THE EFFECT OF DEMAND PATTERN ON THE CHANNEL REDISTRIBUTION Developments in Business Simulation and Experiential Learning, Volume 30, 2003 SIMULATION STUDY OF STOCHASTIC CHANNEL REDISTRIBUTION Yao Dong-Qing Towson University dyao@towson.edu ABSTRACT In this paper, we investigate sales redistribution between direct (i.e., “on-line”) and indirect channels when market demand exhibits a general stochastic pattern. We model the dynamics of the inter-channel diffusion by a system of two differential equations, and then obtain a simple analytical expression of stability conditions for channel redistribution when the market demand is deterministic. The stability conditions are unfortunately unattainable under more realistic market situations where the demand contains uncertain fluctuations. The results from an extensive simulation study indicate that the stability conditions obtained under deterministic market remain applicable to uncertain market situations, including the demand following a stochastic fluctuation and the diffusion parameters following a known distribution. INTRODUCTION The increasing success of electronic commerce has made disintermediation possible in many industries. Consumers now can purchase products either from direct channel or from indirect channel [Majumdar and Venkatram, 1995]. Dell proves the direct model is viable. In many cases it is the best strategy for vendors trying to cut distribution costs. As a result, many manufacturers incorporated direct channel into their distribution channels. For example, HP sold its small and midsize business line of products on its web site, and IBM also rolled out its direct sales web site to its largest accounts [Schwartz and Briody, 1999]. These direct channels exist in parallel with the conventional indirect channels [Sridhar, 1998]. However, the existence of both direct channel and indirect channel arises many interesting questions. For example, does adding additional channel mean more sales? How much should each channel capacity be? How to redistribute channels among distribution channels? In this paper, we will study the channel redistribution on direct and indirect channels. Mainly we will study what effect the different market sizes have on the channel redistribution. In this paper, we will first develop a two- dimensional diffusion model, which mainly consists of a system of two differential equations. We also obtain key results regarding the steady-state redistribution when market size is deterministic. Then, we will study the demand redistribution between direct and indirect channels under stochastic market size. We will choose two common patterns for market size, namely exogenous random disturbance, and stochastic aggregate fluctuation, to study the demand allocation on direct and indirect channels. Since it is difficult to get a closed-form solution when market size is stochastic, we will use simulation to study the effect of market size on channel redistribution. Finally we will assume parameters in differential equations are stochastic, and use simulation to study the demand distribution on two channels. MODEL Consider a simple supply-chain system in an established market, in which a firm is faced with a decision to adopt a direct channel. Suppose the total market size at time t is given as , which is assumed to be continuous and bounded ( ). The primary issue to be examined before a decision on the direct channel can be made pertains to how market sales would redistribute upon the amendment of a direct channel. 0)( >tN N ( ∞<≤ Kt) Upon the addition of a direct channel, there will be demand diffusion between the two channels. For example, the rate of change in sales via direct channel (or indirect channel), denoted as ( x ), will be affected by negative word-of-mouth (WOM) effect )(1 tx& )(2 t& 1ξ ( 2ξ ), positive sales promotion (PRO) effect 1η ( 2η ), and loyal preference ( v ). The loyal preference ( v ) is measured in choice probability for direct (indirect) channel by a loyal customer of the firm. u u In the most recent research by Yao and Liu (2003), the dynamics of the redistribution process is characterized by a system of ordinary differential equations extended from Bass (1969), and Muller (1983). Equation (1) is the ordinary differential equation of the system adopted in their paper. The notations are summarized below: :)(tN the potential market size :)(1 tx sales transacted via direct channel at time t . 63 mailto:dyao@towson.edu Developments in Business Simulation and Experiential Learning, Volume 30, 2003 :)(2 tx sales transacted via indirect channel at time t . 1ξ : diffusion coefficient for negative word-of-mouth moving away from direct channel. 2ξ : diffusion coefficient for negative word-of-mouth moving away from indirect channel. 1η : diffusion coefficient for promotion effect toward direct channel. 2η : diffusion coefficient for promotion effect toward indirect channel. :u coefficient of loyal preference for direct channel. :v coefficient of loyal preference for indirect channel.     +−−+−= +−−+−= • • )())()()(()()( )())()()(()()( 1212222 2211111 tvxtxtxtNtxtx tuxtxtxtNtxtx ηξ ηξ (1) If the market size N(t) is deterministic, then the following proposition and theorem hold. Proposition. If 21ξξ≤uv , then there exists an equilibrium solution )(tx that has the following properties: 1. The equilibrium is feasible: 0)( ≥tx , and )()()( 21 tNtxtx ≤+ 2. The equilibrium is proportionate to the total market:           ∆ =             = 1 1 2 _ 1 _ )( )( )( )( r r tN tx tx tx where uvrr −++=∆ 2121 ξξ 2121 ηηξ ur += 2112 ηξη += vr 3. The cross-channel redistribution ratio is time- invariant: 211 212 2 1 2 1 )( )( ηξη ηηξ + + == v u r r tx tx . Proof: see Yao & Liu (2003). By proposition, the equilibrium redistribution, if it exists, will be solely dependent on the diffusion parameters, regardless of the initial conditions. Theorem. If 1η≥u , 2η≥v Kt ≤ , and , and if the market is continuous and bounded (i.e., is continuous in t, and 02112 >+ ηξηξ )(tN N< )(0 , where K is a positive constant), then the equilibrium redistribution )(tx is asymptotically stable. Proof: see Yao & Liu (2003). A redistribution solution is said to be asymptotically stable if it is stable regardless of the initial status of the system. In other words, an asymptotically stable redistribution will ultimately settle down into a steady state regardless of when to add a direct channel and what volume of existing sales exist at the time of amendment of a direct channel. Intuitively, an equilibrium redistribution can be unstable in the sense that if a change or disturbance occurs to the current equilibrium situation, the system may move away from the equilibrium state and then diverge thereon, which is highly undesirable. The stability conditions are unfortunately unattainable under more realistic market situations where the demand contains uncertain fluctuations and is auto-correlated. In next section, we will use simulation to study if the stability conditions obtained under constant market remain applicable to uncertain market situations, including the demand following a known distribution and stochastic diffusion parameters. SIMULATION STUDY In this section, we investigate, via simulation, on how different time-variant market size affects the channel redistribution. We simulate 1000 times for each case, and collect the sample means of demand on both channels when they are stable, also we will calculate 95% confidence interval (CI) for the redistribution ratio. First we will assume market size is variable with known distribution, and we will consider two cases here, one is exogenous random disturbance which means the demand is deterministic with some disturbance (white noise); the other is stochastic aggregate fluctuation which means the fluctuation could be cumulative. Case 1: Exogenous Random Disturbance, i.e., t cteNNtN ε+−+= − )1()( 10 . Where tε is a white noise, and with known. ),0(~ 2σε Nt 2σ Example 1: we assume the parameters are as following: 3.0c 100 2000 1000 005.0 006.0 01.0 002.0 004.0 03.0 10 22 11 ==== === === σ ηξ ηξ NN v u Figure 3.1 gives the redistribution trajectories for this case. Both demands on direct and indirect channel increases as market size increases. 64 Developments in Business Simulation and Experiential Learning, Volume 30, 2003 Figure 3.1 Increasing Redistribution Solutions 0 100 200 0 1000 2000 3000 4000 Time D em an d X1 X2 Table 3.1 summarizes the cross-channel redistribution ratio when the demands reach equilibrium under different initial conditions. We find the demands on both channel are still stable regardless of the initial demand on both channels. Initial condition 1x 2x 21 / xx 95% CI x1(0)=0, x2(0)=50 281 1492 0.1883 (0.1881, 0.1886) X1(0)=0, x2(0)=300 281 1491 0.1883 (0.1874, 0.1884) X1(0)=0, x2(0)=800 280 1492 0.1883 (0.1882, 0.1885) Table 3.1 The cross-channel redistribution ration on both channels Example 2: 3.0c 200 2000N 1000 005.0 05.0 01.0 005.0 004.0 15.0 10 22 11 ==== === === σ ηξ ηξ N v u This example shows the decreasing channel redistribution. 0 10 20 30 40 50 60 70 80 0 1000 2000 3000 Time D em an d X1 X2 Figure 3.2 Decreasing Redistribution Solutions 65 Developments in Business Simulation and Experiential Learning, Volume 30, 2003 We list more simulation examples in table 3.2. All the simulation results show that the proposition and theorem still hold. It indicates the equilibrium redistribution exists and it is asymptotically stable as long as 1η≥u , 2η≥v even if the market size follows exogenous random disturbance. This simulation results are consistent with Yao & Liu’s (2003). Parameters 1x 2x 21 / xx )( /)( 211 212 ηξη ηηξ + + v u 1.0c 100 01.0 06.0 05.0 006.0 01.0 09.0 2 22 11 == === === σ ηξ ηξ v u 210.3 548.49 0.38339 0.3833 1.0c 100 009.0 3.0 1.0 1.0 2.0 3.0 2 22 11 == === === σ ηξ ηξ v u 831.9 332.3 2.5038 2.5039 3.0c 100 03.0 3.0 08.0 1.0 2.0 1.0 2 22 11 == === === σ ηξ ηξ v u 1767 540 3.27 3.27 3.0c 100 3000 1000 005.0 006.0 01.0 002.0 004.0 03.0 2 0 22 11 = === === === σ ηξ ηξ NN v u 280 1489 0.188 0.188 3.0c 100 005.0 1.0 01.0 005.0 004.0 6.0 2 22 11 == === === σ ηξ ηξ v u 24.5 143 0.17 0.17 Table 3.2 Simulation results of sample means for t tceNNtN ε+−+= − )1()( 2 10 Case 2: Stochastic Aggregate Fluctuation, i.e., )1)(()( 2 10 tc t eNNtN −−++= ε . Where with known. ),0(~ 2σε Nt 2σ Example 3: In this example, we assume 3.0c 100 2000 1000 005.0 006.0 01.0 002.0 004.0 03.0 10 22 11 ==== === === σ ηξ ηξ NN v u Figure 3.3 illustrates the increasing redistribution trajectory for this example. 2001000 3000 2000 1000 0 Time D em an d X1 X2 Figure 3.3 Increasing Redistribution Solutions 66 Developments in Business Simulation and Experiential Learning, Volume 30, 2003 Example 4: In this example, we assume: Figure 3.4 shows the decreasing trajectory solutions for this example. 3.0c 200 2000N 1000 005.0 05.0 01.0 005.0 004.0 15.0 10 22 11 ==== === === σ ηξ ηξ N v u 100500 3000 2000 1000 0 Time D em an d X1 X2 Figure 3.4 Decreasing Redistribution Solutions We list more examples in table 3.3 for case 2. All the simulation results show that the proposition and theorem hold again in this case. It implies even if the market size follows stochastic aggregate fluctuation, the equilibrium redistribution exists and it is asymptotically stable as long as 1η≥u , 2η≥v . Parameters 1x 2x 21 / xx )( /)( 211 212 ηξη ηηξ + + v u 1.0c 100 01.0 06.0 05.0 006.0 01.0 09.0 2 22 11 == === === σ ηξ ηξ v u 209 546 0.3832 0.3833 1.0c 100 009.0 3.0 1.0 1.0 2.0 3.0 2 22 11 == === === σ ηξ ηξ v u 832 333 2.503 2.5039 3.0c 100 03.0 3.0 08.0 1.0 2.0 1.0 2 22 11 == === === σ ηξ ηξ v u 1770 540 3.27 3.27 3.0c 100 3000 1000 005.0 006.0 01.0 002.0 004.0 03.0 2 0 22 11 = === === === σ ηξ ηξ NN v u 280 1490 0.188 0.188 3.0c 100 005.0 1.0 01.0 005.0 004.0 6.0 2 22 11 == === === σ ηξ ηξ v u 24.6 144.6 0.17 0.17 Table 3.3 Simulation results of sample means for )1)(()( 2 10 tc t eNNtN −−++= ε 67 Developments in Business Simulation and Experiential Learning, Volume 30, 2003 Now we examine the model when the parameters are stochastic. When the parameters of the diffusion equations are stochastic, that is:     +−−+−= +−−+−= • • )())()()(()()( )())()()(()(ˆ)( 1 ^ 21 ^ 12 ^ 22 2 ^ 21 ^ 111 txvtxtxtNtxtx txutxtxtNtxtx ηξ ηξ , where , ),(~ 2 11 ^ 1 σξξ N ),(~ 2 12 ^ 2 σξξ N , ),(~ 2 11 ^ 1 σηη N ),(~ 2 12 ^ 2 σηη N , v ),(~ 2 1 ^ σuNu ),(~ 2 1 ^ σvN Here we assume 1σ is smaller enough so that u , almost always holds ∧∧ ≥ 1η ∧∧ ≥ 2ηv Example 5: In this example, we assume 1c 3000 1000 005.0 006.0 01.0 002.0 004.0 03.0 20 22 11 === === === NN v u ηξ ηξ 1882.0 211 212 2 1 = + + = ηξη ηηξ v u x x We list the simulation results in table 3.5, 3.6 when 005.0=σ , 01.0=σ respectively. The results show that the demands on both channels are stable on equilibrium. Simulations indicate that the proposition and theorem still hold even if the market size is not deterministic and the diffusion parameters are stochastic. The simulation study suggests the results given by Yao & Liu under the condition of deterministic demand can be applied to the situation where demand is uncertain. Thus this simulation study is the valuable extension of their research results. Initial condition 1x 2x 21 / xx 95% CI x1(0)=0, x2(0)=50 290 1506 0.1926 ( 0.18495, 0.20047) x1(0)=0, x2(0)=300 284 1480 0.1918 ( 0.18385, 0.20067) x1(0)=0, x2(0)=800 270 1487 0.1815 ( 0.17118, 0.19234) Table 3.4 Redistribution Solutions when 005.0=σ Initial condition 1x 2x 21 / xx 95% CI x1(0)=0, x2(0)=50 282 1450 0.1944 ( 0.17553, 0.21223) x1(0)=0, x2(0)=300 242 1483 0.1628 ( 0.1434, 0.1850) x1(0)=0, x2(0)=800 301 1520 0.1979 ( 0.18096, 0.21627) Table 3.5 Redistribution Solutions when 01.0=σ SUMMARY In this paper, we take the simulation approach to study the effect of market characteristics and parameters variability on the channel redistribution extended from the previous research. Our findings indicate no matter the market size follows exogenous random disturbance, stochastic aggregate fluctuation, or the diffusion parameters vary, the ratio of demand on direct channel to the demand on indirect channel is same when the redistribution is stable. So the results of demand redistribution when market size is deterministic are pretty robust, it really provides managerial implications for the channel of distribution design issue. REFERENCES Bass, F.M (1969) “A New Product Growth Model for Consumer Durables.” Management Science, vol. 15, 1969, 215-227. Majumdar Sumit K. and Ramaswamy Venkatram (1995) “Going Direct to Market: The Influence of Exchange Conditions,” Strategic Management Journal, June 1995, 353-372. Muller, E. (1983), “Trial/Awareness Advertising Decision: A Control Problem with Phase Diagrams with Non- Stationary Boundaries.” J. Econ. Dynamics and Control, 6 (1983), 333-350. Schwartz Ephraim and Briody Dan (1999) “IBM, HP Go to Direct- sales Model.” Infoworld, May 1999. Sridhar Balasubramanian (1998) “ Mail versus Mall: A Strategy Analysis of Competition, Between Marketers and Conventional Retailers.” Marketing Science, vol. 17, issue 3, 1998,181-195. Tsay Andy and Naren Agrawal (2000) “Channel Dynamics Under Price and Service Competition.” Manufacturing and Service Operations Management, 2 (2000), 372- 391. Yao Dong-Qing and John Liu (2003) “Channel Redistribution with Direct Selling.” European Journal of Operational Research, vol. 144, 2003, 646-658. 68 Table of Contents Volume 30, 2003 The Optimal Timing For Introducing Business Simulations Can Handicapped Students Access Your Class Web Site? The Competition Game: Decision Making In A Dynamic Environment Pan-Pacific Enterprises: Strategic Decision Making Simulation Study Of Stochastic Channel Redistribution The Impact Of Business War Games: Quantifying Training Effectiveness Experiential Learning: Introducing Faculty And Staff To A University Leadership Development Program The Feasibility Of The Balanced Scorecard For Business Games A Misuse Of Pims For The Validation Of Marketing Management Simulation Games Incorporating Technology Into The 21st Century Classroom: Are We Facilitating Academic Dishonesty? Improving The Effectiveness Of Peer Evaluations The Use Of A Simulation In An Integrated Mba Curriculum Student Portfolios In Business Education Student Portfolios In Business Education At Ashland University Using SAP ERP Technology To Integrate The Undergraduate Business Curriculum Board Games And Teaching Textile Marketing And Finance Blogging: A New Threat To Student Research? The Way We Talk! Take II Strategic Management: An Evaluation Of The Use Of Three Learning Methods In Hong Kong Adoption Of Discussion-Based Teaching And Assessment In Teaching Strategic Management In Hong Kong The Tobin Q As A Company Performance Indicator Using Representative Nominal Group Technique For Course Review And An Interactive Solicitation Of Ways To Enhance Absel's Image Making Teaching Matter: The Art And Science Of Teaching Business Communication Beyond Sex, Age, And Race: Exploring The Deeper Contents Of Diversity Teaching & Learning The Facilitation Process A Brief On Debriefing: What It Is And What It Isn't Changing Perceptions Of The Importance Of Leadership: The Contribution Of Individual Spirit Harmonics In Leadership Knowledge, Skills And Sustainable Values In Learning Organizations: Some Implications From The Multicultural Virtual Classroom Challenges Of Teaching Undergraduate Organizational Behavior In A Nontraditional Time Format Interactive Online Positioning With The Web-Based Product Positioning Map Graphics Package The Longitudinal Effects Of Entrepreneurship Training On Risk Tolerance: A Look At Similarities And Differences Between Male And Female Undergraduate Students Revisiting Strategy Learning In A Total Enterprise Simulation What Are Simulations For?: Learning Objectives As A Simulation Selection Device Gaming Agency Markets Cooperate For Profits Or Compete For Market? Study Of Oligopolistic Pricing With A Business Game The Design Of A Business Simulation Using A System-Dynamics-Based Approach Modeling The Product Development Function For An Entrepreneurial Firm Simulation Performance And Forecast Accuracy? Is That All? Business Manager Identification Of Competitors In Real World And Simulation Settings Monte Carlo Simulation Analysis On The Costs Reduction Argument Of Interest Rate Swaps Ebiz Game: A Scalable Online Business Simulation Game For Entrepreneurship Training A Model For Online Education Delivery And A Look At Online Delivery Effectiveness Incorporating "Company Reputation" into Total Enterprise Simulations The Genesis And Future Of The Absel "Classicos" Initiative