Hybrid evolutionary algorithm for large-scale project scheduling problems

Highlights

• Proposed a new evolutionary framework for a wide range of project scheduling problems.

• Introduced two heuristics to rectify any infeasible schedule.

• Proposed a new classification technique to determine the hardness of a problem.

• For complex problems, a local search is developed to fine-tune the best solution.

• The proposed algorithm is superior to existing algorithms.

Abstract

The Multi-Mode Resource Constrained Project Scheduling Problem (MMRCPSP) is a challenging NP-hard optimization problem, that schedules activities under a set of resource constraints. Although, over the last few decades, different solution approaches have been proposed, no single algorithm has consistently been the best for a wide range of MMRCPSPs. In this paper, we have proposed an effective hybrid algorithm, in which two multi-operator evolutionary algorithms perform sequentially under two sub-populations, with their sizes dynamically adapted based on their performance during the evolutionary process. In addition, two heuristics are proposed, the first one is based on a linear programming approach with an aim to obtain feasible modes, while the second one is based on a modified forward and backward justification approach with an aim of obtaining feasible schedules. Also, a classification technique is used to determine the complexity of a given problem, based on its resource’s availability. The proposed approach is tested by solving a wide-range of multi-mode resource-constrained project scheduling problems, including available larger test problems, with the results revealing that the proposed method outperforms well-known algorithms.

Introduction

A resource-constrained project scheduling problem (RCPSP) is a challenging optimization problem, which aims to determine a schedule of activities with the best possible allocation of its resources, while satisfying dependency relationships among the activities. This problem is also known as a single mode RCPSP. On the other hand, in a multi-mode RCPSP (MMRCPSP), an activity can be scheduled under different modes, by scheduling the activities and their modes, while minimizing the overall project’s make-span subject to precedence and capacity constraints of renewable and non-renewable resources (Zaman et al., 2018a, Cheng et al., 2015). Depending on the use of non-renewable resources in the early stage of a project, it may be dependent on renewable resources only at a later stage. That means, the allocation of resources (type, mode and their amounts) will vary with time (or as the project progresses).

Although MMRCPSP is more practical than its single-mode counterpart, it is more complex and challenging. According to Chakrabortty, Abbasi, and Ryan (2019), determining a feasible solution for an MMRCPSP with two types of non-renewable resources is NP-complete, and solving such a problem using a computationally efficient approach is very challenging (Shukla, Choudhary, Prakash, Fernandes, & Tiwari, 2013). Nevertheless, the majority of real and hypothetical MMRCPSPs have up to nine non-renewable resources (Peteghem & Vanhoucke, 2014).

Over the last decade, a number of solution approaches have appeared for MMRCPSPs, which can be broadly classified as (i) exact, (ii) heuristic, and (iii) meta-heuristic approaches. For exact methods, linear programming, tree-based branch and bound, and branch and cut, are the most popular approaches (Zaman, Elsayed, Ray, & Sarker, 2018b). Although exact algorithms may give an optimal solution quickly, they are inefficient to solve large-scale problems. Previous research indicated that such approaches are unable to solve MMRCPSP with 20 non-dummy activities and three modes per activity (Chakrabortty et al., 2019).

Many solution approaches have been developed that are based on different variants of heuristic and meta-heuristics for large-scale MMRCPSPs, in which the variable neighbourhood search heuristic (MVNSH) (Chakrabortty et al., 2019), path re-linking (PR) approach (Muritiba, Rodrigues, & da Costa, 2018), Genetic Algorithm (GA), (Peteghem & Vanhoucke, 2010), Differential Evolution (DE)(Damak, Jarboui, Siarry, & Loukil, 2009), and Particle Swarm Optimization (PSO) (Li & Zhang, 2013) have been used due to their flexibility and efficiency. However, no algorithm has consistently performing well for a wide-range of MMRCPSPs (Zaman, Elsayed, Ray, & Sarker, 2016). To improve performance, some hybrid algorithms have been developed, in which a meta-heuristic algorithm is used for global search while a conventional algorithm is used for local search. Of them, Tao and Dong (2018) developed an approach for integrating an alternative project structure into traditional MMRCPSPs, in which a hybrid meta-heuristic based on NSGA-II and a tabu search is used to solve problems. In addition, most of those algorithms used an iterative technique to obtain feasible modes from infeasible ones. However, as obtaining feasible modes is itself NP-hard (Chakrabortty et al., 2019), the iteration process is computationally expensive.

In this research, we propose an evolutionary framework based on two moEAs, two heuristics, and a classification technique, to solve a wide range of large-scale MMRCPSPs. Two moEAs namely (i) a multi-operator GA (moGA) and (ii) a multi-operator DE (moDE), sequentially perform using two sub-populations. Each sub-population consists of two types of individuals, namely as schedules of (i) activities ($\overrightarrow{x}$) and (ii) modes $\left(\overrightarrow{y}\right)$, and the numbers of individuals of each sub-population are dynamically adapted based on the corresponding moEA’s performance. A linear-programming and modified dual justification generation scheme heuristics are used to rectify any infeasible $\overrightarrow{y}$ and $\overrightarrow{x}$, respectively. The classification technique determines the hardness of an MMRCPSP in terms of obtaining feasible solutions. For the difficult problems, nonrenewable resource constraints are relaxed at the initial stage of the search process, while in the later stages, the algorithm fine-tunes the solutions of both $\overrightarrow{x}$ and $\overrightarrow{y}$ by applying neighbourhood search and a re-linking approach, respectively.

The performance of the proposed evolutionary framework is evaluated by solving well-known benchmark sets of MMRCPSPs from both PSPLIB and MMLIB (Peteghem & Vanhoucke, 2014), with the results showing that the proposed algorithm has merit for solving a wide-range of MMRCPSPs, in terms of obtaining high-quality solutions, especially for large-scale MMRCPSPs.

The contributions of this paper are: (i) developing a new multi-method based evolutionary framework for both small and large-scale MMRCPSPs; (ii) introducing a new heuristic that converts an infeasible schedule into a feasible one; (iii) proposing a new classification technique to determine the hardness of a problem, and subsequently, that information is used while solving the problem; (iv) providing in-depth analysis of the algorithm’s components on a wide range of standard benchmarks of MMRCPSPs.

The rest of this paper is organized as follows: Section 2 shows mathematical formulations of MMRCPSP; Section 3 presents a comprehensive literature review; Section 4 describes the proposed evolutionary framework; Section 5 presents the computational results; Section 6 shows comprehensive parametric and statistical tests; and Section 7 presents the conclusion and recommendations for future works.