Optimal pricing and capacity management in service systems with delay-sensitive mixed-risk customers

Abstract

In service systems with delay-sensitive customers, customer decisions to join the queue or balk influence demand, impacting service provider profits. Studies show that customers can be risk-seeking for short waits and risk-averse when waits exceed expectations. Prior research on this mixed-risk attitude is limited, focussing on optimal pricing policies from a social welfare perspective. This study fills this gap by investigating the impact of this behaviour on customer joining strategies and the provider's pricing and capacity policies. We analyse a profit-maximising firm managing an unobservable queue where customers display mixed-risk behaviour towards delay parameterised by a risk-switching point and a degree of risk propensity. Customers join or balk based on expected utility, while the firm determines the service price and rate to maximise expected profit. We derive optimal strategies for both customers and the firm under scenarios with concave or convex capacity costs. Computational experiments show that optimal pricing, capacity, and profit peak at intermediate risk-switching points. For smaller points, optimal price and capacity rise with the customers' degree of risk propensity, while larger points have minimal impact. Numerical exploration emphasises the significance of capacity cost and the effectiveness of leveraging price over controlling capacity for maximising firm profitability.

Keywords:

• Strategic queueing
• mixed-risk behaviour
• delay-sensitive customers
• pricing
• capacity management

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

Parameters

Λ:	=	Customers' potential arrival rate (Market size)
R:	=	Customers' reward upon service completion
c:	=	Customers' waiting cost coefficient
d:	=	Customers' risk-switching point
α:	=	Customers' degree of risk propensity
ν:	=	Provider's idle capacity
$\mathrm{\nu d}:$	=	Customers' normalised risk-switching point
ρ:	=	Provider's utilisation rate

Variables

λ:	=	Customers' effective arrival rate (actual demand)
μ:	=	Provider's service rate (capacity)
p:	=	Provider's price
Functions	=
$W(\cdot ):$	=	Customers' expected delay cost
$U(\cdot ):$	=	Customers' expected utility
$Q(\cdot ):$	=	Provider's capacity cost per unit time
$G(\cdot ):$	=	Provider's expected profit per unit time
$S(\cdot ):$	=	Social welfare per unit time

Regions

$\mathcal{F},\mathcal{P}:$	=	Provider's feasible and profitable region
$\underset{\_}{\mu }:$	=	Provider's minimum capacity in $\mathcal{F}$
$\overline{p}:$	=	Provider's maximum price in $\mathcal{F}$
$\underset{\sim }{\mu },\tilde{\mu }:$	=	Provider's minimum and maximum capacity in $\mathcal{P}$
$\underset{\sim }{p},\tilde{p}:$	=	Provider's minimum and maximum price in $\mathcal{P}$

Data availability statement

Data sharing does not apply to this article as no new data were created or analysed in this study.

Additional information

Funding

This research has been co-financed by the European Union and Greek National funds through the Regional Operational Program of Thessaly (grant no. MIS 5162183) of the Partnership Agreement for the Development Framework 2014–2020.

Notes on contributors

Michalis Deligiannis

Michalis Deligiannis obtained his B.Sc. in Mathematics and his M.Sc. in Statistics and Modeling from Aristotle University of Thessaloniki, Greece, in 2013 and 2016, respectively. He completed his Ph.D. in Operations Management in the Department of Mechanical Engineering at the University of Thessaly, Greece, in 2022. Currently, he is a Postdoctoral Researcher and Lecturer in the Department of Mechanical Engineering at the University of Thessaly. His research interests are in stochastic optimisation, Markovian decision processes, game theory, and machine learning, with a focus on their applications in supply chain management.

Myron Benioudakis

Myron Benioudakis holds a Ph.D. in Production & Operations Management from Athens University of Economics & Business, an M.Sc. in Statistics & Operations Research, and a B.Sc. in Mathematics, both from the National & Kapodistrian University of Athens, Greece. During 2021–2023, he worked as a Postdoctoral Researcher and Lecturer in the Department of Mechanical Engineering at the University of Thessaly, Greece. Currently, he is a Postdoctoral Researcher and Lecturer in the Department of Management Science & Technology at Athens University of Economics & Business, and a Lecturer at Hellenic Open University, Greece. His research interests primarily focus on queueing models with strategic customer behaviour, game theory, stochastic modelling, and applications in supply chain management.

George Liberopoulos

George Liberopoulos (B.S. '85 & M.Eng. '86, Mechanical Engineering, Cornell University; Ph.D. '93, Manufacturing Engineering, Boston University) is a Professor of Production Management and the Director of the Production Management Laboratory in the Department of Mechanical Engineering at the University of Thessaly (UTH), Greece. Before joining UTH, he was a Lecturer in the Department of Mechanical Engineering at Boston University (1993) and a Postdoctoral Research Fellow in Laboratoire d'Informatique de Sorbonne Université/CNRS (LIP6), France (1994–1996). During 2011–2017, he was a member of the Board of Commissioners of Greece's Regulatory Authority for Railways. He is a member of the Scientific Committee of the International Conference on Stochastic Models of Manufacturing and Service Operations and has served on the editorial boards of IISE Transactions, OR Spectrum, and Flexible Services and Manufacturing Journal. His research interests are in performance evaluation and control of manufacturing systems, production planning and scheduling, inventory control and supply chain management, design of electricity markets and service systems, using mathematical and dynamic programming, stochastic modelling and simulation, queueing and game theory, Markovian decision processes, and machine learning.

Apostolos Burnetas

Apostolos Burnetas is a Professor of Stochastic Operations Research in the Department of Mathematics at the National and Kapodistrian University of Athens (NKUA), Greece. He received a Diploma in Electrical Engineering from the National Technical University of Athens in 1986, and an MBA (1992) and Ph.D. (1993) from Rutgers University. Before joining NKUA he was an Assistant Professor (1994–2000) and Associate Professor (2000–2003) in the Department of Operations Research at Case Western Reserve University. He is a member of the Scientific Committee of the Stochastic Modeling Group of INFORMS, an Associate Editor in Operations Research Letters and SN Operations Research Forum, and Area Editor of Operations Research in the Bulletin of the Hellenic Mathematical Society. His research interests are in stochastic modelling and optimisation, queueing theory, inventory and supply chain management, game theory models in queueing and supply chains, optimisation under incomplete information, multi-armed bandit models, Markovian decision processes and reinforcement learning.