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Multi-objective optimization of cost-effective and customer-centric closed-loop supply chain management model in T-environment

Abstract

This article presents one real-life-based cost-effective and customer-centric closed-loop supply chain management model. The review of the existing literature identifies the classical performance indicators to any supply chain management model as the aggregate revenue, the customer satisfaction and the environmental concern. However, this review fails to find a single optimization-based supply chain management model that considers these three indicators, simultaneously. In this article, the proposed model maximizes the customer-satisfaction index and the aggregate revenue both under the environmental considerations via the reverse chain, whereas many existing studies took the reverse chain and the associated subsidies into account; this is the first mathematical model that optimizes the customer-satisfaction index, at the same time. This article employs the T-set that represents the inherent impreciseness to objective functions to the proposed model. The corresponding optimal values are superior than stipulated goals to both the objective functions in T-environment. The managerial insights extracted from sensitivity analysis of parameters suggest the managers to stabilize the environmental concern and the customer satisfaction, while ensuring the cost-effectiveness in real-life-based T-environment. Also, this analysis finds that the subsidy assists any supply chain to sustain, only if it is offered without any break and within the optimally determined bounds.

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References

  1. Agnihotri R, Trainor KJ, Itani OS, Rodriguez M (2017) Examining the role of sales-based CRM technology and social media use on post-sale service behaviors in India. J Bus Res 81:144–154

    Google Scholar 

  2. Ali A, Haseeb M (2019) Radio frequency identification (RFID) technology as a strategic tool towards higher performance of supply chain operations in textile and apparel industry of Malaysia. Uncertain Supply Chain Manag 7:215–226

    Google Scholar 

  3. Alinaghian M, Zamani M (2019) A bi-objective fleet size and mix green inventory routing problem, model and solution method. Soft Comput 23(4):1375–1391. https://doi.org/10.1007/s00500-017-2866-2

    Article  Google Scholar 

  4. Altiparmak F, Mitsuo G, Lin L, Paksoy T (2006) A genetic algorithm approach for multi-objective optimization of supply chain networks. Comput Ind Eng 51(1):196–215

    Google Scholar 

  5. Anderson D (2014) Design for manufacturability: how to use concurrent engineering to rapidly develop low-cost, high-quality products for lean production. CRC Press, Boca Raton

    Google Scholar 

  6. Anderson E, Sullivan M (1993) The antecedents and consequences of CS for firms. Mark Sci 12(2):125–143

    Google Scholar 

  7. Anderson E, Fornell C, Lehmann D (1994) Customer satisfaction, market share, and profitability: findings from Sweden. J Mark 58(3):53–66

    Google Scholar 

  8. Angelova B, Zekiri J (2011) Measuring CS with service quality using American CS model (ACSI model). Int J Acad Res Bus Soc Sci 1(3):232–258

    Google Scholar 

  9. Aquilani B, Silvestri C, Ruggieri A, Gatti C (2017) A systematic literature review on total quality management critical success factors and the identification of new avenues of research. TQM J 29(1):184–213. https://doi.org/10.1108/TQM-01-2016-0003

    Article  Google Scholar 

  10. Attaran M, Sharmin A (2007) Collaborative supply chain management. Bus Process Manag J 13(3):390–404

    Google Scholar 

  11. Aven T (2016) Risk assessment and risk management: review of recent advances on their foundation. Eur J Oper Res 253(1):1–13

    MathSciNet  MATH  Google Scholar 

  12. Bahari F, Elayidom S (2015) An efficient CRM-data mining framework for the prediction of customer behaviour. Procedia Comput Sci 46:725–731

    Google Scholar 

  13. Basu S, Pramanik S, Dey S, Panigrahi G, Jana DK (2019) Fire monitoring in coal mines using wireless underground sensor network and interval type-2 fuzzy logic controller. Int J Coal Sci Technol 6(2):274–285

    Google Scholar 

  14. Bellman RE, Zadeh LA (1970) Decision-making in a fuzzy environment. Manag Sci 17(4):141–164

    MathSciNet  MATH  Google Scholar 

  15. Bloemhof JM, Soysal M (2017) Sustainable food supply chain design. In: Bouchery Y, Corbett C, Fransoo J, Tan T (eds) Sustainable supply chains, springer series in supply chain management, vol 4. Springer, Cham, pp 395–412

    Google Scholar 

  16. Bowen EF, Paul CD, Richard LC, Farukt AC (2009) The role of supply management capabilities in green supply. Prod Oper Manag 10(2):174–189

    Google Scholar 

  17. Brandenburg M, Gerd HJ, Rebs T (2018) Social and environmental dimensions of organizations and supply chains: trade-offs and synergies. Springer, Cham

    Google Scholar 

  18. Campos AC, Mendes J, Valle PO, Scott N (2018) Co-creation of tourist experiences: a literature review. Curr Issues Tour 21(4):369–400

    Google Scholar 

  19. Cardozo RN (1965) An experimental study of customer effort, expectation, and satisfaction. J Mark Res 2(3):244–249

    Google Scholar 

  20. Carter CR, Easton PL (2011) Sustainable supply chain management: evolution and future directions. Int J Phys Distrib Logist Manag 41(1):46–62

    Google Scholar 

  21. Chatterjee K, Pamucar D, Zavadskas EK (2018) Evaluating the performance of suppliers based on using the R’AMATEL-MAIRCA method for green supply chain implementation in electronics industry. J Clean Prod 184:101–129

    Google Scholar 

  22. Chavez R, Yu W, Feng M, Wiengarten F (2016) The effect of customer-centric green supply chain management on operational performance and customer satisfaction. Bus Strategy Environ 25(3):205–220

    Google Scholar 

  23. Chen L, Zhao X, Tang O, Price L, Zhang S, Zhu W (2017) Supply chain collaboration for sustainability: a literature review and future research agenda. Int J Prod Econ 194:73–87

    Google Scholar 

  24. Chin TA, Hamid ABA, Rasli A, Tat HH (2014) A literature analysis on the relationship between external integration, environmental uncertainty and firm performance in Malaysian SMEs. Procedia Soc Behav Sci 130:75–84

    Google Scholar 

  25. Choudhury M, Harrigan P (2014) CRM to social CRM: the integration of new technologies into customer relationship management. J Strateg Mark 22(2):149–176. https://doi.org/10.1080/0965254X.2013.876069

    Article  Google Scholar 

  26. Chuang SH, Lin HN (2013) The roles of infrastructure capability and customer orientation in enhancing customer-information quality in CRM systems: empirical evidence from Taiwan. Int J Inf Manag 33:271–281

    Google Scholar 

  27. Churchill G, Surprenant C (1982) An investigation into the determinants of customer satisfaction. J Mark Res 19(4):491–504

    Google Scholar 

  28. Deloitte (2019) Deloitte touche tohmatsu limited. http://www.deloitte.com. Accessed 21 July 2019

  29. Dubey R, Gunasekaran A, Papadopoulos T, Childe S, Shibin K, Wamba S (2016) Sustainable supply chain management: framework and further research directions. J Clean Prod 142(2):1119–1130

    Google Scholar 

  30. Dubois D, Prade H (2015) The legacy of 50 years of fuzzy sets: a discussion. Fuzzy Sets Syst 281:21–31. https://doi.org/10.1016/j.fss.2015.09.004

    MathSciNet  Article  MATH  Google Scholar 

  31. Dutta A, Jana D (2017) Expectations of the reductions for type-2 trapezoidal fuzzy variables and its application to a multi-objective solid transportation problem via goal programming technique. J Uncertain Anal Appl. https://doi.org/10.1186/s40467-017-0057-4

    Article  Google Scholar 

  32. Ebrahimnejad A, Verdegay JL (2018) Fuzzy sets-based methods and techniques for modern analytics. studies in fuzziness and soft computing. Polish Academy of Sciences, Warsaw

    MATH  Google Scholar 

  33. Elhedhli S, Merrick R (2012) Green supply chain network design to reduce carbon emissions. Transp Res Part D Transp Environ 17(5):370–379

    Google Scholar 

  34. Eltayeb TK, Zailani S, Ramayah T (2011) Green supply chain initiatives among certified companies in Malaysia and environmental sustainability: investigating the outcomes. Resour Conserv Recycl 55(5):495–506

    Google Scholar 

  35. Fernandes AC, Sampaio P, Sameiro M, Truong HQ (2017) Supply chain management and quality management integration: a conceptual model proposal. Int J Qual Reliab Manag 34(1):53–67. https://doi.org/10.1108/IJQRM-03-2015-0041

    Article  Google Scholar 

  36. Flint JD, Christopher BP, Boutin PJ (2011) Customer value anticipation, CS and loyalty: an empirical examination. Ind Mark Manag 40(2):219–230

    Google Scholar 

  37. Fornell C (1992) A national CS barometer: the Swedish experience. J Mark 56(1):6–21

    Google Scholar 

  38. Garai A, Mandal P, Roy TK (2016) Intuitionistic fuzzy T-sets based optimization technique for production-distribution planning in supply chain management. Opsearch 53(4):950–975

    MathSciNet  MATH  Google Scholar 

  39. Garai A, Mandal P, Roy TK (2017) Multipollutant air quality management strategies: t-Sets Based optimization technique under imprecise environment. Int J Fuzzy Syst 19(6):1927–1939

    MathSciNet  Google Scholar 

  40. Garrido-Moreno A, García-Morales LV (2014) Paving the way for CRM success: the mediating role of knowledge management and organizational commitment. Inf Manag 51(8):1031–1042

    Google Scholar 

  41. Gaussin M, Hu G, Abolghasem S, Basu S, Shankar MR, Bidanda B (2013) Assessing the environmental footprint of manufactured products: a survey of current literature. Int J Prod Econ 146(2):515–523

    Google Scholar 

  42. Geng R, Mansouri S, Aktas E (2017) The relationship between green supply chain management and performance: a meta-analysis of empirical evidences in Asian emerging economies. Int J Prod Econ 183(A):245–258

    Google Scholar 

  43. Govindan K, Kaliyan M, Kannan D, Haq A (2014) Barriers analysis for green supply chain management implementation in Indian industries using analytic hierarchy process. Int J Prod Econ 147(B):555–568

    Google Scholar 

  44. Green KW, Zelbst PJ, Meacham J, Bhadauria VS (2012) Green supply chain management practices: impact on performance. Supply Chain Manag Int J 17(3):290–305

    Google Scholar 

  45. Grimm D, Wosten H (2018) Mushroom cultivation in the circular economy. Appl Microbiol Biotechnol 102(18):7795–7803

    Google Scholar 

  46. Guu S-M, Wu Y-K (1999) Two-phase approach for solving the fuzzy linear programming problems. Fuzzy Sets Syst 107(2):191–195

    MathSciNet  MATH  Google Scholar 

  47. Handfield RB, Ernest NL (1999) Introduction to supply chain management. Prentice Hall, Upper Saddle River

    Google Scholar 

  48. Hennig T, Klee A (1997) The impact of customer satisfaction and relationship quality on customer retention: a critical reassessment and model development. Psychol Mark 14(8):737–764

    Google Scholar 

  49. Hsu C-W, Kuo T-C, Chen S, Hu AH (2013) Using DMTEL to develop a carbon management model of supplier selection in green supply chain management. J Clean Prod 56:164–172

    Google Scholar 

  50. Imran M, Hamid S, Aziz A, Hameed W (2019) The contributing factors towards e-logistic customer satisfaction: a mediating role of information technology. Uncertain Supply Chain Manag 7:63–72

    Google Scholar 

  51. Jabbour C, Jabbour A (2016) Green human resource management and green supply chain management: linking two emerging Agendas. J Clean Prod 112:1824–1833

    Google Scholar 

  52. Janssen PB, Johnson MP, Schaltegger S (2015) 20 years of performance measurement in sustainable supply chain management—what has been achieved? Supply Chain Manag Int J 20(6):664–680. https://doi.org/10.1108/SCM-06-2015-02

    Article  Google Scholar 

  53. Jimenez M, Bilbao A (2009) Pareto-optimal solutions in fuzzy multi-objective linear programming. Fuzzy Sets Syst 160(18):2714–2721

    MathSciNet  MATH  Google Scholar 

  54. Jonkman et al (2019) Selecting food process designs from a supply chain perspective. J Food Eng 195:52–60

    Google Scholar 

  55. Kandampully J, Zhang TC, Jaakkola E (2018) Customer experience management in hospitality: a literature synthesis, new understanding and research agenda. Int J Contemp Hosp Manag 30(1):21–56. https://doi.org/10.1108/IJCHM-10-2015-0549

    Article  Google Scholar 

  56. Kannan D, Khodaverdi R, Olfat L, Jafarian A, Diabat A (2013) Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain. J Clean Prod 47:355–367

    Google Scholar 

  57. Kaur J, Sidhu R, Awasthi A, Chauhan S, Goyal S (2018) A DEMATEL based approach for investigating barriers in green supply chain management in Canadian manufacturing firms. Int J Prod Res 56(1-2):312–332

    Google Scholar 

  58. Khansalar E, Bayez MLD, Safari R (2015) Maximizing profit in reverse supply chain. Int Bus Res 8(11):111–118. https://doi.org/10.5539/ibr.v8n11p111

    Article  Google Scholar 

  59. Kronborg JJ, Munksgaard KB, Arlbjorn JS (2013) Chasing value offerings through green supply chain innovation. Eur Bus Rev 25(2):124–146. https://doi.org/10.1108/09555341311302657

    Article  Google Scholar 

  60. Kusi-Sarpong S, Gupta H, Sarkis J (2018) A supply chain sustainability innovation framework and evaluation methodology. Int J Prod Res. https://doi.org/10.1080/00207543.2018.1518607

    Article  Google Scholar 

  61. Li X (2014) Operations management of logistics and supply chain: issues and directions. Discrete Dyn Nat Soc. https://doi.org/10.1155/2014/701938

    Article  Google Scholar 

  62. Markova S, Tatjana P-M (2013) Social media and supply chain. Amfiteatru Econ J 15(33):89–102

    Google Scholar 

  63. Melnyk SA, Flynn BB, Awaysheh A (2018) The best of times and the worst of times: empirical operations and supply chain management research. Int J Prod Res 56(1):164–192

    Google Scholar 

  64. Mumtaz U, Ali Y, Petrillo A (2018) A linear regression approach to evaluate the green supply chain management impact on industrial organizational performance. Sci Total Environ 624:162–169

    Google Scholar 

  65. Oliver Wight (2019) https://www.oliverwight-eame.com. Accessed 21 July 2019

  66. Petljak K, Zulauf K, Stulec I, Seuring S, Wagner R (2018) Green supply chain management in food retailing: survey-based evidence in Croatia. Supply Chain Manag Int J 23(1):1–15. https://doi.org/10.1108/SCM-04-2017-0133

    Article  Google Scholar 

  67. Pramanik S, Jana DK, Mondal SK, Maiti M (2015) A fixed-charge transportation problem in two-stage supply chain network in Gaussian type-2 fuzzy environments. Inf Sci 325:190–214

    MathSciNet  MATH  Google Scholar 

  68. Rajeev A, Pati R, Padhi S, Govindan K (2017) Evolution of sustainability in supply chain management: a literature review. J Clean Prod 162(20):299–314

    Google Scholar 

  69. Rust RT, Zahorik AJ (1993) Customer satisfaction, customer retention, and market share. J Retail 69(2):193–215

    Google Scholar 

  70. Ryu K, Lee HR, Kim WG (2012) The influence of the quality of the physical environment, food, and service on restaurant image, customer perceived value, customer satisfaction, and behavioral intentions. Int J Contemp Hosp Manag 24(2):200–223

    Google Scholar 

  71. Sabri EH, Beamon BM (2000) A multi-objective approach to simultaneous strategic and operational planning in supply chain design. Omega 28(5):581–598

    Google Scholar 

  72. Saeidi SP, Sofian S, Saeidi P, Saeidi SP, Saaeidi SA (2015) How does corporate social responsibility contribute to firm financial performance? the mediating role of competitive advantage, reputation, and customer satisfaction. J Bus Res 68(2):341–350

    Google Scholar 

  73. Sakawa M, Matsui T (2012) An interactive fuzzy satisficing method for multi-objective stochastic integer programming with simple recourse. Appl Math 3:1245–1251

    Google Scholar 

  74. Sakawa M, Yano H, Nishizaki I (2013) Linear and multi-objective programming with fuzzy stochastic extensions. Springer, New York

    MATH  Google Scholar 

  75. Salehi S, Selamat A, Mashiinchi MR, Fujita H (2015) The synergistic combination of particle swarm optimization and fuzzy sets to design granular classifier. Knowl Based Syst 76:200–218

    Google Scholar 

  76. Sarkar B (2019) Mathematical and analytical approach for the management of defective items in a multi-stage production system. J Clean Prod 216:896–919. https://doi.org/10.1016/j.jclepro.2019.01.078

    Article  Google Scholar 

  77. Sarkar B, Guchhait R, Sarkar M, Pareek S, Kim N (2018) Impact of safety factors and setup time reduction in a two-echelon supply chain management. Robot Comput Integr Manuf. https://doi.org/10.1016/j.rcim.2018.05.001

    Article  Google Scholar 

  78. Sarkis J (2003) A strategic decision framework for green supply chain management. J Clean Prod 11(4):397–409

    Google Scholar 

  79. Sarkis J, Qinghua Z, Lai KH (2011) An organizational theoretic review of green supply chain management literature. Int J Prod Econ 130(1):1–15

    Google Scholar 

  80. Sarvestani HK, Zadeh A, Seyfi M, Barzoki MR (2019) Integrated order acceptance and supply chain scheduling problem with supplier selection and due date assignment. Appl Soft Comput 75:72–83. https://doi.org/10.1016/j.asoc.2018.10.045

    Article  Google Scholar 

  81. Secchi E, Roth A, Verma R (2018) The impact of service improvisation competence on customer satisfaction: evidence from the hospitality industry. Prod Oper Manag. https://doi.org/10.1111/poms.12969

    Article  Google Scholar 

  82. Seuring S (2013) A review of modeling approaches for sustainable supply chain management. Decis Support Syst 54(4):1513–1520

    Google Scholar 

  83. Seuring S, Gold S (2013) Sustainability management beyond corporate boundaries: from stakeholders to performance. J Clean Prod 56:1–6

    Google Scholar 

  84. Seuring S, Muller M (2008) From a literature review to a conceptual framework for sustainable supply chain management. J Clean Prod 16(15):1699–1710

    Google Scholar 

  85. Sheu BJ, Chou YH, Hu CC (2005) An integrated logistics operational model for green-supply chain management. Transp Res Part E Logist Transp Rev 41(4):287–313

    Google Scholar 

  86. Siddiqi O (2011) Interrelations between service quality attributes, CS and customer loyalty in the retail banking sector in Bangladesh. Int J Bus Manag 6(3):12–36

    Google Scholar 

  87. Singh R, Sandhu HS, Metri BA, Kaur R (2018) Supply chain management practices, competitive advantage and organizational performance: a confirmatory factor model. Oper Serv Manag. https://doi.org/10.4018/978-1-5225-3909-4.ch054

    Article  Google Scholar 

  88. Srivastava SK (2007) Green supply-chain management: a state-of-the-art literature review. Int J Manag Rev 9(1):53–80

    Google Scholar 

  89. Svensson G (2007) Aspects of sustainable supply chain management (SSCM): conceptual framework and empirical example. Supply Chain Manag Int J 12(4):262–266

    Google Scholar 

  90. Testa F, Iraldo F (2010) Shadows and lights of GSCM (green supply chain management): determinants and effects of these practices based on a multi-national study. J Clean Prod 18(10-11):953–962

    Google Scholar 

  91. Tseng M, Islam M, Karia N, Fauzi F (2019) A literature review on green supply chain management: trends and future challenges. Resour Conserv Recycl 141:145–162

    Google Scholar 

  92. Uygun O, Dede A (2016) Performance evaluation of green supply chain management using integrated fuzzy multi-criteria decision making techniques. Comput Ind Eng 102:502–511

    Google Scholar 

  93. Walker H, Jones N (2012) Sustainable supply chain management across the UK private sector. Supply Chain Manag Int J 17(1):15–28

    Google Scholar 

  94. Wei JT, Lee MC, Chen HK, Wu HH (2013) Customer relationship management in the hairdressing industry: an application of data mining techniques. Expert Syst Appl 40(18):7513–7518

    Google Scholar 

  95. Wei G, Wang H, Zhao X, Lin R (2014) Hesitant triangular fuzzy information aggregation in multi-attribute decision making. J Intell Fuzzy Syst Appl Eng Technol 26(3):1201–1209

    MATH  Google Scholar 

  96. Wieland A, Handfield RB, Durach CF (2016) Mapping the landscape of future research themes in supply chain management. J Bus Logist. https://doi.org/10.1111/jbl.12131

    Article  Google Scholar 

  97. Wu Y-K, Liu CC, Lur YY (2015) Pareto-optimal solution for multi-objective linear programming problems with fuzzy goals. Fuzzy Optim Decis Mak 14(1):43–55

    MathSciNet  MATH  Google Scholar 

  98. Zablah AR, Carlson BD, Donavan DT, Maxham JG, Brown T (2016) A cross-lagged test of the association between CS and employee job satisfaction in a relational context. J Appl Psychol 101(5):743–755

    Google Scholar 

  99. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Google Scholar 

  100. Zhang M, Guo H et al (2019) Linking supply chain quality integration with mass customization and product modularity. Int J Prod Econ. https://doi.org/10.1016/j.ijpe.2017.01.011

    Article  Google Scholar 

  101. Zhou S, Lam J, Zheng WX (2007) Control design for fuzzy systems based on relaxed non-quadratic stability and H∞ performance conditions. IEEE Trans Fuzzy Syst 15(2):188–199

    Google Scholar 

  102. Zhu Q, Sarkis J, Lai KH (2008) Confirmation of a measurement model for green supply chain management practices implementation. Int J Prod Econ 111(2):261–273

    Google Scholar 

  103. Zhu Q, Sarkis J, Lai KH (2013) Institutional-based antecedents and performance outcomes of internal and external green supply chain management practices. J Purch Supply Manag 19(2):106–117

    Google Scholar 

  104. Zimmermann H-J (1976) Description and optimization of fuzzy systems. Int J Gen Syst 2(4):209–215

    MATH  Google Scholar 

  105. Zimmermann H-J (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1(1):45–55

    MathSciNet  MATH  Google Scholar 

  106. Zimmermann H-J (1985) Applications of fuzzy set theory to mathematical programming. Inf Sci 36(1–2):29–58

    MathSciNet  MATH  Google Scholar 

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This study was not funded by any agency or Government. None of the authors received any kind of financial grant or support for this study.

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Correspondence to Arindam Garai.

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Communicated by O. Castillo, D. K. Jana.

Appendices

Appendix A: Symbols and descriptions

See Table 11.

Table 11 Symbols with the descriptions used in the proposed CLSCM model

Appendix B: Drawbacks of classical fuzzy set

The drawbacks of the membership function to classical fuzzy set as well as the redefined membership function in Wu et al. (2015) are as follows (Garai et al. 2016, 2017)

  • Firstly, the employments of redefined membership functions in Wu et al. (2015) were not as per the definitions presented in mathematical model as well as in numerical examples of that article.

  • Secondly, the mathematical model in Wu et al. (2015) employed strictly monotonic redefined membership functions over the entire real line. However, the definitions in that article specified otherwise.

  • In addition, the researchers could only employ the part of redefined membership functions, at which the corresponding objective values lied within the goals and goal plus tolerances, to the minimization type of objective functions. However, so far, researchers did not consider the above prerequisite (in form of constraints) within the mathematical models.

  • Thirdly, Wu et al. (2015) discarded the upper bound at unity to membership functions of fuzzy objective functions. However, the corresponding lower bound to membership functions of fuzzy objective functions remained intact at zero. Here, the authors find this to be arbitrary and biased.

  • Most importantly, the classical fuzzy membership function could provide satisfying solution, only when the extreme ends of imprecise information lied within zero and one. However, Garai et al. (2016, 2017) showed that the proper mathematical representation of impreciseness cannot be confined within any closed and bounded interval of real line, at all times.

Figure 5 illustrates the drawbacks of membership function to classical fuzzy set (Garai et al. 2017).

Fig. 5
figure5

Drawbacks of the membership function to classical fuzzy set

Appendix C: T-set and related definitions

The characteristic function of crisp subset assigns either 1 or 0 to elements of universal set and thereby discriminates between the members and non-members. Again, membership function to fuzzy subset generalizes the characteristic function and the values assigned to elements of universal set fall within the specified range of closed unit interval [0, 1]. However, Garai et al. (2016, 2017) observed that the membership function failed to discriminate between yes and certainly yes by assigning same value ‘unity’ to both cases and failed to discriminate between no and certainly no by assigning same value ‘zero’ to both cases (Garai et al. 2016, 2017).

In other words, elements to universal set must be or be or partly be or not be or never be lying in a subset. The concept of being, partly being or not being is well measured by membership function of fuzzy subset. However, this cannot explain the cases of must being (since membership value is unity, at all times) as well as never being (since membership value is zero, at all times) for elements of fuzzy subset. Garai et al. (2017) presented the following example to illustrate all these as follows

  • The task is to pick up one tall policeman among all policemen for the guard of honour ceremony of the President of USA during the official visit to North Korea. To assign the job solely to some tall person in force is a bold step to North Korea. If one policeman with height of more than 6′ is considered to be tall, he can be assigned unity as the membership value in fuzzy subset of tall policemen.

Assume that Mr. Bansal is 6′4″ tall and Mr. Chowd is 6′6″ tall. In fuzzy set theory, both are given unity as the same membership value. However, if the DM were all machines with zero emotion, this would have been acceptable to choose any one of two persons (or in fact, any policeman taller than 6′). Nevertheless, as human beings, the mind plays key role. DM can have full happiness only after selecting the tallest person. Here, all the persons taller than 6′ are assigned the same membership value (unity). So, DM fail to find the most suitable person in force for the prestigious ceremony.

This marks that the upper bound of membership function of fuzzy subset at unity causes the dilemma.

Consequently, Garai et al. (2016, 2017) introduced the T-characteristic function and then T-set as follows

Definition 1

Let S be universal set and A is any subset of S. Then, T-characteristic function of A is denoted by \( T_{A} \) and is defined as \( T_{A} :S \to IR \). So, this assigns real number \( T_{A} \left( x \right) \) to each element \( x \in S \). Higher the value of \( T_{A} \left( x \right) \), larger the value of membership of \( x \in S \) in A. Here \( IR \) denotes set of real numbers.

Definition 2

Let S be universal set. Then, T-subset A of S is defined as the ordered pair \( A = \{ (x,T_{A} (x)):x \in S\} \), where T-characteristic function \( T_{A} :S \to IR \) assigns real number \( T_{A} \left( x \right) \) as membership value to each \( x \in S. \)

The universal set S is not necessarily an ordered set in imprecise environment. However, the function \( T_{A} :S \to IR \) imposes an ordering over elements of S based on the value of \( T_{A} \left( x \right) \)\( \forall x \in S \).

Definition 3

The union of two T-subsets A and B of S, which is denoted by \( A \cup B \), is defined as \( A \cup B = \{ (x,T_{A \cup B} (x)):x \in S\} \), where T-characteristic function \( T_{A \cup B} (x) \) of \( A \cup B \) is defined as \( T_{A \cup B} (x) = \hbox{max} \{ T_{A} (x),T_{B} (x)\} ,\forall x \in S \).

Definition 4

The intersection of two T-subsets A and B of S, which is denoted by \( A \cap B \), is defined as \( A \cap B = \{ (x,T_{A \cap B} (x)):x \in S\} \), where T-characteristic function \( T_{A \cap B} (x) \) of \( A \cap B \) is defined as \( T_{A \cap B} (x) = \hbox{min} \{ T_{A} (x),T_{B} (x)\} ,\forall x \in S \).

Definition 5

The complement of T-subset A of S, denoted by \( \bar{A}{\text{ or }}A^{c} \), is defined as \( \bar{A} = \{ (x,T_{{\bar{A}}} (x)):x \in S\} \), where T-characteristic function \( T_{A} \left( x \right) \) of \( \bar{A}{\text{ or }}A^{c} \) is defined as \( T_{{\bar{A}}} (x) = 1 - T_{A} (x),\forall x \in S. \)

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Garai, A., Roy, T.K. Multi-objective optimization of cost-effective and customer-centric closed-loop supply chain management model in T-environment. Soft Comput 24, 155–178 (2020). https://doi.org/10.1007/s00500-019-04289-5

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Keywords

  • Customer satisfaction
  • Environmental concerns
  • Closed-loop supply chain
  • T-set
  • Subsidy
  • Multi-objective optimization