Skip to main content

A sustainable production model for waste management with uncertain scrap and recycled material

Abstract

The studies highlightthat only one billion tons were obtained out of four billion tons of solid waste in the world and small value recovered out of it. In such a way, indeed, there is still a wide scope in collecting and converting the waste into value. In the industrial set-up because for various reasons, the production process is not optimum towards the conversion of resources into value. Hence, some defective items are also produced during production, which is a kind of waste for a firm. The firm manages such wastes by selling them into the secondary market at lower prices. To deal with the waste, firm also collects the used products from the customer under the circular economy concept. Later, these used collected products were recycled and used raw material for further production. The way to collect the used product from the market and to use them again as raw material is a remarkable effort to reduce waste. The objective of the research is to optimize the unit time profit and the expected resultant costs by considering the concept of recycling waste in the fuzzy environment for the imperfect production process. The sustainable production model is developed with the system cost with the fuzzy approach and validated the model with an illustration. The unit time profit has been maximized with the help of an analytical approach. A sensitivity analysis is also performed to check the stability of the model and the model is found to be quite stable. The evaluation of the fuzzy system cost may provide the decision maker’s information regarding the system’s behavior uncertainties.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Gupta S, Mohan K, Prasad R, Gupta S, Kansal A (1998) Solid waste management in India: options and opportunities. Resour Conserv Recycl 24(2):137–154

    Article  Google Scholar 

  2. McNicholas G, Cotton M (2019) Stakeholder perceptions of marine plastic waste management in the United Kingdom. Ecol Econ 163:77–87

    Article  Google Scholar 

  3. Wichai-utcha N, Chavalparit O (2019) 3Rs Policy and plastic waste management in Thailand. J Mater Cycles Waste Manage 21(1):10–22

    Article  Google Scholar 

  4. Garg P, Gupta A, Satya S (2006) Vermicomposting of different types of waste using Eisenia foetida: a comparative study. Biores Technol 97(3):391–395

    Article  Google Scholar 

  5. Tsai WT, Chou YH (2004) Government policies for encouraging industrial waste reuse and pollution prevention in Taiwan. J Clean Prod 12(7):725–736

    Article  Google Scholar 

  6. Sarkar B (2019) Mathematical and analytical approach for the management of defective items in a multi-stage production system. J Clean Prod 218:896–919

    Article  Google Scholar 

  7. Wakeyo DB, Singh AP, Avvari M (2019) Influence of abattoir waste management practices on natural environment conservation. Int J Syst Assur Eng Manag 10(5):1010–1022

    Article  Google Scholar 

  8. Teigiserova DA, Hamelin L, Thomsen M (2019) Review of high-value food waste and food residues biorefineries with focus on unavoidable wastes from processing. Resour Conserv Recycl 149:413–426

    Article  Google Scholar 

  9. Sauve G, Van Acker K (2020) The environmental impacts of municipal solid waste landfills in Europe: a life cycle assessment of proper reference cases to support decision making. J Environ Manage 261:110216

    Article  Google Scholar 

  10. Madon I, Drev D, Likar J (2019) Long-term risk assessments comparing environmental performance of different types of sanitary landfills. Waste Manage 96:96–107

    Article  Google Scholar 

  11. Wu Z, Gao G, Wang Y (2019) Effects of soil properties, heavy metals, and PBDEs on microbial community of e-waste contaminated soil. Ecotoxicol Environ Saf 180:705–714

    Article  Google Scholar 

  12. Lestari P, Trihadiningrum Y (2019) The impact of improper solid waste management to plastic pollution in Indonesian coast and marine environment. Mar Pollut Bull 149:110505

    Article  Google Scholar 

  13. Chen D, Ignatius J, Sun D, Zhan S, Zhou C, Marra M, Demirbag M (2019) Reverse logistics pricing strategy for a green supply chain: a view of customers’ environmental awareness. Int J Prod Econ 217:197–210

    Article  Google Scholar 

  14. Rahmasary AN, Robert S, Chang IS, Jing W, Park J, Bluemling B et al (2019) Overcoming the challenges of water, waste and climate change in Asian cities. Environ Manage 63(4):520–535

    Article  Google Scholar 

  15. Zhang A, Venkatesh VG, Liu Y, Wan M, Qu T, Huisingh D (2019) Barriers to smart waste management for a circular economy in China. J Clean Prod 240:118198

    Article  Google Scholar 

  16. Rabbani M, Heidari R, Yazdanparast R (2019) A stochastic multi-period industrial hazardous waste location-routing problem: integrating NSGA-II and Monte Carlo simulation. Eur J Oper Res 272(3):945–961

    MathSciNet  MATH  Article  Google Scholar 

  17. Kushwaha GS, Sharma NK (2016) Green initiatives: a step towards sustainable development and firm’s performance in the automobile industry. J Clean Prod 121:116–129

    Article  Google Scholar 

  18. Sindhu R, Gnansounou E, Rebello S, Binod P, Varjani S, Thakur IS, Pandey A (2019) Conversion of food and kitchen waste to value-added products. J Environ Manage 241:619–630

    Article  Google Scholar 

  19. Duru RU, Ikpeama EE, Ibekwe JA (2019) Challenges and prospects of plastic waste management in Nigeria. Waste Dispos Sustain Energy 1(2):117–126

    Article  Google Scholar 

  20. Banerjee, P., Hazra, A., Ghosh, P., Ganguly, A., Murmu, N. C., & Chatterjee, P. K. (2019). Solid waste management in India: a brief review. Waste management and resource efficiency, 1027–1049.

  21. Manna AK, Dey JK, Mondal SK (2019) Controlling GHG emission from industrial waste perusal of production inventory model with fuzzy pollution parameters. Int J Syst Sci Oper Logist 6(4):368–393

    Google Scholar 

  22. Valenzuela-Levi N (2019) Factors influencing municipal recycling in the Global South: the case of Chile. Resour Conserv Recycl 150:104441

    Article  Google Scholar 

  23. Thürer M, Pan YH, Qu T, Luo H, Li CD, Huang GQ (2019) Internet of Things (IoT) driven kanban system for reverse logistics: solid waste collection. J Intell Manuf 30(7):2621–2630

    Article  Google Scholar 

  24. Green KW, Inman RA, Sower VE, Zelbst PJ (2019) Impact of JIT, TQM and green supply chain practices on environmental sustainability. J Manuf Technol Manag 30:26–47

    Article  Google Scholar 

  25. Lu H, Sidortsov R (2019) Sorting out a problem: a co-production approach to household waste management in Shanghai, China. Waste Manage 95:271–277

    Article  Google Scholar 

  26. Namlis KG, Komilis D (2019) Influence of four socioeconomic indices and the impact of economic crisis on solid waste generation in Europe. Waste Manage 89:190–200

    Article  Google Scholar 

  27. Saeidi-Mobarakeh Z, Tavakkoli-Moghaddam R, Navabakhsh M, Amoozad-Khalili H (2020) A bi-level and robust optimization-based framework for a hazardous waste management problem: a real-world application. J Clean Prod 252:119830

    Article  Google Scholar 

  28. Ghasemzadeh Z, Sadeghieh A, Shishebori D (2021) A stochastic multi-objective closed-loop global supply chain concerning waste management: a case study of the tire industry. Environ Dev Sustain 23(4):5794–5821

    Article  Google Scholar 

  29. Chaturvedi S, Yadav BP, Siddiqui NA, Chaturvedi SK (2020) Mathematical modelling and analysis of plastic waste pollution and its impact on the ocean surface. J Ocean Eng Sci 5(2):136–163

    Article  Google Scholar 

  30. Garg CP, Kashav V (2020) Assessment of sustainable initiatives in the containerized freight railways of india using Fuzzy AHP Framework. Transp Res Procedia 48:522–539

    Article  Google Scholar 

  31. Sivashankari CK (2019) Purchasing inventory models for exponential demand with deteriorating items and discounted cost-in third order equation. Int J Procure Manag 12(3):321–335

    Google Scholar 

  32. Sarc R, Curtis A, Kandlbauer L, Khodier K, Lorber KE, Pomberger R (2019) Digitalisation and intelligent robotics in value chain of circular economy oriented waste management—a review. Waste Manage 95:476–492

    Article  Google Scholar 

  33. Zhao XG, Jiang GW, Li A, Wang L (2016) Economic analysis of waste-to-energy industry in China. Waste Manage 48:604–618

    Article  Google Scholar 

  34. Diaz LA, Lister TE, Parkman JA, Clark GG (2016) Comprehensive process for the recovery of value and critical materials from electronic waste. J Clean Prod 125:236–244

    Article  Google Scholar 

  35. Chen G, Wang X, Li J, Yan B, Wang Y, Wu X, Ma W (2019) Environmental, energy, and economic analysis of integrated treatment of municipal solid waste and sewage sludge: a case study in China. Sci Total Environ 647:1433–1443

    Article  Google Scholar 

  36. Taleizadeh AA, Moshtagh MS, Moon I (2018) Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: game theoretical approach. J Clean Prod 189:406–431

    Article  Google Scholar 

  37. Woon KS, Lo IM (2016) A proposed framework of food waste collection and recycling for renewable biogas fuel production in Hong Kong. Waste Manage 47:3–10

    Article  Google Scholar 

  38. Gradus RH, Nillesen PH, Dijkgraaf E, Van Koppen RJ (2017) A cost-effectiveness analysis for incineration or recycling of Dutch household plastic waste. Ecol Econ 135:22–28

    Article  Google Scholar 

  39. Bundhoo ZM (2018) Solid waste management in least developed countries: current status and challenges faced. J Mater Cycles Waste Manage 20(3):1867–1877

    Article  Google Scholar 

  40. Haywood LK, De Wet B, De Lange W, Oelofse S (2019) Legislative challenges hindering mine waste being reused and repurposed in South Africa. Extr Ind Soc 6(4):1079–1085

    Google Scholar 

  41. Gunarathne ADN, Tennakoon TPYC, Weragoda JR (2019) Challenges and opportunities for the recycling industry in developing countries: the case of Sri Lanka. J Mater Cycles Waste Manage 21(1):181–190

    Article  Google Scholar 

  42. Pivnenko K, Granby K, Eriksson E, Astrup TF (2017) Recycling of plastic waste: screening for brominated flame-retardants (BFRs). Waste Manage 69:101–109

    Article  Google Scholar 

  43. Ikram M, Sroufe R, Mohsin M, Solangi YA, Shah SZA, Shahzad F (2019) Does CSR influence firm performance? A longitudinal study of SME sectors of Pakistan. J Glob Responsib 11:27–53

    Article  Google Scholar 

  44. Fayezi S, Stekelorum R, El Baz J, Laguir I (2019) Paradoxes in supplier’s uptake of GSCM practices: institutional drivers and buyer dependency. J Manuf Technol Manag 31:479–500

    Article  Google Scholar 

  45. Chuang CH, Wang CX, Zhao Y (2014) Closed-loop supply chain models for a high-tech product under alternative reverse channel and collection cost structures. Int J Prod Econ 156:108–123

    Article  Google Scholar 

  46. Tanskanen P (2013) Management and recycling of electronic waste. Acta Mater 61(3):1001–1011

    Article  Google Scholar 

  47. Omar M, Yeo I (2009) A model for a production-repair system under a time varying demand process. Int J Prod Econ 119:17–23

    Article  Google Scholar 

  48. Merrild H, Damgaard A, Christensen TH (2008) Life cycle assessment of waste paper management: the importance of technology data and system boundaries in assessing recycling and incineration. Resour Conserv Recycl 52(12):1391–1398

    Article  Google Scholar 

  49. de Souza Zanuto R, Hassui A, Lima F, Dornfeld DA (2019) Environmental impacts-based milling process planning using a life cycle assessment tool. J Clean Prod 206:349–355

    Article  Google Scholar 

  50. Mishra N, Singh A (2018) Use of twitter data for waste minimization in beef supply chain. Ann Oper Res 270(1–2):337–359

    Article  Google Scholar 

  51. Sarkar B, Saren S (2016) Product inspection policy for an imperfect production system with inspection errors and warranty cost. Eur J Oper Res 248(1):263–271

    MATH  Article  Google Scholar 

  52. Khan SA, Kaviani MA, Galli BJ, Ishtiaq P (2019) Application of continuous improvement techniques to improve organization performance. Int J Lean Six Sigma 10:542–565

    Article  Google Scholar 

  53. Cheikhrouhou N, Sarkar B, Ganguly B, Malik AI, Batista R, Lee YH (2018) Optimization of sample size and order size in an inventory model with quality inspection and return of defective items. Ann Oper Res 271(2):445–467

    MathSciNet  MATH  Article  Google Scholar 

  54. Vannucci M, Colla V (2019) Quality improvement through the preventive detection of potentially defective products in the automotive industry by means of advanced artificial intelligence techniques. Intelligent Decision Technologies 2019. Springer, Singapore, pp 3–12

    Chapter  Google Scholar 

  55. Borenich A, Dickbauer Y, Reimann M, Souza GC (2020) Should a manufacturer sell refurbished returns on the secondary market to incentivize retailers to reduce consumer returns? Eur J Oper Res 282(2):569–579

    MathSciNet  MATH  Article  Google Scholar 

  56. Tighazoui A, Turki S, Sauvey C, Sauer N (2019) Optimal design of a manufacturing-remanufacturing-transport system within a reverse logistics chain. Int J Adv Manuf Technol 101(5–8):1773–1791

    Article  Google Scholar 

  57. Maloney MM, Grimm SD, Anctil R (2020) Atlas international business case: examining globalization and economic indicators for the scrap metal recycling industry. J Account Educ 51:100661

    Article  Google Scholar 

  58. Almazán-Casali S, Alfaro JF, Sikra S (2019) Exploring household willingness to participate in solid waste collection services in Liberia. Habitat Int 84:57–64

    Article  Google Scholar 

  59. Mahpour A (2018) Prioritizing barriers to adopt circular economy in construction and demolition waste management. Resour Conserv Recycl 134:216–227

    Article  Google Scholar 

  60. Noshadravan A, Gaustad G, Kirchain R, Olivetti E (2017) Operational strategies for increasing secondary materials in metals production under uncertainty. J Sustain Metall 3(2):350–361

    Article  Google Scholar 

  61. Jimoda LA, Sulaymon ID, Alade AO, Adebayo GA (2018) Assessment of environmental impact of open burning of scrap tyres on ambient air quality. Int J Environ Sci Technol 15(6):1323–1330

    Article  Google Scholar 

  62. Oyola-Cervantes J, Amaya-Mier R (2019) Reverse logistics network design for large off-the-road scrap tires from mining sites with a single shredding resource scheduling application. Waste Manage 100:219–229

    Article  Google Scholar 

  63. Chaurasia B, Garg D, Agarwal A (2019) Lean six sigma approach: a strategy to enhance performance of first through time and scrap reduction in an automotive industry. Int J Bus Excell 17(1):42–57

    Article  Google Scholar 

  64. Jena SK, Sarmah SP (2016) Price and service co-opetiton under uncertain demand and condition of used items in a remanufacturing system. Int J Prod Econ 173:1–21

    Article  Google Scholar 

  65. Mohan TK, Amit RK (2021) Modeling oligopsony market for end-of-life vehicle recycling. Sustain Prod Consum 25:325–346

  66. Ekvall T (2000) A market-based approach to allocation at open-loop recycling. Resour Conserv Recycl 29(1–2):91–109

    Article  Google Scholar 

  67. Gou Q, Liang L, Huang Z, Xu C (2008) A joint inventory model for an open-loop reverse supply chain. Int J Prod Econ 116(1):28–42

    Article  Google Scholar 

  68. Omar M, Yeo I (2009) A model for a production–repair system under a time-varying demand process. Int J Prod Econ 119(1):17–23

    Article  Google Scholar 

  69. Tseng ML, Tan RR, Siriban-Manalang AB (2013) Sustainable consumption and production for Asia: sustainability through green design and practice. J Clean Prod 40:1–5

    Article  Google Scholar 

  70. Ghare PM, Schrader GH (1963) A model for an exponentially decaying inventory. J Ind Eng 14(1):238–243

    Google Scholar 

  71. Mandal BA, Phaujdar S (1989) An inventory model for deteriorating items and stock-dependent consumption rate. J Oper Res Soc 40(5):483–488

    MATH  Article  Google Scholar 

  72. Tayal S, Singh SR, Sharma R, Chauhan A (2014) Two echelon supply chain model for deteriorating items with effective investment in preservation technology. Int J Math Oper Res 6(1):84–105

    MathSciNet  MATH  Article  Google Scholar 

  73. Chauhan A, Singh AP (2014) Optimal replenishment and ordering policy for time dependent demand and deterioration with discounted cash flow analysis. Int J Math Oper Res 6(4):407–436

    MathSciNet  MATH  Article  Google Scholar 

  74. Tayal S, Singh S, Sharma R (2015) An inventory model for deteriorating items with seasonal products and an option of an alternative market. Uncertain Supply Chain Manag 3(1):69–86

    Article  Google Scholar 

  75. Chauhan A, Singh AP (2015) A note on the inventory models for deteriorating items with Verhulst’s model type demand rate. Int J Oper Res 22(2):243–261

    MathSciNet  MATH  Article  Google Scholar 

  76. Tayal S, Singh SR, Sharma R (2016) An integrated production inventory model for perishable products with trade credit period and investment in preservation technology. Int J Math Oper Res 8(2):137–163

    MathSciNet  MATH  Article  Google Scholar 

  77. Singh SR, Rastogi M, Tayal S (2016) An inventory model for deteriorating items having seasonal and stock-dependent demand with allowable shortages. Proceedings of fifth international conference on soft computing for problem solving. Springer, Singapore, pp 501–513

    Chapter  Google Scholar 

  78. Rastogi M, Singh S, Kushwah P, Tayal S (2017) Two warehouse inventory policy with price dependent demand and deterioration under partial backlogging. Decis Sci Lett 6(1):11–22

    Article  Google Scholar 

  79. Tayal S, Singh SR, Attri AK (2019) Two levels of storage model for deteriorating items, stock dependent demand and partial backlogging with both rented warehouses. Int J Process Manag Benchmarking 9(4):485–498

    Article  Google Scholar 

  80. Rajput N, Pandey RK, Singh AP, Chauhan A (2019) An optimization of fuzzy EOQ model in healthcare industries with three different demand pattern using signed distance technique. Journal MESA 10(2):205–218

    Google Scholar 

  81. Sarkar B, Tayyab M, Kim N, Habib MS (2019) Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through metaheuristic approach. Comput Ind Eng 135:987–1003

    Article  Google Scholar 

  82. Nascimento DLM, Alencastro V, Quelhas OLG, Caiado RGG, Garza-Reyes JA, Rocha-Lona L, Tortorella G (2019) Exploring Industry 4.0 technologies to enable circular economy practices in a manufacturing context: a business model proposal. J Manuf Technol Manag 30(3):607–627

  83. Aydemir-Karadag A (2018) A profit-oriented mathematical model for hazardous waste locating-routing problem. J Clean Prod 202:213–225

    Article  Google Scholar 

  84. Guide VDR Jr, Van Wassenhove LN (2001) Managing product returns for remanufacturing. Prod Oper Manag 10(2):142–155

    Article  Google Scholar 

  85. Luthra S, Govindan K, Kannan D, Mangla SK, Garg CP (2017) An integrated framework for sustainable supplier selection and evaluation in supply chains. J Clean Prod 140:1686–1698

    Article  Google Scholar 

  86. Luthra S, Mangla SK, Shankar R, Prakash Garg C, Jakhar S (2018) Modelling critical success factors for sustainability initiatives in supply chains in Indian context using Grey-DEMATEL. Prod Plan Control 29(9):705–728

    Article  Google Scholar 

  87. Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626

    MathSciNet  MATH  Article  Google Scholar 

  88. Kamble AJ (2017) Some notes on pentagonal fuzzy numbers. Int J Fuzzy Math Arch 13(2):113–121

    Google Scholar 

  89. Mondal SP, Mandal M (2017) Pentagonal fuzzy number, its properties and application in fuzzy equation. Future Comput Inform J 2(2):110–117

    Article  Google Scholar 

  90. Y Zhang, J Zhao (2011). Modeling and solution of the hazardous waste location-routing problem under uncertain conditions. In: ICTE 2011 (pp. 2922–2927).

  91. Hsieh CH, Chen SH (1999) Similarity of generalized fuzzy numbers with graded mean integration representation. In: Proceedings of the 8th international fuzzy systems association world congress, vol 2, pp 551–555. Taipei, Taiwan

  92. Kumar D, Garg CP (2017) Evaluating sustainable supply chain indicators using fuzzy AHP: Case of Indian automotive industry. Benchmarking Int J 24(6). https://doi.org/10.1108/BIJ-11-2015-0111

  93. Garg CP, Sharma A (2020) Sustainable outsourcing partner selection and evaluation using an integrated BWM–VIKOR framework. Environ Dev Sustain 22(2):1529–1557

    Article  Google Scholar 

  94. Pereira T, Neves ASL, Silva FJG, Godina R, Morgado L, Pinto GFL (2020) Production process analysis and improvement of corrugated cardboard industry. Procedia Manuf 51(2020):1395–1402

    Article  Google Scholar 

  95. Pan X, Xie Q, Feng Y (2020) Designing recycling networks for construction and demolition waste based on reserve logistics research field. J Clean Prod 260:120841

    Article  Google Scholar 

  96. Reddy KN, Kumar A (2021) Capacity investment and inventory planning for a hybrid manufacturing–remanufacturing system in the circular economy. Int J Prod Res 59(8):2450–2478

    Article  Google Scholar 

  97. Garg CP (2021) Modeling the e-waste mitigation strategies using Grey-theory and DEMATEL framework. J Clean Prod 281:124035

    Article  Google Scholar 

  98. Yazdani M, Kabirifar K, Frimpong BE, Shariati M, Mirmozaffari M, Boskabadi A (2021) Improving construction and demolition waste collection service in an urban area using a simheuristic approach: a case study in Sydney Australia. J Clean Prod 280:124138

    Article  Google Scholar 

  99. Kazancoglu Y, Ozkan-Ozen YD, Mangla SK, Ram M (2020) Risk assessment for sustainability in e-waste recycling in circular economy. Clean Technol Environ Policy 24:1145–1157

  100. Liu L, Liao W (2021) Optimization and profit distribution in a two-echelon collaborative waste collection routing problem from economic and environmental perspective. Waste Manage 120:400–414

    Article  Google Scholar 

  101. Wang Z, Wang Y, Liu Z, Cheng J, Chen X (2021) Strategic management of product recovery and its environmental impact. Int J Prod Res 59(20):6104–6124

  102. Huang L, Zhen L, Yin L (2020) Waste material recycling and exchanging decisions for industrial symbiosis network optimization. J Clean Prod 276:124073

    Article  Google Scholar 

Download references

Funding

The present work is not funded from any institute.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nagendra Kumar Sharma.

Ethics declarations

Conflict of interest

There is no potential conflict of interest among the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A

Appendix A

Unit time Profit F is convex with respect to the decision variables t3 and t4.

We have unit time profit in fuzzy.

For Case1:

\(\begin{aligned} F(t_{3} ,t{}_{4}) & = \frac{1}{{t_{4} }}\left[\tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)\left\{ {\left( {1 - \frac{{\tilde{y}}}{100}} \right)c_{1} + \frac{{\tilde{y}}}{100}c_{2} } \right\} + c_{s} \tilde{P}t_{2} \frac{{\tilde{x}}}{100} \right. \\ & \quad - (\mu \,\,\tilde{P}t_{2} c_{r} + \tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)\eta \delta (c_{R} - c_{r} )) - \tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)c_{3} - \tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)\eta c_{a} \, \\ & \quad - S_{m} - \left( {h_{m} \frac{{(\tilde{P} - \tilde{D})}}{2}(t_{2} - t_{1} )^{2} + h_{m} \frac{{\tilde{D}}}{2}(t_{3} - t_{2} )^{2} } \right) - T_{p} - \tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)\eta c_{R} - \frac{{h_{r} }}{2}\tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{100}} \right)\eta t_{4} \\ & \quad \left. - c_{s} \tilde{D}(t_{4} - t_{3} ) - (1 - \theta )\tilde{D}(t_{4} - t_{3} )c_{L} \right] \\ \end{aligned}\) and.

For Case 2:

$$\begin{gathered} F(t_{3} ,t{}_{4}) = \frac{1}{{t_{4} }}\left[ {\tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{{100}}} \right)\left\{ {\left( {1 - \frac{{\tilde{y}}}{{100}}} \right)c_{1} + \frac{{\tilde{y}}}{{100}}c_{2} } \right\}} \right. + c_{s} \tilde{P}t_{2} \frac{{\tilde{x}}}{{100}} \hfill \\ - \mu \tilde{P}t_{2} c_{r} - \tilde{P}t_{2} c_{m} \, - \tilde{P}t_{2} \left( {1 - \frac{{\tilde{x}}}{{100}}} \right)c_{3} \hfill \\ \left. { - S_{m} - \left( {h_{m} \frac{{(\tilde{P} - \tilde{D})}}{2}(t_{2} - t_{1} )^{2} + h_{m} \frac{{\tilde{D}}}{2}(t_{3} - t_{2} )^{2} } \right) - c_{s} \frac{{\tilde{D}}}{2}(t_{4} - t_{3} )^{2} - (1 - \theta )\tilde{D}(t_{4} - t_{3} )^{2} c_{L} } \right] \hfill \\ \end{gathered}$$
$$\begin{aligned} F\left(t_{3} ,t{}_{4}\right) & = \frac{1}{{t_{4} }}\left[\left(p_{1} ,p_{2} ,p_{3} ,p_{4} \right)t_{2} \left(1 - \frac{{\left(x_{1} ,x_{2} ,x_{3} ,x_{4} \right)}}{100}\right)\left\{ \left(1 - \frac{{\left(y_{1} ,y_{2} ,y_{3} ,y_{4} \right)}}{100}\right)c_{1} + \frac{{\tilde{y}}}{100}c_{2}\right \} + c_{s} \tilde{P}t_{2} \frac{{\tilde{x}}}{100} \right. \hfill \\ & \quad\mu \tilde{P}t_{2} c_{r} - \tilde{P}t_{2} c_{m} - \tilde{P}t_{2} \left(1 - \frac{{\tilde{x}}}{100}\right)c_{3} - S_{m} - \left(h_{m} \frac{{\left(\tilde{P} - \tilde{D}\right)}}{2}\left(t_{2} - t_{1} \right)^{2} + h_{m} \frac{{\tilde{D}}}{2}\left(t_{3} - t_{2} \right)^{2} \right) \hfill \\ & \quad \left.- c_{s} \frac{{\tilde{D}}}{2}\left(t_{4} - t_{3} \right)^{2} - \left(1 - \theta \right)\tilde{D}\left(t_{4} - t_{3} \right)^{2} c_{L} \right]. \hfill \\ \end{aligned}$$

Here, due to uncertainty \(\tilde{P} = (p_{1} ,p_{2} ,p_{3} ,p_{4} ,p_{5} ),\,\tilde{D} = (d_{1} ,d_{2} ,d_{3} ,d_{4} ,d_{5} ),\,\tilde{x} = (x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} )\,{\text{and}}\,\tilde{y} = (y_{1} ,y_{2} ,y_{3} ,y_{4} ,y_{5} )\) consider as fuzzy number.

Applying the fuzzy operations and GMI technique, defined in The Research Gap Highlights.

$$G(\tilde{F}) = \frac{{\tfrac{1}{2}\int\limits_{0}^{1} {\alpha [F_{L} (\alpha ) + F_{R} (\alpha )]d\alpha } }}{{\int\limits_{0}^{1} {\alpha d\alpha } }}$$

and subsequent the optimal criteria condition defined in Case2: When used items are not collected and all the raw material is purchased fromoutside (open market)

$$\frac{{\partial^{2} F\left( {t_{3} ,t_{4} } \right)}}{{\partial t_{3}^{2} }}\frac{{\partial^{2} F\left( {t_{3} ,t_{4} } \right)}}{{\partial t_{4}^{2} }} - \left( {\frac{{\partial F\left( {t_{3} ,t_{4} } \right)}}{{\partial t_{4} \partial t_{3} }}} \right)^{2} > 0$$

and use of software wolfram Mathematica 11.3,

The condition holds for \(D,P,T_{p} ,x,y,\delta ,\eta ,\theta ,\mu ,c_{1} ,c_{2} ,c_{3} ,c_{sh} , c_{L} ,c_{m} , c_{r} , c_{R} ,h_{m} ,h_{r} \in\).\(Positive Real\mathrm{s}\), and if volume oproduction is more than of demand \(P > D > 0 \,{\text{and}}\left( {D^{2} - DP} \right)\left( {c_{sh} + \left( {1 - \theta } \right)c_{L} } \right)^{2} + 2P\left( {T + c_{6} } \right)h_{m} > 0\) for both cases.

Hence, we achieved the optimal value for decision variable t3 and t4, and unit time profit F.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chauhan, A., Sharma, N.K., Tayal, S. et al. A sustainable production model for waste management with uncertain scrap and recycled material. J Mater Cycles Waste Manag 24, 1797–1817 (2022). https://doi.org/10.1007/s10163-022-01435-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10163-022-01435-4

Keywords

  • Sustainable production
  • Recycling
  • Waste management
  • Fuzzy control
  • Optimization technique