Abstract
We consider a production–inventory queueing system consisting of a queue and an inventory, where there is a single type of product and a single firm. Customers can observe the number of products in the inventory and the number of waiting customers in the queue upon arrival. Customers decide whether to wait for the product or leave without a purchase according to their utility, which depends on the product’s price and estimated waiting time. If the number of products in the inventory is lower than a certain threshold, the firm produces the products. The product’s production time and the customers’ reward from purchasing the product have general distributions. We investigate the customers’ equilibrium strategies, profit maximization and social welfare maximization. Specifically, we show that a customer’s equilibrium strategy exists for a given joint pricing and inventory control. In general, there can exist multiple equilibria. However, if the production time distribution is decreasing mean residual life, the equilibrium is unique. We also present a method for computing equilibrium strategies. In addition, we compute the maximum profit rate and the profit-maximizing solution. We also compute the maximum social benefit rate and the welfare-maximizing solution. Finally, we present various numerical experiments that include comparisons of the maximum profit rate and the maximum social benefit rate, as well as of the profit-maximizing solution and the welfare-maximizing solution.










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References
Benjaafar, S., & Elhafsi, M. (2012). A production–inventory system with both patient and impatient demand classes. IEEE Transactions on Automation Science and Engineering, 9, 148–159.
Benjaafar, S., Gayon, J.-P., & Tepe, S. (2010). Optimal control of a production–inventory system with customer impatience. Operations Research Letters, 38, 267–272.
Cai, X., Li, J., Chen, S., & Tang, X. (2022). Joint pricing and inventory control in a make-to-stock queue with delay-sensitive customers. Journal of the Operational Research Society, 73, 417–429.
Chen, J., Huang, S., Hassin, R., & Zhang, N. (2015). Two backorder compensation mechanisms in inventory systems with impatient customers. Production and Operations Management, 24, 1640–1656.
Chen, X., & Simchi-Levi, D. (2012). Pricing and inventory management. In Ö. Özer & R. Phillips (Eds.), The Oxford handbook of pricing management (pp. 784–822). Oxford University Press.
Chen, Y., & Shi, C. (2019). Joint pricing and inventory management with strategic customers. Operations Research, 67, 1610–1627.
Economopoulos, A. A., Kouikoglou, V. S., & Grigoroudis, E. (2011). The base stock/base backlog control policy for a make-to-stock system with impatient customers. IEEE Transactions on Automation Science and Engineering, 8, 243–249.
Edelson, N. M., & Hildebrand, D. K. (1975). Congestion tolls for Poisson queueing processes. Econometrica, 43, 81–92.
Faaland, B., Mckay, M., & Schmitt, T. (2019). A fixed rate production problem with Poisson demand and lost sales penalties. Production and Operations Management, 28, 516–534.
Gavish, B., & Graves, S. C. (1980). A one-product production/inventory problem under continuous review policy. Operations Research, 28, 1228–1236.
Golrezaei, N., Nazerzadeh, H., & Randhawa, R. (2020). Dynamic pricing for heterogeneous time-sensitive customers. Manufacturing & Service Operations Management, 22, 562–581.
Güller, M. G., Bilgiç, T., & Güllü, R. (2014). Joint inventory and pricing decisions when customers are delay sensitive. International Journal of Production Economics, 157, 302–312.
Hassin, R. (2016). Rational queueing. CRC Press.
Hassin, R., & Haviv, M. (2003). To queue or not to queue: Equilibrium behavior in queueing systems. Boston: Kluwer.
Ioannidis, S., Jouini, O., Economopoulos, A. A., & Kouikoglou, V. S. (2013). Control policies for single-stage production systems with perishable inventory and customer impatience. Annals of Operations Research, 209, 115–138.
Kerner, Y. (2011). Equilibrium joining probabilities for an M/G/1 queue. Games and Economic Behavior, 71, 521–526.
Krishnamoorthy, A., Manikandan, R., & Lakshmy, B. (2015). A revisit to queueing-inventory system with positive service time. Annals of Operations Research, 233, 221–236.
Li, L. (1992). The role of inventory in delivery-time competition. Management Science, 38, 182–197.
Li, N., & Jiang, Z. (2013). Modeling and optimization of a product-service system with additional service capacity and impatient customers. Computers & Operations Research, 40, 1923–1937.
Li, Q., Guo, P., Li, C.-L., & Song, J.-S. (2016). Equilibrium joining strategies and optimal control of a make-to-stock queue. Production and Operations Management, 25, 1513–1527.
Marand, A. J., Li, H., & Thorstenson, A. (2019). Joint inventory control and pricing in a service-inventory system. International Journal of Production Economics, 209, 78–91.
Naor, P. (1969). The regulation of queue size by levying tolls. Econometrica, 37, 15–24.
Ok, E. A. (2007). Real analysis with economic applications. Princeton University Press.
Song, D. P. (2009). Stability and optimization of a production inventory system under prioritized base-stock control. IMA Journal of Management Mathematics, 20, 59–79.
Wang, J., & Zhang, X. (2017). Optimal pricing in a service-inventory system with delay-sensitive customers and lost sales. International Journal of Production Research, 55, 6883–6902.
Wang, J., & Zhang, Z. G. (2018). Strategic joining in an M/M/1 queue with risk-sensitive customers. Journal of the Operational Research Society, 69, 1197–1214.
Whitin, T. M. (1955). Inventory control and price theory. Management Science, 2, 61–68.
Zhang, X., & Wang, J. (2019). Optimal inventory threshold for a dynamic service make-to-stock system with strategic customers. Applied Stochastic Models in Business and Industry, 35, 1103–1123.
Zhao, N., & Lian, Z. (2011). A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129, 225–231.
Acknowledgements
We are grateful to the reviewers for their valuable comments and suggestions. B. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864). J. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01065568).
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Kim, B., Kim, J. & Lee, S. Joint pricing and inventory control for a production–inventory queueing system. Ann Oper Res 331, 787–805 (2023). https://doi.org/10.1007/s10479-022-04948-1
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DOI: https://doi.org/10.1007/s10479-022-04948-1