New exact and heuristic algorithms to solve the prize-collecting job sequencing problem with one common and multiple secondary resources

Highlights

• A scheduling problem with a main resource and preassigned secondary resources.

• A graph reformulation of the problem and labeling algorithms.

• A branch-cut-and-price algorithm to solve exactly the problem.

• An iterated local search to solve heuristically the problem.

• Significant improvement of the literature results.

Abstract

We study the prize-collecting job sequencing problem with one common and multiple secondary resources. In this problem, a set of jobs is given, each with a profit, multiple time windows for its execution, and a duration during which it requires the main resource. Each job also requires a preassigned secondary resource before, during, and after its use of the main resource. The goal is to select and schedule the subset of jobs that maximize the total profit. We present a new mixed integer linear programming formulation of the problem and a branch-cut-and-price algorithm as an exact solution method. We also introduce a heuristic algorithm to tackle larger instances. Extensive numerical experiments show that our exact algorithm can solve to optimality literature instances with up to 500 jobs for a particular dataset and up to 250 jobs for another dataset with different characteristics. Our heuristic builds high-quality solutions in a small computational time. It computes new best-known solutions for most of the larger instances.

Introduction

This work focuses on solving the prize-collecting job sequencing problem with one common and multiple secondary resources (PC-JSOCMSR). This problem was introduced by Horn, Raidl, & Rönnberg (2018). It has application in particle therapy scheduling (Maschler & Raidl, 2020) and in pre-runtime scheduling of avionic systems. We refer to the work Horn, Maschler, Raidl, & Rönnberg (2021a) for details about these real-world applications (see §3.3 of the previously cited work).

The PC-JSOCMSR is defined as follows. We are given a set $\mathcal{J}=⟦1,n⟧$ of non-preemptive jobs, meaning that the execution of a job cannot be interrupted and resumed later (hereafter, $⟦a_1,a_2⟧=[a_1,a_2]\cap \mathbb{Z}$ for every $a_1,a_2\in \mathbb{Z}$ such that $a_1\le a_2$). Each job $j\in \mathcal{J}$ should be entirely scheduled (from its start to its end) within one of the time windows of set ${\mathcal{W}}_j=\left\{T_{jk}\right|k\in ⟦1,w_j⟧\}$ with $T_{jk}=[t_{jk}^1,t_{jk}^2]$. Time windows are indexed in chronological order of the $t_{jk}^1$ values. Denoting $t^1=\underset{j\in \mathcal{J}}{min}\left\{t_{j1}^1\right\}$ and $t^2=\underset{j\in \mathcal{J}}{max}\left\{t_{j{w}_j}^2\right\}$, the planning horizon is defined as the interval $[t^1,t^2]$.

We are also given a set of (renewable) disjunctive resources ${\mathcal{R}}_0=\left\{0\right\}\cup \mathcal{R}$. We refer to resource 0 as the main resource and to $\mathcal{R}=⟦1,m⟧$ as the set of secondary resources. Each job $j\in \mathcal{J}$ requires a single secondary resource $r_j\in \mathcal{R}$ (preassigned to $j$) for $p_j$ consecutive time periods. If the execution of job $j$ starts at time period $t$, then this job requires the main resource (in addition to the resource $r_j$) from time period $t+p_j^{-}$ for $p_j^0$ consecutive time periods (with $p_j^{-}+p_j^0\le p_j$). We denote ${\widehat{p}}_r^{-}={\max }_{j\in \mathcal{J}:r_j=r}\left\{p_j^{-}\right\}$ and ${\widehat{p}}_r^{+}={\max }_{j\in \mathcal{J}:r_j=r}\left\{p_j^{+}\right\}$ where $p_j^{+}=p_j-(p_j^{-}+p_j^0)$. We have $p_j^{-}\ge 0$ and $p_j^{+}\ge 0$ for every job $j$. The order of processing starting times on every secondary resource is a suborder of the order of processing starting times on the main resource.

Scheduling a job $j\in \mathcal{J}$ generates a profit $b_j\ge 0$, whereas not scheduling it generates no profit. The problem is to schedule the jobs while maximizing the total profit. A feasible solution to the problem is a schedule that satisfies the following constraints:

• elementarity: each job is scheduled at most once

• time window: if a job is scheduled, it is scheduled uninterruptedly and entirely (from its start to its end) within one of its time windows

• resource usage: at any time, every resource (main or secondary) is used by at most one job.

Figure 1 displays a feasible solution to an instance with 4 jobs $J_1$, $J_2$, $J_3$, and $J_4$. On the right, the vertical axis lists the resources and the horizontal axis represents time. Each rectangle represents the use of a particular resource by a job during a time interval. In the displayed solution, job $J_2$ is not processed. The starting time of jobs $J_1$, $J_3$, and $J_4$ (i.e., the time at which the preassigned secondary resource starts being used) is equal to 0, 8, and 4 respectively. As $p_{J_4}^{-}=2$, job $J_4$ starts using the main resource at time $6(=4+2)$ for $p_{J_4}^0=3$ time periods. The same is true for the other jobs. The solution value is equal to $b_{J_1}+b_{J_3}+b_{J_4}=8$.

We now outline the main original contributions of this paper:

• We introduce dominance rules that define properties satisfied by at least one optimal schedule.

• We present a pseudo-polynomial time-indexed mixed integer linear programming (MILP) model for the PC-JSOCMSR. Although indexing variables on time may limit a priori its interest, its numerical performance on existing benchmark instances is largely better than an existing compact order-based MILP model introduced by Horn, Raidl, & Rönnberg (2021b), and a significant number of instances are solved to proven optimality in less than two hours.

• We formulate the PC-JSOCMSR as a resource constrained shortest path problem (RCSPP) and describe labeling algorithms to solve it. We discuss different modeling approaches. We present how we use the dominance rules.

• We present a branch-cut-and-price (BCP) algorithm to solve an extended MILP model we introduce for the PC-JSOCMSR. The elementarity constraints are handled in the master problem, which allows to efficiently solve the pricing problem as a RCSPP. Our BCP includes many ingredients found in state-of-the-art BCP algorithms: limited arc memory rank-1 cuts, automatic dual price smoothing stabilization, a procedure for enumerating elementary paths, and a multiphase strong branching procedure.

• We present a heuristic algorithm to initialize the BCP algorithm and to compute high-quality solutions for large-sized instances that remain beyond the reach of an exact algorithm.

Extensive computational experiments on benchmark instances from the literature show that the new exact approaches we present to tackle the PC-JSOCMSR solve to proven optimality instances with up to 250 jobs and up to 500 jobs for some specific instances. They also show that our heuristic algorithm computes high-quality solutions within short running time and outperforms existing heuristics.

The paper is organized as follows. Section 2 presents a brief overview of related literature. Section 3 presents two dominance rules. Section 4 presents two MILP models for the PC-JSOCMSR, one of which is new. Section 5 presents how we model the problem as a RCSPP. Section 6 describes the BCP algorithm. Section 7 presents the heuristic algorithm. Section 8 shows the results of our computational experiments. Section 9 contains final remarks.