# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021079

## Joint optimal pricing and inventory management policy and its sensitivity analysis for perishable products: Lost sale case

 1 School of Automation, Southeast University, Nanjing, Jiangsu 210096, China 2 Industrial Systems Optimization Laboratory, University of Technology of Troyes, Troyes 10004, France

* Corresponding author: Zheng Wang

Received  August 2020 Revised  February 2021 Published  April 2021

Fund Project: This work is supported by National Natural Science Foundation of China under grant 61673109

In the real world, the demand cannot be depicted exactly because of customer behavior cannot be forecasted without error. In this paper, we study the effect of the error of the estimated price-demand parameters by analyzing the sensitivity of the optimal joint pricing and ordering policy on the price-demand parameters based on a periodic-review, multi-period and lost sale inventory model for perishable products with constant quantity decay rate and price-sensitive demand. Firstly, we formulate the joint pricing and inventory control problem and find the optimal ordering quantity and the optimal price for deterministic price-demand function. The optimal solutions show that the retailer tends to set a lower price in early periods of each ordering cycle in order to reduce the inventory holding costs. Furthermore, the sensitivity of the optimal joint pricing and inventory control system with respect to the price-demand parameters is examined analytically and evaluated numerically. The sensitivity analysis reveals that compared to the optimal ordering quantity, the optimal prices are less sensitive in the demand-price parameters. Finally, according to the findings of the sensitivity analysis, a heuristic method of regulating the estimated demand-price parameters is employed to improve the average profit. 185 words.

Citation: Xue Qiao, Zheng Wang, Haoxun Chen. Joint optimal pricing and inventory management policy and its sensitivity analysis for perishable products: Lost sale case. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021079
##### References:
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Whitin, Inventory control and price theory, Management Science, 2 (1955), 1-104.  doi: 10.1287/mnsc.2.1.61.  Google Scholar [24] D. Yao, Joint pricing and inventory control for a stochastic inventory system with Brownian motion demand, IISE Transactions, 49 (2017), 1101-1111.  doi: 10.1080/24725854.2017.1355126.  Google Scholar [25] P. H. Zipkin, Foundations of inventory management, 1$^{st}$ edition, McGraw-Hill, New York, (2000), 82–93. Google Scholar

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##### References:
 [1] E. Borgonovo and L. Peccati, Global sensitivity analysis in inventory management, International Journal of Production Economics, 108 (2007), 302-313.   Google Scholar [2] A. N. Burnetas and C. E. Smith, Adaptive ordering and pricing for perishable products, Oper. Res., 48 (2000), 436-443.  doi: 10.1287/opre.48.3.436.12437.  Google Scholar [3] L. X. Chen, X. Chen, M. F. Keblis and G. Li, Optimal pricing and replenishment policy for deteriorating inventory under stock-level-dependent, time-varying and price-dependent, Computers & Industrial Engineering, 135 (2019), 1294-1299.  doi: 10.1016/j.cie.2018.06.005.  Google Scholar [4] X. Chen, Z. Pang and L. Pan, Coordinating inventory control and pricing strategies for perishable products, Oper. Res., 62 (2014), 284-300.  doi: 10.1287/opre.2014.1261.  Google Scholar [5] X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case, Oper. Res., 52 (2004), 887-896.  doi: 10.1287/opre.1040.0127.  Google Scholar [6] E. P. Chew, C. Lee, R. J. Liu, K. S. Hong and A. M. Zhang, Optimal dynamic pricing and ordering decisions for perishable product, International Journal of Production Research, 157 (2014), 39-48.   Google Scholar [7] L. Feng, J. Zhang and W. Tang, Dynamic joint pricing and production policy for perishable products, Int. Trans. Oper. Res., 25 (2018), 2031-2051.  doi: 10.1111/itor.12239.  Google Scholar [8] A. Gutierrez-Alcoba, R. Rossi, B. Martin-Barragan and E. M. T. Hendrix, A simple heuristic for perishable item inventory control under non-stationary stochastic demand, International Journal of Production Research, 55 (2017), 1885-1897.  doi: 10.1080/00207543.2016.1193248.  Google Scholar [9] Y. He, S. Y. Wang and K. K. Lai, An optimal production-inventory model for deteriorating items with multiple-market demand, European Journal of Operational Research, 203 (2010), 593-600.   Google Scholar [10] A. Herbon and E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, European J. Oper. Res., 260 (2017), 546-556.  doi: 10.1016/j.ejor.2016.12.033.  Google Scholar [11] P. Ignaciuk, LQ optimal and robust control of perishable inventory systems with multiple supply options, IEEE Trans. Automat. Control, 58 (2013), 2108-2113.  doi: 10.1109/TAC.2013.2246093.  Google Scholar [12] P. Ignaciuk and A. Bartoszewicz, Linear-quadratic optimal control of periodic-review perishable inventory systems, IEEE Transactions on Control Systems Technology, 20 (2012), 1400-1407.  doi: 10.1109/TCST.2011.2161086.  Google Scholar [13] P. Ignaciuk and A. Bartoszewicz, LQ optimal sliding-mode supply policy for periodic-review perishable inventory systems, J. Franklin Inst., 349 (2012), 1561-1582.  doi: 10.1016/j.jfranklin.2011.04.003.  Google Scholar [14] O. Kaya and A. L. Polat, Coordinated pricing and inventory decisions for perishable products, OR Spectrum, 39 (2017), 589-606.  doi: 10.1007/s00291-016-0467-6.  Google Scholar [15] D.-H. Kim, Y. Song and H. Xu, A fast estimation procedure for discrete choice random coefficients demand model, Applied Economics, 49 (2017), 5849-5855.  doi: 10.1080/00036846.2017.1349289.  Google Scholar [16] S. Li, J. Zhang and W. Tang, Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products, International Journal of Production Research, 53 (2015), 2937-2950.  doi: 10.1080/00207543.2014.961206.  Google Scholar [17] Z. Lian and L. Liu, A discrete-time model for perishable inventory systems, Ann. Oper. Res., 87 (1999), 103-116.  doi: 10.1023/A:1018960314433.  Google Scholar [18] J. Lu, J. Zhang, X. Jia and G. Wei, Optimal dynamic pricing, preservation technology investment and periodic ordering policies for agricultural products, RAIRO Oper. Res., 53 (2019), 731-747.  doi: 10.1051/ro/2018040.  Google Scholar [19] S. Nahmias, Optimal ordering policies for perishable inventory-Ⅱ, Operations Research, 23 (1975), 735-749.  doi: 10.1287/opre.23.4.735.  Google Scholar [20] N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194.  doi: 10.1287/opre.47.2.183.  Google Scholar [21] Y. Qin, J. Wang and C. Wei, Joint pricing and inventory control for fresh produce and foods with quality and physical quantity deteriorating simultaneously, International Journal of Production Economics, 152 (2014), 42-48.  doi: 10.1016/j.ijpe.2014.01.005.  Google Scholar [22] B. Rabta, Sensitivity analysis in inventory models by means of ergodicity coefficients, International Journal of Production Economics, 188 (2017), 63-71.  doi: 10.1016/j.ijpe.2017.03.014.  Google Scholar [23] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 1-104.  doi: 10.1287/mnsc.2.1.61.  Google Scholar [24] D. Yao, Joint pricing and inventory control for a stochastic inventory system with Brownian motion demand, IISE Transactions, 49 (2017), 1101-1111.  doi: 10.1080/24725854.2017.1355126.  Google Scholar [25] P. H. Zipkin, Foundations of inventory management, 1$^{st}$ edition, McGraw-Hill, New York, (2000), 82–93. Google Scholar
Proof of the zero-inventory property
Repetitive ordering cycles
Change of the optimal ordering cycle
Sensitivity coefficient of the optimal ordering quantity
Sensitivity coefficient of the optimal price
Change of the optimal ordering cycle
Sensitivity coefficient of the optimal ordering quantity
Sensitivity coefficient of the optimal price
Performance of the heuristic regulation method on $s$
Performance of the heuristic regulating method on $\alpha$
Sensitivity analysis on perishability problem
 Paper optimization Backlog or lost sale or no shortage Sensitivity analysis objective Qin et al.[21] Price, order No shortage Decision variables on price-demand parameter and on deterioration function Lu et al.[18] Price, order, quality keeping No shortage Decision variables on all the system parameters Chen et al.[3] Price, order backlog Decision variables on decay rate, demand change, holding cost
 Paper optimization Backlog or lost sale or no shortage Sensitivity analysis objective Qin et al.[21] Price, order No shortage Decision variables on price-demand parameter and on deterioration function Lu et al.[18] Price, order, quality keeping No shortage Decision variables on all the system parameters Chen et al.[3] Price, order backlog Decision variables on decay rate, demand change, holding cost
Sensitivity analysis on perishability problem
 $\Delta s$ $\Delta \alpha$ $TP_{{\rm{new}}}^*$ $\Delta s >0$ $\Delta \alpha=0$ $TP_{{\rm{new}}}^* \le T{P^*}$ $\Delta s <0$ $\Delta \alpha=0$ $TP_{{\rm{new}}}^* \ge T{P^*}$ $\Delta s=0$ $\Delta \alpha >0$ $TP_{{\rm{new}}}^* \ge T{P^*}$ $\Delta s=0$ $\Delta \alpha <0$ $TP_{{\rm{new}}}^* \le T{P^*}$
 $\Delta s$ $\Delta \alpha$ $TP_{{\rm{new}}}^*$ $\Delta s >0$ $\Delta \alpha=0$ $TP_{{\rm{new}}}^* \le T{P^*}$ $\Delta s <0$ $\Delta \alpha=0$ $TP_{{\rm{new}}}^* \ge T{P^*}$ $\Delta s=0$ $\Delta \alpha >0$ $TP_{{\rm{new}}}^* \ge T{P^*}$ $\Delta s=0$ $\Delta \alpha <0$ $TP_{{\rm{new}}}^* \le T{P^*}$
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