Intl. Trans. in Op. Res. 27 (2020) 361–380 DOI: 10.1111/itor.12485 INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH Solving the bi-objective capacitated p-median problem with multilevel capacities using compromise programming and VNS Chandra Ade Irawan a , Arif Imran b and Martino Luis c a Nottingham University Business School China, University of Nottingham Ningbo China, Ningbo, China b Department of Industrial Engineering, Institut Teknologi Nasional, Bandung 40124, Indonesia c Othman Yeop Abdullah Graduate School of Business, Universiti Utara Malaysia, Sintok, Malaysia E-mail: chandra.irawan@nottingham.edu.cn [Irawan]; arifimr@yahoo.com [Imran]; martino@uum.edu.my [Luis] Received 5 February 2017; received in revised form 10 October 2017; accepted 17 October 2017 Abstract A bi-objective optimisation using a compromise programming (CP) approach is proposed for the capacitated p-median problem (CPMP) in the presence of the fixed cost of opening facility and several possible capacities that can be used by potential facilities. As the sum of distances between customers and their facilities and the total fixed cost for opening facilities are important aspects, the model is proposed to deal with those conflicting objectives. We develop a mathematical model using integer linear programming (ILP) to determine the optimal location of open facilities with their optimal capacity. Two approaches are designed to deal with the bi-objective CPMP, namely CP with an exact method and with a variable neighbourhood search (VNS) based matheuristic. New sets of generated instances are used to evaluate the performance of the proposed approaches. The computational experiments show that the proposed approaches produce interesting results. Keywords: capacitated p-median problem; bi-objective; compromise programming; VNS 1. Introduction The aim of the p-median problem (PMP) is to seek the location of p facilities among m discrete potential sites in such a way as to minimise the sum of the distances between customers and their associated facilities. The PMP was originally formulated by ReVelle and Swain (1970). This problem is also known as the minisum location problem which is categorised as NP-hard (Kariv and Hakimi, 1979). In the capacitated version of the p-median problem (CPMP), each customer has a fixed de- mand where each potential facility has a known capacity. Each facility must serve the demand of its customers without violating its capacity. This capacity constraint significantly multiplies the com- plexity of the problem. Therefore, CPMP falls into NP-hard problems (Garey and Johnson, 1979). In many real case applications, when finding the best location for the facilities, the fixed cost for opening facilities is usually taken into account. The fixed cost of a potential facility may be C  2017 The Authors. International Transactions in Operational Research C  2017 International Federation of Operational Research Societies Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148, USA.