Research Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model

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1 Mthemticl Problems in Engineering Volume 4, Article ID 678, pges Reserch Article Competition with Online nd Offline Demnds considering Logistics Costs Bsed on the Hotelling Model Zhi-Hu Hu, Chen Wei, Qing Li, nd Fn Xio Logistics Reserch Center, Shnghi Mritime University, Shnghi 36, Chin College of Informtion Engineering, Shnghi Mritime University, Shnghi 36, Chin Correspondence should be ddressed to Zhi-Hu Hu; Received 8 August 4; Accepted 7 September 4; Published 3 September 4 Acdemic Editor: Tsn-Ming Choi Copyright 4 Zhi-Hu Hu et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Through populr informtion technologies (e.g., cll centers, web portl, ecommerce nd socil medi, etc.), trditionl shops chnge their functions for servicing online demnds while still providing offline sles nd services, which expnd the mrket nd the service cpcity. In the Hotelling model tht formultes the demnd effect by considering just offline demnd, the shops in line city will locte t the center s the result of competition by gmes. The online demnds re met by the delivery logistics services provided by the shops with dditionl cost; the consumers witing time fter their orders lso ffects their choices for shops. The min purpose is to study the effects of the following spects on the shops loction competition: two logistics costs (consumers trvelling cost for offline demnds nd the shops delivery logistics cost for online demnds), the consumers witing cost for online orders, nd the rtios of online demnds to the whole demnds. Therefore, this study primrily contributes to the literture on the formultion of these spects by extending the Hotelling model. These fetures nd effects re demonstrted by experiments using the extended Hotelling models.. Introduction People usully visit nd buy wht they wnt in supermrkets, shopping centers, nd plz, wheres, ccompnied by the informtion technology nd Internet, online business hs n incresing tendency. In 9, the online retil sles of USA grew %, while ll retil sles only grew.5% []. About 54 million people hd the experiences of online shopping, contributing to online retil sles of $55 billion or 6% of totl retil sles. It is forecsted tht the online sles of USA will keep growing t % s n nnul rte through 4 to $5 billion. In Western Europe, online retil sles re expected to grow by % per yer, going from C68 billion in 9 to C4.5 billion in 4. The integrtion of online nd offline undoubtedly brings enterprises grnd profits nd provides multichnnels for consumers. Under the merging of online business, there re more nd more people shopping on the Internet. Without going outside, people cn enjoy the service from shops or other providers ll over the city. Compred to the offline business, shops my suffer n dditionl cost, delivering cost. According to the Hotelling model [], competitive shops often locte in the center of line city, nd they re closed to ech other. However, with the development of online business, orders my come from people ll over the city nd should be met in the shortest time. Obviously, the loctions deduced by thehotellingmodelwillimposegretcostsontheshops for online orders. Concentrted distribution, which works well in the pst time, cn no longer operte perfectly for the following resons. First, remote consumers will wit for longer time to get the ordered goods becuse of long trveling time, which my cuse low stisfction degrees; second, long distnce mkes lrge delivery logistics cost, which crries burdens on shops; third, shops my increse the price to sustin good profit, which my lose some consumers nd form vicious cycle. This work extends the Hotelling model for the offline nd online demnds. This work extends the Hotelling model to study the effects of combintion of offline nd online demnds on the competition of shops. The trditionl Hotelling model

2 Mthemticl Problems in Engineering hs formulted the loction gme considering the pricing problem nd the consumers trvelling logistics costs. For online demnds, the consumers my choose shops bsed on the prices nd the witing time fter orders, lthough the consumers do not py for the trvelling costs by themselves. However, the shops will py for the delivery cost for servicing the consumers tht order goods online. Therefore, the shop s profitshouldbesubtrctedbythelogisticscostthtreltes to the orders nd the distnces between the shops nd consumers. Therefore, the indifferent loction computed in the side of consumers nd the profit computed in the side of shops chllenge the extension of the Hotelling model. Following these clues, new models re developed for nlyzing the effects of logistics cost deduced by online demnds on the shops loction competition. Considering the new sitution tht the combintion of online nd offline business promotes gret success, this pper extends the Hotelling model. The logistics costs, the shops delivery times, nd the consumers witing times re incorported into the extended models. In the fields including online sles nd competition, this study contributes to the literture in the following points. First, bsed on offline business, this work considers the model combining offline nd online business. Second, the logistics costs nd the online demnds re considered in ssessing the loction competition gme between shops. Third, the Hotelling model is extended by simultneously considering offline nd online demnds nd the shops logistics cost for delivery. The rest of the study is orgnized s follows. In Section, there re the studies relted to online nd offline business, competition issues in logistics systems, nd the pplictions of the Hotelling model. Then, in Section 3 the problem is described nd the problem is formulted in Section 4. The developed models re then demonstrted nd nlyzed in Section 5. Finlly, we conclude the study in Section 6.. Relted Studies.. Online nd Offline Business. Since the erly 99s, mny mnufcturers nd retilers hve incorported the Internet into their multichnnel strtegy nd devoted considerble resources to building the online chnnel. Chu et l. [] thoughtthttheinternetsubstntillyreducesserchcosts nd grnts esy ccess to product nd price informtion. Online shopping involves no trvel, product crrying, or restrictions on shopping hours, offering gret ccessibility, convenience, nd time sving. In contrst, offline shopping llows physicl exmintion of the products, interpersonl communiction, nd instnt grtifiction but involves high trvel costs nd serch costs nd often hs restrictions on shopping hours, especilly in countries with strong lbor lws. Shnkr et l. [3] climed tht the online business offers more opportunities for interctive nd personlized mrketing thn offline business. Moreover, the online consumers cn compre lterntives esily especilly for functionl products nd services. Kwon nd Lennon [4] suggested the synergy between online nd offline opertions generted by the semless integrtion between the two chnnels. Kwon nd Lennon believed tht the synergy enriches the consumers experiences with the retiler, strengthens the brnd imge of the retiler, nd cultivtes consumer loylty in both chnnels. The offline nd online business impose complex effects on the consumers loylty. Loureiro nd Roschk [5] devised model tht compres the offline nd online shops nd regrds the consumers ges s modertor. In the offline context, positive emotions predict loylty mong younger consumers but not mong older ones; in the online context, the effect of grphics design on loylty is stronger for younger consumers thn for older ones. Kwon nd Lennon [4] investigtedthe effect of the interply between multichnnel retiler s offline nd online brnd imges on consumers online perceived risk nd online loylty within the frmework of theory of cognitivedissonnce.shnkretl.[3]proposedconceptul frmework nd developed hypotheses bout the effects of the online medium on consumer stisfction nd loylty nd on the reltionships between stisfction nd loylty. Theresultsshowedthtloyltytotheshopsishigherwhen the service is chosen online thn offline. Besides, loylty nd stisfction hve reciprocl reltionship such tht ech positively reinforces the other, nd this reltionship between overll stisfction nd loylty is further strengthened online. The consumers behviors in the online business context re exmined by mny reserchers. Kollmnn et l. [6] identified relevnt shopping motives in multichnnel environments. The empiricl nlysis suggested tht the degree of consumers convenience orienttion in contrst to the degree of risk version nd service orienttion encourges the online chnnel over the offline chnnel. Chu et l. [] studied the moderting effects of household (e.g., shopping frequency) nd product (e.g., sensory nture) chrcteristics on household brnd loylty, size loylty, nd price sensitivity cross online nd offline chnnels for grocery products. Households re more brnd-loyl, more size-loyl but less price sensitive in the online chnnel thn in the offline chnnel. The differences between online nd offline business in brnd loylty nd price sensitivity re lrge for light online shoppers nd smllest for hevy online shoppers. The online nd offline differences in brnd loylty, size loylty, nd price sensitivity re lrge for food products nd for sensory products. Ahn et l. [7] explored some online nd offline fetures of Internet shopping mlls nd their reltionships with the cceptnce behviors of consumers, nd they provide domin-specific, integrtive pproch in evluting the qulity nd ntecedents of user cceptnce for Internet shopping mlls. Technology cceptnce model [8] is vlid in predicting the cceptnce of the Internet shopping mlls nd tht online nd offline fetures hve positive effects on the user cceptnce. Both online nd offline fetures hve greter effects on the usefulness, ttitude, nd intention to use thn either online or offline fetures seprtely. Grewl et l. [9] providednoverviewoffindingsfrom pst reserch in both offline nd online domins nd presents n orgnizing frmework, s well s n gend to dvnce the reserch. The issues on the online nd offline pricing nd promotionl strtegies, s well s coordinting these strtegies, re highlighted. This work lso incorported the

3 Mthemticl Problems in Engineering 3 pricing strtegy in studying the loction competition ffected by logistics costs... Competition in Logistics. The competitions involved in the opertions nd mngement of the supply chins re extensively exmined in literture. Frhni et l. [8] considered tht the competitive supply chin network design ffects itscostswellsitsperformncendssertedthtsupply chins compete to cpture mrket shres for themselves. Brneyet l. [] proposed resource-bsed competition method to ssess nd determine the importnt resources nd opertions of firm. In this strtegic mngement frmework, firms re viewed s entities which possess nd control complex resources. The resources re scrce or unique, substitutble, nd durble, which re thought to contribute to the firm s competitive dvntge. Moreover, the logistic service cn be regrded s specil resource so tht it is lso importnt in the firms decisions. Ppchristos [] concerned the mrket competitions between new nd remnufctured products nd explored the environmentl impct of competition on the shift from products to services with system dynmics model. Rezpour nd Frhni [] designed the network structure of competitive supply chin bsed on bilevel model. The price nd service levels competition in mrket re considered in the model of stochstic price nd service demnds with the presence of existing nd externl rivls. The supply chin network is ssumed to be set t first nd could be djusted for new prices nd services. Ngurney et l. [3] developed gme theory model for supply chin competition in time-sensitive mrkets where consumers re sensitive to the delivery time. An lgorithm is devised to solve the model with equilibrium conditions. The loction gme is typicl kind of competition between firms. Go et l. [4] considered the loction-thenpricegmestwo-stgegmeofn plyers on the grph which is bstrcted from the spred of the firms in the relworld.bsedonthebestresponsedynmicsndthe multinomil logit model, the price nd loction equilibri re built nd shres of n plyers re computed by price equilibrium. The loction nd price equilibri re derived for the competitive irlines in Russin nd Chinese irline mrket. Godinho nd Dis [5] ddressed discrete competitive loction gme in which two decision mkers hve to decide simultneously where to locte their services without knowing the decisions of ech other. Obviously, the decisions mde by one firm will ffect the pyoffs of nother. A model nd lgorithmic pproch re designed to clculte the Nsh equilibri in both preferentil rights overbidding consuming ptterns for frnchisees nd frnchisers. The results re dvntgeous for the frnchiser if the overbidding is possible. Fotkis nd Tzmos [6] studied the fcility loction gmes where number of fcilities re set in metric spce bsed on the loctions reported by the strtegic gents. The gents minimize their connection cost to the fcilities; however, the loctions my be misreported by the gents. The model bsed on the winner-imposing mechnisms is designed to produce the mechnisms for the gents nd fcilities under the presumption tht no gent cn benefit from misreporting his loction. A continuous metric spce is lso considered for the oblivious winner-imposing mechnisms. Some ppers consider the competitions in trnsporttions during the logistics ctivities. For exmple, Reyes [7] studied the trnsshipment problem bsed on the Shpley vlue concept from the coopertive gme theory by considering logistics cost nd qulity. This work studied the loction competition between shops with offline nd online demnds by considering logistics costs. The shops cn gin competition dvntge by improving logistics services..3. Hoteling Model. Since Hotelling [] developed the twostgegmeofsptilcompetitionmodel,lotofrelevnt litertures re grdully emerging. Considering tht no equilibrium price solution exists when both firms re not fr enough from ech other, d Aspremont et l. [8]replcedthe liner trnsporttion cost function with qudrtic function, such tht price equilibrium solution exists everywhere. d Aspremont concluded tht the two firms gthering in the centerwillmketheequilibriumpricezero.anotherwyto solve the equilibrium problem is to chnge the city s shpe to circulr [9]orsqure[]. By introducing consumer differentition, incresing the number of firms, nd relxing the ssumption of demnd elsticity nd product differentition, the Hotelling model is extended by Tbuchi nd Thiees [], Plm et l. [], Eton [3],ndShkedndSutton[4]. The reserches on the pplictions of the Hotelling model cn be divided into two clsses. One solves the problem of selecting loction nd pricing for the product. Wey nd Hong [5] extended the Hotelling model for determining the loctions nd the optiml numbers of plnts for the purpose of using rice strws. Go et l. [4] pplied the Hotelling model to the study of the competition in irline mrket, where irlines first decide plne lloction nd then choose ticket prices. The existence of equilibrium in pure strtegies on the grph for the cse of multiple plys is extensively discussed. Using distnce to mesure product fetures like qulity, for firms, the Hotelling model is used todecidetheproductqulityndbrndpositioningndso forth. Blosh nd Mnceu [6] dded dvertising investment in the Hotelling model nd sserted tht dvertising mkes consumers prefer products, while lso decresing the price. Therefore, dvertising investment is not lwys necessry for firms. Bsed on the Hotelling model, Mriñoso [7] proved tht endogenous switching costs will increse the mrket competition, so tht the firms should produce comptible brnd products. To investigte the effects of the business mode combining offline nd online demnds on the loction competition, the Hotelling model is extended by the following wys. First, in the consumer spects, for online orders, the consumers witing cost is formulted to determine the indifferent loction; second, the profit of firm is subtrcted by the delivery logistics cost for the online demnds; third, the offline nd online demnds re considered in the integrted Hotelling

4 4 Mthemticl Problems in Engineering model for studying the effect of the rtio of online demnd on the competition. 3. Problem Description Hotelling s liner city model ws developed by Hrold Hotellinginhisrticle StbilityinCompetition in99 []. In this model he introduced the notions of loctionl equilibriuminduopolyinwhichtwofirmshvetochoose their loction considering consumers distribution nd trnsporttion costs. Initilly, the model ws developed s gme in which firms first choose loction nd fter selling price for their products. To set their business in the best loction to mximize profits, the firms hve to evlute three key vribles: competitors loction, consumers distribution, nd trnsporttion costs. This model includes two different pproches. First, the sttic pproch consists of single stge,wherefirmschoosetheirloctionndpricessimultneously; second, the dynmic one considers tht the price is set fter determining the loction. The model is bsed on liner city tht consists of only single stright street. For esy comprehension, the Hotelling model is sometimes explined by using the exmple of bech where two ice crem stnds re trying to decide their best loctions. 3.. Sttic Approch. In bech going from west to est, of size [, ], where consumers re distributed evenly, two identicl ice crem stnds (A nd B) with mrginl cost of production, c >, try to determine their best loction. A is locted t distnce from point, while B is locted t distnce b from point. Both ice crem stnds offer the exct sme ice crems, nd therefore consumers utility will begivenonlybythepriceoftheicecremndthedistnce to the stnd. Despite differences in prices (P nd P ), the stnd with the lowest prices will not necessrily ttrct ll thedemndssinceconsumersconsiderthedistncetothe stnds. If both stnds prices re equl, the differentil fctor is how close consumers re to ech stnd. All consumers locted to the left of wouldgotostnda,ndllconsumers loctedtotherightof( b) wouldgotostndb.the remining consumers, locted between both stnds, would go to whichever is the closest. Here, x is the exct middle of tht bech, so consumers to its left would go to stnd A, while consumers to its right would go to stnd B. Two conditions re necessry for profits to be positive nd mximized in both stnds: selling prices must be higher thn mrginl costs; = b. This second condition implies tht both stnds cnnot be locted t the sme point exctly in the middle of the bech. If this position is indifferent for the consumers, ech stnd would decrese its prices to ttrct consumers, nd thus they would enter into price wr. If they hd different mrginl costs, the stnd with the highest mrginlcostwouldbeinclerdisdvntgendwouldend up exiting the mrket, s the other stnd would be ble to push the prices further down nd so ttrct ll consumers. Depending on how prices re set, the prices of both stnds re equl to their mrginl costs. In this pproch the key fctor for product differentition is loction. Ech stnd will therefore set prices tht will be higher thn their mrginl costs nd will choose loction other thn the middle of the bech. 3.. Dynmic Approch. The dynmic pproch hs two stges. At first, the stnds choose their loction nd then the prices re set. With regrd to the previous pproch two dditionl ssumptions re introduced: both stnds hve equl mrginl production cost; totl cost for consumers depends on the price of the goods nd the distnce to the selected stnds (the unit trvelling cost is t). The point of division between the res served by these two stnds is determined by the condition tht t certin point it is indifferent for consumers to choose either one of the stnds. After equting nd equlizing, () isgot.bysolvingit,the demnd functions of the two stnds re (): p t(x ) =p t(( b) x), () D =x= p p t D = x. ( b), By using bckwrd induction the subgme perfect Nsh equilibrium cn be found where both firms mximize their profits. The dynmic pproch hs two opposite effects. First, n incentive for both stnds to locte t the center of the bech increses their mrket shre by reching out the gretest mount of consumers, in wht is known s the demnd effect. And, n incentive for both stnds to locte t opposite extremes exists, in wht is considered to be the strtegic effect. While the first effect will reduce differentition between the stnds, the second one will increse it. This study considers the demnd effect of two shops in line city on their loctions. However, ctivted by the offline nd online sles ptterns in the fst food industry, this work considers dditionl online orders bsed on the Hotelling model. The two shops sell goods t their loctions when consumers come to their shops while they lso ccept online orders nd deliver the goods to the consumers. However, the shops will overtke the delivery logistics costs. In the bove sttic nd dynmic pproches, when the consumers order the goods offline by going to the shops, the consumers themselves must undertke the trvelling cost (t). Therefore, in the proposed problem in this study, s the min difference to the bove Hotelling models, for online demnds the shops undertke the logistics cost (delivery), nd for offline demnds the consumers undertke the logistics costs (trvelling). Under the demnd effect, two shops will locte t thecenterofthecity.theeffectoftheconsidertionofonline demnds nd logistics on the models is focused on in this work. ()

5 Mthemticl Problems in Engineering 5 4. Formultion 4.. Bseline Model Only considering Offline Demnds 4... Shops Locting t the Ends of the Line City. In this nlysis, p i represents the price of shop i, i=,. The demnd of shop i t p i is denoted by D i (p,p ).Thepointx is the indifferent loction of the consumer to buy from shop or shop. And the unit trvel cost for consumer is denoted by t. Assumption. The commodities from both shops re homogeneous; tht is, the commodity cost c is the sme for both shops. Assumption. Without loss of generlity, ssume tht ech consumer purchses single commodity t once. Assumption 3. Shops in the line city re profit-oriented sellers;thtis,profitsoftheshopsshouldbepositive. BsedonAssumptions nd,thepropositionsbelowre proposed. Proposition 4. A consumer chooses the less cost shop in the offlinebselinesitution;thtis,theconsumerontheleftofx chooses shop nd on the right chooses shop. Proposition 4 suggests n economicl consumer behvior tht consumer prefers the less cost commodity. In the offline mode, consumers hve to py for the price of the commodity nd the trvel cost to the shop (no mtter shop or ). Assuming tht the trvel cost is liner with the distnce, x is obtined by p tx=p t( x). (3) Proposition 5. In the offline bseline sitution, the Nsh equilibrium exists in the profits of the shops nd the equilibrium profits re π off =π off =t/. In the context of the line city model, x is between the positions nd. Hence, the demnd D i (p,p ) is determined s D (p,p )=x= p p t, t D (p,p )= x= p (4) p t. t According to the price p i nd D i (p,p ),theprofitπ off i of shop i is evluted by π off (p,p )=(p c)d off (p,p ) = t (p c)(p p t), π off (p,p )=(p c)d off (p,p ) = t (p c)(p p t). (5) To get positive profit s defined in Assumption 3, π off i is positive so tht p i is higher thn c. We get the first order conditions of (5) on the price in the pricing gme between shops nd s π off =p p ct p, π off p =p ct p. Moreover, the second order conditions of (5) re π off p = π off p (6) =. (7) So the Nsh equilibrium profits exists where (8) is stisfied: π off =p p ct p =, π off p =p ct p =. And the equilibrium prices nd profits of both shops re p =p =ct, π off =π off = t (9) Shops Locting t the Arbitrry Positions of the Line City. In the following, the shops cn locte t ny rbitrry positions of the line city. The loction of shop is,while the loction of shop is b. Assumption 6. Without loss of generlity, it is ssumed tht b (the shop is on the left of the shop ). Considering consumer s qudrtic trvelling cost (the trveling cost is td,whered is the trveling distnce from the consumer position to the shop stnd), then the indifferent position x is obtined by p (x ) t=p ( x b) t. () Proposition 7. In the rbitrry position sitution, the Nsh equilibrium exists in the profits of the shops, where p = ct( b)((( b)/3)) nd p = ct( b)(((b )/3)). p nd p re the equilibrium prices of shops nd, respectively. As the result of (), we get the vlue of x by x= b p p t ( b). () Similr to the bseline city sitution, the demnds of both shops re derived s D (p,p )=x= b D (p,p ) = x=b b p p t ( b), p p t ( b). (8) ()

6 6 Mthemticl Problems in Engineering Becuse the positions of the shops re flexible, () is more complex thn(4). Demnd in rbitrry position sitution is determined by five prmeters, tht is, positions, b, prices p, p,ndtrvelcostt. The profits re derived s π (p,p )=(p c)d (p,p ) =(p c)( b π (p,p )=(p c)d (p,p ) =(p c)(b b p p t ( b) ), p p t ( b) ). (3) Anlyzing the first-order nd second-order derivtives of (3) on the prices, the results re presented s π = b p p p t ( b) c t ( b), π =b b p p p t ( b) c t ( b), (4) π p π p = = t ( b), t ( b), (5) According to Assumption 6, the vlues of the first-order derivtives (see (4)) re vrying from positive to negtive while the vlues of the second-order derivtives (see (5)) re negtive.sothenshequilibriumexists(see(6)): π = π =. (6) p p The prices t equilibrium re solved s p p (, b) =ct( b) ( b 3 ), (, b) =ct( b) ( b 3 ). (7) In the bove solution, when the first-order derivtive is derived for the profit to the price, the results will be different for the conditions tht p,p,,b re independent of ech other; nmely, = b = p =. (8) p p p When =b=, the shop loctes t while the shop loctes t ; when = b,twoshopsloctetthesme position. The equilibrium prices under these two conditions re (9)nd()individully: p (, ) =p (, ) =ct, (9) p (, ) =p (, ) =c. () 4.. Extension Model for Online Consumer Demnds. Bsed on the models in Section4. for only considering offline demnds, this section develops models for the online demnds when considering the shops delivery logistics costs. Similr to Section 4., twolevelsreconsidered,firstforthe shops locting t the ends of the line city nd second for deciding loctions for the shops. Two dditionl prmeters re introduced for the online demnds nd delivery logistics: the cost of witing for unit of time is denoted by c w ;ndthe distribution cost for trvelling for unit of time is denoted by c d Shops Locting t the Ends of the Line City Proposition 8. In the online shopping sitution, consumer prefers the less cost shop bsed on the commodity cost nd penlty costs of witing; tht is, if consumers re on the left of the indifferent position x on =(p p )/c w /, they choose shop, while others choose shop. In the perspective of consumers, they choose online shops not ccording to their trvelling costs to the shops, but ccording to the penlty costs deduced by witing times. Therefore, the indifferent position x should stisfy () nd is derived s p c w (x on ) =p c w ( x on ), () x on = p p c w. () Similr to (4), the demnds for the shops re derived s D on = p p c w, D on = p p c w. (3) However, different from the offline mode, the trvel cost for the consumer is substituted by the delivery logistics costs for the shops. Therefore, the profits functions of the shops re different (compring to (5)), s shown in x on = π on (p c c d x) dx =p x cx cd x (p p )/c w (/) =p ( p p c w ) c(p p c w ) cd (p p c w ),

7 Mthemticl Problems in Engineering 7 = (p c c d ( x))dx x on π on =p x cx c d x cd x =(p c c d )( p p c w ) cd ( (p p c w ) ). (p p )/c w (/) (4) Proposition 9. In the online shopping sitution, the Nsh equilibrium exists in the profits of the shops where equilibrium prices re p =p =c(cw /) (c d /). As seen in (4), the profits should be subtrcted by the delivery logistics cost (see c d x nd c d ( x) in the equtions). When p,p,c,c d re independent of ech other, the firstorder derivtives of the profits on the prices re shown in π on = p p c w p c cw c π on = p p c w p c cw c w cd w cd c w (p p c w ), c w (p p c w ). (5) It is found tht the second-order derivtives of the profits onthepricesrenegtive,becusec w nd c d re both positive in (6). So the optiml prices which re equilibrium prices for both shops under the mximiztion of the profits re derived s D on =x on = b p p c w ( b), D on = x on =b b p p c w ( b), = π on (p c ( x) c d )dx x on (p c c d (x ))dx =(p c)( b c d c d b = π on c d ( b x on (p c c d ( b x))dx (p c c d (x ( b)))dx b =(p c)( b (c d b c d ) ( b ( b (c d ( b) p p c w ( b) ) p p c w ( b) ), p p c w ( b) ) p p c w ( b) )) (3) π on p = π on p = c w cd (c w ), (6) c d ( b p p c w ( b) ) ) p =ccw cd, p =ccw cd. (7) 4... Shops Locting t Arbitrry Positions of the Line City Proposition. In the online shopping sitution where shops re t rbitrry positions, the Nsh equilibrium exists in the profits of the shops. In the following, the shop loctions re decisions nd cn locte t rbitrry positions of the line city. Then, the indifferent position x is derived by (8) nd (9). The derivtion processes in (3) (33) re similr to() (7) but the nlytic expressions re more complex: p c w (x on ) =p c w ( x on b), (8) x on = b p p c w ( b), (9) π on p (c d c d b) b cd c d ( b), π on p = ( b = ( b p p c w ( b) ) (p c) c w ( b) c d c w ( b) ( b p p c w ( b) ), (3) p p c w ( b) ) (p c) c w ( b) (c d b c d ) c w ( b) c d c w ( b) ( b p p c w ( b) ), (3)

8 8 Mthemticl Problems in Engineering p =(cw ( b)(3 3b) 3c cd (b) cd (c c d b c d ) c w ) ( b) c w ( b) ( ), 3c w ( b) cd p =cw ( b) cc d b c d (33) Shop profit.8 Shop s equilibrium price (c w ( b)(3 3b) 3c cd (b) cd (c c d b c d ) c w ) ( b) c w ( b) ( ). 3c w ( b) cd 4.3. Model for Online nd Offline Demnds Proposition. In the mixed mode of online nd offline, the profit is the combintion of the online nd offline profits ccording to rte of online consumers nd offline consumers, nd the Nsh equilibrium exists. The shops finlly considered simultneously serve for offline nd online demnds. A demnd hs the possibility of c on to order goods by online methods such tht the possibility of offline demnd is ( c on ). Here, the online nd offline demnds re independent of ech other; the introduction of online chnnel will not increse the totl demnd. Bsed on the models in Sections 4. nd 4.,theprofitmodelforonline nd offline demnds is derived s π =c on π on ( c on )π off, π =c on π on ( c on )π off, where, = π on π on b = (p c c d x) dx x on (p c c d (x ))dx x on (p c c d ( b x))dx (p c c d (x b))dx b π off =x off p π off =( x off )p. (34) When c on is fixed, the equilibrium prices nd profits cn be derived by combining the results in Sections 4. nd Shop price Figure : Reltion between price nd profit for shop when shop uses equilibrium price Shop Shop Shop price Figure : curves of two shops when shop decides its price following pricing of shop. 5. Experimentl Results In the following, the models in Sections 4.., 4.., nd 4.3 re demonstrted. 5.. Online Shops Locting t the Ends of the Line City. The following experiments use the settings c =, c w =., nd c d =.. Figure presents the fct tht the shop s profit vries with the price when the shop s price is the equilibrium price. It is seen tht when the shop devites from the equilibrium price, its profit flls. Figure presents the profit curves for the two shops when the shop determines its price following the pricing decision ofshop.chngingthepricefromtheequilibriumpriceby the shop will incur the profit increse for shop when shop undertkes the following pricing strtegy. Under this strtegy, when shop decreses its price to get competitive position in the price intervl (.4 to.), shop cn get minor dvntge

9 Mthemticl Problems in Engineering 9 Shop price Shop price Figure 3: Reltion between prices of two shops when shop decides its price following shop c d.5.. Figure 5: Impcts of vrying unit witing nd delivery logistics costs on equilibrium profit..5.9 c w c d =. c w = Cost Figure 4: Equilibrium profits vrying with witing cost nd delivery logistics cost Shop Shop Shop position Figure 6: Equilibrium profits vrying with shop s loction when shop loctes t. (b =.). compring to shop while its profit lso decreses. In the other intervls (from. to.4 nd from. to infinite vlues), shop will lose its competitive cpbility. Figure 3 further depicts the fct tht the shop s optiml price is liner to the shop s price under the bove described conditions. Figure 4 presents the result of sensitivity nlysis when the equilibrium profit vries with the cost prmeters (c w nd c d ). In the experiment, one cost vries when the other is set to.. In this study, the delivery logistics cost is liner to the trvelling time from the shops to consumers, wheres the witing cost is liner to the qudric of the witing time (which is the time trvelling from the shop to the consumer). Therefore, the effect of vrying unit delivery logistics cost on the equilibrium profit is lrger thn tht of vrying the unit witing cost. The impcts of vrying the unit witing nd delivery logistics costs on the equilibrium profit re shown in Figure OnlineShopsLoctingtArbitrryPositionsoftheLine City. Under the demnd effect, the two shops will locte t the center of the line city. However, the delivery logistics cost will pull the shops from the center. Figure 6 presents the shops equilibrium profits vrying with shop s loction when shop loctes t. ( =b=.indictes the equilibrium loctions for the two shops). When shop moves to shop, their profits ll decrese, lthough it seems tht shop erns more thn shop. When shop moves to shop, shop with the fixed loction chooses to decrese its price for ttrcting consumer demnds, nd the equilibrium price curves of the two shops re presented in Figure 7. The profit of shop is shown in Figure 8 with vrible loctions of shops nd. When the shop moves towrds the center, the profit decreses until the shop is close enough to nother shop. And there is n interesting phenomenon reveled by the experiment tht when the loction of one shop is fixed, the other shops moving towrd the center

10 Mthemticl Problems in Engineering Price Shop Shop Shop position b Figure 9: Equilibrium profit vrying with loctions of two shops.. Figure 7: Equilibrium prices vrying with shop s loction when shop loctes t. (b =.) b Figure 8: Shop s equilibrium profit vrying with loctions of two shops. promote the profit of the fixed shop. When the shops move closetoechother,thepricesrisebecusetheshopsdonot necessrily ttrct dditionl consumers by decresing the prices. This vriety of price will lso benefit nother fixed shopndledstothepekpointinthesurfceinfigure 8. Furthermore, when the production cost chnges nd the profit does not chnge, they cn be the sme s seen in Figure 8, becuse both shops sell homogeneous products. Figure 9 shows the profit surfces of both shop nd shop from two ngles of view. The intersection of the two surfces isthenshequilibriumprofitlinessocitedwiththeshops, nd it is formed when the distnces of the shops to the two ends of the city re equl. When the reltive loctions of the shops re not equl, the shop t disdvntgeous loction couldmovetotheequilibriumloctiontogetmoreprofitnd simultneously decrese the competitor s profit. Figure depicts the equilibrium profit vrying with the shops loctions nd the unit witing cost, while Figure..8.4 c w =.3 c w =..8.6 b.4. c w =. c w = Figure : Equilibrium profit vrying with shop s loction () nd unit witing cost (c w ). presents the equilibrium profit vrying with the shops loctions nd the unit delivery logistics cost. As comprison result, the effect of unit witing cost on the profit is bigger thn the unit delivery logistics cost. The profit cn be improved by decresing delivery logistics cost nd incresing the delivery service qulity (which helps to decrese the witing cost). 5.3.ShopsLoctingtArbitrryPositionsforMixedOnline nd Offline Demnds. In the following demonstrtion of the model considering online nd offline demnds, the defult prmeters re set s follows: the unit delivery logistics cost for online demnd, c d =.; the unit witing cost for online demnd, c w =.; the unit production cost of goods, c=; the rtio of online demnd, c on =.5;theunittrvellingcost of consumers for offline demnd, t =.. Figures, 3, nd 4 revel the effects of loctions nd rtios of online demnds or unit witing costs on the equilibrium profits. Incresing the rtio of online demnd or unit witing cost improves the equilibrium profits.

11 Mthemticl Problems in Engineering c d =.3 c d = b. c d =. c d = Figure : Equilibrium profit vrying with shop s loction () nd unit delivery logistics cost (c d )..8.4 c on =.5 c on =..8.6 b.4. c on = Figure 3: Equilibrium profit vrying with shops loctions for three rtios of online demnds (c on =.,.5,.8) b Figure : Equilibrium profit vrying with shops loctions (c on =.5) c w =.5.8 c w =..6 b.4. c w = Figure 4: Equilibrium profit vrying with shops loctions for three unit witing costs (c w =.,.5,.8). 6. Conclusion Trditionl merchnts re fcing offline nd online demnds tht re chllenging the business nd service modes. Activted by the new business mode of combining offline nd online sles in the fst food industry, this study exmines the effectsofthemodeonthecompetitionbsedonthehotelling model. Under the demnd effect, the shop stnds will locte t the center in line city in the Hotelling model tht only considers the consumers logistics cost trveling to the shops. In the extended model for the offline nd online demnds, the shops undertke the delivery logistics cost for the online orders, wheres the consumers witing cost will ffect their choices of shops clled. Therefore, the min contributions of this study involve the following fetures: considering online nd offline demnds in n extended Hotelling model, considering consumers trvelling cost to shops for offline demnd nd shops delivery logistics cost for online demnd, nd considering the effect of consumers witing cost for online orders on the indifferent shop loction. As for the future reserch direction, first, the prmeters used in the study should be further verified by empiricl studies. Second, this study does not consider the demnds dependent on the online sles mode, which is prcticl feture of online nd offline business. Third, the formultion for line city could be extended for rel urbn scenrios. Conflict of Interests The uthors declre tht there is no conflict of interests regrding the publiction of this pper. Acknowledgments This study is prtilly supported by the Ntionl Nturl Science Foundtion of Chin (788, 7479, 779, nd 7395), the Ntionl Nturl Science Foundtion of Jingsu (9KJA5), the Science Foundtion of Ministry of Eduction of Chin nd Shnghi (3, 4YZ, 34, 3SG48, nd 9YJA637), the Science nd Technology Commission of Shnghi (ZR48, 556), nd the Grdute Innovtion Foundtion of Shnghi Mritime University (4ycx).

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