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Capacity allocation for demand of different customer-product-combinations with cancellations, no-shows, and overbooking when there is a sequential delivery of service

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Abstract

We consider a problem where different classes of customers can book different types of services in advance and the service company has to respond immediately to the booking request confirming or rejecting it. Due to the possibility of cancellations before the day of service, or no-shows at the day of service, overbooking the given capacity is a viable decision. The objective of the service company is to maximize profit made of class-type specific revenues, refunds for cancellations or no-shows as well as the cost of overtime. For the calculation of the latter, information of the underlying appointment schedule is required. Throughout the paper we will relate the problem to capacity allocation in radiology services. Drawing upon ideas from revenue management, overbooking, and appointment scheduling we model the problem as a Markov decision process in discrete time which due to proper aggregation can be optimally solved with an iterative stochastic dynamic programming approach. In an experimental study we successfully apply the approach to a real world problem with data from the radiology department of a hospital. Furthermore, we compare the optimal policy to four heuristic policies, of whom one is currently in use. We can show that the optimal policy significantly improves the currently used policy and that a nested booking limit type policy closely approximates the optimal policy and is thus recommended for use in practice.

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Acknowledgements

We wish to thank Ernst J. Rummeny and Bernhard Renger from the radiological department of the university hospital ‘Klinikum rechts der Isar’ for supporting this work. Furthermore, we thank an anonymous reviewer for his constructive and insightful remarks which considerably improved this paper. This research was partially funded by the German Research Foundation DFG under grant KO 16180/4-1.

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Authors and Affiliations

  1. TUM School of Management, Technische Universität München, Munich, Germany

    Hans-Jörg Schütz & Rainer Kolisch

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  1. Hans-Jörg Schütz
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  2. Rainer Kolisch
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Correspondence to Rainer Kolisch.

Appendix: Notation

Appendix: Notation

Indices

i=1,…,I :

Patient classes

k=1,…,K :

Examination types

n=N,…,1,0:

Decision periods; 0 = service period

Functions and state variable

G n (i,k):

Expected refund in case of cancellation or no-show

OT(x):

Expected overtime in the service period

U(x,n):

Value function for state x in period n

x=(x 1,…,x I ):

State vector; x i = booked capacity of class i

x ik :

Booked capacity from class i of type k

Z(x,n):

Total number of booked slots in period n

Parameters

a ik :

Percentage of accepted bookings from each class and type at the end of the booking horizon

b ik :

Expected percentage of requests of type k within class i

β ik :

No-show probability of a booking from class i of type k

c ik :

Refund in case of a cancellation of an appointment

C :

Total capacity in slots

d ik :

Refund in case of a no-show from class i of type k

f :

Cost of overtime per slot

l ik :

Capacity requirement of an examination from class i of type k

l max :

Maximum over all l ik for all i and k

p ikn :

Probability of a request from class i of type k in period n

p 0n :

Probability of no event in period n

q ikn :

Probability of a cancellation of a booked appointment

r ik :

Revenue from an examination from class i of type k

s ik :

Cost of rejecting a request from class i of type k

v :

Overbooking pad

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Schütz, HJ., Kolisch, R. Capacity allocation for demand of different customer-product-combinations with cancellations, no-shows, and overbooking when there is a sequential delivery of service. Ann Oper Res 206, 401–423 (2013). https://doi.org/10.1007/s10479-013-1324-5

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  • DOI: https://doi.org/10.1007/s10479-013-1324-5

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