# A Location Routing Problem with Time Windows Consideration: A Metaheuristics Approach
###### Abstract  
A substantial distribution system is achievable to attain through the simultaneous design of the location facility and route generation. This problem is generally called a location routing problem (LRP), where both strategic decision and operational decisions are considered during the decision initial phase. Nevertheless, for some real applications, a consideration of time restriction due to several reasons should be included in the model. Some examples include the distribution system of perishable goods, a company's objective towards a high level of customer service, or emergency/disaster situations. Consequently, this research incorporates the location routing problem (LRP) with time window constraints, including the working hour and also time restriction in each customer. The location routing problem with time windows (LRPTW) could not be solved easily, especially for a large-sized problem and or one which is considered an NP-hard problem. In consideration with its large solution pools, this paper suggests an integration of variable neighborhood search (VNS) with path relinking (PR) to solve LRPTW. The method is first applied in the LRP benchmark to validate the effectiveness and the robustness of the proposed method. Computational results clarify that VNSPR can achieve some of the best and most competitive solutions for the LRP when compared to existing approaches. Further, we developed LRPTW instances from the best solution found in LRP datasets. The proposed VNSPR method is also able to yield nearly optimal solutions for LRPTW instances. location routing problem; time windows; variable neighborhood search; path relinking Article  
A Location Routing Problem with Time Windows Consideration: A Metaheuristics Approach  
[PERSON] [PERSON], [PERSON]. the development of location routing models, which can determine true least-cost solutions to logistical problems while accounting for both strategic policy (facility location) and operational decisions (vehicle routing) [3]. Many variants of LRP have been widely studied depending on type of industry and the problems it is dealing with. The LRP's extensions include pickup and delivery LRP [4; 5; 6], open LRP [7; 8; 9], multi-objectives LRP [10; 11; 12; 13; 14], multi-echelon LRP [15; 11; 16; 17; 18], inventory LRP [19; 20; 21; 22; 23; 24], and LRP with uncertainty [25; 26; 27; 28; 23]. Nevertheless, considering that in practice the customers must be served before a certain time, many studies have added an additional constraint: time windows. The applications for LRP with time windows are quite wide, including for commercial logistics application [27; 29; 30], humanitarian logistics application [31; 32], green and/or sustainable logistics [33; 34; 35], and waste management [36; 37]. [PERSON] [38] introduced a single stage LRP with time windows (SSLRPTW) which combined capacitated facility location problem (CFLP) and VRPTW. The author tried to minimize the cost of routing and depot opening by decomposing the problem into two sub problems: CFLPTW and VRPTW. In the first phase, he solved the CFLPTW by assigning the customers to a depot and in the second phase, he used a tabu search to improve the solution. He extended the Solomon VRPTW benchmark into SSLRPTW benchmark. [PERSON] et al. [28] solved LRPTW with fuzzy travel time. They developed an algorithm called the simulation algorithm by incorporating a simulation module in SA. Small instances problems from [PERSON] et al. [39; 40] and [PERSON] [41] were solved. [PERSON] and [PERSON] [33] introduced the electric location routing problem with time windows and partial recharging (ELRP-TWPR) to model the distribution network using electric vehicle. Similar to [PERSON] [38], they also derived the problem based on VRPTW instances and added some constraints to model the recharging stage of electric vehicles.