Abstract
The modern global economy is becoming more challenging and it is hardly possible to minimize the inventory cost for inventory practitioners in the coming days. Basically, most of the enterprises deal with deteriorating items having flexible demand rate and follow natural idle time in the entire inventory process. Moreover, traditionally most of the research articles have been made under non-stop time frame, but in reality, in a day–night scenario there exists a natural idle time and hence the time consumed for inventory run time may be viewed as single shift or periodic model. Here we formulate an economic order quantity (EOQ) inventory model considering natural idle time and deterioration under some constraints and minimize the average inventory cost. Then, the model is converted into an equivalent fuzzy model, taking the demand and all the cost parameters as linguistic polynomial fuzzy set (LPFS). To defuzzify the model, we have adopted indexing method as well as \(\alpha \)-cut method. To validate the novelty, numerical experimentations have also been analyzed with the help of metaheuristic and evolutionary algorithms like goat search algorithm (GSA) and particle swarm optimization (PSO). Comparative analysis reveals that GSA approach can give finer optimum (− 10 % cost reduction) than other approaches. The main findings of this research give a new technique of (linguistic term) fuzzification–defuzzification of the proposed model and a new solution procedure to optimize the periodic deteriorating inventory model under GSA. To justify this model, sensitivity analysis and graphical illustration have been done. Scopes of future work have been discussed for further improvement of research on optimization problems using metaheuristic algorithms.
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References
Rastogi M, Singh S, Kushwah P and Tayal S 2017 An EOQ model with variable holding cost and partial backlogging under credit limit policy and cash discount. Uncertain Supply Chain Manag. 5(1): 27–42
De S K 2013 EOQ model with natural idle time and wrongly measured demand rate. Int. J. Inv. Control Manag. 3(1–2): 329–354
De S K and Pal M 2016 An intelligent decision for a bi-objective inventory problem. Int. J. Syst. Sci. Oper. Logist. 3(1): 49–62
Mahata P and Mahata G C 2020 Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments. Math. Comput. Modell. Dyn. Syst. 26(4): 374–408
Sen N, Bardhan S and Giri B C 2024 Consignment based integrated inventory model for deteriorating goods with price-and green-sensitive demand. Sadhana 49(1): 1–7. https://doi.org/10.1007/s12046-023-02328-4
Pourahmadi B, Ebrahimnejad S and Rahbari M 2023 Scenario-based robust optimization for online retail orders considering supply chain costs and level of service: a real case study. Sadhana 48(4): 240. https://doi.org/10.1007/s12046-023-02282-1
Khalilpourazari S, Pasandideh S H R and Ghodratnama A 2019 Robust possibilistic programming for multi-item EOQ model with defective supply batches: Whale Optimization and Water Cycle Algorithms. Neural Comput. Appl. 31: 6587–6614. https://doi.org/10.1007/s00521-018-3492-3
Jeshvaghani M D, Amiri M, Khalili-Damghani K and Olfat L 2023 A robust possibilistic multi-echelon multi-product multi-period production-inventory-routing problem considering internal operations of cross-docks: Case study of FMCG supply chain. Comput. Ind. Eng. 179: 109206. https://doi.org/10.1016/j.cie.2023.109206
Wang F F 2023 An efficient optimization procedure for location-inventory problems with (S-1, S) policy and retrial demands. Math. Comput. Simul. 206: 664–688. https://doi.org/10.1016/j.matcom.2022.12.010
Zadeh L A 1965 Fuzzy sets. Inf. Control 8(3): 338–353. https://doi.org/10.1016/S0019-9958(65)90241-x
Yager R R 1981 A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24(2): 143–161. https://doi.org/10.1016/0020-0255(81)90017-7
Bellman R E and Zadeh L A 1970 Decision-making in a fuzzy environment. Manage.Sci. 17(4): B-141-B-164 DOI: https://doi.org/10.1287/mnsc.17.4.B141
Zimmermann H J 1978 Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1(1): 45–55. https://doi.org/10.1016/0165-0114(78)90031-3
Amirfakhrian M 2012 Analyzing the solution of a system of fuzzy linear equations by a fuzzy distance. Soft Compt. 16(6): 1035–1041
Tayyebi J and Hosseinzadeh E 2020 Polynomial form fuzzy numbers and their application in linear programming with fuzzy variables. Ital. J. Pure Appl. Math. 44: 576–588
Nasseri S H, Ebrahimnejad E and Mizuno S 2010 Duality in fuzzy linear programming with symmetric trapezoidal numbers. Appl. Appl. Math. 5(2): 370–385
De S K, Goswami A and Sana S S 2014 An interpolating by pass to Pareto optimality in intuitionistic fuzzy technique for a EOQ model with time sensitive backlogging. Appl. Math. Compt. 230: 664–674. https://doi.org/10.1016/j.amc.2013.12.137
De S K and Sana S S 2013 Fuzzy order quantity inventory model with fuzzy shortage quantity and fuzzy promotional index. Econ. Model. 31: 351–358. https://doi.org/10.1016/j.econmod.2012.11.046
De S and Goswami A 2001 A replenishment policy for items with finite production rate and fuzzy deterioration rate. OPSEARCH. 38: 419–430. https://doi.org/10.1007/BF03398647
Karmakar S, De S K and Goswami A 2017 A deteriorating EOQ model for natural idle time and imprecise demand: Hesitant fuzzy approach. Int. J. Syst. Sci. Oper. Logist 4(4): 297–310
Chan F T and Prakash A 2012 Maintenance policy selection in manufacturing firms using the fuzzy MCDM approach. Int. J. Prod. Res. 50(23): 7044–7056. https://doi.org/10.1080/00207543.2011.653451
De S K and Nandi S 2023 The exact defuzzification method under polynomial approximation of various fuzzy sets. Yugosl. J. Oper. Res. 34(1): 51–72. https://doi.org/10.2298/YJOR2306017D
De S K and Mahata G C 2019 An EPQ model for three-layer supply chain with partial backordering and disruption: Triangular dense fuzzy lock set approach. Sadhana 44(8): 177
Mahata G C, De S K, Bhattacharya K and Maity S 2023 Three-echelon supply chain model in an imperfect production system with inspection error, learning effect, and return policy under fuzzy environment. Int. J. Syst. Sci. Oper. Logist. 10(1): 1962427
De S K, Mahata G C and Maity S 2021 Carbon emission sensitive deteriorating inventory model with trade credit under volumetric fuzzy system. Int. J. Intell. Syst. 36(12): 7563–7590
De S K and Mahata G C 2020 A production inventory supply chain model with partial backordering and disruption under triangular linguistic dense fuzzy lock set approach. Soft Comput. 24(7): 5053–5069
De S K and Mahata G C 2021 A profit jump inventory model for imperfect quality items with receiving reparative batch and order overlapping in dense fuzzy environment. RAIRO-Oper. Res. 55(2): 723–744
De S K and Mahata G C 2021 Solution of an imperfect-quality EOQ model with backorder under fuzzy lock leadership game approach. Int. J. Intell. Syst. 36(1): 421–446
De S K and Mahata G C 2019 A comprehensive study of an economic order quantity model under fuzzy monsoon demand. Sadhana 44(4): 89. https://doi.org/10.1007/s12046-019-1059-3
De S K and Mahata G C 2017 Decision of a fuzzy inventory with fuzzy backorder model under cloudy fuzzy demand rate. Int. J. Appl. Comput. 3: 2593–2609
De S K and Mahata G C 2019 A cloudy fuzzy economic order quantity model for imperfect-quality items with allowable proportionate discounts. J. Ind. Eng. Int. 15(4): 571–583
Karmakar S and De S K 2023 A supply and demand economic order quantity inventory model under Pythagorean fuzzy environment. Sadhana 48(1): 21. https://doi.org/10.1007/s12046-022-02046-3
Singh V P, Sharma K, Singh B, Ebrahimnejad A and Chakraborty D 2023 Fermatean fuzzy vehicle routing problem with profit: solution algorithms, comparisons and developments. Sadhana 48(3): 166. https://doi.org/10.1007/s12046-023-02238-5
Kennedy J and Eberhart R 1995 Particle swarm optimization. IEEE Trans. Neural Networks IEEE In Proceedings of ICNN'95-Int. Conf. on neural networks, Perth, WA, Australia, 4, 1942-1948 IEEE. DOI: https://doi.org/10.1109/ICNN.1995.488968
Shi Y and Eberhart R 1998 A modified particle swarm optimizer. In 1998 IEEE international conference on evolutionary computation proceedings. IEEE world congress on computational intelligence (Cat. No. 98TH8360) (pp. 69-73). IEEE. DOI: https://doi.org/10.1109/ICEC.1998.699146
Gomes H M 2011 Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst. Appl. 38(1): 957–968. https://doi.org/10.1016/j.eswa.2010.07.086
Jain M, Saihjpal V, Singh N and Singh S B 2022 An overview of variants and advancements of PSO algorithm. Appl. Sci. 12(17): 8392. https://doi.org/10.3390/app12178392
Riadi S, Triono S, Syahril S, Nofriansyah D 2019 Effectiveness of Meta-cognitive Learning's Model in Engineering. Int. J. Eng. Advnce. Tech. 9(1): 4438-4443. https://doi.org/10.35940/ijeat.A1457.109119
Jin Y, Wang H and Sun C 2021 Data-driven evolutionary optimization. Stud. Comput. Intell. 23(3): 442–458. https://doi.org/10.1007/978-3-030-74640-7
De S K 2023 The goat search algorithms. Artif. Intell. Rev. 56(8): 8265–8301. https://doi.org/10.1007/s10462-022-10341-y
Liu Z and Nishi T 2023 Data-driven evolutionary computation for service constrained inventory optimization in multi-echelon supply chains. Syst, Complex Intell. https://doi.org/10.1007/s40747-023-01179-0
Perez H D, Hubbs C D, Li C and Grossmann I E 2021 Algorithmic approaches to inventory management optimization. Processes 9(1): 102. https://doi.org/10.3390/pr9010102
Sadeghi A H, Bani E A, Fallahi A and Handfield R 2023 Grey wolf optimizer and whale optimization algorithm for stochastic inventory management of reusable products in a two-level supply chain. IEEE Access 11: 40278–40297. https://doi.org/10.1109/ACCESS.2023.3269292
Dewi S K and Utama D M 2021 A new hybrid whale optimization algorithm for green vehicle routing problem. Syst. Sci. Cont. Eng. 9(1): 61–72. https://doi.org/10.1080/21642583.2020.1863276
Ali H, Das S and Shaikh A A 2023 Investigate an imperfect green production system considering rework policy via Teaching-Learning-Based Optimizer algorithm. Expert Syst. Appl. 214: 119143. https://doi.org/10.1016/j.eswa.2022.119143
Manna A K, Das S, Shaikh A A, Bhunia A K and Moon I 2023 Carbon emission-controlled investment and warranty policy-based production inventory model via meta-heuristic algorithms. Comput. Ind. Eng. 177: 109001. https://doi.org/10.1016/j.cie.2023.109001
Fathi M, Khakifirooz M, Diabat A and Chen H 2021 An integrated queuing-stochastic optimization hybrid genetic algorithm for a location-inventory supply chain network. Int. J. Prod. Econ. 237: 108139. https://doi.org/10.1016/j.ijpe.2021.108139
Utama D M, Maulana S K D B, Baroto T and Dewi S K 2022 Optimizing vendor-buyer inventory model with exponential quality degradation for food product using grey wolf optimizer. Int. J. Food. Syst. Dyn. 13(2): 225–246
Kumar V, Naresh R and Sharma V 2023 Profit based unit commitment problem solution using metaheuristic optimisation approach. Int. J. Syst. Sci. Oper. Logist. 10(1): 2037026. https://doi.org/10.1080/23302674.2022.2037026
Vinod Chandra S S and Anand H S 2022 Nature inspired meta heuristic algorithms for optimization problems. Computing 104(2): 251–269. https://doi.org/10.1007/s00607-021-00955-5
Pasandideh S H R and Niaki S T A 2008 A genetic algorithm approach to optimize a multi-products EPQ model with discrete delivery orders and constrained space. Appl. Math. Comput. 195(2): 506–514. https://doi.org/10.1016/j.amc.2007.05.007
Soni G, Jain V, Chan F T, Niu B and Prakash S 2019 Swarm intelligence approaches in supply chain management: potentials, challenges and future research directions. SCM An Int. J. 24(1): 107–123. https://doi.org/10.1108/SCM-02-2018-0070
Kumar S and Mahapatra R P 2021 Design of multi-warehouse inventory model for an optimal replenishment policy using a rain optimization algorithm. Knowl. Based Syst. 231: 107406. https://doi.org/10.1016/j.knosys.2021.107406
Al-Khazraji H, Cole C and Guo W 2018 Multi-objective particle swarm optimisation approach for production-inventory control systems. J. Modell. Manage. 13(4): 1037–1056. https://doi.org/10.1108/JM2-02-2018-0027
Bhavani G D, Georgise F B, Mahapatra G S and Maneckshaw B 2022 Neutrosophic cost pattern of inventory system with novel demand incorporating deterioration and discount on defective items using particle swarm algorithm. Comput. Intell. Neurosci.. https://doi.org/10.1155/2022/7683417
Yuna F, Erkayman B and Yılmaz M 2023 Inventory control model for intermittent demand: a comparison of metaheuristics. Soft Comput. 27(10): 6487–6505. https://doi.org/10.1007/s00500-023-07871-0
Jafari A, Ganjehlou H G, Darbandi F B, Mohammadi-Ivatloo B and Abapour M 2020 Dynamic and multi-objective reconfiguration of distribution network using a novel hybrid algorithm with parallel processing capability. Appl. Soft Comput. 90: 106146. https://doi.org/10.1016/j.asoc.2020.106146
Khalilpourazari S and Pasandideh S H R 2017 Multi-item EOQ model with nonlinear unit holding cost and partial backordering: Moth-flame optimization algorithm J. . Ind. Prod. Eng. 34(1): 42–51. https://doi.org/10.1080/21681015.2016.1192068
Nia A R, Far M H and Niaki S T A 2015 A hybrid genetic and imperialist competitive algorithm for green vendor managed inventory of multi-item multi-constraint EOQ model under shortage. Appl. Soft Comput. 30: 353–364. https://doi.org/10.1016/j.asoc.2015.02.004
Khalilpourazari S and Pasandideh S H R 2019 Modeling and optimization of multi-item multi-constrained EOQ model for growing items. Knowl. Based Syst. 164: 150–162. https://doi.org/10.1016/j.knosys.2018.10.032
Roozbeh Nia A, Awasthi A and Bhuiyan N 2023 Assessment of coal supply chain under carbon trade policy by extended exergy accounting method. Flexible Serv Manuf J. https://doi.org/10.1007/s10696-023-09502-0
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The authors are thankful to the Editor-In-Chief and anonymous reviewers for their insightful comments and suggestions to improve the quality of the article. No funding was received for this work.
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Appendix I
Appendix I
1.1 A. Abbreviations
- QPSO:
-
Quantum behaved particle swarm optimization
- GWOA:
-
Grey-wolf optimizer algorithm
- TLBO:
-
Teaching-learning based optimizer algorithm
- SSA:
-
Sparrow search algorithm
1.2 B. Calculation of \({\varvec{t}}_{1} + {\varvec{t}}_{2} \approx 1\)
In leap year we have one more day i.e., 366 days in a year. In \(4\) years time increases \(24\) hour. So, in \(1\) year time increases \(6\) hour. Which implies, in \(365\) days time increases \(6\) hours. In one day, time increases \(\frac{6}{365} = .0164\) hours. So, one day is equal to \(24.064\) hours. In this reason for fuzzy model, we take \(t_{1} + t_{2} \approx 1.\)
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Mahato, S., Khan, A. & De, S.K. A study on periodic deteriorating linguistic fuzzy inventory model with natural idle time and imprecise demand using GSA. Sādhanā 49, 177 (2024). https://doi.org/10.1007/s12046-024-02523-x
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DOI: https://doi.org/10.1007/s12046-024-02523-x