SIMULATION STUDY OF STOCHASTIC CHANNEL REDISTRIBUTION

Transcription

1 Developmens in Business Simulaion and Experienial Learning, Volume 3, 3 SIMULATIO STUDY OF STOCHASTIC CHAEL REDISTRIBUTIO Yao Dong-Qing Towson Universiy dyao@owson.edu ABSTRACT In his paper, we invesigae sales redisribuion beween dire (i.e., on-line ) and indire hannels when marke demand exhibis a general sohasi paern. We model he dynamis of he iner-hannel diffusion by a sysem of wo differenial equaions, and hen obain a simple analyial expression of sabiliy ondiions for hannel redisribuion when he marke demand is deerminisi. The sabiliy ondiions are unforunaely unaainable under more realisi marke siuaions where he demand onains unerain fluuaions. The resuls from an exensive simulaion sudy indiae ha he sabiliy ondiions obained under deerminisi marke remain appliable o unerain marke siuaions, inluding he demand following a sohasi fluuaion and he diffusion parameers following a known disribuion. ITRODUCTIO The inreasing suess of eleroni ommere has made disinermediaion possible in many indusries. Consumers now an purhase produs eiher from dire hannel or from indire hannel [Majumdar and Venkaram, 995]. Dell proves he dire model is viable. In many ases i is he bes sraegy for vendors rying o u disribuion oss. As a resul, many manufaurers inorporaed dire hannel ino heir disribuion hannels. For example, HP sold is small and midsize business line of produs on is web sie, and IBM also rolled ou is dire sales web sie o is larges aouns [Shwarz and Briody, 999]. These dire hannels exis in parallel wih he onvenional indire hannels [Sridhar, 998]. However, he exisene of boh dire hannel and indire hannel arises many ineresing quesions. For example, does adding addiional hannel mean more sales? How muh should eah hannel apaiy be? How o redisribue hannels among disribuion hannels? In his paper, we will sudy he hannel redisribuion on dire and indire hannels. Mainly we will sudy wha effe he differen marke sizes have on he hannel redisribuion. In his paper, we will firs develop a wodimensional diffusion model, whih mainly onsiss of a sysem of wo differenial equaions. We also obain key resuls regarding he seady-sae redisribuion when marke size is deerminisi. Then, we will sudy he demand redisribuion beween dire and indire hannels under sohasi marke size. We will hoose wo ommon paerns for marke size, namely exogenous random disurbane, and sohasi aggregae fluuaion, o sudy he demand alloaion on dire and indire hannels. Sine i is diffiul o ge a losed-form soluion when marke size is sohasi, we will use simulaion o sudy he effe of marke size on hannel redisribuion. Finally we will assume parameers in differenial equaions are sohasi, and use simulaion o sudy he demand disribuion on wo hannels. MODEL Consider a simple supply-hain sysem in an esablished marke, in whih a firm is faed wih a deision o adop a dire hannel. Suppose he oal marke size a ime is given as ( >, whih is assumed o be oninuous and bounded ( ( K < ). The primary issue o be examined before a deision on he dire hannel an be made perains o how marke sales would redisribue upon he amendmen of a dire hannel. Upon he addiion of a dire hannel, here will be demand diffusion beween he wo hannels. For example, he rae of hange in sales via dire hannel (or indire hannel), denoed as x& (x ), will be affeed by ( & ( ) negaive word-of-mouh (WOM) effe ( ), posiive sales promoion (PRO) effe η ( η ), and loyal preferene u (v ). The loyal preferene u (v ) is measured in hoie probabiliy for dire (indire hannel by a loyal usomer of he firm. In he mos reen researh by Yao and Liu (3), he dynamis of he redisribuion proess is haraerized by a sysem of ordinary differenial equaions exended from Bass (969), and Muller (983). Equaion () is he ordinary differenial equaion of he sysem adoped in heir paper. The noaions are summarized below: ( : he poenial marke size x ( ) : sales ransaed via dire hannel a ime. 63

2 Developmens in Business Simulaion and Experienial Learning, Volume 3, 3 x ( ) : sales ransaed via indire hannel a ime. : diffusion oeffiien for negaive word-of-mouh moving away from dire hannel. : diffusion oeffiien for negaive word-of-mouh moving away from indire hannel. η : diffusion oeffiien for promoion effe oward dire hannel. η : diffusion oeffiien for promoion effe oward indire hannel. u : oeffiien of loyal preferene for dire hannel. v : oeffiien of loyal preferene for indire hannel. x( = x( + η( ( x( x ( ) + ux ( () x ( = x ( + η ( ( x( x ( ) + vx ( If he marke size ( is deerminisi, hen he following proposiion and heorem hold. uv Proposiion. If, hen here exiss an equilibrium soluion x ( ha has he following properies:. The equilibrium is feasible: x (, and x ( + x ( (. The equilibrium is proporionae o he oal marke: _ x( r ( x( = = _ r x ( where = r + r + uv r = η + uη r = vη + η 3. The ross-hannel redisribuion raio is imeinvarian: x( r η + uη = =. x ( r vη + η Proof: see Yao & Liu (3). By proposiion, he equilibrium redisribuion, if i exiss, will be solely dependen on he diffusion parameers, regardless of he iniial ondiions. v η Theorem. If u η,, and, η + η > and if he marke is oninuous and bounded (i.e., ( is oninuous in, and < ( K, where K is a posiive onsan, hen he equilibrium redisribuion x ( is asympoially sable. Proof: see Yao & Liu (3). A redisribuion soluion is said o be asympoially sable if i is sable regardless of he iniial saus of he sysem. In oher words, an asympoially sable redisribuion will ulimaely sele down ino a seady sae regardless of when o add a dire hannel and wha volume of exising sales exis a he ime of amendmen of a dire hannel. Inuiively, an equilibrium redisribuion an be unsable in he sense ha if a hange or disurbane ours o he urren equilibrium siuaion, he sysem may move away from he equilibrium sae and hen diverge hereon, whih is highly undesirable. The sabiliy ondiions are unforunaely unaainable under more realisi marke siuaions where he demand onains unerain fluuaions and is auo-orrelaed. In nex seion, we will use simulaion o sudy if he sabiliy ondiions obained under onsan marke remain appliable o unerain marke siuaions, inluding he demand following a known disribuion and sohasi diffusion parameers. SIMULATIO STUDY In his seion, we invesigae, via simulaion, on how differen ime-varian marke size affes he hannel redisribuion. We simulae imes for eah ase, and olle he sample means of demand on boh hannels when hey are sable, also we will alulae 95% onfidene inerval (CI) for he redisribuion raio. Firs we will assume marke size is variable wih known disribuion, and we will onsider wo ases here, one is exogenous random disurbane whih means he demand is deerminisi wih some disurbane (whie noise); he oher is sohasi aggregae fluuaion whih means he fluuaion ould be umulaive. Case : Exogenous Random Disurbane, i.e., = + ( e ) + ε (. Where ε is a whie noise, and ε ~ (, σ ) wih σ known. Example : we assume he parameers are as following: =.3 u =.4 η =. v =. =.6 η =.5 = = σ = =.3 Figure 3. gives he redisribuion rajeories for his ase. Boh demands on dire and indire hannel inreases as marke size inreases. 64

3 Developmens in Business Simulaion and Experienial Learning, Volume 3, X X Figure 3. Inreasing Redisribuion Soluions Table 3. summarizes he ross-hannel redisribuion raio when he demands reah equilibrium under differen iniial ondiions. We find he demands on boh hannel are sill sable regardless of he iniial demand on boh hannels. Iniial ondiion x x x / x 95% CI x ()=, x ()= (.88,.886) X ()=, x ()= (.874,.884) X ()=, x ()= (.88,.885) Table 3. The ross-hannel redisribuion raion on boh hannels Example : =.5 v =. = u =.4 η =.5 =.5 η =.5 = σ = =.3 This example shows he dereasing hannel redisribuion. 3 X X Figure 3. Dereasing Redisribuion Soluions 65

4 Developmens in Business Simulaion and Experienial Learning, Volume 3, 3 We lis more simulaion examples in able 3.. All he simulaion resuls show ha he proposiion and heorem sill hold. I indiaes he equilibrium redisribuion exiss and i is asympoially sable as long as u η, v η even if he marke size follows exogenous random disurbane. This simulaion resuls are onsisen wih Yao & Liu s (3). v =.5 σ = Parameers =.9 u =. =.3 v =. σ = v =.8 σ = =.6 η =. =. u =. =. η =.6 =.3 η =.9 =. u =. =.3 v =. =.3 =.3 η =. =.3 η =.3 η =. = = 3 σ = =.6 v =. σ = u =.4 =.6 =.3 η =. η =.5 u =.4 η =.5 =. η =.5 x x x / x ( η + uη ) / ( vη + η ) Table 3. Simulaion resuls of sample means for ( = + ( e ) + ε Case : Sohasi Aggregae Fluuaion, i.e., ( ) = + ( + ε )( e ). Where ε ~ (, σ ) wih σ known. Example 3: In his example, we assume =.3 u =.4 η =. v =. =.6 η =.5 = = σ = =.3 Figure 3.3 illusraes he inreasing redisribuion rajeory for his example. 3 X X Figure 3.3 Inreasing Redisribuion Soluions 66

5 Developmens in Business Simulaion and Experienial Learning, Volume 3, 3 Example 4: In his example, we assume: =.5 v =. = u =.4 η =.5 =.5 η =.5 = σ = =.3 Figure 3.4 shows he dereasing rajeory soluions for his example. 3 X X 5 Figure 3.4 Dereasing Redisribuion Soluions We lis more examples in able 3.3 for ase. All he simulaion resuls show ha he proposiion and heorem hold again in his ase. I implies even if he marke size follows sohasi aggregae fluuaion, he equilibrium redisribuion exiss and i is asympoially sable as long as u η, v η. =.9 v =.5 σ = =.3 v =. σ = =. v =.8 σ = =.3 v =. = =.3 =.6 v =. σ = Parameers u =. =.6 η =. =. u =. =. η =.6 =.3 η =.9 u =. =.3 η =. =.3 η =.3 η =. =.6 η =.5 u =.4 η =. = 3 σ = u =.4 η =.5 =. η =.5 =.3 x x x / x ( η + uη ) / ( vη + η ) Table 3.3 Simulaion resuls of sample means for ( ) = + ( + ε )( e ) 67

6 Developmens in Business Simulaion and Experienial Learning, Volume 3, 3 ow we examine he model when he parameers are sohasi. When he parameers of he diffusion equaions are sohasi, ha is: x = ˆ ( x ( + η( ( x ( x( ) + u x( x( = x( + η( ( x( x( ) + v x(, where ~ (, ), ~ (, ) σ ~ ( η, σ ~ ( u, σ σ η ), η ~ ( η, ) σ u ), v ~ ( v, σ ) Here we assume σ is smaller enough so ha u η, v η almos always holds Example 5: In his example, we assume =.3 v =. = u =.4 η =. =.6 η =.5 = 3 = x η + uη = =.88 x vη + η We lis he simulaion resuls in able 3.5, 3.6 when σ =.5, σ =. respeively. The resuls show ha he demands on boh hannels are sable on equilibrium. Simulaions indiae ha he proposiion and heorem sill hold even if he marke size is no deerminisi and he diffusion parameers are sohasi. The simulaion sudy suggess he resuls given by Yao & Liu under he ondiion of deerminisi demand an be applied o he siuaion where demand is unerain. Thus his simulaion sudy is he valuable exension of heir researh resuls. Iniial ondiion x x x / x 95% CI x ()=, x ()= (.8495,.47) x ()=, x ()= (.8385,.67) x ()=, x ()= (.78,.934) Table 3.4 Redisribuion Soluions when σ =. 5 Iniial ondiion x x x / x 95% CI x ()=, x ()= (.7553,.3) x ()=, x ()= (.434,.85) x ()=, x ()= (.896,.67) Table 3.5 Redisribuion Soluions when σ =. SUMMARY In his paper, we ake he simulaion approah o sudy he effe of marke haraerisis and parameers variabiliy on he hannel redisribuion exended from he previous researh. Our findings indiae no maer he marke size follows exogenous random disurbane, sohasi aggregae fluuaion, or he diffusion parameers vary, he raio of demand on dire hannel o he demand on indire hannel is same when he redisribuion is sable. So he resuls of demand redisribuion when marke size is deerminisi are prey robus, i really provides managerial impliaions for he hannel of disribuion design issue. REFERECES Bass, F.M (969) A ew Produ Growh Model for Consumer Durables. Managemen Siene, vol. 5, 969, 5-7. Majumdar Sumi K. and Ramaswamy Venkaram (995) Going Dire o Marke: The Influene of Exhange Condiions, Sraegi Managemen Journal, June 995, Muller, E. (983), Trial/Awareness Adverising Deision: A Conrol Problem wih Phase Diagrams wih on- Saionary Boundaries. J. Eon. Dynamis and Conrol, 6 (983), Shwarz Ephraim and Briody Dan (999) IBM, HP Go o Dire- sales Model. Infoworld, May 999. Sridhar Balasubramanian (998) Mail versus Mall: A Sraegy Analysis of Compeiion, Beween Markeers and Convenional Reailers. Markeing Siene, vol. 7, issue 3, 998,8-95. Tsay Andy and aren Agrawal () Channel Dynamis Under Prie and Servie Compeiion. Manufauring and Servie Operaions Managemen, (), Yao Dong-Qing and John Liu (3) Channel Redisribuion wih Dire Selling. European Journal of Operaional Researh, vol. 44, 3,