# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018175

## Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects

 1 Ingram School of Engineering, Texas State University, San Marcos, TX 78666, USA 2 School of Management and Engineering, Nanjing University, Nanjing, China 210093 3 Amazon, Seattle, WA 98109, USA

* Corresponding author: Jingquan Li

Received  July 2017 Revised  August 2018 Published  December 2018

This paper studies a single-period inventory-pricing problem with two substitutable products, which is very important in the area of Operations Management but has received little attention. The proposed problem focuses on determining the optimal price of the existing product and the inventory level of the new product. Inspired by practice, the problem considers various pricing strategies for the existing product as well as the cross elasticity of demand between existing and new products. A mathematical model has been developed for different pricing strategies to maximize the expected profit. It has been proven that the objective function is concave and there exists the unique optimal solution. Different sets of computational examples are conducted to show that the optimal pricing and inventory strategy generated by the model can increase profits.

Citation: Zhijie Sasha Dong, Wei Chen, Qing Zhao, Jingquan Li. Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018175
##### References:
 [1] M. Akan, B. Ata and R. C. Savaskan-Ebert, Dynamic pricing of remanufacturable products under demand substitution: a product life cycle model, Annals of Operations Research, 211 (2013), 1-25. doi: 10.1007/s10479-013-1409-1. [2] G. Aydin and E. L. Porteus, Joint inventory and pricing decisions for an assortment, Operations Research, 56 (2008), 1247-1255. doi: 10.1287/opre.1080.0562. [3] D. Honhon, V. Gaur and S. Seshadri, Assortment planning and inventory decisions under stockout-based substitution, Operations Research, 58 (2010), 1364-1379. doi: 10.1287/opre.1090.0805. [4] C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288. doi: 10.1016/j.ejor.2008.02.019. [5] M. Karakul, Joint pricing and procurement of fashion products in the existence of clearance markets, International Journal of Production Economics, 114 (2008), 487-506. doi: 10.1016/j.ijpe.2007.03.026. [6] M. Karakul and L. M. A. Chan, Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem, European Journal of Operational Research, 190 (2008), 179-204. doi: 10.1016/j.ejor.2007.06.026. [7] M. Karakul and L. M. A. Chan, Joint pricing and procurement of substitutable products with random demands - A technical note, European Journal of Operational Research, 201 (2010), 324-328. doi: 10.1016/j.ejor.2009.03.030. [8] M. Khouja, A. Mehrez and G. Rabinowitz, A two-item newsboy problem with substitutability, International Journal of Production Economics, 44 (1996), 267-275. doi: 10.1016/0925-5273(96)80002-V. [9] Y. Lan, Z. Liu and B. Niu, Pricing and design of after-sales service contract: The value of mining asymmetric sales cost information, Asia-Pacific Journal of Operational Research, 34 (2017), 1740002. doi: 10.1142/S0217595917400024. [10] Y. Lan, R. Zhao and W. Tang, A fuzzy supply chain contract problem with pricing and warranty, Fuzzy Systems, 26 (2014), 1527-1538. doi: 10.3233/IFS-130836. [11] X. Li, G. Sun and Y. Li, A multi-period ordering and clearance pricing model considering the competition between new and out-of-season products, Annals of Operations Research, 242 (2016), 207-221. doi: 10.1007/s10479-013-1498-x. [12] S. Mou, D. J. Robb and N. DeHoratius, Retail store operations: Literature review and research directions, European Journal of Operational Research, 265 (2018), 399-422. doi: 10.1016/j.ejor.2017.07.003. [13] M. Nagarajan and S. Rajagopalan, Inventory models for substitutable products: Optimal policies and heuristics, Management Science, 54 (2008), 1453-1466. doi: 10.1287/mnsc.1080.0871. [14] X. A. Pan and D. Honhon, Assortment planning for vertically differentiated products, Production and Operations Management, 21 (2012), 253-275. doi: 10.1111/j.1937-5956.2011.01259.x. [15] A. Sainathan, Pricing and replenishment of competing perishable product variants under dynamic demand substitution, Production and Operations Management, 22 (2013), 1157-1181. doi: 10.1111/poms.12004. [16] Z. Sazvar, S. M. J. Mirzapour Al-e-hashem, K. Govindan and B. Bahli, A novel mathematical model for a multi-period, multi-product optimal ordering problem considering expiry dates in a FEFO system, Transportation Research Part E: Logistics and Transportation Review, 93 (2016), 232-261. doi: 10.1016/j.tre.2016.04.011. [17] H. Shin, S. Park, E. Lee and W. C. Benton, A classification of the literature on the planning of substitutable products, European Journal of Operational Research, 246 (2015), 686-699. doi: 10.1016/j.ejor.2015.04.013.

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##### References:
 [1] M. Akan, B. Ata and R. C. Savaskan-Ebert, Dynamic pricing of remanufacturable products under demand substitution: a product life cycle model, Annals of Operations Research, 211 (2013), 1-25. doi: 10.1007/s10479-013-1409-1. [2] G. Aydin and E. L. Porteus, Joint inventory and pricing decisions for an assortment, Operations Research, 56 (2008), 1247-1255. doi: 10.1287/opre.1080.0562. [3] D. Honhon, V. Gaur and S. Seshadri, Assortment planning and inventory decisions under stockout-based substitution, Operations Research, 58 (2010), 1364-1379. doi: 10.1287/opre.1090.0805. [4] C. C. Hsieh and C. H. Wu, Coordinated decisions for substitutable products in a common retailer supply chain, European Journal of Operational Research, 196 (2009), 273-288. doi: 10.1016/j.ejor.2008.02.019. [5] M. Karakul, Joint pricing and procurement of fashion products in the existence of clearance markets, International Journal of Production Economics, 114 (2008), 487-506. doi: 10.1016/j.ijpe.2007.03.026. [6] M. Karakul and L. M. A. Chan, Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem, European Journal of Operational Research, 190 (2008), 179-204. doi: 10.1016/j.ejor.2007.06.026. [7] M. Karakul and L. M. A. Chan, Joint pricing and procurement of substitutable products with random demands - A technical note, European Journal of Operational Research, 201 (2010), 324-328. doi: 10.1016/j.ejor.2009.03.030. [8] M. Khouja, A. Mehrez and G. Rabinowitz, A two-item newsboy problem with substitutability, International Journal of Production Economics, 44 (1996), 267-275. doi: 10.1016/0925-5273(96)80002-V. [9] Y. Lan, Z. Liu and B. Niu, Pricing and design of after-sales service contract: The value of mining asymmetric sales cost information, Asia-Pacific Journal of Operational Research, 34 (2017), 1740002. doi: 10.1142/S0217595917400024. [10] Y. Lan, R. Zhao and W. Tang, A fuzzy supply chain contract problem with pricing and warranty, Fuzzy Systems, 26 (2014), 1527-1538. doi: 10.3233/IFS-130836. [11] X. Li, G. Sun and Y. Li, A multi-period ordering and clearance pricing model considering the competition between new and out-of-season products, Annals of Operations Research, 242 (2016), 207-221. doi: 10.1007/s10479-013-1498-x. [12] S. Mou, D. J. Robb and N. DeHoratius, Retail store operations: Literature review and research directions, European Journal of Operational Research, 265 (2018), 399-422. doi: 10.1016/j.ejor.2017.07.003. [13] M. Nagarajan and S. Rajagopalan, Inventory models for substitutable products: Optimal policies and heuristics, Management Science, 54 (2008), 1453-1466. doi: 10.1287/mnsc.1080.0871. [14] X. A. Pan and D. Honhon, Assortment planning for vertically differentiated products, Production and Operations Management, 21 (2012), 253-275. doi: 10.1111/j.1937-5956.2011.01259.x. [15] A. Sainathan, Pricing and replenishment of competing perishable product variants under dynamic demand substitution, Production and Operations Management, 22 (2013), 1157-1181. doi: 10.1111/poms.12004. [16] Z. Sazvar, S. M. J. Mirzapour Al-e-hashem, K. Govindan and B. Bahli, A novel mathematical model for a multi-period, multi-product optimal ordering problem considering expiry dates in a FEFO system, Transportation Research Part E: Logistics and Transportation Review, 93 (2016), 232-261. doi: 10.1016/j.tre.2016.04.011. [17] H. Shin, S. Park, E. Lee and W. C. Benton, A classification of the literature on the planning of substitutable products, European Journal of Operational Research, 246 (2015), 686-699. doi: 10.1016/j.ejor.2015.04.013.
Effect of the extant product's price on the new product's order quantity
Effect of the extant product's price on the expected profit
Effect of the inventory level of the existing product on the retailer's optimal policy and the expected profit
 Strategy Variables Values $Q_1$ 160 170 180 190 200 210 220 230 240 250 Unchanged $Q_2$ 840 830 820 810 800 800 800 800 800 800 $s_1 (\$)$12 12 12 12 12 12 12 12 12 12$ EP (\$)$ 5280 5360 5440 5520 5600 5600 5600 5600 5600 5600 $RP (\$)$100.0 100.0 100.0 100.0 100.0 101.0 102.0 103.0 104.0 105.0 Decreased$ Q_2 $840 830 820 810 800 790 780 770 760 746.6$ s_1 (\$)$ 12 12 12 12 12 11.7 11.4 11.1 10.8 10.5 $EP (\$)$5280 5360 5440 5520 5600 5617 5628 5633 5632 5611.4$ RP (\$)$ 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.7
 Strategy Variables Values $Q_1$ 160 170 180 190 200 210 220 230 240 250 Unchanged $Q_2$ 840 830 820 810 800 800 800 800 800 800 $s_1 (\$)$12 12 12 12 12 12 12 12 12 12$ EP (\$)$ 5280 5360 5440 5520 5600 5600 5600 5600 5600 5600 $RP (\$)$100.0 100.0 100.0 100.0 100.0 101.0 102.0 103.0 104.0 105.0 Decreased$ Q_2 $840 830 820 810 800 790 780 770 760 746.6$ s_1 (\$)$ 12 12 12 12 12 11.7 11.4 11.1 10.8 10.5 $EP (\$)$5280 5360 5440 5520 5600 5617 5628 5633 5632 5611.4$ RP (\$)$ 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 99.7
Effect of the salvage value of the existing product on the retailer's optimal policy and the expected profit
 Strategy Variables Values $h_1$ -5 -4 -3 -2 -1 0 1 2 3 4 Unchanged $Q_2$ 800 800 800 800 800 800 800 800 800 800 $s_1 (\$)$12 12 12 12 12 12 12 12 12 12$ EP (\$)$ 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 $RP (\$)$105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 Decreased$ Q_2 $750 750 750 750 750 750 750 750 750 750$ s_1 (\$)$ 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 $EP (\$)$5625 5625 5625 5625 5625 5625 5625 5625 5625 5625$ RP (\$)$ 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
 Strategy Variables Values $h_1$ -5 -4 -3 -2 -1 0 1 2 3 4 Unchanged $Q_2$ 800 800 800 800 800 800 800 800 800 800 $s_1 (\$)$12 12 12 12 12 12 12 12 12 12$ EP (\$)$ 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 $RP (\$)$105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 105.0 Decreased$ Q_2 $750 750 750 750 750 750 750 750 750 750$ s_1 (\$)$ 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 $EP (\$)$5625 5625 5625 5625 5625 5625 5625 5625 5625 5625$ RP (\$)$ 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
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