A robust two layer green supply chain modelling under performance based fuzzy game theoretic approach
Graphical abstract
Note: PCM = Pollution Control Measure.
Introduction
We discuss this section into two different parts namely general overview and specific study.
Modern production system in the world has major concern over pollution by means of carbon emissions due to production and vehicle road transportation. In most of the cases, the problem of such system corresponds the problem of two-layer supply chain. The recent trends of analysing SC problems related to industrial manufacturing are to develop appropriate methodologies based on diversified industrial behaviour and the behaviour of managerial insights with information sharing. Researcher like Seliaman (2013) studied a two-stage supply chain inventory model with full information sharing and two backorders costs using hybrid geometric algebraic method. Khouja (2003) discussed about managerial decisions in a multi-stage multi-customer supply chain. A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain is discussed by Seliaman and Ahmad (2009). An integrated vendor-buyer inventory system was analysed by Teng, Cardenas-Barron, and Lou (2011) without using derivatives for optimality. However, if we take the road truck transportation as the only mode of transporting commodities then limited works are found in the literature. Dubey and Gunasekaran (2015) discussed the role of truck driver on sustainable transportation and logistics. Later, Islam (2017) solved empty truck trips problem at container terminals.
Wiedmann and Minx (2008) gave a definition of ‘Carbon Footprint’. Ebadi, Moharram, Rahnamaee, and Motesadi (2016) determined the ecological footprint of vehicles in Tehran, Iran. Mancini et al. (2016) developed the ecological footprint to refine the carbon footprint calculation. Pamukoglu, Kirkan, and Babalik (2017) did a case study to assess the carbon footprint mass balance in the context of global climate change in Isparta Province of Turkey. Akten and Akyol (2018) studied with determination of environmental perceptions and awareness towards reducing carbon footprint.
To consider the issues of pollution in SC modelling very few research articles have been found in the literature. Most of them came from the policies of carbon footprint or carbon cap taxation also. Xu, Xu, and He (2016) established joint production and pricing decisions for multiple products with cap-and trade and carbon tax regulations. Benjaafar, Li, and Daskin (2010) considered carbon footprint and analysed the management of supply chains with the help of simple models. Bouchery, Ghaffari, Jemai, and Dallery (2012) invented sustainability criteria into inventory models. Subsequently, Chen, Benjaafar, and Elomri (2013) developed the carbon-constrained economic order quantity (EOQ) inventory model efficiently. Aljazzar, Gurtu, and Jaber (2018) established a strategy to reduce carbon emissions from supply chains with delay in payments. Wang, Zhao, and Chen (2018) considered optimal credit period and worked with green consumption policies with cash-credit payments under asymmetric information. Turki, Sauvey, and Rezg (2018) worked on modelling of a (re)manufacturing system with storage facility under carbon cap and trade policy. Tiwari, Ahmed, and Sarkar (2018) analysed multi-item sustainable green production system showing that the trade-credit and partial backordering policy in the chain can minimize environmental pollution.
Moreover, it is a common phenomenon that the sponge iron industry has more harmful effect especially on human health due to aggregated environmental pollution. Dhara (2006) studied risk appraisal in various sponge iron plants in Raigarh district situated in Hyderabad, India. Elgohary and El-Bisy (2015) analysed air pollution in ports and its effect on cargo operation. Rao, Rajasekhar, and Rao (2014) studied detrimental effect of air pollution, corrosion on building materials and historical structures. Aarthi (2017) did a review work on mitigation of air pollution in sponge iron industries.
Indeed, recent modelling on SC problem focuses the strengths of computational methodologies in which game theory approach is one of them. Convexity of the objective function is another essential criterion to reach the target. Preda (1989) introduced convex optimization with nested maxima in matrix game theory and subsequently he made a relation between nonlinear programming and matrix game equivalence via duality theory. Though it has extensive literature to study but not limited to the recent works of Metzger and Rieger (2019) where they analysed an interval valued dominance cores and non-cooperative games with prospect theory players and dominated strategies respectively.
However, performance of the game player might matter the favourable outcomes of the game in any organizational set up. The nature of performances varies with the organizational behaviour even it would be the organization of a SC in production process. Brewer (2010) worked with public service motivation and performance. In view of the production mechanisms, Dubnick (2005) developed a relation between accountability and the promise of managerial performance. Kim (2005) analysed organizational performance with individual level factors in Government sectors. Van Dooren, Bouckaert, and Halligan (2015) studied with performance management in the public sector. A configurational approach on entrepreneurial orientation and small business performance was studied by Wiklund and Shepherd (2005). A comparative study on measurement of business performance in strategy research was done by Venkatraman and Ramanujam (1986). Rosenbusch, Rauch, and Bausch (2013) did a meta-analysis on the mediating role of entrepreneurial orientation in the task environment-performance relationship. Samsonowa (2012) worked on industrial research performance management. Zhu, Johnsson, Varisco, and Schiraldi (2018) did a review paper by analysing the research gap between process industrial needs and ISO 22,400 standard with the help of key performance indicators for manufacturing operations management.
In addition, after the invention of fuzzy set (Zadeh (1965)), numerous research articles have been studied in various field of applications. A procedure for ordering fuzzy numbers has been established by Yager (1981). Through its long journey, De and Beg, 2016a, De and Beg, 2016b investigated new defuzzification procedures for triangular dense fuzzy sets and triangular dense fuzzy neutrosophic sets. Karmakar, De, and Goswami (2017) applied dense fuzzy system first time to make the managerial decision for a sponge iron industry where the pollution due to production has been incorporated by means of a separate pollution function that directly relates to the amount of produced items itself. After the invention of dense fuzzy lock set by De, 2017, Karmakar et al., 2018 invented another model by utilizing the fuzzy lock set into re-manufacturing of waste items into fresh items related to that sponge iron industry also. Though only two research articles were studied yet upon the same SI industry where the major concern was pollution reduction but still it has some challenging situations for transportation and marketing of the finished products which are strongly associated to the managerial/executives’ performances aiming joint pollution and system cost reduction whenever all cost the components, demand rate and production rate assume fuzzy numbers.
Here we shall discuss the literature survey on SC modelling under fuzzy game problem. Seikh, Nayak, and Pal (2015) studied a matrix game where the pay offs follow intuitionistic fuzzy set. The fuzzy game problem has been analysed with a new approach by Thirucheran, Meena kumari, and Lavanya (2017). De, Nayak, Khan, Bhattacharya, and Smarandache (2020) studied a game problem for imperfect production process using neutrosophic fuzzy set. De and Mahata (2020) discussed a leadership game model utilizing fuzzy lock set. In modern Supply Chains, cooperation along the chain is typically observed while competition characterizes the performers belonging to the same stage of the network. In fact, these supply chain networks are basically formed by independent companies collaborating each other to coordinate the process of supply, manufacturing, and logistics and to maximize the overall added value of the whole supply chain network (SCN) for satisfaction of customers' needs. These ideas are discussed in some recent research articles. Stevenson and Spring (2007) discussed a complete understanding of the effects of parametric flexibility across the whole supply chain perspective. Delic and Eyers (2020) studied the effect of additive manufacturing adoption on supply chain flexibility and performance from the automotive industry. Dominguez, Cannella, Barbosa-Póvoa, and Framinan (2018) developed a strategy to implement partial information sharing among supply chain retailers. In a sustainable closed-loop supply chain network design, multi-objective fuzzy robust optimization approach was introduced by Nayeri, Paydar, Asadi-Gangraj, and Emami (2020). Chunhai, Zebin, Thomas, and Zhaoning (2020) studied an inventory model of a deteriorating products considering carbon emissions. Rodríguez-Espíndola, Alem, and Da Silva (2020) jointly developed a shortage risk mitigation model for multi-agency coordination in logistics planning. A study of optimal order policy for a multi-period multi-raw material inventory management problem under carbon emission constraint was discussed by Gao, Chen, Tang, and Zhang (2020).
Quantity discount is a common phenomenon offered by sellers to promote their products ensuring that the customer could purchase a large volume of items. By this way, the buyer receives products with a more favourable price that definitely affects the trade and SCN design. Very few research articles are available along this direction. Song and Gao (2018) developed a supply chain game model where the demand rate is a function of price and greening level, and they analysed the problem under common centralised game and revenue sharing game theoretic approach. Cavone, Dotoli, Epicoco, Morelli, and Seatzu (2020) presented a fuzzy game approach for solving multi-stage, multi-product, multi-item and closed-loop supply chain network design under uncertainty. Lima-Junior and Carpinetti (2017) studied on extensive review paper of modelling on quantitative supply chain performance evaluation.
However, in many cases it is observed that to ensure quality items no quantity discounts are allowed. Especially, in the backlogging model the DM has no pressure to sale the product also. To show the actual novelty and research gap of this study, some recent major supply chain models can be listed over here (shown in Table 1).
From the above study, it is seen that none of the researchers have incorporated two- layer supply chain model with flexible cost components and demand based on performance matrix indices for marketing of finished products and non-cooperative strategic game plan. Thus, in our present study we have introduced an efficient method based on sales executives’ performance matrix having strategic game over the supply chain of a proposed sponge iron industry by constructing an integrated SC cost function aiming to reduce the aggregated pollution level. Moreover, we consider the backorder at buyer’s side to study whether it could be able to reduce pollution in the entire supply chain itself or not.
The organization of this article is expressed as follows: Section 1 includes introduction followed by two subsections namely general overview and specific study. Section 2 includes preliminaries of pollution functions, concept of game theory, performance indices matrix, triangular fuzzy number, strategic normality and managerial implication. Section 3 indicates assumption and notations of the model, a real case study and the problem definition. Section 4 develops formulation of an EPQ model, Section 5 discusses the general fuzzy model followed by several sub models of strategic game and performances with different pollution measures. Section 6 indicates numerical examples and sensitivity analysis. Section 7 includes graphical illustrations; Section 8 gives managerial insights and finally Section 9 gives conclusion of the whole study followed by a scope of future work.
Section snippets
Choice of pollution function [extension of Karmakar et al. (2017)]
Let be the production rate and be the pollution rate. Then, for the absence of environmental pollution, the production rate will decrease exponentially with the maximum consuming capacity of the customers. The production rate will also decrease slowly due to the unavailability of raw materials from the decaying environmental resources or pollutant environment. However, in the absence of production, the pollution rate decreases proportionally and increases exponentially with the
Notations and assumptions
Notations The annual demand rate for the buyer (Consumer) (MT) The annual production for the vendor (Producer) (MT) Set-up cost for the buyer ($) Set-up cost for the vendor ($) Holding cost of raw materials per unit per unit time for the vendor ($) Holding cost per unit per unit time for the vendor ($) Holding cost per unit per unit time for the buyer ($) Backorder cost per unit per unit time for the buyer ($) Pollution cost due to production for the vendor ($) Truck Transportation
Formulation of EPQ model [Extension of Seliaman (2013)]
To construct a best fitted model over the real case study we observe that the extension of Seliaman (2013)’s model might serve this purpose. This extension is done by the insertion of the cost of pollution due to the production and that for transportation alone. Now as per assumptions of Section 3, let the production process starts with zero inventory. The amount of raw materials gradually decreases with the conversion of raw materials to finished products and the number of finished products
Formulation of general fuzzy model
In our proposed study, it is observed that different type of costs, production rate and demand rate may not be fixed due to various external and internal factors such as seasonal change, labour problem, price hike etc. For this reason, we assume all the cost parameters (), production rate (P) and demand rate (D) associated with the model under study are flexible in nature and they might follow triangular fuzzy numbers in the following form:
,
Numerical example
Using the data set of Table 2 and Eq. (7) (Section 3.1) and utilizing the pollution function (2) (Section 2.1) in the main problem (14) we get the optimal results and they are put in Table 3.
Table 3 shows the average SC cost for centralised model when pollution control measures and without pollution control measures are taken care of. The cost is $ 35604.90 with respect to the pollution level 8.605% when PCM is applied. But for without PCM, the pollution raised to 12.825% with the SC cost hike
Graphical illustration
Here we shall draw several figures based on data set obtained on various optimal solutions tables.
Fig. 7 shows the continuous reduction of average SC cost in different scenarios (with PCM) like crisp model, general fuzzy model, fuzzy game model, performance based fuzzy game model and performance based strategic fuzzy game model respectively. We notice that the average SC cost has the lowest minimum at the performance based strategic fuzzy game model with respect to the other models. The cost
Managerial insights
In this article we have studied a pollution sensitive two-layer supply chain model on the basis of performance indices of marketing executives and fuzzy game theory with several strategic plans. We have evaluated several decision variables like performance indices, measures of pollution (%), order quantity (MT), production run time (year), cycle time (year) and average integrated system cost ($) etc. A manager can solve/ tackle any kind of complex situation of an entire industry with proper
Conclusion
We have developed an efficient solution approach of the problems of a joint supply chain of SI industry via case study and modelling. Produced items are pollution sensitive and transportation of finished products might also carry some pollution by means of carbon emissions. Items are sold through three-layer marketing executives (encircling the industry within 200 miles of radius) having differential performances of each individual group and if any shortfall of item appears in the chain, then
Scope of future work
Performance based inventory modelling is the recent trend of analysing any complex situation of internal and external management of industrial affairs. This novel approach can be applied in multicriteria, multi attribute decision making problem also in the study of future research.
CRediT authorship contribution statement
Kousik Bhattacharya: Writing - original draft, Software, Data curation, Visualization. Sujit Kumar De: Conceptualization, Methodology, Investigation, Supervision, Validation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors are thankful to the honourable Editor-in-Chief, Associate Editor and anonymous reviewers for their constructive suggestions to improve the quality of this article.
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