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Allocation of hospital capacity to multiple types of patients


Allocation of a limited capacity of resources among several customer types is a critical decision encountered by many manufacturing and service firms. We tackle this problem by focusing on a hospital setting and formulate a general model that is applicable to various resource allocation problems of a hospital. To this end, we consider a system with multiple customer classes that display different reactions to the delays in service. By adopting a dynamic-programming approach, we show that the optimal policy for a system involving both lost sales and backorders is not simple but exhibits desirable monotonicity properties. Furthermore, we propose a simple threshold heuristic policy that performs well in our experiments.

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We thank Dr Richard Park of Long Island Jewish Hospital, New Hide Park, New York, for his valuable comments and suggestions.

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Proof of Theorem 1: First, define the following function based on (2):

Using an inductive argument, it can be shown that each G t (s t ,x t ) is jointly concave in its arguments, and each f t (s t ) is also concave and decreasing. The details of this argument is standard and can be found in many of the papers mentioned in the literature; in particular, we refer the reader to the proof of Lemma 4.3 and Theorem 4.2 in Porteus (2002).

Note the optimal decision x t *(s t ) is obtained from maximizing a concave function G t (s t ,x t ) with respect to x t . From (3), it suffices to restrict the feasible set of x t to [(Cs t )+,C]. Then, from the definition of G t above and the definition of L(s t ,x t ) in (1),

  1. i)

    Consider the right-hand side of (A1). Since ft+1 is concave, the partial derivative in the last term is decreasing in s t . It follows that ∂G t (s t ,x t )/∂x t is decreasing in s t . Note also that the upper-bound of the feasible region Cs t is decreasing in s t . Therefore, we conclude that x t *(s t ) is decreasing in s t .

  2. ii)

    From (3) and its proof, it can be shown that G t (s t ,x t ) is increasing in x t if x t ⩽(Cs t )+. Thus, the optimal solution x t *(s t ) must satisfy the FOC that the partial derivative ∂G t (s t ,x t )/∂x t at x t =x t *(s t ) is zero.


Now, observe that


This result shows that G t (s t +ɛ,x t ) is increasing with respect to x t when x t =x t *(s t )−ɛ. Thus, we conclude that x t *(s t +ɛ)⩾x t *(s t )−ɛ. □

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Ayvaz, N., Huh, W. Allocation of hospital capacity to multiple types of patients. J Revenue Pricing Manag 9, 386–398 (2010).

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  • hospital resource allocation
  • backorders
  • lost sales
  • dynamic programming