Optimizing a Fuzzy Multi-Item Inventory System and Ordering Cost Depletion Contingent on Lead Time With Carbon Emission Cost


  • R. Vithyadevi Mother Teresa Women’s University, Kodaikanal, Tamilnadu 624101, India; Department of Mathematics, SSM Institute of Engineering and Technology, Dindigul, Tamilnadu 624002, India https://orcid.org/0000-0001-6329-4972
  • K. Annadurai PG and Research Department of Mathematics, M.V. Muthiah Government Arts College for Women, Dindigul, Tamilnadu 624001, India https://orcid.org/0000-0002-4315-6967




Fuzzy multi-item inventory model, Graded mean integration technique, Minimum integrated total cost for multi-item, Optimal order quantity for each item, Kuhn-Tucker method


Multi-item integrated inventory system and ordering cost depletion liable scheduled lead time with carbon emission cost is established in a fuzzy situation. Multiple items can considerably drop total inventory costs for hiring orders aimed at multiple items in single refill demand would drop ordering costs. Owing to the inaccuracy of various parameters and objective is imprecise in the environment. As the development of fuzzy objective is uncertain, the model is formulated as multi-item problems were confident/suspicious profit of the objective with some uncertainty. The model is solved via the graded mean technique with the addition of the Kuhn-Tucker method when the fuzzy equivalent of the problem remains available. An algorithm is established to attain optimal order quantity for each item and then find the minimum integrated total cost for a multi-item inventory system. The evaluation of a fuzzy multi-item inventory system through the crisp multi-item inventory system is completed over mathematical illustrations. Lastly, the graphical demonstration remains offered toward establishing the suggested system. An ending outcome demonstrates that this fuzzy multi-item system is perhaps moderately suitable defining optimal order quantity for each item and then the minimum integrated total cost for the multi-item technique when the lead time is assessed.


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S. S. Ali, H. Barman, R. Kaur, H. Tomaskova and S. K. Roy, Multi-product multi echelon measurements of perishable supply chain: Fuzzy non-linear programming approach, Mathematics 9 (2021), 2093, DOI: 10.3390/math9172093.

L. E. Cárdenas-Barrón, G. Treviño-Garza and H. M. Wee, A simple and better algorithm to solve the vendor managed inventory control system of multi-product multi-constraint economic order quantity model, Expert Systems with Applications 39(3) (2012), 3888 – 3895, DOI: 10.1016/j.eswa.2011.09.057.

S.-H. Chen, Operations on fuzzy numbers with function principle, Tamkang Journal of Management Sciences 6(1) (1985), 13 – 26.

S. K. Das, A fuzzy multi objective inventory model of demand dependent deterioration including lead time, Journal of Fuzzy Extension and Applications 3(1) (2022), 1 – 18, DOI: 10.22105/jfea.2021.306498.1163.

A. Federgruen, H. Groenevelt and H. C. Tijms, Coordinated replenishments in a multi-item inventory system with compound poisson demands, Management Science 30(3) (1984), 263 – 394, DOI: 10.1287/mnsc.30.3.344.

T. Joviani, D. Lesmono and T. Limansyah, A multi-item inventory model with various demand functions considering deterioration and partial backlogging, BAREKENG: Journal of Mathematics and Its Application 17(2) (2023), 1069 – 1080, DOI: 10.30598/barekengvol17iss2pp1069-1080.

A. J. Kamble, Some notes on pentagonal fuzzy numbers, International Journal of Fuzzy Mathematical Archive 13(2) (2017), 113 – 121, DOI: 10.22457/ijfma.v13n2a2.

M. G. Kumar and R. Uthayakumar, Multi-item inventory model with variable backorder and price discount under trade credit policy in stochastic demand, International Journal of Production Research 57(1) (2019), 298 – 320, DOI: 10.1080/00207543.2018.1480839.

Y. Li, X. Xu, X. Zhao, J. H. Y. Yeung and F. Ye, Supply chain coordination with controllable lead time and asymmetric information, European Journal of Operational Research 217(1) (2012), 108 – 119, DOI: 10.1016/j.ejor.2011.09.003.

N. Maheswari, K. R. Balasubramanian and M. Parimaladevi, A cost analysis on multi-item inventory model for factory outlets with investment constraint under ranking asteroid fuzzy set, International Journal of Students’ Research in Technology & Management 10(3) (2022), 12 – 20.

A. K. Maiti, Multi-item fuzzy inventory model for deteriorating items in multi-outlet under single management, Journal of Management Analytics 7(1) (2020), 44 – 68, DOI: 10.1080/23270012.2019.1699873.

A. I. Malik and B. Sarkar, Optimizing a multi-product continuous-review inventory model with uncertain demand, quality improvement, setup cost reduction, and variation control in lead time, IEEE Access 6 (2018), 36176 – 36187, DOI: 10.1109/ACCESS.2018.2849694.

B. Malleeswaran and R. Uthayakumar, An integrated vendor–buyer supply chain model for backorder price discount and price-dependent demand using service level constraints and carbon emission cost, International Journal of Systems Science: Operations & Logistics 9(1) (2020), 111 – 120, DOI: 10.1080/23302674.2020.1833258.

I. Moon and E. A. Silver, The multi-item newsvendor problem with a budget constraint and fixed ordering costs, Journal of the Operational Research Society 51(5) (2000), 602 – 608, DOI: 10.2307/254191.

H. Nagar and P. Surana, Fuzzy inventory model for deteriorating items with fluctuating demand and using inventory parameters as pentagonal fuzzy numbers, Journal of Computer and Mathematical Sciences 6(2) (2015), 55 – 66.

S. H. Nasseri, N. Taghi-Nezhad and A. Ebrahimnejad, A note on ranking fuzzy numbers with an area method using circumcenter of centroids, Fuzzy Information and Engineering 9(2) (2017), 259 – 268, DOI: 10.1016/j.fiae.2017.06.009.

J. C.-H. Pan and J.-S. Yang, A study of an integrated inventory with controllable lead time, International Journal of Production Research 40(5) (2002), 1263 – 1273, DOI: 10.1080/00207540110105680.

L. A. San-José, M. González-De-la-Rosa, J. Sicilia and J. Febles-Acosta, An inventory model for multiple items assuming time-varying demands and limited storage, Optimization Letters 16 (2022), 1935 – 1961, DOI: 10.1007/s11590-021-01815-z.

D. Shin, R. Guchhait, B. Sarkar and M. Mittal, Controllable lead time, service level constraint, and transportation discounts in a continuous review inventory model, RAIRO-Operations Research 50(4-5) (2016), 921 – 934, DOI: 10.1051/ro/2015055.

H. A. Taha, Operations Research: An Introduction, 10th edition, Pearson, Harlow, England, 843 pages (2017).

S. Tiwari, Y. Daryanto and H. M. Wee, Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission, Journal of Cleaner Production 192 (2018), 281 – 292, DOI: 10.1016/j.jclepro.2018.04.261.

R. Vithyadevi and K. Annadurai, Optimization of fuzzy integrated inventory model with ordering cost reduction dependent on lead time, International Journal of Operations Research 18(4) (2021), 79 – 96, URL: http://www.orstw.org.tw/ijor/vol18no4/IJOR2021_vol18_no4_p79_p96.pdf.

R. Vithyadevi and K. Annadurai, Optimization of fuzzy two-level production inventory system with persuasive exertion-reliant demand, Ratio Mathematica 48 (2023), 189 – 212, DOI: 10.23755/rm.v48i0.1152.

J.-S. Yang and J. C.-H. Pan, Just-in-time purchasing: an integrated inventory model involving deterministic variable lead time and quality improvement investment, International Journal Production Research 42(5) (2004), 853 – 863, DOI: 10.1080/00207540310001632448.

H.-J. Zimmerman, Using fuzzy sets in operational research, European Journal of Operational Research 13(3) (1983), 201 – 216, DOI: 10.1016/0377-2217(83)90048-6.




How to Cite

Vithyadevi, R., & Annadurai, K. (2024). Optimizing a Fuzzy Multi-Item Inventory System and Ordering Cost Depletion Contingent on Lead Time With Carbon Emission Cost. Communications in Mathematics and Applications, 14(5), 1693–1725. https://doi.org/10.26713/cma.v14i5.2272



Research Article