A robust-heuristic optimization approach to a green supply chain design with consideration of assorted vehicle types and carbon policies under uncertainty

Homayouni, Zahra; Pishvaee, Mir Saman; Jahani, Hamed; Ivanov, Dmitry Article — Published Version A robust-heuristic optimization approach to a green supply chain design with consideration of assorted vehicle types and carbon policies under uncertainty Annals of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Homayouni, Zahra; Pishvaee, Mir Saman; Jahani, Hamed; Ivanov, Dmitry (2021) : A robust-heuristic optimization approach to a green supply chain design with consideration of assorted vehicle types and carbon policies under uncertainty, Annals of Operations Research, ISSN 1572-9338, Springer US, New York, NY, pp. 1-41, https://doi.org/10.1007/s10479-021-03985-6 This Version is available at: https://hdl.handle.net/10419/287199 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Annals of Operations Research https://doi.org/10.1007/s10479-021-03985-6 ORIGINAL RESEARCH A robust-heuristic optimization approach to a green supply chain design with consideration of assorted vehicle types and carbon policies under uncertainty Zahra Homayouni1·Mir Saman Pishvaee1·Hamed Jahani2·Dmitry Ivanov3 Accepted: 4 February 2021 © The Author(s) 2021 Abstract Adoption of carbon regulation mechanisms facilitates an evolution toward green and sustainable supply chains followed by an increased complexity. Through the development and usage of a multi-choice goal programming model solved by an improved algorithm, this article investigates sustainability strategies for carbon regulations mechanisms. We first propose a sustainable logistics model that considers assorted vehicle types and gas emissions involved with product transportation. We then construct a bi-objective model that minimizes total cost as the first objective function and follows environmental considerations in the second one. With our novel robust-heuristic optimization approach, we seek to support the decisionmakers in comparison and selection of carbon emission policies in supply chains in complex settings with assorted vehicle types, demand and economic uncertainty. We deploy our model in a case-study to evaluate and analyse two carbon reduction policies, i.e., carbon-tax and cap-and-trade policies. The results demonstrate that our robust-heuristic methodology can efficiently deal with demand and economic uncertainty, especially in large-scale problems. Our findings suggest that governmental incentives for a cap-and-trade policy would be more effective for supply chains in lowering pollution by investing in cleaner technologies and adopting greener practices. BDmitry Ivanov [email protected] Zahra Homayouni [email protected] Mir Saman Pishvaee Pishv[email protected] Hamed Jahani [email protected] 1School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran 2School of Accounting, Information Systems and Supply Chain, RMIT University, Melbourne, Australia 3Global Supply Chain and Operations Management, Berlin School of Economics and Law, Berlin, Germany 123 Annals of Operations Research Keywords Robust-heuristic optimization ·Green supply chain ·Sustainable supply chain · Sustainable logistics ·Improved multi-choice goal programming 1 Introduction The integration of sustainability issues into supply chain (SC) management has progressed remarkably, most of it focused on the areas of the green supply chain (GSC) and the sustainable supply chain (SSC) (Tang and Zhou 2012; Golinska-Dawson et al. 2018; Heydari et al. 2020). Increasing concerns about the environmental impacts and international and government regulations have attracted research attention to the GSC problems beyond merely economic aspects (Ivanov et al. 2019). In an GSC, the environmental impacts from SCs need to be minimized complementing total cost minimization (Rezaee et al. 2017). Moreover, social aspects in SCs became a trend and lead to introducing the SSC network (Carter and Rogers 2008; Pavlov et al. 2019). In general, when the financial, environmental and social impacts of the SC are considered simultaneously, the traditional SC shifts toward the SSC. The transition from the traditional goals of the SC to the new sustainable objectives is also identified as the company’s competitive advantage (Dubey et al. 2015; Giannakis and Papadopoulos 2016). Improvements in operating costs efficiency and service levels while paying special attention to the environmental, economic, and social considerations in the SC belong to major requirements to succeed in highly competitive markets (Golinska-Dawson et al. 2018; Brandenburg et al. 2019). Due to environmental pollution and increased global warming, government and international bodies have introduced laws obliging companies to address environmental issues. One of the most important parts of new regulations is reducing carbon emissions/footprint that improves the business’s environmental performance (Golinska and Romano 2012). In an SC, this will bring the integrity of all parts of the SC in social commitments. A carbon footprint reduction project is therefore of a global economic importance (Fahimnia and Jabbarzadeh 2016). A recent European Commission report illustrates that the amount of transport gas emissions has been continually increasing, and if no action is taken, transport emissions could make up more than 30% of total EU gas emissions by the end of 2020. The report also demonstrates that 93–95% of greenhouse gas emissions resulting from transport operations are composed of CO2(Eurostat 2019). The European Union Emissions Trading System (EU-ETS) is a tool that uses a tiered method for calculating emissions, linked to necessary uncertainty ranges that are supported in the EU climate policy’s monitoring and reporting guidelines. The EU-ETS is identified as a powerful system for deducting greenhouse gas emissions especially CO2emissions cost-effectively (EU climate actions 2019). SC’s CO2emissions can be seen in the various components, including raw material procurement, product manufacturing, distribution and retail, and disposal and recycling (Zhen et al. 2019; Golinska et al. 2015). Figure 1illustrates that most of the CO2emissions are accounted in transportation and logistics processes of GSCs and SSCs, according to a survey through 215 SC companies in Europe (GSCmonitor 2015). From GSC and SSC point of view, Mohammed et al. (2017) concede that although the sources of CO2emissions have been broadly investigated in academic research, the decisions regarding types of vehicles used is critical in real-world transportation systems, and this fact was left ignored in most 123 Annals of Operations Research 0% 10% 20% 30% 40% 50% 60% Others Agri. produce/extracon of raw materials Whole life cycle of a product Packaging Recycling Disposal Manufacturing Transportaon for purchasing Intra-Logiscs Transportaon for distribuon 57% 51% 51% 47% 44% 36% 27% 27% 22% 9% Fig. 1 Areas of focus for the calculation of carbon emissions in GSC and SSC companies (Based on GSCmonitor (2015)) of the GSC and SSC models. Moreover, an increasing uncertainty of demand and economic environments represent a research challenge for design of GSC and SSC (Brandenburg and Rebs 2015; Allaoui et al. 2019). Recent literature suggests that instead of choosing an efficient policy from the existing carbon emission plans, an applicable model for the GSC and SSC could be created by considering uncertainty features in the key parameters of the model, especially demand and related costs (He et al. 2019). To this end, we aim to design an GSC by employing a robust-heuristic optimizationmethodtocopewithdemandandeconomicuncertaintyandconsideringassorted vehicletypes.Robust optimization is one of the branches of optimization theory that cope with uncertain optimization problems. In scenario-based robust optimization, this method combines scenario-based description of problem data with the solution formulations like goal programming. This approach tries to generate solutions that are less sensitive to realizations of the model data. Robust optimization has some advantages comparing other approaches such as stochastic programming (Mulvey et al. 1995). We contribute to literature by offering a comprehensive approach to minimize carbon emissions in most of the SC processes and considering diversity of vehicle types and uncertainty in SC costs. We present a bi-objective model that focuses on a specific capacity for an environmentally-friendly SC and carbon emissions. The proposed bi-objective model is converted to a single-objective one that aims to solve the problem by using improved multi-choice goal programming (IMCGP). Since the model is an NP-hard one in largescale problems, for reducing the solving time and complexity of the problem, we employ a heuristic method combined with the improved IMCGP. To test and examine our model, two carbon reduction policies—namely carbon-tax and cap-and-trade—are compared in a real case study. The remainder of this paper is organised as follows. In Sect. 2,wepresentadetailed literature review about the topic. In Sect. 3, the problem is defined and formulated. We explain our novel solution approach in Sect. 4.6. A real case study and related numerical tests are provided in Sect. 5, illustrating the effectiveness of the proposed model. Managerial insights are provided in Sect. 7. In Sect. 8, we conclude with summarizing major insights of our study and outlining possible future research directions. 123 Annals of Operations Research 2 Literature review In this section, we provide a brief review of extant literature in GSC and CO2footprint modeling and sustainability considerations in SC modeling. We also develop a conceptual linkage between these subjects to provide the contributions expected from this study. A summary of recent literature on these topics is presented in Table 1. 2.1 Green supply chain and CO2footprint modeling In recent years, a greater focus on carbon emissions and footprints can be observed in GSC models (Mohammed et al. 2018; Tirkolaee et al. 2020). Carbon footprint measurements are needed to provide a reliable estimate of the total amount of greenhouse gas emissions released in the life cycle of products and services along the SC. These estimations can include various elements from raw material extraction to production, distribution, storage, and recycling (Plassmann et al. 2010). Chen and Chen (2017) study the carbon footprint and allocation of responsibility involved with the production stage. Despite the advantage of optimizing the social value of the GSC, they only consider single-period and single-product network and ignore the transportation and holding processes as the source of carbon emissions. In another study evaluating environmental implications in the SC, Bazan et al. (2015) minimize the total cost of the reverse logistics network, considering carbon footprint and production energy simultaneously. The green concept has also been employed for green inventory routing problems with consideration of several interval fuel consumptions (Franco et al. 2016,2017). In a review article, Dekker et al. (2012) discuss green logistics and the integration of different environmental aspects into green logistics, the most important of which is carbon emissions in transportation. Du et al. (2016) also examine low-carbon production and its implementation in the SC. Their suggested that a commercial system according to energy consumption can still be profitable in the case of low-carbon production. Hao et al. (2017) investigate the amount of greenhouse gas emissions and energy consumption, using a life cycle assessment framework under recycling options. The authors develop a low-carbon design approach to estimate the carbon footprint at each stage of the product life cycle. Aiming to minimize carbon footprint in a reverse logistics network, Kannan et al. (2012) propose a single-product, single-period linear integer programming model. Moreover, Zhao et al. (2013) develop a mathematical model to minimize the level of carbon emissions in transportation system and distribution centers. However, their model lacks the consideration of uncertainty in its parameters—a distinctive and substantial contribution made by our study. Ourliterature analysis demonstrates thatthe consideration of uncertaintyhas been growing for the recent years. For instance, in a study aimed at designing an SSC network under the uncertainty of capacity of suppliers, producers, and warehouses, Shaw et al. (2016) propose a model including greenhouse gas emissions and carbon trading issues. The authors also contemplate demand uncertainty in their model. Aljuneidi and Bulgak (2020)presentan integrated approach to designing a reverse logistics network for a sustainable manufacturing company. Their model minimizes carbon emissions and transportation distances between facilities by considering a hybrid production and reproduction system. Moreover, Reddy et al. (2019) consider a reverse-logistic network design (RLND) and develop a mixed-integer linear programming (MILP) model in a multi-period configuration. Their proposed network focuses on the choice of vehicle type and carbon emissions through operations and transportation by defining a corresponding binary variable that is equal to one when a vehicle type is selected between two nodes. Although their model has the advantage of vehicle type selection, the 123 Annals of Operations Research model is developed for a single period and does not consider any uncertainty in the main parameters of the model. Our review of the literature about GSC models also shows that the majority of the studies develop a multi-objective optimization problem and incorporate uncertainty in a non-linear context. They solve the proposed problem using a single-objective model converted from a multi-objective one (Govindan et al. 2020). 2.2 Sustainability considerations in supply chain modelling We now turn to multi-objective issues in designing SSC. Given the growing concern over sustainability in recent years, researchers and practitioners began to incorporate social and environmental factors into SC design in addition to economic factors. One of the main objective is to develop a model that can simultaneously cover economic, social, and environmental perspectives. While most research has focused on the economic aspects of SCs (Jahani et al. 2019), some recent studies have taken environmental aspects into account (Moreno-Camacho et al. 2019). Researchers believe that the use of sustainable SC management (SSCM) will bring several advantages for the organizations such as reducing environmental risks and pollution (Ansari and Kant 2017) and improving customer relationships (Sauer and Seuring 2019). The sustainability issues can be addressed by defining a multi-objective optimization modelfor an SC. Accordingly, Arampantzi and Minis (2017)consider a multi-objective mathematical framework for an SSC design, including significant decisions in high-performance SC design or redesign. Their model complies with all three goals of the SSC, i.e., economic, social, and environmental areas. Jabbarzadeh et al. (2018) present a hybrid approach to designing an SSC network that is flexible in the face of random disturbances. They also employ a fuzzy c-means clustering method to measure and evaluate suppliers’ sustainability performance. Although a new methodology is developed in their study for the SSC, the model considers a single-period and single-product problem. Considering a multi-product and multi-period reverse SC, John et al. (2017) develop an MILP model by integrating the carbon emission cost of transport activities. This study has ignored carbon emissions generated in the production and holding activities, and the uncertainty features in the main parameters like demand. Taking uncertainty into account is a common characteristic of recent modeling approaches for SSC design. Habibi et al. (2017) propose a multi-objective mathematical model for an SSC network under an uncertain return product parameter. In their RLND model, they propose the first objective as minimization of total cost, including the cost of moving facilities, cost of transfer stations, allocation costs of facilities, shipment costs, recovery activities costs, and penalty costs. The second objective of their model deliberates the minimization of environmental impacts and visual pollution. Zahiri et al. (2018) develop another multi-objective mixed-integer nonlinear programming (MINLP) model for an SSC network under the uncertain demand and supply parameters. They propose the first objective as the minimization of total cost, incorporating inventory costs, location and allocation costs of facilities, manufacturing costs, shipment costs, procurement costs, and fixed ordering costs. The second objective of their model is defined as the minimization of environmental impacts, and the third objective complies with the maximization of an SC’s responsiveness. This research only models a single-product network in which the sources of carbon emissions are ignored in the environmental impacts. In the case of sustainability and the RLND model, many studies can be found regarding a sustainable closed-loop SC network (e.g. Govindan et al. (2016); Soleimani et al. (2017); Sahebjamnia et al. (2018)). Also, some recent motivating studies 123 Annals of Operations Research consider green and sustainable closed-loop SC (see Zhen et al. (2019); Yun et al. (2020); Esmaeili et al. (2020)). 2.3 Contributions of the study The literature review illustrates that although there are many studies in the GSC and SSC fields, there are still some gaps. For example, some studies ignore considering the multiple sources of carbon emissions (see the models provided by Jindal and Sangwan (2017), Chen and Chen (2017), John et al. (2017), Reddy et al. (2019) and Govindan et al. (2020)). Even if some studies consider carbon emissions in various processes of the SC, they ignore the consideration of different types of vehicles in the transportation. We demonstrated the remarkable effect of this assumption in the Introduction Section (see the models developed by Soleimani et al. (2017), Yadollahinia et al. (2018), Yavari and Geraeli (2019), Banasik et al. (2019) and Zahiri et al. (2018)). We also insist on considering uncertainty features in all main parameters of a GSC or SSC model that is ignored in several relevant models (ChibelesMartins et al. 2016; John et al. 2017; Reddy et al. 2019; Franco and Alfonso-Lizarazo 2020). Finally, researchers have had less attention to compare different carbon policies and they usually focused on carbon cap-and-trade policy (see Kaur and Singh (2018) and Gholizadeh et al. (2020)). We contribute to closing these research gaps in multiple ways. Our article investigates sustainability strategies for carbon regulations mechanisms. We first propose a sustainable logistics model that considers assorted vehicle types and gas emissions involved with product transportation. We then construct a bi-objective model that minimizes total cost as the first objective function and follows environmental considerations in the second one. With our novel robust-heuristic optimization approach, we offer a decision-making support in comparison and selection of carbon emission policies in GSCs in complex settings with assorted vehicle types, demand and economic uncertainty. Due to the existence of nonlinear constraints arising from uncertain parameters, in our proposed MINLP model, we employ a robust optimization approach along with a heuristic method which allows reducing the computational time at different levels of uncertainty. Distinctively, in contrast to the consideration of uncertainty in demand and costs in the existing GSC and SSC models, our model has the advantage of contemplating uncertainty in all main costs associated with an SC (i.e. production, transportation, ordering, holding, and shortage costs). We deploy our model in a case-study to evaluate and analyse two carbon reduction policies, i.e., carbon-tax and cap-and-trade policies. The results demonstrate that our robust-heuristic methodology can efficiently deal with demand and economic uncertainty, especially in large-scale problems. Selecting optimal vehicle types with the lowest carbon emissions is another original feature of our study that can help managers to deduct carbon emissions more effectively in their GSCs. Our findings suggest that governmental incentives for a cap-and-trade policy would be more effective for supply chains in lowering pollution by investing in cleaner technologies and adopting greener practices. 3 Problem statement In this study, we consider a CO2footprint network and an emissions-reduction business scenario. We study a multi-period setting for a multi-product and multi-tier SC with multiple suppliers and multi-carrier transport. The problem involves ordering, manufacturing, and 123 Annals of Operations Research Table 1 Recent studies modeling GSC and SSC under uncertainty References Supply chain paradigm CO2footprint process Vehicle type selection Uncertain parameter Objective Function Period Product Method Green Sustainable Eco Env Soc MultiBiSingle Single MultiSingle MultiAkbari and Karimi (2015) *PT***Robust Govindan et al. (2015) * * D * * * Robust hybrid meta-heuristic ChibelesMartins et al. (2016) * * – * * * Simulated annealing meta-heuristic Zhalechian et al. (2016) * * D * * * Self-adaptive genetic algorithm Talaei et al. (2016) * * D-RC-PC * * * Robust fuzzy programming Jindal and Sangwan (2017) * T D * * * Interactive constraint method Chen and Chen (2017) * * P – * * * Particle swarm optimization John et al. (2017) *T – *** Jabbarzadeh et al. (2018) * D* **Robust Golpîra et al. (2017) * D-CT-CS * * * Karush–Kuhn– Tucker Robust Quddus et al. (2017) * D * * * * Heuristic Progressive hedging 123 Annals of Operations Research Table 1 continued References Supply chain paradigm CO2footprint process Vehicle type selection Uncertain parameter Objective Function Period Product Method Green Sustainable Eco Env Soc MultiBiSingle Single MultiSingle MultiSoleimani et al. (2017) * * D * * * GA-fuzzy optimization Zahiri et al. (2018) * D-S * * * Robust Yadollahinia et al. (2018) *D-CF***Robust Heidari-F and Pasandideh (2018) * * * D-S * * * Lagrangian relaxation approach-Robust Banasik et al. (2019) * * D * * * 2-stage stochastic programming Yavari and Geraeli (2019) * D * * * Robust Heuristic Reddy et al. (2019) * P,T * – * * * Three-phase Bender’s decomposition Govindan et al. (2020) * * * T * D * * * Hybrid approach of fuzzy analysis This study * * * P,T,B,S,H * D-PC-, TC-OC-, HC-SC * * * RobustHeuristic Ssupply, Ddemand, PC production cost, TC transportation cost, OC order cost, HC holding cost, SC shortage cost, CF Capacity of facilities, RC re-manufacture capacity, MC manufacture capacity, PT process time, PProduction, TTransportation, BBuyer, SSupplier, HHolding 123 Annals of Operations Research refer to equations without uncertain parameters or variables, whereas control constraints include non-deterministic parameters or variables. According to Mulvey et al. (1995), a typical mathematical formulation for a robust optimization model is represented in Eq. (16). The xvector reflects design variables, and the yvector describes control variables. A,B,andCare the parameter coefficient vectors and band eare the parameter vectors (the right hand side values). Aand bare certain values, while B,C,andeare uncertain. A special understanding of these parameters is known as a scenario that is determined by sindex in these parameters. The probability of each scenario is determined by ps.Thesymbol is employed to represent a set of scenarios. Consequently, the coefficients specifying uncertainty are Csand Bsfor each scenario (s∈). Also, the control variable yis modified after awareness of the scenario and can be replaced with ys regarding scenario s. As a consequence of the uncertainty of the parameters, the model may not be justified for some scenarios. Therefore, ηsis defined to represent the model under unjustified scenario s. Once the model is justified, ηsis equal to zero, and in other situations, it will gain a positive number. Min σ(x,y1,y2,...,ys)+γρ(η1,η 2,...,η s) s.t. Ax =b, Bsx+Csys+ηs=es, x≥0,ys≥0,η s≥0,∀s∈(16) The objective function represented in Eq. (16) includes two terms. The first term calculates therobustnessofthesolutionwhichshowstherisk-aversionlevelof decision-makersandtheir desire for lower costs. The second part calculates the robustness of the model by penalizing the solutions that violate the control constraints. The trade-off between the model robustness and the solution robustness is incorporated using coefficient (weight) γ. To better explain the effect of γ, if we insert a small value for this parameter, the objective function focuses on minimizing the first term, and the probability of obtaining an infeasible solution increases. Whereas, if γis large, the solution tends to be more feasible, but the first part of the objective function (i.e. σ(x,y1,y2,...,ys)) takes higher values. The ξsymbol and the array of ξ= f(x,y)are specified as cost and utility functions, respectively. For each scenario, the high variance for ξs=f(x,ys)determines that the decision is taken at high risk. In other words, a tiny variation in the uncertain parameters can result in large variations in the value of ffunction. Mulvey et al. (1995) employ the terms formulated in Eq. (17) to illustrate the solution’s stability. δis a weight reflecting the solution variance. σ(0)= s∈ psξs+δ s∈ psξs− s∈ p sξ s2 (17) Since the square term of Eq. (17) (i.e. s∈ps(ξs−s∈p sξ s)2) increases the computational time of solving the model, Yu and Li (2000) introduce the absolute value of the term, shown in Eq. (18), to reduce the operations related to the total computational time. σ(0)= s∈ psξs+δ s∈ ps ξs− s∈ p sξ s (18) Dealing with Eq. (18), which contains an absolute value and outlines a non-linear function, two additional variables Q+ sand Q− sare defined to linearize the resultant objective function. 123 Annals of Operations Research If s∈p sξ sis more than ξs,Q− sis returned and otherwise Q+ s. Therefore, Eq. (18)is reformulated as follows: σ(0)= s∈ psξs+δ s∈ ps(Q+ s+Q− s) s.t. ξs− s∈ p sξ s=Q+ s+Q− s,s∈, Q+ s,Q− s≥0,s∈. (19) According to the constraints of Eq. (19), it is clear that one of the values of Q+ sand Q− s is always zero for any δ≥0(Lee2011). Using Eq. (16), the objective function of the final robust optimization model is formulated as follows:  s∈ psξs+δ s∈ ps(Q+ s+Q− s)+γ s∈ psηs(20) Utilizing the abovementioned robust optimization method, the objective functions of our proposed problem are formulated as follows: MinZF = s psOBJ1s+λ s psOBJ1s− s p sOBJ1 s+2θs+ω s psδs (21) MinRF = s psOBJ2s+λ s psOBJ2s− s p sOBJ2 s+2θs(22) Several new constraints should be added to the model, as introduced in Eqs. (23)–(25), where λis the coefficient related to the importance (weight) of optimality robustness and ω reflects the infeasibility weight that a decision-maker sets experimentally. The first and the second terms in Eqs. (21)and(22) indicate the mean and variance of each objective function, respectively. OBJ1s− s psOBJ1s+θs≥0xm ∀s(23) OBJ2s− s psOBJ2s+θs≥0xm ∀s(24) θs≥0 (25) The last term in Eq. (21) measures the model robustness in terms of the infeasibility values of control constraints under each scenario. Constraints (23)to(25) are auxiliary constraints included in the optimization model for converting the nonlinear objective function to a linear one. 5 Solution approach As previously mentioned, the model presented in this paper defines two minimization objective functions with different orientations. Therefore, when seeking interactivity between the objective functions, we employ an IMCGP to solve the problem. The model presented in this section is a combination of two different methods explained in the following subsections. 123 Annals of Operations Research We also compare the results of our solution approach with other methods using a real case study and several related numerical examples to create expected levels for our goals. 5.1 Improved multi-choice goal programming (IMCGP) One of the most effective ways to increase the efficiency of an optimization model is to incorporate the experts’ opinions into the problem. To this end, the goal programming method describes the level of achievement for each of the objective functions according to the experts’ opinions. It is identified as an efficient, well-defined way to solve multi-objective models (Yadollahinia et al. 2018). We employ this method to solve our proposed multi-objective model. Recently, a new multi-choice goal programming (MCGP) approach was proposed by Jadidi et al. (2015). The superiority of this model in comparison with the other MCGP models is the recommended methodology in which the decision-makers can control their priorities more efficiently. The authors present a model that contains the revised goal programming approach (introduced by Chang (2008)) and the original goal programming along with a priority function, taking into account goal efficiency. They believe that sometimes the value of the target function will pass our expectation level, resulting in a penalty for the optimization model. The following equations introduce their model: Max  k (wa kak−wb kβk) s.t. fk(X)=αkfk,min +(1−αk)fk,max +βk(f− k−fk,max)∀k αk≤yk≤1+αk∀k βk+yk≤1∀k yk∈{0,1},0≤αk,β k≤1∀k(26) In Equation set (26), the proposed single-objective optimization model for the MCGP is developed in which kis the number of objective functions that should be converted to maximization functions. The range of [fk,min,fk,max]determines a boundary for the aspiration level ykspecified by the decision-makers ( fk,min ≤yk≤fk,max). αkis a continuous coefficient, valued between 0 and 1, and calculates the normalized distance between the k-th objective function and fk,max (αk=fk,max −fk(X) fk,max−fk,min ). f+ kand f− kindicate the k-th value of the objective function in the desired and undesired conditions, respectively. βkalso defines the normalized distance between the k-th value of the objective function ( fk(X))and fk,max in case fk,max is greater than fk(X)(βk=fk(X)−fk,max f− k−fk,max ). wa kand wb kare the weights specifying the importance of the k-th objective with respect to αkand βk. Jadidi et al. (2015) assume that fk,min =f+ kand divide the range of [f− k,f+ k]into two suboptimal regions of [fk,max,fk,min]and a less favorable boundary of [f− k,fk,max]. They note that one of αkand βkis zero in each of these boundaries. Figure 5illustrates the boundaries introduced by the authors. LDR and MDR stand for the less and more desirable ranges, respectively. The single-objective model introduced in Equation set (26) can be rewritten for our biobjective model as follows: MaxZ =wa 1α1+wa 2α2−wb 1β1−wa 2β2 123 Annals of Operations Research Fig. 5 Relationship between parameters in the proposed IMCGP approach (according to Jadidi et al. (2015)) S.t: Z1=α1f1,min +(1−α1)∗f1,max +β1∗(f− 1−f1,max) Z2=α2f2,min +(1−α2)∗f2,max +β2∗(f− 2−f2,max) α1≤y1≤1+α1 β1+y1≤1 α2≤y2≤1+α2 β2+y2≤1 y1,y2∈{0,1} α1,α 2,β 1,β 2≥0 (27) The values of f+ k,f− k,fk,min and fk,max can be obtained using the abovementioned method for our minimization functions: •To obtain the value of f+ kwe solve a sub-problem with the k-th objective function and all of the constraints to minimize the objective, and the achieved solution is equal to f+ k. •To estimate the value of f− k, a sub-problem is solved to maximize the k-th objective function and the solution is equal to f− k. •The value of fk,min is obtained using a manner similar to f+ k. •To obtain the value of fk,max, a sub-problem with the corresponding objective is solved to maximize the function. fk,max is less than or equal to the achieved solution (based on the decision-makers’ opinion). The pseudo code of the proposed algorithm is presented in ”Appendix B”. 5.2 Heuristic approach Since the computational time for solving an MILNP model increases drastically by adding the linearization binary variables, in this section, we consider a heuristic solution approach for our proposed problem given the circumstances of uncertain data. We introduce our new heuristic method in three steps as shown in Fig. 6. These steps are applied to the MINLP model as a heuristic MINLP (HMINLP) model for medium and large sample sizes. Table 3 introduces each optimization model and the corresponding solution methodology defined in this research. 123 Annals of Operations Research •Relax the constraints including binary variables in the MINLP model by jvt ≥ 0. The resultant model is called relaxed MINLP. •Solve the relaxed MILNP model optimally. •From the solution obtained by the relaxed MINLP, list all the nonzero values of binary variables jvt Step 1 Step 3 Step 2 •Set all non-zero values of jvt as 1 and add these as constraints to the original MINLP. •Solve the new model optimally. Fig. 6 Steps defined for the proposed heuristic solution approach Table 3 Definition of the proposed optimization models and the corresponding solution approaches Model Equations Methodology MINLP (1)–(12)IMCGP MILP (1)–(6), (8)–(15)IMCGP HMNLP (1)–(12) Heuristic method + IMCGP 6 Case study A study on a real-world operational case makes it possible to understand the applicability of our model. We selected a carbon black manufacturer in Iran where the environmental issues are of utmost importance to the government and decision-makers. In this section, initially, some explanations about the firm are presented to specify how the random parameters have been estimated for the relevant processes. We also utilized the opinion of the firm’s experts for a better estimation of the necessary parameters in our proposed model. Carbon black is an essential additive for producing rubber. Although the product is a useful raw material for many industries, it has negative implications for both human health and our climate. Therefore, the design of a holistic SC concerned with environmental issues is crucial for decision-makers. Iran Carbon Company1produces carbon black (industrial carbon black) used by rubber factories. At present, the capacity of carbon black production in this company is over 36 thousand tons of industrial soot. The company’s raw materials include furfural extract, cracked fuel oil (CFO), and fluid catalytic cracking (FCC), which are mainly purchased from Abadan Refinery, Amirkabir, Bandaremam, Shazand, Tabriz, and Jam petrochemical companies. Although the company produces only carbon black, it is able to meet customers’ requirements via several products (e.g. N220, N330, N339, N375, N550, N660) with its high-tech packing system. 1http://www.iran-carbon.com/en/. 123 Annals of Operations Research Table 4 Case study: estimation of the certain parameters introduced in the model Parameter Value Capijt—Capacity of supplier (ton) Uniform (16000,36000) σjv—Capacity of vehicle (ton) Uniform (150,350) Njvt—Total number of vehicle Uniform (15,25) τ(ton)—Amount of permitted carbon gas emissions Uniform (40,50) cp (MIRR/kg)—Price on carbon per unit Uniform (0.6,1.2) Fijvt(kg)—Amount of carbon gas emissions in executing a lot size Uniform (0.1,15) EO t(kg)—Amount of carbon gas emissions due to placing an order Uniform (0.08,0.12) EH t(kg)—Amount of carbon gas emissions due to holding a unit of product Uniform (0.06,0.09) ULit—Lead time upper tolerance Uniform (90,180) ELit—Lead time lower tolerance Uniform (20,80) Livt—Lead time of supplier Uniform (10,50) In this study, we consider three scenarios, namely pessimistic, most likely and optimistic ones. The highest occurrence probability is associated with the most likely scenario. Moreover, the values of the proposed model’s parameters are estimated according to the opinion of the company’s experts. As noted in Sect. 5.1, this study applies an improved MCGP method to solve the proposed model. Hence, the parameters related to this approach (e.g. weights of the objective functions, fk,min,and fk,max) are estimated based on the experts’ opinions as well. Other parameters relative to the proposed solution approach are collected from similar papers, such as Jadidi et al. (2015). Tables 4and 5show the values of certain and uncertain parameters used in our model, respectively. The monetary values are reported with MIRR (million Iranian Rials, the currency of Iran). In addition, the data is collected according to the opinions of the experts working in the carbon black manufacturer. Following the study of Mohammed et al. (2017), three types of vehicles, i.e. light truck, mid-sizetruck,andheavy-dutytruck,arecontemplated.Thevaluesofcarbonemissionfactors and transportation costs for each of these types are given in Table 6.Theδvts parameter was estimated by the carbon emissions caused by each vehicle type in kg per ton of carrying load of carbon black in every km distance. CT jvts parameter was estimated by the average cost of transportation from suppliers using each type of vehicle in MIRR per ton of carbon black carrying load in every km distance. To evaluate the proposed model, several numerical examples have been generated and employed in the sensitivity analysis tests. The model was coded in GAMS software using a computer system with a dual-core 1.40 GHz Pentium CPU and 3 GB of RAM. Numerical examplesarepresentedinfourdifferentproblem sizesintroducedinTable7,eachofwhichhas beenexaminedandtestedatfivedifferentlevelsofuncertainty(penaltycosts P={0.1−1.0}). We solve our proposed MINLP and MILP models, separately, with certain data (using the deterministic solution) and five uncertainty levels of data (using the robust solution), then compare them with our combined robust-heuristic solution, introduced in Sect. 5.2. We also investigated the value of the objective functions (Obj1 and Obj2) in both deterministic and robust cases according to the various penalty costs defined for the availability of data. As it can be seen in Table 10, every objective function value in the deterministic model is greater than the corresponding value of the robust approach for each of the five penalty costs of uncertainty. The values also increase once we have more available data (as 123 Annals of Operations Research Table 5 Case study: estimation of the uncertain parameters introduced in the model under the occurrence of different scenarios Parameter Pessimistic range Expected range Optimistic range Dits (ton)—Demand of products Uniform (2000,3000) Uniform (3000,4000) Uniform (4000,5000) CPijts (MIRR/ton)—Procurement cost of products Uniform (8,10) Uniform (4,8) Uniform (2,4) COijts (MIRR/ton)—Ordering cost of products Uniform (8,12) Uniform (6,8) Uniform (3,6) CHits (MIRR/ton)—Holding inventory cost of products Uniform (8,10) Uniform (5,8) Uniform (2,5) CSits (MIRR/ton)—Shortage cost of products Uniform (100,120) Uniform (80,100) Uniform (60,80) the penalty cost is greater). These trends can be found in any of the MINLP, MILP, and HMINLP models in each of the four problem sizes. Moreover, in each problem size and penalty cost, the target value of the MINLP model is lower than the corresponding one for the MILP and HMINLP models. The values of the objectives increase accordingly for the HMINLP model. Although we observe this slight increase in the values for the HMINLP model (which is not desirable for our minimization problem), the computational times of solving the problems, shown in Table 11, affirm the merit of the proposed heuristic approach, especially for solving the HMINLP model for the bigger problem sizes. Figures 7and 8 illustrate the comparison of the objective functions (reported thoroughly in Tables 10 and 11) in terms of various penalty costs. According to these reasonable trends of the values of objective functions in both deterministic and robust approaches, we can conclude that the models are stable and well-integrated. 6.1 Consideration of different carbon policies We now examine the first objective function of our model, introduced in Eq. (1), to explore the total cost of the SC with respect to several conditions of data availability or uncertainty. Aimed at comparing the carbon-tax policy, in which the carbon price is constant, with the cap-and-trade policy, we explore the effect of changing carbon capacity in each of these policies, separately. Figure 9illustrates the result of the proposed HMINLP model under the condition of certainty and a constant carbon price (cp =10). Figures 10 and 11 show the results of the total cost under uncertainty conditions and this constant price. We focus on investigating the effect of changes in carbon capacity on total costs and carbon emissions in the chain. Figure 9shows that by increasing carbon capacity, the total cost is reduced and carbon emissions increase, but by a specific level of carbon capacity (the point near 45.5 tons). Until this point, the carbon emission curve shows a linear increasing trend per carbon capacity, and consequently, a consistent decline in total cost; however, after this level, more carbon capacity will result in unvarying total cost and carbon emissions. This means that if the carbon capacity is set to this value as its maximum level (45.5 tons), both the minimum total cost will add the highest carbon emissions. Figure 9also determines a trade-off level (e.g. the break-even point 43.8) for the contrary criteria of decision-making—namely total cost, total carbon emission, and carbon capacity. Upon investigation of the model’s formulas, carbon 123 Annals of Operations Research Table 6 Case study: estimation of transportation-related parameters Parameter Type of vehicle Pessimistic range Expected range Optimistic range δvts—Carbon emissions factor caused by vehicles Light truck Uniform(0.048,0.055) Uniform(0.040,0.048) Uniform (0.035,0.040) Mid-size truck Uniform (0.0252,0.0272) Uniform (0.0245,0.0252) Uniform(0.0240,0.0245) Heavy duty truck Uniform (0.297,0.305) Uniform (0.290,0.297) Uniform (0.280,0.290) CT jvts—Transportation cost from suppliers Light truck Uniform(0.110,0.115) Uniform (0.105,0.110) Uniform (0.100,0.105) Mid-size truck Uniform (0.118,0.123) Uniform (0.113,0.118) Uniform (0.108,0.113) Heavy duty truck Uniform (0.125,0.130) Uniform (0.120,0.125) Uniform (0.115,0.120) 123 Annals of Operations Research Table 7 Case study: problem sizes defined for the numerical tests Problem number Problem dimension 1|I|×|J|×|V|×|T|×|S|=8×8×3×3×3 2|I|×|J|×|V|×|T|×|S|=10 ×10 ×5×6×3 3|I|×|J|×|V|×|T|×|S|=15 ×15 ×10 ×5×3 4|I|×|J|×|V|×|T|×|S|=20 ×10 ×15 ×6×3 0 10000000 20000000 30000000 40000000 50000000 60000000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 O OBJ1 Robust optimization penalty cost MINLP MILP HMINLP Fig. 7 Case study: comparing the optimal values of the first objective function for each model and penalty cost 0 1000000 2000000 3000000 4000000 5000000 6000000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 O OBJ2 Robust optimization penalty cost MINLP MILP HMINLP Fig. 8 Case study: comparing the optimal values of the second objective function for each model and penalty cost capture constraint (Eq. 7) is a fundamental limitation for the optimization model and hence directly manipulates the trends shown in Fig. 9. We also can perceive that by reducing carbon capacity from 47 to 42 tons, carbon emissions will decrease by 8%, resulting in an overall cost increase of 0.2%. This affirms that a slight increase in inventory and transportation costs would result in a greater decline in carbon emissions. This would be a beneficial strategy for companies that are under governmental and environmental pressure for their gas emissions pollution. 123 Annals of Operations Research 7.15 7.17 7.19 7.21 7.23 7.25 7.27 7.29 7.31 7.33 7.35 42 42.5 43 43.5 44 44.5 45 45.5 46 46.5 47 Carbon emission (ton) Total cost (MIRR) )ton(Carbon capacity Carbon emission Total cost Fig. 9 Case study: impact of carbon capacity on total cost and carbon emissions under certainty condition and carbon-tax policy It can be seen in Fig. 9that the values of total cost are changing in a limited range (between 7.16 and 7.30). This may occur because the source of uncertainty of all parameters has been assumed as a uniform normal distribution (see Tables 4to 6). It is possible considering different types of distributions for generating the value of the parameters that would lead to significantly different results. Under conditions of uncertainty, Figs. 10 and 11 illustrate similar behaviors as per certain conditions, shown in Fig. 9. By exploring these two charts, we see that while the levels of uncertainty increase (decreasing the penalty cost P), the amount of the total cost increase is negligible for the decision-makers. For instance, in the lowest carbon capacity of 42 tons, by increasing the level of uncertainty by 400% (from P=0.8toP=0.2) the total cost is only boosted by 5% (from 4.55 to 4.77 MIRR), which concludes that the robust strategy outperforms at higher uncertainty levels than at its definitive state. The same justification can be represented for the negligible decrease in carbon emissions, shown in Fig. 11. Figures 12 and 13 demonstrate the relationship between the carbon capacity and total cost in both the definite and non-deterministic (in penalty cost 0.5) conditions, respectively. Each line in Fig. 12 determines the total cost of the chain per different carbon price and carbon capacity. For a fixed carbon price per ton, cp, increasing carbon capacity allows the company to pay less carbon tax, and hence the overall cost decreases. Similar to Fig. 10, these trends indicate that carbon capacity has a reverse relationship with the total cost. As far as the effect of carbon prices on total cost is concerned, we see a specific point for carbon capacity (43 tons). With carbon capacities below this point, any increase in the carbon price leads to an increase in the total cost of the chain; with carbon capacities above this point, the trend is in contrast. This is a noteworthy point that decision-makers can compare with the governmental “cap” on emissions, and if the determined cap is greater (>43 ton), the company is a potential carbon trader. Consequently, the total cost will decrease more with the help of higher carbon 123 Annals of Operations Research An application of assorted vehicle types in our GSC model demonstrated an appropriate effect on both costs and emissions. In general, governments and related organizations should provide the infrastructure required to employ different modes of transport or vehicle types. Our results confirmed that the usage of multi-mode transportation leads to a substantial decrease in the environmental damage caused by the SC. Reduction of costs per vehicle and products handling times for the SC results in a decrease in customs controls for governments and even low rates of theft or damage to the cargo for the insurer companies. Generally, it is suggested that governments define some incentives for SCs to apply multi-mode transportation in their networks. The results of this study regarding the impact of the carbon emissions capacity parameter on both profitability and environmental impacts affirm that the bargaining power of both companies and governments is an important issue in determining the value of this parameter. Our case study was an example for one country; however, in countries with high levels of air pollution, the governments should consider lower values for the carbon emissions capacity because lower values of this parameter lead to a greater decrease in environmental damage caused by the SC. On the other hand, if countries require more production and supply for the products, increasing the value of the carbon cap parameter can be on the government’s agenda (NYT 2019). Despite an examination of our approach by a case study with three scenarios, there are various ways for designing scenarios that can be addressed and compared with this paper. For instance, Fattahi et al. (2017) apply a scenario tree to generate scenarios for stochastic parameters. In their method, in the outset, they consider 200 scenarios and then the scenarios are converted into a scenario tree by reducing the number of scenarios. Given that managers usually develop a limited number of strategies/scenarios to navigate the kinds of extreme events they have recently seen in the real cases, it is suggested to keep the number of scenarios as least as possible. This will also decrease the level of complexity in the understanding of the model’s behaviour in different scenarios. 8 Conclusion In this study, an environmentally-friendly GSC model was introduced to integrate the minimization of economic features with the minimization of environmental impacts of carbon emissions. A framework for planning sustainable logistics was presented in which various vehicle types and gas emissions in transportation, and other SC operations, were considered. Uncertainty was considered in demand and most of the costs related to the operations of the GSC, as well as for the carbon emissions factor caused by every vehicle. The proposed bi-objective multi-supplier, multi-product, multi-carrier, and multi-period model was solved by an improved algorithm for the multi-choice goal programming solution approach. A novel robust-heuristic optimization approach, HMINLP, was also developed to deal with demand and economic uncertainty, especially in larger problem sizes. A real case study of an SC company was introduced to implement the model and test the efficiency of the proposed solution methodology. The results of the numerical tests in the case study demonstrated that the robust-heuristic approach could efficiently mitigate demand and economic uncertainty, and the heuristic solution could decrease computational times substantially for large-scale problems, despite the slight change in the objectives. To achieve environmental sustainability, we compared the carbon-tax policy with the capand-trade policy as they pertain to changing carbon capacity and concluded that, under the 123 Annals of Operations Research carbon-tax policy, an increase in the levels of uncertainty would lead to a negligible increase in total cost. This confirmed that the robust strategy outperformed at higher uncertainty levels than the definitive state. An examination of the cap-and-trade policy demonstrated a lower total cost compared to the other policy and affirmed that decision-makers should select the cap-and-trade policy if both policies are available. The tests on the consideration of various vehicle types confirmed the importance of this assumption in designing the GSC and SSC, regardless of uncertainty or the selection of another solution approach. As for the limitations of our study, we assumed a fixed carbon capacity and price during the time horizon for avoiding further complexity in the model; however, the government can lower the carbon capacity each year to encourage companies through incentives to invest in clean technologies. In such situations, then, the model can be developed with consideration to various carbon capacities and prices for the defined periods. As we pointed out in the Introduction Section, improving service levels beyond reducing operating costs would be a competitive strategy for the GSC and SSC. In our study, we only focused on the costs associated with the SC, however, the trade-off between supply and demand—namely service levels of the SC—in all customer zones could be maximized in one of the objective functions beyond other cost minimization. Several other resilience factors, e.g. capacity disruption, can be included in the problem to configure a more resilient GSC (Ivanov 2018). Incorporating features of responsiveness for an SC, e.g. minimizing the lead-time or total transportation time, would be another direction for future research. In case of the solution approach, a suggestion for future research is the use of meta-heuristic methods and the development of new approaches to refine non-linear, mixed-binary models. Finally, an integration of sustainability with digital supply chains and Industry 4.0 systems can be considered a crucial future research avenue (Mrugalska and Stasiuk-Piekarska 2020; Dolgui et al. 2020;Ivanovetal. 2020). Acknowledgements We thank the guest editor, Dr. Beata Mrugalska, and four anonymous reviewers for their constructive comments that helped to improve this paper immensely. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: Detailed results of case study 123 Annals of Operations Research Table 10 Case study: comparison of the MINLP, MILP and HMINLP models solved by the proposed deterministic, robust, and robust-heuristic solution approaches Problem size Penalty level MINLP MILP HMINLP Deterministic Robust Deterministic Robust Deterministic Robust Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 1 0.1 36103478 3023012 14599510 1618281 36103478 3023012 13937587 1611676 37137462 3194631 14766189 1683264 0.2 15531394 1721575 14827221 1714549 15708711 1790706 0.3 16522759 1831463 15773639 1823988 16711395 1905007 0.4 17577404 1948365 16780467 1940413 17778080 2026603 0.5 18699365 2072729 17851561 2064269 18912851 2155961 0.6 19892942 2205031 18991022 2196031 20120054 2293575 0.7 25654315 2400143 25524005 2389987 26084697 2469831 0.8 29764832 2635412 28983411 2604178 29891011 2691254 0.9 31258951 2764510 30209431 2712105 31705346 2800364 1.0 35024179 2931242 33877516 2901077 35261302 2966114 2 0.1 47632154 4147804 28546021 2431093 47632256 4274308 27855123 2402877 49302148 4491247 31091991 2714126 0.2 30368107 2586269 29633110 2556253 33076586 2887368 0.3 32306497 2751350 31524585 2719418 35187858 3071669 0.4 34368614 2926968 33536792 2892997 37433891 3267733 0.5 36562355 3113796 35677439 3077657 39823288 3476311 0.6 38896123 3312549 37954722 3274103 42365201 3698204 0.7 41975664 3595412 40874365 3478920 45004122 3795102 0.8 43147618 3717962 42951753 3699408 46357880 3977014 0.9 44953620 3865140 43865410 3800225 48034427 4166075 1.0 46145563 4031045 45953170 3992742 50115232 4385044 123 Annals of Operations Research Table 10 continued Problem size Penalty level MINLP MILP HMINLP Deterministic Robust Deterministic Robust Deterministic Robust Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 Obj1 Obj2 3 0.1 651476523 54132410 422958586 32784396 652476523 54862150 419229464 32276075 669547025 56325407 435602289 33515288 0.2 449955942 34877017 445988791 34336250 463406690 35654561 0.3 478676535 37103210 474456161 36527926 492985841 37930385 0.4 509230356 39471500 504740597 38859495 524453022 40351473 0.5 541734421 41990958 536958082 41339889 557928747 42927099 0.6 576313214 44671232 571232002 43978605 593541220 45667127 0.7 589750123 45631422 581342010 45101022 608835401 47341074 0.8 601478659 48562780 599566012 48003301 616745545 48976134 0.9 623647850 49257412 611240201 49011232 637754118 50764253 1.0 631459857 51398652 629945260 50007126 656982575 52976354 4 0.1 968421540 60668501 698277224 42206173 968864253 60785266 693118337 41306983 980035428 61987301 709318886 42218876 0.2 727372109 43964764 721998267 43028107 738873840 43977996 0.3 757679280 45796629 752081529 44820945 769660250 45810413 0.4 789249250 47704822 783418259 46688485 801729427 47719180 0.5 822134635 49692523 816060686 48633838 835134820 49707479 0.6 856390245 51763045 850063215 50660248 869932104 51778624 0.7 879861425 55951310 871596742 55324054 880214557 56958357 0.8 917485035 57325502 909962500 57007432 917358665 59663002 0.9 943654201 58631782 938854041 58004732 946652476 60354168 1.0 952314006 60032115 950102543 59486200 976843241 61432508 Obj1 is total cost (in IRR) and Obj2 is total carbon emissions (in gram) 123 Annals of Operations Research Table 11 Case study: comparison of the CPU computational times (in seconds) for solving the MINLP, MILP, and HMINLP models by the proposed deterministic, robust, and robust-heuristic solution approaches Problem size Penalty level MINLP MILP HMINLP Deterministic Robust Deterministic Robust Deterministic Robust 1 0.1 14511 8175 2200 0.2 14542 8172 2198 0.3 14517 8161 2190 0.4 14516 8170 2190 0.5 14500 14582 8055 8175 2160 2193 0.6 14585 8155 2191 0.7 14509 8160 2190 0.8 14519 8175 2200 0.9 14515 8169 2195 1 14507 8155 2190 2 0.1 43509 29786 7997 0.2 43612 29792 8003 0.3 43568 29798 8007 0.4 43989 29795 7995 0.5 43965 43392 29657 29789 7953 8003 0.6 43995 29785 7990 0.7 43999 29796 8011 0.8 43104 29771 8019 0.9 43997 29801 8016 1 43999 29793 8022 123 Annals of Operations Research Table 11 continued Problem size Penalty level MINLP MILP HMINLP Deterministic Robust Deterministic Robust Deterministic Robust 3 0.1 65647 45319 12170 0.2 65667 45313 12176 0.3 65649 45331 12166 0.4 65662 45318 12168 0.5 64986 65651 45304 45321 12149 12164 0.6 65652 45315 12155 0.7 65661 45332 12153 0.8 65649 45310 12164 0.9 65668 45336 12179 1 65643 45318 12168 4 0.1 116562 67366 18097 0.2 116574 67353 18137 0.3 116573 67357 18145 0.4 116572 67350 18098 0.5 116553 116575 67365 67369 18065 18101 0.6 116566 67359 18078 0.7 116575 67366 18089 0.8 116561 67335 18164 0.9 116569 67369 18178 1 116559 67350 18097 123 Annals of Operations Research Appendix B: Pseudo code of the proposed approach Algorithm 1: The improved multi-choice goal programming algorithm Input the parameters of the model Cope with uncertainty applying the robust optimization method while All the binary variables (Uijvt) get value of zero or one do 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