Meta-heuristics for sustainable supply chain management: a review

ABSTRACT

Due to the complexity and the magnitude of optimisation models that appeared in sustainable supply chain management (SSCM), the use of meta-heuristic algorithms as competent solution approaches is being increased in recent years. Although a massive number of publications exist around SSCM, no extant paper explicitly investigates the role of meta-heuristics in the sustainable (forward) supply chain. To fill this gap, a literature review is provided on meta-heuristic algorithms applied in SSCM by analyzing 160 rigorously selected papers published by the end of 2020. Our statistical analysis ascertains a considerable growth in the number of papers in recent years and reveals the contribution of 50 journals in forming the extant literature. The results also show that in the current literature the use of hybrid meta-heuristics is overtaking pure meta-heuristics, the genetic algorithm (GA) and the non-dominated sorting GA (NSGA-II) are the most-used single- and multi-objective algorithms, the aspects of sustainability are mostly addressed in connection with product distribution and routing of vehicles as pivotal operations in supply chain management, and last but not least, the economic-environmental category of sustainability has been further noticed by the scholars. Finally, a detailed discussion of findings and recommendations for future research are provided.

KEYWORDS:

• Literature review
• meta-heuristics
• logistics
• supply chain management
• sustainability

1. Introduction

Contrary to traditional supply chain management (SCM), which particularly focuses on profitability and financial aspects of a business organisation, sustainable SCM (SSCM) pays attention to environmental and social aspects as well as economic concerns, referred to as the triple-bottom-line (TBL) pillars of sustainability. In this regard, logistics and supply chain (SC) managers and policy-makers are actively engaged to incorporate sustainability issues into the forward SC (Seuring and Müller 2008) or designing closed-loop SC (CLSC) including reverse logistics, recycling, and remanufacturing (Govindan, Soleimani, and Kannan 2015b). To achieve the objectives of SC sustainability, various modeling approaches and solution techniques have been developed in the literature. Seuring (2013) described the SSCM modeling approaches in four categories comprising life-cycle assessment (LCA) based studies, equilibrium models, multi-criteria decision making (MCDM) models, and analytical hierarchy process (AHP) models. In a broader study, Brandenburg et al. (2014) focused on the qualitative models for SSCM including mathematical programming, simulation, heuristic, analytical, and hybrid models.

Over the last few decades, meta-heuristics as a branch of heuristic methods have continuously been receiving much attention due to their strength and efficiency to tackle computationally complex problems and large-sized real-world cases in forward SSCM and reverse logistics. Although a large number of literature reviews and survey papers have been published to address the various aspects of SSCM (Martins and Pato 2019), only a handful number of review papers focus on meta-heuristics for SCM and reverse logistics, and no extant paper explicitly and inclusively reviews the application of meta-heuristic algorithms in forward SSCM. Table 1 summarises review papers discussing meta-heuristics for logistics and SCM.

Table 1. Previous review papers on meta-heuristics in logistics and SCM.

Reference	Scope	N of reviewed papers	Time frame
Turken et al. (2020)	Nature-inspired SC solutions	Not given	Not given-2020
Rachih, Mhada, and Chiheb (2019)	Meta-heuristics in reverse logistics	120	2004–2017
Soni et al. (2019)	Swarm intelligence in SCM	56	1999–2017
Zhang et al. (2015b)	Swarm intelligence in green logistics	115	1995–2014
Griffis, Bell, and Closs (2012)	Meta-heuristics in logistics & SCM	128	1991–2012

In this context, Rachih, Mhada, and Chiheb (2019) conducted a systematic literature review on meta-heuristics for reverse logistics through surveying 120 papers obtained by exploring several electronic databases such as Google Scholar, Scopus, Springer Publishing, and ScienceDirect using the pair of keywords ‘Reverse Logistics’ and ‘Meta-heuristics’ in the title, abstract or keywords of the articles. Taking into account 56 papers adopted from a systematic literature search based on keywords ‘Supply Chain Management’, ‘SCM’, ‘Swarm’, ‘Swarm Intelligence’, ‘Ant Colony Optimisation’, ‘Particle Swarm Optimisation’, ‘Artificial Bee Colony’, ‘Bacterial Foraging’, ‘Cat Swarm Optimisation’, ‘Artificial Immune System’, and ‘Glowworm Swarm Optimisation’ in four platforms, namely, ScienceDirect, Taylor and Francis, Emerald, and Wiley-Interscience, Soni et al. (2019) reviewed the application of swarm intelligence approaches in SCM. In a moderately similar study, Zhang et al. (2015b) merely focused on the use of swarm intelligence in forward and closed-loop green logistics through systematic selection and review of 115 papers acquired by searching the keywords ‘Logistics’ or ‘Supply Chain’ together with ‘Green’ or ‘Environmental’ or ‘Sustainable’ or ‘Closed-loop’ or ‘Reverse’ in IEEE Xplore, ScienceDirect, Scopus, and SpringerLink. Griffis, Bell, and Closs (2012) took into consideration 128 papers applying meta-heuristics in logistics and SCM published in eight management-oriented journals to inform the readers about the types of techniques available to cope with SCM decision problems and to clarify the correlations between these techniques and the problems. Recently, Turken et al. (2020) reviewed nature-inspired supply chain solutions. Although the paper refers to nature-inspired algorithms, the main focus is on the analogies between biological systems and SCs. Overall, looking at these review papers reveals that the previous review studies are either focused on general logistics and SCM without considering sustainability issues (Soni et al. 2019; Griffis, Bell, and Closs 2012), a specific sustainability dimension as environmental in Zhang et al. (2015b), a particular SC structure as reverse logistics in Rachih, Mhada, and Chiheb (2019), or specific types of meta-heuristic algorithms (Soni et al. 2019; Zhang et al. 2015b). Accordingly, a lack of comprehensive literature review on the application of meta-heuristics for sustainable logistics and SCM is utterly appreciated. The visible growth in the number of publications in this research area as well as a variety of applied meta-heuristics in SSCM causes a critical need for a literature review on the application of meta-heuristics for sustainable logistics and SCM to analyze the extant literature, to summarise its current developments, and to provide original research perspectives for scholars and practitioners. To bridge this research gap, we conduct a literature review to mainly answering the following research questions (RQ):

RQ1: What are the types of meta-heuristics applied to solve SSCM optimization problems?

RQ2: What are the most important SSCM problems tackled by meta-heuristics?

RQ3: What is the distribution of sustainability pillars across SSCM problems tackled by meta-heuristics?

The remainder of this paper is organised as follows. In Section 2, the research methodology is described. Section 3 provides a detailed explanation of the material collection and refinement procedure to make the literature database. Section 4 statistically analyzes the papers to bring initial insights into the current status of the literature. Sections 5 and Section 6 present the selected categories to classify the literature and elaborate on the results of the content analysis, respectively. Discussion on the main findings and promising future directions are provided in Section 7. Finally, Section 8 concludes the review.

2. Research methodology

As an integral part of any academic research, the literature review (LR) assists to build the research on the existing knowledge in the field (Webster and Watson 2002). The LR presents a comprehensive overview of the extant literature in a specific area by collecting and synthesising the previous relevant papers (Tranfield, Denyer, and Smart 2003) and aims at providing additional value through reporting the findings, original research gaps, and promising future directions (Wee and Banister 2016). Tremendous speed of knowledge production in business areas that is driven in a fragmented interdisciplinary manner claims an essential need for a systematic methodology to create a rigor, thorough and successful LR (Snyder 2019).

This paper reviews the literature of meta-heuristic-based solution approaches used in SSCM to provide fact-based responses to the aforementioned research questions. Similar to the previous reviews on SSCM (Seuring and Müller 2008; Brandenburg et al. 2014; Rachih, Mhada, and Chiheb 2019), the research methodology applied in this paper includes the following four main steps:

• Material collection. Relevant materials are excerpted from the initially collected papers through an iterative refinement procedure to craft the literature database.

• Descriptive analysis. Initial statistics on the formal aspects of the material are illustrated to clarify the orientation of the literature and to present the contributing journals.

• Category selection. Structural dimensions to classify the reviewed literature are driven deductively or inductively.

• Material evaluation. Through a structured content analysis, the collected material is analyzed and evaluated according to the selected categories.

In the following sections, each step is detailed and related outputs are presented.

3. Material collection

Collecting a comprehensive and reliable list of papers to conduct an authentic LR requires to explicitly define the scope of the literature, the structured search strategy, and the inclusion and exclusion protocols.

3.1. Literature delimitation

Clear specification of search boundaries is a crucial issue in writing an LR. In this line, the following criteria are taken into consideration in this paper:

• This paper reviews the literature of meta-heuristic-based solution approaches used in SSCM.

• Papers focusing on closed-loop SCM and reverse logistics are not included in this review. This enables us to further bind our efforts to those papers focusing on the sustainable forward supply chain. It is worth mentioning that a literature review paper has recently been published on meta-heuristic approaches in reverse logistics (Rachih, Mhada, and Chiheb 2019).

• Papers applying other solution methodologies than meta-heuristics in SSCM (e.g. exact and basic heuristic algorithms) are out of the scope of this review.

• English management-oriented papers published in peer-reviewed scientific journals and indexed in Scopus (www.scopus.com) by the end of 2020 are potential candidates for this review study. Hence, book chapters, conference proceedings, technical reports, and editorials are excluded.

• Following Brandenburg et al. (2014) and Martins and Pato (2019), papers merely addressing the economic dimension of sustainability are excluded. Accordingly, inclusive papers involve either environmental, social, economic-environmental, socio-economic, socio-environmental, or triple-bottom-line pillars of sustainability.

• Following Eskandarpour et al. (2015), papers falling into the field of SSCM due to their inherent characteristics (e.g. renewable energy SCs or electric vehicle-equipped logistic systems) are included only if the environmental or the social impacts have been considered within their optimisation models either in the objective function or in the constraints.

3.2. Search strategy

To craft a comprehensive database of the relevant papers, we firstly conduct a structured keyword search in Scopus academic search engine. Subsequently, the initially collected materials are refined according to the previously mentioned inclusion and exclusion rules through abstract and full-text reading. Ultimately, the remaining list of papers is extended through additional search and backward snowballing and the final literature database is firmed up.

To form the literature database around meta-heuristic algorithms in SSCM, we conducted a structured keyword search by seeking a combination of selected keywords in the title, abstract, and authors’ keywords of the papers. The keyword combination search is inescapable to capture the three main aspects of the subject matter that are meta-heuristic, SC, and sustainability. To refer to the meta-heuristic aspect, we use the variants of the term ‘Meta-heuristics’ as well as the terms to indicate the common meta-heuristics reported in the previous review papers on logistics and SCM (Rachih, Mhada, and Chiheb 2019) as ‘Genetic algorithm’, ‘Swarm’, ‘Ant colony’, ‘Variable neighbourhood search’, ‘Simulated annealing’, ‘Tabu search’, and other popular meta-heuristics in optimisation as ‘Scatter search’, ‘Harmony search’, ‘GRASP’, ‘Evolutionary’, ‘NSGA*’, and ‘MOPSO’. To allude to the SC aspect, we use the terms ‘Supply chain’ and ‘Logistics’ inspired by Seuring and Müller (2008). And finally, to refer to the sustainability aspect, we use the terms ‘Sustainab*’, ‘Environment*’, ‘Green’, ‘Carbon’, ‘Emission’, ‘Social*’, ‘Ethic’, ‘Ecologic*’, and ‘Triple-bottom-line’ mostly adopted form (Seuring and Müller 2008). Note that the Asterisk symbol ‘*’ used at the end of some keywords ensures to find all possible terms that start with the same letters. To make the keyword combination, ‘AND’ operator is used between inter-group keywords and, ‘OR’ operator is applied between intra-group keywords.

Scopus search engine possesses the capability for considering a variety of keyword combinations at the same time. This feature reduces the number of searches to once and prevents to make duplications. Launching the search for the combination of keywords in Scopus resulted in 1,145 English journal papers, which built our initial literature database. Afterward, titles, abstracts, and keywords of the found papers were rigorously reviewed, which led to the exclusion of 876 papers as they were either concerned with closed-loop SCM, reverse logistics, or were not related to SSCM. To produce a top-quality review and according to other previously published LRs in the domain (Seuring and Müller 2008; Hassini, Surti, and Searcy 2012; Brandenburg et al. 2014; Agi, Faramarzi-Oghani, and Hazır 2021), we bound our focus on the journals published by the major publishers in the field that are Elsevier, Taylor and Francis, Informs, Springer, Wiley, and Emerald which left us 174 papers. Through full-text examination of the remaining papers in content analysis procedure, 22 papers are also excluded as they have not explicitly tackled SSCM aspects in their modelling and optimisation procedure. After analyzing the resulting papers of our search, we found the use of two relevant keywords ‘Matheuristic’ and ‘Pollution’ in the keyword list of some papers that have been missed in our structured keyword search. Accordingly, we searched for these two keywords separately in combination with other pre-selected keywords that led us to find eight new papers including three papers applying matheuristics as a type of hybrid meta-heuristics (Talbi 2016) in SSCM, and five papers using meta-heuristics for pollution routing problem (PRP) (Bektaş and Laporte 2011) as a green variant of vehicle routing problems (VRPs) in logistics management. Finally, our literature database was firmed up with 160 papers.

4. Descriptive analysis

In this section, we present the results of our initial statistical analysis to explicitly reveal the current status of the literature and the contributing journals.

4.1. Distribution of reviewed papers per year

This review paper contains 160 primary studies. Figure 1 presents the distribution of the reviewed papers by the year of publication. As can be seen, the number of papers published before 2014 is scant, which is consistent with the findings of Brandenburg et al. (2014) as they reported only a single study on the use of meta-heuristics in SSCM. Since 2014, the number of papers begins to increase and follows a considerable growth in recent years such that almost 50% of all papers have been published in 2019 and 2020. This increasing trend of publication shows that meta-heuristics are receiving more attention in SSCM among scholars and practitioners.

Figure 1. Publishing trend in the area of meta-heuristic-based solution approaches in forward SSCM.

Line chart illustrating the publishing trend since 2007. The number of papers published before 2014 is scant. Since then, the number of papers begins to increase and follows a considerable growth in recent years such that almost 50% of all papers have been published in 2019 and 2020.

4.2. Distribution of reviewed papers by journals

A total of 50 peer-reviewed journals from six different publishers have contributed to the publication of 160 reviewed papers, among which 21 journals published 131 papers altogether, each with minimum of two papers, and the remaining 29 journals published only one paper. Among these journals, the ‘Journal of Cleaner Production’ places at the top of the list with 27 publications, following by the ‘Computers and Industrial Engineering’ with 14 papers, and the ‘International Journal of Production Research’ with 11 papers. Figure 2 shows the distribution of publications by journals. A full list of journals contributed to the extant literature is listed in Appendix 1.

Figure 2. Distribution of publications by journals.

Bar chart showing the number of publications by each contributing journal in the extant literature. Among these journals, the “Journal of Cleaner Production”, the “Computers and Industrial Engineering” and the “International Journal of Production Research” place at the top of the list with 27, 14 and 11 contributions, respectively.

5. Category selection

To efficiently answer the three indicated research questions of this study, we anatomise the extant literature under the following three streams: meta-heuristics, SSC problems, and sustainability. Under the first stream, we aim at identifying the meta-heuristic algorithms applied by the collected materials. This allows answering the first research question (RQ1). Inductively inspired from the literature of meta-heuristics, these algorithms can be categorised in different ways, such as nature-inspired vs. non-nature inspired, single solution-based vs. population-based, dynamic vs. static objective function, one vs. various neighbourhood structures, memory usage vs. memory-less methods (Blum and Roli 2003). However, deductively concluded from our content analysis, we differentiate and discuss the applied meta-heuristics in sustainable SC under the following two classes: pure meta-heuristics, and hybrid meta-heuristics. Pure meta-heuristics refer to the original meta-heuristic algorithms without any hybridisation or evident improvement with other algorithms, while hybrid meta-heuristics refer to the algorithms that combine either two meta-heuristic algorithms or one meta-heuristic with mathematical programming techniques known as matheuristics. Under the second stream, we aim at distinguishing the problems tackled by the literature that aids to uncover the second research question (RQ2). Generally, SCM includes many problems that are discussed in the literature under three decision-making phases: strategic, tactical, and operational (Chopra and Meindl 2018). However, deductively drawn from our content analysis, we classify and detail the collected materials under vehicle routing, location, network design, transportation, inventory management, production planning and scheduling, and partner selection as the main decision problems in forward SSCM, for which the meta-heuristic approaches have been applied. Finally, under the third stream, we identify and discuss the aspects of sustainability spotted by the reviewed papers that serve to respond to the third research question (RQ3). Figure 3 shows the categories and sub-categories under which the extant literature is detailed in the following section.

Figure 3. Selected categories to classify and analyze the literature of interest.

Diagram categorizing the extant literature into three classes of metaheuristics, sustainable supply chain problems and aspects of sustainability.

Figure 4. Distribution of the top-ten meta-heuristics used in the reviewed literature.

Column chart illustrating the frequency of the use of top-ten metaheuristics including GA, NSGA-II, PSO, SA, TS, MOEA, MOPSO, VNS, LNSA, and EA/DE in both pure and hybrid format.

6. Material evaluation through content analysis

As a result of the content analysis, we summarise our major findings from the reviewed literature in the table provided in Appendix 2 which represents for each reference, the main optimisation problem(s), the applied meta-heuristic algorithm(s), and the aspect(s) of sustainability considered. In the following, the results of the content analysis are elaborated under the three selected categories.

6.1. Meta-heuristics

Out of 160 reviewed papers, 31 papers applied more than one meta-heuristic algorithm to solve optimisation problems. The total number of meta-heuristics used across all reviewed papers is 211 including 107 pure meta-heuristics and 104 hybrid meta-heuristics. This statistic shows that hybrid meta-heuristic algorithms are practically receiving much attention as pure meta-heuristics in sustainable forward SCM. According to this fact, we decide to classify the meta-heuristics into pure and hybrid algorithms. Such classification enables us to capture and further discuss the structure of hybrid algorithms and their applications, rather than only focusing on pure algorithms. Also, we find that 72 papers develop multi-objective models requiring multi-objective solution approaches to deal with. To capture and further analyze the type of multi-objective meta-heuristics applied, we extend the classification to make a difference between single- and multi-objective meta-heuristics.

6.1.1. Pure meta-heuristics applied in forward SSCM

Pure meta-heuristics refer to the original meta-heuristic algorithms (e.g. GA, PSO, SA, and multi-objective GA (MOGA)) without any hybridisation or evident improvement with other algorithms. Reviewing the content of the papers manifests that pure meta-heuristics have been used 107 times in the literature of interest including 57 single-objective and 50 multi-objective algorithms.

Among 57 pure single-objective meta-heuristics applied, GA, PSO, SA, TS, and ACO possess the most contributions with the frequency of 23, 6, 6, 4, and 4, respectively. Also, three algorithms namely, memetic algorithm (MA) (Barzinpour and Taki 2018; Hwang et al. 2016), large neighbourhood search (LNS) (Pelletier, Jabali, and Laporte 2019; Franceschetti et al. 2017), and iterated local search (ILS) (Corberán et al. 2018; Kramer et al. 2015a) are applied twice. Other algorithms that are used only once are artificial bee colony (ABC) (Chu et al. 2019), variable neighbourhood search (VNS) (Corberán et al. 2018), harmony search (HS) (Eskandari-Khanghahi et al. 2018), differential evolution (DE) (Wang, Zhang, and Zhu 2017b), artificial immune systems (AIS) (Barzinpour and Taki 2018), red deer algorithm (RDA) (Fathollahi-Fard, Hajiaghaei-Keshteli, and Tavakkoli-Moghaddam 2020; Abdi et al. 2019), and cross-entropy (Fahimnia, Davarzani, and Eshragh 2018).

Among 50 pure multi-objective meta-heuristics employed, non-dominated sorting GA (NSGA-II), multi-objective PSO (MOPSO), and multi-objective evolutionary algorithm (MOEA) possess the most contributions with the frequency of 17, 6, and 5, respectively. Also, multi-objective GA, multi-objective SA, MOEA based on decomposition (MOEA/D), and strength Pareto evolutionary algorithm version 2 (SPEA-II), are applied three times each. Other algorithms that are used only once are multi-objective ACO (Ganji et al. 2020), multi-objective grey wolf optimiser (Heidari, Imani, and Khalilzadeh 2020), multi-objective hyper-heuristic based on decomposition (MOHH/D) (Leng et al. 2020b), MOGA version 2 (MOGA-II) (Validi, Bhattacharya, and Byrne 2014a), non-dominated ranking GA (NRGA) (Abad et al. 2018), multi-objective imperialist competitive algorithm (MOICA) (Eydi and Fathi 2020), multi-objective ABC (Zhang et al. 2016), multi-objective memetic algorithm (Guo et al. 2018), multi-directional local search based on LNS (Eskandarpour, Dejax, and Péton 2021), multi-objective TS based on decomposition (MOTS/D) (Wang et al. 2019a).

6.1.2. Hybrid meta-heuristics applied in forward SSCM

To exploit the complementary character of different optimisation methods, hybrid meta-heuristics have been widely used in the literature of forward SSCM such that 104 hybrid algorithms including 75 single-objective and 29 multi-objective hybrid meta-heuristics are identified in the reviewed papers. In terms of optimisation techniques used in the proposed hybrid algorithms, two types of hybridisation are distinguished. The most-seen hybrid algorithms are designed through the combination of two meta-heuristics, or one meta-heuristic and one heuristic; however, the application of matheuristics as a hybridisation of meta-heuristics with mathematical programming techniques is also evident in some papers. To be more specific and comprehensible, we discuss the hybrid single-objective and multi-objective algorithms separately by emphasising the type of algorithms applied in hybridisation and end the discussion with an introduction to matheuristics and the way they have been crafted in the reviewed literature.

Figure 5 provides an overview of the algorithms used in the hybrid single-objective meta-heuristics developed in the literature. To elude a complicated network, hybridisation between common and more used meta-heuristics is shown in Figure 5-a, and Figure 5-b demonstrates the combination between common meta-heuristics and other algorithms. In these figures, the arc bridging two nodes represents the hybridisation between two algorithms and the number on the arc denotes the frequency of such hybridisation applied in the literature. As can be seen, GA, PSO, VNS, TS, and SA are the algorithms used mainly in the design of hybrid single-objective meta-heuristics, each with 22, 16, 12, 9, and 8 contributions, respectively. To build hybrid single-objective meta-heuristics, GA has been combined with PSO (Biuki, Kazemi, and Alinezhad 2020; Abdi et al. 2019), ACO (Tan et al. 2019), TS (Li et al. 2019b), AIS (Kumar et al. 2016a), imperialist competitive algorithm (ICA) (Nia, Far, and Niaki 2015), variable neighbourhood descent (VND) (Erdem and Koç 2019), and other algorithms such as Keshteli algorithm (Abdi et al. 2019), hill-climbing algorithm (Qin et al. 2019), and so forth (Wang et al. 2019b; Noh and Kim 2019; Zhang et al. 2018a; Zhang et al. 2018b; Dai et al. 2018; Cheng et al. 2016; Yang et al. 2015). For the same objective, the combination of PSO with differential evolution (DE) (Maiyar and Thakkar 2019b; Maiyar and Thakkar 2020), VNS (Govindan, Jafarian, and Nourbakhsh 2019; Zhen et al., 2020), TS (Li, Lim, and Tseng 2019c), GA (Abdi et al. 2019), and other algorithms such as spanning tree-based algorithm (Hong et al. 2018), and so forth (Zhang et al. 2019b; Chu et al. 2019; Rau, Budiman, and Widyadana 2018; De et al. 2016) is evident in the literature. In the same line, combinations between VNS and ACO (Jabir, Panicker, and Sridharan 2020; Jabir, Panicker, and Sridharan 2017), VNS and greedy randomised adaptive search procedure (GRASP) (Ajam, Akbari, and Salman 2019), VNS and SA (Chargui et al. 2020), VNS and ABC (Govindan, Jafarian, and Nourbakhsh 2019), VNS and electromagnetism mechanism algorithm (EMA) (Govindan, Jafarian, and Nourbakhsh 2019), SA and TS (Chargui et al. 2020; Suzuki 2016), SA and GRASP (Chargui et al. 2020), SA and LNS (Koç et al. 2016), and between ACO and ribonucleic acid computing (Zhang et al. 2019a) are interesting and worthy to be noted.

Figure 5. Combination of algorithms used in the literature to craft hybrid single-objective meta-heuristics. Figure (5-a). Hybridisation between common and more used meta-heuristics. Figure (5-b). Combination between common meta-heuristics and other algorithms.

Two graphs presenting how different algorithms are combined to develop hybrid singleobjective metaheuristics. Nodes show the algorithms and edges show the hybridization between them.

Figure 6. Combination of algorithms used in the literature to build hybrid multi-objective metaheuristics.

Graph presenting how different algorithms are combined to develop hybrid multi-objective metaheuristics. Nodes show the algorithms and edges show the hybridization between them.

Figure 6 schematically summarises how different meta-heuristics have been combined in the literature to form the hybrid multi-objective meta-heuristics. As this figure shows, NSGA-II and MOPSO are the two most-used algorithms in the design of hybrid multi-objective meta-heuristics, each with 10 and 7 contributions, respectively. The hybridisation of NSGA-II with other algorithms is seen in the works of Brahami et al. (2020), Wu et al. (2020), Xu et al. (2019), Doolun et al. (2018), Wang et al. (2018), Hassanzadeh and Rasti-Barzoki (2017), Kadziński et al. (2017), and Memari et al. (2016). Also, the capability of MOPSO has been enhanced through the hybridisation with other algorithms in the studies of Chan et al. (2020), Yong et al. (2019), Maiyar and Thakkar (2019a), Canales-Bustos, Santibañez-González, and Candia-Véjar (2017), Kumar et al. (2016b), Validi, Bhattacharya, and Byrne (2014b), and Govindan et al. (2014). Furthermore, combinations between multi-objective EMA and multi-objective VNS (Govindan, Jafarian, and Nourbakhsh 2015a), MOEA and VNS (Zahiri, Zhuang, and Mohammadi 2017), MOEA and local search (Ghannadpour and Zarrabi 2019), neural enhancement for multi-objective (NEMO) and SPEA-II (Kadziński et al. 2017), and between LNS and multi-objective heuristic algorithm (Anderluh et al. 2021) are worthwhile to be noticed.

Hybridisation of meta-heuristics with mathematical programming approaches (e.g. enumerative algorithms, relaxation, and decomposition methods), also known as matheuristics, has been frequently applied in the literature including 9 single-objective and 2 multi-objective matheuristic algorithms. Figure 7 illustrates the meta-heuristics used in combination with mathematical programming approaches in the literature of sustainable forward SCM. In this context, combinations between GA and dynamic programming (Xiao and Konak 2017; Su, Chu, and Wang 2012), GA and quadratic programming (Validi, Bhattacharya, and Byrne 2014a), VNS and integer programming (Wang, Shao, and Zhou 2017a), VND and integer programming (Kramer et al. 2015b), iterated local search and mixed-integer programming (Xiao and Konak 2016), large neighbourhood search and mixed-integer programming (Macrina et al. 2019), GRASP and Benders decomposition (Alkaabneh, Diabat, and Gao 2020), SA and exact algorithm (Wang et al. 2020a), and between simple evolutionary algorithm for multi-objective optimisation version 2 (SEAMO2) and Lagrangian relaxation (Harris, Mumford, and Naim 2014), and MOGA-II and design of experiment (DoE) (Validi, Bhattacharya, and Byrne 2020) are seen.

Figure 7. Matheuristic algorithms used in the reviewed literature.

Graph presenting the link between metaheuristics and mathematical programming techniques to construct matheuristic algorithms.

6.2. SSC problems solved by meta-heuristics

SCM involves many problems that are generally discussed under strategic, tactical, and operational decision-making phases. In the context of forward SSCM, meta-heuristic algorithms have been widely applied to solve a variety of optimisation problems. Following the results of our content analysis, vehicle routing, location/allocation, network design, transportation, inventory management, production planning, and partner selection are respectively the most-studied problems that lie on the interface of forward SSCM and meta-heuristics, and accordingly merit further discussion to clarify the problems and corresponding used algorithms. Alternatively, some problems addressed by a limited number of the reviewed papers do not fall into the seven previously mentioned problem families and can be identified in Appendix 2. The type and number of algorithms used for each category of problems are presented in Table 3.

Table 2. Classification of meta-heuristics for forward SSCM and related number of algorithms used in each class.

	Single-Objective	Multi-Objective	Total
Pure	57	50	107
Hybrid	75	29	104
Total	132	79	211

Table 3. Type and number of algorithms used for each category of problems.

		Problem
Meta-heuristic		VRP	LAP	SCND	TP	IM	PPP	PSP	Other
GA	P	4	7	4	7	2	3	2	4
	H	15	5	1	2	5	5	1	4
NSGA-II	P	9	3	5	1	1	2	–	–
	H	5	2	2	1	–	1	1	–
PSO	P	1	–	2	1	–	1	2	1
	H	11	4	2	2	3	1	–	2
SA	P	3	–	2	1	2	2	–	–
	H	5	2	–	–	1	–	–	3
VNS	P	1	–	–	–	–	–	–	–
	H	11	2	3	–	1	–	–	2
TS	P	3	1	–	–	–	–	–	–
	H	8	1	–	–	1	1	–	1
MOPSO	P	2	1	2	–	1	1	–	1
	H	5	2	2	–	–	1	–	–
ACO	P	4	–	–	–	–	–	–	–
	H	4	–	–	–	–	–	–	–
LNS	P	2	–	–	–	–	–	–	–
	H	5	1	–	–	–	–	–	–
MOEA	P	4	2	–	–	2	–	–	–
	H	1	–	1	–	–	–	–	–
Matheuristic		8	3	–	–	2	–	–	1
Other	P	ABC(1)	AIS(1)	HS(1)	AIS(1)	Cross-Entropy(1)	Cross-Entropy(1)	MOGA(1)	MOEA/D(1)
		ILS(2)	MA(1)	MOABC(1)	Cross-Entropy(1)		MOGA(1)	MOICA(1)
		MOACO(1)	MOEA/D(1)	MOEA/D(1)	MA(1)		MOACO(1)
		MOEA/D(1)	MOHH/D(1)	MOGA(1)	MOMA(1)		MOTS/D(1)
		MOGA-II(1)	MOSA(1)	MOGWO(1)			RDA(1)
		MOHH/D(1)	SPEA-II(1)	MOLNS(1)
		MOSA(2)		MOSA(1)
		MOTS/D(1)		SPEA-II(1)
		NRGA(1)
		RDA(1)
		SPEA-II(2)
	H	ABC(3)	DE(3)	ABC(1)	FA(1)	GRASP(1)	Keshteli(1)	AIS(1)	GRASP(2)
		DE(1)	GrEA(1)	DE(2)	GRASP(1)	HS(1)			EA(1)
		EMA(1)	HS(1)	EMA(1)		ICA(1)
		GrEA(1)	IBEA(1)	MOEMA(1)
		GRASP(1)	MOGA-II(1)	MOVNS(1)
		IBEA(1)	MOVNS(1)	NEMO(2)
		ILS(1)	NSGA-III(1)	SPEA-II(1)
		Keshteli(1)	NSLS(1)	SSA(1)
		MOGA-II(1)	SEAMO2(1)
		MOSEA(1)	SPEA-II(1)
		MOVNS(1)
		NSLS(1)
		NSGA-III(1)
		SPEA-II(1)
		TA(1)
		VND(2)

6.2.1. Vehicle routing problem

The vehicle routing problem (VRP) is one of the broadly studied combinatorial optimisation problems (Ritzinger, Puchinger, and Hartl 2016) that aims at finding optimal paths for a fleet of vehicles (e.g. conventional, electric, drone, ship) for pickup, delivery, and distribution of goods among different entities of an SC under certain constraints (Laporte 1992). In the context of SSC and logistics management, besides the classic cost functions, fuel, energy, emission, and pollution considerations are also noticed as the main objectives of the VRP. The pollution-routing problem (PRP) defined as a VRP with environmental consideration (Bektaş and Laporte 2011), and electric VRP (EVRP) are the common sustainable versions of the VRP. This problem is the most studied optimisation problem in the context of forward SSCM, for which meta-heuristics have been applied.

Out of 160 reviewed papers, 90 papers deal with the VRP either individually as a unique optimisation problem for instance in the works of Giallanza and Puma (2020), Poonthalir, Nadarajan, and Kumar (2020), Xu et al. (2019), Qin et al. (2019), Erdem and Koç (2019), or jointly with other problems in the form of location-routing problem (LRP) (Validi, Bhattacharya, and Byrne 2020; Li et al. 2019b; Quintero-Araujo et al. 2019; Mahmoudsoltani, Shahbandarzadeh, and Moghdani 2018; Simoni et al. 2018), inventory-routing problem (IRP) (Alkaabneh, Diabat, and Gao 2020; Alinaghian and Zamani 2019; Liu and Lin 2019; Cheng et al. 2016), location-inventory-routing problem (LIRP) (Biuki, Kazemi, and Alinezhad 2020; Pourhejazy, Kwon, and Lim 2019; Asadi et al. 2018), production-inventory-routing problem (PIRP) (Chan et al. 2020), SCND and routing (Govindan, Jafarian, and Nourbakhsh 2019; Maiyar and Thakkar 2019a), and production planning and routing (Abdi et al. 2019; Kumar et al. 2016b). To solve routing-related problems, 33 unique meta-heuristics have been used. In this regard, GA is the most-used algorithm that has been applied 19 times including 4 pure and 15 hybrid forms. NSGA-II, PSO, and VNS are placed in the next rankings with the frequency of 14, 12, and 12, respectively. Also, the statistics show the use of 8 matheuristic algorithms. The full list of meta-heuristics used to tackle routing-related problems is shown in Table 3.

6.2.2. Location and allocation

The location-allocation problem (LAP) is one of the key strategic decisions in the SC design. This problem is to find the best location for the facilities and to assign the demand points to the located facilities in a way to satisfy customer needs and to optimise cost or environmental objective functions under specific constraints. In the reviewed literature, the LAP has been studied in 30 papers either separately as a unique optimisation problem as in the works of Sarker, Wu, and Paudel (2019), Teran-Somohano and Smith (2019), and Doolun et al. (2018), or in integration with other problems in the form of LRP (Leng et al. 2020a; Leng et al. 2020b; Validi, Bhattacharya, and Byrne 2020), LARP (Fathollahi-Fard et al. 2019), LIRP (Biuki, Kazemi, and Alinezhad 2020; Karakostas, Sifaleras, and Georgiadis 2020; Pourhejazy, Kwon, and Lim 2019) and so forth. To solve location-related problems, 24 unique meta-heuristics have been used. In this vein, GA is the most-used algorithm that has been applied 12 times including 7 pure and 5 hybrid forms. NSGA-II, PSO, and MOPSO are placed in the next rankings with the frequency of 4, 3 and 3, respectively. Also, the statistics show the use of 3 matheuristic algorithms. The full list of meta-heuristics used to tackle location-related problems is shown in Table 3.

6.2.3. Supply chain network design

Many decisions are made to design an SC network, among which the location of facilities in different tiers is perhaps the most critical one (Farahani et al. 2014). The SCND problem includes many decisions including but not limited to the number, location, and capacity of facilities in different SC stages, supplier selection, technology selection, transportation mode selection, and flow management through the SC facilities (Zhalechian et al. 2016). This problem has been addressed in 25 papers of the reviewed literature for instance in the works of Mogale, Kumar, and Tiwari (2020), Robles, Azzaro-Pantel, and Aguilar-Lasserre (2020), Chalmardi and Camacho-Vallejo (2019), Eskandarpour, Dejax, and Péton (2021), and Govindan, Jafarian, and Nourbakhsh (2019). To solve SCND problems, 22 unique meta-heuristics have been used. In this regard, NSGA-II is the most-used algorithm that has been applied 7 times including 5 pure and 2 hybrid forms. GA, PSO, and MOPSO are placed in the next rankings with the frequency of 5, 4 and 4, respectively. The full list of meta-heuristics used to tackle SCND problem is shown in Table 3.

6.2.4. Transportation problem

The transportation problem aims to find the optimal quantity of products distributed from supply sources to demand points across the supply chain network with the classic goal of minimising the total transportation costs (Winston and Goldberg 2004) and optimising socio-environmental impacts in sustainable logistics and SCM. In the reviewed literature, 15 papers have concerned with the transportation problem either individually in 4 papers (Guo et al. 2018; Chandrasekaran and Ranganathan 2017; Mirkouei et al. 2017; Memari et al. 2016) or jointly with other problems, such as LAP (Sadeghi and Haapala 2019; Maiyar and Thakkar 2019b; Barzinpour and Taki 2018), order consolidation (Salhi et al. 2020), production planning (Wang et al. 2020b; Wang et al. 2020c; Borumand and Beheshtinia 2018), production planning and inventory control (Fahimnia, Davarzani, and Eshragh 2018), production planning and supplier selection (Che 2010) and supplier selection and pricing (Huang et al. 2016) in 11 papers. To solve transportation-related problems, 10 unique meta-heuristics have been used. In this regard, GA is the most-used algorithm that has been applied 9 times including 7 pure and 2 hybrid forms. PSO and NSGA-II are placed in the next rankings with the frequency of 3 and 2, respectively. The full list of meta-heuristics used to tackle transportation-related problems is shown in Table 3.

6.2.5. Inventory management

Many companies hold inventory for different reasons, such as to decouple the various parts of the production process, separate the firm from fluctuations in demand, or hedge against price inflation. The goal of inventory management is to strike a balance between inventory investment and customer service (Chopra and Meindl 2018; Farahani et al. 2015). To achieve this objective, two important questions regarding the quantity and the time of replenishment orders need to be answered expertly and effectively. Out of 160 reviewed papers, 16 papers dealt with inventory decisions either individually (Ganesh Kumar and Uthayakumar 2019; Nia, Far, and Niaki 2015) or in conjunction with other problems, such as vehicle routing problem (Alinaghian and Zamani 2019; Liu and Lin 2019; Rau, Budiman, and Widyadana 2018), vehicle routing and location problem (Biuki, Kazemi, and Alinezhad 2020; Karakostas, Sifaleras, and Georgiadis 2020; Pourhejazy, Kwon, and Lim 2019; Asadi et al. 2018), location problem (Wang et al. 2020a; Dai et al. 2018), and transportation and production planning (Fahimnia, Davarzani, and Eshragh 2018). To solve inventory-related problems 12 unique meta-heuristics have been used. In this regard, GA is the most-used algorithm that has been applied 7 times including 2 pure and 5 hybrid forms. PSO, SA, and MOEA are placed in the next rankings with the frequency of 3, 3, and 2, respectively. Also, the statistics show the use of two matheuristic algorithms. The full list of meta-heuristics used to tackle inventory-related problems is shown in Table 3.

6.2.6. Production planning

Production planning is a central problem for manufacturing companies. Essentially, this problem includes three levels of planning as aggregate planning, material requirements planning (MRP), and detailed work scheduling. This problem attempts to strike a balance between the production capacity and the customer demand in medium-term planning and to determine the quantity and timing of production on the short-term horizon. In the context of forward sustainable logistics and SCM, we notice 14 papers used meta-heuristic algorithms to tackle production planning-related problems. In the reviewed literature, only one paper (De, Das, and Maiti 2018) faces the production planning problem individually, and the remaining papers deal with this problem jointly with the vehicle routing problem (Ganji et al. 2020; Wang et al. 2019a; Abdi et al. 2019; Hassanzadeh and Rasti-Barzoki 2017; Kumar et al. 2016b), vehicle routing and inventory problem (Chan et al. 2020), pricing (Zhang et al. 2020; Hafezalkotob and Zamani 2019), transportation problem (Wang et al. 2020b; Wang et al. 2020c; Borumand and Beheshtinia 2018), transportation and partner selection problem (Che 2010), and with transportation and inventory management (Fahimnia, Davarzani, and Eshragh 2018). To solve production planning-related problems 12 unique meta-heuristics have been used. In this regard, GA is the most-used algorithm that has been applied 8 times including 3 pure and 5 hybrid forms. NSGA-II, PSO, SA, and MOPSO are placed in the next ranking each with three frequencies. The full list of meta-heuristics used to tackle production planning-related problems is shown in Table 3.

6.2.7. Partner selection problem

Companies require to select their service providers, such as suppliers, carriers, and so forth to design supply networks. A partner selection problem aims at finding the best possible partner(s) among the existing alternatives based on some criteria. This problem is observed in 8 papers of the reviewed literature. Wu et al. (2020) applied hybrid NSGA-II for the construction of partner selection criteria in SSC. Eydi and Fathi (2020) developed a multi-objective ICA to solve a supplier and carrier selection problem. Also, the use of the GA, hybrid GA-AIS, PSO, and MOGA is respectively regarded in the works of Fallahpour et al. (2016), Kumar et al. (2016a), Wu and Barnes (2016), and Yeh and Chuang (2011) to tackle the supplier and partner selection problem. Integrated consideration of the partner selection problem with the transportation and pricing problem, and the transportation and production planning problem are noticed in the works of Huang et al. (2016) and Che (2010), respectively, where the former applies GA and the latter uses PSO to solve the problem.

6.3. Aspects of sustainability

Economic, environmental, and social aspects of sustainability are observed in 153, 154, and 37 papers of the reviewed literature, respectively. Economic issues are considered by maximising the profit of SC design and operations for instance in the works of Alkaabneh, Diabat, and Gao (2020), Zhang et al. (2020), Hafezalkotob and Zamani (2019), Barzinpour and Taki (2018), and De, Das, and Maiti (2018), or by minimising or restricting SC costs that typically include but not limited to location, technology selection, transportation, distribution and inventory costs (e.g. Ganji et al. 2020; Mogale, Kumar, and Tiwari 2020; Fathollahi-Fard et al. 2019; Chu et al. 2019; Chalmardi and Camacho-Vallejo 2019). Greenhouse gas and carbon emission as well as fuel and energy consumption are the most noticeable environmental issues considered by the papers dealing with environmental sustainability. In this context, 104 papers concern with minimising or controlling carbon (e.g. Chan et al. 2020; Validi, Bhattacharya, and Byrne 2020; Qin et al. 2019; Yong et al. 2019; Doolun et al. 2018), greenhouse gas (e.g. Heidari, Imani, and Khalilzadeh 2020; Robles, Azzaro-Pantel, and Aguilar-Lasserre 2020; Bravo, Rojas, and Parada 2019; Huang et al. 2016) or dust emission (Kadziński et al. 2017), 32 papers deal with fuel and/or energy consumptions (e.g. Shi et al. 2020; Xu et al. 2019; Ji, Luo, and Peng 2019), and 26 papers take other environmental considerations such as global warming (Miranda-Ackerman, Azzaro-Pantel, and Aguilar-Lasserre 2017), green policies (Wang et al. 2020c), environmental risks (Mahmoudsoltani, Shahbandarzadeh, and Moghdani 2018), and green partner selection (Wu et al. 2020; Wu and Barnes 2016). Customer satisfaction (Ganji et al. 2020; Leng et al. 2020a; Leng et al. 2020b; Tirkolaee et al. 2020; Xu et al. 2019; Ghannadpour and Zarrabi 2019; Bravo, Rojas, and Parada 2019), social benefit maximisation (Shen 2020; Chu et al. 2019; Eskandari-Khanghahi et al. 2018) or social cost minimisation (Zhang et al. 2019b; Teran-Somohano and Smith 2019; Maiyar and Thakkar 2019b), food quality improvement (Chan et al. 2020), reduction of social disturbances such as noise (Anderluh et al. 2021), hazardous materials (Govindan, Jafarian, and Nourbakhsh 2019), risk of accident (Reyes-Rubiano et al. 2020) and safety risk index (Robles, Azzaro-Pantel, and Aguilar-Lasserre 2020), job opportunity creation (Biuki, Kazemi, and Alinezhad 2020; Heidari, Imani, and Khalilzadeh 2020; Govindan, Jafarian, and Nourbakhsh 2019; Tautenhain, Barbosa-Povoa, and Nascimento 2019; Zahiri, Zhuang, and Mohammadi 2017), and balanced workload assignment (Govindan, Jafarian, and Nourbakhsh 2019) are among the most significant social issues considered by the reviewed papers. Table 4 provides a full list of indicators used in the reviewed literature to represent various aspects of sustainability.

Table 4. Full list of sustainability indicators considered in the reviewed literature.

Aspects of sustainability	Indicators	Number of papers
Economic	Cost minimisation	139
	Profit maximisation	13
Environmental	Emission reduction	104
	Energy/Fuel consumption reduction	32
	Environmental considerations	26
Social	Customer satisfaction	20
	Social welfare/cost optimisation	11
	Social disturbance reduction	6
	Job opportunity creation	5
	Working conditions	1

Figure 8 demonstrates the distribution of papers over sustainability categories. As can be seen, out of 160 reviewed papers, 119 papers focus on economic-environmental aspect; simultaneous consideration of economic and social issues are observed in 4 papers (Fathollahi-Fard et al. 2020; Chu et al. 2019; Teran-Somohano and Smith 2019; Moons et al. 2019); pure environmental and social considerations are regarded in 4 papers (Ng, Lam, and Samuel 2019; Ehmke, Campbell, and Thomas 2016; Xiao and Konak 2016; Küçükoğlu et al. 2015) and 2 papers (Ajam, Akbari, and Salman 2019; Cao et al. 2018), respectively; only one paper (Xu et al. 2019) lies into the socio-environmental category; and 30 papers spot the triple-bottom-line pillars of sustainability. This distribution of papers across various categories of sustainability reveals that the economic-environmental category as well as the triple-bottom-line pillars have been further noticed by scholars. More than half of those papers falling into the latter category deal with the muti-objective supply chain network design problem where cost minimisation reflects the financial aspect, emission and fuel consumption minimisation addresses the environmental dimension and finally, the social aspect is covered through consideration of job creation, food quality, customer satisfaction and social welfare. The evolution of these two categories of sustainability in the reviewed literature is shown in Figure 9 and the type and number of meta-heuristics used for the seven major identified problems considering these two categories are illustrated in Table 5. In addition, the consideration of socio-economic factors is seen in the vehicle routing and location problems. For the former problem, ABC, hybrid PSO, hybrid MOSEO and hybrid TA are used, and for the latter, the use of MOEA is observed. The consideration of pure environmental factors is only seen in VRP for which ACO, TS, hybrid SA and hybrid ILS-MIP algorithms are used. The hybrid GRASP-VNS and GA are used to deal with road clearing and relief distribution problem with pure social considerations, respectively. Finally, the consideration of socio-environmental factors is only seen in VRP for which hybrid NSGA-II is applied.

Figure 8. Distribution of papers over sustainability categories (based on Carter and Rogers (2008)).

Venn diagram showing the number of papers in each category of sustainability.

Figure 9. Evolution of two most-noticed categories of sustainability in the reviewed literature.

Line chart showing the ascending trend of publications since 2013 considering economicenvironmental and economic-environmental-social categories of sustainability.

Table 5. Type and number of meta-heuristics used for each category of problems and aspects of sustainability.

	Aspects of sustainability
Problem	Economic-Environmental	Economic-Environmental-Social
VRP	ABC (P:0, H:2); ACO (P:3, H:4); DE (P:0, H:1); GA (P:4, H:11); GRASP (P:0, H:1); ILS (P:2, H:0); Keshteli (P:0, H:1); LNS (P:2, H:4); MOEA (P:2, H:0); MOEA/D (P:1, H:0); MOGA-II (P:1, H:1); MOPSO (P:2, H:4); MOSA (P:1, H:0); MOTS/D (P:1, H:0); MOVNS (P:0, H:1); NSGA-II (P:8, H:2); NRGA (P:1, H:0); PSO (P:0, H:6); RDA (P:1, H:0); SA (P:3, H:5); SPEA-II (P:2, H:0); TS (P:2, H:8); VND (P:0, H:2); VNS (P:1, H:6)	ABC (P:0, H:1); EMA (P:0, H:1); GA (P:0, H:4); GrEA (P:0, H:1); IBEA (P:0, H:1); LNS (P:0, H:1); MOACO (P:1, H:0); MOEA (P:2, H:1); MOHH/D (P:1, H:0); MOPSO (P:1, H:2); MOSA (P:1, H:0); NSGA-II (P:2, H:2); NSGA-III (P:0, H:1); NSLS (P:0, H:1); PSO (P:0, H:3); SPEA-II (P:0, H:1); VNS (P:0, H:4)
LAP	AIS (P:1, H:0); DE (P:0, H:1); GA (P:5, H:3); HS (P:0, H:1); LNS (P:0, H:1); MA (P:1, H:0); MOEA (P:1, H:0); MOEA/D (P:1, H:0); MOGA-II (P:0, H:1); MOPSO (P:1, H:2); MOSA (P:1, H:0); MOVNS (P:0, H:1); NSGA-II (P:3, H:0); PSO (P:1, H:2); SA (P:0, H:2); SEAMO2 (P:0, H:1); SPEA-II (P:1, H:0); TS (P:0, H:1); VNS (P:0, H:3)	DE (P:0, H:2); GA (P:2, H:2); GrEA (P:0, H:1); IBEA (P:0, H:1); MA (P:1, H:0); MOHH/D (P:1, H:0); NSGA-II (P:0, H:2); NSGA-III (P:0, H:1); NSLS (P:0, H:1); PSO (P:0, H:3); SPEA-II (P:0, H:1); TS (P:1, H:0)
SCND	DE (P:0, H:2); GA (P:3, H:0); MOEA/D (P:1, H:0); MOEMA (P:0, H:1); MOGA (P:1, H:0); MOLNS (P:1, H:0); MOPSO (P:2, H:2); MOSA (P:1, H:0); MOVNS (P:0, H:1); NEMO (P:0, H:2); NSGA-II (P:3, H:2); SA (P:1, H:0); SPEA-II (P:1, H:1); SSA (P:0, H:1)	ABC (P:0, H:1); EMA (P:0, H:1); GA (P:1, H:1); HS (P:1, H:0); MOABC (P:1, H:0); MOEA (P:0, H:1); MOGWO (p:1, H:0); NSGA-II (P:2, H:0); PSO (P:1, H:1); SA (P:1, H:0); VNS (P:0, H:3)
IM	Cross-Entropy (P:1, H:0); GA (P:2, H:3); GRASP (P:0, H:1); HS (P:0, H:1); ICA (P:0, H:1); MOEA (P:2, H:0); MOPSO (P:1, H:0); NSGA-II (P:1, H:0); PSO (P:0, H:1); SA (P:2, H:1); TS (P:0, H:1); VNS (P:0, H:1)	GA (P:0, H:2); MOPSO (P:0, H:1); PSO (P:0, H:2)
PPP	Cross-Entropy (P:1, H:0); GA (P:3, H:5); Keshteli (P:0, H:1); MOGA (P:1, H:0); MOPSO (P:0, H:1); MOTS/D (P:1, H:0); NSGA-II (P:1, H:0); PSO (P:1, H:1); RDA (P:1, H:0); SA (P:2, H:0); TS (P:0, H:1)	MOACO (P:1, H:0); MOPSO (P:1, H:1); NSGA-II (P:1, H:0)
TP	AIS (P:1, H:0); Cross-Entropy (P:1, H:0); DE (P:0, H:1); FA (P:0, H:1); GA (P:7, H:2); MA (P:1, H:0); MOMA (P:1, H:0); NSGA-II (P:1, H:1); PSO (P:1, H:1); SA (P:1, H:0)	DE (P:0, H:1); PSO (P:0, H:1)
PSP	AIS (P:0, H:1); GA (P:2, H:1); MOICA (P:1, H:0); MOGA (P:1, H:0); PSO (P:2, H:0)	PSO (P:0, H:1); NSGA-II (P:0, H:1)

7. Discussion and future perspectives

Due to the complexity and the magnitude of the optimisation models arisen in SSCM, the use of meta-heuristic algorithms as competent solution approaches is being increased in recent years. The ability to provide reasonably good solutions for computationally complex and large-scale problems in an acceptable time as well as the ease of the algorithm design comparing to the problem-specific heuristics have made the meta-heuristic algorithms popular among scholars.

Figure 10 facilitates understanding the trend of the leading meta-heuristics applied in the literature of forward SSCM over the last five years. Figure 10-a is dedicated to the five leading single-objective meta-heuristics while Figure 10-b focuses on the two noticeable multi-objective meta-heuristics. As shown by Figure 10-a, the GA is the dominant single-objective meta-heuristic over the five recent years in the literature thanks to its intrinsic attributes, such as the ability to solve a variety of optimisation problems and flexibility to be applied individually or in a hybrid design with other algorithms. A fair growth is also seen in the use of PSO, TS, VNS, and SA in the last two years that seals on the efficiency of these algorithms to tackle different optimisation problems in SSCM. By looking at the time series of the two leading multi-objective meta-heuristics in Figure 10-b, we can deduce an upsurge in the use of NSGA-II to tackle multi-objective forward SSC problems in the recent year while the use of MOPSO is fluctuating in the range of two and three. Although the literature has been saturated somewhat by these leading single- and multi-objective meta-heuristics, there still exists a considerable room to apply less-used meta-heuristics or design new innovative ones to solve SSCM problems for future studies. Comparing meta-heuristics already used in the context of forward SSCM with the available ones demonstrates that the extant literature suffers the absence of many meta-heuristic algorithms including but not limited to the bacterial foraging optimisation algorithm (Passino 2002), shuffled frog leaping algorithm (Eusuff, Lansey, and Pasha 2006), cuckoo search algorithm (Yang and Deb 2010), bat algorithm (Yang and He 2013), flower pollination algorithm (Yang, Karamanoglu, and He 2013), vibration damping optimisation algorithm (Mehdizadeh, Tavakkoli-Moghaddam, and Yazdani 2015), dragonfly algorithm (Mirjalili 2016), social engineering optimiser (Fathollahi-Fard, Hajiaghaei-Keshteli, and Tavakkoli-Moghaddam 2018), and Find-Fix-Finish-Exploit-Analyze (F3EA) metaheuristic (Kashan, Tavakkoli-Moghaddam, and Gen 2019) that can be applied by the future studies as a promising avenue of research.

Figure 10. Trend of the meta-heuristics used in the reviewed literature over the last five years. Figure (10-a). Trend of five leading single-objective meta-heuristics over the last five years. Figure (10-b). Trend of two major multi-objective meta-heuristics over the last five years.

Two line charts showing the trend of the most-used meta-heuristics in the literature over the last five years where GA and NSGA-I are the dominant single-objective and multi-objective algorithms, respectively.

To discuss the status of meta-heuristics in the SSCM literature from another perspective, the trend of pure and hybrid meta-heuristics over the years is illustrated in Figure 11. As can be seen, the use of hybrid meta-heuristics is overtaking pure meta-heuristics. The definite domination of hybrid over pure meta-heuristics in the last two years confirms a huge orientation toward these algorithms such that we expect more growth in the application of these algorithms in SSCM in the recent future. Such tendency is justifiable as hybrid algorithms are generally more efficient than their individual components. To take part in the SSCM literature orientation toward hybrid meta-heuristics, one can develop new hybrid algorithms either for the existing optimisation problems and compare the efficiency of the proposed algorithms with the state-of-the-art ones or apply them to the problems that have rarely been touched by hybrid algorithms. Also, matheuristics as the algorithms involving the features of both exact and meta-heuristic algorithms (Jourdan, Basseur, and Talbi 2009) are finding their place among the scholars in the literature of interest; however, the number of these studies is scant that provide promising opportunities for future research in this domain.

Figure 11. Trend of pure and hybrid meta-heuristics used in the reviewed literature over years.

Line chart showing the comparative trend of pure and hybrid meta-heuristics used in the literature over years where hybrid algorithms are overtaking the pure ones since 2019.

Additionally, to improve the performance of meta-heuristics in terms of convergence speed, solution quality, and robustness, integrating machine learning techniques into these algorithms are receiving more attention (Karimi-Mamaghan et al. 2021). As proposed by Talbi (2021), there are three hierarchical ways to use machine learning in meta-heuristics: problem-level data-driven meta-heuristics, low-level data-driven meta-heuristics, and high-level data-driven meta-heuristics. In this line, one can design and develop data-driven meta-heuristics to solve optimisation problems of SSCM as a trending research opportunity. Furthermore, the combination of simulation with meta-heuristics as a branch of simulation-optimisation methods provides an efficient tool to deal with stochastic optimisation problems that are frequently encountered in the field of SSCM and can be considered as original future research. Moreover, the use of game-theoretic-based meta-heuristics can be an interesting avenue of research to tackle multi-level sustainable supply chain problems.

In the context of SSCM, a variety of optimisation problems have been solved by meta-heuristics. Figure 12 shows the advent and the ups and downs of the six noticeable optimisation problems over the years in the literature of interest. According to this figure, the domination of the VRP is evident over the other problems, which is mostly for the recent importance of distribution and delivery issues in urban areas. Most of the preliminary studies in this scope deal with the pickup and delivery of goods within city areas by conventional vehicles while the recent studies focus on greener deliveries using electric vehicles or a mixed fleet. The LAP and SCND are the two other optimisation problems that have been fairly studied. To extend the extant literature along the optimisation problems, it is recommended to focus on the other significant but rarely studied problems in SCM by taking the aspects of sustainability into account. This can involve supply chain coordination, forecasting (Kantasa-Ard et al. 2021), transportation mode and channel selection, order consolidation, technology and process selection, facility selection and configuration, product design, pricing (Vahedi-Nouri et al. 2021) and advertising. In addition, applying metaheuristic techniques to tackle digital and physical internet SC problems (Chadha, Ülkü, and Venkatadri 2021; Luo, Tian, and Kong 2021) is a promising avenue of research.

Figure 12. Trend of the six most-addressed problems in the reviewed literature over years.

Line chart showing the comparative trend of the six most-addressed problems in the literature over years where the line for VRP is mostly above the others.

Another remarkable issue in the proposed models of the reviewed literature is the consideration of various sustainability aspects. In this concern, economic-environmental and economic-environmental-social are the two most-addressed categories of sustainability. More than 74% of studies focused on the economic-environmental category; however, in modern societies, social aspects (e.g. creating job opportunities, equity among SC entities, equity among employees and workers’ compensation for injuries) are of importance and need to be considered more in future papers. Figure 13 illustrates the number of papers using single-objective and multi-objective meta-heuristics to solve the problems dealing with economic-environmental and economic-environmental-social categories of sustainability, respectively. Although one might firstly realise that considering various dimensions of sustainability will always dominate the use of multi-objective meta-heuristics, the application of single-objective meta-heuristics is more apparent for the studies considering economic-environmental issue. Developing single-objective models and converting multi-objective models into single-objective ones using the state-of-the-art methods are the reasons behind this fact. To enhance the literature in this regard, one can develop multi-objective models following with multi-objective meta-heuristic algorithms as an interesting avenue of research.

Figure 13. Number of papers using single-objective and multi-objective meta-heuristics to solve models considering economic-environmental and economic-environmental-social issues.

Comparative column chart including four columns showing the number of papers using single-objective and multi-objective meta-heuristics to solve models considering economic-environmental and economic-environmental-social issues. The highest belongs to single-objective metaheuristics used in economicenvironmental category.

Last but not least, to validate the proposed models and acknowledge their applicability, 72 papers applied the models on real case studies across the worlds whilst the remaining papers rounded off with randomly generated benchmarks and artificial instances. To step toward the applicability of the models proposed, it is recommended to apply real case studies for model validation in future studies.

8. Conclusion

SSCM optimisation problems are becoming more cumbersome to be solved due to their complicated modeling structures, computational complexity, and large-scale problem size. To cope with these difficulties, the use of meta-heuristics as qualified solution methods is widely penetrating into the literature. To elucidate the true position of meta-heuristics in forward SSCM, this study reviewed a total number of 160 relevant papers selected attentively through a comprehensive structured keyword search from English peer-reviewed journals published by the end of 2020. Our statistical analysis reveals a rapid increase in the number of publications over the recent years together with a fair variety in the contributing journals. To answer the research questions of this study and to scrutinize the literature of interest, we discussed the reviewed papers under three streams: meta-heuristics, SSC problems, and sustainability. Major findings of our content analysis report a considerable growth in the use of hybrid meta-heuristics, introduce GA, NSGA-II, PSO, SA, and VNS as the most-used algorithms, identify vehicle routing, location/allocation, and network design as SC problems for which sustainability aspects have been further noticed, and finally, present economic-environmental category of sustainability as the most-argued one in the literature. In addition, comparing the meta-heuristics applied for sustainable and classic SCM reveals that the use of multi-objective meta-heuristics is more tangible in SSCM as more objectives are typically of interest to consider various aspects of sustainability. Additionally, in order to propose efficient algorithms to cope with more complex SSCM problems in comparison with classic SCM problems, the application of hybrid meta-heuristics is more common in SSCM. In the end, several promising opportunities for future research were recommended to enrich the extant literature.

Despite this research effort was rigorously completed, there are still limitations that leave some opportunities for future research. To extend this study, one can apply a sophisticated bibliometric analysis to provide more statistics regarding the contributing authors and institutes, co-citation analysis, co-word analysis and text mining as in the works of Dolati Neghabadi, Evrard Samuel, and Espinouse (2019), Kazemi, Modak, and Govindan (2019) and Feng, Zhu, and Lai (2017). Driving a detailed analysis of identified hybrid meta-heuristics and further classify them based on the classification proposed by Talbi (2002, 2013, 2016) is also a fruitful avenue of research. Moreover, discussing and further classifying the literature regarding the industrial application of the papers rather than focusing only on optimization problems, modeling issues, and solution approaches represents a promising future direction.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Additional information

Notes on contributors

Sohrab Faramarzi-Oghani

Sohrab Faramarzi-Oghani is an Assistant Professor of Supply Chain Management and the Head of Supply Chain Management and Information Systems department at Rennes School of Business in France. He received his Ph.D. degree in Computer Science from the University of Lille (2018), M.Sc. degrees in Knowledge Integration in Mechnical Production from École Nationale Supérieure d'Arts et Métiers and in Insudtrial and Systems Engineering from the University of Tehran (2015), and B.Sc. degree in Industrial Engineering from the University of Tabriz (2013). His research interests revolve around the application of quantitative models and optimization techniques in sustainable logistics and supply chain management. He is the author of several papers published and presented in international journals and conferences.

Parisa Dolati Neghabadi

Parisa Dolati Neghabadi is a Research Project Manager at Capgemini Engineering and an Adjunct Professor of Supply Chain Management at Rennes School of Business in France. She received her Ph.D. degree in Industrial Engineering from the Grenoble Alpes University (2018), M.Sc. degrees in Knowledge Integration in Mechnical Production from École Nationale Supérieure d'Arts et Métiers and in Insudtrial and Systems Engineering from the University of Tehran (2015), and B.Sc. degree in Industrial Engineering from the University of Tabriz (2012). Her research interests are in the field of logistics and supply network design and optimization, city logistics and ride sharing. She is the author of several papers published and presented in international journals and conferences.

El-Ghazali Talbi

El-Ghazali Talbi received the Master and Ph.D. degrees in Computer Science from the Institut National Polytechnique de Grenoble in France. He is a full Professor at the University of Lille and was the head of DOLPHIN research group from both the Lille's Computer Science laboratory (CRISTAL, Université Lille, CNRS) and INRIA. His current research interests are in the field of multi-objective optimization, parallel algorithms, metaheuristics, combinatorial optimization, hybrid and cooperative optimization, and applications to logistics/transportation, engineering design and networks. Professor Talbi has to his credit more than 170 international publications including journal papers, book chapters and conferences proceedings.

Reza Tavakkoli-Moghaddam

Reza Tavakkoli-Moghaddam is a Professor of Industrial Engineering at the College of Engineering, University of Tehran in Iran. He obtained his Ph.D., M.Sc. and B.Sc. degrees in Industrial Engineering from Swinburne University of Technology in Melbourne (1998), University of Melbourne in Melbourne (1994), and Iran University of Science and Technology in Tehran (1989), respectively. He serves as the Editor-in-Chief of Journal of Industrial Engineering published by the University of Tehran and the Editorial Board member of nine reputable academic journals. He is the recipient of the 2009 and 2011 Distinguished Researcher Awards and the 2010 and 2014 Distinguished Applied Research Awards at University of Tehran, Iran. He has been selected as the National Iranian Distinguished Researcher in 2008 and 2010 by the MSRT (Ministry of Science, Research, and Technology) in Iran. He has obtained the outstanding rank as the top 1% scientist and researcher in the world elite group since 2014. Also, he received the Order of Academic Palms Award as a distinguished educator and scholar for the insignia of Chevalier dans l'Ordre des Palmes Academiques by the Ministry of National Education of France in 2019. He has published 5 books, 25 book chapters, and more than 1000 journal and conference papers.

Notes

1 Note that the bold ‘H’ used before the name of some metaheuristics in the third column of the table stands for the term ‘Hybrid’ and implies that its following algorithm has been applied in a hybrid fashion. In some cases, the bold ‘H’ follows by the name of two algorithms used in a hybrid form; however, once this follows by only the name of a single algorithm, it denotes to the situation where the hybridization has been done between the called algorithm and a less common algorithm, or a local search.

Appendix 1.

Full list of journals contributed to the literature of meta-heuristics for SSCM.

Table

N	Journal	Number of papers published
1	Journal of Cleaner Production	27
2	Computers and Industrial Engineering	14
3	International Journal of Production Research	11
4	European Journal of Operational Research	9
5	Expert Systems with Applications	9
6	International Journal of Production Economics	8
7	Transportation Research Part E Logistics and Transportation Review	7
8	Computers and Operations Research	7
9	Transportation Research Part D Transport and Environment	6
10	Annals of Operations Research	4
11	Applied Energy	4
12	Industrial Management and Data Systems	4
13	Applied Soft Computing Journal	3
14	Resources, Conservation and Recycling	3
15	Transportation Research Part B Methodological	3
16	Applied Mathematical Modelling	2
17	Applied Mathematics and Computation	2
18	Energy Conversion and Management	2
19	International Transactions in Operational Research	2
20	Journal of Intelligent Manufacturing	2
21	Soft Computing	2
22	Asia Pacific Journal of Marketing and Logistics	1
23	Case Studies on Transport Policy	1
24	Computers and Chemical Engineering	1
25	Engineering Applications of Artificial Intelligence	1
26	Enterprise Information Systems	1
27	Environmental Impact Assessment Review	1
28	Environmental Processes	1
29	Environmental Science and Pollution Research	1
30	International Journal of Energy Research	1
31	International Journal of Logistics Research and Applications	1
32	International Journal of Precision Engineering and Manufacturing	1
33	Journal of Ambient Intelligence and Humanized Computing	1
34	Journal of Engineering, Design and Technology	1
35	Journal of Environmental Engineering and Landscape Management	1
36	Journal of Heuristics	1
37	Journal of Industrial Engineering International	1
38	Journal of Modelling in Management	1
39	Knowledge-Based Systems	1
40	KSCE Journal of Civil Engineering	1
41	Kybernetes	1
42	Neural Computing and Applications	1
43	Omega	1
44	OPSEARCH	1
45	Production and Operations Management	1
46	Production Planning and Control	1
47	Swarm and Evolutionary Computation	1
48	Transportation Letters	1
49	Transportation Research Part A: Policy and Practice	1
50	Transportation Science	1

Appendix 2.

Summary of the reviewed papers in terms of problems, meta-heuristics, and sustainability aspects.¹

Table

			Sustainability
Reference	Problem	Meta-heuristic	Eco.	Env.	Soc.
Alkaabneh, Diabat, and Gao (2020)	IRP	H.GRASP-BD	✓	✓
Biuki, Kazemi, and Alinezhad (2020)	LIRP	H.GA-PSO (2)	✓	✓	✓
Brahami et al. (2020)	SCND	H.NSGA-II	✓	✓
Chan et al. (2020)	PIRP	H.MOPSO	✓	✓	✓
Chen, Dan, and Shi (2020)	VRP	H.VNS	✓	✓
Fathollahi-Fard et al. (2020)	HHRP	H.MOSEO	✓		✓
Ganji et al. (2020)	PPP	MOACO	✓	✓	✓
	VRP	MOPSO
		NSGA-II
Giallanza and Puma (2020)	VRP	NSGA-II	✓	✓
Heidari, Imani, and Khalilzadeh (2020)	SCND	MOGWO	✓	✓	✓
		NSGA-II
Jabir, Panicker, and Sridharan (2020)	VRP	ACO	✓	✓
		H.ACO-VNS
Karakostas, Sifaleras, and Georgiadis (2020)	LIRP	H.VNS	✓	✓
Leng et al. (2020a)	LRP	H.NSGA-II	✓	✓	✓
		H.NSGA-III
		H.SPEA-II
		H.NSLS
		H.GrEA
		H.IBEA
Leng et al. (2020b)	LRP	MOHH/D	✓	✓	✓
Li et al. (2020a)	VRP	GA	✓	✓
		H.GA-TS
Li et al. (2020b)	VRP	ACO	✓	✓
Liu et al. (2020)	VRP	SA	✓	✓
Mogale, Kumar, and Tiwari (2020)	SCND	PSO	✓	✓
		H.PSO
Poonthalir, Nadarajan, and Kumar (2020)	VRP	H.PSO	✓	✓
Rasi and Sohanian (2020)	SCND	MOGA	✓	✓
		MOPSO
Reyes-Rubiano et al. (2020)	VRP	H.VNS	✓	✓	✓
Robles, Azzaro-Pantel, and Aguilar-Lasserre (2020)	SCND	GA	✓	✓	✓
		NSGA-II
Salhi et al. (2020)	OC&TP	H.GRASP	✓	✓
Shen (2020)	SCND	H.GA	✓	✓	✓
Shi et al. (2020)	VRP	H.TS-SA	✓	✓
Tirkolaee et al. (2020)	PRP	MOSA	✓	✓	✓
		NSGA-II
Validi, Bhattacharya, and Byrne (2020)	LRP	H.MOGA-II-DOE	✓	✓
Wang et al. (2020a)	LIP	H.SA-Exact Algorithm	✓	✓
Wang et al. (2020b)	PPP TP	H.GA	✓	✓
Wang et al. (2020c)	PPP TP	H.GA	✓	✓
Wang et al. (2020d)	VRP	H.NSGA-II	✓	✓
Wu et al. (2020)	PSP	H.NSGA-II	✓	✓	✓
Zhang et al. (2020)	PPP, PP	H.GA	✓	✓
Abdi et al. (2019)	PPP, VRP	SA	✓	✓
		RDA
		H.GA-PSO
		H.GA-Keshteli
Ajam, Akbari, and Salman (2019)	RCP	H.GRASP-VNS			✓
Alinaghian and Zamani (2019)	IRP	MOEA	✓	✓
Anderluh et al. (2021)	VRP	H.LNS	✓	✓	✓
Asghari and Al-e-hashem (2020)	VRP	NSGA-II	✓	✓
Bravo, Rojas, and Parada (2019)	PRP	MOEA	✓	✓	✓
Chalmardi and Camacho-Vallejo (2019)	SCND	SA	✓	✓
Chargui et al. (2020)	CGP	H.VNS-SA	✓	✓
		H.GRASP-SA
		H.TS-SA
Chiang et al. (2019)	VRP	GA	✓	✓
Chu et al. (2019)	VRP	ABC	✓		✓
		H.PSO (2)
Erdem and Koç (2019)	VRP	H.GA-VND	✓	✓
Eskandarpour, Dejax, and Péton (2021)	SCND	MOLNS	✓	✓
Eydi and Fathi (2020)	SSP, CSP	MOICA	✓	✓
Fathollahi-Fard et al. (2019)	LARP	MOSA	✓	✓
Ganesh Kumar and Uthayakumar (2019)	IM	GA	✓	✓
Ghannadpour and Zarrabi (2019)	VRP	H.MOEA	✓	✓	✓
Govindan, Jafarian, and Nourbakhsh (2019)	SCND, VRP	H.PSO-VNS	✓	✓	✓
		H.EMA-VNS
		H.ABC-VNS
Hafezalkotob and Zamani (2019)	PPP, PP	GA	✓	✓
Ji, Luo, and Peng (2019)	ECAP	MOEA/D	✓	✓
Karbassi Yazdi et al. (2019)	SRSP	PSO	✓	✓
Li, Wang, and Tan (2019a)	SCND	H.SSA-DE	✓	✓
Li et al. (2019b)	LRP	H.GA-TS	✓	✓
Li, Lim, and Tseng (2019c)	VRP	H.PSO-TS	✓	✓
Liu and Lin (2019)	IRP	H.TS	✓	✓
Liu et al. (2019)	VRP	ACO	✓	✓
Macrina et al. (2019)	VRP	H.LNS-MIP	✓	✓
Maiyar & Thakkar (2019a)	SCND, VRP	H.MOPSO-DE	✓	✓
Maiyar & Thakkar (2019b)	LP, TP	H.PSO-DE	✓	✓	✓
Maiyar and Thakkar (2020)	LP, TP	H.PSO-DE	✓	✓
Moons et al. (2019)	VRP	H.TA	✓		✓
Ng, Lam, and Samuel (2019)	VRP	ACO		✓
Noh and Kim (2019)	SCCC	GA	✓	✓
		H.GA
Pelletier, Jabali, and Laporte (2019)	VRP	LNS	✓	✓
Pourhejazy, Kwon, and Lim (2019)	LIRP	MOEA	✓	✓
Qin et al. (2019)	VRP	H.GA	✓	✓
Quintero-Araujo et al. (2019)	LRP	H.VNS	✓	✓
Rauniyar, Nath, and Muhuri (2019)	PRP	NSGA-II	✓	✓
		SPEA-II
Sadeghi and Haapala (2019)	LP, TP	GA	✓	✓
Sarker, Wu, and Paudel (2019)	LAP	GA	✓	✓
Tan et al. (2019)	PRP	H.ACO-GA	✓	✓
Tautenhain, Barbosa-Povoa, and Nascimento (2019)	SCND	Heuristic/D	✓	✓	✓
Teran-Somohano and Smith (2019)	LP	MOEA	✓		✓
Wang et al. (2019a)	MSP, VRP	H.TS	✓	✓
		MOTS/D
Wang et al. (2019b)	VRP	H.GA	✓	✓
Xu et al. (2019)	VRP	H.NSGA-II		✓	✓
Yong et al. (2019)	VRP	H.MOPSO	✓	✓	✓
Zhang et al. (2019a)	VRP	H.ACO	✓	✓
Zhang et al. (2019b)	CSR	H.PSO	✓	✓	✓
Zhen et al. (2020)	VRP	H.PSO-VNS	✓	✓
Abad et al. (2018)	PRP	NSGA-II	✓	✓
		NRGA
		MOPSO
Asadi et al. (2018)	LIRP	NSGA-II	✓	✓
		MOPSO
Barzinpour and Taki (2018)	LP, TP	GA	✓	✓
		AIS
		MA
Borumand and Beheshtinia (2018)	PPP, TP	GA	✓	✓
Cao et al. (2018)	RDP	GA			✓
Corberán et al. (2018)	PRP	VNS	✓	✓
		ILS
Dai et al. (2018)	LP, IM	H.GA	✓	✓
		H.HS
De, Das, and Maiti (2018)	PPP	MOGA	✓	✓
Doolun et al. (2018)	LAP	H.NSGA-II-DE	✓	✓	✓
Eskandari-Khanghahi et al. (2018)	SCND	SA	✓	✓	✓
		HS
Fahimnia, Davarzani, and Eshragh (2018)	TP, PPP, IM	GA	✓	✓
		SA
		Cross-Entropy
Guo et al. (2018)	TP	MOMA	✓	✓
Hong et al. (2018)	SCCP	PSO	✓	✓
		H.PSO
Kesharwani, Sun, and Dagli (2018)	SCND	PSO	✓	✓	✓
Lee et al. (2018)	VSO	MOPSO	✓	✓
Mahmoudsoltani, Shahbandarzadeh, and Moghdani (2018)	LRP	NSGA-II	✓	✓
		SPEA-II
		MOEA/D
Niu et al. (2018)	VRP	H.TS	✓	✓
Rahbari, Naderi, and Mohammadi (2018)	IRP	SA	✓	✓
Rau, Budiman, and Widyadana (2018)	IRP	H.PSO	✓	✓
Simoni et al. (2018)	LRP	GA	✓	✓
Wang et al. (2018)	VRP	H.NSGA-II	✓	✓
Zhang et al. (2018a)	LRP	H.GA	✓	✓
Zhang et al. (2018b)	LIIP	H.GA	✓	✓
Azadeh et al. (2017)	SCND	MOEA/D	✓	✓
		NSGA-II
		MOPSO
Canales-Bustos, Santibañez-González, and Candia-Véjar (2017)	SCND	H.MOPSO	✓	✓
Chandrasekaran and Ranganathan (2017)	TP	GA	✓	✓
Franceschetti et al. (2017)	PRP	LNS	✓	✓
Gupta et al. (2017)	VRP	MOEA	✓	✓	✓
Hassanzadeh and Rasti-Barzoki (2017)	SCS, VRP	H.NSGA-II	✓	✓	✓
Jabir, Panicker, and Sridharan (2017)	VRP	H.ACO-VNS	✓	✓
Kadziński et al. (2017)	SCND	NSGA-II	✓	✓
		SPEA-II
		H.NEMO-NSGA-II
		H.NEMO-SPEA-II
Kumar et al. (2017)	SCND	NSGA-II	✓	✓
Miranda-Ackerman, Azzaro-Pantel, and Aguilar-Lasserre (2017)	SCND	GA	✓	✓
Mirkouei et al. (2017)	TP	GA	✓	✓
Musavi and Bozorgi-Amiri (2017)	HLSP	NSGA-II	✓	✓
Velazquez Abad, Cherrett, and Waterson (2017)	TSP	H.EA	✓	✓
Wang, Shao, and Zhou (2017a)	VRP	H.VNS-IP	✓	✓
Wang, Zhang, and Zhu (2017b)	P&A	DE	✓	✓
Xiao and Konak (2017)	VRP	H.GA-DP	✓	✓	✓
Zahiri, Zhuang, and Mohammadi (2017)	SCND	H.MOEA	✓	✓	✓
Cheng et al. (2016)	IRP	H.GA	✓	✓
Chibeles-Martins et al. (2016)	SCND	MOSA	✓	✓
De et al. (2016)	SRSP	H.PSO	✓	✓
Ehmke, Campbell, and Thomas (2016)	VRP	TS		✓
Fallahpour et al. (2016)	SSP	GA	✓	✓
Huang et al. (2016)	TP, SSP, PP	GA	✓	✓
Hwang et al. (2016)	LP	GA	✓	✓	✓
		TS
		MA
Koç et al. (2016)	LRP	H.LNS-SA	✓	✓
Kumar et al. (2016a)	SSP	H.GA-AIS	✓	✓
Kumar et al. (2016b)	PPP, PRP	H.MOPSO	✓	✓
		NSGA-II
Memari et al. (2016)	TP	NSGA-II	✓	✓
		H.FA-NSGA-II
Suzuki (2016)	PRP	H.SA-TS	✓	✓
Wu and Barnes (2016)	PSP	PSO	✓	✓
Xiao and Konak (2016)	VRP	H.ILS-MIP		✓
Yin and Chuang (2016)	VRP	H.ABC	✓	✓
Zhang et al. (2016)	SCND	MOABC	✓	✓	✓
Diabat and Al-Salem (2015)	LRP	GA	✓	✓
Goeke and Schneider (2015)	VRP	H.LNS	✓	✓
Govindan, Jafarian, and Nourbakhsh (2015a)	SCND	H.MOEMA-MOVNS	✓	✓
Kramer et al. (2015a)	PRP	ILS	✓	✓
Kramer et al. (2015b)	PRP	H.VND-IP	✓	✓
Küçükoğlu et al. (2015)	VRP	H.SA		✓
Nia, Far, and Niaki (2015)	IM	H.GA-ICA	✓	✓
Yang et al. (2015)	VRP	H.GA	✓	✓	✓
Zhang et al. (2015a)	VRP	TS	✓	✓
Ćirović et al. (2014)	VRP	H.SA	✓	✓
Govindan et al. (2014)	LRP	H.MOPSO-MOVNS	✓	✓
Harris, Mumford, and Naim (2014)	LAP	H.SEAMO2-LR	✓	✓
Koç et al. (2014)	PRP	H.LNS	✓	✓
Validi, Bhattacharya, and Byrne (2014a)	VRP	NSGA-II	✓	✓
		MOGA-II
		H.GA-QP
Validi, Bhattacharya, and Byrne (2014b)	LRP	H.MOPSO	✓	✓
Zhang et al. (2014)	VRP	H.ABC	✓	✓
Bashiri, Mirzaei, and Randall (2013)	LP	GA	✓	✓	✓
Jabali, Van Woensel, and De Kok (2012)	VRP	TS	✓	✓
Su, Chu, and Wang (2012)	PDP	H.GA-DP	✓	✓
Yeh and Chuang (2011)	PSP	MOGA	✓	✓
Che (2010)	SSP, PPP, TP	PSO	✓	✓
Ayoub et al. (2009)	SCND	GA	✓	✓
Ayoub et al. (2007)	SCND	Ga	✓	✓

Abbreviation used in column ‘Problem’

CGP	=	Container Grouping Problem
CSP	=	Carrier Selection Problem
CSR	=	Cold Storage Reconstruction
ECAP	=	Express Cabinet Assignment Problem
HHRP	=	Home Healthcare Routing Problem
HLSP	=	Hub Location Scheduling Problem
IM	=	Inventory Management
IRP	=	Inventory Routing Problem
LAP	=	Location Allocation Problem
LARP	=	Location Allocation Routing Problem
LIP	=	Location Inventory Problem
LIIP	=	Logistics Infrastructure Investment Problem
LIRP	=	Location Inventory Routing Problem
LP	=	Location Problem
LRP	=	Location Routing Problem
MSP	=	Machine Scheduling Problem
OC&TP	=	Order Consolidation & Transshipment Problem
P&A	=	Pricing & Advertising
PIRP	=	Production Inventory Routing Problem
PP	=	Pricing Problem
PPP	=	Production Planning Problem
PRP	=	Pollution Routing Problem
PSP	=	Partner Selection Problem
RCP	=	Road Clearing Problem
RDP	=	Relief Distribution Problem
SCCC	=	Supply Chain Coordination Contract
SCCP	=	Supply Chain Configuration Problem
SCND	=	Supply Chain Network Design
SCS	=	Supply Chain Scheduling
SRSP	=	Ship Routing Scheduling Problem
SSP	=	Supplier Selection Problem
PDP	=	Product Design Problem
TP	=	Transportation Problem
TSP	=	Technology Selection Problem
VRP	=	Vehicle Routing Problem
VSO	=	Vessel Speed Optimization

Abbreviation used in column ‘Meta-heuristic’

ABC	=	Artificial Bee Colony
ACO	=	Ant Colony Optimization
AIS	=	Artificial Immune Systems
BD	=	Benders Decomposition
DE	=	Differential Evolution
DoE	=	Design of Experiment
DP	=	Dynamic Programming
EA	=	Evolutionary Algorithm
EMA	=	Electromagnetism Mechanism Algorithm
FA	=	Firefly Algorithm
GA	=	Genetic Algorithm
GRASP	=	Greedy Random Adaptive Search Procedure
GrEA	=	Grid-based Evolutionary Algorithm
HS	=	Harmony Search
IBEA	=	Indicator-based Evolutionary Algorithm
ICA	=	Imperial Competitive Algorithm
ILS	=	Iterated Local Search
IP	=	Integer Programming
LNS	=	Large Neighborhood Search
LR	=	Lagrangian Relaxation
LS	=	Local Search
MA	=	Memetic Algorithm
MDLNS	=	Multi-directional LNS
MIP	=	Mixed Integer Programming
MO	=	Multi-Objective
MOEA	=	Multi-Objective Evolutionary Algorithm
MOGA-II	=	Multi-Objective GA version 2
MOGWO	=	Multi-Objective Grey Wolf Optimizer
MOHH	=	Multi-Objective Hyper Heuristic
MOSA	=	Multi-Objective Simulated Annealing
MOSEO	=	Multi-Objective Social Engineering Optimizer
NEMO	=	Neural Enhancement for Multi-Objective
NRGA	=	Non-dominated Ranking GA
NSGA-II	=	Non-dominated Sorting GA
NSLS	=	Non-dominated Sorting & Local Search
PSO	=	Particle Swarm Optimization
QP	=	Quadratic Programming
RDA	=	Red Deer Algorithm
SA	=	Simulated Annealing
SEAMO2	=	Simple Evolutionary Algorithm for Multi-objective Optimization version 2
SPEA-II	=	Strength Pareto Evolutionary Algorithm version 2
SSA	=	Slap Swarm Algorithm
TA	=	Travel Algorithm
TS	=	Tabu Search
VND	=	Variable Neighborhood Descent
VNS	=	Variable Neighborhood Search