ORIGINAL ARTICLE Two meta-heuristics for a multi-period minisum location–relocation problem with line restriction N. Javadian & R. Tavakkoli-Moghaddam & M. Amiri-Aref & S. Shiripour Received: 21 February 2011 /Accepted: 18 November 2013 /Published online: 18 December 2013 # Springer-Verlag London 2013 Abstract This paper investigates a multiperiod rectilinear distance minisum location problem, as a mixed-integer non- linear programming (MINLP) model with a line-shaped barrier restriction, in which the starting point of the barrier uniformly distributed in the plane. The objective function of this model is to minimize the sum of the costs associated with the expected weighted barrier distance of the new facility from the existing facilities and the costs incurred by location- dependent relocation during the planning horizon. Then, a lower bound based on the forbidden region is presented. To show the validation of the presented model, a number of numerical examples are illustrated. The associated results show that the optimization software is effective for small-sized prob- lems. However, the optimization software is unable to find an optimum solution for large-sized problems in a reasonable time. Thus, two meta-heuristics, namely genetic algorithm (GA) and imperialist competitive algorithm (ICA), are proposed. Finally, the associated results are compared and discussed. Keywords Dynamic minisum problem . Rectilinear distance . Location-dependent relocation . Probabilistic line barrier . Meta-heuristics 1 Introduction Facility location is one of the well-established research areas in operations research. Despite the study of the facility location historically returns to Galileo, the modern-day facility location problem, classically referred to as median or minisum problem, is generally called a Weber problem, in honor of the German economist Alfred Weber. He studied the problem of locating a central warehouse to minimize the total travel distance between the warehouse and its customers. In location problems, a number of restrictions are considered by many researchers. Most of the common restricted planar location problems fall into one of the following three categories explained below, in which establishing and/or traveling through some area is not permitted: (1) forbidden regions (e.g., national parks or other protected areas), where establishing of a facility is prohibited but traveling through the regions is permit- ted; (2) congested regions (e.g., big lakes or forest), where establishing of a facility is prohibited but traveling through the region is possible with a penalty; and (3) barrier regions (e.g., military areas, mountain ranges, big rivers), where both establishing and traveling are forbidden. Hamacher and Nickel [1] surveyed location problems with forbidden regions extensively. For better understanding, see Table 1. The majority of the recent location problems have been developed for a single period planning horizon, in which some parameters (e.g., existing facility locations and demands) are assumed as constant for the entire planning horizon. However, decision on opening a facility can be affected by the environ- mental behavior of the existing facilities during the planning horizon and can be changed frequently over the period. For example, the facility should be opened in a given period, and then it should be closed in the next period. Note that opening and closing may impose fixed and variable costs. So, consid- ering discreteness, time-dependent parameters can strongly influence on strategic decision making processes. It is gener- ally true that in dynamic facility location problems, customers have to be served during a long period of time, considering some aspects (e.g., demand and cost that can be changed over N. Javadian : M. Amiri-Aref : S. Shiripour Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran R. Tavakkoli-Moghaddam (*) School of Industrial Engineering and Center of Excellence for Intelligence Based Experimental Mechanics, College of Engineering, University of Tehran, Tehran, Iran e-mail: tavakoli@ut.ac.ir Int J Adv Manuf Technol (2014) 71:1033–1048 DOI 10.1007/s00170-013-5511-y