# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021221
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## Selling by clicks or leasing by bricks? A dynamic game for pricing durable products in a dual-channel supply chain

 1 Department of Industrial Engineering, Faculty of Engineering, Alzahra University, Tehran, Iran 2 School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

* Corresponding author: Maryam Esmaeili

Received  May 2021 Revised  October 2021 Early access December 2021

In this paper, we discuss if and which pricing policies by a manufacturer who sells its products online motivate a retailer as an independent part to enter the market to provide selling and leasing options through a brick store. Moreover, the impact of online shopping preferences and brand image on end-user behavior is examined, and different consumption patterns are considered. For this purpose, a dynamic game is applied to model a supply chain consisting of one manufacturer and one retailer. The model aims to specify the optimal pricing policies in the second-hand market and according to physical utility associated with depreciation, brand image, and online shopping preferences for different end-users in an infinite time horizon. Markov perfect equilibria are considered as the solution concept to predict the behavior of end-users in the long term. The results revealed that enriching brand image always benefits the manufacturer and the retailer, while it does not mean there is the same optimal brand image level for both manufacturer and retailer. Besides, the improvement of physical utility makes more demand for leasing products and motivates the retailer to be active in the market. Notably, online shopping preferences play a prominent role in market segmentation and retailer decision as a result. Also, growing production costs have a significant reverse effect on the profitability of both manufacturer and retailer. Therefore, the manufacturer must focus on economic production.

Citation: Solaleh Sadat Kalantari, Maryam Esmaeili, Ata Allah Taleizadeh. Selling by clicks or leasing by bricks? A dynamic game for pricing durable products in a dual-channel supply chain. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021221
##### References:
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Sudha, A relationship between brand image and brand loyalty, EPRA International Journal of Economic and Business Review, 4 (2016), 139-145. [31] I. Ofosu-Boateng and S. D. Agyei, Influence of branding and advertising on consumers'purchase decision of inter-city transport service in accra ghana: A case of state transport company, European Journal of Management and Marketing Studies, (2020), 553–578. [32] A. Paul, M. Pervin, S. K. Roy, et al., Effect of price-sensitive demand and default risk on optimal credit period and cycle time for a deteriorating inventory model, RAIRO Oper. Res., 55 (2021), S2575–S2592. doi: 10.1051/ro/2020108. [33] M. Pervin, S. K. Roy and G. W. Weber, An integrated vendor-buyer model with quadratic demand under inspection policy and preservation technology, Hacettepe Journal of Mathematics and Statistics, 49 (2020), 1168-1189.  doi: 10.15672/hujms.476056. [34] M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price-and stock-sensitive demand, J. Ind. Manag. Optim., 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019. [35] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, J. Ind. Manag. Optim., 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098. [36] B$\breve{A}$Rbulescu Po and Enache Nlpic, Influence factors for leasing market contracts, Management & Marketing Journal, 14 (2016) [37] J. Rogers and A. Rodrigues, Product leasing: A strategy to allow manufacturers and customers to benefit from elongation of product life, 2015. [38] S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, J. Ind. Manag. Optim., 16 (2020), 553-578.  doi: 10.3934/jimo.2018167. [39] R. Sellers-Rubio, F. Mas-Ruiz and F. 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[44] L. Wang, H. Song and D. Zhang, Pricing decisions for complementary products in a fuzzy dual-channel supply chain, J. Ind. Manag. Optim., 15 (2019), 343-364.  doi: 10.3934/jimo.2018046. [45] Y. Wang and D. S. Richardson, To buy or to lease: The advantages and costs of leasing versus buying scientific instruments for academic core facilities, EMBO Reports, 21 (2020), e49971. doi: 10.15252/embr.202049971. [46] T. Xiao and J. Shi, Pricing and supply priority in a dual-channel supply chain, European J. Oper. Res., 254 (2016), 813-823.  doi: 10.1016/j.ejor.2016.04.018. [47] Y. Xiong, W. Yan and K. Fernandes, Bricks vs. clicks: The impact of manufacturer encroachment with a dealer leasing and selling of durable goods, European Journal of Operational Research, 217 (2012), 75-83.  doi: 10.1016/j.ejor.2011.08.012. [48] W. Yan, Y. Li, Y. Wu, et al., A rising e-channel tide lifts all boats? The impact of manufacturer multichannel encroachment on traditional selling and leasing, Discrete Dyn. Nat. Soc., 2016 (2016), Art. ID 2898021, 18 pp. doi: 10.1155/2016/2898021. [49] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039. [50] W. S. Yoo and E. Lee, Internet channel entry: A strategic analysis of mixed channel structures, Marketing Science, 30 (2011), 29-41.  doi: 10.1287/mksc.1100.0586. [51] D. Z. Yu, T. Cheong and D. Sun, Impact of supply chain power and drop-shipping on a manufacturer's optimal distribution channel strategy, European J. Oper. Res., 259 (2017), 554-563.  doi: 10.1016/j.ejor.2016.11.025. [52] Y. Yu, L. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656.  doi: 10.1080/01605682.2019.1621221. [53] Y. Zare Mehrjerdi and R. 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Ji, Coordinating a dual-channel supply chain with conditional value-at-risk under uncertainties of yield and demand, Computers & Industrial Engineering, 139 (2020), 106181.  doi: 10.1016/j.cie.2019.106181.

show all references

##### References:
 [1] S. Agarwal, B. W. Ambrose and H. Huang, The term structure of lease rates with endogenous default triggers and tenant capital structure: Theory and evidence, Journal of Financial and Quantitative Analysis, 46 (2011), 553-584.  doi: 10.1017/S0022109010000839. [2] V. V. Agrawal, M. Ferguson and L. B. Toktay, Is leasing greener than selling?, Management Science, 58 (2012), 523-533.  doi: 10.1287/mnsc.1110.1428. [3] A. Andrikopoulos and R. N. Markellos, Dynamic interaction between markets for leasing and selling automobiles, Journal of Banking & Finance, 50 (2015), 260-270.  doi: 10.1016/j.jbankfin.2014.01.032. [4] H. Barman, M. Pervin and S. K. Roy, Back-ordered inventory model with inflation in a cloudy-fuzzy environment, J. Ind. Manag. Optim., 17 (2021), 1913-1941.  doi: 10.3934/jimo.2020052. [5] T. Bașar and G. Zaccour, Handbook of Dynamic Game Theory, Springer, 2018. doi: 10.1007/978-3-319-44374-4. [6] R. Batarfi, M. Y. Jaber and S. Zanoni, Dual-channel supply chain: A strategy to maximize profit, Appl. Math. Model., 40 (2016), 9454-9473.  doi: 10.1016/j.apm.2016.06.008. [7] A. M. Batkovskiy, E. G. Semenova, V. Y. Trofimets, et al., Computer modeling of leasing operations, Indian Journal of Science and Technology, 9 (2016). [8] S. R. Bhaskaran and S. M. Gilbert, Implications of channel structure for leasing or selling durable goods, Marketing Science, 28 (2009), 918-934.  doi: 10.1287/mksc.1080.0458. [9] I. Brocas, J. D. Carrillo and A. Sachdeva, The path to equilibrium in sequential and simultaneous games: A mousetracking study, Journal of Economic Theory, 178 (2018), 246-274.  doi: 10.1016/j.jet.2018.09.011. [10] K. Cao, J. Wang and G. Dou, Optimal trade-in strategy of retailers with online and offline sales channels, Computers & Industrial Engineering, 123 (2018), 148-156.  doi: 10.1016/j.cie.2018.05.028. [11] J. Chen, L. Liang and D.-Q. Yao, Price and quality decisions in dual-channel supply chains, European J. Oper. Res., 259 (2017), 935-948.  doi: 10.1016/j.ejor.2016.11.016. [12] S. K. Das, M. Pervin, S. K. Roy, et al., Multi-objective solid transportation-location problem with variable carbon emission in inventory management: A hybrid approach, Annals of Operations Research, (2021), 1–27. doi: 10.1007/s10479-020-03809-z. [13] S. K. Das and S. K. Roy, Effect of variable carbon emission in a multi-objective transportation-$p$-facility location problem under neutrosophic environment, Computers & Industrial Engineering, 132 (2019), 311-324.  doi: 10.1016/j.cie.2019.04.037. [14] S. K. Das, S. K. Roy and G. W. Weber, Heuristic approaches for solid transportation-$p$-facility location problem, CEJOR Cent. Eur. J. Oper. Res., 28 (2020), 939-961.  doi: 10.1007/s10100-019-00610-7. [15] S. K. Das, S. K. Roy and G. W. Weber, An exact and a heuristic approach for the transportation-$p$-facility location problem, Comput. Manag. Sci., 17 (2020), 389-407.  doi: 10.1007/s10287-020-00363-8. [16] J. Guo, Y. Zhou and B. Li, The optimal pricing and service strategies of a dual-channel retailer under free riding, J. Ind. Manag. Optim., 2021. [17] M. Hamidi, H. Liao and F. Szidarovszky, Non-cooperative and cooperative game-theoretic models for usage-based lease contracts, European J. Oper. Res., 255 (2016), 163-174.  doi: 10.1016/j.ejor.2016.04.064. [18] E. D. Hess, Best Buy Co., Inc., Darden Business Publishing Cases, 2017. doi: 10.1108/case.darden.2016.000040. [19] C.-C. Hsieh, Y.-L. Chang and C.-H. Wu, Competitive pricing and ordering decisions in a multiple-channel supply chain, International Journal of Production Economics, 154 (2014), 156-165.  doi: 10.1016/j.ijpe.2014.04.024. [20] S. Huang, Y. Yang and K. Anderson, A theory of finitely durable goods monopoly with used-goods market and transaction costs, Management Science, 47 (2001), 1515-1532.  doi: 10.1287/mnsc.47.11.1515.10250. [21] M. Lai, H. Yang and E. Cao, Optimal decisions for a dual-channel supply chain under information asymmetry, J. Ind. Manag. Optim., 14 (2018), 1023-1040.  doi: 10.3934/jimo.2017088. [22] C.-Y. Lee, Advertising, its determinants, and market structure, Review of Industrial Organization, 21 (2002), 89-101.  doi: 10.1023/A:1016020026207. [23] M. Lei, H. Liu and H. Deng, Demand information sharing and channel choice in a dual-channel supply chain with multiple retailers, International Journal of Production Research, 52 (2014), 6792-6818.  doi: 10.1080/00207543.2014.918286. [24] K. Li, T. Zhou and B. Liu, The comparison between selling and leasing for new and remanufactured products with quality level in the electric vehicle industry, J. Ind. Manag. Optim., 17 (2021), 1505-1529.  doi: 10.3934/jimo.2020032. [25] R. Lotfi, N. Mardani and G. W. Weber, Robust bi-level programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534.  doi: 10.1002/er.6332. [26] R. Lotfi, Y. Z. Mehrjerdi and M. S. Pishvaee, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numer. Algebra Control Optim., 11 (2021), 221-253.  doi: 10.3934/naco.2020023. [27] R. Lotfi, Z. Yadegari, S. H. Hosseini, et al., A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020158. [28] E. Maskin and J. Tirole, A theory of dynamic oligopoly, Ⅱ: Price competition, kinked demand curves, and Edgeworth cycles, Econometrica, 56 (1988), 571-599.  doi: 10.2307/1911701. [29] G. R. Nasiri, M. Kalantari and B. Karimi, Fast-moving consumer goods network design with pricing policy in an uncertain environment with correlated demands, Computers & Industrial Engineering, 153 (2021), 106997.  doi: 10.1016/j.cie.2020.106997. [30] R. Natarajan and V. Sudha, A relationship between brand image and brand loyalty, EPRA International Journal of Economic and Business Review, 4 (2016), 139-145. [31] I. Ofosu-Boateng and S. D. Agyei, Influence of branding and advertising on consumers'purchase decision of inter-city transport service in accra ghana: A case of state transport company, European Journal of Management and Marketing Studies, (2020), 553–578. [32] A. Paul, M. Pervin, S. K. Roy, et al., Effect of price-sensitive demand and default risk on optimal credit period and cycle time for a deteriorating inventory model, RAIRO Oper. Res., 55 (2021), S2575–S2592. doi: 10.1051/ro/2020108. [33] M. Pervin, S. K. Roy and G. W. Weber, An integrated vendor-buyer model with quadratic demand under inspection policy and preservation technology, Hacettepe Journal of Mathematics and Statistics, 49 (2020), 1168-1189.  doi: 10.15672/hujms.476056. [34] M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price-and stock-sensitive demand, J. Ind. Manag. Optim., 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019. [35] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, J. Ind. Manag. Optim., 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098. [36] B$\breve{A}$Rbulescu Po and Enache Nlpic, Influence factors for leasing market contracts, Management & Marketing Journal, 14 (2016) [37] J. Rogers and A. Rodrigues, Product leasing: A strategy to allow manufacturers and customers to benefit from elongation of product life, 2015. [38] S. K. Roy, M. Pervin and G. W. Weber, A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy, J. Ind. Manag. Optim., 16 (2020), 553-578.  doi: 10.3934/jimo.2018167. [39] R. Sellers-Rubio, F. Mas-Ruiz and F. Sancho-Esper, Reputation and advertising of collective brand members in the wine industry: The moderating role of market share, Journal of Wine Economics, 16 (2021), 169-188.  doi: 10.1017/jwe.2020.52. [40] F. Soleimani, Optimal pricing decisions in a fuzzy dual-channel supply chain, Soft Computing, 20 (2016), 689-696.  doi: 10.1007/s00500-014-1532-1. [41] H. Sun, Y. Wan and Y. Li, Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours, J. Ind. Manag. Optim., 17 (2021), 601-631.  doi: 10.3934/jimo.2019125. [42] P. van Loon, C. Delagarde and L. N. Van Wassenhove, Leasing or buying white goods: Comparing manufacturer profitability versus cost to consumer, International Journal of Production Research, 58 (2020), 1092-1106. [43] C. Wang, M. Leng and L. Liang, Choosing an online retail channel for a manufacturer: Direct sales or consignment?, International Journal of Production Economics, 195 (2018), 338-358.  doi: 10.1016/j.ijpe.2017.11.001. [44] L. Wang, H. Song and D. Zhang, Pricing decisions for complementary products in a fuzzy dual-channel supply chain, J. Ind. Manag. Optim., 15 (2019), 343-364.  doi: 10.3934/jimo.2018046. [45] Y. Wang and D. S. Richardson, To buy or to lease: The advantages and costs of leasing versus buying scientific instruments for academic core facilities, EMBO Reports, 21 (2020), e49971. doi: 10.15252/embr.202049971. [46] T. Xiao and J. Shi, Pricing and supply priority in a dual-channel supply chain, European J. Oper. Res., 254 (2016), 813-823.  doi: 10.1016/j.ejor.2016.04.018. [47] Y. Xiong, W. Yan and K. Fernandes, Bricks vs. clicks: The impact of manufacturer encroachment with a dealer leasing and selling of durable goods, European Journal of Operational Research, 217 (2012), 75-83.  doi: 10.1016/j.ejor.2011.08.012. [48] W. Yan, Y. Li, Y. Wu, et al., A rising e-channel tide lifts all boats? The impact of manufacturer multichannel encroachment on traditional selling and leasing, Discrete Dyn. Nat. Soc., 2016 (2016), Art. ID 2898021, 18 pp. doi: 10.1155/2016/2898021. [49] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039. [50] W. S. Yoo and E. Lee, Internet channel entry: A strategic analysis of mixed channel structures, Marketing Science, 30 (2011), 29-41.  doi: 10.1287/mksc.1100.0586. [51] D. Z. Yu, T. Cheong and D. Sun, Impact of supply chain power and drop-shipping on a manufacturer's optimal distribution channel strategy, European J. Oper. Res., 259 (2017), 554-563.  doi: 10.1016/j.ejor.2016.11.025. [52] Y. Yu, L. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656.  doi: 10.1080/01605682.2019.1621221. [53] Y. Zare Mehrjerdi and R. 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The outline of the proposed model
Consumption pattern of each classification under various levels of brand image
The amounts of selling and leasing prices under various levels of brand image
The amounts of marginal profit for the manufacturer and retailer under various levels of brand image
Consumption pattern of in each classification under various physical utilities
The amounts of selling and leasing prices under various physical utilities
The amounts of marginal profit for the manufacturer and the retailer under various physical utilities
Consumption pattern of each classification under various production costs
The amounts of selling and leasing prices under various production costs
The amounts of marginal profit for the manufacturer and the retailer under various production costs
Consumption pattern in each classification under various dealing costs
The amounts of selling and leasing prices under various dealing costs
The amounts of marginal profit for the manufacturer and retailer under various dealing costs
Consumption pattern of each classification under various response levels
The amounts of selling and leasing prices under various response levels
The amounts of marginal profit per product for the manufacture under various response levels
The amounts of marginal profit per product for the manufacturer and retailer based on various alpha
of dual-channel studies and transaction ways for durable products studies in the literature with the present study
 The previous studies Transaction Ways Dynamic nature More than one channel Game theory approach Second hand Market Durable Products Deprecia tion Brand Image Leasing selling Cao et al., 2018 [10] $\star$ $\star$ $\star$ Wang et al., 2018 [43] $\star$ $\star$ $\star$ Wang et al., 2019 [44] $\star$ $\star$ $\star$ Lai et al., 2018 [21] $\star$ $\star$ $\star$ Soleimani, 2016 [40] $\star$ $\star$ $\star$ Yu et al., 2020[52] $\star$ $\star$ $\star$ Zhu et al., 2020 [57] $\star$ $\star$ $\star$ Sun et al., 2021 [41] $\star$ $\star$ $\star$ Zhang and Xiao, 2013 [54] $\star$ $\star$ $\star$ Yang et al., 2018 [49] $\star$ $\star$ $\star$ Guo et al., 2021 [16] $\star$ $\star$ $\star$ Hsieh et al., 2014 [19] $\star$ $\star$ $\star$ Yoo and Lee, 2011[50] $\star$ $\star$ $\star$ Xiong et al., 2012 [47] $\star$ $\star$ $\star$ $\star$ $\star$ Xiao and Shi, 2016 [46] $\star$ $\star$ $\star$ Zhao et al., 2017 [56] $\star$ $\star$ $\star$ Chen et al., 20171 [11] $\star$ $\star$ $\star$ Nasiri et al., 2021 [29] $\star$ $\star$ Agarwal et al., 2011 [1] $\star$ $\star$ $\star$ Rogers and Rodrigues, 2015 [37] $\star$ $\star$ $\star$ Agrawal et al., 2012 [2] $\star$ $\star$ $\star$ Andriko poulos and Markellos, 2015 [3] $\star$ $\star$ $\star$ $\star$ Li et al., 2021 [24] $\star$ $\star$ $\star$ $\star$ Barbulescu and Enache, 2016 [36] $\star$ $\star$ Yan et al., 2016 [48] $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ hamidi et al., 2016 [17] $\star$ $\star$ $\star$ This paper $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$
 The previous studies Transaction Ways Dynamic nature More than one channel Game theory approach Second hand Market Durable Products Deprecia tion Brand Image Leasing selling Cao et al., 2018 [10] $\star$ $\star$ $\star$ Wang et al., 2018 [43] $\star$ $\star$ $\star$ Wang et al., 2019 [44] $\star$ $\star$ $\star$ Lai et al., 2018 [21] $\star$ $\star$ $\star$ Soleimani, 2016 [40] $\star$ $\star$ $\star$ Yu et al., 2020[52] $\star$ $\star$ $\star$ Zhu et al., 2020 [57] $\star$ $\star$ $\star$ Sun et al., 2021 [41] $\star$ $\star$ $\star$ Zhang and Xiao, 2013 [54] $\star$ $\star$ $\star$ Yang et al., 2018 [49] $\star$ $\star$ $\star$ Guo et al., 2021 [16] $\star$ $\star$ $\star$ Hsieh et al., 2014 [19] $\star$ $\star$ $\star$ Yoo and Lee, 2011[50] $\star$ $\star$ $\star$ Xiong et al., 2012 [47] $\star$ $\star$ $\star$ $\star$ $\star$ Xiao and Shi, 2016 [46] $\star$ $\star$ $\star$ Zhao et al., 2017 [56] $\star$ $\star$ $\star$ Chen et al., 20171 [11] $\star$ $\star$ $\star$ Nasiri et al., 2021 [29] $\star$ $\star$ Agarwal et al., 2011 [1] $\star$ $\star$ $\star$ Rogers and Rodrigues, 2015 [37] $\star$ $\star$ $\star$ Agrawal et al., 2012 [2] $\star$ $\star$ $\star$ Andriko poulos and Markellos, 2015 [3] $\star$ $\star$ $\star$ $\star$ Li et al., 2021 [24] $\star$ $\star$ $\star$ $\star$ Barbulescu and Enache, 2016 [36] $\star$ $\star$ Yan et al., 2016 [48] $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ hamidi et al., 2016 [17] $\star$ $\star$ $\star$ This paper $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$
The payoff matrix for end-user $\theta$ with $N = 2$
 $N^{t}(\theta)$ $S^{t}(\theta)$ $L^{t}(\theta)$ $W^{t}(\theta)$ $N^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}+p_{1}^{t}-d_{2}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}+p_{1}^{t}-d_{2}$ $p_{1}^{t}-d_{2}$ $S^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$ $L^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$ $W^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$
 $N^{t}(\theta)$ $S^{t}(\theta)$ $L^{t}(\theta)$ $W^{t}(\theta)$ $N^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}+p_{1}^{t}-d_{2}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}+p_{1}^{t}-d_{2}$ $p_{1}^{t}-d_{2}$ $S^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$ $L^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$ $W^{t-1}(\theta)$ $\Delta_{1}\theta+\lambda_{1}\varphi^{t}-p_{0}^{t}$ $\Delta_{2}\theta+\lambda_{3}\varphi^{t}-p_{1}^{t}$ $\Delta_{1}\theta+\lambda_{2}\varphi^{t}-r^{t}$ $0$
End-users policy in various classification
 Policy Interval $LL$ $(\theta_{1},1)$ $NS$ $(\theta_{2},\theta_{1})$ $SS$ $(\theta_{3},\theta_{2})$ $WW$ $(0,\theta_{3})$
 Policy Interval $LL$ $(\theta_{1},1)$ $NS$ $(\theta_{2},\theta_{1})$ $SS$ $(\theta_{3},\theta_{2})$ $WW$ $(0,\theta_{3})$
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