Location-allocation Model for the Supply Chain of Multi-Product and Multi-Period Oil and Gas Industries with Consideration of Sustainability Under Uncertainty

Abstract

Oil and gas products are routinely transported from producers to distributors, and then from distributors to retailers. In the process, operating costs, transportation expenses, and environmental impacts often increase due to time constraints, high transportation frequency, limited inventory storage, and other factors. To reduce costs, meet end-customer demands, and create competitive advantages, sustainable development (SD) has become a key consideration for supply chain (SC) managers, prompting a shift towards more sustainable supply networks. This study aims to design a sustainable multi-period, multi-product oil and gas SC consisting of producers, distributors, and retailers under uncertain conditions. To address uncertainty in demand, production, transportation, holding costs, and facility capacity, and to maximize net present value (NPV) while considering social impacts, the study develops a Robust Counterpart Optimization model based on the robust optimization method. The model is implemented in the programming environment of SAS Studio 3.5. Numerical results provide valuable insights into strategic and planning decisions for SC design. Finally, sensitivity analyses are conducted on key parameters, with the changes in objective functions being thoroughly examined.

Keywords

supply chain robust counterpart optimization model location-allocation uncertainty sustainability

1. Introduction

The logistics network of a supply chain (SC), which encompasses manufacturers, suppliers, production and distribution centers, and warehouses, includes all functions and parties involved, directly or indirectly, in fulfilling customer requests. This network involves suppliers, manufacturers, transporters, retailers, and warehouses (Govindan et al., 2022; Zhao et al., 2024). To achieve efficiency and effectiveness within the network, many businesses and industries are compelled to invest in SCs (Armin Jabbarzadeh & Sabouhi, 2018). Supply chain network design (SCND) refers to an integrative network of facilities that carry out the procurement of materials, their transformation into intermediates, and the distribution of finished products to end customers. This process involves managing the flow of materials, resources, and information within companies. In the supply chain, the primary objective is often to minimize costs and/or optimize net benefits (Chen et al., 2023; Foroozesh et al., 2023; Tsiakis et al., 2001) Businesses have recently become increasingly aware of this concept, recognizing it as a mechanism to create a competitive advantage. SCND involves two levels of decision-making. The first-level decisions, taken at the strategic level, are costly to modify and focus on defining the network configuration. This includes optimizing the number of facilities and selecting the best locations for the network. The second-level decisions pertain to the operational aspects of the supply chain, such as allocating demands, managing material flows, and coordinating facilities (Yu and Solvang, 2020). It is well-established that production and economic systems cannot be isolated from environmental considerations. Modern ecological economic theory emphasizes the growing impact of human activities on the natural environment, underscoring the need for sustainable practices (Genovese et al., 2017; Hadi-Vencheh et al., 2024). Therefore, the emphasis on sustainability has become even more critical in the present time. The ability of a business or firm to make supply chain decisions without compromising the future condition of three key dimensions–society, the environment, and business stability (economic)–is referred to as sustainability (Tavana et al., 2023; Tirkolaee et al., 2020). With the growing global population, the concept of sustainability in SCND is increasingly seen as a crucial issue for organizations, governments, and environmentalists. Sustainable development (SD), which encompasses social, economic, and environmental sustainability, provides a framework to achieve social justice, foster economic growth, and promote environmental stewardship (Jacobs, 1999). In sustainable systems, it is essential to possess the capacity to address challenges and effectively respond to unpredictable disturbances. Although sustainable supply chains may not always be the most cost-effective option, they offer resilience and the ability to adapt to uncertain business environments. Sustainable supply chain management (SSCM), as a strategic process, aims to integrate environmental considerations into organizations by mitigating the unintended adverse impacts of consumption and production processes (Genovese et al., 2017; Liu et al., 2021). In recent decades, theories of SSCM, primarily grounded in operational considerations and product life-cycle influences, have been emerging. A sustainable supply chain network design (SSCND) seeks to model the optimal supply chain configuration that enables businesses and firms to maximize long-term benefits across all three pillars of sustainability-economic, social, and environmental (Govindan et al., 2015). Although SSCND is a critical activity, there is a lack of extensive research, particularly on the three fundamental pillars of sustainability. Additionally, there is an opportunity to develop quantitative models for SSCND that can effectively support SD. The integration of sustainability into supply chain management (SCM) has become a necessity for businesses, attracting significant attention from industries and academics to address economic, environmental, and social considerations. With constantly evolving standards, supply chain goals are no longer just a source of competitive advantage in target markets but a fundamental requirement for businesses (Shafiee et al., 2021). In a highly competitive environment, SC activities must be efficiently managed to achieve the following objectives: (i) minimizing costs, inventories, delivery delays, investments, and environmental and social impacts, and (ii) maximizing customer service levels, deliveries, profits, net present value (NPV), production, and return on investment (ROI) (Hosseinkhani et al., 2014; Tsiakis et al., Aug. 2001). Therefore, appropriate sustainability indicators, which play a leading role in responsible business management, should be incorporated into the framework of the entire system. This means involving all activities within the respective supply chains (Marzban et al., 2022). To design an effective SC, the primary objective of management must be achieved: ensuring proper allocation of resources to the right place at the right time, thereby reducing total system costs and delivering satisfactory services to potential customers. The location of facilities is one of the most strategic and challenging decisions, as it significantly impacts the efficiency of the supply chain. Total chain costs and service levels are largely influenced by the size, number, and location of facilities. Consequently, a considerable portion of research has focused on improving SC efficiency in relation to facility location (Saidi-Mehrabad et al., 2017).

Uncertainty and risk have always been critical issues in SCM. Several complexities, such as uncertainties in external supplier and customer interfaces, large-scale distribution networks, physical production flows, and the nonlinear dynamics of internal data flows, make supply chain operations a challenging task (Arman et al., 2024; Tsiakis et al., 2001). Supply chain risk (SCR), a significant concern for enterprises, broadly encompasses various potential threats and disruptions, such as economic downturns, supplier bankruptcies, climate change, and political instability. These risks can lead to severe disruptions in supply chain operations. Previous studies have demonstrated that SCRs can have serious negative impacts on the operations of affected businesses, potentially jeopardizing their overall performance and competitiveness (Arman, 2023; Bavarsad et al., 2014). It is true that when risk mitigation measures fail to help a business avoid disruptions, the organization may face even greater SC disruptions or other adverse side effects. A key characteristic of supply chain risk is its ability to extend beyond the boundaries of a single business. These boundary-spanning flows can themselves become sources of supply chain risks (Juttner, 2005). According to the classification provided by Tang (2006) supply chain risks can be categorized into two types: disruption risks and operational risks. Disruption risks arise from major man-made or natural disasters, such as economic crises, floods, tsunamis, or earthquakes. In contrast, operational risks are associated with uncertainties inherent in a SC, including demand, supply, and cost uncertainties. Given the interdependencies between organizations and their SCs, risk can flow in both directions: a business may be at risk due to its SC, or the SC may be at risk due to the business. Furthermore, the close correlation among SC members means that a risk or disruption in any segment of the SC can impact the entire chain, compromising its overall performance. For instance, the bankruptcy of a supplier not only affects subsequent links in the chain but also has repercussions for all members involved, disrupting their operations (Christopher and Peck, 2004). A study (Monfared et al., 2018) examined the identification and mitigation of disruptive risks within the Foolad Steel Company's SC, focusing on the graphite electrode, a critical material. Using the Analytic Network Process (ANP), the risks were evaluated, with the most pressing risk being supplier non-flexibility. Other risks, such as long delivery times, low product quality, and price hikes, were also considered. To address these, a multi-objective model was developed to minimize each identified risk, with solutions derived using both the Absolute Priority Method and Goal Programming. The effectiveness of these methods was compared based on the outcomes.

To overcome SCRs, organizations and businesses must implement appropriate strategies to manage them, highlighting the importance of supply chain risk management (SCRM). Specifically, two effective measures can address unforeseen events in SCs: (i) designing SCs with built-in risk tolerance and (ii) containing damage once adverse events occur. Both approaches require a thorough understanding of potential unwanted events in SCs, along with their associated impacts and consequences (Hosseinkhani and Kargari, 2022; Viswanadham and Gaonkar, 2008). It is worth noting that SCM has always emphasized risk management. However, the concept of SCRM has recently gained increasing attention due to the growing complexity of supply chains.

SCs must be supported by innovation, which significantly influences SSCM activities (Gurzawska, 2020). Innovation drives the development of sustainable SCs through sustainability-oriented measures. Research has shown that innovation is crucial for achieving competitive advantage and ensuring business sustainability (Hanaysha et al., 2022; Nidumolu et al., 2009; Pagell & Wu, 2009). Sustainable innovation can enhance productivity, increase market share, introduce novelty to existing product lines or processes, and facilitate expansion into new market sectors (Surya et al., 2021). Given the multi-actor nature of SCs, innovation must focus on the interaction between process design and participants. This approach should extend beyond individual firms and encompass the entire SC.

2. Literature Reviews

2.1. Theoretical Underpinnings and General Findings

To justify the need for a literature review, a comprehensive examination of key contributions in the field was conducted. Grosvold et al. (2014) identified specific SSCM inter-organizational measures, including codes of conduct, supplier education, certifications, changes in production processes, the development of emission-reduction technologies, and more effective raw material utilization. Similarly, Dubey et al. (2017) outlined major SSCM drivers, such as green warehousing, continuous environmental improvement, strategic supplier collaboration, logistics optimization, economic stability, ethical practices, and sustainable corporate strategies. Wagner and Bode (2008) emphasized that supply-side risks, such as issues related to supplier quality, defaults, and delivery failures, significantly affect SC performance, particularly in terms of delivery speed, dependability, order fulfillment, and customer satisfaction. Schonsleben (2007) and Ziegenbein (2007) further supported this by stating that a broad range of operational and supply chain indicators-such as product quality, delivery lead time, delivery reliability, operational cost, and flexibility-can be adversely impacted by SCRs. The study (Homayouni et al., 2023) developed a robust-heuristic optimization framework for designing green SC under uncertainty, emphasizing the integration of carbon regulation mechanisms. A bi-objective model was formulated to minimize total cost while simultaneously considering environmental objectives, such as emissions from diverse transportation modes. By employing a multi-choice goal programming model enhanced with heuristic algorithms, the study addressed the challenges posed by economic and demand uncertainty in complex supply chain networks (SCNs). A case study was conducted to compare the effectiveness of carbon-tax and cap-and-trade policies, revealing that cap-and-trade mechanisms, when supported by governmental incentives, were more effective in promoting cleaner technologies and sustainable logistics practices. The findings underscored the potential of robust-heuristic approaches to improve SC decision-making under uncertain and regulatory-intensive environments, aligning closely with infrastructure-oriented projects that require both cost control and environmental responsibility. Rice and Caniato (2003) quantified the effects of disruptions, noting that these could cost firms between USD 50 and100 per day within their supply networks. Bavarsad et al. (2014), in a study specific to Iran's automotive industry, reported that SCRs had a significant negative effect on organizational performance, with macroeconomics risks being the most detrimental. Finance indices were shown to be more vulnerable to SCRs than other organizational elements. The study (Amirian et al., 2022) conducted a systematic review of several studies at the intersection of sustainability, reliability, and SCND. Their analysis highlighted the evolution of SC paradigms, encompassing green SCs, socially responsible SCs, sustainable supply chains, and reliable SCs. The study emphasizes that integrating sustainability and reliability into SCND is a burgeoning area of research, particularly within developing nations and manufacturing sectors. Notably, most reviewed studies focused on forward-flow SCs, indicating a gap in reverse logistics and closed-loop systems. The authors proposed a comprehensive framework for sustainable and reliable SCs, detailing conceptual elements, assessment tools, and implementation contexts. This framework serves as a valuable guide for practitioners and policymakers aiming to enhance SC resilience and sustainability. On the topic of innovation, Hanaysha et al. (2022) indicated that both product and service innovations positively influence business sustainability, with innovation capabilities enhancing competitive strength. The study recommended adopting innovative marketing approaches by capitalizing on internet technologies and ensuring efficient operational processes to reduce environmental stress and costs. Lambert et al. (1998) highlighted the role of data integration through information technology in enabling collaborative innovation within SCs. Mangun and Thruston (2002) indicated that coordinated innovation in product design can substantially enhance sustainable development outcomes. Pagell and Wu (2009) introduced the concept of ‘innovation and the design of capabilities’ within SSCM frameworks, underscoring its strategic importance. Gao et al. (2017), conducted a conceptual analysis linking innovation, supply chains, and sustainability, thereby contributing to a more integrated understanding of SSCM. The study (Mousavi et al., 2023) presented a robust SCN model aimed at balancing cost-efficiency and environmental sustainability under uncertainty. Implemented in LINGO and integrated with Excel, the model used the Bertsimas and Sim method to manage uncertainty in product return rates. Simulations across eleven protection levels–each repeated 10,000 times–quantified risk by tracking constraint violations, revealing that higher protection reduced risk but increased costs. Environmental compliance introduced added expenses, yet was justified by broader social responsibility. Practically, the study underscored the need to align strategic SCN decisions with environmental planning, especially in regulation-sensitive projects. It also introduced new supplier selection indicators relevant to infrastructure-focused, multi-project supply chains. Finally, the model's scalability to multi-tier networks supports its application in complex supply systems like substation development.

2.2. Specific Studies Related to the Research Problem

Several recent studies have provided direct insight into SC modeling and sustainability under uncertainty–particularly within petrochemical and industrial contexts. The study (Ziaei & Jabbarzadeh, 2021) presented a multi-objective robust optimization model for the green location-routing planning of multi-modal transportation systems under uncertainty, with specific attention to hazardous material logistics. Utilizing polyhedral uncertainty sets, the model effectively addressed variability in accident probabilities, emission factors, and infrastructure costs. The model aimed to concurrently minimize transportation costs, carbon emissions, and public safety risks, demonstrating its applicability in complex logistics scenarios such as petroleum product distribution. Importantly, the robust optimization approach ensured solution reliability under uncertain parameters, with findings indicating measurable reductions in both emissions and risk levels per marginal cost increase. This work is particularly relevant to industrial SCs involving environmental constraints and operational uncertainties, aligning well with multi-project infrastructure development contexts such as substation construction. Lababidi et al. (2004) developed an optimization model for the SC of a petrochemical company operating under uncertain economic and operational conditions. Two approaches–introducing deviations in a deterministic model and applying scenario-based stochastic analysis–were applied to assess the impact of uncertainty. The proposed objective function aimed at optimizing resource utilization by reducing total production costs, raw material procurement, lost demand, backlog, storage penalties, and transportation costs. The study concluded that deterministic optimization models might lead to suboptimal planning outcomes when market demands are uncertain, whereas the stochastic approach effectively managed these uncertainties, yielding production plans with a notably low Expected Value of Perfect Information (EVPI). The findings underscored the critical influence of demand uncertainty on production decision-making in the oil and gas industry. To tackle demand uncertainty in the tactical planning of refined products distribution within a downstream oil supply chain, Lima et al. (2019) proposed a two-stage Adjustable Robust Optimization (ARO) model. Their findings indicated that, while the non-adjustable model was the most conservative and the stochastic model the least, the adjustable approach outperformed both. Specifically, it achieved the highest service level in meeting demand requirements, demonstrating superior effectiveness in managing uncertainty. Rabbani et al. (2020) developed a sustainable, multi-objective, and multi-period location-allocation SCN model that incorporated various levels of vehicle fleet technology, backorders from customers with different priority levels, and distinct CO₂ emissions for each potential facility location to design a green SC network. To handle uncertain parameters, they applied the Hybrid Robust Possibilistic Programming-II (HRPP-II) approach. Additionally, the improved Augmented ɛ-Constraint (AUGMECON) method was employed to obtain Pareto-optimal solutions. Their results revealed that while adopting a pessimistic outlook could enhance robustness across scenarios, it might also result in increased costs. To achieve SD goals in the petrochemical industry, Sangbor et al. (2022) identified and prioritized key components enabling SSCM using the Meta-Synthesis method. The extracted components were systematically categorized using the Graph Theory and Matrix Approach (GTMA) into six areas: "Corporate Management," "Supply Chain Continuity," "Supply Chain Management," "Supply Chain Partnership," "Supply Chain Features," and "Employees." According to data analysis, "Supply Chain Continuity" was identified as the top priority for planning. Consequently, the development of partnerships in SCs, increased trust between companies, and knowledge sharing among SC partners should be considered by policymakers and managers. Patidar and Agrawal (2020) targeted the agri-fresh food supply chain (AFSC) in India, proposing two mathematical models–a discrete dynamic (multi-period) mixed-integer linear programming (MILP) model and an extended version of the developed formulation–aimed at reducing post-harvest losses and distribution costs. Their multi-echelon, multi-product models facilitated optimal hub placement and infrastructure development, particularly for perishable products. Mastrocinque et al. (2020) applied the Triple Bottom Line (TBL) principles and the Analytic Hierarchy Process (AHP) to develop a decision-making framework for sustainable SC development in the renewable energy sector. The framework was validated using case studies from seven European nations, demonstrating its utility for sustainable investment decisions. To optimize the financial, social, and environmental impacts of a Sustainable Closed-loop Supply Chain (SCLSC), Nayeri et al. (2020) developed a multi-objective mathematical model for a water tank. Given the inherent uncertainty in configuring the SCLSC network, a Fuzzy Robust Optimization (FRO) approach was applied. The results indicated that the proposed model selected optimal transportation modes and suppliers, determined product flows between facilities, and identified the appropriate number of facilities to be established. Furthermore, the FRO model outperformed alternative approaches in terms of sustainability metrics. Gao and Cao (2020) established a novel multi-objective, scenario-based optimization model to support SD and address the integration of used-product recovery within existing forward logistics networks. This model aimed to maximize expected job opportunities and monetary profits while minimizing the total expected costs associated with carbon emissions. Accordingly, the study emphasized the importance of carefully determining the disposal threshold value in decision-making processes. To address the dual challenge of minimizing CO₂ emissions and operational costs in sustainable supply chains, Mogale et al. (2020) developed a bi-objective decision support model incorporating key problem-specific features, including multi-echelon, multi-modal, multi-period transportation, multiple sourcing and distribution strategies, emission constraints, capacitated warehouses, and heterogeneous vehicle fleets. Sensitivity analysis, conducted using two Pareto-based multi-objective algorithms across various realistic scenarios, highlighted the need for policymakers to strategically determine the number and location of warehouses in both producing and consuming regions to effectively balance environmental and economic objectives. Hendalianpour et al. (2022) explored the transition from single- to multi-channel SC networks in the oil, gas, and petrochemical sectors, focusing on sustainable location-allocation models for multi-product, multi-period SCs under uncertainty. They developed a multi-objective mathematical model aimed to minimize costs while improve sustainability and customer satisfaction. The model was validated using a real-world petrochemical supply chain case study and was solved through a hybrid Benders Decomposition and Lagrangian Relaxation algorithm. Findings indicated that improving service levels and adopting strategic resource allocation significantly contributed to improved economic and environmental performance while maintaining cost efficiency. Jabbarzadeh et al. (2016) presented a hybrid robust-stochastic optimization model, coupled with a Lagrangian Relaxation solution method, to design a SC resilient to supply/demand interruptions and facility disruptions. The model's performance, evaluated through Monte Carlo simulation, demonstrated that SC resilience could be significantly improved with minimal modifications to the existing configuration and only a slight increase in costs. Moreover, initial capital investment was identified as a critical factor in developing resilience and alleviating strategic costs, while excessive budget injections did not necessarily lead to additional cost savings. To minimize costs and reduce emissions in the petrochemical industry, Vadian et al. (2022) established a closed-loop green SC model that incorporated both direct and reverse logistics. The fuzzy mathematical programming approach has proved effective in reducing environmental pollutants, waste, and transportation costs. Given the increasing frequency of supply and demand disruptions, SC resilience has become essential for maintaining competitiveness. As supplier selection is a critical source of external risk, Grey Relational Analysis (GRA), based on the linguistic evaluation of supplier ratings, has been suggested as a useful tool for prioritizing suppliers based on multiple qualitative and quantitative criteria. Rajesh and Ravi (2015) investigated supplier selection using GRA within resilient SCs and concluded that this method outperformed traditional AHP and ANP methods by effectively integrating qualitative and quantitative evaluations. The study (Emami et al., 2021) presented a three-tier location-allocation model tailored to project-based SCs, with a focus on substation development projects. It made a significant contribution to the modeling of complex, multi-level SCs involving multiple decision-making layers. The proposed model aimed to determine the optimal locations for suppliers and warehouses while efficiently managing the flow of critical equipment across various project sites. It accounted for key cost components, including warehouse rental, production, and transportation expenses, with the dual objectives of minimizing total SC costs and maximizing demand satisfaction. The model was validated through a real-world case study using a multi-objective particle swarm optimization (MOPSO) algorithm, providing practical insights and decision-support tools for project portfolio managers. The work (Hosseini Shekarabi et al., 2025) developed an extended robust optimization framework for designing sustainable and resilient SCNs under disruption and uncertainty, with a focus on perishable goods. The study introduced a novel axis-shift robust (AxShR) method integrated with a p-robust framework, enabling strategic decisions in facility selection, capacity planning, and flow optimization across uncertain environments. The model was enhanced by multi-cut Benders decomposition for computational efficiency and incorporated artificial intelligence for disruption prediction using risk indicators. Results showed that applying multiple resilience strategies could eliminate unmet demand under moderate disruptions and reduce shortages by up to 86% during severe disruptions. Moreover, the model achieved a balance between profitability and risk mitigation, and consistently outperformed traditional methods in terms of lower average performance gaps and higher minimum profits. These findings underscore the model's relevance to complex, multi-level SCs in industrial sectors, supporting robust decision-making in both strategic design and sustainability-driven operations. The research (Chen and Liu, 2023) developed a globalized robust goal programming model to address the SC planning challenges in biomass-based power generation under uncertainty. Focusing on a multi-period, multi-feedstock, and multi-technology context, the model integrated economic, environmental, and social objectives using priority-based goal constraints. Key uncertainties–particularly unit emissions and social performance scores–were characterized using inner-outer uncertainty sets, enabling more flexible and realistic planning in complex, sustainability-driven SCs. The resulting mixed-integer linear programming (MILP) model transformed globalized robust constraints into tractable forms, making it computationally viable for large-scale issues. The findings demonstrated that while environmental and social goals could be consistently met by adjusting model parameters, economic objectives were more sensitive to uncertainty. Notably, the model achieved a 42.5% higher realized economic profit on average compared to traditional robust goal programming approaches, affirming its robustness and practical applicability. This methodological framework, although applied to biomass-based energy, offers valuable insights for multi-project infrastructure SCs-particularly where trade-offs among cost, sustainability, and risk resilience are critical.

3. Problem Description

The growing social and environmental concerns over the impacts of the oil and gas supply chain on the natural environment have led to increasing pressure to enhance the sustainable performance of the product life cycle, from "farm to people." Despite the importance of SCM and its growing complexity, the oil and gas industries are still in the early stages of efficiently managing their SCs. Given the complexity and inflexibility of the industry's SC, there is significant potential for cost reduction, particularly in the logistics sector. Although many studies have been conducted on SC sustainability within the industry, measuring sustainability in the context of equipment location-allocation for distribution centers remains a challenge. Providing the necessary devices and equipment for transporting fossil resources to refineries requires specific modes of transportation. If this equipment can be provided more quickly and with better quality, it will have a greater positive impact on SCs, helping to meet the organization's goals. Opening new production sites or establishing distribution centers in strategic locations closer to dispersed customers can be an effective measure for reducing transportation costs and lead times. However, acquiring such facilities in these industries, if feasible, requires a significant investment and often leads to higher inventory and operating costs (Hussain et al., 2006). It has been recognized that improving supply chain efficiencies offers significant opportunities for cost savings, as well as for addressing environmental and social concerns. Despite the critical importance of the oil and gas industries in the modern world, this topic has received limited attention in operations and SCM literature. While there is substantial emphasis on SD in polluting industries, there remains a gap in adopting a comprehensive, integrated approach that considers economic, environmental, and social aspects, particularly in the context of parameter uncertainty. The objective of the present work is to develop a location-allocation model for the SC, encompassing suppliers (offshore platforms), producers (gas refinery sites), and consumers (gas distribution and export centers), within the multi-product and multi-period oil and gas industries, while considering sustainability under uncertainty. In other words, the focus is on the production and distribution network of the South Pars Gas Complex (SPGC), aiming to make strategic decisions, such as selecting optimal locations for gas platform construction, as well as mid-term tactical decisions, such as optimizing the flow of materials across the supply chain under uncertainty.

The objective of the present study is to develop a location-allocation model for the SC, encompassing suppliers (offshore platforms), producers (gas refinery sites), and consumers (gas distribution and export centers), within the multi-product, multi-period oil and gas industry, while incorporating sustainability considerations under uncertainty. Specifically, the focus is on the production and distribution network of the South Pars Gas Complex (SPGC), with the aim of supporting strategic decisions–such as selecting optimal locations for gas platform construction–as well as mid-term tactical decisions, such as optimizing material flows across the supply chain under uncertain conditions. To address the identified challenges and guide the development of a sustainable supply chain model for the oil and gas industry, this study is driven by the following key research questions:

How can a multi-objective, location-allocation model be developed to optimize economic, environmental, and social performance in the oil and gas supply chain under parameter uncertainty?

What are the optimal locations for production and distribution centers in the SPGC network to minimize total supply chain costs and greenhouse gas emissions while ensuring timely delivery of products?

How can sustainability indicators (e.g., emission thresholds, social equity, energy efficiency) be integrated into the strategic and tactical planning of the oil and gas supply chain?

What trade-offs exist between cost efficiency and sustainability objectives (environmental and social) in designing a multi-period, multi-product oil and gas supply chain?

This approach aims to achieve economic efficiency, establish a balance between pollutant levels in the industry, and consider social aspects. The main assumptions considered in this work are as follows:

The supply chain operates with three echelons: manufacturers, distributors, and retailers.

The model is multi-product, multi-period, and multi-objective;

The demand for products in the market is certain;

Product storage and maintenance costs are accounted for at both production and distribution sites;

The network consists of a fixed number of production and distribution sites;

The production and distribution sites have predetermined capacities;

Inventory is managed over multiple time periods, ensuring no shortages at production sites;

Transportation equipment capacities are predetermined;

Greenhouse gas emissions from production processes and vehicles must not exceed a specified threshold;

Sustainability criteria are incorporated into the decision-making process to transform the model into a sustainable supply chain.

The key research contributions include:

The development of a multi-objective mixed-integer nonlinear programming (MINLP) model tailored to the specific characteristics of the oil and gas industry;

The integration of environmental, economic, and social sustainability dimensions into the SCND model, reflecting contemporary sustainability priority;

The provision of managerial insights to support sustainable supply chain design in the oil and gas sector.

4. Research Methodology

4.1. Production Scenarios

The proposed mathematical model addresses a multi-objective, multi-period, and multi-product problem by providing an algorithm for managing a multi-level SCN across production, distribution, and sales centers. The model incorporates sustainability analysis under uncertainty, along with inventory allocation and control policies within the network.

In the supply chain, materials flow through a predefined sequence within the network and are then distributed by sales centers to customers. Typically, the supply chain consists of multiple layers, including production plants, distribution centers, and customers. While planning is an ongoing process over time, various factors, such as costs, may be overlooked when planning for different periods. However, the model is executed through multi-period programming. It is assumed that production levels remain stable throughout the planning horizon, and the sales network does not face any demand-side shocks.

The proposed model incorporates sustainable development goals that address the three dimensions of sustainability: economic, environmental, and social aspects. From an economic perspective, the model focuses on profit, defined as the sum of revenues minus expenses across all periods. In terms of environmental and social factors, the model aims to simultaneously minimize environmental impacts, such as CO₂ emissions, while also increasing job opportunities at production and distribution centers.

4.2. Modelling Approach

4.2.1. Indexes and Parameters, Decision Variables and Objective Functions

To design the model, a mixed nonlinear programing approach is presented, which meets all assumptions and constraints while aiming to minimize monetary, environmental and social costs. Each mathematical model consists of sets, parameters and decision variables, as outlined in Tables 1 to 3.

Table 1.
Sets Definition.

$i \in I$ : A set of potential oil and gas production locations

$j \in J$ : A set of potential oil and gas distribution locations

$m \in M$ : A set of product sales markets

$p \in P$ : A set of final products (oil and gas derivatives)

$s \in S$ : A set of scenarios

$t \in T$ : A set of time periods

Table 2.

Model Parameters.

$B_{i}$	: Fixed cost related to opening (building) production center i
$C R_{i}$	: Variable cost related to opening (building) production center i
$C O_{p i t}$	: Production cost per unit of product p in production center i in time period t
$C \ln_{p i}$	: Cost of holding product p in production center i
$B_{j}$	: Fixed cost related to opening distribution center j
$C R_{j}$	: Variable cost related to opening distribution center j
$C \ln_{p j}$	: Cost of holding product p in distribution center i
$π_{p j t}$	: Inventory shortage cost of product p in distribution center j in time period t
$C a p_{i p}$	: Capacity of production center i to produce product p
$C a p_{j p}$	: Capacity of distribution center i for product p
${Pr}_{m p}^{G a s}$	: Selling price per unit of gas product p in market m
${Pr}_{m p}^{O i l}$	: Selling price per unit of oil product p in market m
$C r_{p i t}$	: Cost of achieving sustainable criteria Cr per unit of product p in production center i in time period t
$d_{i j}$	: Distance between production center i and distribution center j
$d_{i m}$	: Distance between distribution center i and retailer m
$T C_{i j}$	: Transportation cost from production center i to distribution center j
$T C_{j m}$	: Transportation cost from distribution center j to retailer m
${Pr}_{p i t}^{S}$	: Amount of produced product p in production center i in time period t in scenario s
$\ln_{p i t}$	: Amount of held product p in production center i in time period t
$D e_{j p t}^{S}$	: Demand amount of distribution center j for product p in time period t in scenario s
$Q I_{p j t}$	: Amount of held product p in distribution center i in time period t
${\tilde{D e}}_{m p t}^{s}$	: Demand amount of retailer m for product p in time period t in scenario s
$N_{j p t}$	: Non-satisfied demand number of distribution center j for product p in time period t
$E N_{p i}^{S}$	: Amount of petroleum products and gas emissions per unit of product p in production center i in scenario s
$S O_{i t}$	: Amount of increased satisfaction associated with sustainability elements of the social dimension resulting from the number of created job opportunities for production center i in time period t
$U R_{p i}$	: Utilization coefficient of raw materials containing contaminants in production phase of product p in production center i
$J o b_{i}$	: Satisfaction percentage with created job opportunities for opening production unit i in scenario s
$J o b_{j}$	: Satisfaction percentage with created job opportunities for opening distribution center i in scenario s
$M a x \ln$	: Maximum capacity of holding product in production / distribution center in time period
$M a x π$	: Maximum product capacity with allowable shortage in distribution center in time period
$P R_{S}$	: Occurrence probability of scenario s
$\ln c_{t}^{S}$	: Total income in time period t in scenario s
$C o s t_{t}^{S}$	: Total cost in time period t in scenario s
$I_{t}$	: Discount rate over time t
$δ_{j o b}$	: Total number of created job possibilities
$θ^{max}$	: Maximum acceptable amount of greenhouse gas (GHG) emissions in supply chain
$φ$	: Amount of greenhouse gas (GHG) produced by vehicle
$β$	: Vehicle capacity
$M$	: A very large number
$Γ_{i}$	: Coefficient stability of robust model
$Z_{o}$	: Value for objective function of robust model

Table 3.

Decision Variables.

$Q Z_{p i j t}$	: Amount of product p transported from production center i to distribution center j in time period t
$S a_{p j m t}$	: Amount of product p sold (sent out) by distribution center j to retailer m in time period t
$X_{p t}$	: Binary variable; 1 if product p is produced in time period t; 0, otherwise
$Y_{i t}$	: Binary variable; 1 if production center i is opened in time period t; 0, otherwise
$W_{j t}$	: Binary variable; 1 if distribution center j is opened in time period t; 0, otherwise

In this work, the goal of the sustainable supply chain is to (i) maximize the NPV, (ii) minimize total environmental impacts, and (iii) maximize social benefits resulting from any created job opportunities. Hence, the mathematical model, comprising three objective functions, evaluates monetary, environmental and social costs as follows:

\begin{aligned} Max Z_{1} (NPV) & = P R_{s} \cdot (\sum_{t} I_{t} \cdot {Inc}_{t}^{S} - \sum_{t} I_{t} \cdot {Cost}_{t}^{S}) \end{aligned}

(1)

\begin{aligned} {Inc}_{t}^{S} & = \sum_{i} \sum_{j} \sum_{m} \sum_{p} \sum_{t} {Pr}_{mpt}^{Gas} \cdot Q Z_{pijt} + \sum_{i} \sum_{j} \sum_{m} \sum_{p} \sum_{t} {Pr}_{mpt}^{Oil} \cdot Q Z_{pijt} \end{aligned}

(1-a)

\begin{aligned} C o s t_{t}^{S} & = ⌈ \sum_{i} \sum_{t} (B_{i} + C R_{i}) \cdot Y_{i t} + \sum_{i} \sum_{p} \sum_{t} C O_{p i t} \cdot X_{p t} + \sum_{p} \sum_{i} \sum_{t} I n_{p i t} \cdot C I n_{p i} + \sum_{j} \sum_{t} (B_{j} + C R_{j}) \\ \cdot W_{j t} + \sum_{i} \sum_{j} \sum_{p} \sum_{t} d_{i j} \cdot T C_{i j} \cdot Q Z_{p i j t} + \sum_{j} \sum_{p} \sum_{m} \sum_{t} d_{j m} \cdot T C_{j m} \cdot S a_{p j m t} + \sum_{p} \sum_{j} \sum_{t} C I n_{p j} \\ \cdot Q I_{j p t} + \sum_{i} \sum_{p} \sum_{t} \sum_{s} C r_{p i t} \cdot P r_{i p t}^{S} + \sum_{p} \sum_{j} \sum_{m} \sum_{t} I_{t} \cdot S a_{p j m t} + \sum_{p} \sum_{j} \sum_{t} N_{j p t} \cdot π_{j p t} ⌉ \end{aligned}

(1-b)

\begin{aligned} Min Z_{2} & = P R_{s} . [\sum_{i} \sum_{p} \sum_{s} \sum_{t} (X_{pt} . U R_{pi} \cdot {EN}_{pi}^{s})] \end{aligned}

(2)

\begin{aligned} Max Z_{3} & = \sum_{i} \sum_{j} \sum_{t} \sum_{s} {SO}_{it} \cdot [({Job}_{i} \cdot Y_{it}) . ({Job}_{j} \cdot W_{jt})] \end{aligned}

(3)

In this regard, the formulation of the mathematical models includes objective functions, described in Equations (1) to (3), and constraints, as outlined in Equations (4) to (22). The first objective function (Equation (1)) aims to maximize the NPV, which is calculated as the total revenue minus expenses across all periods of the supply chain. The discount rate is applied in financial modeling to determine the present value of income and expense flows over time. Equation (1-a) represents the gross incomes generated from the sale of oil and gas derivatives in each period. Equation (1-b) pursues to minimize the total cost of the system in each period, encompassing fixed and variable costs associated with building production centers, producing products, and maintaining inventory at production centers. It also includes the fixed and variable costs of opening distribution centers, shipping products from the production centers to distribution centers, transporting products from distribution centers to sales market, maintaining inventory at distribution centers; achieving sustainability goals, addressing inventory shortages at distribution centers, and providing sales discounts. Equation (2), representing the second objective function of the problem, minimizes environmental loads over the given time period. This objective function includes the environmental loads associated with material production. The third objective (Equation 3) also aims to maximize the cumulative social benefit resulting from job creation at both production and distribution centers. The social benefit is represented as a function of the number of jobs created (Job Opportunities_it), weighted by a satisfaction coefficient (SO_it). This coefficient captures the relative importance or perceived utility of those jobs in each time period t and location i. This approach mirrors the "weighted employment impact" metric commonly used in social sustainability assessments, where jobs are not merely counted but are adjusted for quality, local economic impact, and alignment with sustainability goals. For example, jobs created in economically deprived or high-unemployment regions may be assigned a higher social benefit coefficient, reflecting their greater social value.

4.2.2. Constraints of the Model

The model's constraints are illustrated and described as follows:

\begin{aligned} {Pr}_{pit}^{S} & \leq Y_{it} \cdot Ca p_{ip} \forall i . p . t . s \end{aligned}

(4)

\begin{aligned} Q Z_{pijt} & \leq W_{jt} \cdot Ca p_{jp} \forall i . j . p . t \end{aligned}

(5)

\begin{aligned} Q Z_{pijt} & \leq {Pr}_{pit}^{S} + Q I_{pjt} \forall i . j . p . t . s \end{aligned}

(6)

\begin{aligned} Q I_{pjt} + N_{jpt} & \leq 1 \forall j . p . t \end{aligned}

(7)

\begin{aligned} I n_{pit} & \leq MaxIn \forall i . p . t \end{aligned}

(8)

\begin{aligned} Q I_{pjt} & \leq MaxIn \forall j . p . t \end{aligned}

(9)

\begin{aligned} N_{jpt} & \leq Max π \forall j . p . t \end{aligned}

(10)

\begin{aligned} S a_{pjmt} & \geq {\tilde{De}}_{mpt}^{s} \forall j . m . p . s . t \end{aligned}

(11)

\begin{aligned} Q Z_{pijt} & \leq W_{jt} \cdot M \forall i . j . p . t \end{aligned}

(12)

\begin{aligned} Q Z_{pijt} & \leq β \forall i . j . p . t \end{aligned}

(13)

\begin{aligned} S a_{pjmt} & \leq β \forall m . j . p . t \end{aligned}

(14)

\begin{aligned} {EN}_{pi}^{S} \cdot Y_{it} & < θ^{max} \forall i . t \end{aligned}

(15)

\begin{aligned} φ \cdot d_{ij} & \cdot Q Z_{pijt} + φ \cdot d_{jm} \cdot S a_{pjmt} < θ^{max} \forall i . j . m . p . t \end{aligned}

(16)

\begin{aligned} {Job}_{i}^{S} & \cdot Y_{it} + {Job}_{j}^{S} \cdot W_{jt} > δ_{job} \forall i . j . S . t \end{aligned}

(17)

\begin{aligned} Q Z_{pijt} & = {De}_{jpt}^{s} \forall i . j . m . p . t \end{aligned}

(18)

\begin{aligned} Q Z_{pijt} + Q I_{pj (t - 1)} & = S a_{pjmt} + Q I_{pjt} \forall i . j . p . m . t \end{aligned}

(19)

\begin{aligned} Q Z_{pijt} & = S a_{pjmt} \forall i . j . p . m . t \end{aligned}

(20)

\begin{aligned} Q Z_{pijt} . S a_{pjmt}, NPV & \geq 0 \forall i . j . p . m . t \end{aligned}

(21)

\begin{aligned} X_{pt} . Y_{it} . W_{jt} & \in 0.1 \forall i . j . p . t \end{aligned}

(22)

Conditional on the described objective functions, these constraints (Equations (4) to (22)) pertain to the developed model. Each factory operates with a limited capacity for each product and period. Equation (4) ensures that, once operational, the production centers maintain that the production volume of a given product does not exceed the manufacturing capacity of the respective center. Similarly, Equation (5) specifies that the capacity of a distribution center is determined by the quantity of the product transferred from the production centers and the end-customers’ demands, ensuring that the maximum inventory at the distribution site does not exceed the center's capacity. Equation (6) assures that the production centers fulfill all retailer demands without facing inventory shortages. Furthermore, Equation (7) indicates that inventory and shortage items are mutually exclusive, meaning they cannot occur simultaneously. Equations (8) and (9) define the maximum storage capacity for the specified product at the production centers and distribution centers, respectively. Additionally, Equation (10) specifies the maximum allowable product shortage over the given time period. According to Equation (11), the inventory at a given distribution center must be sufficient to meet the retailer's demand for the specified product over the desired period. Equation (12) states that if a distribution center is not established, no product can be shipped to it. Equations (13) and (14) pertain to vehicle capacity, ensuring it is greater than or equal to the corresponding order quantity. According to Equations (15) and (16) limit the amount of GHG emissions generated during product manufacturing at the production centers and by the vehicles, ensuring emissions do not exceed specified thresholds. Equation (17) also indicates the number of job opportunities created by the establishment of both production and distribution centers. It should be noted that this level must exceed the predicted amount. Besides, Equation (18) states that since the producer will never face an inventory shortage, the distributor's demand must always be satisfied. Equation (19) logically indicates that, for every product in every period, the quantity of the product transferred from the producer's facilities at the beginning of the period, combined with the inventory carried over from the previous period, must equal the sum of the quantity shipped to the retailer and the remaining inventory at the end of the period. Furthermore, Equation (20) ensures the balance of product flow throughout all design periods within the entire supply chain. Finally, Equations (21) and (22) define the range of the decision variables.

4.2.3. Robust Counterpart Optimization Approach

Due to inaccuracies, the inability to predict future events, and persistent variability, decision-making processes often face inevitable uncertainties. There are three major approaches to incorporating uncertainty parameters in mathematical models (Rabbani et al., 2020). In this context, robust optimization models are frequently applied in the fields of ethics and operations research, as they reduce the risk of malfunctions or misuse. Robustness refers to a system's ability to maintain performance despite perturbations, in contrast to sensitivity, which describes a system's responsiveness to such perturbations. Robust optimization is often regarded as an overly optimistic yet complementary approach for addressing probable planning scenarios and conducting sensitivity analysis. Unlike stochastic optimization, robust optimization represents uncertain variables in interval form. Various mathematical models propose different robust optimization approaches that can be applied to decision-making problems. These approaches are designed to avoid increasing the complexity of robust models while effectively addressing uncertainties. In other words, in terms of complexity, both the robust model and the deterministic model are equivalent (Khorasani, 2018). Based on the above, Robust Counterpart Optimization has been proposed to address uncertainties in the input data of SCs under risk. This robust approach considers parameter uncertainties in both the objective functions and constraints.

According to Pishvaee et al. (2011), the correlated uncertain linear optimization problem that comprises a collection of linear optimization problems can be determined as follows:

\begin{aligned} \begin{array}{l} min  cx + d \\ s . t . \\ Ax \leq b \\ c, d, A, b \in U \end{array} \end{aligned}

(23)

Where, the parameters A, b, c, and d vary within a desired uncertainty set U, which is defined as a specified closed and bounded box. The general form of this box can be represented as follows:

\begin{aligned} u_{Box} = ξ \in R | ξ_{t} - {\bar{ξ}}_{t} | \leq ρ G_{t} t = 1.2. \dots . n \end{aligned}

(24)

Where, $\bar{ξ_{t}}$ stands for the nominal value of the $ξ_{t}$ , which is a parameter of the vector $ξ$ (defined as an n-dimension vector). The term $G_{t} \geq 0$ indicates the "uncertainty scale'', and the positive numbers $ρ$ represent the "uncertainty level''. It should be noted that a specific case of interest is $G_{t} = \bar{ξ_{t}}$ , corresponding to a situation where the box includes values of $ξ_{t}$ whose relative deviation (RD) from the normal data equals $ρ$ . According to Ben-Tal et al. (2005), in this case (a closed-bounded box with $ξ > 0$ ), the robust counterpart model can be converted into a tractable equivalent model where corner points of the given interval are used instead of u_Box. Consequently, the following inequality (Equation 25), where the parameter di is an uncertain variable, will be replaced by Equation (26).

\begin{aligned} \begin{array}{l} a_{i} x \geq d_{i} \forall i \in 1, 2, \dots, n_{d} \\ \forall d \in u_{Box}^{d} u_{Box}^{d} = d \in R : | d_{i} - d_{i} | \leq ε_{d} G_{t}^{d}, t = 1, \dots, n_{d} \end{array} \end{aligned}

(25)

\begin{aligned} a_{i} x \geq {\bar{d}}_{i} + ε_{d} G_{t}^{d} \forall i \in 1, \dots, n_{d} \end{aligned}

(26)

As mentioned, the production costs, holding costs at production/distribution centers, variable costs for starting up both production and distribution facilities, shortage costs, selling prices of gas and oil products, quantities of held products, quantities of petroleum products, gas emissions, total revenues and expenses, the amount of the produced GHG emissions, and levels of created job opportunities are considered as uncertain parameters due to their significant fluctuations throughout the SC. Accordingly, the model's accuracy is enhanced using the robust counterpart model to address the uncertain optimization problem. This adaptation modifies the objective functions (Equations (1 to 3)) and their respective constrains (Equations (4), and (6), (15 to 17)) into their robust counterparts: objective functions (Equations (27 to 29)) and constrains (Equations (30 to 34)).

\begin{aligned} Max Z_{1} (NPV) & = P R_{s} \cdot (\sum_{t} I_{t} \cdot ({\tilde{Inc}}_{t}^{s} + Γ_{t}^{s}) - \sum_{t} I_{t} \cdot ({\tilde{Cost}}_{t}^{s} + Γ_{t}^{s})) \end{aligned}

(27)

\begin{aligned} {Inc}_{t}^{S} & = \sum_{i} \sum_{j} \sum_{m} \sum_{p} \sum_{t} ({\tilde{Pr}}_{mpt}^{Gas} + Γ_{mpt}^{Gas}) \cdot Q Z_{pijt} + \sum_{i} \sum_{j} \sum_{m} \sum_{p} \sum_{t} ({\tilde{Pr}}_{mpt}^{Oil} + Γ_{mpt}^{Oil}) \cdot Q Z_{pijt} \end{aligned}

(27-a)

\begin{aligned} {Cost}_{t}^{S} & = ⌈ \sum_{i} \sum_{t} B_{i} \cdot Y_{it} + \sum_{i} \sum_{t} (\tilde{C R_{i}} + Γ_{i}) \cdot Y_{it} + \sum_{i} \sum_{p} \sum_{t} ({\tilde{CO}}_{pit} + Γ_{pit}) \cdot X_{pt} + \sum_{p} \sum_{i} \sum_{t} I n_{pit} \\ \cdot ({\tilde{CIn}}_{pi} + Γ_{pi}) + \sum_{j} \sum_{t} B_{i} \cdot W_{jt} + \sum_{j} \sum_{t} ({\tilde{CR}}_{j} + Γ_{j}) . W_{jt} + \sum_{i} \sum_{j} \sum_{p} \sum_{t} d_{ij} \cdot T C_{ij} \cdot Q Z_{pijt} \\ + \sum_{j} \sum_{p} \sum_{m} \sum_{t} d_{jm} \cdot T C_{jm} \cdot S a_{pjmt} + \sum_{p} \sum_{j} \sum_{t} ({\tilde{CIn}}_{pi} + Γ_{pi}) \cdot Q I_{jpt} + \sum_{i} \sum_{p} \sum_{t} \sum_{s} C r_{pit} \\ \cdot ({\tilde{Pr}}_{pit}^{S} + Γ_{ipt}^{s}) + \sum_{p} \sum_{j} \sum_{m} \sum_{t} I_{t} \cdot S a_{pjmt} + \sum_{p} \sum_{j} \sum_{t} N_{jpt} \cdot ({\tilde{π}}_{jpt} + Γ_{jpt}) ⌉ \end{aligned}

(27-b)

\begin{aligned} Min Z_{2} & = \sum_{i} \sum_{p} \sum_{s} \sum_{t} X_{pt} \cdot U R_{pi} \cdot ({\tilde{EN}}_{pi}^{s} + Γ_{pi}^{s}) \end{aligned}

(28)

\begin{aligned} Max Z_{3} & = \sum_{i} \sum_{j} \sum_{t} \sum_{s} ({\tilde{SO}}_{it}^{S} + Γ_{it}^{s}) \cdot {Job}_{i}^{S} \cdot Y_{it} + \sum_{i} \sum_{j} \sum_{t} \sum_{s} ({\tilde{SO}}_{it}^{S} + Γ_{it}^{s}) . {Job}_{j}^{S} \cdot W_{jt} \end{aligned}

(29)

The first objective function, which minimizes total costs (maximizes profits), is calculated as the sum of total revenues minus expenses across all periods in the SC. Considering the uncertainty in income and expense flows, this first objective function is reformulated in a robust form (Equation 27). The robust form of gross incomes derived from the sales of oil and gas derivatives in each period is expressed in Equation. (27-a). Additionally, Equation. (27-b) aims to minimize the total variable costs, including the costs of building production centers, producing products, and maintaining (holding) inventory in the production centers. It also accounts for the variable costs associated with opening distribution centers, maintaining inventory at the distribution centers, and addressing inventory shortages at the distribution centers. Due to uncertainties in expense streams, these parameters are expressed in a robust form in Equation (27-b). The second objective function minimizes the environmental impacts of production and transportation systems. These impacts arise from factors such as equipment failures, fossil fuel consumption, the use of gas filter-free device, disuse of dual-fuel vehicles, rainfall patterns (e.g., light and low-intensity rainfalls), and exceeding the manufacturing capacity of production centers. Considering the uncertainty in these factors, they are described in a robust form in Equation (28). The third objective function enhances social aspects by promoting job creation opportunities in production and distribution centers. These opportunities are influenced by uncertainties related to bureaucracy, employment laws and regulations, poor organizational structures, and other factors. To address these uncertainties, the parameters are presented in a robust form in Equation (29).

4.2.3.1. Constraints of the robust model

Conditional on the newly expressed objective functions, the following new constraints are related to the developed model (Equations (30 to 35)). Other constraints remain unchanged. Equation (30) declares that the production quantity cannot not be definitively determined, making overproduction almost inevitable.

\begin{aligned} ({\tilde{Pr}}_{pit}^{S} + Γ_{1}) = Y_{it} \cdot Ca p_{ip} \forall i . p . t . s \end{aligned}

(30)

The production centers will control the rate of the product supply flows. Equation (31) states that it is unlikely to satisfy all demands of the given centers due to inevitable frequent issues such as equipment failures or a shortage of raw materials.

\begin{aligned} Q Z_{pijt} & = ({\tilde{Pr}}_{pit}^{S} + Γ_{2}) + Q I_{pjt} \forall i . j . p . t . s \end{aligned}

(31)

The levels of the GHG emissions released during production at the given producer's centers may exceed a certain limit, as shown in Equation (32). According to Equation (33), GHG emissions may also surpass a specific threshold in the cases of equipment failures, fossil fuel consumption, gas filter-free device uses, dual-fuel vehicle disuse, and so on. Moreover, Equation (34) reflects the level of job creation, which is contingent upon the aforementioned considerations. It states that the exact number of job opportunities created cannot be precisely determined, and as a result, the actual levels of employment may be lower than the expected levels.

\begin{aligned} ({EN}_{pi}^{S} + Γ_{3}) \cdot Y_{it} = θ^{max} \forall i . t \end{aligned}

(32)

\begin{aligned} \begin{array}{l} (φ + Γ_{4}) \cdot (d_{ij} \cdot Q Z_{pijt} + d_{jm} \cdot S a_{pjmt}) = θ^{max} \forall i . j . m . p . t \\ {Job}_{i}^{S} \cdot Y_{it} + {Job}_{j}^{S} \cdot W_{jt} = (δ_{job} + Γ_{5}) \forall i . j . S . t \end{array} \end{aligned}

(33)

\begin{aligned} Q Z_{pijt} . S a_{pjmt}, NPV, Γ_{t}^{s}, Γ_{mpt}^{Gas}, Γ_{mpt}^{Oil}, Γ_{i}, Γ_{j}, Γ_{pit}, Γ_{pi}, Γ_{ipt}^{s}, Γ_{jpt}, Γ_{pi}^{s}, Γ_{it}^{s}, \geq 0 \forall i . j . p . m . t \end{aligned}

(34)

4.3. Solution Approach

The model presented in this research features three objective functions with diverse directions (two maximization and one minimization functions). To solve the mathematical model of the assignment problem, three variations were designed and implemented, corresponding to small, medium, and large sizes. The number of small-scale outputs was initially limited to 1. while the maximum number of compounds on the large scale was set to 10. This approach was also employed to outgoing vehicles. Given the complexity, constraints, and the number of variables, the proposed mathematical model was coded in the programming environment of web-based SAS Studio 3.5. According to the given constraints and the three optimization objective functions, the robust and optimal values for the response levels were determined to optimize operations management policies (Tables 4 to 6).

Table 4.
Problem Size.

Scenarios Time periods Final products Potential retailers Potential locations

Distribution Production

5 12 5 5 4 3

Table 5.

Data on transportation mode (Mohammed et al., 2017).

Scale	Capacity (L)*	Greenhouse gas (GHG) emission factor (mL/km)
Small	50000	0.0480
Medium	75000	0.0252
Large	100000	0.2870

* Liter

Table 6.

the Distribution Function of the Parameters.

Row	Indices	Parameter	Distribution function
1	$B_{i}$	Fixed cost related to opening (building) production center i	$U (50, 100)$
2	$C R_{i}$	Variable cost related to opening (building) production center i	$U (60, 120)$
3	$C O_{pit}$	Production cost per unit of product p in production center i in time period t	$U (25, 50)$
4	$CI n_{pi}$	Cost of holding product p in production center i	$U (20, 40)$
5	$B_{j}$	Fixed cost related to opening distribution center j	$U (30, 80)$
6	$C R_{j}$	Variable cost related to opening distribution center j	$U (10, 40)$
7	$CI n_{pj}$	Cost of holding product p in distribution center i	$U (10, 50)$
8	$π_{pjt}$	Inventory shortage cost of product p in distribution center j in time period t	$U (50, 100)$
9	$Ca p_{ip}$	Capacity of production center i to produce product p	$U (50, 100)$
10	$Ca p_{jp}$	Capacity of distribution center i for product p	$U (200, 500)$
11	${Pr}_{mp}^{Gas}$	Selling price per unit of gas product p in market m	$U (200, 500)$
12	${Pr}_{mp}^{Oil}$	Selling price per unit of oil product p in market m	$U (200, 500)$
13	$C r_{pit}$	Cost of achieving sustainable criteria Cr per unit of product p in production center i in time period t	$U (5, 25)$
14	$d_{ij}$	Distance between production center i and distribution center j	$U (5, 25)$
15	$d_{jm}$	Distance between distribution center i and retailer m	$U (2000, 10000)$
16	$T C_{ij}$	Transportation cost from production center i to distribution center j	$U (2000, 10000)$
17	$T C_{jm}$	Transportation cost from distribution center j to retailer m	$U (2000, 10000)$
18	${Pr}_{pit}^{S}$	Amount of produced product p in production center i in time period t in scenario s	$U (2000, 10000)$
19	$I n_{pit}$	Amount of held product p in production center i in time period t	$U (350, 750)$
20	${De}_{jpt}^{S}$	Demand amount of distribution center j for product p in time period t in scenario s	$U (100, 500)$
21	$Q I_{pjt}$	Amount of held product p in distribution center i in time period t	$U (100, 500)$
22	${\tilde{De}}_{mpt}^{s}$	Demand amount of retailer m for product p in time period t in scenario s	$U (100, 500)$
23	$N_{jpt}$	Non-satisfied demand number of distribution center j for product p in time period t	$U (10, 50)$
24	${EN}_{pi}^{S}$	Amount of petroleum products and gas emissions per unit of product p in production center i in scenario s	$U (0.1, 0.9)$
25	${SO}_{it}^{S}$	Amount of increased satisfaction associated with sustainability elements of the social dimension resulting from number of created job opportunities for production center i in time period t	$U (0.1, 0.9)$
26	$U R_{pi}$	Utilization coefficient of raw materials containing contaminants in production phase of product p in production center i	$U (0.1, 0.9)$
27	${Job}_{i}^{S}$	Satisfaction percentage with created job opportunities for opening production unit i in scenario s	$U (5, 75)$
28	${Job}_{j}^{S}$	Satisfaction percentage with created job opportunities for opening distribution center i in scenario s	$U (10, 80)$
29	$MaxIn$	Maximum capacity of holding product in production / distribution center in time period	$U (5, 50)$
30	$Max π$	Maximum product capacity with allowable shortage in distribution center in time period	$U (5, 50)$
31	$P R_{s}$	Occurrence probability of scenario s	$U (0.1, 0.9)$
32	${Inc}_{t}^{S}$	Total income in time period t in scenario s	$U (10000, 50000)$
33	${Cost}_{t}^{S}$	Total cost in time period t in scenario s	$U (10000, 50000)$
34	$I_{t}$	Discount rate over time t	$U (0.1, 0.9)$
35	$δ_{job}$	Total number of created job possibilities	$U (10, 80)$
36	$θ^{max}$	Maximum acceptable amount of greenhouse gas (GHG) emissions in supply chain	$U (0.1, 0.9)$
37	$φ$	Amount of greenhouse gas (GHG) produced by vehicle	$U (0.01, 0.95)$
38	$β$	Vehicle capacity	$U (100, 500)$

Note: Parameter values in Rows 25-28 were estimated using a combination of expert judgment from industrial HR managers, literature-informed scoring frameworks, and internal data projections. Normalization and regional weighting techniques were applied to align with social sustainability modeling practices. All inputs were subject to uncertainty and were treated as such in the robust model.

4.4. Input Data

GHG Emissions Data : To assess the performance of the proposed model, test problems of various sizes were developed. The number of facilities (problem size) is depicted in Table 4. The GHG emission values per transportation mode were derived from reputable sources aligned with the IPCC Guidelines and Transportation Sector Emissions Factors provided by prior research (Mohammed et al., 2017; Tsiakis et al., 2001; Vadian et al., 2022). Specifically:

The emission factor values (in mL/km) for small, medium, and large vehicles (Table 5) were calibrated based on fuel consumption rates (L/km), engine types, and the presence or absence of emission-reduction technologies;

The vehicle types reflect commonly used categories in oil and gas product logistics (e.g., tanker trucks, semi-trailers), and the corresponding emissions per kilometer were obtained from region-specific environmental impact reports and validated against peer-reviewed studies.

To ensure robustness, these GHG values were incorporated into the model with uncertainty bounds and included in the robust counterpart optimization formulation. This allows the model to account for fluctuations in actual emissions due to factors like traffic, fuel type variability, and equipment efficiency.

Job Satisfaction and Social Impact Metrics: Social benefit metrics, particularly the job satisfaction coefficient (SO_it) and job opportunity weightings, were derived using a contextual expert judgment and literature-informed scoring methodology. The process was as follows:

Expert Elicitation: Industrial operations managers and regional HR specialists in the oil and gas sector were consulted to assess job satisfaction multipliers under different employment scenarios (permanent vs. contract jobs, regional development needs, etc.);

Normalization: The job satisfaction levels were mapped to a standardized scale between 0.5 and 1.0 to reflect low to high satisfaction, ensuring consistency with prior sustainable SC models (Liu et al., 2021; Sangbor et al., 2022);

Regional Weighting: Higher satisfaction scores were assigned to jobs created in underserved regions or in high-skilled roles. This reflects their stronger contribution to the social sustainability objective in Equation (3).

The parameter values in Table 6 (Rows 25–28) reflect this combination of quantitative HR data (e.g., projected jobs per facility size) and qualitative assessments (e.g., job quality, long-term community impact). All values were treated as uncertain and modeled accordingly in the robust formulation to account for fluctuations in satisfaction due to policy, workforce demographics, or economic changes.

4.5. Results

The objective function for a model is a mathematical expression that represents the measure of performance for the problem in terms of the decision variables. In a mathematical model, the number of objective functions can range from 1 to 5. The optimal value, or the best solution, is determined through the optimization process. Optimization problems can involve seeking a maximum or minimum value, and they may be single-objective or multi-objective. Optimizing multi-objective problems (MOPs) involves optimizing more than one objective function simultaneously. Optimal values of the decision variables and the given objective functions are presented in Tables 7 to 9.

Table 7.
Optimal Objective Function Values of the Given Multi-Objective Problem.

Objective function Lower bound Upper bound Optimal value

The first 4,987,125 5,856,802 5,651,584

The second 0.22 0.15 0.19

The third 0.72 0.92 0.85

Objective function	Lower bound	Upper bound	Optimal value
The first	4,987,125	5,856,802	5,651,584
The second	0.22	0.15	0.19
The third	0.72	0.92	0.85

Table 8.

Continuous decision variables of the proposed multi-objective optimization problem.

Time period												$S a_{p j m t}$	$Q Z_{p i j t}$	Product	Market	Distribution	Production
12	11	10	9	8	7	6	5	4	3	2	1			Product	Market	Distribution	Production
58	85	14	30	73	130	53	7	58	8	83	54			1	1	1	1
211	63	16	64	16	182	59	7	54	20	138	58			2	2	2
132	85	6	36	36	164	76	20	29	73	116	83			3	3	3
217	69	6	8	50	51	94	10	8	18	68	61			4	4	4
245	78	15	65	48	181	89	15	1	39	213	56			1	1	1	2
135	62	4	53	54	114	57	14	30	73	130	53			2	2	2
61	96	17	37	22	104	54	16	64	16	182	59			3	3	3
208	69	4	43	34	143	84	6	36	36	164	76			4	4	4
177	54	20	43	64	175	75	6	8	50	51	94			1	1	1	3
152	52	15	51	75	211	75	15	65	48	181	89			2	2	2
164	82	3	52	38	163	66	4	53	54	114	57			3	3	3
112	97	18	16	25	55	67	17	37	22	104	54			4	4	4
200	66	19	65	36	209	96	4	43	34	143	84			1	1	1	4
118	88	12	16	25	105	73	20	43	64	175	75			2	2	2
227	65	8	33	23	215	88	15	51	75	211	75			3	3	3
117	57	9	19	12	188	60	3	52	38	163	66			4	4	4
210	76	15	22	14	171	55	18	16	25	55	67			1	1	1	1
211	70	6	1	40	128	61	19	65	36	209	96			2	2	2
61	99	1	75	7	64	78	12	16	25	105	73			3	3	3
131	65	19	42	75	250	84	8	33	23	215	88			4	4	4
244	80	20	38	12	183	97	9	19	12	188	60			1	1	1	2
133	79	16	42	13	96	53	15	22	14	171	55			2	2	2
117	81	8	15	51	163	81	6	1	40	128	61			3	3	3
83	54	1	24	69	106	97	1	75	7	64	78			4	4	4
138	58	13	5	64	81	82	19	42	75	250	84			1	1	1	3
116	83	17	37	22	104	54	20	38	12	183	97			2	2	2
68	61	4	43	34	143	84	16	42	13	96	53			3	3	3
213	56	20	43	64	175	75	8	15	51	163	81			4	4	4
130	53	15	51	75	211	75	1	24	69	106	97			1	1	1	4
182	59	3	52	38	163	66	13	5	64	81	82			2	2	2
164	76	18	16	25	55	67	12	4	44	197	53			3	3	3
51	94	19	65	36	209	96	8	23	60	58	85			4	4	4
181	89	12	16	25	105	73	9	14	60	211	63			1	1	1	1
114	57	8	33	23	215	88	10	55	36	132	85			2	2	2
104	54	9	19	12	188	60	7	73	69	217	69			3	3	3
143	84	15	22	14	171	55	5	29	40	245	78			4	4	4
175	75	6	1	40	128	61	3	9	76	135	62			1	1	1	2
78	3	1	75	7	64	78	3	64	64	61	96			2	2	2
84	19	19	42	75	250	84	19	38	11	208	69			3	3	3
97	10	20	38	12	183	97	10	55	20	177	54			4	4	4
53	15	16	42	13	96	53	15	55	11	152	52			1	1	1	3
81	7	8	15	51	163	81	7	34	43	164	82			2	2	2
97	1	1	24	69	106	97	1	53	5	112	97			3	3	3
82	13	13	5	64	81	82	13	51	38	200	66			4	4	4
53	8	12	4	44	197	53	8	43	79	118	88			1	1	1	4
85	13	8	23	60	58	85	13	16	55	227	65			2	2	2
63	17	9	14	60	211	63	17	41	70	117	57			3	3	3
85	3	10	55	36	132	85	3	60	40	210	76			4	4	4
69	12	7	73	69	217	69	12	34	24	211	70			1	1	1	1
78	12	5	29	40	245	78	12	59	15	61	99			2	2	2
62	11	3	9	76	135	62	11	55	77	131	65			3	3	3
96	10	3	64	64	61	96	10	77	61	244	80			4	4	4
69	10	19	38	11	208	69	10	25	77	133	79			1	1	1	2
54	8	10	55	20	177	54	8	28	54	117	81			2	2	2
52	7	15	55	11	152	52	7	58	8	83	54			3	3	3
175	75	16	182	59	7	54	7	54	20	138	58			4	4	4
211	75	36	164	76	20	29	20	29	73	116	83			1	1	1	3
163	66	50	51	94	10	8	10	8	18	68	61			2	2	2
55	67	48	181	89	15	1	15	1	39	213	56			3	3	3
209	96	54	114	57	14	30	14	30	73	130	53			4	4	4
105	73	22	104	54	16	64	16	64	16	182	59			1	1	1	4
215	88	34	143	84	6	36	6	36	36	164	76			2	2	2
188	60	64	175	75	6	8	6	8	50	51	94	3	3	3
171	55	75	211	75	15	65	15	65	48	181	89	4	4	4
128	61	38	163	66	4	53	4	53	54	114	57	1	1	1	1
64	78	25	55	67	17	37	17	37	22	104	54	2	2	2
250	84	36	209	96	4	43	4	43	34	143	84	3	3	3
16	42	25	105	73	20	43	20	43	64	175	75	4	4	4
8	15	23	215	88	15	51	15	51	75	211	75	1	1	1	2
1	24	12	188	60	3	52	3	52	38	163	66	2	2	2
13	5	14	171	55	18	16	18	16	25	55	67	3	3	3
12	4	40	128	61	19	65	19	65	36	209	96	4	4	4
8	23	7	64	78	12	16	12	16	25	105	73	1	1	1	3
9	14	75	250	84	8	33	15	1	39	213	56	2	2	2
10	55	12	183	97	9	19	14	30	73	130	53	3	3	3
7	73	13	96	53	15	22	16	64	16	182	59	4	4	4
5	29	51	163	81	6	1	6	36	36	164	76	1	1	1	4
3	9	69	106	97	1	75	6	8	50	51	94	2	2	2
3	64	64	81	82	19	42	15	65	48	181	89	3	3	3
19	38	22	104	54	20	38	4	53	54	114	57			4		4	4

Table 9.

Binary Decision Variables of the Proposed Multi-Objective Optimization Problem.

Time period
12	11	10	9	8	7	6	5	4	3	2	1	$W_{j t}$	$Y_{i t}$	$X_{p t}$	Product	Distribution	Production
0	1	1	1	1	0	0	1	1	0	0	1				1	1	1
1	0	0	1	1	0	0	0	0	0	0	1				2	2
1	0	0	0	0	0	0	1	1	0	0	1				3	3
1	1	0	1	1	1	0	1	0	1	0	1				4	4
1	1	1	1	1	0	0	1	1	0	0	1				1	1	2
1	0	0	1	1	0	0	0	1	0	0	1				2	2
1	0	0	0	0	1	1	0	1	0	0	1				3	3
1	0	0	1	1	0	0	0	1	0	0	1				4	4
1	0	0	1	1	0	0	1	1	0	0	1				1	1	3
0	1	1	1	0	1	0	1	0	1	1					2	2
0	0	1	0	0	1	0	1	0	0	0					3	3
0	1	0	1	0	1	1	0	0	0	1					4	4
1	1	0	0	0	0	1	1	0	1	0					1	1	4
1	1	0	1	1	0	1	1	0	1	1					2	2
1	1	0	0	1	0	0	1	0	1	0					3	3
1	1	1	1	1	0	0	0	0	1	1					4	4

Based on Table 7, the first objective function yielded a value of 5,651,584 units. An appropriate optimal value for the NPV was achieved in accordance with the values of the second and third objective functions, which focused on minimizing environmental pollutants and maximizing the social dimension, specifically job opportunities created in the production and distribution centers, respectively.

Each objective function includes parameters (inputs) and decision variables. According to the constraints defined in the study, the capacities of the production and distribution locations, among other factors, were used to determine the decision variables of the problem. Table 8 lists the values of decision variables that led to these objective functions. It shows the results of solving the problem in three dimensions (small, medium and large). The decision variables examined included the distributed product p from production center i to distribution center j over the time period t (QZ_pijt) and the product sold at distribution center j to market m over the time period t (Sa_pjmt). Table 9 depicts the binary variables related to the distribution and production centers, as well as the products (as shown in Table 3). Based on the obtained results, achieving the optimal state in all three problem sizes is crucial. According to the results of the algorithm employed, small problems in the model can reach the optimal state with very high speed and accuracy.

4.6. Sensitivity analysis

This section analyzes the impact of three key parameters on the objective functions across four different scenarios: (i) I_t (discount rate over time t), (ii) UR_pi (utilization coefficient of raw materials containing contaminants in production phase of product p at production center i), and (iii) SO_it (increase in satisfaction associated with the number of job opportunities created at production center i during time period t). These parameters have been identified as significant factors based on the results obtained from solving the model. Each sensitivity analysis was conducted in four modes, altering the values of these parameters in the research data to evaluate their impact on the objective functions. Furthermore, to evaluate the impacts of these changes on the objective functions, the average of the non-dominated solutions identified by the algorithm has been utilized.

Sensitivity analysis on I_t _: Table 10 presents the results of the sensitivity analysis conducted on the discount rate parameter over time; meanwhile, Figure 1 illustrates the behavior of the objective functions across the four cases. The discount rate is a central economic parameter in the model, particularly in objective function 1, which aims to maximize the NPV across the planning horizon. Its inclusion was justified by the following considerations:

The discount rate directly affects how future revenues and costs are evaluated, impacting strategic decisions such as facility location, production planning, and investment timing;

In industries like oil and gas, which involve long-term capital-intensive infrastructure, the discount rate significantly influences profitability and financial sustainability;

Sensitivity to this parameter offers managers insights into the financial resilience of network design under varying economic conditions (e.g., inflation, interest rates, or policy changes).

Figure 1.

Behavior of the Objective Functions in the Sensitivity Analysis on the Discount Rate Parameter Over Time (I_t).

Table 10.

Sensitivity Analysis on the Discount Rate Parameter Over Time.

Case No.	I_t	First objective function
1	30	657065
2	37	658839
3	48	663829
4	85	664418

In this analysis, a wide range of values (10%–45%) was selected to simulate real-world variability across different fiscal environments. The results demonstrated the robustness of the model and how profitability shifted under different economic assumptions. The I_t increased from 10% to 45%, resulting in two distinct effects on the objective functions. As I_t rose, the NPV increased due to the corresponding growth in demand, as reflected in the first objective function. Based on these results, it is suggested to relevant managers that increasing the discount rate can enhance the NPV by driving a growth in demand. This increase in demand not only boosts the NPV but also reduces maintenance and other related costs.

Sensitivity analysis on UR_pi: The raw material utilization coefficient represents the efficiency of converting raw materials into final products, with a particular focus on those that contribute to pollutant generation (e.g., CO₂ emissions). This parameter was chosen for the following reasons:

It is directly linked to objective function 2, which seeks to minimize environmental impacts;

The coefficient influences the amount of waste and emissions, especially in scenarios where polluting raw materials (e.g., fossil-based inputs) are used inefficiently;

In practical terms, this parameter reflects technological capability, process optimization, and material selection policies, all of which are crucial levers in environmental sustainability strategies.

Varying the UR_pi across a realistic spectrum (5% to 20%) enabled us to observe the sensitivity of both economic and environmental objectives to technological inefficiencies. As the results in Table 11 and Figure 2 showed, higher inefficiencies significantly rose system costs and emissions, thereby reinforcing the need for green process innovation. This highlighted the significant role of this parameter in achieving an optimal system configuration. Notably, the impact of this increase on the second objective function–representing the costs associated with transportation–exceeded its effect on the total costs in the system, as captured by the first objective function.

Figure 2.

Behavior of objective functions for sensitivity analysis on the consumption coefficient of raw materials containing pollutants.

Table 11.

Sensitivity Analysis on the Consumption Coefficient of raw Materials Containing Pollutants in Product Production.

Case No.	UR_pi	First objective function	Second objective function
1	5	6533185	5
2	10	6577123	10
3	15	6594257	14
4	20	6617833	18

Sensitivity analysis on SO_it: Table 12 depicts the sensitivity analysis on the increase in social satisfaction of sustainability, resulting from the number of job opportunities created at each production center and its impact on the model's objective functions. The behavior of the objective functions is depicted in Figure 3. In this analysis, the social satisfaction of sustainability, which was linked to the number of job opportunities created throughout the region in each period, was assumed to increase from 45% to 80%. The effects of this increase on the objective functions were measured. Increasing the level of social satisfaction of sustainability led to a reduction in processing and transfer costs, but it also increased shortage costs. Given the uncertainty in demand and the significant impact of this factor on system costs, the use of a dynamic model like the one developed in this research provides valuable insights for managers.

Figure 3.

Behavior of the objective functions for the sensitivity analysis on the social satisfaction of sustainability.

Table 12.

Sensitivity Analysis on Social Satisfaction of Sustainability.

Case No.	SO_it	First objective function	Third objective function
1	50	6174493	82
2	55	6272704	87
3	65	6353318	88
4	75	6458219	89

4. Conclusion

SCM was analyzed through the development of a multi-period, multi-product model addressing a location-allocation problem within a SC. The model integrated monetary, environmental, and social dimensions to establish a sustainable network under uncertain conditions. Environmental considerations were incorporated by accounting for varying levels of vehicle fleets and GHG emissions. Additionally, the model included the creation of job opportunities at production and distribution centers, contributing to customer satisfaction and reflecting the social perspective. To address the inherent uncertainty in parameters, the Robust Counterpart Optimization approach was employed. It was concluded that achieving the optimal solution in all three problems was crucial. The results demonstrated that smaller-scale problems in the model could achieve the optimal solution with exceptional speed and high accuracy. According to the sensitivity analyses on the discount rate (I_t), the utilization coefficient of raw materials containing contaminants in the production phase of products in production centers (UR_pi), and the level of increased satisfaction associated with the number of job opportunities created in production centers (SO_it), an increase in these parameters was found to elevate all objective functions. Therefore, it is advisable for industries and organizations to focus on mitigating these parameters to reduce monetary and environmental costs effectively. To achieve sustainable management, managers are encouraged to adopt a dynamic model that accounts for demand uncertainty and the significant influence of the social satisfaction parameter on system costs.

It is noteworthy that, like any modeling approach, this study has limitations. Model scalability may pose challenges when extended to highly complex, large-scale systems, and the results depend heavily on the quality and granularity of the available input data. Additionally, real-world implementation may require adjustments to account for regional policy constraints and logistical variances.

Future research should focus on enhancing model scalability through hierarchical or decentralized frameworks, improving data integration with real-time systems, and expanding the model to incorporate emerging concepts in environmental and social sustainability. Special emphasis should also be placed on the development of social impact assessment frameworks, which remain underexplored and deserve greater attention.

6. Managerial Implications

This study offers key insights for supply chain managers in the oil and gas sector aiming to enhance sustainability and efficiency. The integrated model supports strategic decision-making under uncertainty by balancing economic, environmental, and social objectives.

Robust Decision-Making: The use of Robust Counterpart Optimization enables managers to develop flexible strategies that remain effective amid demand and cost uncertainties.

Financial Optimization: An increase in the discount rate (I_t) boosts NPV through higher demand, suggesting that demand-stimulation strategies can enhance profitability and reduce operational costs.

Environmental Management: Limiting the use of pollutant-laden raw materials (UR_pi) is crucial for improving efficiency and minimizing environmental impacts.

Social Sustainability: Job creation (SO_it) enhances social satisfaction but affects system costs. Workforce planning should align with sustainability and CSR objectives.

Scalable Application: The model's efficiency in small-scale scenarios supports its gradual deployment across regions or product lines before full-scale implementation.

Footnotes

ORCID iDs

Sajad Taghizadeh Irani

Hosein Arman

Abdollah Hadi-Vencheh

References

Amirian

Amiri

Taghavifard

M. T.

(2022). The emergence of a sustainable and reliable supply chain paradigm in supply chain network design. Complexity, 2022, 1–29. https://doi.org/10.1155/2022/9415465

Arman

(2023). Fuzzy analytic hierarchy process for pentagonal fuzzy numbers and its application in sustainable supplier selection. J Clean Prod, 409(Suppl. C), 137190. https://doi.org/10.1016/j.jclepro.2023.137190

Arman

Hadi-Vencheh

Golmohammadi

A.-M.

Dehghani

Nadimi-Shahraki

M. H.

(2024). Optimal locating by integrating volumetric fuzzy sets and geographic coordinate system: An application to healthcare. Comput Oper Res, 161(Suppl. C), 106377. https://doi.org/10.1016/j.cor.2023.106377

Armin Jabbarzadeh

B. F.

Sabouhi

(2018). Resilient and sustainable supply chain design: Sustainability analysis under disruption risks. Int J Prod Res, 56(17), 5945–5968. https://doi.org/10.1080/00207543.2018.1461950

Bavarsad

Boshagh

Kayedian

(2014). A study on supply chain risk factors and their impact on organizational performance. Int J Oper Logist Manag, 3(3), 192–211.

Ben-Tal

Golany

Nemirovski

Vial

J.-P.

(2005). Retailer-supplier flexible commitments contracts: A robust optimization approach. Manuf Serv Oper Manag, 7(3), 169–271. https://doi.org/10.1287/msom.1050.0081

Chen

Hendalianpour

Feylizadeh

M. R.

(2023). Factors affecting the use of blockchain technology in humanitarian supply chain: A novel fuzzy large-scale group-DEMATEL. Gr Decis Negot, 32(2), 359–394. https://doi.org/10.1007/s10726-022-09811-z

Chen

Liu

(2023). Designing globalized robust supply chain network for sustainable biomass-based power generation problem. J Clean Prod, 413, 137403. https://doi.org/10.1016/j.jclepro.2023.137403

Christopher

Peck

(2004). Building the resilient supply chain. Int J Logist Manag, 15(2), 1–13. https://doi.org/10.1108/09574090410700275

10.

Dubey

Gunasekaran

Papadopoulos

Childe

S. J.

Shibin

K. T.

Wamba

S. F.

(2017). Sustainable supply chain management: Framework and further research directions. J Clean Prod, 142, 1119–1130. https://doi.org/10.1016/j.jclepro.2016.03.117

11.

Emami

Kamranrad

Jokar

(2021). Location and allocation in multi-level supply chain network of projects. J. Qual. Eng. Prod. Optim, 6(1), 2423–3781. https://doi.org/10.22070/JQEPO.2021.13527.1173

12.

Foroozesh

Karimi

Mousavi

S. M.

Mojtahedi

(2023). A hybrid decision-making method using robust programming and interval-valued fuzzy sets for sustainable-resilient supply chain network design considering circular economy and technology levels. J Ind Inf Integr, 33, 100440. https://doi.org/10.1016/j.jii.2023.100440

13.

Gao

Cao

(2020). A novel multi-objective scenario-based optimization model for sustainable reverse logistics supply chain network redesign considering facility reconstruction. J Clean Prod, 270, 122405. https://doi.org/10.1016/j.jclepro.2020.122405

14.

Gao

Ruan

Y. Z.

(2017). From a systematic literature review to integrated definition for sustainable supply chain innovation (SSCI). J Clean Prod, 142(Part 4), 1518–1538. https://doi.org/10.1016/j.jclepro.2016.11.153

15.

Genovese

Acquaye

A. A.

Figueroa

Koh

S. C. L.

(2017). Sustainable supply chain management and the transition towards a circular economy: Evidence and some applications. Omega, 66(Part PB), 344–357. https://doi.org/10.1016/j.omega.2015.05.015

16.

Govindan

Jafarian

Nourbakhsh

(2015). Bi-objective integrating sustainable order allocation and sustainable supply chain network strategic design with stochastic demand using a novel robust hybrid multi-objective metaheuristic. Comput Oper Res, 62, 112–130. https://doi.org/10.1016/j.cor.2014.12.014

17.

Govindan

Kannan

Jørgensen

T. B.

Nielsen

T. S.

(2022). Supply chain 4.0 performance measurement: A systematic literature review, framework development, and empirical evidence. Transp Res Part E Logist Transp Rev, 164, 102725. https://doi.org/10.1016/j.tre.2022.102725

18.

Grosvold

Hoejmose

S. U.

Roehrich

J. K.

(2014). Squaring the circle: Management, measurement and performance of sustainability in supply chains. Supply Chain Manag, 19(3), 292–305. https://doi.org/10.1108/SCM-12-2013-0440

19.

Gurzawska

(2020). Towards responsible and sustainable supply chains–innovation, multi-stakeholder approach and governance. Philos Manag, 19(3), 267–295. https://doi.org/10.1007/s40926-019-00114-z

20.

Hadi-Vencheh

Khodadadipour

Tan

Arman

Roubaud

(2024). Cross-efficiency analysis of energy sector using stochastic DEA: Considering pollutant emissions. J Environ Manage, 364, 121319. https://doi.org/10.1016/j.jenvman.2024.121319

21.

Hanaysha

J. R.

Al-Shaikh

M. E.

Joghee

Alzoubi

H. M.

(2022). Impact of innovation capabilities on business sustainability in small and Medium enterprises. FIIB Bus Rev, 11(1), 67–78. https://doi.org/10.1177/23197145211042232

22.

Hendalianpour

Fakhrabadi

Sangari

Razmi

(2022). A combined benders decomposition and lagrangian relaxation algorithm for optimizing a multi-product, multi-level omni-channel distribution system. Sci Iran E, 29(1), 355––3371. https://doi.org/10.24200/sci.2020.53644.3349

23.

Homayouni

Pishvaee

M. S.

Jahani

Ivanov

(2023). A robust-heuristic optimization approach to a green supply chain design with consideration of assorted vehicle types and carbon policies under uncertainty. Ann Oper Res, 324(1), 395–435. https://doi.org/10.1007/s10479-021-03985-6

24.

Hosseini Shekarabi

S. A.

Kiani Mavi

Macau

F. R.

(2025). An extended robust optimisation approach for sustainable and resilient supply chain network design: A case of perishable products. Eng Appl Artif Intell, 152, 110846. https://doi.org/10.1016/j.engappai.2025.110846

25.

Hosseinkhani

Kargari

(2022). Production of high-quality drinking water from chillers and air conditioning units’ condensates using UV/GAC/MF/NF hybrid system. J Clean Prod, 368, 133177. https://doi.org/10.1016/j.jclepro.2022.133177

26.

Hosseinkhani

Kargari

Sanaeepur

(2014). Facilitated transport of CO2 through co(II)-S-EPDM ionomer membrane. J Memb Sci, 469, 151–161. https://doi.org/10.1016/j.memsci.2014.06.021

27.

Hussain

Assavapokee

Khumawala

(2006). Supply chain management in the petroleum industry: Challenges and opportunities. Int J Glob Logist Supply Chain Manag, 1(2), 90–97.

28.

Jabbarzadeh

Fahimnia

Sheu

J.-B.

Moghadam

H. S.

(2016). Designing a supply chain resilient to major disruptions and supply/demand interruptions. Transp Res Part B Methodol, 94, 121–149. https://doi.org/10.1016/j.trb.2016.09.004

29.

Jacobs

(1999). Sustainable development as a contested concept. In Fairness and futurity: Essays on environmental sustainability and social justice (pp. 21–45). Oxford University Press. https://doi.org/10.1093/0198294891.003.0002

30.

Juttner

(2005). Supply chain risk management: Understanding the business requirements from a practitioner perspective. Int J Logist Manag, 16(1), 120–141. https://doi.org/10.1108/09574090510617385

31.

Khorasani

S. T.

(2018). A robust optimization model for supply chain in Agile and flexible mode based on variables of uncertainty. Glob J Flex Syst Manag, 19(3), 239–253. https://doi.org/10.1007/s40171-018-0191-y

32.

Lababidi

H. M. S.

Ahmed

M. A.

Alatiqi

I. M.

Al-Enzi

A. F.

(2004). Optimizing the supply chain of a petrochemical company under uncertain operating and economic conditions. Ind & Eng Chem Res, 43(1), 63–73. https://doi.org/10.1021/ie030555d

33.

Lambert

D. M.

Cooper

M. C.

Pagh

J. D.

(1998). Supply chain management: Implementation issues and research opportunities. Int J Logist Manag, 9(2), 1–20. https://doi.org/10.1108/09574099810805807

34.

Lima

Relvas

Barbosa-Póvoa

Morales

J. M.

(2019). Adjustable robust optimization for planning logistics operations in downstream oil networks. Processes, 7(8), 1–31. https://doi.org/10.3390/pr7080507

35.

Liu

Hendalianpour

Hamzehlou

Feylizadeh

M. R.

Razmi

(2021). Identify and rank the challenges of implementing sustainable supply chain blockchain technology using the Bayesian best worst method. Technol Econ Dev Econ, 27(3), 656–680. https://doi.org/10.3846/tede.2021.14421

36.

Mangun

Thurston

(2002). Incorporating component reuse, remanufacture, and recycle into product portfolio design. IEEE Trans Eng Manag, 49(4), 479–490. https://doi.org/10.1109/TEM.2002.807292

37.

Marzban

Shafiee

Mozaffari

M. R.

(2022). Four-Stage supply chain design for perishable products and evaluate it by considering the triple dimensions of sustainability. J Syst Manag, 8(4), 109–132. https://doi.org/10.30495/JSM.2022.1966734.1684

38.

Mastrocinque

Ramírez

F. J.

Honrubia-Escribano

Pham

D. T.

(2020). An AHP-based multi-criteria model for sustainable supply chain development in the renewable energy sector. Expert Syst Appl, 150, 113321. https://doi.org/10.1016/j.eswa.2020.113321

39.

Mogale

N. C. D. G.

Tiwari

M. K.

(2020). Modelling of sustainable food grain supply chain distribution system: A bi-objective approach. Int J Prod Res, 58(18), 5521–5544. https://doi.org/10.1080/00207543.2019.1669840

40.

Mohammed

Selim

S. Z.

Hassan

Syed

M. N.

(2017). Multi-period planning of closed-loop supply chain with carbon policies under uncertainty. Transp Res Part D Transp Environ, 51, 146–172. https://doi.org/10.1016/j.trd.2016.10.033

41.

Monfared

Arman

M. H.

Barati

(2018). Combination of the analytic network process method and multi-objective decision-making in order to predict and reduce the future risks of suppliers. J Ind Manag Perspect, 8(2), 111–134. https://jimp.sbu.ac.ir/article_87175.html

42.

Mousavi

S. F.

Apornak

Pourhassan

M. R.

(2023). Robust optimization model to improve supply chain network productivity under uncertainty. J Appl Res Ind Eng, 10(2), 273–285. https://doi.org/10.22105/jarie.2022.316357.1402

43.

Nayeri

Paydar

M. M.

Asadi-Gangraj

Emami

(2020). Multi-objective fuzzy robust optimization approach to sustainable closed-loop supply chain network design. Comput Ind Eng, 148, 106716. https://doi.org/10.1016/j.cie.2020.106716

44.

Nidumolu

Prahalad

C. K.

Rangaswami

M. R.

(2009). Why sustainability is now the key driver of innovation. Harvard Business Review, 87(9), 57–64.

45.

Pagell

(2009). Building a more complete theory of sustainable supply chain management using case studies of 10 exemplars. Supply Chain Manag, 45(2), 37–56. https://doi.org/10.1111/j.1745-493X.2009.03162.x

46.

Patidar

Agrawal

(2020). A mathematical model formulation to design a traditional Indian agri-fresh food supply chain: A case study problem. Benchmarking An Int. J, 27(8), 2341–2363. https://doi.org/10.1108/BIJ-01-2020-0013

47.

Pishvaee

M. S.

Rabbani

Torabi

S. A.

(2011). A robust optimization approach to closed-loop supply chain network design under uncertainty. Appl Math Model, 35(2), 637–649. https://doi.org/10.1016/j.apm.2010.07.013

48.

Rabbani

Hosseini-Mokhallesun

S. A. A.

Ordibazar

AH.

Farrokhi-Asl

(2020). Masoud rabbani seyed ali akbar hosseini-mokhallesun and H. Farrokhi-asl, ‘A hybrid robust possibilistic approach for a sustainable supply chain location-allocation network design’. Int J Syst. Sci Oper & Logist, 7(1), 60–75. https://doi.org/10.1080/23302674.2018.1506061

49.

Rajesh

Ravi

(2015). Supplier selection in resilient supply chains: A grey relational analysis approach. J Clean Prod, 86, 343–359. https://doi.org/10.1016/j.jclepro.2014.08.054

50.

Rice

J. B.

Caniato

(2003). Building a secure and resilient supply network. Supply Chain Management Review, 7(5), 22–30.

51.

Saidi-Mehrabad

Aazami

Goli

(2017). A location-allocation model in the multi-level supply chain with multi-objective evolutionary approach. J Ind Syst Eng, 10(3), 140–160.

52.

Sangbor

A. M.

Safi

M. R.

Azar

Rabieh

(2022). Identifying and prioritizing sustainable supply chain management enablers in the petrochemical industry by combined approach of meta-synthesis method and graph theory and matrix approach (GTMA). Ind Manag Stud, 20(46), 1–34. https://doi.org/10.22054/JIMS.2022.39256.2254

53.

Schonsleben

(2007). Integral logistics management - Operations and supply chain management in comprehensive value-added networks (3rd ed). Auerbach Publications.

54.

Shafiee

Kazemi

Jafarnejad Chaghooshi

Sazvar

Amoozad Mahdiraji

(2021). A robust multi-objective optimization model for inventory and production management with environmental and social consideration: A real case of dairy industry. J Clean Prod, 294, 126230. https://doi.org/10.1016/j.jclepro.2021.126230

55.

Surya

Menne

Sabhan

Suriani

Abubakar

Idris

(2021). Economic growth, increasing productivity of SMEs, and open innovation. J Open Innov Technol Mark Complex, 7(1), 1–37. https://doi.org/10.3390/joitmc7010020

56.

Tang

(2006). Perspectives in supply chain risk management. Int J Prod Econ, 103(2), 451–488. https://doi.org/10.1016/j.ijpe.2005.12.006

57.

Tavana

Arman

Hadi-Vencheh

Mansoori

(2023). A fuzzy multi-objective optimization model for sustainable location planning using volumetric fuzzy sets. Ann Oper Res, 1–29. https://doi.org/10.1007/s10479-023-05505-0

58.

Tirkolaee

E. B.

Mardani

Dashtian

Soltani

Weber

G.-W.

(2020). A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design. J Clean Prod, 250, 119517. https://doi.org/10.1016/j.jclepro.2019.119517

59.

Tsiakis

Shah

Pantelides

C. C.

(2001). Design of multi-echelon supply chain networks under demand uncertainty. Ind Eng Chem Res, 40(16), 3585–3604. https://doi.org/10.1021/ie0100030

60.

Vadian

Movahedi

M. M.

Abri

A. G.

Lotfi

M. R.

(2022). The impact of petrochemical industry economic activities on environmental factors using a fuzzy mathematical programming model. Int J Nonlinear Anal. Appl, 13(1), 539–553. https://doi.org/10.22075/ijnaa.2021.23377.2525

61.

Viswanadham

Gaonkar

R. S.

(2008). Risk management in global supply chain networks. In Tang

C. S.

Teo

C.-P.

Wei

K.-K.

(Eds.), Supply chain analysis: A handbook on the interaction of information, system and optimization (pp. 201–222). Springer US. https://doi.org/10.1007/978-0-387-75240-2_8

62.

Wagner

S. M.

Bode

(2008). An empirical examination of organizational performance along several dimensions of risk. J Bus Logist, 29(1), 307–325. https://doi.org/10.1002/j.2158-1592.2008.tb00081.x

63.

Solvang

W. D.

(2020). A fuzzy-stochastic multi-objective model for sustainable planning of a closed-loop supply chain considering mixed uncertainty and network flexibility. J Clean Prod, 266, 121702. https://doi.org/10.1016/j.jclepro.2020.121702

64.

Zhao

Hendalianpour

Liu

(2024). Blockchain technology in omnichannel retailing: A novel fuzzy large-scale group-DEMATEL & ordinal priority approach. Expert Syst Appl, 249, 123485. https://doi.org/10.1016/j.eswa.2024.123485

65.

Ziaei

Jabbarzadeh

(2021). A multi-objective robust optimization approach for green location-routing planning of multi-modal transportation systems under uncertainty. J Clean Prod, 291, 125293. https://doi.org/10.1016/j.jclepro.2020.125293

66.

Ziegenbein

(2007). Supply Chain Risiken: Identifikation, Bewertung und Steuerung. vdf Hochschulverlag AG.

$i \in I$	: A set of potential oil and gas production locations
$j \in J$	: A set of potential oil and gas distribution locations
$m \in M$	: A set of product sales markets
$p \in P$	: A set of final products (oil and gas derivatives)
$s \in S$	: A set of scenarios
$t \in T$	: A set of time periods

Scenarios	Time periods	Final products	Potential retailers	Potential locations
				Distribution	Production
5	12	5	5	4	3

Location-allocation Model for the Supply Chain of Multi-Product and Multi-Period Oil and Gas Industries with Consideration of Sustainability Under Uncertainty

Abstract

Keywords

1. Introduction

2. Literature Reviews

2.1. Theoretical Underpinnings and General Findings

2.2. Specific Studies Related to the Research Problem

3. Problem Description

4. Research Methodology

4.1. Production Scenarios

4.2. Modelling Approach

4.2.1. Indexes and Parameters, Decision Variables and Objective Functions

Table 4. Problem Size. Scenarios Time periods Final products Potential retailers Potential locations Distribution Production 5 12 5 5 4 3

4.5. Results

Table 7. Optimal Objective Function Values of the Given Multi-Objective Problem. Objective function Lower bound Upper bound Optimal value The first 4,987,125 5,856,802 5,651,584 The second 0.22 0.15 0.19 The third 0.72 0.92 0.85

6. Managerial Implications

Footnotes

ORCID iDs

References

Table 4.
Problem Size.

Scenarios Time periods Final products Potential retailers Potential locations

Distribution Production

5 12 5 5 4 3

Table 7.
Optimal Objective Function Values of the Given Multi-Objective Problem.

Objective function Lower bound Upper bound Optimal value

The first 4,987,125 5,856,802 5,651,584

The second 0.22 0.15 0.19

The third 0.72 0.92 0.85