1 Introduction

Purchasing and procurements are important activities in every organization. Supplier Selection and Order Allocation (SSOA) are prominent elements of purchasing and procurement (Bohner and Minner 2017). Both qualitative and quantitative factors such as quality, cost, and delivery time should be considered in the supplier selection problem (Arabsheybani et al. 2018). Therefore, supplier selection is a Multi-Criteria Decision-Making (MCDM) problem (Cheraghalipour and Farsad 2018; Moheb-Alizadeh and Handfield 2019).

Cost saving and minimization of risks can be achieved by using suitable supplier selection methods (Çebi and Otay 2016; Arabsheybani et al. 2018). Some authors have combined supplier selection and order allocation together to solve these two problems simultaneously (e.g., Babbar and Amin 2018). Supplier selection and order allocation are very important in green supply chain management considering sustainability and environmental factors (Hamdan and Cheaitou 2017a). Sustainable supplier selection encompasses cost, environmental, and social criteria and supplier’s performance history (Hamdan and Cheaitou 2017b; Ghadimi et al. 2018).

Supplier selection is a strategic process in organizations, and plays a critical role in the success of them (Razaei et al. 2020). Offering quantity discount is an important feature in the selection of the best suppliers. Therefore, each company can achieve low cost while allocating large volume orders to the suppliers (Alegoz and Yapicioglu 2019).

In this paper, the related peer-reviewed journal papers have been found and reviewed via search in globally recognised databases such as ScienceDirect (Elsevier), Taylor and Francis, and Google Scholar. The main keyword is “supplier selection and order allocation” which is used to search related papers published between 2015 and 2020. As a result, 92 articles are analysed. The majority of other literature review papers in this field just have focused on supplier selection, and order allocation has been ignored. In addition, they have been written some years ago, and there is a need to have an updated literature review paper about supplier selection and order allocation. The structure of our paper is new among those literature review papers. Considering uncertainty in the reviewed papers and reviewing the applied operations research techniques are other main characteristics of our paper.

The other parts of this paper are as follows. The taxonomy and classification of the literature are provided in Section 2. Then, some observations and discussions are mentioned in Section 3. In addition, the related conclusions and future research avenues are provided in Section 4.

2 Taxonomy

Two dimensions are utilized in this review paper to categorize the papers. The first one is the problem domain. Besides, the second one is the operations research (optimization) methods. This classification is useful to analyze the supplier selection and order allocation problem based on both conceptual and mathematical viewpoints.

2.1 Problem domain

The problem domain comprises three subsections: Literature Reviews (LR), Deterministic Optimization (DO) models, and Uncertain Optimization (UO) models. Table 1 includes the related papers.

Table 1 Problem domain and related references
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2.1.1 Literature reviews

Some authors have published literature review papers in the field of supplier selection and order allocation. Govindan et al. (2015) reviewed green purchasing and green supplier selection process of some articles published between 1997 and 2011. They found that Analytic Hierarchy Process (AHP) is the most popular MCDM method for assessing green suppliers. In addition, Fuzzy AHP is very popular in the environmental management systems. Yildiz and Yayla (2015) reviewed 91 articles that have been published between 2001 and 2014 about supplier selection. They stated that quality and cost are the most significant criteria in the supplier selection problem.

Wetzstein et al. (2016) reviewed several papers in the supplier selection field published between 1990 and 2015. They mentioned that there are future research avenues in considering green and sustainable factors. Karsak and Dursun (2016) reviewed 149 articles published between 2001 and 2013 concentrating nondeterministic analytical methods (i.e., stochastic/fuzzy) under imprecise data.

Simić et al. (2017) examined the last 50 years (50th anniversary of fuzzy sets theory established by Lotfiali Askar Zadeh in 1965) of articles in supplier selection and evaluation that are based on fuzzy sets theory, fuzzy models, and fuzzy hybridization. The authors combined individual and integrated approaches to effectively review the fuzzy supplier selection methods. The authors selected 54 papers published in the reputable journals.

Alkahtani and Kaid (2018) studied some journal papers published between 1995 and 2018 focusing on supplier selection. They provided information about the trends, the research gaps, and the selection criteria in the supplier selection field. Ocampo et al. (2018) reviewed 240 articles from peer-reviewed journals published between 2006 and 2016 focusing on the applications of different approaches for supplier selection and evaluation which include individual and hybrid methods. The authors indicated that the novel methods in the literature include uncertainty, risk analysis, and sustainability factors.

Aouadni et al. (2019) reviewed 270 articles published between 2000 and 2017 about supplier selection and order allocation. In their paper, about 17 %, 9 %, and 7 % of the reviewed papers were about AHP, TOPSIS, and ANP methods, respectively. They mentioned that fuzzy multiple-objective programming is a popular method in this area. In addition, some papers have used genetic algorithm to determine the orders. Chai and Ngai (2019) reviewed SSOA papers published among 2013 and 2018. They brought to light that MCDM methods and optimization are the most popular techniques for supplier selection and order allocation.

2.1.2 Deterministic optimization models

In this part, deterministic optimization methods for supplier selection and order allocation are discussed. We provide some information about the publications that have received several citations. Other publications are written and mentioned in the following table.

Sodenkamp et al. (2016) used a novel approach (trade-off mechanism) because the current multi-objective methods were not capable to create positive and negative performance synergies. Bohner and Minner (2017) disscussed a mixed-integer linear programming model for solving the intricate issue of supplier selection by having a backup supplier who is not cost effective. However, it minimizes the risk related to the stock out condition. Nourmohamadi Shalke et al. (2018) studied purchasing decision-making through a TOPSIS method and a multi-choice goal programming model. In addition, they developed rough set theory and grey system. Moheb-Alizadeh and Handfield (2019) constructed a multi-objective optimization model for a manufacturer of automobile transmission systems for order allocation considering minimization of the CO2 emissions.

Nazeri et al. (2019) proposed a multi-objective model to test effective ranking in military’s SSOA for outsourcing of hazardous materials. Wang et al. (2020) implemented carbon emission trading schemes using analytic network process-integer programming model and stipulated approach. They minimized cost and carbon emissions. The deterministic optimization models of supplier selection and order allocation are classified in Table 2 according to the multiple elements (sets) including parts, products, periods, suppliers, and scenarios. In some papers, it has been assumed that the parts can be assembled to make a product. Products usually represent the final products that can be sold in the markets. In addition, some papers have considered different periods such as months in their mathematical models. Besides, multiple potential suppliers have been considered by some authors. Furthermore, some authors have assumed different scenarios to analyze the problem under uncertainty.

Table 2 Deterministic optimization models according to multiple elements
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2.1.3 Uncertain optimization models

In this part, we mention some important publications that have developed uncertain optimization models and have received several citations. Other papers are mentioned in the following two tables.

The paper of Azadnia et al. (2015) has been cited by more than 200 papers in Google Scholar. In this article, the authors introduced sustainable supplier selection by adding occupational health and safety management system. Those sub-criteria are important components in sustainable supplier criteria. The authors utilized a fuzzy AHP and a rule based weighted fuzzy approach. They selected a sustainable supplier using a multi-product lot sizing order model. Çebi and Otay (2016) proposed a two-stage fuzzy method including fuzzy MULTIMOORA and fuzzy goal programming for the supplier selection and order allocation problem. They considered green supplier selection in the beverage industry. Govindan and Sivakumar (2016) studied the selection of the best supplier by minimizing the greenhouse gas emissions using a fuzzy TOPSIS and a multi-objective method. They determined the ranks of the green suppliers, and they classified the potential suppliers. Pazhani et al. (2016) proposed a mathematical model to find the optimal inventory level and showed that cost could be minimized if the transportation cost is considered in the objective. They also discussed the benefits of the integrated inventory management system approach comparing the sequential approach for solving the supplier selection problem.

Ghorabaee et al. (2017) stated that a novel EDAS technique and interval type-2 fuzzy sets lead to a good multi-criteria green supplier selection model. Hamdan and Cheaitou (2017b) proposed a model to choose a green supplier and allocate the orders using a multi-objective optimization model. For solving the problem, the authors utilized AHP, Fuzzy TOPSIS, and multi-objective optimization techniques. Arabsheybani et al. (2108) considered some risk factors and enhanced FMEA method to determine the risk and price criteria.

Babbar and Amin (2018) proposed a novel two-stage QFD model and an optimization model for SSOA. They solved the optimization model using GAMS software. Their method can handle the vagueness and uncertainty considering qualitative and quantitative criteria. Ahmadi and Amin (2019) introduced the supplier selection and order allocation in closed-loop network of cellular phone industry in Toronto Canada. They developed a fuzzy based solution approach using IBM ILOG CPLEX 12.7.1.0 software. Hosseini et al. (2019) developed a bi-objective mixed-integer programming model (stochastic) for the SSOA problem. They illustrated the application of the model in the automotive industry. Bektur (2020) introduced F-AHP and F-PROMETHEE to study uncertainty in the decision-making environment. Govindan et al. (2020) described sustainability through incorporating circular supplier selection and order allocation. They combined all activities such as waste reduction in transportation. Hasan et al. (2020) developed a Decision Support System (DSS) for companies operating under logistics industry 4.0.

Jia et al. (2020) developed a robust optimization goal programming model for a steel company. They solved the problem by CPLEX, and optimized it considering the total cost, CO2 emission, and environmental objectives. Kaur and Singh (2020) proposed a model with consideration of risks and disruption (both natural and man made), suitable for industry 4.0 environment. The uncertain optimization models are categorized in Table 3 according to the multiple elements. In addition, the uncertainty sources are shown in Table 4; Fig. 1. Demand, capacity, and cost are the major sources of uncertainty in the reviewed papers.

Table 3 Uncertain optimization models according to multiple elements
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Table 4 The references and uncertainty sources
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Fig. 1
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Sources of uncertainty

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2.2 Operations research techniques

We divide the references according to the operations research (optimization) techniques in Table 5. Several authors have utilized hybrid techniques in this field. Table 6 includes the types of the objective functions in different papers. In addition, single objective and multi-objective functions are illustrated in Fig. 2.

Table 5 Arrangement of the papers according to the operations research methods
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Table 6 Single objective and multi-objective models
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Fig. 2
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Mono-objective and multi-objective models

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3 Observations and related recommendations

In this part, observations and recommendations according to the reviewed papers are provided.

3.1 The most popular domain

In this paper, we discussed three domains of supplier selection and order allocation including literature reviews, deterministic optimization models, and uncertain optimization models. The most popular domain is the uncertain optimization models (73 % of the papers). Deterministic optimization models (17 % of the papers), and Literature reviews (10 % of the papers) domains have the next ranks.

3.2 The most popular uncertainty source

According to Table 4; Fig. 1, the most popular sources of uncertainty are demand, capacity, and cost, respectively. Among them, demand has been considered more than the other factors (29 % of the papers). This parameter usually affects the order allocation significantly, and it is considered as one of the constraints of the optimization models.

3.3 The most popular technique

Based on the information in Table 5, Fuzzy TOPSIS, Fuzzy-multi objective programming, Stochastic programming, and Mixed-integer linear programming are the most popular techniques in the literature of SSOA. Fuzzy TOPSIS is a useful technique to determine the weights (importance) of the suppliers. Fuzzy sets theory enables researchers to consider uncertainty in the parameters. Fuzzy multi-objective programming considers uncertainty and the effects of some objectives on the problem. There are several stochastic programing models in the literature which are based on the probabilities in the SSOA problem. Mixed-integer linear programming also have been utilized in the literature because it can handle both non-negative and 0–1 variables.

3.4 The most popular multi-objective method

There are several techniques for solving multi-objective problems. Based on our observation, weighted-sums method is the most popular one. Several authors also have utilized goal programming and ɛ-constraints method to solve multi-objective SSOA problems.

3.5 The most popular applications

The applications of the models have been categorized in Table 7. Several authors have considered case studies. “Automotive industry” is a popular application in the supplier selection and order allocation field.

Table 7 Applications of the models
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3.6 The list of publications

Table 8 includes the information related to the names of the journals. These journals have published papers related to SSOA. “Journal of Cleaner Production”, “Computers & Industrial Engineering”, “International Journal of Production Research”, “International Journal of Production Economics”, and “Expert Systems with Applications” have published several papers in this field.

Table 8 The publications list
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3.7 Classification of the articles based on year

Table 9 includes the classification of the papers based on year and the mentioned three domains. The journal papers from 2015 to 2020 have been examined in this research. It is noticeable that our paper has been written in 2020. Therefore, the number of the published journal papers in 2020 are limited in Table 9.

Table 9 Classification of the papers based on year
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4 Conclusions

In this research, three problem domains including literature reviews, deterministic optimization models, and uncertain optimization models have been considered, and the related papers have been gathered and analyzed. In addition, these papers (92 publications between 2015 and 2020) have been classified according to the operations research methods. Furthermore, observations have been provided and discussed. We have observed that most of the mathematical models in SSOA belong to the uncertain optimization models category. Supplier selection and order allocation methods may create competitive advantages for companies, and at the same time, poor selection of the suppliers may result in the failure of the companies. The basic criteria for supplier selection include cost, quality, and time. Recently, more green and environmental factors such as minimization of carbon emissions have been considered in the SSOA process. There are numerous directions for future research in the SSOA problem. Some of them are as follow:

  1. (i)

    Usually a few sources of uncertainty have been considered in the optimization models of order It is useful to consider several sources of uncertainty simultaneously using advanced methods such as robust optimization.

  2. (ii)

    Most of the SSOA papers have focused on manufacturing systems such as automobile It is valuable to consider SSOA in service industry such as healthcare systems (e.g., hospitals).

  3. (iii)

    Fuzzy sets theory and fuzzy logic have been combined with other techniques to handle However, there are some practical challenges in applying these methods in More case studies can be considered in this area to show and discuss the applications of these methods.

  4. (iv)

    Under special circumstances such as COVID-19, the normal supplier selection and order allocation may not lead to excellent Developing new methods for these situations can be an avenue of future research.

  5. (v)

    There are several parameters such as cost, capacity, and demand in the optimization models of the order These parameters can be estimated using advanced forecasting techniques such as machine learning, deep learning, and neural To our knowledge, this area of research is new, and has not been explored in the SSOA papers.