Robust design of service systems with immobile servers, general arrival and service patterns, and demand uncertainty

Abstract

This paper addresses the problem of robustly designing a service system consisting of immobile servers, each modelled as a G/G/1 queuing system, when the arrival rates are not known with certainty. The problem involves locating service centers, determining their capacities and assigning customers to them to minimize the total cost, which includes the setup, access and waiting costs. Besides the nominal problem, two robust problems with budget and ball uncertainty sets are considered. A piecewise-linear approximation is applied to handle the nonlinear waiting cost, which enables all the problems to be tightly approximated as mixed-integer quadratic programs. We also propose a Lagrangian approach that is capable of finding high-quality solutions and strong bounds for instances of practical sizes. Numerical experiments were conducted to validate the proposed models and solution methods and to study the effect of the problem parameters, the uncertainty set size and the objective function approximations on the optimal solution.

Keywords:

• Service systems
• network design
• quadratic programming
• Lagrangian relaxation
• piecewise-linear approximation

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement (DAS)

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

1 It should be noted that once this assignment is determined, the optimal service rate in a SC can be easily found using first-order optimality conditions, i.e. by equating to zero the partial derivative of the cost function with respect to its service rate variable ${\mu }_j$.

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [RGPIN-2018-04745].

Notes on contributors

Ahmed Saif

Ahmed Saif is an Associate Professor in the Department of Industrial Engineering, Dalhousie University (Canada). He holds a PhD in Management Sciences from the University of Waterloo, a MSc in Engineering Systems and Management and an MBA. His research interests include decision making under uncertainty, large scale optimization and nonlinear optimization, with applications in supply chain management, energy and finance.

Nazanin Madani

Nazanin Madani holds a MASc in Industrial Engineering from Dalhousie University and a MSc degree in Applied Mathematics from KNTU (Iran). Her research interests include nonlinear and robust optimization.