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doi: 10.3934/jimo.2021202
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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

T. AraujoJ. R. CopulskyJ. L. HayesS. 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.  Google Scholar

[4]

W. Arens and L. B. Courtland, Contemporary Advertising, 5$^{nd}$ edition, McGraw-Hill Education, 1994. Google Scholar

[5]

F. M. BassN. BruceS. 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.  Google Scholar

[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.  Google Scholar

[7]

R. Bellman and S. Dreyfus, Functional approximations and dynamic programming, Math. Tables Aids Comput., (13) (1959), 247–251. doi: 10.2307/2002797.  Google Scholar

[8]

C. Beltran-RoyoL. 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.  Google Scholar

[9]

C. Beltran-RoyoH. ZhangL. 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.  Google Scholar

[10]

D. P. Bertsekas, Dynamic Programming and Optimal Control, 4$^{th}$ edition, Athena Scientific, 2012.  Google Scholar

[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.  Google Scholar

[12]

P. Doyle and J. Saunders, Multiproduct advertising budgeting, Marketing Science, 9 (1990), 97-113.  doi: 10.1287/mksc.9.2.97.  Google Scholar

[13]

R. DuQ. 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.  Google Scholar

[14]

M. Fischer, S. Albers, N. Wagner and M. Frie, Dynamic marketing budget allocation across countries, products, and marketing activities, J. Marketing Research, 2009. Google Scholar

[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.  Google Scholar

[16]

T. P. HsiehC. 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.  Google Scholar

[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.  Google Scholar

[18]

A. KaulS. AggarwalM. 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.  Google Scholar

[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.  Google Scholar

[20]

P. Kotler and K. L. Keller, Marketing Management, 14$^{th}$ edition, Prentice Hall, 2012. Google Scholar

[21]

N. K. KwakC. 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.  Google Scholar

[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.  Google Scholar

[23]

X. LiY. Li and W. Cao, Cooperative advertising models in O2O supply chains, International J. Production Economics, 215 (2019), 144-152.   Google Scholar

[24]

F. LuJ. 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.  Google Scholar

[25]

P. ManikA. GuptaP. 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.  Google Scholar

[26]

M. MemarpourE. HassannayebiN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. A. NaikK. 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.  Google Scholar

[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.  Google Scholar

[31]

B. Pérez-GladishI. GonzálezA. 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.   Google Scholar

[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.   Google Scholar

[33]

W. B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, John Wiley & Sons, 2007. doi: 10.1002/9780470182963.  Google Scholar

[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.  Google Scholar

[35]

M. L. Puterman, Markov decision processes, Handbooks Oper. Res. Management Sci., 2 (1990), 331-434.   Google Scholar

[36]

J. Z. Sissors and R. B. Baron, Advertising Media Planning, 7$^{th}$ edition, Mc Graw hill Publishin, 2010. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

R. Van der WurffP. Bakker and R. G. Picards, Economic growth and advertising expenditures in different media in different countries, J. Media Economics, 21 (2008), 28-52.   Google Scholar

[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.  Google Scholar

[42]

X. Wang, F. Li and F. Jia, Optimal advertising budget allocation across markets with different goals and various constraints, Complexity, 2020, 2020. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. AraujoJ. R. CopulskyJ. L. HayesS. 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.  Google Scholar

[4]

W. Arens and L. B. Courtland, Contemporary Advertising, 5$^{nd}$ edition, McGraw-Hill Education, 1994. Google Scholar

[5]

F. M. BassN. BruceS. 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.  Google Scholar

[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.  Google Scholar

[7]

R. Bellman and S. Dreyfus, Functional approximations and dynamic programming, Math. Tables Aids Comput., (13) (1959), 247–251. doi: 10.2307/2002797.  Google Scholar

[8]

C. Beltran-RoyoL. 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.  Google Scholar

[9]

C. Beltran-RoyoH. ZhangL. 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.  Google Scholar

[10]

D. P. Bertsekas, Dynamic Programming and Optimal Control, 4$^{th}$ edition, Athena Scientific, 2012.  Google Scholar

[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.  Google Scholar

[12]

P. Doyle and J. Saunders, Multiproduct advertising budgeting, Marketing Science, 9 (1990), 97-113.  doi: 10.1287/mksc.9.2.97.  Google Scholar

[13]

R. DuQ. 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.  Google Scholar

[14]

M. Fischer, S. Albers, N. Wagner and M. Frie, Dynamic marketing budget allocation across countries, products, and marketing activities, J. Marketing Research, 2009. Google Scholar

[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.  Google Scholar

[16]

T. P. HsiehC. 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.  Google Scholar

[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.  Google Scholar

[18]

A. KaulS. AggarwalM. 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.  Google Scholar

[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.  Google Scholar

[20]

P. Kotler and K. L. Keller, Marketing Management, 14$^{th}$ edition, Prentice Hall, 2012. Google Scholar

[21]

N. K. KwakC. 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.  Google Scholar

[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.  Google Scholar

[23]

X. LiY. Li and W. Cao, Cooperative advertising models in O2O supply chains, International J. Production Economics, 215 (2019), 144-152.   Google Scholar

[24]

F. LuJ. 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.  Google Scholar

[25]

P. ManikA. GuptaP. 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.  Google Scholar

[26]

M. MemarpourE. HassannayebiN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. A. NaikK. 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.  Google Scholar

[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.  Google Scholar

[31]

B. Pérez-GladishI. GonzálezA. 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.   Google Scholar

[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.   Google Scholar

[33]

W. B. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, John Wiley & Sons, 2007. doi: 10.1002/9780470182963.  Google Scholar

[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.  Google Scholar

[35]

M. L. Puterman, Markov decision processes, Handbooks Oper. Res. Management Sci., 2 (1990), 331-434.   Google Scholar

[36]

J. Z. Sissors and R. B. Baron, Advertising Media Planning, 7$^{th}$ edition, Mc Graw hill Publishin, 2010. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

R. Van der WurffP. Bakker and R. G. Picards, Economic growth and advertising expenditures in different media in different countries, J. Media Economics, 21 (2008), 28-52.   Google Scholar

[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.  Google Scholar

[42]

X. Wang, F. Li and F. Jia, Optimal advertising budget allocation across markets with different goals and various constraints, Complexity, 2020, 2020. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

Figure 1.  Costs and budget percentages of each media in the agency's advertising packages
Figure 2.  The $ BCG $ matrix of the company studied and its parameters
Figure 3.  Appropriate number of iterations to get a converged answer
Figure 4.  Sales volume with different number of parameters $ a $ and $ b $
Figure 5.  Percentage of selected advertising packages in each period
Figure 6.  Costs and additional value in each period
Figure 7.  Percentage of selected advertising packages in different budget levels
Figure 8.  A comparison between different policies
Figure 9.  Comparison between the results by our proposed approach – ADP – and a genetic algorithm
Figure 10.  Classical life cycle curves
Figure 11.  Price forecast regression in 12 future decision periods
Figure 12.  An example of initial chromosomes for two products
Figure 13.  A four-point crossover example
Figure 14.  A point average crossover example
Figure 15.  A strand crossover example
Table 1.  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
Table 2.  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
Table 3.  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
Table 4.  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.
Table 5.  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
Table 6.  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
Table 7.  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
Table 8.  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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