Supplier selection and order allocation model with disruption and environmental risks in centralized supply chain

Abstract

Supply chain managers have realized that competition between supply chains has replaced competition between companies. In addition, with increasing disruptions and uncertainty in planning, companies need to be able to make informed decisions at risk. Coordination in the resilient supply chain and appropriate selection of suppliers play a key role in risky situations. In previous research, mainly the impact of resilience strategies in the decentralized supply chain has been investigated and ignored the reliability of suppliers in the decision-making process. Therefore, we provide an effective framework for selecting reliable suppliers and order allocation, which increases the supply chain's benefits by considering the risk reduction strategies and coordination between the buyer and the supplier. Thus, we optimized the problem of supplier selection and order allocation in a centralized supply chain using mixed-integer nonlinear programming models and risk reduction strategies. These strategies are protected suppliers, back-up suppliers, reserving additional capacity, emergency stock, and geographical separation. Also, by considering the failure mode and effects analysis technique and the risk priority number constraint, suppliers' reliability has been considered. A numerical example is solved with the exact method. In addition, the application of the proposed models in a case study has been investigated by the Grasshopper optimization algorithm. Based on the sensitivity analysis results, we found that the simultaneous use of risk reduction strategies in the models significantly reduces supply chain costs and increases its benefits. Also, considering the reliability constraints causes supply chain managers to choose suppliers with more desirable reliability.

1 Introduction

With the expansion of competition in today's dynamic and competitive environments, supply chain management has considered it necessary to pay attention to all supply chain components at decision levels. Organizations realized that effective interaction and communication with top suppliers and decision-making in supplier selection are important parts of production and procurement management (Pazhani et al. 2016; Rezaei et al. 2020a, b). In fact, the supply chain's efficiency depends on the flexibility and coordination of its various components (Esmaeili-Najafabadi et al. 2019). Moreover, one of the effective strategic factors in supply chain management is selecting suitable suppliers and reviewing related processes. The optimal order allocation process is an important issue that affects the results of the supplier selection process and their performance (Vahidi et al. 2018; Alejo-Reyes et al. 2021). Therefore, sourcing and inventory management through the proper selection of suppliers supporting the company's long-term competitive strategy and position are considered two most critical factors in a supply chain. As a result, studying supplier selection and optimal order allocation can play a key role in reducing supply chain management costs.

In addition, uncertainties and risks in today's competitive environment add another complexity element to decision making. The wide range of risks in the supply chain may have a negative impact on supply chain performance. The occurrence of disruption risk considerably affects the entire supply chain and leads to economic, social effects and reduces market share (Tolooie et al. 2020; abazari et al. 2020). Therefore, providing the appropriate flexibility and reliability to protect buyers from disruption requires a more secure supply system (Torabi et al. 2015). Such a system's main purpose is to balance costs and meet demand (Esmaeili-Najafabadi et al. 2019). Kamalahmadi and Parast (2017) divide supply chain risk into two categories: disruption and environmental risks. In fact, the risk of disruption refers to the risk that disrupts an individual supplier and the environmental risk makes a number of suppliers in a particular area inaccessible. Similarly, by increasing the likelihood of uncertainties and risks in each component of the supply chain, creating a suitable level of flexibility and reliability to deal with risks is critical. Therefore, in order to overcome supply chain risks, organizations must use appropriate strategies (Torabi et al. 2015). Moreover, due to the close relationship between the members of the supply chain with each other and decentralized nature of them, the occurrence of disruption or, in other words, risk in each part of the supply chain affects the whole chain and disrupts its performance (Baghalian et al. 2013). Thus, removing cost reduction or ignoring efficiency is not the key to managing supply chains under disruptions, but achieving supply chains that are both efficient in stable environments and reliable and flexible in unstable environments (Tolooie et al. 2020).

Therefore, this study contributes to the literature in several ways. First, we solve the problem of supplier selection and order allocation in a centralized supply chain, while in previous studies it is very rare. Second, we compared five different resilience strategies by developing three mathematical models, which have received less attention in previous studies. Third, we propose an effective framework for selecting reliable suppliers to support the resilience strategies. In recent years, the use of effective methods that provide a reasonable trade-off between cost and reliability for consumers has been considered by manufacturers (Azadeh et al. 2015). This issue is more important under disruption conditions and affects companies' strategic decisions. However, in previous studies, the combination of risk management methods and mathematical planning in a resilient supply chain has been ignored by researchers. Thus, we used the failure mode and effects analysis (FMEA) method to select reliable suppliers and then optimally allocate demand to them. Finally, we investigated the application of the proposed models using a metaheuristic algorithm in a real study.

This way, researchers intend to answer the following questions in this research:

• 1. What is the impact of budget and RPN constraints?

• 2. Which supplier is selected as the main supplier and which one is the back-up supplier?

• 3. How much restorative capacity is allocated to the main suppliers?

• 4. What percentage of unmet demand is met by main and back-up suppliers under each disruption scenarios?

To answer the research questions, we optimized the problem of supplier selection and order allocation in a centralized supply chain using mixed-integer nonlinear programming (MINLP) models. Disruption and environmental risks are considered to two reasons for suppliers’ inaccessibility. Moreover, due to the importance of suppliers' reliability, the FMEA method is implemented to evaluate each supplier's risk. The results of the FMEA method will be influential in the selection of suppliers. Suppliers with less risk priority number (RPN) value are selected as the potential suppliers. Then we expand three extensions of the Model, for the five redundancy strategies that also consider RPN value of each supplier and evaluate the supply chain profit of each extended model. In fact, we intend to show how adding risk mitigation strategies can maintain supply chain benefits by reducing the impact of disruptions. In the first model, reserving extra capacity strategy is considered for suppliers, and the model decides under RPN and budget constraints which suppliers use the reserving extra capacity strategy. Selected suppliers are the main suppliers in the next developed models. The second model considers the strategies of back-up suppliers and Emergency stock. Finally, in the last model, the strategies of protected suppliers and geographical segmentation are added.

The rest of this research is structured as follows. The next section provides a literature review of the research. In the third section, the problem and developed models are discussed. The research methodology is given in the fourth section. Accordingly, the results and related discussion along with the conclusion are discussed in the last sections, respectively.

2 Literature review

In this section, the research literature is divided into two sections. In the first section, a review of supplier selection and order allocation is provided, and in the second section, studies related to risk management and supply chain disruption are presented. Subsequently, the contributions of the present study are described.

2.1 Supplier selection and order allocation

Studies on supplier selection problems are generally divided into two main categories (Ware et al. 2014): descriptive models (using multiple attribute decision-making methods) and quantitative models (using mathematical modeling), which are discussed following.

Lo et al. (2018) solved the problem of green supplier selection and demand allocation by a combination of best–worst method (BWM) and mathematical programming. Mohammed et al. (2019) studied the problem of supplier selection and order allocation by considering sustainability criteria and an optimization approach. In the paper of Kellner and Utz (2019), by considering the objective functions of cost, sustainability, and risk, supplier selection and order allocation problem have been investigated. Mari et al. (2019) developed the qualitative criteria for selecting resilient suppliers in a fuzzy environment and responded to the orders by presenting a multi-objective mathematical model. Rezaei et al. (2020a, b) developed the problem of lean supplier selection and order allocation by combining decision-making methods and mathematical modeling. Khoshfetrat et al. (2020) integrated sustainable supplier selection and order allocation with inflation and risk characteristics in a fuzzy environment. The issue of selecting green suppliers and determining the optimal order was investigated by Feng and Gong (2020) by combining the linguistic entropy weight method and multi-objective planning. Similarly, the mentioned approaches also applied in the paper of Beiki et al. (2021) to choose sustainable suppliers and allocate demand to them. Mohammed et al. (2021) provided a combined framework of decision-making and mathematical programming methods to select the green-resilient suppliers and assign demands to them.

As mentioned, it is important to the issue of supplier selection and order allocation, taking into account the benefits/costs of the entire supply chain. Therefore, some of the researchers have integrated the issue of supplier selection and demand allocation. For example, Trivedi et al. (2017) optimized profit by using the mathematical modeling approach and providing the optimal product combination, simultaneous supplier selection, and order allocation. In the paper of Hu et al. (2018), a mutual decision model is developed to simultaneously determine suppliers' optimal selection and demand allocation. Mirzaee et al. (2018) proposed the issue of supplier selection and order allocation by considering discounts, budget, and resource constraints. In the paper of Moheb-Alizadeh and Handfield (2019), the issue of sustainable supplier selection and order allocation by considering discounts and multilateral transportation has been investigated. Jia et al. (2020) examined buyer decisions to achieve sustainable procurement by optimizing cost, ${\text{CO}}_{{2}}$ emissions, quality, and society objectives.

2.2 Supply disruption/risk management

In recent decades, research on supply chain risk management has been considered. Torabi et al. (2015) examined the issue of supplier selection and order allocation in the resilient supply chain under operational and disruption risks. Hasani and Khosrojerdi (2016) presented six risk reduction strategies to solve the problem of supplier selection and order allocation in the global supply chain. In the study of Thangam (2017), a mathematical model is proposed to optimize the supplier and retailer inventory system. This model takes into account the risks of transportation and its management while minimizing the total cost. Kamalahmadi and Parast (2017) investigated the problem of supplier selection and demand allocation in the supply chain under disruption and environmental risks. In this study, the combination of three risk reduction strategies on the performance of proposed models and supply chain costs is measured. Rezapour et al. (2017) studied supply chains that face supplier disruption and intense competition. They reduced the impact of disruptions by using emergency inventory, multiple sourcing, and excess capacity storage strategies. In the study of Arabshaibani et al. (2018), a multi-product and multi-period model with discount consideration is addressed to solve the problem of supplier selection and order allocation. In the study of Dehghani et al. (2018) a scenario-based robust model is proposed for designing a resilient supply chain under uncertainty and risks. They increased supply resilience and flexibility using three strategies to reduce risk and control inherent uncertainties. Vahidi et al. (2018) proposed a scenario-based model including sourcing, protected suppliers, and excess capacity strategies. Gong and You (2018) designed an appropriate framework for the resilient design of performance systems in the face of disruptions. In this regard, they used a multi-objective programming model and five risk reduction strategies. Jabbarzadeh et al. (2018) examined purchasing decisions and dealing with disruption by designing a sustainable supply chain network and using resilience strategies. The use of surplus inventory and protected suppliers policies to deal with disruption risk and optimize supplier selection and order allocation in a centralized supply chain was studied by Esmaeili-Najafabadi et al. (2019). Hosseini et al. (2019) examined the issue of supplier selection and demand allocation by considering the risk of disruption and presented important decisions for reactive and proactive response before and after the occurrence of the disruption. Sawik (2019) proposed an optimization approach for the multi-tier supply chain in disruption conditions. This approach has been developed in the two periods and controls the disruptions using the selection of primary and recovery suppliers and plants. Yavari and Zaker (2020) proposed a green-resilient closed-loop network for perishable products considering economic and environmental objectives. They studied the impact of disruption in the main and power network by considering five risk reduction strategies. Rezaei et al. (2020a, b) formulated a two-stage stochastic planning model under conditions of uncertainty and disruption risk. Uncertain suppliers and reserving from backup suppliers strategies were used to deal with risk. Bakhtiari Tavana et al. (2020) used a two-stage stochastic programming approach to the sustainable supplier selection and order allocation under disruption risk. The strategies of multiple sourcing and surplus production capacity are used to mitigate the risk. Sharifi et al. (2020) formulated resilience indicators for the optimal design of sustainable biofuel supply chain networks under uncertainty. Tolooie et al. (2020) investigated the issue of resilient supply chain network design at the risk of facility disruption and uncertain demand. Hasani et al. (2021) used a robust multi-objective planning approach to design a green-resilient supply chain network to balance economic, environmental and resilience issues. Kaur and Singh (2021) presented a combined framework of supplier selection and order allocation under disruption risk and Industry 4.0 environment criteria. In this study, an emergency inventory maintenance strategy is proposed to reduce and control the disruption's impact. Esmaeili-Najafabadi et al. (2021) examined the issue of supplier selection and order allocation under the local and regional disruption. They used value at risk and conditional value at risk tools to evaluate the proposed model.

Table 1 summarizes some of the recent research on supplier selection and order allocation under supply chain risks.

Table 1 Comparison of previous studies on the purchasing process under risk

In previous studies, mainly the impact of resilience strategies in the decentralized supply chain has been investigated, and this issue can affect the results of disruption risk control. Moreover, there is no effective framework that considers the reliability of suppliers in the supplier selection and order allocation process. Therefore, we optimized the problem of supplier selection and order allocation in a centralized supply chain by considering the risk of disruption and environment using three mixed-integer nonlinear programming models. The three models increase the supply chain benefits by considering the five risk reduction strategies and coordination between the buyer and the suppliers. Also, by considering the FMEA technique and RPN constraint, the reliability of suppliers has been considered. Therefore according to the previous literature gaps, the current study has the following contributions:

• Coordination between suppliers and the buyer is considered by maximizing the entire supply chain's profit instead of maximizing the buyer’s profit alone.

• Using the FMEA method to determine potential suppliers and calculate the RPN value of each supplier in order to use in RPN constraints.

• The three proposed models consider and compare different strategies to reduce the impact of risks (Protected suppliers, back-up suppliers, reserving additional capacity, emergency stock, and geographical separation).

• Satisfying the resilient criteria including responsiveness, risk reduction, back-up supplier contracting, cooperation, restorative capacity, and surplus inventory.

• Implementing the models in a real case

3 Problem description

In this study, the issue of a two-tier centralized supply chain is investigated by considering a single buyer and several suppliers. Suppliers include main and back-up suppliers, and disruption and environmental risks are considered to two reasons for suppliers’ inaccessibility. Natural disasters such as earthquakes, floods, hurricanes, etc. are considered as the environmental risk of each region and disruption of infrastructure, labor strikes, bankruptcy, currency fluctuations and political changes due to the risk of disruption of each supplier. To mitigate the effect of disruptions, several protective risk strategies have been considered. Five strategies used to handle both risks including protected suppliers, back-up suppliers, reserving additional capacity, emergency stock for back-up suppliers and geographical separation. Since the main suppliers lose part of their production capacity due to disruptions, they are considered as protected suppliers by reserving an additional capacity policy. These two strategies are presented in a general model and then developed by two models. In the second model, the strategy of back-up suppliers and maintenance of additional inventory is added and in the final model, the strategy of geographical separation is also considered. The objective function optimizes the profit of suppliers and the buyer, which leads to the optimization of the entire supply chain's profit. Also, due to suppliers' level of reliability is important, then the FMEA method is implemented to evaluate each supplier's risk. The FMEA method results will be influential in the selection of suppliers and added to the model as a constraint.

3.1 Assumption

The proposed models are based on the following assumptions:

• 1. The models are a multi-product, and its parameters are deterministic.

• 2. 2. None of the discounts are considered, and the purchase cost is fixed.

• 3. 3. Disruptions of suppliers are independent, and they do not have a ripple effect on each other.

• 4. 4. Proposed models for supplier selection and demand allocation are presented in the medium-term.

• 5. 5. In the latter model, due to geographical segmentation, the cost of maintaining additional inventory and the cost of the contract with back-up suppliers increases by 20%.

The next section presents the notations and parameters that are used in the models.

4 Notation

Notations and parameters used in the models are presented below:

Notations:

$i \in I$: The set of suppliers.

$i \in \overline{I}$: The set of main suppliers.

$i \in I^{\prime}$: The set of back-up suppliers.

$j \in J$: The set of products.

$s \in S$: The set of scenarios.

Parameters:

$O_{ij}$: Ordering cost of product j from supplier i.

$\omega_{ij}$: Unit price of product j from supplier i.

$e_{ij}$: Unit cost of holding the emergency stock for product j at supplier i.

$\alpha_{ij}$: Unit cost of extra capacity of product j from supplier i.

$C_{ij}$: Unit cost of product j for supplier i.

$\delta_{j}$: Unit penalty charge for product j.

$Pr_{j}$: Unit selling price of product j.

$F_{i}$: The cost of contract with back-up supplier i.

$bh_{i}^{{}}$: The buyer’s holding cost for product from the supplier i.

$vh_{i}^{{}}$: The i _th supplier’s holding cost.

$\rho_{i}$: The lost capacity rate of supplier i (Note that this parameter is a fraction of the local disruption probability for supplier i).

$Q_{j}$: Quantity ordered of product j.

$D_{j}$: Demand of product j.

$\theta_{i}$: The local disruption probability for supplier i.

$\theta^{*}$: The environmental disruption probability.

$\beta_{s}$: Disruption probability of scenario s.

$r_{ij}$: The portion of capacity allocated to product j of supplier i.

$cap_{i}$: The capacity of supplier i.

$Ec_{i}^{\max }$: The maximum capacity of supplier i.

$RPN_{i}$: The RPN value of each supplier.

$R^{\max } \,$: Upper bound of RPN value (Note that this parameter is a fraction of the total RPN value of each supplier).

$I_{s}$: Suppliers that are not disrupted under scenario s.

n: The upper bound of number of main suppliers.

B: The budget.

M: A big number.

Decision variables:

$x_{i}$: $\left\{ \begin{gathered} 1{\text{ if supplier i is main }} \hfill \\ 0{\text{ otherwise}} \hfill \\ \end{gathered} \right.$

$\gamma_{j}$: $\left\{ \begin{gathered} 1{\text{ If strategies are activated}} \hfill \\ 0{\text{ otherwise (If strategies are not activated)}} \hfill \\ \end{gathered} \right.$

$y_{ij}^{s}$: The portion of demands for product j allocated to back-up supplier i under scenario s.

$w_{ij}^{s}$: The portion of demands for product j allocated to main supplier i under scenario s.

$z_{ij}^{s}$: The emergency stock of product j that is holding at back-up supplier I (Note that this variable is a fraction of the i-th supplier capacity).

$ec_{i}^{s}$: The extra capacity of main supplier i under scenario s.

$U_{j}^{s}$: Unmet demand of product j under scenario s.

4.1 Computation of disruption probability

The impact of risks and disruptions on parts supply and the procurement process can have far-reaching effects. Some of the risks, called disruption risk, that result from the impact of the disruption on one supplier. If we assume that the risk of disruption for the supplier i is $\theta_{i}$, then it will be able to deliver orders with a probability of (1-$\theta_{i}$). Another type of risk that is usually less probability to occur but has far-reaching and long-term consequences is called environmental risk. This risk can make all suppliers in an area inaccessible at the same time.

In order to calculate the probability of each scenario, we define the binary parameters ${\text{Re}}_{s,r}$ as the state of the region under the scenario and $S_{s,i}$ as the status of supplier i under scenario s. If region or supplier is available, the parameters will be one and otherwise zero. Also, as mentioned earlier, the parameter $\theta_{r}^{*}$ shows the environmental risk of each region. Thus the probability of each scenario is calculated as Eq. (1) (Kamalahmadi and Parast 2017):

$$B_{s} = \prod\nolimits_{r \in R} {\left[ {\left( {1 - {\text{Re}}_{s,r} } \right)\theta_{r}^{*} + {\text{Re}}_{s,r} \left( {1 - \theta_{r}^{*} } \right)\prod\nolimits_{i \in I} {\left( {(1 - S_{s,i} )\theta_{i} + S_{s,i} (1 - \theta_{i} )} \right)} } \right]}$$

(1)

4.2 Models formulation

This section is divided into three sections to show the differences between the models. The first section clarifies the reserving extra capacity model for supplier selection and order allocation under disruption and environmental risks and extends this model. In the next section, the first model is developed and the strategies of back-up suppliers and maintaining additional inventory are added to it. Finally, in the last part, the developed model is presented considering all the strategies.

4.2.1 Reserving additional capacity model

In this model, it is assumed that all suppliers are located in one geographical area and the environmental disruption in this area is high. This environmental disruption, along with the disruption risk of any supplier, causes suppliers in the area to lose much of their capacity. Hence the multi-sourcing policy and the reserving additional capacity are considered for the region's suppliers. This strategy allows selected suppliers to be used as primary and protected suppliers in future models. So in the following, the model is presented to increase supply chain profits.

Buyer’s profit:

$${\text{G}}^{b} = \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} \left( {1 - U_{j}^{s} } \right)Pr_{j} } } - \left[ \begin{gathered} \sum\limits_{i \in I} {\sum\limits_{j \in J} {{\raise0.7ex\hbox{${D_{j} O_{ij} x_{i} }$} \!\mathord{\left/ {\vphantom {{D_{j} O_{ij} x_{i} } {Q_{j} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${Q_{j} }$}}} } + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} w_{ij}^{s} \omega_{ij} + } } } \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} w_{ij}^{s} bh_{i}^{{}} }$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} w_{ij}^{s} bh_{i}^{{}} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } \hfill \\ + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right]$$

(2)

Profit of the i-th supplier:

$${\text{G}}_{i}^{v} = \sum\limits_{j \in J} {\sum\limits_{s \in S} {\left( {D_{j} w_{ij}^{s} \omega_{ij} - C_{ij} D_{j} w_{ij}^{s} - {\raise0.7ex\hbox{${Q_{j} w_{ij}^{s} vh_{i}^{{}} }$} \!\mathord{\left/ {\vphantom {{Q_{j} w_{ij}^{s} vh_{i}^{{}} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \right)} }$$

(3)

In the Eqs. (2 and 3), the economic order quantity (EOQ) concepts are used to the problem of selection (Esmaeili-Najafabadi et al. 2019). So, the total profit of supply chain is:

$$\begin{gathered} {\text{G}}^{sc} {\text{ = G}}^{b} + \beta_{s} \sum\limits_{i \in I} {{\text{G}}^{v} } = \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} \left( {1 - U_{j}^{s} } \right)Pr_{j} } } \hfill \\ - \left[ \begin{gathered} \sum\limits_{i \in I} {\sum\limits_{j \in J} {{\raise0.7ex\hbox{${D_{j} O_{ij} x_{i} }$} \!\mathord{\left/ {\vphantom {{D_{j} O_{ij} x_{i} } {Q_{j} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${Q_{j} }$}}} } + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )}$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } \hfill \\ + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(4)

According to the article of Esmaeili-Najafabadi et al. (2019), the above equation is convex with respect to $Q_{j}$. Therefore, in order to find the maximum value of the supply chain profit, we derive from Eq. (4) with respect to $Q_{j}$ and set it to zero.

$$\frac{{\partial G^{sc} }}{{\partial Q_{j} }} = D_{j} \sum\limits_{i \in I} {{\raise0.7ex\hbox{${O_{ij} x_{i} }$} \!\mathord{\left/ {\vphantom {{O_{ij} x_{i} } {Q_{j}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${Q_{j}^{2} }$}}} - \sum\limits_{i \in I} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )}$} \!\mathord{\left/ {\vphantom {{\beta_{s} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} }$$

(5)

$$\Rightarrow Q_{j} = \sqrt {\frac{{2D_{j} \sum\limits_{i \in I} {O_{ij} x_{i} } }}{{\sum\limits_{i \in I} {\sum\limits_{s \in S} {\beta_{s} w_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right)} } }}}$$

(6)

Consequently, the final model can be formulated as following:

$${\text{Max G}}^{sc} { = }\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} \left( {1 - U_{j}^{s} } \right)Pr_{j} } } - \left[ \begin{gathered} \sum\limits_{j \in J} {\sqrt {2D_{j} \sum\limits_{i \in I} {O_{ij} x_{i} } \times \sum\limits_{i \in I} {\sum\limits_{s \in S} {\beta_{s} w_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right)} } } } + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } \hfill \\ + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right]$$

(7)

s.t.

$$\sum\limits_{i \in I} {(w_{ij}^{s} + ec_{ij}^{s} ) + U_{j}^{s} = 1} \, \forall j \in J,s \in S$$

(8)

$$ec_{ij}^{s} \le {\text{x}}_{i} \, \forall i \in I,j \in J,s \in S$$

(9)

$$ec_{ij}^{s} D_{j}^{{}} \le r_{ij} Ec_{i}^{\max } \, \forall i \in I,j \in J,s \in S$$

(10)

$$D_{j} w_{ij}^{s} \le \left( {r_{ij} cap_{i} (1 - \rho_{i} )} \right){\text{x}}_{i} \, \forall i \in I,j \in J,s \in S$$

(11)

$$\sum\limits_{i \in I} {{\text{x}}_{i} } \le n \,$$

(12)

$$\sum\limits_{i \in I} {RPN_{i} } \times x_{i} \le R^{\max } \,$$

(13)

$$\left[ \begin{gathered} \sum\limits_{j \in J} {\sqrt {2D_{j} \sum\limits_{i \in I} {O_{ij} x_{i} } \times \sum\limits_{i \in I} {\sum\limits_{s \in S} {\beta_{s} w_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right)} } } } + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } \hfill \\ + \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right] \le B \,$$

(14)

$$x_{i} \in \left\{ {0,\left. 1 \right\}} \right., \, w_{ij}^{s} ,U_{j}^{s} ,ec_{ij}^{s} \ge 0 \, \forall i \in I,j \in J,s \in S$$

(15)

The objective function (7) is to maximize the supply chain profit. This function includes the cost of ordering, purchasing, reserving extra capacity and the penalty of unmet demands. The constraint (8) shows the allocation of orders between suppliers. The constraint (9) is satisfied when the ordered products are allocated to the selected suppliers. The constraint (10) ensures that extra capacity cannot be more than the extra capacity available to suppliers. The constraint (11) shows that demand met by additional capacity cannot exceed the capacity of each selected supplier. The constraint (12) shows the upper bound of number of main suppliers. The constraints (13) and (14) show the constraint of the RPN and the budget limitation. The constraint (15) specifies that the variables of the problem are nonnegative.

4.2.2 Back-up and emergency stock model

In this section, the previous model has been developed and the policy of using back-up suppliers and emergency inventory has been added. It is noteworthy that in this model, all main and back-up suppliers are located in the same geographical area. As mentioned in the previous section, this area has a high environmental risk and the back-up suppliers are completely disrupted after the disruption. In this case, the back-up suppliers can only use the emergency inventory to meet the buyer's demand. Given these assumptions, the model presented is as follows:

Buyer’s profit:

$$\begin{gathered} {\text{G}}^{b} = \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} w_{ij}^{s} Pr_{j} } } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} y_{ij}^{s} Pr_{j} } } } - \sum\limits_{i \in I} {\sum\limits_{j \in J} {{\raise0.7ex\hbox{${D_{j} O_{ij} x_{i} }$} \!\mathord{\left/ {\vphantom {{D_{j} O_{ij} x_{i} } {Q_{j} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${Q_{j} }$}}} } - \sum\limits_{{i \in I^{\prime}}} {F_{i} x_{i} } \hfill \\ - \left[ \begin{gathered} \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} w_{ij}^{s} \omega_{ij} + } } } \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} y_{ij}^{s} \omega_{ij} + } } } \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} w_{ij}^{s} bh_{i}^{{}} }$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} w_{ij}^{s} bh_{i}^{{}} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} y_{ij}^{s} bh_{i}^{{}} }$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} y_{ij}^{s} bh_{i}^{{}} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } \hfill \\ + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} e_{ij}^{{}} r_{ij} cap_{i}^{{}} z_{ij}^{s} } } } + \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(16)

Profit of the i-th supplier:

$${\text{G}}_{i}^{v} = \sum\limits_{j \in J} {\sum\limits_{s \in S} {\left( {D_{j} (w_{ij}^{s} + y_{ij}^{s} )\omega_{ij} - C_{ij} D_{j} (w_{ij}^{s} + y_{ij}^{s} ) - {\raise0.7ex\hbox{${Q_{j} (w_{ij}^{s} + y_{ij}^{s} )vh_{i}^{{}} }$} \!\mathord{\left/ {\vphantom {{Q_{j} (w_{ij}^{s} + y_{ij}^{s} )vh_{i}^{{}} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \right)} }$$

(17)

Therefore, the total supply chain profit is:

$$\begin{gathered} {\text{G}}^{sc} {\text{ = G}}^{b} + \beta_{s} \sum\limits_{i \in I} {{\text{G}}^{v} } = \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} w_{ij}^{s} Pr_{j} } } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} y_{ij}^{s} Pr_{j} } } } - \sum\limits_{i \in I} {\sum\limits_{j \in J} {{\raise0.7ex\hbox{${D_{j} O_{ij} x_{i} }$} \!\mathord{\left/ {\vphantom {{D_{j} O_{ij} x_{i} } {Q_{j} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${Q_{j} }$}}} } - \sum\limits_{{i \in I^{\prime}}} {F_{i} x_{i} } \hfill \\ - \left[ \begin{gathered} \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )}$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} w_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {{\raise0.7ex\hbox{${\beta_{s} Q_{j} y_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )}$} \!\mathord{\left/ {\vphantom {{\beta_{s} Q_{j} y_{ij}^{s} (bh_{i}^{{}} + vh_{i}^{{}} )} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} } } + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } \hfill \\ + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} } } } C_{ij} D_{j} y_{ij}^{s} + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} + } } } \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} e_{ij}^{{}} r_{ij} cap_{i}^{{}} z_{ij}^{s} } } } + \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(18)

As mentioned, the total benefit should be derived from Q and then the final model is presented as following:

$$\begin{gathered} {\text{Max G}}^{sc} = \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} w_{ij}^{s} Pr_{j} } } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} D_{j} y_{ij}^{s} Pr_{j} } } } - \sum\limits_{{i \in I^{\prime}}} {F_{i} x_{i} } \hfill \\ - \left[ \begin{gathered} \sum\limits_{j \in J} {\sqrt {2D_{j} \sum\limits_{i \in I} {O_{ij} x_{i} } \times \sum\limits_{{i \in \overline{I}}} {\sum\limits_{s \in S} {\beta_{s} w_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right) + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{s \in S} {\beta_{s} y_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right)} } } } } } + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } \hfill \\ + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} } } } C_{ij} D_{j} y_{ij}^{s} + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} } } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} e_{ij}^{{}} r_{ij} cap_{i}^{{}} z_{ij}^{s} } } } + \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(19)

s.t.

Constraint (9)-(11) and (13)

$$x_{i} \le \sum\limits_{j \in J} {\gamma_{j} } \, \forall i \in I^{\prime}$$

(20)

$$z_{ij}^{s} \le {\text{x}}_{i} \, \forall i \in I^{\prime},j \in J,s \in S$$

(21)

$$U_{j}^{s} + \sum\limits_{{i \in \overline{I}}} {(w_{ij}^{s} + ec_{ij}^{s} ) + \sum\limits_{{i \in I^{\prime}}} {y_{ij}^{s} = 1} } \, \forall j \in J,s \in S$$

(22)

$$z_{ij}^{s} \le r_{ij} cap_{i} \, \forall i \in I^{\prime},j \in J,s \in S$$

(23)

$$D_{j} y_{ij}^{s} \le \left( {r_{ij} cap_{i} z_{ij}^{s} } \right){\text{x}}_{i} \, \forall i \in I^{\prime},j \in J,s \in S$$

(24)

$$\sum\limits_{{i \in \overline{I}}} {\left( {r_{ij} cap_{i} (1 - \rho_{i} ) + ec_{ij}^{s} } \right) \ge D_{j} - M \times \gamma_{j} } \, \forall j \in J,s \in S$$

(25)

$$\sum\limits_{{i \in \overline{I}}} {\left( {r_{ij} cap_{i} (1 - \rho_{i} ) + ec_{ij}^{s} } \right) < D_{j} + M \times (1 - \gamma_{j} )} \, \forall j \in J,s \in S$$

(26)

$$\left[ \begin{gathered} \sum\limits_{j \in J} {\sqrt {2D_{j} \sum\limits_{i \in I} {O_{ij} x_{i} } \times \sum\limits_{{i \in \overline{I}}} {\sum\limits_{s \in S} {\beta_{s} w_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right) + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{s \in S} {\beta_{s} y_{ij}^{s} \left( {bh_{i}^{{}} + vh_{i}^{{}} } \right)} } } } } } \hfill \\ + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} C_{ij} D_{j} w_{ij}^{s} } } } + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} } } } C_{ij} D_{j} y_{ij}^{s} + \sum\limits_{{i \in \overline{I}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} ec_{ij}^{s} D_{j} \alpha_{ij} } } } \hfill \\ + \sum\limits_{{i \in I^{\prime}}} {\sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} e_{ij}^{{}} r_{ij} cap_{i}^{{}} z_{ij}^{s} } } } + \sum\limits_{j \in J} {\sum\limits_{s \in S} {\beta_{s} \delta_{j} U_{j}^{s} D_{j} } } + \sum\limits_{{i \in I^{\prime}}} {F_{i} x_{i} } \hfill \\ \end{gathered} \right] \le B \,$$

(27)

$$x_{i} ,\gamma_{j} \in \left\{ {0,\left. 1 \right\}} \right., \, y_{ij}^{s} ,w_{ij}^{s} ,z_{ij}^{s} ,U_{j}^{s} ,ec_{ij}^{s} \ge 0 \, \forall i \in I,j \in J,s \in S$$

(28)

The objective function (19) is to maximize the supply chain profit. This function includes the cost of ordering, purchasing, keeping an emergency inventory, reserving extra capacity, contract with back-up suppliers, and the penalty of unmet demands. The constraint (20) indicates how the reserving emergency inventory strategy for back-up suppliers is activated. The Constraint (21) guarantees that emergency stock will be allocated when the back-up supplier is selected. The constraint (22) shows the allocation of orders between main and back up suppliers. The constraint (23) ensures that emergency stock cannot be more than the capacity available to suppliers. The constraint (24) shows that demand met by emergency inventory cannot exceed each back-up suppliers' capacity. The constraints (25) and (26) show that risk reduction strategies are activated if demand exceeds main suppliers' capacity. The constraint (27) shows the budget limitation. The constraint (28) indicates that the variables of the problem are nonnegative.

4.2.3 Segmentation region model

In this section, the proposed model includes geographic segmentation strategy in addition to previous strategies. In fact, in this model, two separate geographical areas are considered. The first geographical area is the same area as the previous models and as mentioned, this area has a high risk of disruption and environment and causes the complete disruption of back-up suppliers. Therefore, in order to use the main capacity of the back-up suppliers, these suppliers are located in another geographical area that has a low environmental risk. Therefore the back-up suppliers lose part of their main capacity after the disruption. Also back-up suppliers can also take advantage of additional inventory policy. Locating back-up suppliers in a low-risk environment and using an additional inventory maintenance policy will result in a protected back-up policy. In this model, the capacity constraint of back-up suppliers and how to calculate the probability of disruption scenarios has changed.

$${\text{Max G}}^{sc}$$

s.t.

Constraint (9–11 and 13).

Constraint (20–23)

$$D_{j} y_{ij}^{s} \le \left( {r_{ij} cap_{i} (1 - \rho_{i} ) + z_{ij}^{s} } \right){\text{x}}_{i} \, \forall i \in I^{\prime},j \in J,s \in S$$

(29)

Constraint (25–28).

5 Solution methods

The proposed methodology consists of three steps for supplier risk assessment presented by Hu et al. (2009). The first phase refers to the identification of criteria in an appropriate structure based on the FMEA method. The second phase involves calculations related to the analytic hierarchy process (AHP) and determines the importance of each criterion. Finally, by combining the two methods, the final risk score of each supplier is evaluated and calculated. Moreover small size of the problems is solved with exact method and due to the long time, which is required for solving large size problem with exact methods; GOA method is employed for solving our model in large size.

5.1 Failure modes and effects analysis (FMEA)

In recent years, various methods for risk assessment have been developed. One of these methods is the FMEA method. This method first used in the aerospace industry to systematically analyze failure modes and their effects in military products, and then developed in the automotive industry (Puente et al. 2002). The most important purpose of applying the FMEA method is to identify potential failure modes in system components, determine the causes, evaluate their effects on system performance and finally determine the ways in which reduced the chances of occurrence and consequences and increased the ability to detect failure modes (Stamatis 2003). In the conventional FMEA method, the RPN is used for risk calculation, and it calculated by Multiplying three risk factors: Occurrence (O), Severity (S), and Detection (D) that each of these is rated with numbers between 1 and 10 (Liu et al. 2013).

$$RPN = (S) \times (O) \times (D)$$

(30)

In this study, we applied this technique for evaluating the risk of each supplier. Suppliers with less RPN value are selected as the potential suppliers and RPN value each supplier used in RPN constraint.

5.2 Identifying the risk criteria of FMEA

First, we using the criteria extracted from articles Abbasgholizadeh Rahimi et al. (2015) and Zammori and Gabbrielli (2012) establish the hierarchical structure based on the purpose of the problem and its application in FMEA method (according to Fig. 1). In this structure, considering the purpose of the problem, each of the risk factors is divided into criteria to make a more appropriate comparison. Therefore, the severity factor is presented with product losses, reduced quality, damages and failure costs criteria, and occurrence and detection factors are presented with frequency, infrastructure failure, method of systematic detection and hardness of proactive inspection, respectively.

Fig. 1

Hierarchy structure for supplier risk assessment

Descriptions related to each of the mentioned criteria are given in Tables 2, 3 and 4, so that they can be used to make appropriate scoring. Table 5 also shows how to score the criteria.

Table 2 FMEA severity assessment criteria

Table 3 FMEA occurrence assessment criteria

Table 4 FMEA detection assessment criteria

Table 5 The scale of FMEA criteria

5.3 Determining the weights of risk criteria of FMEA by AHP

The AHP was utilized to determine the weights of eight criteria by three decision makers (DM). The comparison judgments of the eight main criteria concerning the overall goal are performed. Finally, the weights obtained are used in the final calculations. A brief description of the AHP method is provided below.

5.3.1 Analytic hierarchy process (AHP)

The AHP method for calculating the weights of a set of factors and prioritizing them was introduced by Saaty (1980). This method is calculated in four steps:

• Step 1 Establishing the hierarchical structure:

At this stage, according to the problem under study, the decision criteria are arranged in a hierarchical structure. In hierarchical structure design, the goal is shown at the highest level, the criteria at the second level, and the alternatives at the last level.

• Step 2 Constructing a pairwise comparison matrix:

In this step, the elements of each level are compared to each of the higher level elements using pairwise comparison matrix. Of course, it should be noted that the formation of a pairwise comparison matrix is necessary for qualitative criteria, and for quantitative criteria, it is enough for their normalized values to enter the decision matrix.

• Step 3 Calculating the local weights of criteria and alternatives:

At this stage, each element's local weight relative to the higher level is extracted by completing the pairwise comparisons matrix and performing the relevant calculations.

• Step 4 Calculating the consistency:

This step must be performed simultaneously with the third step. According to this step, if the rate of inconsistency is greater than 0.1, each pair comparison matrix must be modified by the decision-maker.

• Step 5 Calculating the final weights of the alternatives:

The final weights of the alternatives are obtained by multiplying the local weight of each alternative respect to the criterion in the local weight of the relevant criterion.

5.4 The formula for S-RPN

Finally, the weights of the 8 criteria were determined using AHP, and the formation of S-RPN was used to calculate the risks with regard to each supplier. The formula of calculation for S-RPN is shown in Eq. (31) (Hu et al. 2009).

$$(S - RPN)_{i} = W_{{(S_{1} )}} \times S_{{i(S_{1} )}} + W_{{(S_{2} )}} \times S_{{i(S_{2} )}} + ... + W_{{(O_{1} )}} \times S_{{i(O_{1} )}} + ... + W_{{(D_{2} )}} \times S_{{i(D_{1} )}}$$

(31)

5.5 Grasshopper metaheuristic algorithm (GOA)

The grasshopper metaheuristic algorithm (GOA) was proposed by Saremi et al. (2017) by mimicking the behavior of grasshopper's populations. The main components of this algorithm include social interaction $(S_{i} )$, gravity $(G_{i} )$ and wind trend $(A_{i} )$. In this method, social interaction is the basis of the search process and is defined as follows.

$$S_{i} = \sum\limits_{j = 1,j \ne i}^{N} {s(d_{ij} )\hat{d}_{ij} }$$

(32)

$d_{ij}$ is the distance between grasshopper i and j and it calculated as $d_{ij} = \left| {x_{j} - x_{i} } \right|$. $\hat{d}_{ij} = \frac{{x_{j} - x_{i} }}{{d_{ij} }}$ is a unit vector from distance between grasshopper i and j. s is a function of social interaction affecting the behavior of grasshoppers, which is defined as Eq. (33):

$$s(r) = fe^{{\frac{ - r}{l}}} - e^{ - r}$$

(33)

The parameter f is the gravitational intensity, and the l is the attractive length scale. This function creates the force of gravity and repulsion between the grasshoppers and changing it will significantly impact the behavior of the grasshoppers. Therefore, setting the swarm model is important for designing the GOA algorithm. This model simulates the interaction between grasshoppers and is presented as follows:

$$X_{i}^{d} = c\left( {\sum\limits_{j = 1,j \ne i}^{N} {c\frac{{ub_{d} - lb_{d} }}{s}s\left( {\left| {x_{j}^{d} - x_{i}^{d} } \right|} \right)\frac{{x_{j} - x_{i} }}{{d_{ij} }}} } \right) + \hat{T}_{d}$$

(34)

Parameters $ub_{d}$ and $lb_{d}$ show the upper and the lower limit of d -th dimension, respectively, $\hat{T}_{d}$ is the value of d -th dimension in the best solution, and parameter c is defined as the main parameter of controlling the repulsion and attraction zone as follows:

$$c = c_{\max } - l\frac{{c_{\max } - c_{\min } }}{L}$$

(35)

L is the maximum number of iterations and l is the present iteration. The parameters $c_{\max }$ and $c_{\min }$ are equal to 1 and $0.00001$ respectively.

6 Numerical example

6.1 Computational experiment

This section examines the performance of the proposed MINLP model for risk management in the supplier selection and demand allocation process. For this purpose, first a numerical example in small size is solved using the exact method and its results are discussed and evaluated. The issue has been investigated in small size with 5 suppliers and 2 products and its input data are given in Appendix 1. It should be noted that the RPN values used in this size have been calculated using case study data. Exact method is implemented using solver BARON in GAMS24.1.2 running on laptop with Core i5 CPU and 4 GB of RAM. Also the problem is examined in a case study and the size of the problem increases to 15 suppliers and 8 products. Due to the increase dimensions of the problem more time is required to solve it with the exact method. So it is necessary to use the meta-heuristic method, and then the GOA has been used for this purpose.

6.2 Parameters tuning

In order to use meta-heuristic algorithms more effectively, it is necessary to estimate its parameters at the appropriate level. For this purpose, the Taguchi method is used to adjust the parameters. Taguchi results for tuning some GOA parameters such as iteration and number of grasshoppers are presented in Table 6.

Table 6 State table and Taguchi analyze parameters

According to the number of factors (parameters) and their state, it is necessary to design 9 experiments. These experiments and their results are reviewed in the Minitab software for model 1 and 3 and then the results are shown in Fig. 2a and b. For the analysis of Fig. 2a and b, if the objective minimizes the variables, the lowest state is used and if the objective maximizes the variables, the highest state is used. According to the nature of the experiment, we intend to have the shortest distance from the ideal value, so the purpose of the experiment is to minimize. According to Fig. 2a, the best parameters which help to the efficiency of the algorithm for first model, are selected to use in the algorithm that is iteration = 50 and npop = 50. Moreover, according to Fig. 2b best parameters is iteration = 60 and npop = 50.

Fig. 2

a Taguchi’s result for model 1 b Taguchi’s result for model 3

6.3 Model validation

In order to ensure the validity of the model, a sensitivity analysis is performed on the demand and the capacity of each supplier examines its impact on unmet demand and the number of suppliers. Obviously, by increasing the demand, unmet demand or extra capacity and emergency inventory will increase and by decreasing the demand will be expected that unmet demand or extra capacity and emergency inventory will decrease. Figure 3 indicates the sensitivity analysis for increasing and decreasing the demand. It shows that as demand increases in the first model, unmet demand increases. Also, in the second and third models, emergency inventory maintenance is increased by increasing the number of backup suppliers. However, if demand decreases, the opposite will happen (see Fig. 3). Moreover, Fig. 4 indicates the sensitivity analysis for increasing and decreasing the capacity of each supplier. It shows that as capacity increases in the first model, unmet demand decreases. Also, in the second and third models, emergency inventory maintenance is decreased by decreasing the number of back-up suppliers. However, if capacity decreases, the opposite will happen.

Fig. 3

increase and decrease of the demand

Fig. 4

increase and decrease of the capacity

7 Result and discussion

In this section, the results of the FMEA method are presented and then each model is solved in small and large sizes. First, solving the models in small size is presented and then it solved by the GOA, and the results are compared. Finally, the case study is solved with a GOA and the results are presented.

7.1 The result of FMEA

As mentioned, in order to calculate the risk values of each supplier, it is necessary to form a matrix of pairwise comparisons. Therefore, the three industrial experts as decision-makers performed pairwise comparisons according to the hierarchical structure given in the Appendix 2. All these experts are familiar with risk management and its tools within their expertise field and they are faced with the issue of supplier selection. Moreover, we considered the importance of all three decision-makers the same. The matrix resulting from the aggregation of decision makers' opinions is given in Tables 7, 8, 9 and 10. We aggregated the all preferences expressed by the three decision-makers by the geometric mean method. In the following, the weights of each of the risk sub-factors are calculated according to AHP method and are given in Table 1. Finally, potential suppliers of an Automotive parts company selected as a case study are scored based on each of the sub-risk factors, and the final values are calculated according to Eq. (31) and presented in Table 12.

Table 7 Pairwise comparison matrix of FMEA risk factors

Table 8 Pairwise comparison matrix of risk sub-factors with regard to severity

Table 9 Pairwise comparison matrix of risk sub-factors with regard to occurrence

Table 10 Pairwise comparison matrix of risk sub-factors with regard to detection

Table 11 Weights of risk sub-factors

Table 12 Final weights of suppliers with regard to risk sub-factors

7.2 The result of small size

7.2.1 Reserving additional capacity model

In order to solve the models, it is first necessary to calculate the probability of disruption scenario. Therefore, this value is calculated by Eq. (1) and shown in Table 13.

Table 13 Scenario calculation

The results of solving the model are given in Table 14 and Fig. 5. Figure 5 shows the schematic view of the chain and the values of the variables.

Table 14 Solution results for the small size of the first model

Fig. 5

Schematic view of the results according to the small size solution of the first model

According to the results shown in Figs. 6, 7, 8, 9 in case only the reserving extra capacity strategy is used, the proposed model selects the suppliers based on the balance of the RPN value, purchase cost and local disruption. In addition, suppliers 2 and 4 have lower reserving extra capacity cost and local disruption than other suppliers. Supplier 7 has responded to more parts of the demand with extra capacity because it has more lost capacity due to the higher risk of disruption (see Fig. 9).

Fig. 6

Basic features of suppliers (RPN and disruption probability)

Fig. 7

Basic features of suppliers (RPN and purchase and reserving costs)

Fig. 8

Basic features of suppliers (purchase and reserving costs and disruption probability)

Fig. 9

The amount of demand that met by selected suppliers

7.2.2 Back-up and emergency stock model

As mentioned, the suppliers selected in the first model are considered as the main suppliers and the rest are considered as the back-up suppliers. The results of the model are shown in Table 15 and Fig. 10.

Table 15 Solution results for the small size of the second model

Fig. 10

Schematic view of the results according to the small size solution of the second model

According to the results of the second model, the selected main suppliers have an appropriate balance in the RPN value, purchase cost and local disruption (see Figs. 6, 7, 8). Moreover, the maintenance cost and RPN value for supplier 7 is higher than suppliers 2, 4, so the additional capacity allocated to it in the previous model was allocated to the backup supplier as an emergency inventory (see Fig. 11). In this case, a more appropriate balance is achieved in the chain and its benefit is increased. Also, among back-up suppliers, supplier 6 have been selected, which has less RPN value and cost of contract and purchase, which shows that RPN and costs have been key elements in selecting a supplier.

Fig. 11

The amount of demand that met by selected suppliers

7.2.3 Segmentation region model

The results of the model are shown in Table 16 and Fig. 12.

Table 16 Solution results for the small size of the third model

Fig. 12

Schematic view of the results according to the small size solution of the third model

According to the results shown in this section, the environmental disruption of the regions has acted as a key element and the back-up suppliers have responded to a larger share of the demand (see Fig. 13). Also, as shown in Figs. 6, 7, 8, the selected suppliers (main and back-up) have an appropriate balance of the RPN value, purchase cost, and local disruption. In addition, among the main suppliers, only suppliers 2,4 have been selected because it is in a better balance of the RPN value, purchase cost, and local disruption than supplier 7. Also, in this model, the back-up suppliers do not lose their main capacity, therefore the main suppliers don't keep extra capacity, which has led to an increase in the supply chain's profit.

Fig. 13

The amount of demand that met by selected suppliers

Chopra and Sodhi (2014) in a survey study showed that risk management and mitigation strategies are not cost-effective due to high investment requirements. However, the results of the current study show that the use of risk reduction strategies not only protect the supply chain from risks and uncertainties, but also increases profit of supply chain by increasing resilience and reliability. In fact the use of disruptive supply chain risk reduction strategies enables the supply chain to a certain level of confidence is able to meet the demand when the disruption occurs. As shown in Fig. 14, the use of strategies in each model has increased the profitability of the chain. These results show that the use of the third model, which is able to consider several strategies, significantly increases the benefit of the chain.

Fig. 14

Comparison of expected total profit of each model

7.3 Comparison of exact and metaheuristic methods

In order to validate the GOA for the proposed models, we solve two problems of small size by it. Therefore, in this section, the results of solving the first and third models in small size are presented and compared with the results of exact solution (according to Tables 17 and 18). The convergence diagrams of the results are shown in Figs. 15 and 16. Due to the acceptable difference between the results of the meta-heuristic method and the exact method, the relevant algorithm can be used to solve models in large size.

Table 17 Comparison of the results of solving the first model in small size using exact and GOA methods

Table 18 Comparison of the results of solving the third model in small size using exact and GOA methods

Fig. 15

Convergence diagram of the first model in small size

Fig. 16

Convergence diagram of the third model in small size

7.4 Case study

In this section, in order to evaluate the performance of the proposed models, a case study of Iran is presented and solved. The company under study is an automotive parts company. The company has 15 suppliers to supply 8 pieces of its products due to unforeseen disruptions. These suppliers are selected and ordered according to their current situation. The problem faced the company is that it wants to decide which of the suppliers is the main supplier and which one is the back-up, according to the conditions of each supplier in relation to the risk of disruption and environmental risk. Suppliers are segregated in different geographical areas, and the environmental risk of each area according to the decision-maker opinion is 0.3, 0.05 and 0.07, respectively. Therefore, according to the first model, the main suppliers and the optimal values of their extra capacity are determined. Then, considering the third model and the strategy of using back-up suppliers and emergency inventory, back-up suppliers and their emergency inventory are also determined. The case study also intends to respond to a higher level of demand by increasing the reliability of suppliers. Therefore, using the proposed research method, each supplier's RPN values are calculated (see Table 12). The rest of the input information is given in Appendix 3. It should be noted that the ordering cost is equal to 10% of the purchase cost and the available budget is 200,000,000 currency units.

Due to the large size of the case study problem the GOA has been used to solve it. The results of the solution are given in Tables 19 and 20. The convergence diagrams of the first and second models are also shown in Figs. 17 and 18, respectively.

Table 19 Solution results for first model

Table 20 Solution results for third model

Fig. 17

Convergence diagram of the first model in large size

Fig. 18

Convergence diagram of the third model in large size

According to the results, the selected suppliers are at an appropriate level of reliability, cost and risk of disruption. In the first model, due to the loss of part of the capacity of the main suppliers, most of the demand remains unmet. But in the next model, using the other strategy, the supply chain will be able to respond to demand appropriately and make the right decision in selecting suppliers, taking into account reliability, cost, and environmental and disruption risks. This will significantly increase the benefits of the supply chain (see Fig. 19).

Fig. 19

Comparison of total profit of each model

8 Sensitivity analysis

This section is divided into several sub-sections. The first part examines the changes in the parameter θ and its effect on the solution results. In the second subsection, the proposed models are solved by ignoring the RPN constraint, and the results are presented. Then, in the third and fourth parts, changes in the inventory maintenance cost parameter and changes related to the cost of reserving additional capacity are examined, respectively. Finally, in the last section, the effects of changes in the lost capacity rate and the maximum capacity of the main supplier presented separately and in combination.

8.1 Change in the parameter θ

In this section, in order to examine the effect of the disruption probability parameter, we examine its changes in values less than 0.1 and more than 0.2 in each of the proposed models. The results are listed in the next section, respectively.

8.1.1 Reserving additional capacity model

First, we set the parameter θ for each of the suppliers in values less than 0.1, and then we solved the model for values greater than 0.2. According to the results shown in Fig. 20, when the risk of disruption is small, and the values are less than 0.1, the amount of additional capacity has decreased compared to the optimal state. Also the key parameters in selecting suppliers are the amount of RPN and costs. As the parameter θ decreases, suppliers are able to maintain their main capacity at a better level and reserve less additional capacity.

Fig. 20

The amount of demand that met by selected suppliers

Moreover, when the risk of disruption of each supplier increases, the key parameters in selecting suppliers are the costs and disruption risk. Because in this situation, a large part of the main capacity of suppliers has been lost and they need to maintain additional capacity, so the model tends to select suppliers with the risk of disruption and lower cost so that it can meet higher demand by gaining more profit. As shown in Fig. 20, the main suppliers responded to a larger share of demand using additional capacity, and suppliers who had a higher risk of disruption were only able to respond a small share of demand with their main capacity.

Also, if a disruption risk is less than 0.1, the model can fully meet the demand, but in quantities greater than 0.2, more than part of the demand is not met. Therefore, as shown in Fig. 21, the profit of the supply chain in $\theta \le 0.1$ has increased and in $\theta \ge 0.2$ has decreased.

Fig. 21

Comparison of expected total profit

8.1.2 Back-up and emergency stock model

In this section, the results of changing this parameter and its effect on the variables and the objective function in the second model are examined. In this model, by changing θ to values less than 0.1, the key parameters in selecting suppliers are RPN value and costs. By reducing the risk of disruption, the model can fully respond to demand, so backup suppliers are not selected (see Fig. 22). Moreover, as the risk of each supplier's disruption increases, more suppliers are selected to meet demand. As the risk of disruption increases, suppliers' main capacity is disrupted and it is necessary to use risk reduction strategies to increase chain profits.

Fig. 22

The amount of demand that met by selected suppliers

In the second model, the benefit of the chain is increased by reducing the risk of disruption and not using back-up suppliers. Also with the increased risk of disruption and the use of backup suppliers, a part of the demand that was not answered has been met, so the benefit of the supply chain is improved compared to the first model (compare Figs. 21 and 23).

Fig. 23

Comparison of expected total profit

8.1.3 Segmentation region model

Finally, in this section, the results of parameter θ and its effect on the variables and the objective function in the last model are examined. In the latter model, by reducing the risk of disruption, the model tends to use the main capacity of the main suppliers and, as shown in Fig. 24, the excess capacity is less allocated to the main suppliers. Also, because back-up suppliers are at lower environmental risk, the model tends to meet unmet demand after disruption. It should also be noted that the selected suppliers are in equilibrium conditions in terms of RPN value, θ and cost. When the risk of disruption increases, the proposed model tends to respond to a larger share of demand using backup suppliers. Because these suppliers are in good condition in terms of environmental risk and their main capacity is less disturbed. They are also well-balanced in terms of RPN value, θ and cost parameters.

Fig. 24

The amount of demand that met by selected suppliers

As previously explained, in this model, the benefit of the chain has increased compared to the previous case. This suggests that the use of risk reduction strategies at each level of the risk disruption parameter enables the supply chain to get greater benefits (see Fig. 25).

Fig. 25

Comparison of expected total profit

8.2 RPN constraint

In order to investigate the effect of RPN values constraint, we solve this model by keeping the rest of the parameters constant and without RPN values constraint to investigate the impact of RPN constraint on supplier selection and supply chain benefits. The results show that if this constraint is ignored in proposed models, they tend to select and allocate processes based on the purchase and reserving costs, and local disruption. Of course, the cost parameter has the most impact parameter in selecting suppliers and assigning orders to them. Ignoring the RPN value constraint has led to demand for suppliers (2,4,7 and 13) with low local disruption and the lowest purchase cost (see Fig. 8).

Paying attention to the cost of purchasing as the most effective factor in selecting and allocating suppliers has led to a slight increase in the profit of the supply chain if the RPN constraint is ignored in model 1, 2, However, considering this limitation in the third model creates a balance in the key parameters of the problem and increases the benefit of the supply chain (see Fig. 26). Because if the RPN constraint is taken into account, suppliers must be selected with consideration to reasonable purchase cost and the appropriate RPN, which is the reason for this difference in the profit of the supply chain. However, due to the high importance of suppliers' RPN values in reliability and their timely response to demand, we can ignore this small difference in profit, increase the level of reliability and responsiveness of the supply chain. Therefore, the use of RPN values constraint enables the model to take into account the RPN values of each supplier, respond to the appropriate level of demand and manage the costs of supply chain by creating a balance between cost and disruption mitigation.

Fig. 26

Comparison of expected total profit in the presence and absence of RPN constraint

8.3 Change in the cost of pre-positioning the emergency inventory

In this section, the impact of increasing and decreasing the cost of emergency inventory maintenance by back-up suppliers in the second and third models is investigated. For this purpose, we have increased the maintenance cost of emergency inventory and then reduced it, and the results are presented in Figs. 27 and 28.

Fig. 27

Effect of change in the cost of pre-positioning the emergency inventory in second model

Fig. 28

Effect of change in the cost of pre-positioning the emergency inventory in third model

As the results show, with the reduction of emergency inventory maintenance costs, a larger share of demand has been met by emergency stock of back-up suppliers. However, in order to improve the profitability of the supply chain, the use of excess capacity by main suppliers has been reduced. Moreover, with the increase in the emergency inventory cost, the tendency of models to use the emergency inventory maintenance strategy has decreased and the model shows the tendency to use more of the main suppliers and use the surplus capacity strategy.

8.4 Change in the cost of extra capacity

In this section, in order to investigate the effect of increasing and decreasing the cost of reservation of excess capacity, first the amounts of cost of excess capacity are increased and then decreased. As the cost of excess capacity decreases, the results of the variables remain unchanged, and the benefit of the supply chain increases by reducing the cost of excess capacity. As shown in Fig. 29, as the cost of reserving excess capacity increases, the model tends to use less of the reserving extra capacity strategy. With the reduction of the use of surplus capacity strategy, the use of back-up suppliers and emergency inventory maintenance has been considered. As in the third model, most of the demand is met by the strategy of using back-up suppliers and emergency inventory maintenance (see Fig. 30).

Fig. 29

Effect of change in the cost of extra capacity in second model

Fig. 30

Effect of change in the cost of extra capacity in third model

8.5 Changes in the lost capacity rate and extra capacity of supplier

In this section, the effects of changes the lost capacity rate and the maximum capacity of main suppliers presented separately and in combination. According to Fig. 31, the effect of reducing the rate of lost capacity of main suppliers is examined. As shown in the figure, by reducing the value of parameter $\rho_{i}$, the profit of the entire supply chain will increase. In fact, in this case, a large share of demand is met by the main suppliers and therefore, fewer backup suppliers are selected. In this way, the contract, inventory holding and purchase costs associated with them are reduced, and the total profit is increased.

Fig. 31

The effects of changes in the lost capacity rate of main suppliers

Figure 32 shows the effect of increasing the excess capacity of the main suppliers on the objective function. According to this figure, if the excess capacity of the supplier increases, the total profit will increase according to the second model, while it will decrease in the third model. Increasing the extra capacity in the second model compensates the share of unmet demand and reduces costs. In the third model, increasing excess capacity prevents the use of other strategies and reduces supply chain profits.

Fig. 32

The effects of changes in the maximum capacity of main suppliers

The effect of simultaneous change of both parameters is shown in Fig. 33. Increasing excess capacity and decreasing the rate of lost capacity of the main suppliers show an increasing trend in the objective function.

Fig. 33

The combined effect of the lost capacity rate and maximum capacity of main suppliers

9 Discussion and managerial insights

Some researchers have examined the issue of supplier selection and order allocation under risk condition, such as Sawik (2013) and Torabi et al. (2015) addressed the supplier selection and order allocation problem with disruption risks. They developed some protection strategies including the selection of a number of suppliers to be protected against disruptions and allocation of emergency inventories to be pre-positioned at the protected suppliers in order to decrease the disruption risks. These papers did not address the integration of decisions in a centralized supply chain. Also Esmaeili-Najafabadi et al. (2019) presented a model to optimize supplier selection and order allocation simultaneously in centralized supply chains considering disruption risks. Despite the impact of disruption on the capacity and reliability of suppliers, they are not addressed in these papers, so that the capacity and reliability of suppliers remain constant after the disruption. Therefore, in this study, the strategy of reserving extra capacity with RPN limitation has been considered.

In the previous section, a good understanding of the behavior of objective functions and variables was obtained by changing the parameters. Therefore, based on the findings and results of the analysis, some economic and managerial insights are presented as follows.

• 1. Managers believe that using resilience strategies to reduce and control supply chain risks requires high investment and is not cost-effective. However, we have shown in this study that by properly controlling risks and increasing the level of access to supply chain components using strategies, the level of demand response is increased and thus the expected profit of the supply chain is met. In fact, the use of disruption strategies enables the supply chain to meet a certain level of confidence in the event of a disruption.

• 2. Consider the coordination between supply chain components in this study; In addition to effects such as increasing product quality, reducing inventory maintenance costs, reducing lead-time, etc., led to an increase in the level of responsiveness and accessibility in the event of disruption, which resulted increase in the profit of the entire supply chain.

• 3. In cases where the risk of disruption is low, it is better to choose suppliers who have more reliability and a good balance in the costs of purchase, inventory, and capacity reservation. Moreover, in cases where the risk of disruption is high, it is necessary to pay attention to suppliers who have the lowest probability of disruption so that they lose less of their capacity and are able to respond to demand at a higher level. Also, by considering costs as the next priority in selecting suppliers, the chain's profitability can be improved to an acceptable level.

• 4. If supplier reliability (RPN constraint) is ignored, the choice of suppliers is based on costs. Paying attention to the cost of purchasing as the most effective factor in selecting and allocating suppliers has led to a slight increase in the profit of the supply chain. However, considering this limitation in the third model creates a balance in the key parameters of the problem and increases the benefit of the supply chain (see Fig. 25). Because if the RPN constraint is taken into account, suppliers must be selected with consideration to reasonable purchase cost and the appropriate RPN, which is the reason for this difference in the profit of the supply chain. Therefore, the use of RPN values constraint enables the model to take into account the RPN values of each supplier, respond to the appropriate level of demand and manage the costs of supply chain by creating a balance between cost and disruption mitigation.

• 5. With the increase in emergency inventory costs, it is better to make a better choice in terms of cost among back-up suppliers or to allocate less emergency inventory to them. While reducing the cost of emergency inventory and increasing the cost of capacity reservation, the use of backup suppliers creates more benefits for the supply chain.

• 6. As a comparison between the results of the models, the use of a combination of surplus inventory maintenance, back-up suppliers and additional capacity with geographical segmentation strategy has a significant impact on controlling the negative effects of risks and supply chain costs.

• 7. The use of some resilient supply chain strategies, such as the pre-positioning inventory strategy, is conflict with the lean supply chain. In this study, by studying and using other strategies, we showed that paying attention to the geographical segmentation strategy causes a lower level of emergency inventory to be maintained. This suggests that the combination of the use of strategies enables the supply chain to not only creating an appropriate level of flexibility but also do not distance itself from leanness as much as possible.

• 8. The results of sensitivity analysis showed that reducing the rate of lost capacity of main suppliers plays a significant role in supply chain profitability. Therefore, managers are advised to improve the level of lost capacity under disruption by using fortification of suppliers and implementing suppliers’ business continuity plans.

• 9. Under conditions where managers are not able to use the main capacity of the backup suppliers (similar to the third model), they can increase the share of the total supply chain profit by allocating more excess capacity to the main suppliers.

• 10. Continuing and integrated cooperation with main suppliers will bring wide benefits to centralized supply chains. Improving the resilience of key suppliers and allocating additional capacity to them will enable managers to maintain their cooperation with them under conditions of disruption and reduce the use of backup suppliers while improving supply chain profits.

10 Conclusion

Each component of the supply chain affects each other and considering the coordination between the components of the supply chain leads to improved performance and increased competitive advantage. In this research, the issue of supplier selection and order allocation under the conditions of disruption and environmental risk in a centralized supply chain has been investigated. For this purpose, three nonlinear programming models have been proposed with considering five risk reduction strategies that optimize the chain profit. Five strategies used to cope with both risks are protected suppliers, back-up suppliers, reserving additional capacity, surplus stock and geographical separation. Since the main suppliers lose part of their production capacity due to disruption, they are considered as protected suppliers by reserving extra capacity policy. These two strategies are presented in the form of a general model and then the two models are developed. In the second model, the strategy of back-up suppliers and maintenance of additional inventory is added and in the final model, the strategy of geographical segmentation is also considered. Also, due to the importance of the supplier availability level, FMEA method has been used to calculate their RPN value and the results of which have been used in the proposed models. The study shows that the use of risk reduction strategies leads to control of supply chain risks and consequently increases profits. Moreover, the disruption risk, RPN value, and costs are the key factor in selecting suppliers and order allocation in the basic and extended models. As compared to the models, when a disruption risk is small, suppliers with a low cost are selected, and orders are allocated among them. When the disruption probability is increased, the disruption probability and RPN are the key selection parameter. Considering the uncertainty in the model parameters and the dynamic nature of the studies, offering discounts to improve the procurement process and adding operational risk to the present study are some of the suggestions for future. Moreover in this study, the FMEA method was used, and future researchers can use other methods of risk management such as failure mode effects and criticality analysis (FMECA), hazard and operability study (HAZOP) (mainly applicable in the petrochemical industry), and fault tree analysis (FTA) to assess risk and reliability of suppliers.

Appendices

Appendix 1

See Table

Table 21 The value of the parameters in small sizes

21

Appendix 2

See Tables

Table 22 Pairwise comparison matrix of FMEA risk factors of first DM

22,

Table 23 Pairwise comparison matrix of risk sub-factors with regard to severity of first DM

23,

Table 24 Pairwise comparison matrix of risk sub-factors with regard to occurrence of first DM

24,

Table 25 Pairwise comparison matrix of risk sub-factors with regard to detection of first DM

25,

Table 26 Final weights of suppliers with regard to risk sub-factors of first DM

26,

Table 27 Pairwise comparison matrix of FMEA risk factors of second DM

27,

Table 28 Pairwise comparison matrix of risk sub-factors with regard to severity of second DM

28,

Table 29 Pairwise comparison matrix of risk sub-factors with regard to occurrence of second DM

29,

Table 30 Pairwise comparison matrix of risk sub-factors with regard to detection of second DM

30,

Table 31 Final weights of suppliers with regard to risk sub-factors of second DM

31,

Table 32 Pairwise comparison matrix of FMEA risk factors of third DM

32,

Table 33 Pairwise comparison matrix of risk sub-factors with regard to severity of third DM

33,

Table 34 Pairwise comparison matrix of risk sub-factors with regard to occurrence of third DM

34,

Table 35 Pairwise comparison matrix of risk sub-factors with regard to detection of third DM

35,

Table 36 Final weights of suppliers with regard to risk sub-factors of third DM

36

Appendix 3

See Tables

Table 37 The value of the parameters in case study

37,

Table 38 The value of the parameters in case study

38,

Table 39 The value of the $(C_{ij} ,\alpha_{ij} ,e_{ij} ) \times 10^{3}$ parameters in case study

39