 i the larget and that of the line w A =w A +Δw i the allet. 4.. The Unit Wholeale Price of the Supplier. The anufacturer coonality trategy depend on the unit wholeale price. In thi ubection we focu on the unit wholeale price deciion. Conidering the reaction function of the anufacturer the profit function in different region (I II and III) can be expreed a follow: π I (w Aw A ) ={q (w A c A )(q q c B w A +w A ) +(w A c A )[(c B +w A )q w A q ]} III = {(w A w A ) w A < in { (w A B 1 ) A 1 w A }} [q (q q )] 1 (15)

6 6 Matheatical Proble in Engineering π II π III (w A )= [ w A +(c A +q )w A c A q ] (q ) (16) (w A )= [ w A +(c A +q )w A c A q ] (q ). (17) The upplier will axiize her profit acro all feaible region. Fro (15) (17) we can derive the following. Propoition 5. (i) The upplier ell two kind of coponent A to the anufacturer if π I (wi A wi A ) ax {πii with the SPNE unit wholeale price (w I A wi A )= { where w I A =(q (w I A1 wi A1 ) { {(w A0 w A0 ) (w II A )πiii (w III A )} (18) if (wi A1 wi A1 ) I (w I A wi A ) if wi A1 +Δw w I A1 w I A1 < (wi A1 B 1) A 1 w I A <w A0 (w I A3 wi A3 ) if wi A1 +Δw <w I A1 w I A1 (wi A1 B 1) A 1 w I A3 <w A0 otherwie ) w I A1 =wi A3 = (c A +q w I A1 = (q +c A c B ) w I A3 =wi A3 +Δw w A0 = (B 1 +A 1 Δw ) (1 A 1 ) w A0 = (Δw +B 1 ) (1 A 1 ) w I A =A 1w I A +B 1 (B 1 +c A c A c B +q q ) +A 1 ( B 1 q ((A 1 q +c Aq +q A 1 q )) 1 ; c Aq +c B q )) (19) (0) (1) (ii) if π II (wii A ) ax{πi (wi A wi A ) πiii (w III A )} theupplier only offer low-quality coponent A with the SPNE unit wholeale price w II A =(c A +q )/; theunitwholealeprice of the high-quality one would be any value that atifie w II A > ax{w II A +Δw q /q wii A +(q q q )}; (iii) if π III (w III A ) ax{πi (wi A wi A ) πii (wii A )} theupplier only offer high-quality coponent A with the SPNE unit wholeale price w III A = (c A +q )/; theunitwholeale price of the low-quality one hould atify w III ax{ q /q wiii A ( q q q ) B 1 +A 1 w III A > A }. Fro Propoition 5 we know that if the upplier provide two quality level of coponent A he deterine the unit wholeale price (w I A1 wi A1 ) if (wi A1 wi A1 ) I.Ifthe upplier can get a higher profit fro only providing one coponent he will try to induce the anufacturer to ue a coon coponent. The unit wholeale price of a coon coponent increae with it unit production cot and the quality level of the low-end product. If the quality of coponent B increae the anufacturer can get a higher unit profit o the upplier can increae the unit wholeale price to gain a larger unit profit. We copare the equilibriu outcoe when the anufacturer ue coon coponent which i uarized in Corollary 6. Corollary 6. (i) w III A D if c A/c A > q >wii A ;(ii)d +D <D + /q ;(iii)theuppliercangaina higher profit under trategy than under trategy only if q (q c A ) q (q c A). Fro Corollary 6weknowthattheunitwholealeprice for the coon high-quality coponent i higher than that for the coon low-quality one. When uing a coon coponent the deand of high-end product under trategie and are identical uch that the total deand depend on the low-end one. The retail price of the lowend product increae with the unit production cot of the low-quality coponent A uch that ore conuer do not buy any product. On the other hand a higher product quality for the low-end product attract ore conuer. A a reult the total deand under trategy i higher than that under trategy iftherelativeunitproductioncotoflow-quality coponent A i ufficiently low. Since the optial wholeale price of the upplier under dedicated coponent are coplex we illutrate the onotonicity of the unit wholeale price by eploying nuerical exaple. Figure 3 7 how how the quality and the unit production cot ay affect the optial wholeale price and the coonality trategy. ere the default value of the paraeter are ued a: c A =1.5 c A =.5 c B = q A = 0 q A =5 q B =1 q B =18 ω A =0.6andω B = 0.4. Fro Figure 3 and 4 we know that the upplier profit under trategy i higher than that under trategy which iplie that the upplier ha no incentive to offer a unit wholeale price to induce trategy. Intuitivelythe lowquality coponent A ha a lower cot which i beneficial to

7 Matheatical Proble in Engineering π III ( wa III ) π I ( wi A wi A ) w A Profit π II ( wii A ) q A Wholeale price w A 1.6 Figure 3: The upplier profit veru the quality of coponent A in egent q B =18 q B =0 c A Figure 6: The unit wholeale price veru q B and c A π III ( wa III ) Profit π I ( wi A wi A ) c A π II ( wii A ) Figure 4: The upplier profit veru the unit production cot of high-quality coponent A. Wholeale price wa wa q B Wholeale price w A c B = c B =.3 w A q A Figure 5: The unit wholeale price veru q A and c B. the upplier. On the other hand the quality of the high-end product under trategy i lower than that under trategy which i harful to the upplier due to a lower arket deand. Moreover uing trategy decreae product differentiation which increae the price copetition. A a reultrelativetotrategy the negative effect of uing trategy on the upplier outweigh it poitive effect. The upplier will encourage the anufacturer to ue trategy or. c A = 1.5 c A = Figure 7: The unit wholeale price veru q B and c A. Figure 3 iplie that the upplier will induce trategy if the quality of the low-quality coponent A i ufficiently low;otherwietheupplierwillinducetrategy. Intuitively under trategy a higher quality of coponent A in egent increae the upplier profit becaue it reduce the quality diadvantage. Moreover product differentiation under trategy i larger than under trategy. A a reult the upplier profit under trategy i higher if the quality of coponent A in egent i ufficiently high. Figure 4 iplie that the upplier ha an incentive to induce trategy if the unit production cot of coponent A in egent i ufficiently low. Intuitively the negative effect of the unit production cot c A on the upplier under trategy i larger than that under trategy uch that the upplier can obtain a higher profit under trategy if it i very all. Moreover a pointed out earlier the upplier prefer trategy to trategy. Figure 5 7 illutratetheipactofthequalityandunit production cot on the optial wholeale price. In Figure 5 the firt interval atche trategy while thefollowing one relate to trategy. Figure 5 iplie that when the quality

8 8 Matheatical Proble in Engineering of coponent A in egent i ufficiently high the upplier will trategically increae the unit wholeale price of coponent A in egent and decreae that in egent which induce the anufacturer to chooe trategy. In addition we find that the unit wholeale price doe not increae with the unit production cot of coponent B. In Figure 6 the two interval atch trategie and repectively. The upplier will be reluctant to induce the anufacturer to ue when the unit production cot of coponent A in egent i ufficiently high. When the unit production cot increae the upplier charge a higher wholeale price for high-quality coponent A. In order to encourage the anufacturer to ue trategy the upplier alo raie the unit wholeale price of the low-quality coponent. The firt interval relate to trategy while the econd one correpond to trategy in Figure 7. When the quality of coponent B in egent i high enough the upplier will induce the anufacturer to ue trategy ;the upplier trategically decreae the unit wholeale price for the high-quality coponent A to induce trategy ; however whether to increae the unit wholeale price for the lowquality coponent A depend on it unit production cot. 5. Equilibriu Outcoe under Coonality Strategy Firt (CSF) Scenario Under CSF cenario the upplier et the unit wholeale price after the product line configuration i deigned [8]. Specifically the tie equence of the gae i a follow. (i) The anufacturer chooe coonality trategy: or. (ii) Oberving coonality trategy the upplier deterine the unit wholeale price. (iii) Oberving the unit wholeale price the anufacturer et the retail price. Siilar to Propoition 5 wecanhowthatthespne deciion are the ae a thoe under WPF cenario if the anufacturer ue a coon coponent A;thatitheunit and that under trategy i w III A. Under trategy theunitwholealeprice are (w I A1 wi A1 ).Notethatiftheanufactureruetrategy the upplier charge the ae wholeale price a that under trategy ; if the anufacturer ue trategy he will pay ore for the high-quality coponent A. Corollary 7 give out the condition under which the equilibriu deciion under WPF and CSF cenario are identical. wholeale price under trategy i w II A Corollary 7. If c A [q q c B Δw +c A (q + c A c B )A 1 +B 1 q ]productlinedeignandpricing deciion under WPF and CSF cenario are identical. Corollary 7 iplie that the tie equence of deciion doe not affect the equilibriu deciion if the unit production cot for the low-quality coponent A i ediu. The ain caue i that within the range the upplier can optiize her wholeale price under WPF cenario becaue the anufacturer ha an incentive to chooe trategy. Thu thetieequenceofdeciiondoenotaffectthepricing deciion under trategy. Moreover we know that the tie equence doe not change the pricing deciion under the other trategie. A a conequence the pricing deciion and coonality trategie under two cenario are identical. owever if the unit production cot c A i either too high or low the tie equence affect the pricing deciion a well a coonality trategy. Under CSF cenario the anufacturer profit are π =(q π π [4 (q c A (q c B) +q (16q (q q )) 1 =(q [4c B +(q c A (q (16q =(q [(q c A)(q q )) q (8c B 7q +3q )] ) (4q c A )(q q )) (q q )) 1 8c B 3q )] c B) +q (c A +c B q ) +c A (c A +c B q )] c A (c A q (16q (q q )) 1. +c B q c A q )) () ere the upercript repreent CSF cenario. Define Δπ 1 =π π Δπ =π π and Δπ 3 = π π.whenbothδπ 1 and Δπ are negative the anufacturer will ue trategy. Propoition 8 uarize the coonality trategy. Propoition 8. Under CSF cenario we have the following. (i) The anufacturer chooe trategy for c A [c A1 in{c A1+ c A3 }]. (ii)theanufacturerchooetrategy if c A± exit and c A [ax{c A c A3 } c A+ ]. (iii) Otherwie the anufacturer chooe trategy where c A1± =c A +c B q +q ±(q q c B) (q q ) (q q )

9 Matheatical Proble in Engineering 9 c A± = B ± B +4A q c A3 =q (q q q q c A ) q (q q ) (q q ) q (3) π π π A =q {q [c B (3q 6c B (q +3(q c A c B q +c A (q +q 4q ) q q )(q ) (q (q q ) q )(q q ) q )] q q )} Figure 8: The upplier profit veru the unit production cot of the low-quality coponent A c A B =q q (c A +c B )(q q ). (4) π π Propoition 8 identifie the condition under which the anufacturer chooe a coonality trategy. Fro Propoition 8 we can ee that trategy i poible under CSF cenario. Thi reult i inconitent with that under WPF cenario in a tatitical ene (ee Figure 3 and 4) which iplie that the tie equence of deciion affect the exitence of trategy. Specifically trategy i ore prevalent under CSF cenario. To better undertand the effect of the tie equence of deciion on the player profit we depict Figure 8 and 9 where the default value are the ae a in Figure 3 7. Fro Figure 8 and 9 we ee that the upplier profit under WPF cenario i higher than that under CSF cenario; however the anufacturer profit under CSF cenario i higher. That i there exit a firt-over advantage for the two player. Specifically under WPF cenario the upplier can trategically offer a unit wholeale price to induce the coonality trategy that axiize her profit. Under CSF cenario the anufacturer can ue the retail price (o order quantity) to avoid the upplier holdup behavior and chooe the coonality trategy to axiize hi profit. We exaine the other default value of paraeter and find that the ain inight i unchanged. 6. Concluion Mot paper on coonality trategy in product line deign only conider anufacturer level. In reality the upplier ay take the anufacturer behavior into conideration to decide the unit wholeale price. The vertical interaction between the upplier and the anufacturer play an iportant role in aking coonality trategy for the anufacturer. In thi paper we develop two gae odel to invetigate how the vertical interaction affect coonality trategy and tudy how the upplier trategically decide the unit wholeale π Figure 9: The anufacturer profit veru the unit production cot of low-quality coponent A. price. ere the anufacturer provide low-end product and high-end product following quality egentation. Under WPF cenario the upplier firt decide the unit wholeale price and then the anufacturer deterine coonality trategy. We explore how the upplier trategically offer a unit wholeale price to induce the anufacturer to chooe the optial coonality trategy for the upplier. We give the wholeale price range of each coonality trategy and find that (i) the total deand of uing coon low-quality coponent A ay be higher than that of uing coon high-quality coponent A; (ii) the upplier can achieve a higher profit under trategy than under trategy becaue the price copetition i weaker; (iii) coonality equilibriu i either trategy or trategy ; (iv)when the quality of coponent A in egent and the unit productioncotofthehigh-qualitycoponenta are ufficiently low the upplier will trategically decreae the unit wholeale price of the high-quality coponent A and increae that of thelow-qualityonetoinducetrategy. Under CSF cenario the anufacturer firt decide coonality trategy and then the upplier offer the unit wholeale price. ere the coonality trategy i regarded a a trategy tool of the anufacturer. We give the unit production cot range of each coonality trategy and find that when the unit production cot of the low-quality coponent A i ediu the equilibriu outcoe i the ae a that under WPF cenario. In addition we find that a c A

10 10 Matheatical Proble in Engineering firt-over advantage exit for the two player; that i the upplier prefer WPF cenario while the anufacturer prefer CSF cenario. Thi paper doe not conider the effect of coonality on arginal cot. Greater coonality can reduce arginal cot [9]. One can extend to invetigate how the cot reduction effectofcoonalityaffectthecoonalitytrategyina upply chain. We aue that the anufacturer purchae two type of coponent A frotheaeupplier.soetie high-quality and low-quality coponent are provided by different upplier where the upplier play a price gae. In thi cae the coonality trategy ay be intereting. Appendix Proof of Propoition 1. The eian atrix of π (θ 1θ w A w A ) over (θ 1 θ ) i ( (q q ) 0 0 q ) whichinegatively definite becaue of q > q.solvingthefirt- order condition π (θ 1θ w A w A )/ θ 1 = 0 and π (θ 1θ w A w A )/ θ =0for (θ 1 θ )wehave 1 (w A w A )= c B +q θ (w A w A )= q θ q +w A w A (q q ) +w A q. (A.1) Inerting θ 1 and θ into D D andp =q θ 1 + q (θ θ 1 ) p = θ q we obtain the deand and retail price of the anufacturer. Siilarly we can obtain the reaction function when the anufacturer ue a coon coponent. Proof of ea. Fro D (w Aw A )>0and D w A )>0wehave (q q ) (w A w A ) >c B > q Siilarly we have the following: q q >c B > (q q q q >c B > (q q )w A q q w A q. )w A w A. (w A (A.) (A.3) By conidering (A.) and (A.3) togetherwe can have theunit wholeale price interval. Proof of ea 3. Differentiating π (w Aw A ) repect to w A wehave π (w Aw A ) w A = c B q +q +w A w A (q q ) = D (w Aw A )<0 and π (w Aw A )/ w A =1/[(q Siilarly uing ea and q how the other reult. Proof of Propoition 4. Fro q that F 1 (Δw) =((q q (q q +[(q (4(q )Δw +[c B (q )Δw q )(q q q q with (A.4) )] > 0. >c B wecan =q ()and(4)itfollow q )(q q )) 1 q )] ) c B ](q q )) (A.5) where Δw = w A w A. By differentiating F 1 (Δw) with repect to Δw twice we have F 1 (Δw)/ Δw =1/[(q q )] > 0. SolvingF 1(Δw) = 0 wecanobtaintwopoitive olution Δw ± where Δw ± =((q ±(q q c B )(q q ) q (q q ) 1. c B ) (q q )(q q )) (A.6) Solving the firt-order condition F 1 (Δw)/ Δw = 0 for ΔwwehaveΔw 1 =q q c B.Froea weknow that Δw < Δw 1.SoF 1 (Δw) 0 i equivalent to Δw Δw. Uing q =q weobtainf 0 =F 4q q (q )(q )where q q F 0 (w A w A ) =q q (q q )w A (q (c B +w A )w A +c B q q +c B q w A +(q (q q )(q q (q q ) +q q )q q )q w A q )q (A.7)

11 Matheatical Proble in Engineering 11 which i a convex function of w A. F 0i equivalent to F 0 0.TherearetwoolutionforF 0 (w A w A ) = 0 w A±.Fro F 0 (w A w A )/ w A =0wecanhavew A1 = q (c B+w A )/q.froea weeew A <q (c B+ w A )/q. Further it follow that F (w A w A ) 0 i equivalent to w A w A =A 1 w A +B 1.Siilarlyfro q q >c B q =q q +q andea it follow that F 3 (w A w A ) 0i equivalent to w A w A where w A i given by Propoition 4. Uing q q =q q q q =q q q q =q =q (A.8) we can find that w A w A =Δw w A =w A andw A = w A interect in one point (w A0 w A0 ).Froq >q and q > q weeea 1 < 1.Nowwehowthat 1/A 1 > q /q (q q )(q atified becaue of q q + (q <(q +q >1 which i equivalent to q q + ) < q. Thi inequality i q q )(q )+(q q ) q )+(q q )=q. (A.9) Thu we can differentiate the feaible region for the unit wholealepriceintothethreeregion:ι II and III. Further we can coplete the proof of Propoition 4. Proof of Propoition 5. Fro (15)itfollowthatπ I (w Aw A ) i concave over (w A w A ) becaue the eian atrix i negatively definite. Solving the firt-order condition π I / w A = 0 and π I / w A = 0weobtainw I A1 = (c A +q )/ w I A1 =(q +c A c B )/ when we ignore the unit wholeale price contraint. Solving w A =(w A B 1 )/A 1 and w A =w A +Δw we obtain w A0 and w A0 givenbypropoition 5. Fro Propoition 4weknowthatonlywhen(w I A1 wi A1 ) Ithe anufacturer will buy two quality level for coponent A.If w I A1 +Δw w I A1 and wi A1 <(wi A1 B 1)/A 1 theupplier ha equilibriu wholeale price at the boundary w A = A 1 w A +B 1 becauef (w A w A ) decreae with w A. Inerting w A = A 1 w A +B 1 into π I (w Aw A )wehave π I (w A) which i a concave of w A.Byolvingthefirt-order condition we have w I A.IfwI A <w A0theunitwholeale price are (A 1 w I A +B 1w I A ). Siilarly we can obtain the other olution; thu in region I the unit wholeale price for the upplier are ) given by Propoition 5. (w I A wi A Fro (16) and (17) itfollowthatπ II (w A) i a concave function of w A while π III (w A ) i a concave function of w A. So the optial wholeale price in the two region are w II A = (c A +q )/ and wiii A =(c A +q )/. Furtherorewe can have π III (w III A )=(q c A ) (8q ) c A) π II (w II A )=(q (8q ). (A.10) Proof of Corollary 6. Part (i) i obviou. Part (ii) i a follow: by ubtituting w II A and wiii A into D (w A) and D (w A) we can obtain D = (c B q (q q )(c A + q ))/[4q (q q )] and D = (c B q (q q )(c A+q ))/[4q (q q )].Froq q =q q it follow that D < D i equivalent to c A /c A > q /q.uingd =D wecancopletetheproof. Fro (A.10) we can how part (iii). Proof of Corollary 7. Fro Propoition 5 wecanhavethat w I A1 +Δw w I A1 i equivalent to (q q c B) Δw + c A c A while w I A1 (w I A1 B 1)/A 1 i equal to (c A +q c B)A 1 +B 1 q c A.Sowehave c A [(q q c B ) Δw +c A (c A +q c B)A 1 +B 1 q ] if (w I A1 wi A1 ) I. (A.11) Fro Propoition 5 we ee that the SPNE deciion under two cenario are identical if the anufacturer ue coon coponent A. We can coplete the proof. Proof of Propoition 8. Uing q obtain Δπ 1 = c B (4q q 3q ) 16 (q q )(q q ) =q and q +((c A c A ) (q q c A +c A ) c B (c A c A +3q 3q )) (16 (q q )) 1 + 4q q 16 3q =q we (A.1)

12 1 Matheatical Proble in Engineering which i a concave function of c A becaue of q >q c B >0.SolvingΔπ 1 =0for c A wehave q q c A1± =c A +c B q +q ±(q q c B) (q q ) (q q ). Thu we ee Δπ 1 0for c A [c A1 c A1+ ]. Uing q =q and q =q weobtain Δπ = A +B c A q 16q q (q q q (q )(q q )c A which i a concave function of c A becaue of q Solving Δπ =0for c A weobtain c A± = B ± B +4A q q q q (q q.note (A.13) q ) (A.14) >q. (q q ) ). (A.15) Thu we have Δπ 0 when c A± exit and c A [c A c A+ ];otherwieδπ <0. Uing q =q q +q weobtain Δπ 3 =(q [c A c Aq +q +q q c A q c A ) (16q q ) 1 (q q )] (A.16) which i an increaing and concave function of c A becaue of w I A1 >c A.SolvingΔπ 3 =0for c A weobtainc A3 =q (q c A ) q /q. Further we ee that Δπ 3 0 i equivalent to c A c A3. Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper. Acknowledgent Thi reearch wa upported in part by (i) China National Fund for Ditinguihed Young Scientit under Grant and (ii) the National Natural Science Foundation of China under Grant Reference [1] P. Deai S. Kekre S. Radhakrihnan and K. Srinivaan Product differentiation and coonality in deign: balancing revenue and cot driver Manageent Sciencevol.47 no.1pp [] A. K. Chakravarty and N. Balakrihnan Achieving product variety through optial choice of odule variation IIE Tranactionvol.33no.7pp [3] U. W. Thoneann and M.. Brandeau Optial coonality in coponent deign Operation Reearch vol.58no.1pp [4] M. Fiher K. Rada and K. Ulrich Coponent haring in the anageent of product variety: a tudy of autootive braking yte Manageent Science vol. 45 no. 3 pp [5] F. Berntein A. G. Kökand.Xie Theroleofcoponent coonality in product aortent deciion Manufacturing & Service Operation Manageent vol.13no.pp [6] P. C. Verhoef K.. Pauwel and M. A. Tuk Aeing conequence of coponent haring acro brand in the vertical product line in the autootive arket Product Innovation Manageentvol.9no.4pp [7] D. P. Rutenberg Deign coonality to reduce ulti-ite inventory: optial depth of a product line Operation Reearchvol.19no.pp [8] K.KiD.ChhajedandY.iu Cancoonalityrelievecannibalization in product line deign? Marketing Sciencevol.3 no.3pp [9]. S. eee and J. M. Swainathan Product line deign with coponent coonality and cot-reduction effort Manufacturing & Service Operation Manageentvol.8no.pp [10] R. Subraanian M. E. Ferguon and. Beril Toktay Reanufacturing and the coponent coonality deciion Production and Operation Manageentvol.no.1pp [11] E. abro The cot effect of coponent coonality: a literature review through a anageent-accounting len Manufacturing & Service Operation Manageentvol.6no.4 pp [1] S. K. Fixon Modularity and coonality reearch: pat developent and future opportunitie Concurrent Engineering Reearch and Application vol. 15 no. pp [13] P. acourbe A odel of product line deign and introduction equence with reervation utility European Operational Reearchvol.0no.pp [14] G. Van Ryzin and S. Mahajan On the relationhip between inventory cot and variety benefit in retail aortent Manageent Science vol. 45 no. 11 pp [15] U. W. Thoneann and J. R. Bradley The effect of product variety on upply-chain perforance European Operational Reearchvol.143no.3pp [16] T. Xiao and T. Xu Pricing and product line trategy in a upply chainwithrik-avereplayer International Production Econoicvol.156pp [17] Y. C. iu and T.. Cui The length of product line in ditribution channel Marketing Science vol. 9 no. 3 pp [18] S. Rajagopalan and N. Xia Product variety pricing and differentiation in a upply chain European Operational Reearchvol.17no.1pp [19]T.XiaoT.M.ChoiandT.C.Cheng Productvarietyand channel tructure trategy for a retailer-stackelberg upply chain European Operational Reearchvol.33no.1 pp

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