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Pricing and Capacity Decisions for a Service Facility: Stability and Multiple Local Optima

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  • Shaler Stidham

    (Department of Operations Research CB# 3180, Smith Building, University of North Carolina, Chapel Hill, North Carolina 27599-3180)

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    Abstract

    We consider a model for optimal pricing and capacity for a service facility. The problem is formulated as one of optimal design of a single-server queueing system, in which the design variables are the service rate and the arrival rate (equivalently, the price charged for admission). The model is a variant of one introduced by Dewan and Mendelson. We allow for an upper bound on the arrival rate and consider slightly more general user value functions. We show that an optimal solution may not lie in the interior of the feasible region and thus may not be characterized by the first-order differential conditions. Moreover, the first-order conditions typically have several solutions, some of which may be relative minima and produce a negative value of the objective function (customer value minus the sum of expected delay cost and capacity cost per unit time). We also examine the stability of the equilibrium arrival rate and the convergence of a dynamic adaptive algorithm for finding the optimal service rate, in the context of a model in which the distribution of customers' value of service is uniform. We show that the equilibrium arrival rate is stable if and only if the service rate is above a threshold value, which depends on the price charged for admission and the parameters of the uniform distribution of value of service. The dynamic, adaptive algorithm always converges to a relative maximum of the objective function if the service rate can be adjusted every time the arrival rate changes. Otherwise, the algorithm will start to diverge if and when the service rate ever falls below the threshold value associated with stability of the equilibrium arrival rate.

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    File URL: http://dx.doi.org/10.1287/mnsc.38.8.1121
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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 38 (1992)
    Issue (Month): 8 (August)
    Pages: 1121-1139

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    Handle: RePEc:inm:ormnsc:v:38:y:1992:i:8:p:1121-1139

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    Keywords: optimal pricing; optimal capacity; service facility; optimal design of queues; stable equilibrium;

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    Cited by:
    1. Grossman, Thomas A. & Brandeau, Margaret L., 2002. "Optimal pricing for service facilities with self-optimizing customers," European Journal of Operational Research, Elsevier, vol. 141(1), pages 39-57, August.
    2. Wang, E. T. G., 2000. "Information and incentives in computing services supply: The effect of limited system choices," European Journal of Operational Research, Elsevier, vol. 125(3), pages 503-518, September.
    3. Aktaran-KalaycI, Tûba & Ayhan, Hayriye, 2009. "Sensitivity of optimal prices to system parameters in a steady-state service facility," European Journal of Operational Research, Elsevier, vol. 193(1), pages 120-128, February.
    4. Bitran, Gabriel R. & Rocha e Oliveira, Paulo & Schilkrut, Ariel, 2008. "Managing customer relationships through price and service quality," IESE Research Papers D/750, IESE Business School.
    5. Eric Friedman, 1998. "Learnability of a class of Non-atomic Games arising on the Internet," Departmental Working Papers 199824, Rutgers University, Department of Economics.
    6. Friedman, Eric J., 2002. "Strategic properties of heterogeneous serial cost sharing," Mathematical Social Sciences, Elsevier, vol. 44(2), pages 145-154, November.
    7. Mei Xue & Patrick T. Harker, 2003. "Service Co-Production, Customer Efficiency and Market Competition," Center for Financial Institutions Working Papers 03-03, Wharton School Center for Financial Institutions, University of Pennsylvania.
    8. Pekka Ilmakunnas, 2002. "Strategic behavior in a service industry," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 23(2), pages 69-82.
    9. So, Kut C. & Song, Jing-Sheng, 1998. "Price, delivery time guarantees and capacity selection," European Journal of Operational Research, Elsevier, vol. 111(1), pages 28-49, November.
    10. Wang, Shoou-Jiun & Batta, Rajan & Rump, Christopher M., 2005. "Stability of a crime level equilibrium," Socio-Economic Planning Sciences, Elsevier, vol. 39(3), pages 229-244, September.
    11. Jayaswal, Sachin & Jewkes, Elizabeth & Ray, Saibal, 2011. "Product differentiation and operations strategy in a capacitated environment," European Journal of Operational Research, Elsevier, vol. 210(3), pages 716-728, May.
    12. Ke-Wei Huang & Arun Sundararajan, 2006. "Pricing Digital Goods: Discontinuous Costs and Shared Infrastructure," Working Papers 06-11, NET Institute, revised Sep 2006.
    13. Zhou, Wenhui & Lian, Zhaotong & Wu, Jinbiao, 2014. "When should service firms provide free experience service?," European Journal of Operational Research, Elsevier, vol. 234(3), pages 830-838.
    14. Jiang, Bin, 2005. "The decision-making on an in-house logistic division's operation strategies," International Journal of Production Economics, Elsevier, vol. 96(1), pages 37-46, April.
    15. Son, Jae-Dong, 2007. "Customer selection problem with profit from a sideline," European Journal of Operational Research, Elsevier, vol. 176(2), pages 1084-1102, January.

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