Minimizing the number of vehicles in periodic scheduling: The non-Euclidean case

Abstract

In this paper we consider the problem of minimizing the number of vehicles needed to meet a fixed periodically repeating set of tasks where set-up times between tasks do not satisfy the triangle inequality. We reduce this problem to finding the minimal length cycle-cover in a graph. In a special case, where the set-up times satisfy the triangle inequality, we reduce the scheduling problem to the assignment problem.

Introduction

In this paper we consider the problem of minimizing the number of vehicles needed to meet a fixed periodically repeating set of tasks. For example, if we interpret tasks as flights and vehicles as airplanes, the resulting problem is to minimize the number of airplanes needed to fly a fixed daily repeating schedule of flights. We will focus on the general problem where distances and set-up times between the flights do not necessarily satisfy the triangle inequality. The simplest example arises when some direct flights are impracticable. Another example is when the set-up times include the deadhead time and other delays, in which case they do not necessarily satisfy the triangle inequality.

We term this problem the periodic vehicle scheduling problem with non-Euclidean distance metrics. We reduce the scheduling problem to finding the minimal length cycle-cover in a finite graph. In a special case – where the set-up times do satisfy the triangle inequality, we reduce the problem to the assignment problem.

Various versions of the airplane scheduling problem, with Euclidean distance metrics, have been intensively studied in the literature. Dantzig and Fulkerson [1]solved the problem of minimizing the number of vehicles to meet a fixed finite-horizon schedule. Ford and Fulkerson [2]showed how to reduce that problem to finding a minimum cardinality chain-cover in an acyclic graph. Dantzig [3], as described by Simpson [4]and Orlin [5]considered the problem of minimizing the number of airplanes to meet a fixed periodic schedule under the added restriction that the final flight schedule consists of daily-repeating flights. Orlin [5]solved the problem without the restriction that the final flight schedule repeats daily. (The interested reader is referred to [5]for more discussions and references on periodic airplane scheduling.)

All the mentioned scheduling problems were converted to various network flow problems. However, none of the above authors studied the non-Euclidean case of the problem in spite of its evident practical applications.

Another periodical vehicle scheduling problem has been studied by Karzanov and Livshits [6]. These authors have investigated the problem of minimizing the number of robots needed to meet a periodic flowshop schedule under the added restriction that each task is to be carried out by one robot only. They have efficiently solved the problem by elegantly reducing it to the assignment problem.

A direct comparison of the Karzanov–Livshits and Orlin scheduling models reveals that they are two specific examples of periodic vehicle scheduling each one treating different carriers (robots vs. airplanes) and stipulating distinct physical conditions. The two studies differ with respect to the modelling technique and mathematical analysis, and they are not reducible to each other. The common result emerging from these past studies, is that both periodic scheduling problems can be efficiently solved using infinite periodic graphs extending the Ford–Fulkerson finite graph model [2].

In this paper we extend further the latter approach and suggest an infinite periodic graph model which permits to treat the vehicle scheduling problem with non-Euclidean distances. The main theoretical result of the paper is that, using the graph approach, we solve, in polynomial time, this new class of vehicle scheduling problems. Notice that a straightforward attempt to use the known Orlin technique [5]for solving the vehicle scheduling problems with non-Euclidean distances fails since the task instances in the Orlin model must constitute a periodic partially ordered set; however, the latter assumption is immediately violated as far as the distances do not satisfy the triangle inequality (see [5]for the terminology and illustrative examples).

The present paper improves, also, some known results for periodic scheduling problems. First, we can relax the Karzanov–Livshits restriction [6]that each task is to be carried out by one vehicle only. Then the resulting problem, though becoming somewhat more complicated, remains to be polynomially solvable whereas the relaxation can lead to a lesser number of needed vehicles (see Example 3 in Section 4).

Further, our graph model permits to improve Orlin's computational results: in the case of the Euclidean distances, we reduce Orlin's airplane scheduling problem to the assignment problem rather than to the minimal length cycle-cover of a finite graph.

In Section 2we give a description of the periodic vehicle scheduling problem. In Section 3we present a graph construction permitting to treat the non-Euclidean distances. In 4 Reduction of the chain-cover problem to a cycle-cover problem, 5 Reduction of the cycle-cover problem to an assignment problemwe convert the scheduling problem to a network flow problem, and discuss its relationship with the assignment problem.