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Item New formulations for multiple traveler minimum latency problem with time windows(2022) Uzun, Gozde Onder; Kara, Imdat; ABH-1078-2021In this paper, new mathematical models for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows (kLPTW), named as M2 and M4 are developed. These models are computationally compared with existing models named as M1 and M3 for kLPTW in terms of CPU times and percentage deviation from linear programming relaxation values. A short summary of the computational analysis is given in table A below. In Table A, k is the number of travelers. The first column under the number of traveler cell shows the average CPU times of problems solved in time limit and the second column shows the average percentage deviations. We observed that, our formulations are superior than the existing formulations for all the problems for both kLPTW types with respect to each performance criteria. Purpose: The aim of this study is to develop new mathematical formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows. Theory and Methods: Based on the mixed integer linear programming, mathematical models with polynomial number of decision variables and constraints are developed. Benchmark instances from the literature are solved with existing formulations and proposed new formulations by using CPLEX 12.5.0.1. CPU times and percentage deviation from linear programming relaxation values are considered as performance criteria. Results: We solved 125 problems with varying number of nodes and time windows. In all the problem solved proposed formulations are better than the existing formulations in terms of both of the performance criteria. Conclusion: The proposed formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows are superior than the existing formulations and able to solve the problems up to 100 nodes with narrow time windows. Proposed formulations may be used to solve small and moderate real-life problems very easily. They may also be used for testing the performance of the heuristics constructed for kLPTW.Item New formulations for the traveling repairman problem with time windows(2021) Uzun, Gozde Onder; Kara, Imdat; ABH-1078-2021The Traveling Repairman Problem (TRP) is one of the most important variants of the Traveling Salesman Problem (TSP). The objective function of TRP is to find a Hamiltonian path or tour starting from the origin while minimizing the total latency (waiting or delay time) for all customers. The latency of a customer is defined as the time passed from the beginning of a tour (or path) until a customer?s service is completed. TRP with time windows (TRPTW) is the case where the earliest and latest times for visiting each customer are restricted by prescribed time windows. The literature on TRPTW is scarce. We only found one formulation for TRPTW and one formulation for its variant. In this paper, we propose four new mathematical models for TRPTW with O(n2) binary variables and O(n2) constraints. We computationally analyze the performance of existing and new formulations by solving symmetric and asymmetric benchmark instances with CPLEX 12.5.0.1 and compare the results in terms of CPU times and optimality gap. We observed that our two formulations were extremely faster than existing formulations, and they could optimally solve symmetric instances up to 150 nodes and asymmetric instances up to 131 nodes within seconds.