Exact solution approaches for non-Hamiltonian vehicle routing problems

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Paternotte, Hande Yaman
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Bilkent University
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In this thesis, we study di erent non-Hamiltonian vehicle routing problem variants and concentrate on developing e cient optimization algorithms to solve them. First, we consider the split delivery vehicle routing problem (SDVRP).We provide a vehicle-indexed ow formulation for the problem, and then, a relaxation obtained by aggregating the vehicle-indexed variables over all vehicles. This relaxation may have optimal solutions where several vehicles exchange loads at some customers. We cut-o such solutions either by extending the formulation locally with vehicle-indexed variables or by node splitting. We compare these approaches using instances from the literature and new randomly generated instances. Additionally, we introduce two new extensions of the SDVRP by restricting the number of splits and by relaxing the depot return requirement, and modify our algorithms to handle these extensions. Second, we focus on a problem unifying the notion of coverage and routing. In some real-life applications, it may not be viable to visit every single customer separately due to resource limitations or e ciency concerns. In such cases, utilizing the notion of coverage; i.e., satisfying the demand of multiple customers by visiting a single customer location, may be advantageous. With this motivation, we study the time constrained maximal covering salesman problem (TCMCSP) in which the aim is to nd a tour visiting a subset of customers so that the amount of demand covered within a limited time is maximized. We provide ow and cut formulations and derive valid inequalities. Since the connectivity constraints and the proposed valid inequalities are exponential in the size of the problem, we devise di erent branch-and-cut schemes. Computational experiments performed on a set of problem instances demonstrate the e ectiveness of the proposed valid inequalities in terms of strengthening the linear relaxation bounds as well as speeding up the solution procedure. Moreover, the results indicate the superiority of using a branch-and-cut methodology over a ow-based formulation. Finally, we discuss the relation between the problem parameters and the structure of optimal solutions based on the results of our experiments. Third, we study the vehicle routing problem with roaming delivery locations (VRPRDL) in which a customer order has to be delivered to the trunk of the customer's car during the time that the car is parked at one of the locations in the (known) customer's travel itinerary. We formulate the problem as a set covering problem and develop a branch-and-price algorithm for its solution. The algorithm can also be used for solving a more general variant in which a hybrid delivery strategy is considered that allows a delivery to either a customer's home or to the trunk of the customer's car. We evaluate the e ectiveness of the many algorithmic features incorporated in the algorithm in an extensive computational study and analyze the bene ts of these innovative delivery strategies. The computational results show that employing the hybrid delivery strategy results in average cost savings of nearly 20% for the instances in our test set.Finally, we consider the dynamic version of the VRPRDL in which customer itineraries may change during the execution of the planned delivery schedule, which can become infeasible or suboptimal as a result. We refer to this problem as the dynamic VRPRDL (D-VRPRDL) and propose an iterative solution framework in which the previously planned vehicle routes are re-optimized whenever an itinerary update is revealed. We use the branch-and-price algorithm developed for the static VRPRDL both for solving the planning problem (to obtain an initial delivery schedule) and for solving the re-optimization problems. Since many re-optimization problems may have to be solved during the execution stage, it is critical to produce solutions to these problems quickly. To this end, we devise heuristic procedures through which the columns generated during the previous branch-and-price executions can be utilized when solving a re-optimization problem. In this way, we may be able to save time that would otherwise be spent in generating columns which have already been (partially) generated when solving the previous problems, and nd optimal solutions or at least solutions of good quality reasonably quickly. We perform preliminary computational experiments and report the results.

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