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Publication# A ride time-oriented scheduling algorithm for dial-a-ride problems

Abstract

This paper offers a new algorithm to efficiently optimize scheduling decisions for dial-a-ride problems (DARPs), including problem variants considering electric and autonomous vehicles (e-ADARPs). The scheduling heuristic, based on linear programming theory, aims at finding minimal user ride time schedules in worst-case quadratic time. The algorithm can either return feasible routes or it can return incorrect infeasibility declarations, on which feasibility can be recovered through a specifically-designed heuristic. The algorithm is furthermore supplemented by a battery management algorithm that can be used to determine charging decisions for electric and autonomous vehicle fleets. Timing solutions from the proposed scheduling algorithm are obtained on millions of routes extracted from DARP and e-ADARP benchmark instances. They are compared to those obtained from a linear program, as well as to popular scheduling procedures from the DARP literature. The proposed procedure mostly yields optimal solutions, with nearly-optimal solutions occurring in only 27 out of 21.5 million cases on DARP instances. Additionally, it demonstrates an average speed improvement of around 60% compared to a linear program and performs comparably to benchmark scheduling approaches from the DARP literature, while outperforming them in solution quality.

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Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

Linear programming relaxation

In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form The relaxation of the original integer program instead uses a collection of linear constraints The resulting relaxation is a linear program, hence the name.

Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.

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