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## An application of approximate dynamic programming in multi-period multi-product advertising budgeting

 1 Department of Industrial Engineering, Ferdowsi University of Mashhad, Iran, Azadi Square, Mashhad, Iran 2 Faculty of Management, Economics and Social Sciences, University of Cologne, Cologne, Germany

*Corresponding author: Hossein Neghabi

Received  March 2021 Revised  August 2021 Early access November 2021

Fund Project: The second author is supported by Iran's National Elites Foundation

Advertising has always been considered a key part of marketing strategy and played a prominent role in the success or failure of products. This paper investigates a multi-product and multi-period advertising budget allocation, determining the amount of advertising budget for each product through the time horizon. Imperative factors including life cycle stage, $BCG$ matrix class, competitors' reactions, and budget constraints affect the joint chain of decisions for all products to maximize the total profits. To do so, we define a stochastic sequential resource allocation problem and use an approximate dynamic programming ($ADP$) algorithm to alleviate the huge size of the problem and multi-dimensional uncertainties of the environment. These uncertainties are the reactions of competitors based on the current status of the market and our decisions, as well as the stochastic effectiveness (rewards) of the taken action. We apply an approximate value iteration ($AVI$) algorithm on a numerical example and compare the results with four different policies to highlight our managerial contributions. In the end, the validity of our proposed approach is assessed against a genetic algorithm. To do so, we simplify the environment by fixing the competitor's reaction and considering a deterministic environment.

Citation: Majid Khalilzadeh, Hossein Neghabi, Ramin Ahadi. An application of approximate dynamic programming in multi-period multi-product advertising budgeting. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021202
##### References:
 [1] V. S. Abedi, Allocation of advertising budget between multiple channels to support sales in multiple markets, J. Operational Research Society, 68 (2017), 134-146.  doi: 10.1057/s41274-016-0026-1. [2] A. Albadvi and H. Koosha, A robust optimization approach to allocation of marketing budgets, Management Decision, 49 (2011), 601-621.  doi: 10.1108/00251741111126512. [3] T. Araujo, J. R. Copulsky, J. L. Hayes, S. J. Kim and J. Srivastava, From purchasing exposure to fostering engagement: Brand–consumer experiences in the emerging computational advertising landscape, J. Advertising, 49 (2020), 428-445.  doi: 10.1080/00913367.2020.1795756. [4] W. Arens and L. B. Courtland, Contemporary Advertising, 5$^{nd}$ edition, McGraw-Hill Education, 1994. [5] F. M. Bass, N. Bruce, S. Majumdar and B. P. S. Murthi, Wearout effects of different advertising themes: A dynamic bayesian model of the advertising-sales relationship, Marketing Science, 26 (2007), 179-195.  doi: 10.1287/mksc.1060.0208. [6] F. M. Bass and R. T. Lonsdale, An exploration of linear programming in media selection, Mathematical Models in Marketing, 132 (1976), 137-139.  doi: 10.1007/978-3-642-51565-1_46. [7] R. Bellman and S. Dreyfus, Functional approximations and dynamic programming, Math. Tables Aids Comput., (13) (1959), 247–251. doi: 10.2307/2002797. [8] C. Beltran-Royo, L. F. Escudero and H. Zhang, Multiperiod multiproduct advertising budgeting: Stochastic optimization modeling, Omega, 59 (2016), 26-39.  doi: 10.1016/j.omega.2015.02.013. [9] C. Beltran-Royo, H. Zhang, L. A. Blanco and J. Almagro, Multistage multiproduct advertising budgeting, European J. Oper. Res., 225 (2013), 179-188.  doi: 10.1016/j.ejor.2012.09.022. [10] D. P. Bertsekas, Dynamic Programming and Optimal Control, 4$^{th}$ edition, Athena Scientific, 2012. [11] P. J. Danaher and R. Rust, Determining the optimal return on investment for an advertising campaign, European J. Oper. Res., 95 (1996), 511-521.  doi: 10.1016/0377-2217(95)00319-3. [12] P. Doyle and J. Saunders, Multiproduct advertising budgeting, Marketing Science, 9 (1990), 97-113.  doi: 10.1287/mksc.9.2.97. [13] R. Du, Q. Hu and S. Ai, Stochastic optimal budget decision for advertising considering uncertain sales responses, European J. Oper. Res., 183 (2007), 1042-1054.  doi: 10.1016/j.ejor.2006.02.031. [14] M. Fischer, S. Albers, N. Wagner and M. Frie, Dynamic marketing budget allocation across countries, products, and marketing activities, J. Marketing Research, 2009. [15] H. K. Gajjar and K. G. Adil, A dynamic programming heuristic for retail shelf space allocation problem, Asia-Pac. J. Oper. Res., 28 (2011), 183-199.  doi: 10.1142/S0217595911003120. [16] T. P. Hsieh, C. Y. Dye and K. K. Lai, A dynamic advertising problem when demand is sensitive to the credit period and stock of advertising goodwill, J. Operational Research Society, 71 (2020), 948-966.  doi: 10.1080/01605682.2019.1595189. [17] K. F. Jea, J. Y. Wang and C. W. Hsu, Two-agent advertisement scheduling on physical books to maximize the total profit, Asia-Pac. J. Oper. Res., 36 (2019), 1950014. doi: 10.1142/S0217595919500143. [18] A. Kaul, S. Aggarwal, M. Krishnamoorthy and P. C. Jha, Multi-period media planning for multi-products incorporating segment specific and mass media, Ann. Oper. Res., 269 (2018), 317-359.  doi: 10.1007/s10479-018-2771-9. [19] H. Koosha and A. Albadvi, Allocation of marketing budgets to maximize customer equity, Operational Research, 20 (2020), 561-583.  doi: 10.1007/s12351-017-0356-z. [20] P. Kotler and K. L. Keller, Marketing Management, 14$^{th}$ edition, Prentice Hall, 2012. [21] N. K. Kwak, C. W. Lee and J. H. Kim, An MCDM model for media selection in the dual consumer/industrial market, European J. Operational Research, 166 (2005), 255-256.  doi: 10.1016/j.ejor.2004.02.016. [22] G. Li and B. Sun, Optimal dynamic pricing for used products in remanufacturing over an infinite horizon, Asia-Pac. J. Oper. Res., 31 (2014), 1450012.  doi: 10.1142/S0217595914500122. [23] X. Li, Y. Li and W. Cao, Cooperative advertising models in O2O supply chains, International J. Production Economics, 215 (2019), 144-152. [24] F. Lu, J. Zhang and W. Tang, Wholesale price contract versus consignment contract in a supply chain considering dynamic advertising, Int. Trans. Oper. Res., 26 (2019), 1977-2003.  doi: 10.1111/itor.12388. [25] P. Manik, A. Gupta, P. C. Jha and K. Govindan, A goal programming model for selection and scheduling of advertisements on online news media, Asia-Pac. J. Oper. Res., 33 (2016), 1650012.  doi: 10.1142/S0217595916500123. [26] M. Memarpour, E. Hassannayebi, N. F. Miab and A. Farjad, Dynamic allocation of promotional budgets based on maximizing customer equity, Operational Research, 21 (2021), 2365-2389.  doi: 10.1007/s12351-019-00510-3. [27] H. I. Mesak and H. Zhang, Optimal advertising pulsation policies: A dynamic programming approach, J. Operational Research Society, 52 (2001), 1244-1255.  doi: 10.1057/palgrave.jors.2601219. [28] A. Mihiotis and I. Tsakiris, A mathematical programming study of advertising allocation problem, Appl. Math. Comput., 148 (2004), 373-379.  doi: 10.1016/S0096-3003(02)00853-6. [29] P. A. Naik, K. Raman and R. S. Winer, Planning marketing-mix strategies in the presence of interaction effects, Marketing Science, 24 (2005), 25-34.  doi: 10.1287/mksc.1040.0083. [30] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1962), 167-168.  doi: 10.1007/978-3-642-51565-1_54. [31] B. Pérez-Gladish, I. González, A. Bilbao-Terol and M. Arenas-Parra, Planning a TV advertising campaign: A crisp multiobjective programming model from fuzzy basic data, Omega, 38 (2010), 84-94. [32] W. B. Powell, An operational planning model for the dynamic vehicle allocation problem with uncertain demands, Transportation Research Part B: Methodological, 21 (1987), 217-232. [33] W. B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, John Wiley & Sons, 2007. doi: 10.1002/9780470182963. [34] A. Prasad and S. P. Sethi, Competitive advertising under uncertainty: A stochastic differential game approach, J. Optim. Theory Appl., 123 (2004), 163-185.  doi: 10.1023/B:JOTA.0000043996.62867.20. [35] M. L. Puterman, Markov decision processes, Handbooks Oper. Res. Management Sci., 2 (1990), 331-434. [36] J. Z. Sissors and R. B. Baron, Advertising Media Planning, 7$^{th}$ edition, Mc Graw hill Publishin, 2010. [37] S. Sriram and M. U. Kalwani, Optimal advertising and promotion budgets in dynamic markets with brand equity as a mediating variable, Management Science, 53 (2007), 46-60. [38] R. S. Sutton and A. G. Barto, Toward a modern theory of adaptive networks: Expectation and prediction, Psychological Review, 88 (1981), 135-170.  doi: 10.1037/0033-295X.88.2.135. [39] J. N. Tsitsiklis, Asynchronous stochastic approximation and Q-learning, 32nd IEEE Conference on Decision and Control, 16 (1994), 185-202.  doi: 10.1109/CDC.1993.325119. [40] R. Van der Wurff, P. Bakker and R. G. Picards, Economic growth and advertising expenditures in different media in different countries, J. Media Economics, 21 (2008), 28-52. [41] M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381.  doi: 10.1287/opre.5.3.370. [42] X. Wang, F. Li and F. Jia, Optimal advertising budget allocation across markets with different goals and various constraints, Complexity, 2020, 2020. [43] C. Yang and Y. Xiong, Nonparametric advertising budget allocation with inventory constraint, European J. Oper. Res., 285 (2020), 631-641.  doi: 10.1016/j.ejor.2020.02.005. [44] Y. Yang, B. Feng, J. Salminen and B. J. Jansen, Optimal advertising for a generalized Vidale–Wolfe response model, Electronic Commerce Research, 285 (2021), 1–31. doi: 10.1007/s10660-021-09468-x. [45] T. Zhao, W. Zhang, H. Zhao and Z. Jin, A reinforcement learning-based framework for the generation and evolution of adaptation rules, In 2017 IEEE International Conference on Autonomic Computing (ICAC), (2017), 103–112.

show all references

##### References:
 [1] V. S. Abedi, Allocation of advertising budget between multiple channels to support sales in multiple markets, J. Operational Research Society, 68 (2017), 134-146.  doi: 10.1057/s41274-016-0026-1. [2] A. Albadvi and H. Koosha, A robust optimization approach to allocation of marketing budgets, Management Decision, 49 (2011), 601-621.  doi: 10.1108/00251741111126512. [3] T. Araujo, J. R. Copulsky, J. L. Hayes, S. J. Kim and J. Srivastava, From purchasing exposure to fostering engagement: Brand–consumer experiences in the emerging computational advertising landscape, J. Advertising, 49 (2020), 428-445.  doi: 10.1080/00913367.2020.1795756. [4] W. Arens and L. B. Courtland, Contemporary Advertising, 5$^{nd}$ edition, McGraw-Hill Education, 1994. [5] F. M. Bass, N. Bruce, S. Majumdar and B. P. S. Murthi, Wearout effects of different advertising themes: A dynamic bayesian model of the advertising-sales relationship, Marketing Science, 26 (2007), 179-195.  doi: 10.1287/mksc.1060.0208. [6] F. M. Bass and R. T. Lonsdale, An exploration of linear programming in media selection, Mathematical Models in Marketing, 132 (1976), 137-139.  doi: 10.1007/978-3-642-51565-1_46. [7] R. Bellman and S. Dreyfus, Functional approximations and dynamic programming, Math. Tables Aids Comput., (13) (1959), 247–251. doi: 10.2307/2002797. [8] C. Beltran-Royo, L. F. Escudero and H. Zhang, Multiperiod multiproduct advertising budgeting: Stochastic optimization modeling, Omega, 59 (2016), 26-39.  doi: 10.1016/j.omega.2015.02.013. [9] C. Beltran-Royo, H. Zhang, L. A. Blanco and J. Almagro, Multistage multiproduct advertising budgeting, European J. Oper. Res., 225 (2013), 179-188.  doi: 10.1016/j.ejor.2012.09.022. [10] D. P. Bertsekas, Dynamic Programming and Optimal Control, 4$^{th}$ edition, Athena Scientific, 2012. [11] P. J. Danaher and R. Rust, Determining the optimal return on investment for an advertising campaign, European J. Oper. Res., 95 (1996), 511-521.  doi: 10.1016/0377-2217(95)00319-3. [12] P. Doyle and J. Saunders, Multiproduct advertising budgeting, Marketing Science, 9 (1990), 97-113.  doi: 10.1287/mksc.9.2.97. [13] R. Du, Q. Hu and S. Ai, Stochastic optimal budget decision for advertising considering uncertain sales responses, European J. Oper. Res., 183 (2007), 1042-1054.  doi: 10.1016/j.ejor.2006.02.031. [14] M. Fischer, S. Albers, N. Wagner and M. Frie, Dynamic marketing budget allocation across countries, products, and marketing activities, J. Marketing Research, 2009. [15] H. K. Gajjar and K. G. Adil, A dynamic programming heuristic for retail shelf space allocation problem, Asia-Pac. J. Oper. Res., 28 (2011), 183-199.  doi: 10.1142/S0217595911003120. [16] T. P. Hsieh, C. Y. Dye and K. K. Lai, A dynamic advertising problem when demand is sensitive to the credit period and stock of advertising goodwill, J. Operational Research Society, 71 (2020), 948-966.  doi: 10.1080/01605682.2019.1595189. [17] K. F. Jea, J. Y. Wang and C. W. Hsu, Two-agent advertisement scheduling on physical books to maximize the total profit, Asia-Pac. J. Oper. Res., 36 (2019), 1950014. doi: 10.1142/S0217595919500143. [18] A. Kaul, S. Aggarwal, M. Krishnamoorthy and P. C. Jha, Multi-period media planning for multi-products incorporating segment specific and mass media, Ann. Oper. Res., 269 (2018), 317-359.  doi: 10.1007/s10479-018-2771-9. [19] H. Koosha and A. Albadvi, Allocation of marketing budgets to maximize customer equity, Operational Research, 20 (2020), 561-583.  doi: 10.1007/s12351-017-0356-z. [20] P. Kotler and K. L. Keller, Marketing Management, 14$^{th}$ edition, Prentice Hall, 2012. [21] N. K. Kwak, C. W. Lee and J. H. Kim, An MCDM model for media selection in the dual consumer/industrial market, European J. Operational Research, 166 (2005), 255-256.  doi: 10.1016/j.ejor.2004.02.016. [22] G. Li and B. Sun, Optimal dynamic pricing for used products in remanufacturing over an infinite horizon, Asia-Pac. J. Oper. Res., 31 (2014), 1450012.  doi: 10.1142/S0217595914500122. [23] X. Li, Y. Li and W. Cao, Cooperative advertising models in O2O supply chains, International J. Production Economics, 215 (2019), 144-152. [24] F. Lu, J. Zhang and W. Tang, Wholesale price contract versus consignment contract in a supply chain considering dynamic advertising, Int. Trans. Oper. Res., 26 (2019), 1977-2003.  doi: 10.1111/itor.12388. [25] P. Manik, A. Gupta, P. C. Jha and K. Govindan, A goal programming model for selection and scheduling of advertisements on online news media, Asia-Pac. J. Oper. Res., 33 (2016), 1650012.  doi: 10.1142/S0217595916500123. [26] M. Memarpour, E. Hassannayebi, N. F. Miab and A. Farjad, Dynamic allocation of promotional budgets based on maximizing customer equity, Operational Research, 21 (2021), 2365-2389.  doi: 10.1007/s12351-019-00510-3. [27] H. I. Mesak and H. Zhang, Optimal advertising pulsation policies: A dynamic programming approach, J. Operational Research Society, 52 (2001), 1244-1255.  doi: 10.1057/palgrave.jors.2601219. [28] A. Mihiotis and I. Tsakiris, A mathematical programming study of advertising allocation problem, Appl. Math. Comput., 148 (2004), 373-379.  doi: 10.1016/S0096-3003(02)00853-6. [29] P. A. Naik, K. Raman and R. S. Winer, Planning marketing-mix strategies in the presence of interaction effects, Marketing Science, 24 (2005), 25-34.  doi: 10.1287/mksc.1040.0083. [30] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1962), 167-168.  doi: 10.1007/978-3-642-51565-1_54. [31] B. Pérez-Gladish, I. González, A. Bilbao-Terol and M. Arenas-Parra, Planning a TV advertising campaign: A crisp multiobjective programming model from fuzzy basic data, Omega, 38 (2010), 84-94. [32] W. B. Powell, An operational planning model for the dynamic vehicle allocation problem with uncertain demands, Transportation Research Part B: Methodological, 21 (1987), 217-232. [33] W. B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, John Wiley & Sons, 2007. doi: 10.1002/9780470182963. [34] A. Prasad and S. P. Sethi, Competitive advertising under uncertainty: A stochastic differential game approach, J. Optim. Theory Appl., 123 (2004), 163-185.  doi: 10.1023/B:JOTA.0000043996.62867.20. [35] M. L. Puterman, Markov decision processes, Handbooks Oper. Res. Management Sci., 2 (1990), 331-434. [36] J. Z. Sissors and R. B. Baron, Advertising Media Planning, 7$^{th}$ edition, Mc Graw hill Publishin, 2010. [37] S. Sriram and M. U. Kalwani, Optimal advertising and promotion budgets in dynamic markets with brand equity as a mediating variable, Management Science, 53 (2007), 46-60. [38] R. S. Sutton and A. G. Barto, Toward a modern theory of adaptive networks: Expectation and prediction, Psychological Review, 88 (1981), 135-170.  doi: 10.1037/0033-295X.88.2.135. [39] J. N. Tsitsiklis, Asynchronous stochastic approximation and Q-learning, 32nd IEEE Conference on Decision and Control, 16 (1994), 185-202.  doi: 10.1109/CDC.1993.325119. [40] R. Van der Wurff, P. Bakker and R. G. Picards, Economic growth and advertising expenditures in different media in different countries, J. Media Economics, 21 (2008), 28-52. [41] M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381.  doi: 10.1287/opre.5.3.370. [42] X. Wang, F. Li and F. Jia, Optimal advertising budget allocation across markets with different goals and various constraints, Complexity, 2020, 2020. [43] C. Yang and Y. Xiong, Nonparametric advertising budget allocation with inventory constraint, European J. Oper. Res., 285 (2020), 631-641.  doi: 10.1016/j.ejor.2020.02.005. [44] Y. Yang, B. Feng, J. Salminen and B. J. Jansen, Optimal advertising for a generalized Vidale–Wolfe response model, Electronic Commerce Research, 285 (2021), 1–31. doi: 10.1007/s10660-021-09468-x. [45] T. Zhao, W. Zhang, H. Zhao and Z. Jin, A reinforcement learning-based framework for the generation and evolution of adaptation rules, In 2017 IEEE International Conference on Autonomic Computing (ICAC), (2017), 103–112.
Costs and budget percentages of each media in the agency's advertising packages
The $BCG$ matrix of the company studied and its parameters
Appropriate number of iterations to get a converged answer
Sales volume with different number of parameters $a$ and $b$
Percentage of selected advertising packages in each period
Costs and additional value in each period
Percentage of selected advertising packages in different budget levels
A comparison between different policies
Comparison between the results by our proposed approach – ADP – and a genetic algorithm
Classical life cycle curves
Price forecast regression in 12 future decision periods
An example of initial chromosomes for two products
A four-point crossover example
A point average crossover example
A strand crossover example
Effectiveness of advertising packages according to the system state for product 1
 Product 1 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ Product Int 1.12 1.26 1.12 1.21 1.20 1.08 1.02 1.20 1.01 1.15 0.98 1.08 life Gr 1.24 1.51 1.30 1.35 1.25 1.38 1.10 1.43 1.09 1.42 0.98 1.18 cycle Ma 1.00 1.20 0.98 1.16 0.96 1.15 0.94 1.13 0.93 1.10 0.80 0.93 Dec 0.90 1.12 0.93 1.05 0.90 1.05 0.88 1.04 0.87 1.01 0.65 0.90 Competitive H-Def 0.83 1.02 0.88 0.93 0.84 0.92 0.79 0.92 0.78 0.87 0.71 0.84 strategy H-Off 0.80 0.86 0.70 0.90 0.68 0.81 0.54 0.89 0.53 0.88 0.55 0.70 L-Def 0.82 0.95 0.80 0.92 0.77 0.88 0.68 0.92 0.67 0.89 0.64 0.78 L-Off 0.80 0.93 0.78 0.90 0.75 0.85 0.65 0.89 0.64 0.86 0.62 0.76 BCG Qus 1.12 1.26 1.11 1.22 1.06 1.16 1.04 1.10 1.01 1.09 0.89 1.09 Matrix Str 1.25 1.52 1.30 1.36 1.25 1.36 1.11 1.44 1.09 1.37 0.95 1.15 C-Co 1.00 1.29 1.00 1.19 0.98 1.21 0.95 1.18 0.93 1.12 0.89 0.98 Dg 1.00 1.04 0.96 1.00 0.94 0.98 0.90 0.95 0.88 0.95 0.72 0.80 Int: Introduction, Gr: Growth, Ma: Maturity, Dec: Decline, H-Def: High defensive H-Off: High offensive L-def: Low defensive L-Off: Low offensive Qus: Question marks Str: Stars C-Co: Cash cows Dg: Dogs
 Product 1 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ Product Int 1.12 1.26 1.12 1.21 1.20 1.08 1.02 1.20 1.01 1.15 0.98 1.08 life Gr 1.24 1.51 1.30 1.35 1.25 1.38 1.10 1.43 1.09 1.42 0.98 1.18 cycle Ma 1.00 1.20 0.98 1.16 0.96 1.15 0.94 1.13 0.93 1.10 0.80 0.93 Dec 0.90 1.12 0.93 1.05 0.90 1.05 0.88 1.04 0.87 1.01 0.65 0.90 Competitive H-Def 0.83 1.02 0.88 0.93 0.84 0.92 0.79 0.92 0.78 0.87 0.71 0.84 strategy H-Off 0.80 0.86 0.70 0.90 0.68 0.81 0.54 0.89 0.53 0.88 0.55 0.70 L-Def 0.82 0.95 0.80 0.92 0.77 0.88 0.68 0.92 0.67 0.89 0.64 0.78 L-Off 0.80 0.93 0.78 0.90 0.75 0.85 0.65 0.89 0.64 0.86 0.62 0.76 BCG Qus 1.12 1.26 1.11 1.22 1.06 1.16 1.04 1.10 1.01 1.09 0.89 1.09 Matrix Str 1.25 1.52 1.30 1.36 1.25 1.36 1.11 1.44 1.09 1.37 0.95 1.15 C-Co 1.00 1.29 1.00 1.19 0.98 1.21 0.95 1.18 0.93 1.12 0.89 0.98 Dg 1.00 1.04 0.96 1.00 0.94 0.98 0.90 0.95 0.88 0.95 0.72 0.80 Int: Introduction, Gr: Growth, Ma: Maturity, Dec: Decline, H-Def: High defensive H-Off: High offensive L-def: Low defensive L-Off: Low offensive Qus: Question marks Str: Stars C-Co: Cash cows Dg: Dogs
Effectiveness of advertising packages according to the system state for product 2
 Product 2 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ Product Int 1.17 1.25 1.14 1.18 1.09 1.20 1.04 1.18 1.03 1.12 0.96 1.03 life Gr 1.32 1.40 1.24 1.42 1.25 1.38 1.12 1.40 1.11 1.39 0.99 1.10 cycle Ma 1.10 1.28 1.14 1.16 1.09 1.17 1.00 1.18 1.00 1.15 0.87 1.00 Dec 0.98 1.03 0.94 1.03 0.90 1.00 0.85 1.00 0.80 1.00 0.77 0.92 Competitive H-Def 0.80 1.04 0.90 0.90 0.86 0.89 0.78 0.92 0.80 0.88 0.73 0.84 strategy H-Off 0.75 0.88 0.73 0.84 0.70 0.82 0.60 0.84 0.55 0.85 0.60 0.69 L-Def 0.77 0.99 0.80 0.90 0.81 0.86 0.76 0.86 0.75 0.83 0.68 0.78 L-Off 0.74 0.94 0.70 0.90 0.75 0.82 0.74 0.80 0.70 0.77 0.62 0.76 BCG Qus 1.18 1.30 1.15 1.20 1.12 1.19 1.06 1.16 1.03 1.12 0.92 1.08 Matrix Str 1.30 1.45 1.25 1.43 1.26 1.39 1.13 1.31 1.12 1.21 0.98 1.05 C-Co 1.14 1.24 1.14 1.18 1.12 1.17 1.06 1.15 1.04 1.12 0.90 0.99 Dg 1.00 1.10 0.98 1.05 0.96 1.04 0.90 0.99 0.91 0.95 0.68 0.88
 Product 2 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ $r_1$ $r_2$ Product Int 1.17 1.25 1.14 1.18 1.09 1.20 1.04 1.18 1.03 1.12 0.96 1.03 life Gr 1.32 1.40 1.24 1.42 1.25 1.38 1.12 1.40 1.11 1.39 0.99 1.10 cycle Ma 1.10 1.28 1.14 1.16 1.09 1.17 1.00 1.18 1.00 1.15 0.87 1.00 Dec 0.98 1.03 0.94 1.03 0.90 1.00 0.85 1.00 0.80 1.00 0.77 0.92 Competitive H-Def 0.80 1.04 0.90 0.90 0.86 0.89 0.78 0.92 0.80 0.88 0.73 0.84 strategy H-Off 0.75 0.88 0.73 0.84 0.70 0.82 0.60 0.84 0.55 0.85 0.60 0.69 L-Def 0.77 0.99 0.80 0.90 0.81 0.86 0.76 0.86 0.75 0.83 0.68 0.78 L-Off 0.74 0.94 0.70 0.90 0.75 0.82 0.74 0.80 0.70 0.77 0.62 0.76 BCG Qus 1.18 1.30 1.15 1.20 1.12 1.19 1.06 1.16 1.03 1.12 0.92 1.08 Matrix Str 1.30 1.45 1.25 1.43 1.26 1.39 1.13 1.31 1.12 1.21 0.98 1.05 C-Co 1.14 1.24 1.14 1.18 1.12 1.17 1.06 1.15 1.04 1.12 0.90 0.99 Dg 1.00 1.10 0.98 1.05 0.96 1.04 0.90 0.99 0.91 0.95 0.68 0.88
Competitors' reactions probabilities
 Product 1 Product 2 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction AP 1 AP 2 AP 3 AP 4 AP 5 Inaction H-Def 0.15 0.2 0.22 0.21 0.25 0.3 0.18 0.2 0.21 0.23 0.26 0.32 H-Off 0.35 0.3 0.22 0.14 0.12 0.3 0.3 0.25 0.19 0.12 0.1 0.16 L-Def 0.3 0.3 0.3 0.34 0.36 0.2 0.2 0.35 0.33 0.3 0.24 0.23 L-Off 0.2 0.2 0.26 0.31 0.27 0.2 0.32 0.2 0.27 0.35 0.4 0.29
 Product 1 Product 2 AP 1 AP 2 AP 3 AP 4 AP 5 Inaction AP 1 AP 2 AP 3 AP 4 AP 5 Inaction H-Def 0.15 0.2 0.22 0.21 0.25 0.3 0.18 0.2 0.21 0.23 0.26 0.32 H-Off 0.35 0.3 0.22 0.14 0.12 0.3 0.3 0.25 0.19 0.12 0.1 0.16 L-Def 0.3 0.3 0.3 0.34 0.36 0.2 0.2 0.35 0.33 0.3 0.24 0.23 L-Off 0.2 0.2 0.26 0.31 0.27 0.2 0.32 0.2 0.27 0.35 0.4 0.29
The market volume over the past year and estimation for future periods
 $Periods$ 1 2 3 4 5 6 7 8 9 10 11 12 MV 3.18* 2.36 2.15 2.24 2.13 1.90 1.64 1.49 1.36 1.37 1.53 2.27 MV(LY) 2.88* 2.07 1.85 1.98 1.88 1.70 1.50 1.45 1.43 1.49 1.69 2.47 $*$:$\times {10^4}$, MV: Market value, MV(LY): Market volume last year.
 $Periods$ 1 2 3 4 5 6 7 8 9 10 11 12 MV 3.18* 2.36 2.15 2.24 2.13 1.90 1.64 1.49 1.36 1.37 1.53 2.27 MV(LY) 2.88* 2.07 1.85 1.98 1.88 1.70 1.50 1.45 1.43 1.49 1.69 2.47 $*$:$\times {10^4}$, MV: Market value, MV(LY): Market volume last year.
Policy 1 based on budget and life cycle stages
 Budget bound Product 1 Product 2 Int Gr Ma Dec Int Gr Ma Dec 0-20 AP 5 AP 4 AP 3 AP 2 AP 5 AP 4 AP 3 AP 3 20-40 AP 4 AP 3 AP 2 AP 2 AP 4 AP 3 AP 3 AP 2 40-60 AP 3 AP 2 AP 2 AP 1 AP 3 AP 3 AP 2 AP 1 60-80 AP 2 AP 1 AP 1 AP 1 AP 2 AP 2 AP 1 AP 1 80-100 AP 2 AP 1 AP 1 AP 1 AP 1 AP 1 AP 1 AP 1
 Budget bound Product 1 Product 2 Int Gr Ma Dec Int Gr Ma Dec 0-20 AP 5 AP 4 AP 3 AP 2 AP 5 AP 4 AP 3 AP 3 20-40 AP 4 AP 3 AP 2 AP 2 AP 4 AP 3 AP 3 AP 2 40-60 AP 3 AP 2 AP 2 AP 1 AP 3 AP 3 AP 2 AP 1 60-80 AP 2 AP 1 AP 1 AP 1 AP 2 AP 2 AP 1 AP 1 80-100 AP 2 AP 1 AP 1 AP 1 AP 1 AP 1 AP 1 AP 1
Policy 2 based on budget and competitors' reaction
 Budget bound Product 1 Product 2 H-Def H-Off L-Def L-Off H-Def H-Off L-Def L-Off 0-20 AP 5 AP 4 AP 5 AP 4 AP 5 AP 3 AP 4 AP 4 20-40 AP 5 AP 3 AP 4 AP 3 AP 4 AP 3 AP 3 AP 3 40-60 AP 4 AP 2 AP 3 AP 3 AP 3 AP 2 AP 2 AP 3 60-80 AP 3 AP 2 AP 2 AP 2 AP 2 AP 1 AP 2 AP 1 80-100 AP 2 AP 1 AP 2 AP 2 AP 1 AP 1 AP 1 AP 1
 Budget bound Product 1 Product 2 H-Def H-Off L-Def L-Off H-Def H-Off L-Def L-Off 0-20 AP 5 AP 4 AP 5 AP 4 AP 5 AP 3 AP 4 AP 4 20-40 AP 5 AP 3 AP 4 AP 3 AP 4 AP 3 AP 3 AP 3 40-60 AP 4 AP 2 AP 3 AP 3 AP 3 AP 2 AP 2 AP 3 60-80 AP 3 AP 2 AP 2 AP 2 AP 2 AP 1 AP 2 AP 1 80-100 AP 2 AP 1 AP 2 AP 2 AP 1 AP 1 AP 1 AP 1
Policy 3 based on budget and $BCG$ matrix class
 Budget bound Product 1 Product 2 Qus Str C-Co Dg Qus Str C-Co Dg 0-20 AP 5 AP 5 AP 5 AP 5 AP 2 AP 2 AP 5 AP 5 20-40 AP 5 AP 4 AP 5 AP 4 AP 1 AP 1 AP 4 AP 4 40-60 AP 4 AP 3 AP 4 AP 2 AP 1 AP 1 AP 4 AP 3 60-80 AP 3 AP 2 AP 3 AP 1 AP 1 AP 1 AP 3 AP 2 80-100 AP 2 AP 1 AP 2 AP 1 AP 1 AP 1 AP 2 AP 1
 Budget bound Product 1 Product 2 Qus Str C-Co Dg Qus Str C-Co Dg 0-20 AP 5 AP 5 AP 5 AP 5 AP 2 AP 2 AP 5 AP 5 20-40 AP 5 AP 4 AP 5 AP 4 AP 1 AP 1 AP 4 AP 4 40-60 AP 4 AP 3 AP 4 AP 2 AP 1 AP 1 AP 4 AP 3 60-80 AP 3 AP 2 AP 3 AP 1 AP 1 AP 1 AP 3 AP 2 80-100 AP 2 AP 1 AP 2 AP 1 AP 1 AP 1 AP 2 AP 1
Policy 4 based on budget and price product
 Budget bound Product 1 (price) Product 2 (price) [0, 1.67] [1.67, 1.73] [1.73, $+ \infty$) [0, 2.4] [2.4, 2.47] [2.47, $+ \infty$) 0-20 AP 5 AP 4 AP 3 AP 5 AP 5 AP 3 20-40 AP 5 AP 4 AP 2 AP 5 AP 4 AP 3 40-60 AP 4 AP 3 AP 2 AP 4 AP 3 AP 2 60-80 AP 4 AP 3 AP 2 AP 3 AP 2 AP 1 80-100 AP 3 AP 2 AP 1 AP 3 AP 1 AP 1
 Budget bound Product 1 (price) Product 2 (price) [0, 1.67] [1.67, 1.73] [1.73, $+ \infty$) [0, 2.4] [2.4, 2.47] [2.47, $+ \infty$) 0-20 AP 5 AP 4 AP 3 AP 5 AP 5 AP 3 20-40 AP 5 AP 4 AP 2 AP 5 AP 4 AP 3 40-60 AP 4 AP 3 AP 2 AP 4 AP 3 AP 2 60-80 AP 4 AP 3 AP 2 AP 3 AP 2 AP 1 80-100 AP 3 AP 2 AP 1 AP 3 AP 1 AP 1
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