Abstract
The capacitated p-median transportation inventory problem with heterogeneous fleet (CLITraP-HTF) aims to determine an optimal solution to a transportation problem subject to location-allocation, inventory management and transportation decisions. The novelty of CLITraP-HTF is to design a supply chain that solves all these decisions at the same time. Optimizing the CLITraP-HTF is a challenge because of the high dimension of the decision variables that lead to a large and complex search space. The contribution of this paper is to develop a dimensionality-reduction procedure (DRP) to reduce the CLITraP-HTF complexity and help to solve it. The proposed DRP is a mathematical proof to demonstrate that the inventory management and transportation decisions can be solved before the optimization procedure, thus reducing the complexity of the CLITraP-HTF by greatly narrowing its number of decision variables such that the remaining problem to solve is the well-known capacitated p-median problem (CPMP). The conclusion is that the proposed DRP helps to solve the CLITraP-HTF because the CPMP can be and has been solved by applying different algorithms and heuristic methods.
Highlights
The capacitated p-median transportation inventory problem with heterogeneous fleet model (CLITraP-HTF) is a non-linear-mixed-integer problem (MINLP)
The CLITraP-HTF is used to design the supply chain network (SCN) of any product that aims to determine an optimal solution to a transportation problem subject to extra constraints that locate supply and customers facilities, to manage the inventory of all facilities in a SCN, and to manage a fleet of vehicles characterized by different capacities and costs for the distribution of a product
Since Kariv and Hakimi [1] prove that the p-median problem (PMP) is a non-deterministic polynomial-time hardness (NP-hard) problem, and this paper proves the capacitated p-median problem (CPMP) is a subproblem of the CLITraP-HTF, it is possible to conclude that the CLITraP-HTF is a NP-hard problem too
Summary
The capacitated p-median transportation inventory problem with heterogeneous fleet model (CLITraP-HTF) is a non-linear-mixed-integer problem (MINLP). One option to solve a high dimensionality optimization problem, such as the CLITraP-HTF, in polynomial time is to sacrifice optimality by finding a feasible solution with a heuristic method, but an optimal solution is probably not achieved Another option is to relax the complexity of the problem by reducing the number of decision variables to solve with a dimensionality-reduction procedure (DRP) [11]. The DRP developed in this paper is a mathematical proof (Section 3) that helps to solve the CLITraP-HTF, the distribution between facilities is always made with a single type of vehicle, the one with the cheapest cost, and by sending only one shipment every replenishment period When solving these transportation decisions, the replenishment period decisions (one for each connection between facilities), the number of shipments between facilities per order or per replenishment period using the chosen vehicle type, the investment decisions, and the level of service decision variables are solved, all before the application of an optimization methodology. The LIRP published in Carmona-Benitez et al [14] is not a routing problem because it does not solve routing decisions, it is a transportation problem, because it solves transportation decisions, reason why, in this paper, the CLITraP-HTF is classified as a transportation problem
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