A genetic programming hyper-heuristic approach for the multi-skill resource constrained project scheduling problem
Introduction
As one of the most important issues in decision-making of project management, resource-constrained project scheduling problem (RCPSP) is widely encountered in modern manufacturing systems and industrial processes, such as semiconductor wafer fabrication (Wang, Zhang & Wang, 2018), automobile assembly process (Bartels & Zimmermann, 2015), cloud workflows scheduling (Arabnejad, Bubendorfer & Ng, 2017), etc. In RCPSP, a set of tasks are linked by a task-on-node network and finish-to-start precedence relation, without preemption. The tasks are performed over time with limited resources and precedence constraints. The RCPSP aims to minimize the makespan of the project.
Since the RCPSP has been proved to be NP-hard (Blazewicz, Lenstra & Kan, 1983), the existing exact algorithms (Demeulemeester and Herroelen, 1992, Zhu, Bard and Yu, 2006) can only solve the small-scale problems within an acceptable time. For the large-scale problems, using heuristics to address the RCPSP aroused the attention, which can obtain near-optimal solution. Generally speaking, heuristics can be categorized into constructive and meta-heuristic methods. In the first category, heuristic algorithms based on priority rules are commonly used, which mainly employ the schedule generation scheme (SGS) as decoding strategy to address the RCPSP. There are two kinds of commonly used SGSs: the serial method and the parallel method ( Kolisch, 1996). By using the SGS, a great number of priority rules have been proposed, such as latest start time (LST) (Kolisch & Rainer, 1995), resource scheduling method (RSM) (Shaffer, Ritter & Meyer, 1965), minimum slack (MSLK) and latest finish time (LFT) (Davis & Patterson, 1975), etc. In recent years, meta-heuristic method has become a research hotspot because of its outstanding global search performance in solving complicated optimization problems. As a typical meta-heuristic method, various genetic algorithms (GAs) have been proposed for the RCPSP by hybridizing with other different evolutionary strategies (Alcaraz, Maroto and Ruiz, 2003, Hartmann, 2002, Valls, Ballestín and Quintanilla, 2008). After this, many other mete-heuristics have also been applied to address the RCPSP, including tabu search (TS) (Nonobe & Ibaraki, 2002), simulated annealing (SA) (Bouleimen & Lecocq, 2003), particle swarm optimization (PSO) (Jia & Seo, 2013), shuffled frog-leaping algorithm (SFLA) (Fang & Wang, 2012) and harmony search (HS) (Giran, Temur & Bekdaş, 2017), etc.
With the in-depth research on the RCPSP, researchers not only devote themselves to designing effective methods to obtain the optimal solution, but also combining classical RCPSP with practical application. Especially, in the process of scheduling the resources like manpower or multipurpose machine, each resource masters several different skills. Based on this characteristic, the multi-skill resource constrained project scheduling problem (MS-RCPSP) is proposed to make it be more close to the real-world situation.
Compared to the RCPSP, the study focused on the multi-skilled human resources is relatively scant. In earlier research, there is lack of unified model of the MS-RCPSP, researchers handle the problem with various objectives and datasets. In order to minimize the makespan of the project, Bellenguez-Morineau and Neron (2007) proposed a branch-and-bound method, while Almeida, Correia and Saldanha-da-Gama, 2016, Almeida, Correia and Saldanha-da-Gama, 2018 developed priority-based heuristics and a biased random-key genetic algorithm. To minimize the total staffing cost, Li and Womer (2009) presented a hybrid benders decomposition algorithm which combines mixed-integer linear programming and constraint programming. By considering the multi-skilled human workforce with heterogeneous and static efficiencies in the simultaneous scheduling of multiple projects, Heimerl and Kolisch (2010) proposed a mixed-integer linear programming. Besides, Kolisch and Heimerl (2012) developed a meta-heuristic consisting of a GA and a TS to address integrated scheduling and staffing IT projects with human resources having multiple skills. To minimize the maximal lateness of the project, Drezet and Billaut (2008) proposed a linear programming formulation and designed some greedy algorithms based on tabu search to generate an initial solution. To maximize a weighted average of economic gains and strategic gains, Gutjahr et al., 2008, Gutjahr et al., 2010 developed new model for project portfolio selection by considering competence development. In order to get close to the actual environment, Al-Anzi, Al-Zame and Allahverdi (2010) added the proficiency of a staff in certain skill in the model, and developed a lower bound using the linear programming scheme.
Although the problem with multi-skilled human resources is various, it is hard to find datasets that could be regarded as a benchmark. Recently, Myszkowski, Skowronski, Olech and Oslizlo (2015) generated a dataset derived from real-world production management and presented a hybrid ant colony optimization (HACO). In the generated dataset, each resource masters several skills with certain familiarity levels and each task requires a skill with certain type at a minimum level, which is a little different from the previous models that each task requires multiple skills. Based on the dataset, many other meta-heuristic methods are gradually proposed to address the problem, including teaching-learning-based optimization (TLBO) (Zheng, Wang & Zheng, 2017), knowledge-based fruit fly optimization algorithm (KBFOA) (Zheng, Wang & Zheng, 2015). However, the dataset used above is defined using calendar restrictions. To make it more general, the dataset is redefined in Myszkowski, Skowronski and Sikora (2015), and has been used as the benchmark to test the performanc of the greedy randomized adaptive search procedure (GRASP) (Myszkowski & Siemieński, 2016) and the hybrid differential evolution and greedy algorithm (DEGR) (Myszkowski, Olech, Laszczyk & Skowroński, 2017). In the previous works, researches mainly focused on minimizing the makespan or cost of the project (Zheng et al., 2017). However, it is difficult to find the exact solutions of the MS-RCPSP, since it has been proven to be an NP-hard problem in Myszkowski, Skowronski, Olech and Oslizlo (2015). In this paper, the MS-RCPSP with the objective of minimizing the makespan is investigated.
As an effective search methodology, hyper-heuristic automatically selects, combines or generates several simple low-level heuristics to handle complicated optimization problems (Burke et al., 2010). Over the past few years, hyper-heuristic attracts a growing number of attentions. Typically, Lin, Wang and Li, (2017) proposed a backtracking search hyper-heuristic to address the distributed assembly flow-shop scheduling problem. In Asta, Karapetyan, Kheiri, Ozcan and Parkes (2016), Asta et al. combined Monte–Carlo and hyper-heuristic method for the multi-mode resource-constrained multi-project scheduling problem. To handle combinatorial optimization problems, Sabar, Ayob, Kendall and Qu (2015) designed a novel hyper-heuristic which employs the gene expression programming as the high-level strategy. In addition, a hyper-heuristic algorithm based on harmony search was developed for examination timetabling problems (Anwar, Khader, Al-Betar & Awadallah, 2013). However, to the best of our knowledge, there is no hyper-heuristic based approach has been applied to address the MS-RCPSP.
In this paper, an effective genetic programming based hyper-heuristic (GP-HH) is designed for the MS-RCPSP. In GP-HH, genetic programming (GP) is employed as the high-level strategy to manage several designed low-level heuristics, rather than to improve the scheme by adjusting the schedule. As a classic algorithm, GP is proposed by Koza (1992) in the 1990s, and has demonstrated great performance compared with other methods (Dumic, Sisejkovic, Coric and Jakobovic, 2018, Durasevic, Jakobovic and Knezevic, 2016, Oliveira, Souza, Goues and Camilo-Junior, 2018, Santos et al., 2015). In particular, GP can evolve functions of arbitrary complexity since it adopts variable size LISP-tree (van Lon, Branke & Holvoet, 2017) to represent the solution. Based on this characteristic, the design of low-level heuristics will be more flexible. According to previous literature, GP-HH has been employed to handle a wide range of problems, including dynamic job shop scheduling (Park, Mei, Nguyen, Chen & Zhang, 2018), multidimensional knapsack problem (Drake, Hyde, Ibrahim & Ozcan, 2014), storage location assignment problem (Xie, Mei, Ernst & Song, 2014), etc. The influence of the parameter setting is also investigated in this paper. Moreover, experiments and comparisons are carried out to verify the effectiveness of the GP-HH.
The remainder of this paper is organized as follows: The description of the MS-RCPSP is presented in Section 2. The formulation of the GP is introduced in Section 3. The framework of the GP-HH is detailed in Section 4. The computational results of the benchmark and the comparison to some existing algorithms are discussed in Section 5. At last, Section 6 concludes this study.
Section snippets
Multi-skill resource constrained project scheduling problem
As a practical extension of the RCPSP, the skills domain is introduced into the MS-RCPSP, which can be generalized into two sub-problems: task scheduling problem and resource assigning problem. The problem can be depicted in Fig. 1. In this section, the description of the problem is presented at first. Then, the mathematical model of the MS-RCPSP is introduced and an example is given to further illustrate the problem.
Genetic programming
GP is an efficient algorithm which is developed on the basis of the genetic algorithm (GA). Similar with GA, GP contains following phases: initialization, selection, crossover, mutation. However, GP uses nonlinear encoding strategy by adopting variable size LISP-tree to represent the solution. The aim of GP is to find the tree with the best fitness value.
In GP, a feasible solution is represented by a binary tree which consists of function set and terminal set
Solution encoding and decoding schemes
Different from the commonly used encoding scheme in the literature (Zheng et al., 2017) which employs two lists including task list and resource list to represent solutions, each solution in GP-HH is encoded with a single task sequence vector, and each element of the solution denotes a specific task. For the project given in Table 2 and Fig. 2, a feasible solution π can be illustrated as in Fig. 6. To generate feasible schedules for MS-RCPSP efficiently, the idea of repair method as used in (
Computational results and comparison
To evaluate the performance of the proposed GP–HH for addressing the MS–RCPSP, numerical experiments are carried out on the new intelligent multi-objective project scheduling environment (iMOPSE) benchmark dataset which is generated by Myszkowski, Skowronski and Sikora (2015). The iMOPSE consists of 36 project instances with the size ranging from 100 tasks to 200 tasks. The GP-HH algorithm is implemented in Visual C++ 6.0 and operated on a core i5-7360 U processor with 2.3 GHz and 8 GB RAM. For
Conclusion
For multi-skill resource constrained project scheduling problem (MS–RCPSP), this paper proposes a genetic programming hyper-heuristic algorithm to minimizing the makespan. To the best of our knowledge, this is the first research work to apply the hyper-heuristic approach to address the MS–RCPSP. The main contributions of this paper are summarized as follows: (1) ten simple heuristics rules are employed to construct a set of low-level heuristics; (2) the genetic programming is utilized as the
Conflict of interest statement
The authors declared that they have no conflicts of interest to this work.
CRediT authorship contribution statement
Jian Lin: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Supervision, Resources, Investigation. Lei Zhu: Methodology, Formal analysis, Conceptualization, Writing - original draft, Writing - review & editing, Data curation. Kaizhou Gao: Supervision, Writing - review & editing, Software, Validation, Investigation, Resources.
Acknowledgement
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. The authors also sincerely thank Professor Paweł B. Myszkowski for providing the dataset and the detailed results. This work is part of a project supported by the National Natural Science Foundation of China (Grant Nos. 61973267, 61503331, 71671160 and 61503330), the Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY19F030007 and LY19G010004) and the Zhejiang Key Laboratory
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