An accelerated Benders decomposition algorithm for

Transportation Research Part E 67 (2014) 14–38

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

An accelerated Benders decomposition algorithm for sustainable supply chain network design under uncertainty: A case study of medical needle and syringe supply chain M.S. Pishvaee a,⇑, J. Razmi b, S.A. Torabi b a b

School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 21 September 2013 Received in revised form 24 January 2014 Accepted 4 April 2014

Keywords: Supply chain management Sustainability Benders decomposition Possibilistic programming Social responsibility Life cycle assessment

a b s t r a c t This paper proposes a multi-objective possibilistic programming model to design a sustainable medical supply chain network under uncertainty considering conflicting economic, environmental and social objectives. Effective social and environmental life cycle assessment-based methods are incorporated in the model to estimate the relevant environmental and social impacts. An accelerated Benders decomposition algorithm utilizing three efficient acceleration mechanisms is devised to cope with computational complexity of solving the proposed model. Computational analysis is also provided by using a medical industrial case study to present the significance of the proposed model as well as the efficiency of the accelerated Benders decomposition algorithm. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The concern about environmental and social impacts of business activities has led to development of a new paradigm called ‘‘sustainable development’’. As early as 1987, the World Commission on Environment and Development (WCED) defined the sustainable development as ‘‘Development that meets the needs of the present generation without compromising the ability of future generations to meet their own needs’’ (WCED, 1987). Fifteen years later, at World summit on Sustainable Development 2002, it has been said that sustainable development is not only balancing the economic benefits with environment protection, but also depends on another ‘‘pillar’’ named social development (White and Lee, 2009). Despite its simple definition, sustainable development is a complex concept that the implementation of its principles, both at macro and micro-levels, encounters significant difficulties. To deploy sustainability into business environment, it is not sufficient to only control the level of sustainability within the boundary of a corporation’s ownership, rather the level of sustainability should be assured at the whole supply chain network (SCN) (Cruz, 2009). Despite the importance of supply chain sustainability, the relevant literature suffers from lack of decision making and planning tools and techniques. Carter and Rogers (2008) define the sustainable supply chain management (SSCM) as the strategic and transparent integration of organizational goals at economical (Eco), environmental (Env) and social (Soc) aspects and the achievement of them through a systemic inter-organizational coordination to improve the overall long-term performance of the whole supply chain (SC). Additionally, they provide a framework for SSCM that emphasizes on integration of strategic decisions with the concept ⇑ Corresponding author. Tel.: +98 21 73225000; fax: +98 21 73225098. E-mail addresses: [email protected] (M.S. Pishvaee), [email protected] (J. Razmi), [email protected] (S.A. Torabi). http://dx.doi.org/10.1016/j.tre.2014.04.001 1366-5545/Ó 2014 Elsevier Ltd. All rights reserved.

M.S. Pishvaee et al. / Transportation Research Part E 67 (2014) 14–38

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of sustainability and also managing the uncertainty through the SC to assure the planned level for sustainability under dynamic conditions in practice. Supply chain network design (SCND), as the most important strategic level decision in the SCM field, plays a significant role in the overall sustainability of a SC. In general, SCND decisions include determining the number, location, capacity and the technology of the required network facilities and the aggregate quantities of flow between them to accurately meet the demand side needs (Simchi-Levi et al., 2004). To respond the need of sustainability, a number of research works have ever been presented in the context of SCND problem. Nevertheless, the literature on sustainable SCND that covers all the three aspects of sustainability (i.e., Eco, Env and Soc aspects) is very scarce. Recently, Seuring (2013) presents a comprehensive review on modeling approaches for sustainable supply chain management problems to support a number of future research directions. In this paper, we survey specific strategic network design problems for sustainable supply chains. The relevant literature can be classified into the two major groups: green SCND and socially responsible or sustainable SCND (SSCND). It should be noted that these two groups are not fully separated from each other and have some common areas and similarities. For more information, interested readers can consult with Ahi and Searcy (2013) which provides a comparative literature analysis on green and sustainable supply chain management. Env or green supply chain management (GSCM) addresses the incorporation of Env aspect into the SCM to account for environmental factors in every decision making process across SCN (Srivastara, 2007). Notably, GSCM does not cover the Soc dimension of sustainability. Reverse SC (logistics) network design, as the more traditional part of GSCM, has a larger literature than the other parts. Reverse logistics includes all the activities and issues related to recovery, recycling or safe disposal of the used products (Fleischmann et al., 1997). The proper establishment of the reverse SCN may help firms to reduce the negative Env impact of end-of-life (EOL) products and to gain more Eco benefits by recapturing the value of used products and enhancing their green image in the market. The literature of reverse SCND includes a variety of network design models from simple single product facility location models (e.g. Marín and Pelegrin, 1998) to complex multi-period (e.g. Mansour and Zarei, 2007) and multi-objective (MO) network design models (e.g. Du and Evans, 2008; Fonseca et al., 2010). However, to avoid the sub-optimality resulting from the separated design of the forward and reverse SCNs, a number of researches have focused on developing models for integrated forward/reverse SCND in the recent years (e.g. Lee and Dong, 2008; Pishvaee et al., 2010). The dynamic and complex nature of a SC results in a high degree of uncertainty that influences the effectiveness of SC planning decisions, and the strategic level decisions in particular (Klibi et al., 2010; Peidro et al., 2010). Since the quantity and quality of returned products taint with higher degree of uncertainty compared to new products, uncertainty is significantly more important factor in reverse SC planning problems (Fleischmann et al., 2001). To cope with this issue in a reverse SCND problem, several authors have proposed a number of stochastic programming (SP) models (e.g. Salema et al., 2007; El-Sayed et al., 2010). However, there are several major drawbacks in using SP approaches, especially for strategic level decisions, such as high complexity and unavailability of sufficient historical data. To avoid this, a number of authors have applied possibilistic programming (e.g. Qin and Ji, 2010; Pishvaee and Torabi, 2010) and robust optimization (e.g. Pishvaee et al., 2011) models for reverse and closed-loop SCND problems under epistemic uncertainty (i.e., lack of knowledge about the precise values of input data). Quariguasi Frota Neto et al. (2008) addressed Env impacts in a European pulp and paper logistics network design problem. However, the proposed model is only able to optimize the quantity of flow between facilities in a simple logistics network and ignores the other SCND decisions such as the location and number of required facilities at different echelons. A somehow similar work is presented by Quariguasi Frota Neto et al. (2009) for ELV electrical and electronic equipment recycling network. Due to importance of controlling emissions (especially carbon emission) across SCs, a number of authors (e.g. Wang et al., 2011; Pishvaee et al., 2012a) employed CO2 emission index to measure and model Env impact in SCND problem. Chaabane et al. (2012) also used carbon dioxide equivalent index to model the emissions across supply chain in a SCND problem as well as analyzing the effects of emission trading scheme. By the aid of more comprehensive Env impact assessment methods, i.e., life cycle assessment (LCA)-based methods, Hugo and Pistikopoulos (2005) and Pishvaee and Razmi (2012) developed bi-objective optimization models to consider Env burden in forward and forward/reverse SCND problems, respectively. Stochastic (e.g. Guillén-Gosálbez and Grossmann, 2010) and fuzzy programming (e.g. Pishvaee and Razmi, 2012) models have also used in the recent literature to deal with uncertainty in Env SCND problem. Apart from the importance of Env impact of SCNs, the level of social responsibility (SR) should be also considered according to the triple lines of sustainability. Most of the operations research-based works in the context of SCND and also other areas such as Env management and SCM did not focus on social dimension (White and Lee, 2009). Therefore, the body of literature is very thin in this part. Dehghanian and Mansour (2009) maximized SR besides minimizing the total costs and Env impact in a tire recycling network by developing a MO mixed integer programming (MIP) model. Cruz (2008, 2009) provided frameworks for modeling and analyzing socially responsible SCN and the behavior of manufacturers, retailers and customers within the SCN. Last but not the least, Pishvaee et al. (2012b) proposed various robust possibilistic programming models to formulate a socially responsible SCND problem under epistemic uncertainty. It should be noted that in the past, sustainability only would address the Eco and Env issues while SR refers only to Soc aspects (e.g. human rights). However, during the last decade these two notions have grown into convergence and now many scientists believe that SR and sustainability are synonyms (Ciliberti et al., 2008). But still sustainability pays more attention to Env dimension compared to SR that considers Env concerns as a subset of social issues. To fill the gap of literature on SSCND, this paper proposes a comprehensive and practical, but tractable, multi-objective possibilistic programming (MOPP) model based on the credibility measure for a real-life SSCND problem under supply,

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process and demand uncertainties. As a theoretical contribution in the area of fuzzy/possibilistic programming, this paper describes how to deal with equality chance constraints based on credibility measure. Two LCA-based impact assessment methods are integrated in the proposed model to formulate and measure the Env and Soc impacts of different SCN configurations. To the best of our knowledge, this is the first time that the social life cycle assessment method is successfully applied in the area of SCM and extended in some technical aspects to formulate and calculate the social impacts of supply chain planning related decisions. Considering the effects of imprecise parameters in Soc and Env performances tainted with epistemic uncertainty is the other factor that differentiates this paper form the ones existed in the literature. Additionally, the developed model is able to integrate the design of reverse and forward SCNs to support the ‘‘cradle-to-grave’’ perspective and to gain savings through sharing the resources and infrastructures. The proposed model is used for designing a real medical needle and syringe supply chain as a case study which can prove the practical value of this research. Notably, medical needle and syringe is known as a strategic medical device in national health systems. To solve the developed model, this paper proposes a powerful Benders decomposition algorithm benefiting from several acceleration methods to significantly speed up the convergence of the algorithm. During the development of the accelerated Benders decomposition algorithm, two propositions and one corollary are also analytically proven which can be valuable from theoretical point of view. The rest of this paper is organized as follows. In the next section the concerned problem is defined based on a real industrial case. The proposed MOPP model and the method used to cope with the imprecise parameters as well as the methods used to assess the Env and Soc impacts are described in Section 3. The proposed solution method including an accelerated Benders decomposition algorithm is elaborated in Section 4. The usefulness of the proposed possibilistic programming model and the performance of the developed solution method are investigated through Section 5 by the aid of the data extracted from the studied case. Finally, Section 6 concludes this paper and offers some directions for future research.

2. Problem definition The SSCND problem described in this section is formed based on a real medical device industrial case. The case study is an Iranian single-use medical needle and syringe (SMNS) manufacturer named AVAPezeshk (AVAP) (www.avapezeshk.com). SMNS is a strategic medical device in the health system according to its important role in vaccination and treatment process. As the World Health Organization (WHO) reported, approximately 16 billion injections are administered around the world annually (WHO, 2005). The EOL SMNS is a potentially infectious waste and has significant Env and Soc impacts. Annually, 8–16 million hepatitis B, 2.3–4.7 million hepatitis C and 80,000–160,000 human immunodeficiency virus (HIV) infections are estimated to occur from reused unsterilized needles and syringes including the ones resulted from needle sticks (i.e., accidental piercing by a needle) (WHO, 2005; Hanson and Hitchcock, 2009). Therefore, it can be concluded that the EOL management of SMNS is very critical. According to this importance, WHO provides a policy paper for management of infectious wastes (e.g. needles and syringes) to reduce burden of disease resulted from infectious wastes (WHO, 2004). In this policy paper WHO committed to support countries in development and implementation of national plans, policies and legislation for health-care waste management. Now, AVAP has the highest market share in Iran (about 70%) and also a production plant with about 600 million production capacity per year to satisfy the customers’ demand. In the recent years, AVAP received more significant orders from geographically dispersed customers including considerable orders from the three neighbor countries. Therefore, the current SCN cannot satisfy the customers’ demand completely. Additionally, in 2009 annual strategy review, AVAP strategic committee decided to make a fair balance between all the stakeholders’ expectations and not only satisfy the shareholders’ gain. To this aim the strategic objectives were set in the way that could cover three major aspects: (1) profit, (2) people and (3) planet. These three aspects are very similar to – even the same as – sustainability aspects. As a result, a number of objectives are inserted in AVAP strategic map to reduce the Env impact and increase the SR of the company. These strategic objectives should also be deployed at the whole SC. Therefore, the AVAP SCN should be redesigned to satisfy new demands while accounting for Env and Soc issues alongside the Eco objectives. As it was mentioned before, SMNS cannot be discarded as usual garbage because of possible infections. Thus, managing the EOL SMNS that results in design and establishment of reverse SC should be considered in AVAP SCN redesign problem. Nevertheless, the Env impact of other phases of the SMNS life cycle should not be neglected in the concerned problem. To cope with EOL SMNS, three major options are available: (1) incineration methods (e.g. rotary kiln and cement incinerator), which is one of the mostly used options because of its low cost and capability of energy recovery, but at the same time this method has significant Env burden such as air pollution (Hanson and Hitchcock, 2009), (2) non-incineration methods (e.g. steam autoclave with sanitary landfill and microwave disinfection), which is also capable of energy recovery with various efficiencies (Zhao et al., 2009), and (3) recycling, that generally known as a forbidden option for medical waste management according to possible infection of medical wastes (see Zhao et al., 2009), but recently the usefulness of this method has been proved by a research in India under supervision of WHO (2005). According to the context of the studied case, three EOL options including (1) incineration at cement incinerators, (2) safe landfill and (3) recycling at plastic and steel recycling centers are available to handle the used SMNS. SMNS is a widely used product in health system and the quality and safety degree of this product can influence human health considerably. Regarding the Soc impacts, the configuration of AVAP SC can cause significant effects on different Soc stakeholders such as consumers and local society. Therefore, to achieve sustainability, the Soc impact of different SCN configuration options should also be taken into account. Notably, this paper is the first one in the literature that considers the selection


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Fig. 1. The underlying structure of the concerned supply chain network.

of best EOL options for SMNS according to sustainability performance (i.e., Eco, Env and Soc performances) of the whole forward-reverse supply chain. The structure of the studied SCN is illustrated in Fig. 1. This SC includes both forward and reverse networks. Through the forward network, new products are manufactured by production centers (PCs) and shipped to customers to fully satisfy their demands (i.e., shortage is not allowed). In the reverse chain, the returned EOL products are first fully collected in collection/ disassembly centers (CCs) and secondly shipped to incineration, landfill or recycling centers. The amount of returned products is determined as a predefined percent of demand of each customer zone. Without loss of generality, a single product is moved through the forward and reverse networks by ‘‘pull’’ and ‘‘push’’ mechanisms, respectively. Since the production technology (PT) and capacity level of facilities (CF) have significant effect on Soc, Env and Eco performances, the determination of PT and CF are also considered as output decisions in the studied problem. The selection of PT and CF according to their effects on the level of sustainability performance of the concerned SC is one of eminence properties of the discussed problem. The main problem to be addressed by this research is to (re)design the AVAP SCN, i.e. determining the number, location and capacity of PCs and CCs, selection of the best production technologies for PCs and the best EOL options as well as the material flow quantities between network facilities, with the aim of forming a reasonable balance between the three dimensions of sustainability. Notably, establishing a PC and CC in a joint location may lead to cost saving because of sharing the required resources and utilizing common infrastructures (see Pishvaee et al., 2010). This issue also affects the Env and Soc performances of the SC. It is worth noting that although, the network studied and modeled in this research is formed based on a SMNS SC, however, the proposed model has a somehow general form and it can be used to (re)design of SCN of other types of products such as medical devices, particularly the category of sharps and single-use devices (see WHO, 2004; Hanson and Hitchcock, 2009), or more different products such as paper (e.g. Pati et al., 2008) with some little modifications. Lack of knowledge about the real value of input parameters mostly originate from dynamic nature of SC, strategic horizon of SCND problem and unavailability or incompleteness of required data leading to a high degree of uncertainty in such a problem. For example, in the concerned problem there is no historical data on the quantity of return products and only partial data is available on customers’ demand. Even though the historical data is available, the behavior of parameters may not necessarily comply with their historical pattern in the future according to dynamic nature and strategic horizon of SCND problem (see Pishvaee et al., 2011). For instance, some parameters such as transportation costs have dynamic and uncertain values over the problem horizon according to fluctuations in oil prices and economic crises. Notably, the degree of uncertainty increases in the SCND problem when reverse SC as well as Env and Soc issues are included in this problem (see Fleischmann et al., 2001; Erol et al., 2011; Pishvaee et al., 2012b). Noteworthy, using of probability distributions is impossible in the above-mentioned situations according to lack of required historical data and impreciseness of available data. Therefore, as an appropriate alternative, possibility distributions (Zadeh, 1978) are applied in this research to formulate the imprecise parameters by relying on both available objective (but not sufficient) data and experts subjective opinions. Particularly, trapezoidal possibility distributions, as a more general form compared to triangular form, are adopted to illustrate the uncertain parameters. 3. Model formulation Before the presentation of the mathematical model and to provide concise picture for the proposed model, the verbal description of the model is illustrated at below.

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Minimization of Total Cost ¼ Fixed opening costs þ Variable transportation and processing costs Sav ings from integrating facilities Minimization of Env Impact ¼ Damage to human health þ Damage to ecosystem þ Damage to resources Maximization of SR ¼ Created job opportunities Consumer risk Damage to workers health þ Value of local dev elopment Subject to: – – – – –

Satisfaction of customers’ demand and return. Flow balance at network facilities. Meeting capacity constraints. Logical constraints related to the different capacity levels and production technology types. Non-negativity and binary constraints on decision variables.

As it can be seen from the verbal description, the model includes three objectives covering the three aspects of sustainability. The cost objective (the first objective) can be simply modeled and calculated according to numerous works and methods existed in the literature. However, the estimation and modeling of Env and Soc impacts that form the second and third objective functions (OFs) is a hard and complex task with respect to scarce literature and available experiences in the context of SCM. Therefore, the methods applied to model these OFs are first described and explained in Sections 3.1 and 3.2 before elaborating the mathematical model.

3.1. Environmental impact assessment (EIA) To achieve a sustainable SCN, we need to have methods and tools to measure the environmental impact (EI) of different SCND decisions. Each product has different EIs in its life cycle stages. Focusing on the whole life cycle of a product, as a systemic approach that supports the cradle-to-grave perspective, provides an appropriate framework to discover opportunities for improving the efficiency and effectiveness of the concerned system. Now, LCA is the most credible methodology used to quantify and assess the EI of a product. The importance of this methodology made the International Standard Organization (ISO) to develop ISO 14000 series on LCA that its framework is now widely accepted by researchers and practitioners (Rebitzera et al., 2004). Despite the advantages of LCA, the direct use of this methodology requires a complex, costly and time consuming process that its results cannot be directly used and needs to be weighted and interpreted (Chiu et al., 2008; Goedkoop and Spriensma, 2001). To escape from the aforementioned complexities, a number of methods have been developed as standardized and simplified versions of the LCA. By the aid of these methods amateurs can also assess the EI of a product in a reasonable time with a little knowledge on Env issues. To select an appropriate method for EI assessment (EIA) in the concerned problem, the most credible EIA methods are studied and analyzed in this research. The list of these methods and some of their characteristics are illustrated in Table 1. These methods are formed based on LCA methodology and most of them classify and standardize EIs in mid-point or/and end-point impact categories. Some of the methods also provide normalization and weighting mechanisms to quantify the results and present the final result as a number. Among the studied EIA methods, ReCiPe 2008 is selected to estimate the EI of SCND decisions because of the following advantages: (1) the method is able to assess EI based on both mid-point and end-point impacts; (2) as the method is one of the most recently developed ones, it is benefited from the latest advances in the area of Env sciences; (3) ReCiPe is one of the most comprehensive EIA methods which appropriately covers most of the

Table 1 Characteristics of credible EIA methods. EIA methods

CML2001 (Guinèe et al., 2001) Eco-indicator 99 (Goedkoop and Spriensma, 2001) EDIP 2003 (Hauschild and Potting, 2005) EPS 2000 (Steen, 1999) IMPACT 2002+ (Jolliet et al., 2003) Ecological Scarcity (Brand et al., 1998) TRACI (Bare et al., 2003) ReCiPe 2008 (Goedkoop et al., 2009) a

Covering midpoint impact categories p p p p p p p pa

Providing weighting method

Requiring goal setting

Providing normalization method p p p

p p

p

p p p

p p

p

p

p⁄

p

p

Covering end-point impact categories p

The method is able to assess EI based on both mid-point and end-point impacts.


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possible mid-point and end-point impacts; (4) since ReCiPe is formed based on CML and Eco-indicator 99, it contains the advantages of both methods; and (5) despite the methods such as Ecological Scarcity it does not require goal setting. ReCiPe is a LCA-based damage oriented method that firstly calculates the amount of life cycle inventories (i.e. the materials and processes of each life cycle stage) and their EI in 18 mid-point impact categories and then aggregates the results in three end-point impact categories including (1) human health, (2) ecosystem diversity and (3) resource availability. Finally, by the aid of a weighting method, the final result is presented in the form of a single number. Originally, ReCiPe provides three different weighting methods according to different cultural perspectives; however, usually the ‘‘average’’ version is used as the more moderate method to aggregate the results. In the ‘‘average’’ version, human health, ecosystem diversity and resource availability contribute 40%, 40% and 20% in the final score, respectively. For more information about the ReCiPe method the readers can consult with Goedkoop et al. (2009). Despite the complex technical details of ReCiPe, the application of this method is somehow simple for the end users. To apply the ReCiPe method, users can apply the following four steps. At the first step, the scope and mission of the concerned system and the purpose of using ReCiPe should be defined. The scope of the studied system is illustrated in Fig. 1 and the mission of this system is to satisfy the customers’ demand by producing and distributing SMNS packages and handling the EOL products through the reverse network in an effective and efficient way. ReCiPe is used in this system to measure the EI of different SCN configurations. Secondly, the life cycle stages should be defined and at the third step, the amount of inventories in each stage must be determined. The life cycle and the corresponding inventories for the concerned SMNS supply chain have been depicted in Fig. 2. At the fourth step, the final score is calculated via multiplying the amounts of inventories by the corresponding Env indicator values and adding up the subsidiary results. Indicator values are accessible from standard databases provided based on ReCiPe method. Here, we use the database of ECO-it software (http:// www.pre.nl/eco-it) to calculate the scores. ECO-it is developed to support the implementation of ReCiPe and all the above-mentioned steps can be implemented in the context of this software. For example, consider the environmental impact (EI) of steel recycling (per each used needle and syringe) reported in Table E5 at the Electronic Supplementary Material. This EI is calculated as follows.

EI of recycling the steel part of one used product ¼ imprecise amount of recycled steel ðfuzzy numberÞ corresponding indicator ¼ ð0:065; 0:07; 0:075; 0:08Þ ½kg ð70Þ ½millipoints per kg ¼ ð0:00455; 0:0049; 0:00525; 0:0056Þ ½millipoints

3.2. Social impact assessment (SIA) Social issues form the third pillar of sustainability. Measuring and controlling the social impact (SI) is a multi-stakeholder and multidisciplinary issue because of extensive scope and complex nature of SIs (Pishvaee et al., 2012a). Accordingly, it would be impossible to measure all SIs of an activity or process. Recently, ISO developed the ‘‘International Guidance Standard on Social Responsibility-ISO 26000’’ (ISO, 2010) to provide a comprehensive framework for SR. This standard classifies Soc issues into seven core subjects including (1) organizational governance, (2) human rights, (3) labor practices, (4) the environment, (5) fair operating practices, (6) consumer issues, and (7) community involvement and development. As it was mentioned in Section 1, SR also includes Env issues (see core subject #4), however with respect to high importance of environment in sustainability paradigm, Env issues are considered as a separate part from the Soc issues in this study. Notably, ISO 26000 emphasizes that national and autochthonous considerations may affect the measurement indices, goal setting and the scope of the seven core subjects. A number of methods and guidelines have been developed by researchers and practitioners to support and simplify the measurement and implementation of SR. Here, to select an appropriate method for assessing the SI in the studied problem, the most relevant, popular and credible methods and guidelines are analyzed and investigated based on ISO 26000 core subjects (see Table 2). Among the studied methods, we select the ‘‘Guidelines for Social Life Cycle Assessment of Products’’ (GSLCAP) (Benoıˆt and Mazijn, 2009) as a basis to assess the SIs in the concerned problem. GSLCAP has the following advantages compared to other investigated methods: (1) GSLCAP is a product oriented (vs. organization oriented) SIA method that has formed based on LCA and therefore, it complies appropriately with the logic of SC and the applied EIA method (i.e., ReCiPe) and reduces the difficulty of model design and formulation; (2) this method covers the Soc issues appropriately and also does not address the Env and organizational subjects (see Table 2), thus, it is more compatible with sustainability paradigm and Soc concerns through SC; and (3) as it is one of the most recently developed frameworks, it is benefited from the latest advances in the area of SIA. GSLCAP is the only product oriented SIA framework and mainly relies on the concept of Soc LCA (see Benoıˆt et al., 2010) and stakeholder theory. SLCA is the complement of Env LCA (ELCA) and its process is also somehow similar to ELCA. However, despite of ELCA, in SLCA autochthonous considerations and subjective data play important roles. The architecture of GSLCAP is designed based on stakeholder categories and (the related) impact categories. Since GSLCAP is a recently developed method, it still needs more developments in different parts, especially to provide normalization and weighting mechanisms. Here, we have used GSLCAP to assess the SI in the studied problem with some little modifications and extensions. To

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Fig. 2. The life cycle stages and corresponding inventories.

apply GSLCAP, firstly, the scope and purpose of SIA should be defined. Here, the purpose is the measurement of SI of different SCN configurations and the scope is the same as the scope defined in Section 3.1, while implementing ReCiPe. Secondly, the life cycle stages and the corresponding problem decisions (i.e., the concerned SCND problem decisions) should be determine. This issue is illustrated in Table 3. Thirdly, the relevant stakeholder categories and social/socio-economic subcategories and the corresponding impact mechanisms, that represent the impact of problem decisions in each subcategory, should be defined. At the fourth step, some indicators must be determined to measure the impact of problem decisions on subcategories. GSLCAP introduce five stakeholder categories including workers (employees), local community, society, consumers and value chain actors. A number of social/socio-economic subcategories are assigned to each stakeholder category. Totally, GSLCAP proposes 31 subcategories. Due to space limitation, here, we only mention and study those subcategories affected by the

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M.S. Pishvaee et al. / Transportation Research Part E 67 (2014) 14–38 Table 2 Comparison of SIA methods and guidelines. SIA methods and guidelines

Organizational governance

SA8000 (SAI, 2008) GRI (GRI, 2011) ETI (ETI, 2009) FLA (FLA, 2011) GC (UNGC, 2007) GSLCAP (Benoıˆt and Mazijn, 2009)

p p

Human rights p pp p p pp p

Labor practices pp pp pp pp pp pp

The environment pp

pp

Fair operating practices p pp

p pp

Consumer issues pp

Community involvement and development p pp

pp

pp

p pp [ ] Partial coverage. [ ] Full coverage.

Table 3 Life cycle stages and the corresponding decisions. Life cycle stages

Production stage Distribution stage Collection stage EOL stage

Problem decisions Material flow between facilities

Processing technology at facilities

Capacity of facilities

Locations of facilities

Amount of production

Production technology

Capacity level of production centers

Location of production centers

Capacity level of collection centers

Location of collection centers

Amount of delivered products to customers Amount of Collected EOL products at collection centers Amount of EOL products processed by each EOL option

concerned network design decisions. The complete list of subcategories and their relevancies with problem decisions is provided at the Electronic Supplementary Material. In the employees category only one subcategory, i.e. the safety and health of workers, is influenced by the problem’s decisions. The type of PT affects this subcategory as it can change the risk of working at PCs. The number of lost days caused from work damages is used as an indicator to measure this impact. This indicator is one of the credible measures applied in the literature (see GRI, 2011; Krajnc and Glavic, 2003) to represent the impact of business activities on health and safety of employees. Problem decisions influence the local community (the second stakeholder category) at ‘‘delocalization and migration’’ and ‘‘local employment’’ impact categories by creating job opportunities. It is obvious that more job opportunities enhance the level of local employment rate and prevent undesirable delocalization and migration. The number of created job opportunities is known as a popular SR indicator (see Dehghanian and Mansour, 2009; GRI, 2011). As a contribution in the field of SR measurement in this research we have modified this well-known indicator via multiplying the number of created job opportunities in each region by the regional unemployment rate. This means that creating a job opportunity in a region with higher unemployment rate leads to more Soc value. According to the society stakeholder category, problem decisions influence only the ‘‘contribution to economic development’’ subcategory. The establishment of PCs and CCs affects the economic development of the society. To illustrate the relevant impact, the economic value of maximum producible number of products by each established facility during the problem horizon can be applied as an indicator. However, with respect to importance of balanced economic development in sustainability literature (see OECD, 2010), this indicator is also modified in this research by adding the local development rate as a coefficient beside it. In this manner, more importance is given to less development areas and therefore balanced development is assured. Notably, the two latter issues, i.e., the issue of job creation and balanced development, are among the most critical Soc issues in Iran as the government has paid significant attention to them at the ‘‘Fifth development plan of Islamic Republic of Iran’’. In the consumer stakeholder category, the ‘‘health and safety’’ and ‘‘EOL responsibility’’ subcategories are influenced by the problem’s decisions. PT affects the health and safety of consumers as it plays an important role in the quality of product, particularly the needle part. To measure this impact, several indicators such as number of relevant customer complaints, number of damaged customers, number of risky (unsafe) products and the level of product risk are proposed in the literature (see Dehghanian and Mansour, 2009; GRI, 2011). Here, the average percentage of risky products (for each type of PT) is selected as an indicator. Regarding the ‘‘EOL responsibility’’ subcategory, since it is assumed that all the return products should be collected, the SC is fully responsible for collecting EOL products and therefore there is no need to assign an indicator in this area. In the last stakeholder category, i.e. the ‘‘value chain actors’’, problem’s decisions do not directly influence any of the relevant subcategories. It should be noted that all the data needed to insert as an input parameter in the employed SR measures are accessible objective data and the collection of these data is very simple for most of companies. For example,

22


consider the number of created (direct) job opportunities for each new established plant. This parameter can be simply defined by the personnel needed to operate a new plant under a specific capacity and production technology. Some other parameters such as unemployment rate at each region and the regional development rate are also provided annually in almost all countries by governmental organizations. At the fifth step, the measured impacts should be normalized and weighted to facilitate the calculation of the total SI. Notably, GSLCAP does not prescribe any normalization method; therefore, a method is used to normalize the SR indicators at this stage. The normalization method employed in this research is the same as the normalization method applied by Human Development Index (HDI) (UNDP, 2013). Accordingly, if simax ; simin and si represent the maximum possible, minimum possible and actual value of the concerned indicator, the following formulations are used for normalization of indicators when the minimization and maximization of them are desirable, respectively.

si simin simax simin si si ¼ max simax simin

sinor ¼

sinor

ð1Þ ð2Þ

Finally, to specify the importance weight of indicators, a number of multi-criteria decision making methods can be used. Herein, we have used the linear programming model in an analytical hierarchy process (AHP) structure to determine the importance weights. For more information and details about this method interested readers can refer to Chandran et al. (2005). The method is implemented by the aid of a balanced multi-dimensional expert group (including AVAP managers and a number of external experts). Accordingly, the weight vector (0.128, 0.148, 0.148, 0.576) is resulted from the method for health and safety of workers, employment and delocalization, balanced economic development and consumer health and safety, respectively. 3.3. Proposed mathematical model The indices, parameters and variables used to formulate the problem mathematically are described below. It should be noted that symbols with a tilde on indicate parameters tainted with epistemic uncertainty. Indices i j k l m n o p q t (ei, ek)

index of candidate locations for production centers i = 1, . . . , I index of fixed locations of customers j = 1, . . . , J index of candidate locations for collection centers k = 1, . . . , K index of existing steel recycling centers l = 1, . . . , L index of existing plastic recycling centers m = 1, . . . , M index of existing incineration centers (cement plants) n = 1, . . . , N index of existing safe landfill centers o = 1, . . . , O index of capacity levels available for production centers p = 1, . . . , P index of capacity levels available for collection centers q = 1, . . . , Q index of different production technologies available for production centers t = 1, . . . , T pair indices of joint potential locations between collection and production centers ei i, ek k

Technical parameters ~ d demand of customer zone j j ~j x returned products from customer zone j (usually considered as a percentage of satisfied demand) p~ pi capacity of production center i with level p g~ qk capacity of collection center k with level q ~ dl capacity of steel recycling center l ~fm capacity of plastic recycling center m ~ nn capacity of incineration center n v~ o capacity of landfill center o Economic (cost) parameters ~f pt fixed cost of opening production center i with capacity level p and technology t i g~qk fixed cost of opening collection center k with capacity level q ~cij transportation cost per product unit from production center i to customer j ~jk a transportation cost per used product unit from customer j to collection center k ~ b transportation cost per steel part of used product unit from collection center k to steel recycling center l kl ~ h transportation cost per plastic parts of used product unit from collection center k to plastic recycling center m km ~rkn transportation cost per used product unit from collection center k to incineration center n ~eko transportation cost per used product unit from collection center k to landfill center o

23


q~ ti ~k u

~l b s~m ~ hn t~o sv cpqt ei;ek

manufacturing cost per unit of product at production center i with technology t processing cost per unit of used product at collection center k processing cost per steel part of used product unit at steel recycling center l processing cost per plastic part of used product unit at plastic recycling center m processing cost per used product unit at incineration center n processing cost per used product unit at landfill center o fixed saving opening cost fraction when production center ei i with capacity level p and technology t and collection center ek k with capacity level q are opened jointly (in a single location)

Environmental parameters ef pt environmental impact per production of one product by production technology t e et ij environmental impact of shipping one product from plant i to customer j f ec jk environmental impact of shipping one used product from customer j to collection center k ef n kn environmental impact of shipping one collected used product from collection center k to incineration center n f ed environmental impact of shipping one collected used product from collection center k to landfill center o ko e kl es environmental impact of shipping steel part of used product unit from collection center k to steel recycling center l e el environmental impact of shipping plastic part of used product unit from collection center k to plastic recycling km center m ef o environmental impact per handling one collected used product at collection centers e er environmental impact of incinerating one used product f ee environmental impact of recycling the steel part of one used product ef v environmental impact of recycling the plastic part of one used product ef a environmental impact of land-filling one f used product f ex pt environmental impact associated with establishing production center i with capacity level p and technology t i f ey qk environmental impact associated with establishing collection center k with capacity level q sv epqt saving fraction of environmental impact when production center ei i with capacity level p and technology t ei;ek and collection center ek k with capacity level q are opened jointly (in a single location) Social parameters et ld average number of annual per capita lost days caused from work’s damages when production technology t is employed f pr t average fraction of potentially hazardous (risky) products when technology t is employed f pt jcp number of created job opportunities if a production center is opened at location i with capacity level p and i technology t fq jcc number of created job opportunities if a collection center is opened at location k with capacity level q k upi unemployment rate at location i uck unemployment rate at location k vfp pi economic value of production center i with capacity level p vfc qk economic value of collection center k with capacity level q edpi level of regional development at location i edck level of regional development at location k sv jpqt fraction of decrease in number of created job opportunities when production center ei i with capacity level p ei;ek and technology t and collection center ek k with capacity level q are opened jointly (in a single location) P ws e t value of social impact related to ‘‘health and safety of workers’’ subcategory e ld ¼ i;p;t xpt si i P pr t e pr t value of social impact related to ‘‘health and safety of consumers’’ subcategory si ¼ i;j;t uij f P e jc pqt pt q f pt up þ P yq jcc f q uc P f pt fq si jcp ¼ i;p;t xpt value of social impact k i k;q k ei;ek;q;p;t sv jei;ek jcp ei upei þ jcc ek ucek xei yek i i k e pt si

related to ‘‘employment’’ and ‘‘delocalization’’ subcategories P P ¼ i;p;t xpt vfp pi ð1 edpi Þ þ k;q yqk vfc q ð1 edck Þ value of social impact related to ‘‘balanced economic i

e ws si max e pr si max e jc si max e pt si max e ws si min e pr si min e jc si

development’’ subcategory maximum possible value of social impact related to ‘‘health and safety of workers’’ subcategory maximum possible value of social impact related to ‘‘health and safety of consumers’’ subcategory maximum possible value of social impact related to ‘‘employment’’ and ‘‘delocalization’’ subcategories maximum possible value of social impact related to ‘‘balanced economic development’’ subcategory minimum possible value of social impact related to ‘‘health and safety of workers’’ subcategory minimum possible value of social impact related to ‘‘health and safety of consumers’’ subcategory minimum possible value of social impact related to ‘‘employment’’ and ‘‘delocalization’’ subcategories

min

(continued on next page)

24


e pt si min ws wp wc wt

maximum possible importance weight importance weight importance weight importance weight

value of social impact related to ‘‘balanced economic development’’ subcategory of social impact indicator related to worker’s health and safety subcategory of social impact indicator related to consumer’s health and safety subcategory of social impact indicator related to employment and delocalization subcategories of social impact indicator related to (balanced) economic development subcategory

Decision variables utij quantity of products produced at production center i with technology t and shipped to customer j sjk quantity of used products shipped customer j to collection center k vkl quantity of steel part of used products shipped from collection center k to steel recycling center l wkm quantity of plastic part of used products shipped from collection center k to plastic recycling center m zkn quantity of used products shipped from collection center k to incineration center n cko quantity of used products shipped from collection center k to landfill center n 1 if a production center with capacity level p and technology t is opened at location i xpt ¼ i 0 otherwise 1 if a production center with capacity level q is opened at location k q ¼ yk 0 otherwise In terms of the above notations, the considered sustainable supply chain network design problem can be formulated as follows.

Min W 1 ¼

XXX pt pt XX q q XXX XX XX ~f x þ ~ Þv ~l þ b ~jk Þsjk þ ~k þ a q~ ti þ ~cij utij þ ðu ðb g~k yk þ kl kl i i p

i

t

q

k

i

t

j

j

k

XX XX XX ~ Þw þ ~m þ h ~o þ ~eko Þcko þ ðs ð~hn þ ~r kn Þzkn þ ðt km km k

m

ei

p

n

k

k

XXXXX ~pt ~q pt q sv cpqt ei;ek f ei þ g ek xei yek Min W 2 ¼

t

XXX p

i

ek

ð3Þ

q

pt f ex pt i xi þ

t

m

ei

p

XX

f ey qk yqk þ

q

k

XXX XX XX e ij Þut þ e kl Þv kl p t þ et oþ f ec jk Þsjk þ ee þ es ð ef ð ef ðf ij i

j

t

k

n

XXXXX pt f f q pt q sv epqt ei;ek ex ei þ ey ek xei yek t

ek

l

o

j

k

XX XX XX f Þc e þ ef ea þ ed þ ð ef v þ ele km Þwkm þ ð er ðf n kn Þzkn þ ko ko k

k

k

k

l

o

e jc þ wt si e pt ws si e ws wp si e pr Max W 3 ¼ wc si nor nor nor nor P P P pt f pt P P q fq i p t xi jcp i upi þ k q yk jcc k uc k ¼ wc wc e jc si e jc si max min P P P P P pqt pt q f pt fq e jc ei p t ek q sv jei;ek jcp ei upei þ jcc ek uc ek xei yek þ si min þ wt e jc si e jc si max min P P P pt p P P q q e t xpt e pt e ws P P P ld f f si i p t xi v p i ð1 edpi Þ þ k q yk v c k ð1 edck Þ si min i p t max i þ ws þ wp pt pt e e e ws si e ws si si si max max min min t t e pr P P P f si max i j t pr uij pr pr e e si si max

s:t:

XX ~; utij P d j

ð4Þ

q

ð5Þ

min

8j;

ð6Þ

t

i X ~ j; sjk ¼ x

8j;

ð7Þ

k

X X X X sjk ¼ cko þ zkn þ v kl ; j

X

v kl

o

X ¼ wkm ;

l

m

j

p

X X pt p ~i ; utij 6 xi p

n

8k;

ð8Þ

l

8k; 8i; t;

ð9Þ ð10Þ


X X q q ~k ; sjk 6 yk g

8k;

25

ð11Þ

q

j

X

cko 6 v~ o ; 8o;

ð12Þ

k

X

v kl 6 ~dl ; 8l;

ð13Þ

k

X wkm 6 ~fm ;

8m;

ð14Þ

k

X zkn 6 ~nn ;

8n;

k X pt X xi 6 1; p

ð15Þ

8i;

ð16Þ

t

X q yk 6 1;

8k;

ð17Þ

q q pt q xpt 8i; k; ei; ek; p; q; t; i ; yk ; xei ; yek 2 f0; 1g

ð18Þ

utij ; sjk ; v kl ; wkm ; cko ; zkn P 0;

ð19Þ

8i; j; k; l; m; n; o; t:

OFs (3) and (4) minimize the total cost and Env impact and OF (5) maximizes the SR of the studied SCN, respectively. Constraints (6) and (7) ensure that the demands of all customers are satisfied and all the returned products are collected from customers. Constraints (8) and (9) assure the flow balance at CCs. Since three EOL options are taken into account in the studied problem, the collected used products should be sent to incineration or landfill centers or being disassembled into plastic and steel parts and then being sent to plastic and steel recycling centers (constraints 8). Therefore, the number of plastic and steel parts sent to recycling centers should be equal (constraints 9), because they are disassembled from one used product. Eqs. (10)–(15) are capacity constraints at different facilities. Constraints (10) and (11) prohibit the units of products and returned products from being transferred to PCs and CCs which are not opened, respectively. Eqs. (16) and (17) ensure that only one capacity level is assigned to opened PCs and CCs and only one type of technology is assigned to each PC. Finally, Constraints (18) and (19) enforce the binary and non-negativity restrictions on decision variables. As it can be seen from q the model, the quadratic terms xpt ei yek destroy the linearity of the three OFs. Here, to escape from complexity of solving such non-linear model directly, these non-linear terms are converted into the linear ones by defining new binary variables and adding several constraints to the model as follow. Let Wpqt ei;ek be an auxiliary variable which is defined as follows: pt q Wpqt ei;ek ¼ xei yek ; pqt ei;ek

W

2 f0; 1g;

8ei; ek; p; q; t; 8ei; ek; p; q; t:

ð20Þ

q To linearize the model, the non-linear terms in the objective functions (i.e., xpt ei yek ) should be replaced with the new binpqt ary variable (i.e., Wei;ek ). Additionally, to limit the value of these auxiliary variables into reasonable values, the following constraints are added to the original model.

pt q 2Wpqt ei;ek 6 xei þ yek ;

8ei; ek; p; q; t;

pt q Wpqt ei;ek P xei þ yek 1;

8ei; ek; p; q; t:

ð21Þ ð22Þ

3.4. Credibility-based fuzzy chance constrained programming approach As mentioned in Section 2, according to the epistemic uncertainty that we encounter in the concerned problem, possibility distributions are applied to model the imprecise parameters. Consequently, possibilistic programming (PP) approaches should be used to cope with these possibilistic data. Notably, PP and flexible programming are the two main branches of fuzzy mathematical programming. The first one is used to deal with lack of knowledge about the exact values of the model parameters (epistemic uncertainty) and the second is applied to handle flexible target value of goals and soft constraints elasticity (fuzziness/vagueness) (Inuiguchi and Ramik, 2000; Mula et al., 2006). Here, a credibility-based fuzzy programming (CFP) approach (Liu and Liu, 2002; Liu, 2004) as one of the most confident PP approaches in the literature is employed to handle the imprecise parameters in the proposed model. CFP is a credible PP approach that relies on strong mathematical concepts, i.e., the expected value (EV) of a possibilistic number and the credibility measure, and enables the decision maker (DM) to control the confidence level of constraints’ satisfactions besides supporting various kinds of possibilistic numbers such as triangular and trapezoidal forms. It should be noted that despite the possibility and necessity measures that have no self-duality property, the credibility measure is a self-dual measure (Li and Liu, 2006). In other words, if the credibility value of a fuzzy event achieves 1, DM believes the fuzzy event will surely happen; however, when the corresponding possibility measure achieves 1, the fuzzy event may fail to happen and when its necessity measure is equal to 0 the fuzzy event may hold. Among the available CFP models, the new PP method proposed by Pishvaee et al. (2012b) is used to convert the concerned model into its crisp counterpart. This PP approach is actually a combination of two credibility measure based approaches, i.e., the fuzzy chance constrained programming (see Liu and Iwamura, 1998) and the EV (see Liu and Liu,

26


2002) models as it applies the EV to convert the possibilistic OFs into their crisp counterparts and the chance constrained programming approach to transform the possibilistic chance constraints including imprecise parameters into their crisp counterparts. Assume that ~ n is a fuzzy variable with membership function l(x), and let r be a real number. According to Liu and Liu (2002), the credibility measure can be defined as follows:

Crf~n 6 rg ¼

1 ðPosf~n 6 rg þ Necf~n 6 rgÞ: 2

n 6 rg ¼ supx6r Noteworthy, since Posf~ as follows:

Crf~n 6 rg ¼

ð23Þ

lðxÞ and Necf~n 6 rg ¼ 1 supx>r lðxÞ, the credibility measure can also be defined

1 sup lðxÞ þ 1 sup lðxÞ : 2 x6r x>r

ð24Þ

n can be determined based on credibility measure as follows (Liu and Liu, 2002): Also, the EV of ~

E½~n ¼

Z

1

Crf~n P rgdr

0

Z

0

Crf~n 6 rgdr:

ð25Þ

1

n be a trapezoidal fuzzy number denoted by four prominent points as ~ n ¼ ðnð1Þ ; nð2Þ ; nð3Þ ; nð4Þ Þ. According to Eq. Now, let ~ ~ is (n(1) + n(2) + n(3) + n(4))/4 and the corresponding credibility measures are as follows: (25), the EV of n

8 0; > > > rn > ð1Þ > > > > 2ðnð2Þ nð1Þ Þ ; < Crf~n 6 rg ¼ 12 ; > > r2nð3Þ þnð4Þ > > ; > > 2ðnð4Þ nð3Þ Þ > > : 1; 8 1; > > > > 2nð2Þ nð1Þ r > > ; > > < 2ðnð2Þ nð1Þ Þ Crf~n P rg ¼ 12 ; > > nð4Þ r > > ; > > 2ðnð4Þ nð3Þ Þ > > : 0;

r 2 ð1; nð1Þ ; r 2 ðnð1Þ ; nð2Þ ; r 2 ðnð2Þ ; nð3Þ ;

ð26-1Þ

r 2 ðnð3Þ ; nð4Þ ; r 2 ðnð4Þ ; þ1; r 2 ð1; nð1Þ ; r 2 ðnð1Þ ; nð2Þ ; r 2 ðnð2Þ ; nð3Þ ;

ð26-2Þ

r 2 ðnð3Þ ; nð4Þ ; r 2 ðnð4Þ ; þ1;

Consequently, based on (26-1) and (26-2), it can be shown that if a > 0.5 then (Pishvaee et al., 2012b):

Crf~n 6 rg P a () r P ð2 2aÞnð3Þ þ ð2a 1Þnð4Þ ; Crf~n P rg P a () r 6 ð2a 1Þnð1Þ þ ð2 2aÞnð2Þ :

ð27 - 1Þ ð27-2Þ

Eqs. (27-1) and (27-2) can be applied directly to covert the fuzzy inequality chance constraints, including constraints (6) and (10)–(15), into their equivalent crisp ones. Additionally, the definition for EV (based on credibility measure) can be used to convert the three OFs into their equivalent crisp terms. But constraint (7) is an equality and cannot be treated by the above-mentioned formulations. As the credibility literature shows, there is no method or formulation to cope with equality chance constraints. Here, a definition is proposed for dealing with equality chance constraints. It can be seen from Eqs. (26-1) and (26-2) that if n(2) 6 r 6 n(3) then Crf~ n 6 rg ¼ 0:5 and Crf~ n P rg ¼ 0:5. In this state, it can be said that r and ~ n are indifferent or approximately equal. Definition 1. The real number r and trapezoidal fuzzy number ~ n are indifferent or approximately equal if and only if n(2) 6 r 6 n(3). That is:

~n r () nð2Þ 6 r 6 nð3Þ :

ð28Þ

According to Eq. (28), the equality chance constraints such as constraint (7) can be converted into their crisp counterparts. With respect to space limitation here we avoid presenting the crisp equivalent model; however, interested readers can refer to the Electronic Supplementary Material (Part I).

4. Solution approach To solve the proposed model, first a method should be provided to deal with multiple objectives of the model. Generally, three distinct categories of methods including (1) priori, (2) interactive and (3) posteriori approaches are available in the literature of MO optimization (Hwang and Masud, 1979). Here, a posteriori method is proposed to cope with multiple


27

objectives. The main advantage of posteriori methods against the priori and interactive methods is the ability of this approach in providing a universal image from the Pareto-optimal set for the DM. Thus, DM can choose the most preferred solution more confidently with respect to more available information. The proposed posteriori method uses a fuzzy linear membership function to normalize each OF and enable the DM to adjust the satisfaction degree of each objective. The steps of the proposed solution method can be summarized as follows: Step 1: Determine the minimum acceptable confidence level of chance constraints (i.e., a1j ; a2i ; a3k ; a4o ; a5l ; a6m ; a7n ). Step 2: Determine the positive ideal solution (PIS) and negative ideal solution (NIS) for each OF. To achieve the PISs and PIS PIS PIS the corresponding OF values, i.e., W PIS , W PIS and W PIS , the crisp equivalent model should be solved for 1 ; x1 2 ; x2 3 ; x3 each OF separately, and then the NISs can be specified as follows.

wNIS ; h ¼ max wh xh h¼1;2;3

h ¼ 1; 2;

wNIS ; h ¼ min wh xh h¼1;2;3

h ¼ 3:

Step 3: Specify a linear fuzzy membership function for each OF as follows:

l1 ðxÞ ¼

l3 ðxÞ ¼

8 > > > 1;
> > : 0; 8 > 1; > > < > > > :

NIS ; if W PIS 1 6 W1 6 W1

if W 1 > W NIS 1

l2 ðxÞ ¼

8 > > > 1; < > > > :

W NIS 2 W 2 PIS W NIS 2 W 2

0;

if W 2 < W PIS 2 PIS ; if W NIS 2 6 W2 6 W2

if W 2 > W NIS 2

if W 3 > W PIS 3

W 3 W NIS 3 NIS W PIS 3 W 3

NIS ; if W PIS 3 6 W3 6 W3

if W 3 < W NIS 3

0;

which lh(x) indicates the satisfaction degree of hth OF. Step 4: Convert the crisp equivalent MO model into a single-objective one by the aid of following formulation.

max HðxÞ ¼

X

.h lh ðxÞ; s:t: x 2 FðxÞ:

ð29Þ

h

where .h represents the importance weight of OFs and F(x) indicates the feasible region of the crisp equivalent MO model. As can be seen from the above formulation, a weighted sum fuzzy aggregation function is used to aggregate the three OFs. Despite some methods (e.g. Lai and Hwang, 1993) that may lead to weakly efficient solutions, this method – as it relies on weighted sum method (see Ehrgott, 2005) – ensures the achievement of efficient solution. Step 5: Determine the interested range for importance weight of OFs (.h) and segment the range of OFs into several (usually equal) parts. Then, use the resulted grid points iteratively as the value of importance weights to achieve different balanced and unbalanced efficient solutions over the interested range via solving the resulted single-objective model. Step 6: If DM is satisfied with one of the obtained solutions, stop and select the most preferred one, otherwise form a new range between the two more preferred solutions and go to step 5 to generate new efficient solutions. Also, DM may be interested in changing the value of confidence level of possibilistic constraints. In this case, the algorithm should be restarted from step 1. 4.1. Accelerated Benders decomposition algorithm Our initial tests showed that the implementation of the above-mentioned algorithm required significant computation time during step 5. This observation results from the complex structure of the developed model. From analytical viewpoint, it can be also shown that the model is strongly NP-hard. Analytical investigations about this issue are provided for interested readers at the Electronic Supplementary Material. To alleviate the model’s computational complexity, in this section an accelerated Benders decomposition algorithm (BDA) is developed. BDA that was first introduced by Benders (1962), is now known as an efficient algorithm to solve large-scale MIP problems. In BDA instead of solving the original complex MIP problem, the problem is decomposed into a pure integer programming (master problem) and a linear programming (sub-problem) problems. These two problems are solved iteratively by using the solution of one in the other, while the optimal solution being achieved. BDA has significant advantages compared to other solution methods (e.g. metaheuristic methods) such as: (1) it relies on strong algebra concepts, (2) the convergence of this algorithm and achievement of optimal solution is analytically proven, (3) the DM can adjust the optimality gap precisely when it is needed and (4) other efficient solution methods can be employed while solving the decomposed problems within a BDA. These advantages caused the usage of this algorithm in different contexts such as energy management (e.g. Zhang and Ponnambalam, 2006), scheduling (e.g. Rekik et al., 2008) and SCM (e.g. Üster and Agrahari, 2011). To develop a BDA for the concerned model, the dual subproblem (DSP) and master problem (MP) should be formulated first. To work more convenient, we consider the compact form of the original model (the aggregated single-objective model) which has been presented in the Electronic Supplemen-

28


pqt pt q q pqt tary Material. Let fix the binary variables to given values xpt i ¼ xi ; yk ¼ yk ; Wei;ek ¼ Wei;ek . Then, the Benders primal subproblem (PSP) is formulated as follows:

Max PSP ¼

XXX XX XX XX XX XX ct tij utij þ atjk sjk þ btkl v kl þ htkm wkm þ tt kn zkn þ qtko cko i

s:t:

t

j

j

XX utij P dtj ;

k

k

l

m

k

n

k

8j;

k

ð30Þ

o

ð31Þ

t

i X sjk 6 rt j ;

8j;

ð32Þ

k

X sjk P nt j ;

8j;

ð33Þ

k X X X X sjk ¼ cko þ zkn þ v kl ; o

j

X

v kl ¼

n

X wkm ;

8k;

ð34Þ

l

8k;

ð35Þ

m

l X pt p X xi pt i ; utij 6 j

p

j

q

8i; t;

X X q q k gt k ; sjk 6 y X

ð36Þ

8k;

ð37Þ

cko 6 vto ; 8o;

ð38Þ

k

X

v kl 6 dtl ; 8l;

ð39Þ

k

X wkm 6 ft m ;

8m;

ð40Þ

k

X zkn 6 nt n ; k utij ; sjk ;

8n;

ð41Þ

v kl ; wkm ; cko ; zkn P 0; 8i; j; k; l; m; n; o; t:

ð42Þ

If #

represents the dual variables of the constraints of the Benders PSP, then the DSP that produces a lower bound for the objective function of original model at each iteration, is formulated as follows:

Min DSP ¼

X j

þ

dt j #1j þ

X

v

þ

#4k

j

m

l

þ

i

t

p

q

k

X X X þ dtl #9l þ ft m #10 ntn #11 m þ n

to #8o

1 t s:t: #6t i #j P ct ij ;

#1j

! ! X XX X pt p 6t X X q q 7 xi pti #i þ k gtk #k y nt j #3j þ

j

X o

#2j

rtj #2j

ð43Þ

n

8i; j; t; #7k

ð44Þ

8j; k;

P at jk ;

ð45Þ

k; l; #5k #4k þ #9l P bt kl ; 5 # P ht ; k; m; #10 km m k 4 k; n; #11 n #k P tt kn ; 8 4 k; o; #o #k P qt ko ;

8

ð46Þ

8

ð47Þ

8

ð48Þ

8

ð49Þ

7 8 9 10 11 #1j ; #2j ; #3j ; #6t i ; #k ; #o ; #l ; #m ; #n P 0;

#4k ; #5k free;

8i; j; k; l; m; n; o; t;

ð50Þ

8k:

ð51Þ

Now according to DSP’s solution, the MP that produces an upper bound for the objective function of original model at each iteration, is represented as follows:

Max MP ¼ C þ s:t: C 6

XXX p

i

XXX pt pt XX q q XXXXX pqt pqt fti xi þ gtk yk st ei;ek Wei;ek þ constant i

p

t

p pt #^6t i pt i xi þ

XX

t

k

k

q

p

ei

#^7k gt qk yqk

q

m

ð52Þ

q

ek

X X X X dt j #^1j þ rtj #^2j nt j #^3j þ vto #^8o j

j

X X X þ dt l #^9l þ ftm #^10 nt n #^11 m þ n l

t

o

j

ð53Þ

n

XX X X X X XXX p pt dtj #^1j þ rt j #^2j ntj #^3j þ vto #^8o #^6t #^7k gtqk yqk i p t i xi þ i

p

t

k

q

j

j

X X X þ dt l #^9l þ ftm #^10 nt n #^11 m þ n P 0; l

m

n

j

o

ð54Þ


XX p

xpt i 6 1;

8i;

29

ð55Þ

t

X q yk 6 1;

8k;

ð56Þ

q pt q 2Wpqt ei;ek 6 xei þ yek ;

W

pqt ei;ek

P

xpt ei

þ

q pt q xpt i ; yk ; xei ; yek ;

yqek

8ei; ek; p; q; t;

1;

pqt ei;ek

W

C P 0:

8ei; ek; p; q; t;

2 f0; 1g 8i; k; ei; ek; p; q; t;

ð57Þ ð58Þ ð59Þ ð60Þ

In MP, Eqs. (52) and (53) represent the optimality and feasibility cuts and #^

and #^

indicate the extreme points and rays (resulted from solving DSP), respectively. Usually, at the outset of the BDA the upper bound (UB) and lower bound (LB) are assumed to be equal to +1 and 1, respectively. However, if it is possible, it is more desirable to determine a tighter interval at the beginning of the algorithm. As it can be seen from model (29), the maximum possible value of the aggregated OF is 1 (when all the OFs are fully satisfied) and the minimum possible value is equal to 0. Therefore, the UB and LB can be set to 1 and 0, respectively at the beginning of the algorithm. The BDA in its primary form, may require a large number of iterations for being converged, especially for those complex MIP problems such as the one considered in this research. Consequently, to improve the convergence of the devised BDA, several acceleration strategies suggested in the literature are introduced and employed. 4.1.1. Valid inequalities One of the reasons for slow convergence of a BDA is the low quality of MP solutions at the primary iterations of the algorithm (Saharidis et al., 2011). To avoid this inefficiency, some valid inequalities (constraints) may be added to MP in order to restrict the feasible region and produce high quality solutions. These valid inequalities should contain useful information from the original problem. Here, based on demand and return of customers, two valid inequalities can be added to MP as follows:

XXX

ptpi xpti P

i

p

k

q

XX

t

X dt j ;

ð61Þ

j

X gtqk yqk P ntj :

ð62Þ

j

Constraints (61) and (62) ensure that the capacity of established PCs and CCs are at least equal to summation of customers’ demand and return, respectively. Indeed, adding these constraints into MP prevents the establishment of insufficient number of PCs and CCs (especially at primary iterations) and hence improve the quality of MP solutions. Proposition 1. For given binary variables, i.e. xpt ; yqk ; xpt ; yqek ; Wpqt , which satisfy the constraints (55)–(59) as well as (61) and i ei ei;ek (62), the Benders primal sub-problem (PSP) (30)–(42) is always feasible and bounded.

Proof. The PSP (30)–(42) is extracted from the (compact) original model (E.1)–(E.18) presented in the Electronic Supplementary Material by fixing its binary variables; therefore, if the (compact) original model is feasible and bounded for given binary variables satisfying constraints (55)–(59) and (61) and (62), then the PSP would be also feasible and bounded. If the capacity of the established production and collection centers can satisfy (is greater than) the demands and returns of customers and constraints (E.13)–(E.17) are also satisfied, then the (compact) original model would be always feasible. In other words, for any binary variables with afore-mentioned properties, the original model becomes a multi-stage transportation problem with sufficient resources that is always feasible and bounded. Constraints (E.13)–(E.17) are the same as constraints (55)–(59) in the compact form of the original model. Constraints (61) and (62) also ensure that the capacity of the established PCs and CCs can satisfy customers’ demand and return constraints. Also, even with ignoring the above descriptions, it is obvious that the objective function of the original model, when its feasible space is not empty, can only vary between 0 and 1, i.e., it is bounded. Thus, the PSP is always feasible and bounded for given binary variables satisfying constraints (55)–(59)and (61) and (62). h Proposition 2. Dual sub-problem (DSP) (43)–(51) is always feasible and bounded and it has at least one optimal solution for any q pt q pqt given binary variables, i.e. xpt i ; yk ; xei ; yek ; Wei;ek , satisfying constraints (55)–(59) and Eqs. (61) and (62). Proof. If DSP is unbounded then PSP would be infeasible and if DSP is infeasible then primal sub-problem would be unbounded or infeasible (weak duality theorem). Proposition 1 proved that PSP is feasible and bounded; therefore DSP is also feasible and bounded. Additionally, since both PSP and DSP are bounded and feasible, the problem has at least one optimal solution (strong duality theorem). h

30


Corollary 1. Since the DSP is always feasible and bounded for any given binary variables satisfying constraints (55)–(59) and Eqs. (61) and (62), there is no need to feasibility cuts (54) in master problem and in each iteration only optimality cuts would be added. According to Propositions 1 and 2 and Corollary 1, it can be concluded that when valid inequalities (61) and (62) are added, only optimality cuts will be added to MP in each iteration and therefore the convergence of BDA will be accelerated. As it was mentioned before, a feasible solution for binary variables is needed at the outset of the BDA. Since adding valid inequalities (61) and (62) ensures that MP generates feasible solution, the following model can be used to generate a (good) feasible solution at the outset of the algorithm.

Max IN ¼

XXX pt pt XX q q XXXXX pqt pqt fti xi þ gt k yk stei;ek Wei;ek þ constant p

i

s.t.

t

q

k

ei

p

t

ek

ð63Þ

q

(55)–(59) and (61) and (62).

4.1.2. Disaggregation of Benders cut This kind of acceleration method was firstly introduced by Dogan and Goetschalckx (1999) while solving a multi period production–distribution planning problem via BDA. Thereafter, this method was used for solving complex network design problems in SC context (e.g. Üster et al., 2007) and other areas (e.g. Gzara and Erkut, 2011). The idea of this method is disaggregation of Benders cut based on disaggregation of PSP. Therefore, the structure of PSP would be capable to be separated into two or more independent sub-problems. According to this capability, multiple Benders cuts -generated by multiple DSPs- would be added to MP in each iteration and thus, the solution space of MP will be restricted more effectively. To apply this method on the concerned problem, the PSP should be disaggregated. Accordingly, PSP (30)–(42) can be separated into the following sub-problems that the first one is related to the forward SC and the second one is associated with the reverse SC. First primal sub-problem (forward chain):

Max PSP1 ¼

XXX cttij utij i

ð64Þ

t

j

s:t: ð31Þ and ð36Þ utij P 0;

8i; j; t:

ð65Þ

Second primal sub-problem (reverse chain):

Max PSP2 ¼

XX XX XX XX XX atjk sjk þ btkl v kl þ htkm wkm þ tt kn zkn þ qtko cko j

k

k

l

k

m

k

n

ð66Þ

o

k

s:t: ð32Þ—ð35Þ and ð37Þ—ð41Þ sjk ; v kl ; wkm ; cko ; zkn P 0;

8j; k; l; m; n; o:

ð67Þ

Based on PSP1 and PSP2, related dual sub-problems can now be defined as follows: First dual sub-problem:

Min DSP1 ¼

X

dt j #1j

! XX X pt p xi pti #6t þ i

j

s:t:

i

1 t #6t i #j P ct ij ;

#1j ; #6t i P 0;

t

p

ð68Þ

8i; j; t;

8i; j; t:

Second dual sub-problem:

Min DSP2 ¼

X

rtj #2j

! X X X q q 7 X X X k gtk #k þ y nt j #3j þ vto #8o þ dtl #9l þ ftm #10 m

j

j

X þ ntn #11 n

q

k

o

l

m

n

s:t:

#2j #1j þ #4k þ #7k P at jk ;

8j; k;

k; l; #5k #4k þ #9l P btkl ; 5 #10 # P ht ; k; m; km m k 4 k; n; #11 n #k P tt kn ; k; o; #8o #4k P qtko ;

8

ð69Þ

8

8

8

11 #2j ; #3j ; #7k ; #8o ; #9l ; #10 m ; #n P 0;

#4k ; #5k

free;

8k:

8j; k; l; m; n; o;

31


With the aid of solutions obtained by DSP1 and DSP2, the MP can be rewritten as follows:

Max MSP ¼ C1 þ C2 þ

XXX pt pt XX q q XXXXX pqt pqt fti xi þ gt k yk stei;ek Wei;ek þ constant i

p

t

k

q

ei

p

t

ek

s:t: ð55Þ—ð59Þ and ð61Þ and ð62Þ XXX X p pt C1 6 dt j #^1j #^6t i p t i xi i

p

k

q

t

ð70Þ

q

ð71Þ

j

XX X X X X X X C2 6 rt j #^2j nt j #^3j þ vto #^8o þ dtl #^9l þ ftm #^10 nt n #^11 #^7k gtqk yqk þ m þ n j

o

j

C1 ; C2 P 0:

l

m

ð72Þ

n

ð73Þ

As it can be seen from MP formulation, primary optimality cuts (53) are disaggregated into two optimality cuts (71) and (72) based on DSP1 and DSP2. 4.1.3. Local branching Computational complexity of MP is one of main reasons of tardy convergence of BDA. DSP is a linear programming model that usually can be solved immediately; however, MP is an integer programming (IP) model and appending of new optimality or feasibility cuts in each iteration makes it even more complex. Therefore, solving the MP, especially at middle and terminating iterations, requires significant time. To overcome this problem, Geoffrion and Graves (1974) showed that there is no need to acquire the optimal solution of MP at each iteration, but a near optimal solution could be also useful. To implement this idea they proposed a modified model for MP. Based on this idea, Rei et al. (2009) applied a local branching (LA) method to solve complex MP in BDA more efficiently. LA strategy was previously developed by Fischetti and Lodi (2003) to deal with complex IP models. In this method, the feasible region of MP is restricted by determining a reasonable neighborhood around the previous solution and then solving the restricted MP to find an optimal or a good feasible solution. By the aid of this strategy, complex MP can be efficiently solved within the restricted feasible region and significant time can be saved. Assume that yr is the solution obtained by MP at iteration r, now the following constraint can be added to MP to restrict the feasible region in the next iteration.

Dðyr ; yrþ1 Þ 6 c; where D is an appropriate distance function and c is a positive number (usually integer) with a reasonable value. To apply LA

q r method to our BDA, Hamming (1950) distance function is employed here. Let X r :¼ i; p; t : xpt i ¼ 1 and y :¼ i; p; t : yk ¼ 1 , then the following constraint can be added to MP (55)–(59), (61) and (62) and (70)–(73) to restrict the feasible region:

X

X pt X X q 1 xpt xi þ 1 yqk þ yk 6 c; þ i

i;p;t2X r

i;p;tRX r

k;q2Y r

ð74Þ

k;qRY r

which 1 6 c 6 jIj + jKj. It should be noted that selecting excessive small values for c results in infeasibility of MP; therefore this parameter should be determined conservatively or dynamically being varied while the MP is solved. We have used a conservative value for c while applying LA on the proposed BDA. To know how to dynamically vary the value c during the algorithm, interested readers can consult with Rei et al. (2009). Now, according to given descriptions about the employed acceleration methods, the accelerated BDA (ABDA) can be summarized as follows: Proposed accelerated Benders decomposition algorithm 1. Set LB = 0 and UB = 1. q b pqt 2. Solve model (55)–(59) and (61)–(63) to find an initial feasible value for binary variables xpt i ; yk ; W ei;ek . 3. Solve the disaggregated dual sub-problems (68) and (69) to find the optimal value of dual variables #^

. P P P pt pt pqt qk ei;p;t;ek;q stpqt 4. If LB < DSP1 þ DSP2 þ i;p;t fti xi þ k;q gtqk y ei;ek Wei;ek þ constant, then set LB ¼ DSP1 þ DSP2 þ P P P pt pt q q pqt pqt ft gt y st W þ constant. x þ i;p;t

i

i

k;q

k k

ei;p;t;ek;q

ei;ek

ei;ek

5. Form the master problem (55)–(59) and (61) and (62)and (70)–(74) by adding optimality cuts (71) and (72) and updating constraint (74). 6. Solve the master problem (55)–(59), (61) to (62) and (70)–(74) and set UB = MSP⁄. 7. If UB = LB (UB LB 6 e), then stop the algorithm and report the optimal (desirable) solution and relevant objective function value; else, go to step 3.

5. Implementation and evaluation In this section, the proposed SSCND model and the proposed solution algorithm is implemented for the studied case, i.e. AVAP SC, and the corresponding numerical results are evaluated and analyzed. As illustrated in Section 2, the concerned SMNS SC includes four stages. The first stage is dedicated to PCs. In this stage, AVAP has currently one active plant with

32

Table 4 Computational results under different importance weights of objective functions (OFs). Importance weight of OFs (.1, .2, .3)

Aggregated OF value H(x)

Cost (Million Rials) (w1)

Environmental impact (Pt) (w2)

Social responsibility (w3)

CC (*)

PC (*, #)

(1, 0, 0)

1

1609998

108985600

0.608

0 0%

Varamin (2) Saveh (1) Semnan (1) Arak (2)

Varamin (1, 1) Semnan (1, 1) Zanjan (1, 1) Ashtian (2, l)

(0.9, 0.05, 0.05)

0.916

1616617

100317500

0.646

6619 0.4%

Varamin (2) Semnan (1) Zanjan (2) Najafabad (2)

Varamin (1, 3) Semnan (1, 1) Zanjan (1, 1) Ashtian (2, 4)

(0.85, 0.1, 0.05)

0.902

1649022

84859030

0.647

39024 2.4%

Varamin (1) Semnan (1) Zanjan (1) Najafabad (1) Salafchegan (1)


(0.85, 0.05, 0.1)

0.881

1631241

103672300

0.728

21243 1.3%

Varamin (2) Semnan (1) Zanjan (2) Najafabad (2)


(0.8, 0.1, 0.1)

0.868

1663334

82256750

0.668

53336 3.3%

Saveh (1) Semnan (1) Zanjan (1) Najafabad (1) Salafchegan (1)

Saveh (1, 3) Semnan (1, 3) Zanjan (1, 3) Ashtian (2, 4)

(0.8, 0.15, 0.05)

0.904

1657402

81856210

0.647

47404 2.9%

Varamin (1) Semnan (1) Zanjan (1) Najafabad (1) Salafchegan (1)


(0.8, 0.05, 0.15)

0.852

1631846

103499500

0.729

21848 1.3%

Varamin (1) Semnan (1) Arak (l) Zanjan (1) Najafabad (1)


OF values

Location and type of established facilities


* Capacity level. # Type of production technology.

Price of environmental and social protection (Million Rials) & (%)


33

Fig. 3. The reciprocal performance of model objective functions.

600 million production capacity per year. Seven other candidate locations are considered for establishing new plants. Four types of PTs and two capacity levels are available for each candidate PC. At the second stage, 23 customer zones including 3 foreign and 20 domestic customers, should be supplied. Eleven candidate locations are also considered for CCs with two available capacity levels at the third stage. Five joint locations there exist among the candidate ones for PCs and CCs that may result in cost saving and also different impacts on Env and Soc performances when a PC and a CC are established in the same location. Finally, the fourth stage is related to facilities used to deal with EOL products. Four types of facilities including five incineration, eight safe landfill, five plastic recycling and seven steel recycling centers, are existed at this stage. As mentioned before, linear possibilistic distributions in the form of trapezoidal fuzzy numbers (TFNs) are used to illustrate the imprecise parameters. The four prominent values of TFNs have been determined at experts and AVAP managers consensus sessions based on their subjective knowledge and available objective data. Due to space limitation, most of used data in this case study have been provided at the Electronic Supplementary Material. According to DMs’ (AVAP managers) opinion, higher weight is given to economic objective (EO) compared to Env and Soc objectives while implementing the model. Particularly, the range 0.8–1 is assigned to importance weight of EO for implementing the proposed posteriori fuzzy solution algorithm and the range of 0–0.15 is also dedicated to other two OFs (i.e., Env and Soc OFs). The step size 0.05 is selected and the minimum confidence level of chance constraints is also set to 0.9. Notably, GAMS 22.9.2 optimization software is used to implement the model and solution approach and all the empirical experiments are carried out by a Pentium dual-core 1.40 GHz computer with 3 GB RAM. Table 4 reports the results obtained under aforementioned conditions. As it can be seen form Table 4, the three OFs are in conflict with each other as the one gets more desirable values, the others fall into more undesirable values. The pairwise conflicts of OFs and their reciprocal behavior are presented in Fig. 3 more clearly. According to such confliction between OFs it can be concluded that the company should pay more to protect Env and Soc aspects compared to a situation in which only economic criterion is taken into account. The corresponding price is reported in the sixth column of Table 4 for different importance weights of OFs. The protection price varies from 0.4% to 3.3% of the cost value of the most cost-optimum solution (i.e., the solution obtained when the model is solved only according to cost OF). When the protection price is equal to 3.3% of the cost value of the most cost optimum solution, the satisfaction degree of the Env and Soc objectives are equal to 0.91 and 0.67, respectively. This means that with only 3.3% increase on the total cost a desirable level of Env and Soc protection can be achieved. The value of protection price increases when more importance is given to Env and Soc objectives; however, as the numerical results indicated the Env objective imposes higher costs than the Soc OF. Additionally, locations, number and capacities of established facilities as well as the selected PTs show that Env and Soc OFs push the model for more decentralized SCN and the former also prefers more environmentally friendly PTs. On the other

34


Fig. 4. The illustration of SC network nodes under different importance weight of OFs.

hand, cost OF has a tendency toward a centralized SCN and less expensive PTs. For example, the illustration of SC network nodes under two different importance weight vector of OFs, i.e., (1, 0, 0) and (0.8, 0.15, 0.05), is given in Fig. 4. As Fig. 4 shows, in the most cost-optimum solution, i.e., (1, 0, 0), less collection centers are opened compared to the network formed for (0.8, 0.15, 0.05) importance weight vector. The location of collection centers also shows that when importance weight vector is set to (0.8, 0.15, 0.05), more sporadic locations are selected for collection centers. The number and locations of production centers are similar for both of the above-mentioned importance weight vectors; however, the cost-optimum solution prefers to use the first PT as the most cost-efficient one while the other solution uses the third and fourth PTs as the more

35


environmentally friendly ones. Moreover, the solution obtained for importance weight vector (0.8, 0.15, 0.05) employs recycling to cope with EOL products while the cost-optimum solution prefers to use safe landfill as the EOL option. The usage portion of EOL options at different importance weight vectors of OFs is presented in Fig. 5. Recycling, as the most environmentally friendly EOL option, is used more when more importance is given to second OF. On the other hand, safe landfill and incineration are more cost efficient options.

Fig. 5. Usage portion of EOL options under different importance weight of OFs.

Table 5 Computational results under different confidence levels. Confidence level values a


OF values Cost (Million Rials)

Environmental impact (Pt)

Social responsibility

1 0.9 0.8 0.7

1 0.916 0.917 0.920

1636661 1616617 1528704 1487777

101987800 100317500 96519000 93775850

0.643 0.646 0.657 0.668

Price of environmental and social protection (Million Rials) & (%) 12771-0.7% 6619-0.4% 16207-1% 15767-1%

Table 6 Comparative results among ABDA and CPLEX. Problem size jIj * jJj * jKj * jLj * jMj * jNj * jOj

Importance weight of OFs (.1, .2, .3)


ABDA

8 * 23 * 11 * 7 * 5 * 5 * 8 (Studied case)

(1, 0, 0) (0.9, 0.05, 0.05) (0.85, 0.1, 0.05) (0.85, 0.05, 0.1) (0.8, 0.1, 0.1) (0.8, 0.15, 0.05) (0.8, 0.05, 0.15)

1 0.916 0.902 0.881 0.868 0.904 0.852

11 15 15 12 15 15 13

1 0.929 0.914 0.885 0.870 0.903 0.865

16 22 22 25 31 27 19

1 0.967 0.954 0.955 0.943 0.946 0.946

25 28 48 30 47 63 28

Number of iterations

Total spent time for generating the seven Pareto-optimal solutions: 12 * 30 * 15 * 9 * 7 * 7 * 8

(1, 0, 0) (0.9, 0.05, 0.05) (0.85, 0.1, 0.05) (0.85, 0.05, 0.1) (0.8, 0.1, 0.1) (0.8, 0.15, 0.05) (0.8, 0.05, 0.15)

Total spent time for generating the seven Pareto-optimal solutions: 16 * 46 * 22 * 10 * 8 * 8 * 10

(1, 0, 0) (0.9, 0.05, 0.05) (0.85, 0.1, 0.05) (0.85, 0.05, 0.1) (0.8, 0.1, 0.1) (0.8, 0.15, 0.05) (0.8, 0.05, 0.15)

Total spent time for generating the seven Pareto-optimal solutions:

CPLEX CPU time (s)

CPU time (s)

34 128 49 90 94 56 75

58 1016 452 1610 131 355 825

526

4447

106 327 478 328 661 372 118

809 3020 1818 2300 4570 1522 2966

2390

17,005

63 236 1409 332 1003 585 94

502 1306 1759 499 1519 706 65

3722

6456

36


Fig. 6. The convergence progression of the proposed ABDA.

Furthermore, Table 5 illustrates the performance of the proposed model under various minimum acceptable confidence level of chance constraints. As the value of confidence levels increases, more resources should be consumed to satisfy the chance constraints. Thus, the cost and Env OFs are raised when the confidence levels increase; however this excessive cost also results in more immunized solution against uncertainty with higher degree of robustness. To evaluate and analyze the performance of the proposed ABDA, this algorithm is coded in GAMS 22.9.2 optimization software and CONOPT and CPLEX 11 solvers are used to solve the linear DSP model and integer MP model, respectively. The original MILP model is also solved by CPLEX 11. In addition to studied case, two other test problems are employed in our numerical experiments. The required parameters for these test problems are produced randomly in the range of case study’s parameters; however, the sizes of test problems are larger than the studied case. Notably, the optimality gap of ABDA is set equal to zero. The related experimental results are reported in Table 6. As the experimental results indicate, the proposed ABDA is significantly more time efficient compared to CPLEX 11 solver. Except one instance (i.e., the last one), ABDA finds the final Pareto-optimal solution in a less time. In average, ADBA is 8.5, 7.12 and 1.7 times faster than CPLEX in the first (i.e., the studied case), second and third problems, respectively. Therefore, it can be concluded that the proposed ABDA can achieve the exact optimal solution in a reasonable time and the usage of this algorithm for the concerned problem is quite acceptable. Moreover, the use of three acceleration methods is significantly effective in obtaining the above-mentioned results. As our initial tests show the three applied acceleration methods play important role in rapid convergence of the proposed ABDA. The visual representation of the ABDA convergence is provided at Fig. 6 for one of the instances.

6. Conclusions Sustainability of SCs has become more important in recent years due to increasing concerns about the Soc and Env impacts of business processes. Public demand and governmental forces also intensify the need for sustainability in today’s business environment. To move forward the literature in this area, this paper proposes a multi-objective possibilistic programming model for a real medial needle and syringe SSCND problem under epistemic uncertainty of input data. A credibility measure based approach is used to convert the original possibilistic model into its crisp counterpart. LCA-based impact assessment methods are provided to model and measure Soc and Env impacts of SCND decisions. To deal with multiple conflicting objectives of the proposed model, a posteriori fuzzy solution approach is developed. Since the resulted single-objective model is strongly NP-hard, an accelerated Benders decomposition algorithm utilizing three efficient acceleration methods is devised to solve the model in an efficient way. The proposed model and the designed solution approach are validated by using the data provided from a medical supply chain. Empirical results show the applicability of the developed model as well as the computational efficacy of the proposed accelerated BDA. Particularly, this paper includes some contributions from both practical and theoretical points of view. Regarding the practical aspect, this paper proposes a powerful mathematical model for designing a sustainable medical needle and syringe supply chain as an important strategic medical device in health systems. The model is able to manage the Env and Soc burdens of medical needle and syringe, especially at EOL stage, as well as the inherent epistemic uncertainty which affects the Eco, Env and Soc supply chain performances. From the theoretical point of view, this paper extends the recently developed social LCA-based impact assessment method, i.e., GSLCAP, to model the Soc impacts of design decisions beside the application of ReCiPe, as a credible EIA method, to estimate


37

the Env burdens. The development of an efficient accelerated Benders decomposition algorithm which includes the proof of two considerable propositions beside describing a method for dealing with equality chance constraints based on credibility measure are other major theoretical contributions of this paper. Many extensions on the presented work could be aimed for future researches. Addressing tactical and operational SC planning decisions under sustainability paradigm is an important need to cover the whole SSCM scope. To respond to this need, we require flexible social and Env impact assessment methods. Since the literature on SIA methods is very thin, developing such methods and frameworks is another critical need in this area. Regarding the solution approach, developing other decomposition-based methods (e.g. cross decomposition algorithms) can be an attractive future research direction with significant practical applications because of decomposable structure of SCND problems. 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An accelerated Benders decomposition algorithm for

An accelerated Benders decomposition algorithm for

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