Energetic reasoning and mixed-integer linear programming for scheduling with a continuous resource and linear efficiency functions

Abstract

International audienceThis paper addresses a scheduling problem with a continuously-divisible, cumulative and renewable resource with limited capacity. During its processing, each task consumes a part of this resource, which lies between a minimum and a maximum requirement. A task is finished when a certain amount of energy is received by it within its time window. This energy is received via the resource and an amount of resource is converted into an amount of energy with a non-decreasing and continuous function. The goal is to find a feasible schedule, which is already NP-complete, and then to minimize the resource consumption. For the case where all functions are linear, we present two new Mixed Integer Linear Programs (MILP), as well as improvements of an existing formulation. We also present a detailed version of the adaptation of the well-known "left-shift/right-shift" satisfiability test for the cumulative constraint and the associated time-window adjustments to our problem. For this test, three ways of computing relevant intervals are described. Finally, a hybrid branch-and-bound using both the satisfiability test and the MILP is presented with a new heuristic for choosing the variable on which the branching is done. Computational experiments on randomly generated instances are reported in order to compare all of these solution methods

Cumulative scheduling with variable task profiles and concave piecewise linear processing rate functions

“…In particular, we compute w(i, t 1 , t 2 ) and β(i, t 1 , t 2 ) for task 3 We now describe how this quantity is used to perform time-bound adjustments. These adjustments are similar to the ones for linear functions described in [12], so we just briefly present them. We start by defining some notation.…”

“…3 gives the shape of the processing rate function of a fuel cell [7]. An extension of the work of [2] to linear processing rate functions as well as several solution methods was proposed in [12,13].…”

“…In [12], the authors use linear functions to approximate real-world ones. Despite the fact that there exist many real-world processing functions, which are concave [8,11], approximating a function by a concave piecewise linear one is always at least as good as an approximation by a linear function (see Example 2 below).…”

“…Despite the fact that there exist many real-world processing functions, which are concave [8,11], approximating a function by a concave piecewise linear one is always at least as good as an approximation by a linear function (see Example 2 below). Furthermore, the authors of [12] prove the NP-hardness of their problem using a reduction from the cumulative problem. This proof is still valid in our case and then the CECSP with concave functions is NP-hard.…”

Editorial “Project Management and Scheduling”

“…Nattaf et al (2016) address a scheduling problem with a continuously divisible, cumulative and renewable resource with limited capacity. To minimize the resource consumption the authors propose three mixed-integer mathematical programs and an adapted satisfiability test for the cumulative constraint.…”