# American Institute of Mathematical Sciences

January  2011, 7(1): 117-137. doi: 10.3934/jimo.2011.7.117

## The optimal production and sales policy for a new product with negative word-of-mouth

 1 School of Software, Dongguan University of Technology, Dongguan 523808, Guangdong, China 2 Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China, China, China

Received  December 2009 Revised  October 2010 Published  January 2011

In this paper, we consider a firm which maximizes its profit by determining the production and sales policies for a new product during the lifetime of the product. Because of capacity constraint, we extend Bass demand process a more general case including the negative effectiveness of word-of-mouth. We analyze the production and sales policies in two cases: strong negative word-of-mouth and weak negative word-of-mouth. In the case of strong negative word-of-mouth, we show that myopic policy is optimal under some mild conditions. In the case of weak negative word-of-mouth, we show that build-up policy is optimal in a special case of negligible holding cost and discount rate. However, for positive holding cost and discount rate, we compare myopic policy with build-up policy by numerical examples, and show that the build-up policy is no longer a robust approximation to the optimal policy.
Citation: Xiaoming Yan, Ping Cao, Minghui Zhang, Ke Liu. The optimal production and sales policy for a new product with negative word-of-mouth. Journal of Industrial and Management Optimization, 2011, 7 (1) : 117-137. doi: 10.3934/jimo.2011.7.117
##### References:
 [1] F. M. Bass, A new product growth for model consumer durables, Management Science, 15 (1969), 215-227. doi: 10.1287/mnsc.15.5.215. [2] F. M. Bass, T. V. Krishnan and D. C. Jain, Why the bass model fits without decision variables, Marketing Science, 13 (1984), 203-223. doi: 10.1287/mksc.13.3.203. [3] J. G. Blodgett, D. H. Granbois and R. G. Walters, The effects of perceived justice on complainants' negative word-of-mouth behavior and repatronage intentions, Journal of Retailing, 69 (1983), 399-428. doi: 10.1016/0022-4359(93)90015-B. [4] P. F. Bone, Word-of-Mouth effects on short-term and long-term product judgments, Journal of Business Research, 32 (1995), 213-223. doi: 10.1016/0148-2963(94)00047-I. [5] R. Cellini and L. Lambertini, Advertising in a differential oligopoly game, Journal of Optimization Theory and Applications, 116 (2003), 61-81. doi: 10.1023/A:1022158102252. [6] D. Charlett, R. Garland and N. Marr, How damaging is negative word of mouth? Marketing Bulletin, 6 (1995), 42-50. [7] G. Cheryl, Simultaneous and dynamic price, production, inventory and capacity decisions, European Journal of Operational Research, 35 (1988), 426-441. doi: 10.1016/0377-2217(88)90232-9. [8] C. Dellarocas, The digitization of word of mouth: promise and challenges of online feedback mechanisms, Management Science, 49 (2003), 1407-1424. doi: 10.1287/mnsc.49.10.1407.17308. [9] E. J. Dockner and G. E. Fruchter, Dynamic strategic pricing and speed of diffusion, Journal of Optimization Theory and Applications, 123 (2004), 331-348. doi: 10.1007/s10957-004-5152-2. [10] R. J. Dolan and A. P. Jeuland, Experience curves and dynamic demand models: implications for optimal pricing strategies, Journal of Marketing, 45 (1984), 52-63. doi: 10.2307/1251720. [11] J. A. Dodson and E. Muller, Models of new product diffusion through advertising and word-of-mouth, Management Science, 24 (1978), 1568-1578. doi: 10.1287/mnsc.24.15.1568. [12] R. Easta, K. Hammondb and W. Lomax, Measuring the impact of positive and negative word of mouth on brand purchase probability, International Journal of Research in Marketing, 25 (2008), 215-24. doi: 10.1016/j.ijresmar.2008.04.001. [13] R. Easta, K. Hammondb and M. Wright, The relative incidence of positive and negative word of mouth: A multi-category study, International Journal of Research in Marketing, 24 (2007), 175-184. doi: 10.1016/j.ijresmar.2006.12.004. [14] Z. Griliches, Hybrid corn: an exploration in the economics of technoligical change, Econometrica, 25 (1957), 501-522. doi: 10.2307/1905380. [15] R. F. Hartl, A. J. Novak and S. P. Sethi, Optimal pricing of a product diffusing in rich and poor populations, Journal of Optimization Theory and Applications, 117 (2003), 349-375. doi: 10.1023/A:1023635807884. [16] X. He and S. P. Sethi, Dynamic slotting and pricing decisions in a durable product supply chain, Journal of Optimization Theory and Applications, 137 (2008), 363-379. doi: 10.1007/s10957-007-9330-x. [17] D. Heyman and M. J. Sobel, "Stochastic Models in Operations Research Volume II: Stochastic Optimization," 1st edition, McGraw-Hill Book Company, New York, 1984. [18] T. Ho, S. Savin and C. Terwiesch, Managing demand and sale dynamics in new product diffusion under supply constraint, Management Science, 48 (2002), 187-206. doi: 10.1287/mnsc.48.2.187.257. [19] D. Jain, V. Mahajan and E. Muller, Innovation diffusion in the presence of supply restrictions, Marketing Science, 10 (1991), 83-90. doi: 10.1287/mksc.10.1.83. [20] S. JØrgensen, S. Taboubi and G. Zaccour, Cooperative advertising in a marketing channel, Journal of Optimization Theory and Applications, 110 (2001), 145-158. doi: 10.1023/A:1017547630113. [21] S. Kalish, Monopolist pricing with dynamic demand and production cost, Marketing Science, 2 (1983), 135-159. doi: 10.1287/mksc.2.2.135. [22] S. Kalish, A new product adoption model with price, advertising and uncertainty, Management Science, 31 (1985), 1569-1585. doi: 10.1287/mnsc.31.12.1569. [23] T. V. Krishnan, F. M. Bass and D. C. Jain, Optimal pricing strategy for new products, Management Science, 45 (1999), 1650-1663. doi: 10.1287/mnsc.45.12.1650. [24] T. V. Krishnan and D. C. Jain, Optimal dynamic advertising policy for new products, Management Science, 52 (2006), 1957-1969. doi: 10.1287/mnsc.1060.0585. [25] S. Kumar and J. M. Swaminathan, Diffusion of innovations under supply constraints, Operations Research, 51 (2003), 866-879. doi: 10.1287/opre.51.6.866.24918. [26] S. Kumar and S. P. Sethi, Dynamic pricing and advertising for web content providers, European Journal of Operational Research, 197 (2009), 924-944. doi: 10.1016/j.ejor.2007.12.038. [27] R. N. Laczniak, T. E. DeCarlo and S. N. Ramaswami, Consumers' responses to negative word-of-mouth communication: An attribution theory perspective, Journal of Consumer Psychology, 11 (2001), 57-73. doi: 10.1207/S15327663JCP1101_5. [28] G. T. Lau and S. Ng, Individual and situational factors influencing negative word-of-mouth behaviour, Canadian Journal of Administrative Sciences, 18 (2009), 163-178. doi: 10.1111/j.1936-4490.2001.tb00253.x. [29] X. Luo, Quantifying the long-term impact of negative word of mouth on cash flows and stock prices, Marketing Science, 28 (2009), 148-165. doi: 10.1287/mksc.1080.0389. [30] E. Mansfield, Technical change and the rate of imitation, Econometrica, 29 (1961), 741-766. doi: 10.2307/1911817. [31] V. Mahajan, E. Muller and F. M. Bass, New product diffusion models in marketing: A review and directions for research, Journal of Marketing Research, 54 (1990), 1-26. [32] V. Mahajan, E. Muller and R. A. Kerin, Introduction strategy for new products with positive and negative word-of-mouth, Management Science, 30 (1984), 1389-1404. doi: 10.1287/mnsc.30.12.1389. [33] V. Mahajan, E. Muller and Y. Wind, "New-Product Diffusion Models," 1st edition, Kluwer Academic Publisher, Boston, MA, 2000. [34] H. I. Mesak and J. W. Clark, Monopolist optimum pricing and advertising policies for diffusion models of new product innovations, Optimal Control Applications and Methods, 19 (1998), 111-136. doi: 10.1002/(SICI)1099-1514(199803/04)19:2<111::AID-OCA622>3.0.CO;2-M. [35] M. L. Richins, Negative word-of-mouth by dissatisfied consumers: a pilot study, Journal of Marketing, 47 (1983), 68-78. doi: 10.2307/3203428. [36] B. Robinson and C. Lakhani, Dynamic price models for new product planning, Management Science, 21 (1975), 1113-1122. doi: 10.1287/mnsc.21.10.1113. [37] S. P. Sethi and G. L. Thompson, "Optimal Control Theory," 2nd edition, Kluwer Academic Publisher, Boston, MA, 2000. [38] H. Simon and K. H. Sebastian, Diffusion and advertising: the german telephone campaign, Management Science, 33 (1987), 451-466. doi: 10.1287/mnsc.33.4.451. [39] J. Singh, Voice, exit, and negative word-of-mouth behaviors: An investigation across three service categories, Journal of the Academy of Marketing Science, 18 (1990), 1-15. doi: 10.1007/BF02729758. [40] G. L. Thomposon, S. P. Sethi and J-T. Teng, Strong planning and forecast horizons for a model with simultaneous price and production decisions, European Journal of Operational Research, 16 (1984), 378-388. doi: 10.1016/0377-2217(84)90292-3. [41] F. V. Wangenheim, Postswitching negative word of mouth, Journal of Service Research, 8 (2005), 67-78. doi: 10.1177/1094670505276684. [42] I. M. Wetzer, M. Zeelenberg and R. Pieters, Never eat in that restaurant, i did!: exploring why people engage in negative word-of-mouth communication, Psychology and Marketing, 24 (2007), 661-680. doi: 10.1002/mar.20178. [43] X. Yan and K. Liu, Optimal control problems for a new product with word-of-mouth, International Journal of Production Economics, 119 (2009), 402-414. doi: 10.1016/j.ijpe.2009.03.011.

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##### References:
 [1] F. M. Bass, A new product growth for model consumer durables, Management Science, 15 (1969), 215-227. doi: 10.1287/mnsc.15.5.215. [2] F. M. Bass, T. V. Krishnan and D. C. Jain, Why the bass model fits without decision variables, Marketing Science, 13 (1984), 203-223. doi: 10.1287/mksc.13.3.203. [3] J. G. Blodgett, D. H. Granbois and R. G. Walters, The effects of perceived justice on complainants' negative word-of-mouth behavior and repatronage intentions, Journal of Retailing, 69 (1983), 399-428. doi: 10.1016/0022-4359(93)90015-B. [4] P. F. Bone, Word-of-Mouth effects on short-term and long-term product judgments, Journal of Business Research, 32 (1995), 213-223. doi: 10.1016/0148-2963(94)00047-I. [5] R. Cellini and L. Lambertini, Advertising in a differential oligopoly game, Journal of Optimization Theory and Applications, 116 (2003), 61-81. doi: 10.1023/A:1022158102252. [6] D. Charlett, R. Garland and N. Marr, How damaging is negative word of mouth? Marketing Bulletin, 6 (1995), 42-50. [7] G. Cheryl, Simultaneous and dynamic price, production, inventory and capacity decisions, European Journal of Operational Research, 35 (1988), 426-441. doi: 10.1016/0377-2217(88)90232-9. [8] C. Dellarocas, The digitization of word of mouth: promise and challenges of online feedback mechanisms, Management Science, 49 (2003), 1407-1424. doi: 10.1287/mnsc.49.10.1407.17308. [9] E. J. Dockner and G. E. Fruchter, Dynamic strategic pricing and speed of diffusion, Journal of Optimization Theory and Applications, 123 (2004), 331-348. doi: 10.1007/s10957-004-5152-2. [10] R. J. Dolan and A. P. Jeuland, Experience curves and dynamic demand models: implications for optimal pricing strategies, Journal of Marketing, 45 (1984), 52-63. doi: 10.2307/1251720. [11] J. A. Dodson and E. Muller, Models of new product diffusion through advertising and word-of-mouth, Management Science, 24 (1978), 1568-1578. doi: 10.1287/mnsc.24.15.1568. [12] R. Easta, K. Hammondb and W. Lomax, Measuring the impact of positive and negative word of mouth on brand purchase probability, International Journal of Research in Marketing, 25 (2008), 215-24. doi: 10.1016/j.ijresmar.2008.04.001. [13] R. Easta, K. Hammondb and M. Wright, The relative incidence of positive and negative word of mouth: A multi-category study, International Journal of Research in Marketing, 24 (2007), 175-184. doi: 10.1016/j.ijresmar.2006.12.004. [14] Z. Griliches, Hybrid corn: an exploration in the economics of technoligical change, Econometrica, 25 (1957), 501-522. doi: 10.2307/1905380. [15] R. F. Hartl, A. J. Novak and S. P. Sethi, Optimal pricing of a product diffusing in rich and poor populations, Journal of Optimization Theory and Applications, 117 (2003), 349-375. doi: 10.1023/A:1023635807884. [16] X. He and S. P. Sethi, Dynamic slotting and pricing decisions in a durable product supply chain, Journal of Optimization Theory and Applications, 137 (2008), 363-379. doi: 10.1007/s10957-007-9330-x. [17] D. Heyman and M. J. Sobel, "Stochastic Models in Operations Research Volume II: Stochastic Optimization," 1st edition, McGraw-Hill Book Company, New York, 1984. [18] T. Ho, S. Savin and C. Terwiesch, Managing demand and sale dynamics in new product diffusion under supply constraint, Management Science, 48 (2002), 187-206. doi: 10.1287/mnsc.48.2.187.257. [19] D. Jain, V. Mahajan and E. Muller, Innovation diffusion in the presence of supply restrictions, Marketing Science, 10 (1991), 83-90. doi: 10.1287/mksc.10.1.83. [20] S. JØrgensen, S. Taboubi and G. Zaccour, Cooperative advertising in a marketing channel, Journal of Optimization Theory and Applications, 110 (2001), 145-158. doi: 10.1023/A:1017547630113. [21] S. Kalish, Monopolist pricing with dynamic demand and production cost, Marketing Science, 2 (1983), 135-159. doi: 10.1287/mksc.2.2.135. [22] S. Kalish, A new product adoption model with price, advertising and uncertainty, Management Science, 31 (1985), 1569-1585. doi: 10.1287/mnsc.31.12.1569. [23] T. V. Krishnan, F. M. Bass and D. C. Jain, Optimal pricing strategy for new products, Management Science, 45 (1999), 1650-1663. doi: 10.1287/mnsc.45.12.1650. [24] T. V. Krishnan and D. C. Jain, Optimal dynamic advertising policy for new products, Management Science, 52 (2006), 1957-1969. doi: 10.1287/mnsc.1060.0585. [25] S. Kumar and J. M. Swaminathan, Diffusion of innovations under supply constraints, Operations Research, 51 (2003), 866-879. doi: 10.1287/opre.51.6.866.24918. [26] S. Kumar and S. P. Sethi, Dynamic pricing and advertising for web content providers, European Journal of Operational Research, 197 (2009), 924-944. doi: 10.1016/j.ejor.2007.12.038. [27] R. N. Laczniak, T. E. DeCarlo and S. N. Ramaswami, Consumers' responses to negative word-of-mouth communication: An attribution theory perspective, Journal of Consumer Psychology, 11 (2001), 57-73. doi: 10.1207/S15327663JCP1101_5. [28] G. T. Lau and S. Ng, Individual and situational factors influencing negative word-of-mouth behaviour, Canadian Journal of Administrative Sciences, 18 (2009), 163-178. doi: 10.1111/j.1936-4490.2001.tb00253.x. [29] X. Luo, Quantifying the long-term impact of negative word of mouth on cash flows and stock prices, Marketing Science, 28 (2009), 148-165. doi: 10.1287/mksc.1080.0389. [30] E. Mansfield, Technical change and the rate of imitation, Econometrica, 29 (1961), 741-766. doi: 10.2307/1911817. [31] V. Mahajan, E. Muller and F. M. Bass, New product diffusion models in marketing: A review and directions for research, Journal of Marketing Research, 54 (1990), 1-26. [32] V. Mahajan, E. Muller and R. A. Kerin, Introduction strategy for new products with positive and negative word-of-mouth, Management Science, 30 (1984), 1389-1404. doi: 10.1287/mnsc.30.12.1389. [33] V. Mahajan, E. Muller and Y. Wind, "New-Product Diffusion Models," 1st edition, Kluwer Academic Publisher, Boston, MA, 2000. [34] H. I. Mesak and J. W. Clark, Monopolist optimum pricing and advertising policies for diffusion models of new product innovations, Optimal Control Applications and Methods, 19 (1998), 111-136. doi: 10.1002/(SICI)1099-1514(199803/04)19:2<111::AID-OCA622>3.0.CO;2-M. [35] M. L. Richins, Negative word-of-mouth by dissatisfied consumers: a pilot study, Journal of Marketing, 47 (1983), 68-78. doi: 10.2307/3203428. [36] B. Robinson and C. Lakhani, Dynamic price models for new product planning, Management Science, 21 (1975), 1113-1122. doi: 10.1287/mnsc.21.10.1113. [37] S. P. Sethi and G. L. Thompson, "Optimal Control Theory," 2nd edition, Kluwer Academic Publisher, Boston, MA, 2000. [38] H. Simon and K. H. Sebastian, Diffusion and advertising: the german telephone campaign, Management Science, 33 (1987), 451-466. doi: 10.1287/mnsc.33.4.451. [39] J. Singh, Voice, exit, and negative word-of-mouth behaviors: An investigation across three service categories, Journal of the Academy of Marketing Science, 18 (1990), 1-15. doi: 10.1007/BF02729758. [40] G. L. Thomposon, S. P. Sethi and J-T. Teng, Strong planning and forecast horizons for a model with simultaneous price and production decisions, European Journal of Operational Research, 16 (1984), 378-388. doi: 10.1016/0377-2217(84)90292-3. [41] F. V. Wangenheim, Postswitching negative word of mouth, Journal of Service Research, 8 (2005), 67-78. doi: 10.1177/1094670505276684. [42] I. M. Wetzer, M. Zeelenberg and R. Pieters, Never eat in that restaurant, i did!: exploring why people engage in negative word-of-mouth communication, Psychology and Marketing, 24 (2007), 661-680. doi: 10.1002/mar.20178. [43] X. Yan and K. Liu, Optimal control problems for a new product with word-of-mouth, International Journal of Production Economics, 119 (2009), 402-414. doi: 10.1016/j.ijpe.2009.03.011.
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