A robust, sustainable, resilient, and responsive model for forward/reverse logistics network design with a new approach based on horizontal collaboration

Abstract

Optimizing the strategic and tactical decisions regarding sustainability and resilience is important because they affect the performance of the supply chain over a long period. This study proposes a new multi-objective model for resilient, sustainable, and responsive forward/reverse logistics network design considering cost, social responsibility, CO₂ emission, water consumption, responsiveness time, and collaboration risk. The most significant impacts on costs are identified using the design of experiments, and a robust optimization method is applied to deal with operational risk. In addition, horizontal collaboration for the first time as a quantitative model is suggested to deal with the disruption risk. Also, a multi-objective solving algorithm with a new aggregation model is developed, and the efficiency of the proposed solution method is shown. The analysis of results based on the iron and steel industries shows the efficiency of resilience strategies and the improvement of costs, sustainability and responsiveness indicators. Also, the findings demonstrate that collaboration has a higher performance than fortification. Finally, managers and supply chain decision-makers are provided with managerial implications to make more effective decisions about the supply chain network design, technology selection, supply chain sustainability, and collaborators selection, using the presented results.

Introduction

One of the most important strategic decisions of the supply chain (SC) is supply chain network design (SCND). SCND is mainly related to the location of facilities, the flows between them, and the capacities of required facilities (Pishvaee et al., 2010). Since establishing a new facility and changing the existing facilities need a lot of investment and are influential in other SC decisions, this issue has become one of the most critical cases in the field of SC. On the other hand, reverse logistics, due to environmental impacts and resource savings, have attracted considerable attention in recent decades. As SCs become more complex, they also face various risks that are very important in SCND for cost savings (Zhalechian et al., 2018). Risks in the SC include operational risks (changes in demand, cost, and supply of raw materials) and disruption risks (technological threats related to equipment failure or disruption risks resulting from natural disasters or human threats including strikes and war) (Snyder et al., 2016; Zhalechian et al., 2018).

SC resilience refers to a return to normal conditions and the ability to recover the desired capabilities after a disruption occurs. There are several ways to increase SC resilience, for example, multi-resource, fortification facilities, capacity increasing, inventory, and business plan development (Sabouhi et al., 2018). Another way to increase resilience is collaboration in the SC (Duong & Chong, 2020). Collaboration is a process through which two or more companies work together to achieve common goals and mutual benefits in planning and performing SC management operations (Cao & Zhang, 2011). There are several alternatives to collaboration in the SC, such as collaboration in information sharing, collaboration in setting incentives in the SC, collaboration in decisions making, sharing of resources and capabilities, and collaboration in knowledge sharing (Badea et al., 2014).

Collaboration in the SC leads to faster service improvements, better SC performance, and faster recovery of capabilities after disruption (Duong & Chong, 2020; Zhong et al., 2022). Collaboration can also increase responsiveness (Heaslip et al., 2012). Asset sharing and collaboration can reduce costs after disruption (Baharmand et al., 2017) and find optimal and faster solutions to deal with SC disruption (Maon et al., 2009). SC members can also use collaboration to increase the amount of responsiveness and ability to better performance recovery processes when disrupted (Scholten & Schilder, 2015). Accurate and appropriate information is highly significant for planning after a disruption, so information sharing and collaborating in this area will help manage disruptions in the SC (Scholten & Schilder, 2015).

Environmental pollution and greenhouse gas emissions and their destructive effects on the environment, as well as human rights and the optimal use of resources, have made researchers pay attention to the issue of SC sustainability. Decision-makers and SC managers must pay attention to social and environmental problems to maintain their competitive advantage and consider environmental and social concerns in their activities (Tirkolaee et al., 2021). Researchers in sustainable SCND have made efforts to configure the SC network so that the economic, social, and environmental aspects are as desirable as possible (Govindan et al., 2015). Also, due to the importance of this problem, authors consider sustainability and resilience in the SC simultaneously.

According to the explanations above, considering all pillars of sustainability (economic, social responsibility, and environmental) and responsiveness to survive and increase the SC's competitiveness and SC risks to prevent unwanted costs in the future is essential. On the other hand, considering decisions related to sustainability, resilience, responsiveness, and collaboration in SCND is a significant issue since they will be effective in the SC for a long time. Due to this great importance, this study proposes a mixed-integer linear programming model for SCND to simultaneously consider the sustainability, robustness, responsiveness, and resilience under operational and disruption risks. Along with traditional strategies to increase network efficiency, it applies horizontal collaboration as a new strategy to the mathematical model with the evaluation of its role in the strategic decisions of SC and responsiveness, resilience, and cost reduction and the balance between risk and collaboration performance in the SC network. As a new concept, the forward and reverse physical flow of products and the flow of horizontal collaboration in the SC are considered simultaneously. This study also develops a solution method for the proposed multi-objective model. Considering the role of the iron and steel industry in increasing environmental pollution and social responsibility, it finally applies the proposed model in this industry and performs relevant analyses. According to the provided explanations, the most critical questions of this research are as follows.

• Which facilities can we establish from the capacity and technology aspects, and how is the flow of products between them?

• How can we balance collaboration risk and SC network performance?

• Which manufacturers and distributors can collaborate horizontally?

• What effects do collaboration and fortification strategies have on SC responsiveness, sustainability, and total costs?

The remainder of the study is divided as follows. Section 2 reviews the related literature under various sub-sections. Section 3 presents research assumptions, mathematical models, and robust optimization method. After that, Sect. 4 explains the multi-objective solution method, and Sect. 5 introduces a case study on the iron and steel SC with related data. Section 6 analyses the results to illustrate the efficiency of the proposed mathematical model and solving algorithm. Finally, Sect. 7 concludes the study and proposes future research.

Literature review

This section provides a literature review on collaboration in SCND, the association of risk with collaboration, sustainable logistics and SCND, resilience-sustainable SCND, and SC responsiveness. It also describes research gaps and innovations.

Collaboration in SCND

One of the first studies on the impact of collaboration in the SC network is proposed by Groothedde et al. (2005). They showed that collaboration could reduce costs and improve SC performance. They have applied no mathematical model. Other researchers, such as Pan et al. (2013), Tang et al. (2016), and Habibi et al. (2018), provided a mathematical model to examine the role of collaboration in locating the hub in distribution chains. Also, Pan et al. (2014), Ouhader and El Kyal (2017), and Aloui et al. (2021) studied the economic and sustainability effects of asset sharing in the SC network using the mathematical model. Arslan et al. (2020) developed a model for network design considering strategic alliances. Guo et al. (2022) applied cost-sharing strategy and lateral collaboration for sustainable SC design.

Relationship between risk and collaboration

Shahbaz et al. (2019) categorized SC risks into three parts: outside the SC, inside the organization, and outside the organization and within the SC. They considered the collaboration risks in the group of risks within the SC and between organizations. Das and Teng (1998) classified collaboration risks into two parts: communication risk and performance risk. Communication risk means the risk of communication with other companies, for example, information theft and partner skills. Therefore, many SC members avoid sharing assets, such as knowledge and physical ones. Tang et al. (2016) also state that many companies are sensitive to sharing knowledge and critical assets. Mafini and Muposhi (2017) introduced the disclosure of sensitive information, unreliable partners, lack of technical knowledge, and inadequate partner assets as collaborative risks. On the other hand, lack of supply, unmet demand, research, and development risks are performance risks that working together can reduce them. Sharing risk among partners can decrease performance risks (Difrancesco et al., 2021).

Also, some researchers have introduced the collaboration concept to reduce operational and disruption risks in the SC. For example, Duong and Chong (2020) proposed various mechanisms such as collaboration agreements to reduce disruption risk, collaboration to establish decision-making systems, and collaboration in information and technology sharing to deal with disruption. Peng and Pang (2020), Oh et al. (2021), and Difrancesco et al. (2021) suggested different solutions for reducing risk through collaboration. Rajagopal et al. (2017) reviewed and compared various decision models according to different risk reduction strategies, for example, collaboration and risk mitigation contracts, resilience strategies, and robust programming for the SC.

A review of the literature revealed that authors applied collaboration to reduce risk or considered collaboration a factor in creating risks such as disclosure of information and opportunism of individuals.

Sustainable logistics and SCND

In the last decade, sustainability has drawn the researchers' attention, and some authors, such as Joshi (2022) and Yozgat and Erol (2022), reviewed sustainability in logistics and SCND. Regarding sustainable logistics network design, Mondal et al. (2021) applied a multi-item and multi-objective model for a sustainable solid transportation problem. Their model minimized the costs and time of product transportation and maximized the created jobs in a fuzzy environment. Midya et al. (2021) proposed a multi-stage model with cost, time, and CO₂ emissions objectives for transportation in a green SC under an intuitionistic fuzzy environment. Mondal and Roy (2021) formulated a model for opened and closed-loop sustainable SC, considering the cost, total backlog amount, time, and job opportunity objectives. They designed the transportation problem in the first and the vehicle routing problem in the second stage under uncertainty. Mondal and Roy (2022) proposed a new multi-objective model and multi-criteria decision-making method in a new uncertain environment and considered CO₂ emission and job opportunities created during transportation. Ghosh et al. (2022) developed a sustainable multi-objective model for waste management in solid transportation problems and considered offset policy, cap-and-trade, and carbon tax strategies for carbon restriction. Tirkolaee and Aydin (2022) proposed a bi-level decision support system. They designed a SC in the first level and transportation network in the second level for perishable products. Also, they developed a solution for the proposed model based on fuzzy goal programming and possibilistic linear programming.

A review of the literature on sustainability in SCND shows that researchers have considered various indicators for SCND in mathematical models. CO₂ emission is one of the most significant environmental indicators for production activities in the SC (Alinezhad et al., 2022; Babaeinesami et al., 2022; Zhalechian et al., 2018). Guo et al. (2021) have considered water consumption and pollution. Also, several authors (Pourmehdi et al., 2020; Taleizadeh et al., 2019) have considered energy consumption a sustainability indicator.

Regarding social criteria, one of the most critical indicators is job opportunities (Goli et al., 2020; Soleimani et al., 2022). The number of days lost due to damage to workers (Pourmehdi et al., 2020; Sahebjamnia et al., 2018), facility location in less developed areas (Pishvaee et al., 2014; Rahmani-Ahranjani et al., 2017; Sherafati et al., 2019) and factors influencing customer health (Pishvaee et al., 2014; Taleizadeh et al., 2019) are the other social criteria. Also, ISO 26000 (2010) proposed other factors related to social responsibility. This standard provides organizations with guidance on social responsibility. Table 1 summarizes the core subjects in the ISO 26000 standard and the issues considered in this study.

Table 1.

The subjects of ISO 26000

Subjects	Sub Issues	This study
Human rights	Due diligence
	Risk situations	*
	Cultural and Economic rights
	Rights at work
Community development and involvement	Community involvement and Employment creation	*
	Education and culture
	Social investment	*
	Priority to local supplier
	Justice in creating jobs	*
Environment	Water consumption	*
	Energy consumption
	water recycling
	Pollution environment	*
	Product recycling	*
	Sustainable resource use
Organizational governance	–
Fair operating practices	Fair competition
	Anti-corruption
	Political involvement
	Respect for property rights
Consumer issues	Responsiveness	*
	Consumers safety
	Consumers health
	Consumer support
	Education
	Access to essential services
	Collect used products	*
Labor practices	Employment	*
	Conditions of work
	Safety	*
	Accident for workers	*
	Training in the workplace

Resilience and sustainable SCND

Due to the growing importance of sustainability and resilience in the SC, some researchers have considered this issue simultaneously. Jabbarzadeh et al. (2018) considered additional capacity, multiple sources, and parallel resources strategies for resilience-sustainable network design. Hosseini-Motlagh et al. (2020) developed a multi-objective model for developing a resilient electricity network according to social responsibility criteria. Hasani et al. (2021) modeled a green and resilient SC network and considered parallel sources, fortification of equipment, and similar resource strategies for resilience. Sazvar et al. (2021) presented a model for designing a sustainable resilience network considering the capacity planning approach. Sabouhi et al. (2021) and Fazli-Khalaf et al. (2021) developed a model for SCND by determining sustainability decisions and resilience strategies. Vali-Siar and Roghanian (2022) formulated a model for developing a resilient closed-loop sustainable network by considering environmental factors, job creation, and damage reduction to workers' health. They also applied capacity expansion, dual-channel distribution, and inventory strategies in resilient SCND.

SC responsiveness

Pishvaee et al. (2010) developed a model for designing a network with forward and reverse physical flow of products and responsiveness maximization according to the rate of returned products and customer demand. Ramezani et al. (2013) considered the fulfillment of customer demand in the forward flow and the collection of products in the reverse flow by assigning weights to the objective function. Dubey et al. (2015) developed a model for designing a sustainable and responsive network whose goal is to respond appropriately in uncertain situations. Fattahi et al. (2017) addressed the relationship between demand and the length of the order period in a multi-period model. Sabouhi et al. (2020) for each customer considered a specific service level that the meeting of demand must be greater than the desired service level. Goli et al. (2020) considered the lead time for delivering and manufacturing perishable products in the closed-loop SC and proposed a new hybrid algorithm to solve the problem. Vali-Siar and Roghanian (2022) examined the responsiveness in forward and reverse logistics. They applied some constraints that guarantee the satisfaction of customer demand. Shaw et al. (2022) proposed a multi-objective model under triangular type-2 fuzzy number for the location-allocation problem where the cost, affected coverage area, and the shortest time are optimized.

Research gap and innovations

Table 2 reports a review of the literature. As can be understood from Table 2, researchers have recently considered collaboration in the SC to reduce risk and costs that needs development, especially in quantitative models and related analysis. Considering resilience, sustainability, and responsiveness strategies simultaneously and examining the effects of one of the strategies on others needs more attention. Although some researchers, including Duong and Chong (2020), have addressed theoretical concepts on the benefits of collaboration in decreasing the disruption risk, there have been few mathematical models. Although many studies have investigated sustainability indicators, there are many indicators, for example, water consumption, to be considered by researchers (Table 1).

Table 2.

Summary of the most important studies related to this study

References	Risk			Objective function									Flow
				Cost					Sustainability
	Inside organization	Outside the organization and inside the SC	Outside SC	Transportation	Inventory	Handling	Fix cost	Collaboration	Environment	Water consumption	Social responsibility	Responsiveness or time	Forward	Reverse
Groothedd et al. (2005)				*	*	*							*
Pan et al. (2013)			*	*									*
Ramezani et al. (2013)	*			*		*	*	*	*			*	*	*
Pan et al. (2014)			*	*	*	*							*
Tang et al. (2016)				*		*							*
Ouhader and El Kayel (2017)			*	*		*	*						*
Habibi et al. (2018)				*			*						*
Jabbarzadeh et al. (2018)	*		*	*	*	*	*
Sherafati et al. (2019)	*		*	*	*		*		*		*		*	*
Arslan et al. (2020)				*			*	*					*
Goli et al. (2020)				*		*	*		*		*	*	*	*
Aloui et al. (2021)			*	*		*	*						*	*
Hasani et al. (2021)	*		*	*		*	*		*				*
Sabouhi et al. (2021)	*		*	*	*	*	*		*		*		*
Sazvar et al. (2021)	*		*	*	*	*	*		*		*	*	*
Midya et al. (2021)	*			*			*		*			*	*
Mondal and Roy (2021)	*		*	*	*	*					*	*	*	*
Tirkolaee and Aydin (2022)	*		*	*	*	*	*		*		*		*
Shaw et al. (2022)	*		*	*			*					*	*
Vali-Siar and Roghanian (2022)	*		*	*		*	*		*		*		*	*
Babaeinesami et al. (2022)			*	*		*	*		*				*	*
Guo et al. (2022)			*	*			*	*	*				*
This study	*	*	*	*		*	*	*	*	*	*	*	*	*

References	Decision variable						Type of collaboration			Type of asset sharing							Resilience strategy
	Quantity of shipment	Routing	Collaborator	Flow of product	Facility location	Technology selection	Horizontal	Vertical	Hybrid	Information	Knowledge and experience	Facility	Maintenance capability	Vehicle	Human resource	Capacity
Groothedd et al. (2005)	*						*					*
Pan et al. (2013)	*				*		*					*
Ramezani et al. (2013)				*	*
Pan et al. (2014)	*				*		*					*		*
Tang et al. (2016)				*	*		*							*
Ouhader and El Kayel (2017)	*	*		*	*		*					*		*
Habibi et al. (2018)				*	*		*					*
Jabbarzadeh et al. (2018)				*	*
Sherafati et al. (2019)				*	*
Arslan et al. (2020)			*	*	*		*									*
Goli et al. (2020)				*	*	*
Aloui et al. (2021)	*	*		*	*		*					*		*
Hasani et al. (2021)				*	*												*
Sabuhi et al. (2021)				*	*												*
Sazvar et al. (2021)				*	*												*
Midya et al. (2021)	*			*	*
Mondal and Roy (2021)		*		*
Tirkolaee and Aydin (2022)				*	*	*
Shaw et al. (2022)				*	*
Vali-Siar and Roghanian (2022)				*	*	*											*
Babaeinesami et al. (2022)				*	*		*
Guo et al. (2022)			*	*	*		*	*				*		*
This study			*	*	*	*	*				*		*			*	*

Although different authors have considered disruption and operational risks separately, the simultaneous consideration of the risks and collaboration risks needs more attention. Also, examining the effect of the risks, such as disruption, on sustainability requires further studies in this field.

This study develops the literature by proposing a mathematical model based on cost, responsiveness, and sustainability objectives, and it considers operational, disruption, and collaboration risk. Also, it presents a novel model considering horizontal collaboration to reduce costs and deal with the disruption risk and provides a multi-objective solution method. The present study applies the proposed model to the iron and steel industry because it is one of the industries that play a critical role in environmental pollution, water consumption, and job creation. Therefore, the contribution of this research can be summarized as follows.

• This study proposes a new model for SCND by simultaneous consideration of responsiveness, environment, social responsibility, and resilience.

• For the first time, the collaboration strategy is quantified as a resilience strategy to deal with disruption risk and increases responsiveness.

• Asset-sharing decisions, such as knowledge and maintenance capabilities, are considered in strategic SC decisions.

• Through collaboration in forward/reverse logistics network design, this study introduces a new problem: The collaborative forward/reverse logistics network design (CFRLND) problem.

• Significant parameters on operational risks with the design of experiments are determined, and operational, disruptions, and collaboration risks are considered simultaneously,

• A multi-objective solving algorithm is developed, and the proposed model is applied in the iron and steel industry.

The framework of the study and implementation of the mathematical model is described in Fig. 1

Fig. 1.

Framework of study and implementation of the mathematical model

Research assumptions and mathematical model

This study applies mixed-integer linear programming to design the iron and steel forward/reverse logistics network. The network includes primary steel manufacturing plants, consumable product manufacturing plants, distribution centers in the forward direction, and collection centers for scrap in the reverse path. In this network, at first, the steel in the primary steel manufacturing plants is produced by the scrap collected from different parts of consumption; and the steel is transferred to the consumable product manufacturing plant. After the production of new products, these products are distributed among the consumption points by the distribution centers. Since there is always some waste in the production of products, these wastes are transferred to collection centers. Products are collected from consumption parts and transferred to the collection centers and then to the primary steel manufacturing plant to manufacture new steel (Fig. 2). This network can be applied in different metal SCs, such as lead production in the battery industry and so on. Other assumptions of the mathematical model are as follows.

1. Direct delivery of products from primary steel or consumable product manufacturing plants to distribution centers is not considered.

2. Various levels of facility fortification are considered to increase SC resilience.

3. Companies use different mechanisms to deal with disruption, such as creating collaboration agreements to reduce risk, collaboration to create decision-making systems, sharing information, technology, and so on.

4. The SC network includes one product since the studied network is the iron and steel network.

5. Various levels of collaboration are determined according to the number of collaborating companies. Therefore, these levels are determined based on the high and low levels for the number of collaborating companies.

6. Collaboration causes collaboration costs; on the other hand, it also influences reducing variable production costs.

7. Due to the collaboration between distribution centers, it is assumed that in case of disruption, to deal with lost sales, it is possible to send the product between distribution centers.

8. Due to the fortification level of distribution centers, a high fortification level is considered a reliable distribution center because the probability of a capacity reduction in them is low, and other distribution centers are unreliable distribution centers. In the disruption, we can transfer the product from reliable to unreliable distribution centers.

9. The capacity reduction percentage in facilities is determined based on the fortification level or collaboration level under different scenarios.

10. The location of consumables product manufacturing plants is predetermined.

Fig. 2.

Schematic diagram for forward/reverse logistics, and time

The symbols used in modeling are as follows.

Set

A: Set of candidate points for the primary steel manufacturing plants $(a,a^{{}^{\prime }}\in A)$

B: Set of fixed points for consumable product manufacturing plants $(b,b^{{}^{\prime }}\in B)$

C: Set of capacities for primary steel manufacturing plants $(c\in C)$

D: Set of candidate points for collection centers $(d\in D)$

O: Set of technologies for primary steel manufacturing plants $(o\in O)$

T: Set of candidate points for reliable distribution centers $(t\in T)$

H: Set of candidate points for unreliable distribution centers $(h\in H)$

L: Set of fixed points for consumption markets $(l\in L)$

G: Set of fortification levels $(g\in G)$

I: Set of collaboration levels for primary steel manufacturing plants $(i\in I)$

J: Set of collaboration levels for consumable product manufacturing plants $(j\in J)$

P: Set of collaborating primary steel manufacturing plants $(p\in P)$

Q: Set of collaborating consumable product manufacturing plants $(q\in Q)$

S: Set of scenarios $(s\in S)$

Parameters

${\rho }_s$: The probability of scenario s.

${\mathit{Afi}}_{\mathit{ag}}^{\mathit{co}}$: Fixed cost of establishing primary steel manufacturing plant a with technology o, capacity c, and fortification level g.

${\mathit{Tfi}}_t$: Fixed cost of opening reliable distribution center t.

${\mathit{Hfi}}_h$: Fixed cost of opening unreliable distribution center h.

${\mathit{Dfi}}_d^g$: Fixed cost of opening collection center d with fortification level g.

${\mathit{Aca}}_{\mathit{ai}}^{\mathit{co}}$: Maximum capacity of primary steel manufacturing plant a with capacity level c, technology o, and collaboration level i.

${\mathit{Bca}}_b$: Maximum capacity of consumable product manufacturing plant b.

${\mathit{Tca}}_t$: Maximum capacity of reliable distribution center t.

${\mathit{Hca}}_h$: Maximum capacity of unreliable distribution center h.

${\mathit{Dca}}_d$: Maximum capacity of collection center d.

${\delta }_{\mathit{xy}}$: Distance between two facilities x and y.

${\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}}$: Manufacturing cost per unit of steel in plant a, using technology o, capacity c, and collaboration level i.

${\mathit{Bo}}_b^j$: Production cost per unit of product in plant b when the level of collaboration in the plant is j.

${\mathit{To}}_t$: Operating cost of each product unit in reliable distribution center t.

${\mathit{Ho}}_h$: Operating cost of each product unit in unreliable distribution center h.

${\mathit{Do}}_d$: Operating cost of each product unit in collection center d.

ϐ: Cost of transporting each unit per kilometer.

${\mathit{En}}_o$: Environmental effects of the production of each unit of steel production in the primary steel manufacturing plant using technology o.

${\mathit{Wc}}_o$: Water consumption per unit of steel production in the primary steel manufacturing plant using technology o.

$\omega$: Environmental effects of transporting each unit of product per kilometer.

${\beta }_b$: Rate of production waste in plant b.

${\mathit{dl}}_l$: Demand in the consumption market l.

${\mathit{Rl}}_l$: Return rate of products in consumption market l.

${\mathit{Ajo}}_{\mathit{agi}}^{\mathit{oc}}$: Fixed-job opportunities created if plant a is established with technology o, capacity c, fortification level g, and collaboration level i.

${\mathit{Tjo}}_{\mathrm{t}}$: Fixed-job opportunities created in reliable distribution center t.

${\mathit{Hjo}}_h$: Fixed-job opportunities created in unreliable distribution center h.

${\mathit{Djo}}_d^g$: Fixed-job opportunities created in collection centers d with fortification level g.

${\mathit{Adl}}_{\mathit{agi}}^{\mathit{oc}}$: Average days lost due to accidents in plant a with technology o, capacity c, fortification level g, and collaboration level i.

${\mathit{Tdl}}_t$: Average days lost due to accidents in reliable distribution center t.

${\mathit{Hdl}}_h:$ Average days lost due to accidents in unreliable distribution center h.

${\mathit{Ddl}}_d^g:$ Average days lost due to accidents in collection center d with fortification level g.

${\mathit{Alj}}_{\mathit{agi}}^{\mathit{oc}}$: Job opportunities lost due to disruption in plant a with technology o, capacity c, fortification level g, and collaboration level i.

${\mathit{Tlj}}_t$: Job opportunities lost due to disruption in reliable distribution center t.

${\mathit{Hlj}}_h$: Job opportunities lost due to disruption in unreliable distribution center h.

${\mathit{Dlj}}_d^g$: Job opportunities lost due to disruption in collection centers d with fortification level g.

$\theta$: Transfer capacity of each shipment between facilities.

$\mathit{kt}$: Time of transporting each unit of product per kilometer.

${\mathit{At}}_{\mathit{ai}}^g$: Process time per unit of product in plant a with fortification level g and collaboration level i.

${\mathit{Bt}}_{\mathit{bj}}:$ Process time per unit of product in plant b with collaboration level j.

${\mathit{Tt}}_t$: Process time per unit of product in reliable distribution center t.

${\mathit{Ht}}_h$: Process time per unit of product in unreliable distribution center h.

${\mathit{Dt}}_d^g$: Process time per unit of product in collection centers d with fortification level g.

${\mathit{Ar}}_{\mathit{ai}}^g$: Percentage of capacity reduction due to disruption in plant a with fortification level g and collaboration level i.

${\mathit{Br}}_{\mathit{bj}}$: Percentage of capacity reduction due to disruption in plant b with collaboration level j.

${\mathit{Tr}}_t$: Percentage of capacity reduction due to disruption in reliable distribution center t.

${\mathit{Hr}}_h:$ Percentage of capacity reduction due to disruption in unreliable distribution center h.

${\mathit{Dr}}_d^g$: Percentage of capacity reduction due to disruption in collection centers d with fortification level g.

${\mathit{ax}}_{\mathit{max}}$: Maximum possible for collaboration between steel manufacturing plants.

${\mathit{ai}}_{\mathit{min}}$: Minimum possible for collaboration between steel manufacturing plants.

${\mathit{bx}}_{\mathit{max}}$: Maximum possible for collaboration between consumable product manufacturing plants.

${\mathit{bi}}_{\mathit{min}}$: Minimum possible for establishing collaboration between consumable product manufacturing plants.

$la\left(i\right):$ Lower bound for collaboration level i.

$lb(j)$: Lower bound for collaboration level j.

$ua\left(i\right):$ Upper bound for collaboration level i.

$ub\left(j\right):$ Upper bound for collaboration level j.

${\nu }_1,{\nu }_2$: Importance weight of collaboration risk related to collaboration between steel and consumable product manufacturing plants.

${\mathit{Acl}}_a^p$: Cost of collaboration between plants a when the number of collaboration plants is p.

${\mathit{Bcl}}_b^q:$ Cost of collaboration between plants b when the number of collaboration plants is q.

Decision variables.

$y_{\mathit{agi}}^{\mathit{oc}}$: If plant in location a with technology o, capacity c, fortification level g, and collaboration level i is established 1, and otherwise 0.

$r_t$: If a reliable distribution center is established in location t 1, and otherwise 0.

$u_h$: If an unreliable distribution center is established in location h 1, and otherwise 0.

$x_d^g$: If the collection center with fortification level g is established in location d 1, and otherwise 0.

${\mathit{ca}}_{a{a}^{{}^{\prime }}}$: If plant a collaborates horizontally with plant ${\mathrm{a}}^{{}^{\prime }}$ 1, and otherwise 0.

${\mathit{cb}}_{b{b}^{{}^{\prime }}}$: If plant b collaborates horizontally with plant ${\mathrm{b}}^{{}^{\prime }}$ 1, and otherwise 0.

${\mathit{na}}_a^p$: If the number of plants collaborating with plant a is p 1, and otherwise 0.

${\mathit{nb}}_b^q$: If the number of plants collaborating with plant b is q 1, and otherwise 0.

${\mathit{ea}}_a^i$: If plant a is assigned to collaboration level i 1, and otherwise 0.

${\mathit{eb}}_b^j$: If plant b is assigned to collaboration level j 1, and otherwise 0.

${\mathit{Am}}_a$: Quantity of scrap entering the primary steel manufacturing plant a.

${\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}$: Quantity of steel shipped from plant a with collaboration level i, capacity c, and technology o to plant b.

${\mathit{Tm}}_{\mathit{bt}}^j$: Quantity of product shipped from the production plant b with collaboration level j to the distribution center t.

${\mathit{Hm}}_{\mathit{bh}}^j$: Quantity of product shipped from plant b with collaboration level j to distribution center h.

${\mathit{km}}_{\mathit{tl}}$: Quantity of product shipped from distribution center t to the consumption market l.

${\mathit{Um}}_{\mathit{hl}}$: Quantity of product shipped from distributor center h to the consumption market l.

${\mathit{Rm}}_{\mathit{th}}$: Quantity of product shipped from distribution center t to distribution center h in the event of a disruption.

${\mathit{Lm}}_{\mathit{ld}}$: Quantity of product shipped from the consumption market l to the collection center d.

${\mathit{Dm}}_{\mathit{da}}$: Quantity of product shipped from the collection center d to the primary steel manufacturing plant a.

${\mathit{Vm}}_{\mathit{bd}}^j$: Quantity of product shipped from the plant b with collaboration level j to the collection center d.

Deterministic mathematical model

The deterministic model is as follows:

$$\begin{array}{cc}\hfill Min{z}_1& =\sum _o\sum _a\sum _c\sum _g\sum _{i}Af{i}_{\mathit{ag}}^{\mathit{co}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _{t}Tf{i}_t{r}_t+\sum _{h}Hf{i}_h{u}_h+\sum _g\sum _{d}Df{i}_d^g{x}_d^g\hfill \\ \hfill & +\sum _a\sum _o\sum _c\sum _b\sum _{i}B{m}_{\mathit{aib}}^{\mathit{oc}}({\delta }_{\mathit{ab}}\beta +A{o}_{\mathit{ai}}^{\mathit{oc}})+\sum _b\sum _j\sum _t({\delta }_{\mathit{bt}}\beta +B{o}_b^j)T{m}_{\mathit{bt}}^j\hfill \\ \hfill & +\sum _b\sum _j\sum _h({\delta }_{\mathit{bh}}\beta +B{o}_b^j)H{m}_{\mathit{bh}}^j+\sum _b\sum _j\sum _d({\delta }_{\mathit{bd}}\beta +B{o}_b^j)V{m}_{\mathit{bd}}^j\hfill \\ \hfill & +\sum _t\sum _l\left({\delta }_{\mathit{tl}}\beta +T{o}_t\right)k{m}_{\mathit{tl}}+\sum _h\sum _l\left({\delta }_{\mathit{hl}}\beta +H{o}_h\right)U{m}_{\mathit{hl}}\hfill \\ \hfill & +\sum _t\sum _h\left({\delta }_{\mathit{th}}\beta +T{o}_t\right)R{m}_{\mathit{th}}+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta L{m}_{\mathit{ld}}+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +D{o}_d)D{m}_{\mathit{da}}\hfill \\ \hfill & +\sum _p\sum _{a}n{a}_a^{p}Ac{l}_a^p+\sum _q\sum _{b}n{b}_b^{q}Bc{l}_b^q\hfill \\ \hfill \end{array}$$ 1

$$\begin{array}{cc}\hfill Min{z}_2^1& =\sum _a\sum _o\sum _c\sum _b\sum _{i}B{m}_{\mathit{aib}}^{\mathit{oc}}E{n}_o+\sum _a\sum _o\sum _c\sum _b\sum _i{\delta }_{\mathit{ab}}\omega B{m}_{\mathit{aib}}^{\mathit{oc}}\hfill \\ \hfill & +\sum _b\sum _j\sum _t{\delta }_{\mathit{bt}}\omega T{m}_{\mathit{bt}}^j+\sum _b\sum _j\sum _h{\delta }_{\mathit{bh}}\omega H{m}_{\mathit{bh}}^j\hfill \\ \hfill & +\sum _t\sum _l{\delta }_{\mathit{tl}}\omega k{m}_{\mathit{tl}}+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega U{m}_{\mathit{hl}}\hfill \\ \hfill & +\sum _t\sum _h{\delta }_{\mathit{th}}\omega R{m}_{\mathit{th}}+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega L{m}_{\mathit{ld}}\hfill \\ \hfill & +\sum _b\sum _j\sum _d{\delta }_{\mathit{bd}}\omega V{m}_{\mathit{bd}}^j+\sum _d\sum _a{\delta }_{\mathit{da}}\omega D{m}_{\mathit{da}}\hfill \\ \hfill \end{array}$$ 2

$$Min{z}_2^2=\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}{\mathit{Wc}}_o$$ 3

$$Max{z}_2^3=\sum _o\sum _a\sum _c\sum _g\sum _{i}Aj{o}_{\mathit{agi}}^{\mathit{oc}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _{t}Tj{o}_t{r}_t+\sum _{h}Hj{o}_h{u}_h+\sum _g\sum _{d}Dj{o}_d^g{x}_{d-}^g\left(\sum _o\sum _a\sum _c\sum _g\sum _{i}Al{j}_{\mathit{agi}}^{\mathit{oc}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _{t}Tl{j}_t{r}_t+\sum _{h}Hl{j}_h{u}_h+\sum _g\sum _{d}Dl{j}_d^g{x}_d^g\right)$$ 4

$$Min{z}_2^4=\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Adl}}_{\mathit{agi}}^{\mathit{oc}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tdl}}_t{r}_t+\sum _h{\mathit{Hdl}}_h{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g$$ 5

$$Max{z}_2^5=\sum _o\sum _a\sum _c\sum _g\sum _i{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{r}_t+\sum _h{u}_h+\sum _g\sum _d{x}_d^g$$ 6

$$\begin{array}{cc}\hfill Min{z}_3& =\sum _a\sum _c\sum _o\sum _i\sum _b\sum _g\left(A{t}_{\mathit{ai}}^g+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\theta }\right)B{m}_{\mathit{aib}}^{\mathit{oc}}\hfill \\ \hfill & +\sum _b\sum _j\sum _t\left(B{t}_{\mathit{bj}}+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\theta }\right)T{m}_{\mathit{bt}}^j+\sum _b\sum _j\sum _h\left(B{t}_{\mathit{bj}}+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }\right)H{m}_{\mathit{bh}}^j\hfill \\ \hfill & +\sum _b\sum _j\sum _d\left(B{t}_{\mathit{bj}}+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }\right)V{m}_{\mathit{bd}}^j+\sum _h\sum _l\left(H{t}_h+{\delta }_{\mathit{hl}}\ast \frac{\mathit{St}}{\theta }\right)U{m}_{\mathit{hl}}\hfill \\ \hfill & +\sum _t\sum _l\left(T{t}_t+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }\right)k{m}_{\mathit{tl}}+\sum _t\sum _h\left(T{t}_t+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }\right)R{m}_{\mathit{th}}\hfill \\ \hfill & +\sum _l\sum _d\left({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }\right)L{m}_{\mathit{ld}}+\sum _g\sum _d\sum _a\left(D{t}_d^g+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }\right)D{m}_{\mathit{da}}\hfill \\ \hfill \end{array}$$ 7

$$Min{z}_4={\nu }_1\frac{{\sum }_a{{\sum }_{a^{{}^{\prime }}\left(a,\ne ,a^{{}^{\prime }}\right)}ca}_{a{a}^{{}^{\prime }}}-{\mathit{ai}}_{\mathit{min}}}{{\mathit{ax}}_{\mathit{max}}-{\mathit{ai}}_{\mathit{min}}}+{\nu }_2\frac{{\sum }_b{\sum }_{b^{{}^{\prime }}\left(b,\ne ,b^{{}^{\prime }}\right)}{\mathit{cb}}_{b{b}^{{}^{\prime }}}-{\mathit{bi}}_{\mathit{min}}}{{\mathit{bx}}_{\mathit{max}}-{\mathit{bi}}_{\mathit{min}}}$$ 8

$$\sum _t{\mathit{km}}_{\mathit{tl}}{+\sum _h{\mathit{Um}}_{\mathit{hl}}=dl}_l\forall \left(l,\in ,L\right)$$ 9

$$\sum _{d}L{m}_{\mathit{ld}}=\left(\sum _{t}k{m}_{\mathit{tl}}+\sum _{h}U{m}_{\mathit{hl}}\right)\ast R{l}_l\forall \left(l\in L\right)$$ 10

$$\sum _j\sum _b{\mathit{Tm}}_{\mathit{bt}}^j=\sum _l{\mathit{km}}_{\mathit{tl}}+\sum _h{\mathit{Rm}}_{\mathit{th}}\forall \left(t,\in ,T\right)$$ 11

$$\sum _j\sum _b{\mathit{Hm}}_{\mathit{bh}}^j+\sum _t{\mathit{Rm}}_{\mathit{th}}=\sum _l{\mathit{Um}}_{\mathit{hl}}\forall (h\in H)$$ 12

$${\beta }_b\sum _a\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}=\sum _d\sum _j{\mathit{Vm}}_{\mathit{bd}}^j\forall (b\in B)$$ 13

$$\left(1,-,{\beta }_b\right)\sum _a\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}=\sum _j\sum _t{\mathit{Tm}}_{\mathit{bt}}^j+\sum _j\sum _h{\mathit{Hm}}_{\mathit{bh}}^j\forall (b\in B)$$ 14

$$\sum _d{\mathit{Dm}}_{\mathit{da}}+{\mathit{Am}}_a=\sum _b\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}\forall (a\in A)$$ 15

$$\sum _d{\mathit{Dm}}_{\mathit{da}}=\sum _b\sum _j{\mathit{Vm}}_{\mathit{bd}}^j+\sum _l{\mathit{Lm}}_{\mathit{ld}}\forall \left(d,\in ,D\right)$$ 16

$$\sum _b{\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}}\le {\mathit{Aca}}_{\mathit{ai}}^{\mathit{co}}\sum _g{y}_{\mathit{agi}}^{\mathit{oc}}\left(1,-,{\mathrm{Ar}}_{\mathrm{ai}}^{\mathrm{g}}\right)\forall \left(a,\in ,A,,,i,\in ,I,,,c,\in ,C,,,o,\in ,O\right)$$ 17

$$\sum _b\sum _j{\mathit{Tm}}_{\mathit{bt}}^j\le {\mathit{Tca}}_t{r}_t\left(1,-,{\mathit{Tr}}_t\right)\forall \left(t,\in ,T\right)$$ 18

$$\sum _b\sum _j{\mathit{Hm}}_{\mathit{bh}}^j\le {\mathit{Hca}}_h{u}_h\left(1,-,{\mathit{Hr}}_h\right)\forall \left(h,\in ,H\right)$$ 19

$$\sum _t{\mathit{Rm}}_{\mathit{th}}\le {\mathit{Hca}}_h{u}_h{\mathit{Hr}}_h\forall \left(h,\in ,H\right)$$ 20

$$\sum _a{\mathit{Dm}}_{\mathit{da}}\le {\mathit{Dca}}_d\sum _g{x}_d^g\left(1,-,{\mathrm{Dr}}_{\mathrm{d}}^{\mathrm{g}}\right)\forall \left(d,\in ,D\right)$$ 21

$$\sum _t{\mathit{Tm}}_{\mathit{bt}}^j+\sum _h{\mathit{Hm}}_{\mathit{bh}}^j+\sum _d{\mathit{Vm}}_{\mathit{bd}}^j\le {\mathit{eb}}_b^j{\mathit{Cb}}_b(1-{\mathit{Br}}_{\mathit{bj}})\forall \left(b,\in ,B,,,j,\in ,J\right)$$ 22

$$\sum _t{r}_t\ge 1\forall \left(t,\in ,T\right)$$ 23

$$\sum _g\sum _o\sum _c\sum _i{y}_{\mathit{agi}}^{\mathit{oc}}=1\forall \left(a,\in ,A\right)$$ 24

$$\sum _{a^{{}^{\prime }}(a^{{}^{\prime }}\ne a)}{\mathit{ca}}_{a{a}^{{}^{\prime }}}\le \sum _{p}p\ast {\mathit{na}}_a^p\forall \left(a,\left(a,\ne ,a^{{}^{\prime }}\right),\in ,A\right)$$ 25

$$\sum _{b^{{}^{\prime }}(b^{{}^{\prime }}\ne b)}{\mathit{ca}}_{b{b}^{{}^{\prime }}}\le \sum _{q}q\ast {\mathit{nb}}_b^q\forall \left(b,\left(b,\ne ,b^{{}^{\prime }}\right),\in ,B\right)$$ 26

$$\sum _i{\mathit{ea}}_a^i\le 1\forall \left(a,\in ,A\right)$$ 27

$$\sum _j{\mathit{eb}}_b^j\le 1\forall \left(b,\in ,B\right)$$ 28

$$\sum _i{\mathit{ea}}_a^i\le \sum _p{\mathit{na}}_a^p\forall \left(a,\in ,A\right)$$ 29

$$\sum _j{\mathit{eb}}_b^j\le \sum _q{\mathit{nb}}_b^q\forall \left(b,\in ,B\right)$$ 30

$$\sum _{i}la\left(i\right){\mathit{ea}}_a^i\le \sum _{p}p\ast {\mathit{na}}_a^p\forall \left(a,\in ,A\right)$$ 31

$$\sum _{p}p\ast {\mathit{na}}_a^p\le \sum _{i}ua\left(i\right){\mathit{ea}}_a^i\forall \left(a,\in ,A\right)$$ 32

$$\sum _{j}lb\left(j\right){\mathit{eb}}_b^j\le \sum _{q}q\ast {\mathit{nb}}_b^q\forall \left(b,\in ,B\right)$$ 33

$$\sum _{q}q\ast {\mathit{nb}}_b^q\le \sum _{j}ub\left(j\right){\mathit{eb}}_b^j\forall \left(b,\in ,B\right)$$ 34

$${\mathit{ca}}_{a{a}^{{}^{\prime }}}+{\mathit{ca}}_{a^{{}^{\prime }}{a}^{{}^{\prime }{}^{\prime }}}-{\mathit{ca}}_{a{a}^{{}^{\prime }{}^{\prime }}}\le 1\forall \left(a,,,a^{{}^{\prime }},,,a^{{}^{\prime }{}^{\prime }},\in ,A\right)$$ 35

$${\mathit{cb}}_{b{b}^{{}^{\prime }}}+{\mathit{cb}}_{b^{{}^{\prime }}{b}^{{}^{\prime }{}^{\prime }}}-{\mathit{cb}}_{b{b}^{{}^{\prime }{}^{\prime }}}\le 1\forall (b,b^{{}^{\prime }},b^{{}^{\prime }{}^{\prime }}\in B)$$ 36

$$\sum _g\sum _o\sum _c\sum _i{y}_{\mathit{agi}}^{\mathit{oc}}\ge \sum _i{\mathit{ea}}_a^i\forall \left(a,\in ,A\right)$$ 37

$$y_{\mathit{agi}}^{\mathit{oc}}\le {\mathit{ea}}_a^i\forall \left(a,\in ,A,,,i,\in ,I,,,c,\in ,C,,,o,\in ,O,,,g,\in ,G\right)$$ 38

$$y_{\mathit{agi}}^{\mathit{oc}},r_t,u_h,x_d^gϵ\left\{0,1\right\}$$ 39

$${\mathit{Bm}}_{\mathit{aib}}^{\mathit{oc}},{\mathit{Tm}}_{\mathit{bt}}^j,{\mathit{Hm}}_{\mathit{bh}}^j,{\mathit{Vm}}_{\mathit{bd}}^j,{\mathit{Um}}_{\mathit{hl}},{\mathit{km}}_{\mathit{tl}},{\mathit{Rm}}_{\mathit{th}},{\mathit{Lm}}_{\mathit{ld}},{\mathit{Dm}}_{\mathit{da}}\ge 0$$ 40

In the first objective function (Eq. 1), apart from four sentences reflecting the fixed costs of establishing the facility, the other ones are related to the costs of production and transportation in the iron and steel SC network and collaboration costs in the manufacturing plants a and b. In the second objective function (Eq. 2), in the first part, the amount of carbon dioxide emissions from production in the primary steel manufacturing plant is minimized, and carbon dioxide minimization from transportation is in other phrases. The third objective function (Eq. 3) considers the minimum rate of water consumption in steel plants. Objective function (4) maximizes the job opportunities created and minimizes the number of job opportunities lost due to disruption. Objective function (5) minimizes the number of lost working days. In the sixth objective function (Eq. 6), to reduce network risk and also justice in creating jobs, the number of facilities launched is maximized. Equation (7) minimizes the total forward and reverse flow time. Finally, Eq. (8) states that risk reduction associated with collaboration, such as information disclosure, necessitates collaboration reduction. For this purpose, weights are considered for collaboration in primary steel and consumable product manufacturing plants b.

Constraint (9) indicates that the number of products shipped from reliable and unreliable distributors must meet the demand of the consumption market. Constraint (10) states that the quantity of scrap collected from the consumer market should be equal to the amount of steel entering the consumer market at the rate of return of products. Constraints (11–16) reflect the balance between steel input and output to the facilities. Constraints (17–22) reflect the capacity limitation for different facilities. Constraint (23) indicates the establishing of at least one reliable distribution center. Constraint (24) states the deployment of only one steel plant with the required technology, capacity, collaboration, and fortification levels in each potential location. Constraints (25) and (26) indicate the number of collaborating companies with primary steel and consumable product manufacturing plants. Constraints (27) and (28) guarantee allocating plants to only one collaboration level. Constraints (29) and (30) show the relationship between the allocation variable and the number of collaborator plants. Constraints (31–34) determine the collaboration level based on various lower and upper bounds. Constraints (35) and (36) show if a company has a relationship with a second company that has a relationship with a third company, the first company also has a relationship with the third company. Constraint (37) shows that a company can have a collaborator after establishment, and Constraints (38) states that if a plant is established, it should be allocated to a collaboration level. Constraints (39) and (40) show the binary and continuous variables.

Robust mathematical model

A robust model allows the flexibility to increase the efficiency of the forward/reverse logistics under uncertain conditions. In this study, due to the variability of parameters and to deal with operational risk, the robust method proposed by Mulvey et al. (1995) is applied. In this method, the number of parameter deviations, such as demand and rate of return variability, can be considered to reduce operational risk, and solutions become less sensitive to the uncertain condition in the future. Another advantage is that, this method has penalty functions for the optimal and feasible solution. Also, in this model, the parameters of the robust model can be selected by the decision-maker and can be weighted according to preference between optimality and feasibility. To describe the robust mathematical model, consider the following model.

$$min\pi \left({\tau }_1,,,{\tau }_2,{,\cdots \tau }_s\right)+\phi ({\vartheta }_1,{\vartheta }_2{,\cdots \vartheta }_s)$$ 41

$$E\tau =\mho$$ 42

$$M\gamma +N_S{\tau }_S+{\vartheta }_S={\zeta }_{S}sϵS$$ 43

$${\tau }_S,\gamma ,{\vartheta }_S\ge 0sϵS$$ 44

S = {1,2,…s} represents the set of scenarios, in which occurrence probability is indicated by ${\rho }_s$. $\left({\tau }_1,,,{\tau }_2,{,\cdots \tau }_s\right)$ are control variables, and the set $({\vartheta }_1,{\vartheta }_2{,\cdots \vartheta }_s)$ shows the amount of infeasibility in the constraint. If z is a cost function represented by ${\mathrm{z}}_{\mathrm{s}}=\mathrm{f}\left({\tau }_{\mathrm{s}},,,\gamma \right)$, then it will be ${\mathrm{z}}_{\mathrm{s}}=\mathrm{f}\left({\tau }_{\mathrm{s}},,,\gamma \right)$ according to the set of scenarios. Mulvey et al. (1995) considered the variance value Eq. (45) in the mathematical model.

$$\pi \left(0\right)=\sum _{sϵS}{{\rho }_{s}z}_s+\lambda \sum _{sϵS}{\rho }_s{(z_s-\sum _{sϵs}{{\rho }_{s}z}_s)}^2$$ 45

$\lambda$ is the weight related to the deviations in the objective function. Due to the complexity of Eq. (45), Yu and Li (2000) proposed another method to calculate deviations (Eq. (46)).

And Eq. (46) is converted into a linear form with Eq. (47) and (48).${\alpha }_s$ is the coefficient used to linearize the objective functions

$$\pi \left(0\right)=\sum _{sϵS}{{\rho }_{s}z}_s+\lambda \sum _{sϵS}{\rho }_s\left|z_s,-,\sum _{sϵS},{{\rho }_{s}z}_s\right|$$ 46

$$\pi \left(0\right)=\sum _{sϵS}{{\rho }_{s}z}_s+\lambda \sum _{sϵS}{\rho }_s[(z_s-\sum _{sϵs}{{\rho }_{s}z}_s)+2{\alpha }_s]$$ 47

$$z_s-\sum _{sϵS}{{\rho }_{s}z}_s+{\alpha }_s\ge 0forallsϵS$$ 48

Finally, the objective function becomes Eq. (49). In this objective function $\phi {\sum }_{sϵS}{\rho }_s{\vartheta }_S$ is the amount of the penalty cost for infeasibility.

$$Min\pi \left(0\right)=\sum _{sϵS}{\rho }_s{z}_s+\lambda \sum _{sϵS}{\rho }_s\left[\left(z_s-\sum _{sϵs}{\rho }_s{z}_s\right)+2{\alpha }_s\right]+\phi \sum _{sϵS}{\rho }_s{\vartheta }_S$$ 49

The constraints and objective functions of the robust mathematical model are as follows:

$$Min{z}_1=\sum _o\sum _a\sum _c\sum _g\sum _i{{\mathit{Afi}}_{\mathit{ag}}^{\mathit{co}}y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tfi}}_t{r}_t+\sum _h{\mathit{Hfi}}_h{u}_h+\sum _g{\sum _d{\mathit{Dfi}}_d^{g}x}_d^g+\sum _p\sum _a{\mathit{na}}_a^p{\mathit{Acl}}_a^p+\sum _q\sum _b{\mathit{nb}}_b^q{\mathit{Bcl}}_b^q+\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}({\delta }_{\mathit{ab}}\beta +{\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}})+\sum _b{\sum }_j\sum _t{(\delta }_{\mathit{bt}}\beta +{\mathit{Bo}}_b^j){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{(\delta }_{\mathit{bh}}\beta +{\mathit{Bo}}_b^j){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b{\sum }_j\sum _d{(\delta }_{\mathit{bd}}\beta +{\mathit{Bo}}_b^j){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _t\sum _l\left({\delta }_{\mathit{tl}},\beta ,+,{\mathit{To}}_t\right){\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l\left({\delta }_{\mathit{hl}},\beta ,+,{\mathit{Ho}}_h\right){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h\left({\delta }_{\mathit{th}},\beta ,+,{\mathit{To}}_t\right){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta {\mathit{lm}}_{\mathit{ld}}^s+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +{\mathit{Do}}_d){\mathit{Dm}}_{\mathit{da}}^s)+{\lambda }_1\sum _s{\rho }_s[(\sum _a\sum _o\sum _c\sum _b\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}({\delta }_{\mathit{ab}}\beta +{\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}})+\sum _b{\sum }_j\sum _t{(\delta }_{\mathit{bt}}\beta +{\mathit{Bo}}_b^j){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{(\delta }_{\mathit{bh}}\beta +{\mathit{Bo}}_b^j){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b{\sum }_j\sum _d{(\delta }_{\mathit{bd}}\beta +{\mathit{Bo}}_b^j){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _t\sum _l\left({\delta }_{\mathit{tl}},\beta ,+,{\mathit{To}}_t\right){\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l\left({\delta }_{\mathit{hl}},\beta ,+,{\mathit{Ho}}_h\right){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h\left({\delta }_{\mathit{th}},\beta ,+,{\mathit{To}}_t\right){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta {\mathit{lm}}_{\mathit{ld}}^s+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +{\mathit{Do}}_d){\mathit{Dm}}_{\mathit{da}}^s)-\sum _s{\rho }_{\mathrm{s}}(\sum _a\sum _o\sum _c\sum _b\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}({\delta }_{\mathit{ab}}\beta +{\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}})+\sum _b{\sum }_j\sum _t{(\delta }_{\mathit{bt}}\beta +{\mathit{Bo}}_b^j){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{(\delta }_{\mathit{bh}}\beta +{\mathit{Bo}}_b^j){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b{\sum }_j\sum _d{(\delta }_{\mathit{bd}}\beta +{\mathit{Bo}}_b^j){\mathit{Vm}}_{\mathit{bd}}^{\mathrm{j}s}+\sum _t\sum _l\left({\delta }_{\mathit{tl}},\beta ,+,{\mathit{To}}_t\right){\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l\left({\delta }_{\mathit{hl}},\beta ,+,{\mathit{Ho}}_h\right){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h\left({\delta }_{\mathit{th}},\beta ,+,{\mathit{To}}_t\right){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta {\mathit{lm}}_{\mathit{ld}}^s+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +{\mathit{Do}}_d){\mathit{Dm}}_{\mathit{da}}^s)+2{\alpha }_{1s}]+{\phi }_1\sum _s\sum _l{\rho }_s{\vartheta }_{s1}^l+{\phi }_2\sum _s\sum _l{\rho }_s{\vartheta }_{s2}^l$$ 50

$$Min{z}_2^1=\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{En}}_o+\sum _a\sum _o\sum _c\sum _b{\sum }_i{\delta }_{\mathit{ab}}\omega {\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b{\sum }_j\sum _t{\delta }_{\mathit{bt}}\omega {\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{\delta }_{\mathit{bh}}\omega {\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t\sum _l{\delta }_{\mathit{tl}}\omega {\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega {\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h{\delta }_{\mathit{th}}\omega {\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega {\mathit{lm}}_{\mathit{ld}}^s+\sum _b{\sum }_j\sum _d{\delta }_{\mathit{bd}}\omega {\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _d\sum _a{\delta }_{\mathit{da}}\omega {\mathit{Dm}}_{\mathit{da}}^s)+{\lambda }_2\sum _s{\rho }_s[(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{En}}_o+\sum _a\sum _o\sum _c\sum _b{\sum }_i{\delta }_{\mathit{ab}}\omega {\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b{\sum }_j\sum _t{\delta }_{\mathit{bt}}\omega {\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{\delta }_{\mathit{bh}}\omega {\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t\sum _l{\delta }_{\mathit{tl}}\omega {\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega {\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h{\delta }_{\mathit{th}}\omega {\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega {\mathit{lm}}_{\mathit{ld}}^s+\sum _b{\sum }_j\sum _d{\delta }_{\mathit{bd}}\omega {\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _d\sum _a{\delta }_{\mathit{da}}\omega {\mathit{Dm}}_{\mathit{da}}^s)-\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{En}}_o+\sum _a\sum _o\sum _c\sum _b{\sum }_i{\delta }_{\mathit{ab}}\omega {\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b{\sum }_j\sum _t{\delta }_{\mathit{bt}}\omega {\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{\delta }_{\mathit{bh}}\omega {\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t\sum _l{\delta }_{\mathit{tl}}\omega {\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega {\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h{\delta }_{\mathit{th}}\omega {\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega {\mathit{lm}}_{\mathit{ld}}^s+\sum _b{\sum }_j\sum _d{\delta }_{\mathit{bd}}\omega {\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _d\sum _a{\delta }_{\mathit{da}}\omega {\mathit{Dm}}_{\mathit{da}}^s)+2{\alpha }_{2s}]$$ 51

$$Min{z}_2^2=\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{Wc}}_o)+{\lambda }_3\sum _s{\rho }_s[(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{Wc}}_o)-\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{Wc}}_o)+2{\alpha }_{3s}]$$ 52

$$Max{z}_2^3=\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Ajo}}_{\mathit{agi}}^{\mathit{oc}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tjo}}_t{r}_t+\sum _h{\mathit{Hjo}}_h{u}_h+\sum _g{\sum _d{\mathit{Djo}}_d^{g}x}_d^g-(\sum _s{\rho }_s(\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tlj}}_t^s{r}_t+\sum _h{\mathit{Hlj}}_h^s{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g)+{\lambda }_4\sum _s{\rho }_s[(\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tlj}}_t^s{r}_t+\sum _h{\mathit{Hlj}}_h^s{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g)-\sum _s{\rho }_s(\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tlj}}_t^s{r}_t+\sum _h{\mathit{Hlj}}_h^s{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g)+2{\alpha }_{4s}])$$ 53

$$Min{z}_3=\sum _s{\rho }_s(\sum _a\sum _c\sum _o\sum _i\sum _b\sum _g{(At}_{\mathit{ai}}^{\mathit{gs}}+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b\sum _j\sum _t({\mathit{Bt}}_{\mathit{bj}}^s+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b\sum _j\sum _h{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b\sum _j\sum _d{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _h\sum _l{(Ht}_h^s+{\delta }_{\mathit{hl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _l({\mathit{Tt}}_t^s+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{km}}_{\mathit{tl}}^s+\sum _t\sum _h({\mathit{Tt}}_t^s+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }){\mathit{lm}}_{\mathit{ld}}^s+\sum _g\sum _d\sum _a{(Dt}_d^{\mathit{sg}}+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Dm}}_{\mathit{da}}^s)+{\lambda }_5\sum _s{\rho }_s[(\sum _a\sum _c\sum _o\sum _i\sum _b\sum _g{(At}_{\mathit{ai}}^{\mathit{gs}}+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b\sum _j\sum _t({\mathit{Bt}}_{\mathit{bj}}^s+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b\sum _j\sum _h{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b\sum _j\sum _d{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _h\sum _l{(Ht}_h^s+{\delta }_{\mathit{hl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _l({\mathit{Tt}}_t^s+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{km}}_{\mathit{tl}}^s+\sum _t\sum _h({\mathit{Tt}}_t^s+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }){\mathit{lm}}_{\mathit{ld}}^s+\sum _g\sum _d\sum _a{(Dt}_d^{\mathit{sg}}+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Dm}}_{\mathit{da}}^s)-(\sum _s{\rho }_s(\sum _a\sum _c\sum _o\sum _i\sum _b\sum _g{(At}_{\mathit{ai}}^{\mathit{gs}}+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b\sum _j\sum _t({\mathit{Bt}}_{\mathit{bj}}^s+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b\sum _j\sum _h{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b\sum _j\sum _d{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _h\sum _l{(Ht}_h^s+{\delta }_{\mathit{hl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _l({\mathit{Tt}}_t^s+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{km}}_{\mathit{tl}}^s+\sum _t\sum _h({\mathit{Tt}}_t^s+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }){\mathit{lm}}_{\mathit{ld}}^s+\sum _g\sum _d\sum _a{(Dt}_d^{\mathit{sg}}+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Dm}}_{\mathit{da}}^s)+2{\alpha }_{5s}$$ 54

And objective functions ($z_2^4$), ($z_2^5$), and ($z_4$)

$$\sum _t{\mathit{km}}_{\mathit{tl}}^s+\sum _h{\mathit{Um}}_{\mathit{hl}}^s+{\vartheta }_{s1}^l={\mathit{dl}}_l^s\forall \left(l,\in ,L,,,s,\in ,S\right)$$ 55

$$\sum _d{\mathit{Lm}}_{\mathit{ld}}^s+{\vartheta }_{s2}^l=(\sum _t{\mathit{km}}_{\mathit{tl}}^s+\sum _h{\mathit{Um}}_{\mathit{hl}}^s)\ast {\mathit{Rl}}_l^s\forall \left(l,\in ,L,,,s,\in ,S\right)$$ 56

$$\sum _j\sum _b{\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}=\sum _l{\mathit{km}}_{\mathit{tl}}^s+\sum _h{\mathit{Rm}}_{\mathit{th}}^s\forall \left(t,\in ,T,,,s,\in ,S\right)$$ 57

$$\sum _j\sum _b{\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t{\mathit{Rm}}_{\mathit{th}}^s=\sum _l{\mathit{Um}}_{\mathit{hl}}^s+\sum _t{\mathit{Rm}}_{\mathit{th}}^s\forall \left(h,\in ,H,,,s,\in ,S\right)$$ 58

$${\beta }_b\sum _a\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}=\sum _d\sum _j{\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}\forall \left(b,\in ,B,,,s,\in ,S\right)$$ 59

$$\left(1,-,{\beta }_b\right)\sum _a\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}=\sum _j\sum _t{\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _j\sum _h{\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}\forall \left(b,\in ,B,,,s,\in ,S\right)$$ 60

$$\sum _d{\mathit{Dm}}_{\mathit{da}}^s+{\mathit{Am}}_a^s=\sum _b\sum _o\sum _c\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}\forall \left(a,\in ,A,,,s,\in ,S\right)$$ 61

$$\sum _a{\mathit{Dm}}_{\mathit{da}}^s=\sum _b\sum _j{\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _l{\mathit{Lm}}_{\mathit{ld}}^s\forall \left(d,\in ,D,,,s,\in ,S\right)$$ 62

$$\sum _b{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}\le {\mathit{Aca}}_{\mathit{ai}}^{\mathit{co}}\sum _g{y}_{\mathit{agi}}^{\mathit{oc}}\left(1,-,{\mathit{Ar}}_{\mathit{ai}}^{\mathit{gs}}\right)\forall \left(a,\in ,A,,,i,\in ,I,,,c,\in ,C,,,o,\in ,O,,,s,\in ,S\right)$$ 63

$$\sum _b\sum _j{\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}\le {\mathit{Tca}}_t{r}_t\left(1,-,{\mathit{Tr}}_t^s\right)\forall \left(t,\in ,T,,,s,\in ,S\right)$$ 64

$$\sum _b\sum _j{\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}\le {\mathit{Hca}}_h{u}_h\left(1,-,{\mathit{Hr}}_h^s\right)\forall \left(h,\in ,H,,,s,\in ,S\right)$$ 65

$$\sum _t{\mathit{Rm}}_{\mathit{th}}^s\le {\mathit{Hca}}_h{u}_h{\mathit{Hr}}_h^s\forall \left(h,\in ,H,,,s,\in ,S\right)$$ 66

$$\sum _a{\mathit{Dm}}_{\mathit{da}}^s\le {\mathit{Dca}}_d\sum _g{x}_d^g\left(1,-,{\mathit{Dr}}_d^g\right)\forall \left(d,\in ,D,,,s,\in ,S\right)$$ 67

$$\sum _t{\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _h{\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _d{\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}\le {\mathit{eb}}_b^j{\mathit{Cb}}_b\left(1,-,{\mathit{Br}}_{\mathit{bj}}^s\right)\forall \left(b,\in ,B,,,j,\in ,J,,,s,\in ,S\right)$$ 68

$${\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}},{\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}},{\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}},{\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}},{\mathit{Um}}_{\mathit{hl}}^s,{\mathit{Rm}}_{\mathit{th}}^s,{\mathit{lm}}_{\mathit{ld}}^s,{\mathit{Dm}}_{\mathit{da}}^s,{\mathit{Am}}_a^s\ge 0$$ 69

Subject to constraint (23–39).

According to Eq. (48), linearization constraints (70–74) are added.

$$(\sum _a\sum _o\sum _c\sum _b\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}({\delta }_{\mathit{ab}}\beta +{\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}})+\sum _b{\sum }_j\sum _t{(\delta }_{\mathit{bt}}\beta +{\mathit{Bo}}_b^j){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{(\delta }_{\mathit{bh}}\beta +{\mathit{bp}}_b^j){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b{\sum }_j\sum _d{(\delta }_{\mathit{bd}}\beta +{\mathit{Bo}}_b^j){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _t\sum _l\left({\delta }_{\mathit{tl}},\beta ,+,{\mathit{To}}_t\right){\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l\left({\delta }_{\mathit{hl}},\beta ,+,{\mathit{Ho}}_h\right){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h\left({\delta }_{\mathit{th}},\beta ,+,{\mathit{To}}_t\right){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta {\mathit{lm}}_{\mathit{ld}}^s+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +{\mathit{Do}}_d){\mathit{Dm}}_{\mathit{da}}^s)-\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b\sum _i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}({\delta }_{\mathit{ab}}\beta +{\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}})+\sum _b{\sum }_j\sum _t{(\delta }_{\mathit{bt}}\beta +{\mathit{Bo}}_b^j){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{(\delta }_{\mathit{bh}}\beta +{\mathit{bp}}_b^j){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b{\sum }_j\sum _d{(\delta }_{\mathit{bd}}\beta +{\mathit{Bo}}_b^j){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _t\sum _l\left({\delta }_{\mathit{tl}},\beta ,+,{\mathit{To}}_t\right){\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l\left({\delta }_{\mathit{hl}},\beta ,+,{\mathit{Ho}}_h\right){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h\left({\delta }_{\mathit{th}},\beta ,+,{\mathit{To}}_t\right){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\beta {\mathit{lm}}_{\mathit{ld}}^s+\sum _d\sum _a({\delta }_{\mathit{da}}\beta +{\mathit{Do}}_d){\mathit{Dm}}_{\mathit{da}}^s)+{\alpha }_{1s}\ge 0\forall (s\in S)$$ 70

$$(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{En}}_o+\sum _a\sum _o\sum _c\sum _b{\sum }_i{\delta }_{\mathit{ab}}\omega {\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b{\sum }_j\sum _t{\delta }_{\mathit{bt}}\omega {\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{\delta }_{\mathit{bh}}\omega {\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t\sum _l{\delta }_{\mathit{tl}}\omega {\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega {\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h{\delta }_{\mathit{th}}\omega {\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega {\mathit{lm}}_{\mathit{ld}}^s+\sum _b{\sum }_j\sum _d{\delta }_{\mathit{bd}}\omega {\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _d\sum _a{\delta }_{\mathit{da}}\omega {\mathit{Dm}}_{\mathit{da}}^s)-\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{En}}_o+\sum _a\sum _o\sum _c\sum _b{\sum }_i{\delta }_{\mathit{ab}}\omega {\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b{\sum }_j\sum _t{\delta }_{\mathit{bt}}\omega {\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b{\sum }_j\sum _h{\delta }_{\mathit{bh}}\omega {\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _t\sum _l{\delta }_{\mathit{tl}}\omega {\mathit{km}}_{\mathit{tl}}^s+\sum _h\sum _l{\delta }_{\mathit{hl}}\omega {\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _h{\delta }_{\mathit{th}}\omega {\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d{\delta }_{\mathit{ld}}\omega {\mathit{lm}}_{\mathit{ld}}^s+\sum _b{\sum }_j\sum _d{\delta }_{\mathit{bd}}\omega {\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _d\sum _a{\delta }_{\mathit{da}}\omega {\mathit{Dm}}_{\mathit{da}}^s)+{\alpha }_{2s}\ge 0\forall \left(s,\in ,S\right)$$ 71

$$\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{Wc}}_o-\sum _s{\rho }_s(\sum _a\sum _o\sum _c\sum _b{\sum }_i{\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}{\mathit{Wc}}_o)+{\alpha }_{3s}\ge 0\forall (s\in S)$$ 72

$$(\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tlj}}_t^s{r}_t+\sum _h{\mathit{Hlj}}_h^s{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g)-\sum _s{\rho }_s(\sum _o\sum _a\sum _c\sum _g\sum _i{\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}{y}_{\mathit{agi}}^{\mathit{oc}}+\sum _t{\mathit{Tlj}}_t^s{r}_t+\sum _h{\mathit{Hlj}}_h^s{u}_h+\sum _g{\sum _d{\mathit{Ddl}}_d^{g}x}_d^g)+{\alpha }_{4s}\ge 0\forall \left(s,\in ,S\right)$$ 73

$$(\sum _a\sum _c\sum _o\sum _i\sum _j\sum _g{(At}_{\mathit{ai}}^{\mathit{gs}}+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\partial }){\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b\sum _j\sum _t({\mathit{Bt}}_{\mathit{bj}}^s+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\partial }){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b\sum _j\sum _h{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b\sum _j\sum _d{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _h\sum _l{(Ht}_h^s+{\delta }_{\mathit{hl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _l({\mathit{Tt}}_t^s+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{km}}_{\mathit{tl}}^s+\sum _t\sum _h({\mathit{Tt}}_t^s+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }){\mathit{lm}}_{\mathit{ld}}^s+\sum _g\sum _d\sum _a{(td}_d^{\mathit{sg}}+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Dm}}_{\mathit{da}}^s)-(\sum _s{\rho }_s(\sum _a\sum _c\sum _o\sum _i\sum _j\sum _g{(At}_{\mathit{ai}}^{\mathit{gs}}+{\delta }_{\mathit{ab}}\ast \frac{\mathit{kt}}{\partial }){\mathit{Bm}}_{\mathit{aib}}^{\mathit{ocs}}+\sum _b\sum _j\sum _t({\mathit{Bt}}_{\mathit{bj}}^s+{\delta }_{\mathit{bt}}\ast \frac{\mathit{kt}}{\partial }){\mathit{Tm}}_{\mathit{bt}}^{\mathit{js}}+\sum _b\sum _j\sum _h{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bh}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Hm}}_{\mathit{bh}}^{\mathit{js}}+\sum _b\sum _j\sum _d{(Bt}_{\mathit{bj}}^s+{\delta }_{\mathit{bd}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Vm}}_{\mathit{bd}}^{\mathit{js}}+\sum _h\sum _l{(Ht}_h^s+{\delta }_{\mathit{hl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Um}}_{\mathit{hl}}^s+\sum _t\sum _l({\mathit{Tt}}_t^s+{\delta }_{\mathit{tl}}\ast \frac{\mathit{kt}}{\theta }){\mathit{km}}_{\mathit{tl}}^s+\sum _t\sum _h({\mathit{Tt}}_t^s+{\delta }_{\mathit{th}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Rm}}_{\mathit{th}}^s+\sum _l\sum _d({\delta }_{\mathit{ld}}\ast \frac{\mathit{kt}}{\theta }){\mathit{lm}}_{\mathit{ld}}^s+\sum _g\sum _d\sum _a{(td}_d^{\mathit{sg}}+{\delta }_{\mathit{da}}\ast \frac{\mathit{kt}}{\theta }){\mathit{Dm}}_{\mathit{da}}^s)+{\alpha }_{5s}\ge 0\forall (s\in S)$$ 74

The multi-objective solution method

The suggested method is a combination of the methods of Parra et al. (2005), Jimenez et al. (2007), and Torabi and Hassini (2008), (the TH method), and is applied by some researchers such as Pishvaee and Torabi (2010). It has the advantage that it does not increase the number of objective functions and constraints and maintains the linear state of the problem; Here, to reduce the limitations of the proposed method, we considered changes in that method and customized it by changing aggregation model and considering other hypotheses for the model. The steps of the proposed method are as follows.

• Step 1 Due to a large number of objective functions and the difficulty of deciding on objectives and considering the four areas of economic, sustainability, responsiveness, and collaboration risk, normalization using the method of Pishvaee et al. (2014) is applied.

  $$\text{Max}{z}_1=\frac{max(z_1)-z_1}{max(z_1)-min(z_1)}$$ 75

  $$\begin{array}{cc}\hfill \text{Max}{z}_2=& w{s}_1\frac{max\left(z_2^1\right)-z_2^1}{max(z_2^1)-min(z_2^1)}+w{s}_2\frac{max\left(z_2^2\right)-z_2^2}{max(z_2^2)-min(z_2^2)}+w{s}_3\frac{z_2^3-min\left(z_2^3\right)}{max(z_2^3)-min(z_2^3)}\hfill \\ \hfill & +w{s}_4\frac{max(z_2^4)-z_2^4}{max(z_2^4)-min(z_2^4)}+w{s}_5\frac{z_2^5-min(z_2^5)}{max(z_2^5)-min(z_2^5)}\hfill \\ \hfill \end{array}$$ 76

  $$\text{Max}{z}_3=\frac{max\left(z_3\right)-z_3}{max\left(z_3\right)-min\left(z_3\right)}$$ 77

  $$\text{Max}{z}_4=\frac{max\left(z_4\right)-z_4}{max\left(z_4\right)-min\left(z_4\right)}$$ 78

  $\left(w{s}_1-w{s}_5\right)$ represent the normalization weights related to sustainability objective functions (CO₂ emissions, water consumption, job creation, damage to personnel, and justice in job creation).

• Step 2 Determine the ideal and anti-ideal solutions.

  For each objective function to obtain the ideal solutions ${\text{z}}_{\text{k}}^{\text{id}}$, each objective is solved separately, and the ideal solution is obtained. The worst value from the other objectives is considered the anti-ideal value for that objective function. For example, in the first objective function, the anti-ideal value is calculated as follows.

  $${\tau }_1^{n-id}=\text{min}\left(Z_1\left({\tau }_2^{\mathit{id}}\right),Z_1\left({\tau }_3^{\mathit{id}}\right),Z_1\left({\tau }_4^{\mathit{id}}\right)\right)$$ 79
• Step 3 Determine the utility function for objectives

  $${\mu }_k\left(\tau \right)=\left\{\begin{array}{cc}1\hfill & if{z}_k>z_k^{\mathit{id}}\\ \frac{z_k-z_k^{n-id}}{z_k^{\mathit{id}}-z_k^{n-id}}\hfill & if{z}_k^{n-id}\le z_k\le z_k^{\mathit{id}}\\ 0\hfill & if{z}_k<z_k^{n-id}\end{array}\right)$$ 80

• Step 4 Convert a multi-objective model to a single-objective.

  Torabi and Hassini (2008) have proposed the following method.

  $$\begin{array}{c}max\left(\eta \right){\sigma }_0+\left(1-\eta \right)\sum _{k}w{f}_k{\mu }_k\left(\tau \right)\\ s.t\\ {\mu }_k\left(\tau \right)\ge {\sigma }_0\forall k\\ \tau \epsilon F_{\tau }\\ {\sigma }_0\epsilon \left\{0,1\right\}\\ \end{array}$$ 81
  In Eq. (81), ${\sigma }_0$ indicates the minimum satisfaction level of the objective functions, and η is also considered the compensation coefficient between the objectives. $w{f}_k$ represents the importance weights related to the final objective function k (cost, sustainability, responsiveness, and collaboration risks). In the proposed model in the fourth step, the difference between the minimum satisfaction and different objectives is not considered. To solve this issue and to get closer the model to the actual situation, weights are considered for each of the values of the minimum satisfaction level, and a new aggregation model is introduced. So, the new aggregation model is as fallow.

  $$\begin{array}{c}max\left(\eta \right)\sum _{k}w{f}_k{\sigma }_{0k}+\left(1-\eta \right)\sum _{k}w{f}_k{\mu }_k\left(\tau \right)\\ s.t\\ {\mu }_k\left(\tau \right)\ge {\sigma }_{0k}\forall k\\ \tau \epsilon F_{\tau }\\ {\sigma }_{0k}\epsilon \left\{0,1\right\}\\ \end{array}$$ 82

• Step 5 The coefficient of compensation and the coefficients of the aggregation model are considered by the decision-maker, and the relevant single-objective model is solved. If the decision-maker is unsatisfied, the values of σ and $\mathrm{\eta }$ are changed, and the single-objective model is solved.

Case study

Iron and steel have been widely used by humans. Today, the steel industry is estimated to produce about 4% to 5% of all man-made greenhouse gases (Smale et al., 2006). The average CO₂ production per ton of steel is about 1.9 tons. Due to the production of more than ($1.3E^9$) tons of steel, more than ($2E^9$) tons of CO₂ is emitted into the environment. Most CO₂ emissions are related to crude steel production (Smale et al., 2006).

Steel production technology

Today, crude steel is usually produced by two technologies: blast furnaces and direct reduction. (Strezov et al., 2013) identified the factors and indicators related to the sustainable development of the iron and steel industry. These sustainability indicators include greenhouse gases and water consumption. The results are summarized in Table 3. Also, CO₂ emissions in transportation methods are reported in Table 4. Candidate locations for the facility are shown In Fig. 3, and other required data for a steel SC in Iran are reported in Table 5.

Table 3.

The sustainability parameters in two methods of steel production (Strezov et al., 2013)

Sustainability factor	Blast furnace	Direct reduction
CO2 emission (t CO2/t)	2.1	1.1
Water consumption (m3/t)	2.6	1.4

Table 4.

CO₂ emissions in transportations methods (Piecyk, and McKinno, 2010)

Type of transportation	g CO2/tone-km
road	62
railroad	22
Marine	31
Combining rail and road	26
Air transportation	602
Pipeline	5

Fig. 3.

Facilities candidate locations in the case study

Table 5.

Range of parameters

Parameter	Value	Parameter	Value
${\mathit{Afi}}_{\mathit{ag}}^{1\mathit{o}}$	Uniform (300,000–960,000)	${\mathit{Alj}}_{\mathit{agi}}^{\mathit{ocs}}$	Uniform(15–80)
${\mathit{Afi}}_{\mathit{ag}}^{2\mathit{o}}$	$\mathrm{\nabla }{\mathbf{\ast }\mathit{A}\mathit{f}\mathit{i}}_{\mathit{ag}}^{1\mathit{o}}$	${\mathit{Hlj}}_{\mathit{h}}$	Uniform(2–9)
$\mathrm{\nabla }$ l	Uniform (0.5–0.7)	${\mathit{Tlj}}_{\mathit{t}}$	Uniform(1–4)
${\mathit{Tfi}}_{\mathit{t}}$	Uniform (5000–11,000)	${\mathit{Dlj}}_{\mathit{d}}^{\mathit{gs}}$	Uniform(2–9)
${\mathit{Hfi}}_{\mathit{h}}$	Uniform (4000–8000)	${\mathit{Ao}}_{\mathit{ai}}^{\mathit{oc}}$	Uniform (60–120)
${\mathit{Dfi}}_{\mathit{d}}^{\mathit{g}}$	Uniform (4000–8000)	${\mathit{Bo}}_{\mathit{b}}^{\mathit{j}}$	Uniform (25–35)
${\mathit{Aca}}_{\mathit{ai}}^{1\mathit{o}}$	Uniform(1500,3000)	${\mathit{Ho}}_{\mathit{h}}$	Uniform (2–5)
${\mathit{Aca}}_{\mathit{ai}}^{2\mathit{o}}$	Uniform(750–1500)	${\mathit{To}}_{\mathit{t}}$	Uniform (4–7)
${\mathit{Bca}}_{\mathit{b}}$	Uniform(1500–3000)	${\mathit{Do}}_{\mathit{d}}$	Uniform (2–5)
${\mathit{Tca}}_{\mathit{t}}$	Uniform (800–1800)	${\mathit{\beta }}_{\mathit{b}}$	Uniform (0.85–0.97)
${\mathit{Hca}}_{\mathit{h}}$	Uniform (800–1800)	$\mathit{kt}$	Uniform (1–1.5)
${\mathit{Dca}}_{\mathit{d}}$	Uniform (800–1800)	${\mathit{At}}_{\mathit{ai}}^{\mathit{gs}}$	Uniform (0.6–0.8)
${\mathit{\delta }}_{\mathit{ij}}$	Uniform (20–1000)	${\mathit{Bt}}_{\mathit{bj}}^{\mathit{s}}$	Uniform (0.5–0.7)
${\mathit{Ar}}_{\mathit{ai}}^{\mathit{gs}}$,${\mathit{Br}}_{\mathit{bj}}^{\mathit{s}}$	Uniform (0–1)	${\mathit{Tt}}_{\mathit{t}}^{\mathit{s}}$	Uniform (0.1–0.2)
${\mathit{Hr}}_{\mathit{h}}^{\mathit{s}},{\mathit{Dr}}_{\mathit{d}}^{\mathit{gs}}$	Uniform (0–1)	${\mathit{Dt}}_{\mathit{d}}^{\mathit{gs}}$	Uniform (0.15–0.25)
${\mathit{Ajo}}_{\mathit{agi}}^{\mathit{oc}}$	Uniform (200,900)	${\mathit{Adl}}_{\mathit{agi}}^{\mathit{oc}}$	Uniform (15–100)
${\mathit{Hjo}}_{\mathit{h}}$	Uniform (15–25)	${\mathit{Hdl}}_{\mathit{h}}$	Uniform (4–6)
${\mathit{Tjo}}_{\mathit{t}}$	Uniform (15–30)	${\mathit{Tdl}}_{\mathit{t}}$	Uniform (3–5)
${\mathit{Djo}}_{\mathit{d}}^{\mathit{g}}$	Uniform (15–25)	${\mathit{Ddl}}_{\mathit{d}}^{\mathit{g}}$	Uniform (3–5)

Analysis of results

To cope with the operational risks, first, by the fractional design of the experiment, demand, return rate, shipping cost, production cost in the primary steel manufacturing plant, and production costs in consumable production plants factors are considered (respectively, factors A to E). The effect of each factor on the total cost by changing 10% is shown in Fig. 4.

Fig. 4.

Minitab software output to examine the effect of factors on cost

As per Fig. 4, the most significant effect of the variability of the factors is related to the demand and rate of return. So, the variability, by experts, is identified and considered by different scenarios in the mathematical model.

As mentioned in the first step of the solution method, for the sustainability objective function, the weight of 0.3 for CO₂ emission, 0.2 for water consumption, 0.3 for the job opportunity, 0.2 for damages to personnel, and 0.05 for facilities dispersion and creating justice in creating jobs, is considered by experts of iron and steel industry. Also, for objective function related to collaboration risks, a weight of 0.5 is considered for collaboration in primary steel and consumable manufacturing plants. Next, according to the second step, the ideal and anti-ideal values of the objective functions are calculated. After converting the multi-objective model to a single-objective model using the new aggregation model, the requirements for the decision-maker are provided by the model. To implement the proposed mathematical model, the software Gams is used. An example of calculations performed on a computer with a Core-i5 processor and 4 GB of RAM is reported in Table 6.

Table 6.

Result of the multi-objective model for different values of parameters

$\eta$	${\sigma }_{0k}$				Proposed aggregation model				TH model
					Cost utility	Sustainability utility	Responsiveness utility	Collaboration risk utility	Cost utility	Sustainability utility	Responsiveness utility	Collaboration risk utility
0.2	0.6	0.6	0.6	0.6	0.628	0.6	0.817	1	0.628	0.6	0.817	1
	0.6	0.6	0.6	0.5	0.622	0.622	0.827	0.8	0.635	0.568	0.867	0.95
	0.6	0.5	0.6	0.5	0.642	0.571	0.867	0.8	0.635	0.568	0.867	0.95
	0.4	0.7	0.5	0.8	0.401	0.7	0.803	0.917	0.641	0.571	0.868	0.91
	0.8	0.5	0.5	0.4	0.8	0.538	0.752	0.5	0.641	0.571	0.868	0.91
	0.3	0.7	0.8	0.9	0.346	0.7	0.869	1	0.641	0.572	0.867	0.8
	0.5	0.7	0.4	0.1	0.5	0.7	0.4	0.25	0.638	0.587	0.852	0.85
	0.1	0.7	0.9	0.9	0.351	0.7	0.909	1	0.638	0.587	0.852	0.85
	0.7	0.4	0.8	0.4	0.7	0.523	0.891	0.917	0.641	0.571	0.868	0.91
	0.8	0.4	0.7	0.3	0.8	0.540	0.738	0.5	0.641	0.572	0.867	0.8
0.1	0.6	0.6	0.6	0.6	0.620	0.6	0.828	1	0.620	0.6	0.828	1
	0.6	0.6	0.6	0.5	0.622	0.622	0.827	0.8	0.634	0.572	0.867	0.95
	0.6	0.5	0.6	0.5	0.642	0.571	0.867	0.8	0.634	0.572	0.867	0.95
	0.4	0.7	0.5	0.8	0.401	0.703	0.795	0.917	0.641	0.570	0.869	0.91
	0.8	0.5	0.5	0.4	0.8	0.534	0.760	0.5	0.641	0.570	0.869	0.91
	0.3	0.7	0.8	0.9	0.349	0.703	0.866	1	0.642	0.571	0.867	0.8
	0.5	0.7	0.4	0.1	0.5	0.7	0.4	0.25	0.639	0.581	0.859	0.85
	0.1	0.7	0.9	0.9	0.349	0.7	0.911	1	0.639	0.581	0.859	0.85
	0.7	0.4	0.8	0.4	0.7	0.525	0.868	0.917	0.641	0.570	0.869	0.91
	0.8	0.4	0.7	0.3	0.8	0.537	0.777	0.5	0.642	0.571	0.867	0.8

As shown in Table 6, the proposed method provides more balanced solutions. It happens because the TH method considers the lowest value of the satisfaction level for ${\sigma }_0({\sigma }_0=min\left({\sigma }_{0k}\right))$, but this method for each objective function considers the level of satisfaction separately. As reported in Table 6, if all objective functions have the same level of satisfaction, the answer is the same for both methods. Also, when the minimum satisfaction level of the objective function is important, the proposed method performs better than the TH method.

Validation of the robust model

To perform the sensitivity analysis in the proposed robust model, the relationships between infeasibility cost and unsatisfied demand also, infeasibility cost, and total cost are evaluated. As shown in Fig. 5 and Fig. 6, by increasing the infeasibility cost, the total cost exponentially increased, and the unsatisfied demand exponentially decreased, which is consistent with the results presented in the paper by Mulvey et al. (1995).

Fig. 5.

Relationship between Infeasibility cost and total shortage

Fig. 6.

Relationship between Infeasibility cost and total cost

Robust modeling is looking for a solution that is less sensitive to changing parameters (optimality robustness) and remains feasible due to different scenarios (model robustness), as can be seen from Fig. 7, considering the More infeasibility and variance cost the robustness value and the total costs increases. These results are entirely consistent with the characteristics of the robust method proposed by (Mulvey et al., 1995) and indicate the validity of the presented model. So, the decision-maker according to the degree of uncertainty and variability of the data can decide regarding these parameters.

Fig. 7.

Total cost and shortage for different $\mathit{\lambda }$ and $\phi$

To compare the result of the Robust and deterministic model, some Scenarios, according to the probability of their occurrence, are selected randomly. And the results for the three minimum satisfaction levels (${\sigma }_{0\text{k}}$) are reported in Table 7. In all three cases, the robust model provides less average and standard deviation of cost than the deterministic model. It happens, because the shortage cost is considered in the network, and deviations are not taken into account in the deterministic model; therefore, the probability of shortage and the total cost in the deterministic model increase.

Table 7.

Values of the deterministic and robust model in different scenarios and satisfaction levels

Number	Scenario	${\mathrm{\sigma }}_{0\text{k}}$(0.6,0.6,0.6,0.6)		${\mathrm{\sigma }}_{0\text{k}}$(0.5,0.5,0.5,0.5)		${\mathrm{\sigma }}_{0\text{k}}$(0.4,0.4,0.4,0.4)
		Deterministic	Robust	Deterministic	Robust	Deterministic	Robust
1	3	3,569,687	3,523,022	3,478,537	3,445,084	3,462,880	3,432,012
2	2	3,679,157	3,622,022	3,577,537	3,544,084	3,561,880	3,531,012
3	5	3,375,206	3,421,798	3,292,942	3,366,587	3,285,952	3,354,898
4	4	3,477,615	3,692,016	3,389,782	3,462,418	3,373,910	3,441,069
5	3	3,569,687	3,523,022	3,478,537	3,445,084	3,462,880	3,432,012
6	1	3,818,187	3,770,522	3,726,037	3,692,584	3,710,380	3,679,512
7	3	3,569,687	3,523,022	3,478,537	3,445,084	3,462,880	3,432,012
8	2	3,679,157	3,622,022	3,577,537	3,544,084	3,561,880	3,531,012
9	1	3,818,187	3,770,522	3,726,037	3,692,584	3,710,380	3,679,512
10	2	3,679,157	3,622,022	3,577,537	3,544,084	3,561,880	3,541,012
Mean		3,623,573	3,608,999	3,530,302	3,518,168	3,515,490	3,505,406
SD		132,936	108,074	129,399	102,453	127,850	103,032

Sensitivity analysis on changes in demand

To test the validity of the proposed model, a sensitivity analysis is performed on some of the parameters of the proposed model. For this purpose, by considering the fixed capacity for the facility, the demand rate was changed from –15% to + 15%, and Fig. 8 shows the change in the total cost and CO₂ emission in the entire network. When the demand increases, the model tends to meet the increased demand, so the cost of transportation, production, and facilities establishing increases and the total cost also increases. On the other hand, due to the rise in transportation and production to meet the demand, CO₂ emission increases.

Fig. 8.

Sensitivity analysis on changes in demand

Sensitivity analysis on capacity reduction duo to disruption

To investigate the role of disruption in the network, the percentage of capacity reduction due to disruption is changed from 0 to 50%, considering constant demand. As reported in Fig. 9, when the percentage of the capacity reduction increases, the total cost increases because to meet the demand, new facilities must be established. On the other hand, the transportation in the network duo to disruption rises, which causes an increase in costs. Also, by decreasing capacity duo to disruption, collaboration in the SC network to meet demand and the product transferred between distribution centers increase.

Fig. 9.

Sensitivity analysis on capacity reduction due to disruption

Sensitivity analysis on capacity

By changing the capacity of steel production factories and collection centers from 0 to 25%, the change in the total cost is shown in Fig. 10. According to Fig. 10, when capacity assuming constant establishing costs increases, the total cost increases; because the fixed and transportation costs in the network are decreased. It is also seen from Fig. 10 that the cost reduction for steel manufacturers is more than collection centers because the fixed cost of these factories is more than collection centers.

Fig. 10.

Sensitivity analysis on capacity

Analysis of proposed resilient strategies

Figure 11 compares the total SC network cost of different resilience strategies. As shown in Fig. 11, applying the fortification and collaboration strategies reduces the total costs, and their simultaneous use reduces costs by about 11%. The reason for this decrease can be that the disruption reduces the capacity, which in case of customer's unmet demands will face a shortage cost, and timely supply of customer demand will require the establishment of new facilities and increase the quantity of transportation in the network. In any case, costs increase; therefore, applying disruption strategies will help reduce costs. Disruption also makes scrap products unavailable to manufacturers, which is another factor in growing network costs.

Fig. 11.

Relationship between Resilience strategies and total cost

The minimum utility value for the cost function is 0.65 to investigate the effects of resilience strategies on the utility of sustainability, and finally, Fig. 12 reports its effect extent. Resilience strategies also improve sustainability since, disruption reduces capacity, more transportation must take place, and therefore the degree of CO₂ emission increases. Also, job loss resulting from disruption will make resilient strategies effective in increasing sustainability indicators. For this case study, applying resilience strategies can improve the utility function by about 13%. Also, considering the minimum utility of 0.65 for costs, the effects of resilience strategies on SC responsiveness are reported in Fig. 13. It is evident from Fig. 13 that, using resilience strategies such as collaboration can improve responsiveness by 24%. The reduced disruption caused by the collaboration strategy in the SC is the reason behind this. For example, companies can share knowledge or maintenance capability in disruptive conditions and improve the delay time.

Fig. 12.

Relationship between Resilience strategies and sustainability

Fig. 13.

Relationship between resilience strategies and total responsiveness

Considering two objectives as essential objective functions and setting ${\sigma }_{0\text{k}}=0.5$ for the other objectives can determine the relationship between the two objectives. As shown in Fig. 14, an increase in the utility of one objective function decreases the utility of another. According to sustainability and responsiveness indicators, the decision-maker can make the desired decisions about the structure of the network, the degree of collaboration among its members, and the selection of technologies.

Fig. 14.

The reciprocal performance of objective functions

Managerial implications

Given the implementation of the steel industry and sensitivity analysis, managerial implications are as follows:

• The analysis of the robust model shows that it is possible to judge the parameters of the robust model according to shortage and total costs. Therefore, when the uncertainty in the parameters, such as demand for steel, is high, it is possible to avoid additional costs in the future by assigning more weight to cost variance.

• Managers can reduce the total cost by using the proposed strategies since they usually keep some inventory for critical times to prevent shortages and loss of the organization's reputation and meet customer demand on time. Therefore, they can apply collaboration and fortification strategies to deal with the risk of disruption in steel SC.

• According to the trade-off between the risk of collaboration and the benefits created by sharing assets, managers and investors in this area can apply the proposed model to decide about collaboration and asset sharing (for example, sharing maintenance capabilities, spare parts, condition monitoring equipment, and sharing information and expert personnel in iron and steel SC). The current research can provide a favorable perspective for SC managers to design and redesign the SC network.

• Since as the number of scraps increases, the number of required facilities and the number of related jobs opportunity increases, considering this point by managers and creating incentive policies to increase the collection of scraps can lead to an increase in sustainability.

• The research results show that collaboration and fortification strategies can improve the amount of CO₂ emission and social responsibility parameters of the SC, and the efficiency of collaboration is higher than fortification. Therefore, managers and decision-makers can increase the SC responsiveness rate and improve the sustainability capabilities of the SC by using horizontal collaboration in disruption. Also, due to the lower cost of the collaboration strategy, it has a higher priority than fortification.

• The more the level of collaboration in primary steel and consumable manufacturing plants increases, the risk of opportunism and the disclosure of sensitive information will increase. Therefore, they can decide on communication in the SC considering the balance between the risks and benefits of collaboration when collaboration has risks for the organization.

• The higher the steel demand, the greater possibility of not meeting demand, which causes significant damage to the company's reputation and the profitability of the SC. A collaboration strategy can create more capacity and increase the production rate since the effects of disruption become less. By applying the collaboration and fortification strategies, it is possible to meet more additional customers' demands at a more appropriate speed that simultaneously improves the profitability and responsiveness of the SC.

• The results from this research enable decision makers, especially managers of the iron and steel SC, to make more informed decisions about technology type, capacity and location of facilities and flow among them, colleague selection through the balance between risk and performance of the network, increasing SC responsiveness by resilience strategies, and improving SC sustainability.

• Reducing environmental impacts and considering social responsibility may increase costs, but reduced water resources and increased greenhouse gases can have irreparable effects. Also, according to the sensitivity analysis, when demand increases, CO₂ emission increases. Therefore, government awards in this field can effectively reduce the environmental effects of steel producers.

Conclusions and future research

Operational risks, such as changes in demand, and disruption risks, can affect SC performance. For example, in recent years, the covid-19 disease has confirmed this issue. Organizations have also found that they can benefit from a collaborative policy to deal with risks. This study proposes a model to design the forward/reverse logistics network and considers operational and disruption risks. Through the design of experiment, the proposed model identifies the influential factors in operational risk and applies a robust model to address the operational risks. This study also uses fortification and collaboration strategies to deal with the disruption risk. It also considers job creation, job loss due to disruption, environmental pollution, water consumption, and responsiveness of the network as objective functions in the model, and the steel industry, as one of the most influential industries in environmental pollution and job creation, is considered a case study.

The results of the analyses showed the efficiency of the proposed strategies in increasing SC performance, so that the simultaneous use of the collaboration and fortification strategy reduces costs by 11%, improves the sustainability function by 13%, and the SC responsiveness by 24%. Also, the results of the solution algorithm demonstrated the efficiency of the proposed method for solving the multi-objective model. Accordingly, the proposed study gives the manager and the decision-makers more flexibility to make decisions regarding sustainability indicators, SC responsiveness, and the final structure of the forward and reverse logistics in selecting the collaborator and the required technologies.

In future research, various indicators in SC sustainability, for example, energy and land consumption or other social responsibility issues proposed in ISO 26000 (2010), can be considered. Also, strategies to deal with disruption risk, for example, inventory (Sabouhi et al., 2018), can be addressed to make a comparison with the proposed ones. Applying other robust optimization methods (Lotfi et al., 2022) or other methods of dealing with uncertainty, such as grey systems (Kanan et al., 2022; Roy et al., 2017) and comparing the performance of the desired models with the model presented is one of the opportunities for future research. Furthermore, this study considers horizontal collaboration in the SCND. Future studies can address the use of lateral collaboration. Given the high computational complexity of the presented mathematical model resulting from a combination of binary and continuous variables, the development of new exact solution methods, such as the benders decomposition algorithm (Reddy et al., 2022) or new meta-heuristic algorithms (Hashim et al., 2022; Tanhaeean et al., 2022) for the presented model can be an attractive research topic.