Skip to main content
SpringerLink
Log in

Robust possibilistic programming for multi-item EOQ model with defective supply batches: Whale Optimization and Water Cycle Algorithms

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

This paper proposes a new mathematical model for multi-product economic order quantity model with imperfect supply batches. The supply batch is inspected upon arrival using “all or None” policy and if found defective, the whole batch will be rejected. In this paper, the goal is to determine optimal order quantity and backordering size for each product. To develop a realistic mathematical model of the problem, three robust possibilistic programming (RPP) approaches are developed to deal with uncertainty in main parameters of the model. Due to the complexity of the proposed RPP models, two novel meta-heuristic algorithms named water cycle and whale optimization algorithms are proposed to solve the RPP models. Various test problems are solved to evaluate the performance of the two novel meta-heuristic algorithms using different measures. Also, single-factor ANOVA and Tukey’s HSD test are utilized to compare the effectiveness of the two meta-heuristic algorithms. Applicability and efficiency of the RPP models are compared to the Basic Possibilistic Chance Constrained Programming (BPCCP) model within different realizations. The simulation results revealed that the RPP models perform significantly better than the BPCCP model. At the end, sensitivity analyses are carried out to determine the effect of any change in the main parameters of the mathematical model on the objective function value to determine the most critical parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Source: Eskandar et al. [47]

Fig. 5

Source: Mirjalili and Lewis [46]

Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Salameh MK, Jaber MY (2000) Economic production quantity model for items with imperfect quality. Int J Prod Econ 64(1):59–64. https://doi.org/10.1016/s0925-5273(99)00044-4

    Article  Google Scholar 

  2. Cárdenas-Barrón LE (2000) Observation on:“economic production quantity model for items with imperfect quality”[Int J Prod Econ 64:59–64]. Int J Prod Econ 67(2):201

  3. Goyal SK, Cárdenas-Barrón LE (2002) Note on: economic production quantity model for items with imperfect quality–a practical approach. Int J Prod Econ 77(1):85–87

    Article  Google Scholar 

  4. Hayek PA, Salameh MK (2001) Production lot sizing with the reworking of imperfect quality items produced. Prod Plan Control 12(6):584–590

    Article  Google Scholar 

  5. Chung KJ, Huang YF (2006) Retailer’s optimal cycle times in the EOQ model with imperfect quality and a permissible credit period. Qual Quant 40(1):59–77

    Article  MathSciNet  Google Scholar 

  6. Eroglu A, Ozdemir G (2007) An economic order quantity model with defective items and shortages. Int J Prod Econ 106(2):544–549

    Article  Google Scholar 

  7. Cárdenas-Barrón LE (2009) Economic production quantity with rework process at a single-stage manufacturing system with planned backorders. Comput Ind Eng 57(3):1105–1113

    Article  Google Scholar 

  8. Khan M, Jaber MY, Bonney M (2011) An economic order quantity (EOQ) for items with imperfect quality and inspection errors. Int J Prod Econ 133(1):113–118

    Article  Google Scholar 

  9. Yassine A, Maddah B, Salameh M (2012) Disaggregation and consolidation of imperfect quality shipments in an extended EPQ model. Int J Prod Econ 135(1):345–352

    Article  Google Scholar 

  10. Ouyang LY, Chang CT, Shum P (2012) The EOQ with defective items and partially permissible delay in payments linked to order quantity derived algebraically. Central Eur J Oper Res 20(1):141–160

    Article  MathSciNet  MATH  Google Scholar 

  11. Hsu JT, Hsu LF (2013) An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns. Int J Prod Econ 143(1):162–170

    Article  Google Scholar 

  12. Nasr WW, Maddah B, Salameh MK (2013) EOQ with a correlated binomial supply. Int J Prod Econ 144(1):248–255

    Article  Google Scholar 

  13. Skouri K, Konstantaras I, Lagodimos AG, Papachristos S (2014) An EOQ model with backorders and rejection of defective supply batches. Int J Prod Econ 155:148–154

    Article  Google Scholar 

  14. Mukhopadhyay A, Goswami A (2014) Economic production quantity models for imperfect items with pollution costs. Syst Sci Control Eng Open Access J 2(1):368–378

    Article  Google Scholar 

  15. Pasandideh SHR, Niaki STA, Niknamfar AH (2014) Lexicographic max–min approach for an integrated vendor-managed inventory problem. Knowl-Based Syst 59:58–65

    Article  Google Scholar 

  16. Hsu LF, Hsu JT (2016) Economic production quantity (EPQ) models under an imperfect production process with shortages backordered. Int J Syst Sci 47(4):852–867

    Article  MathSciNet  MATH  Google Scholar 

  17. Kumar RS, Goswami A (2015) A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints. Appl Soft Comput 34:838–850

    Article  Google Scholar 

  18. Pasandideh SHR, Niaki STA, Gharaei A (2015) Optimization of a multiproduct economic production quantity problem with stochastic constraints using sequential quadratic programming. Knowl-Based Syst 84:98–107

    Article  Google Scholar 

  19. Khalilpourazari S, Pasandideh SHR (2016) Bi-objective optimization of multi-product EPQ model with backorders, rework process and random defective rate. In: 2016 12th international conference on industrial engineering (ICIE). IEEE pp 36–40

  20. Mondal S, Maiti M (2003) Multi-item fuzzy EOQ models using genetic algorithm. Comput Ind Eng 44(1):105–117

    Article  Google Scholar 

  21. Chang HC (2004) An application of fuzzy sets theory to the EOQ model with imperfect quality items. Comput Oper Res 31(12):2079–2092

    Article  MathSciNet  MATH  Google Scholar 

  22. Papachristos S, Konstantaras I (2006) Economic ordering quantity models for items with imperfect quality. Int J Prod Econ 100(1):148–154

    Article  Google Scholar 

  23. Wee HM, Yu J, Chen MC (2007) Optimal inventory model for items with imperfect quality and shortage backordering. Omega 35(1):7–11

    Article  Google Scholar 

  24. Baykasoğlu A, Göçken T (2007) Solution of a fully fuzzy multi-item economic order quantity problem by using fuzzy ranking functions. En Optim 39(8):919–939

    Article  MathSciNet  Google Scholar 

  25. Konstantaras I, Goyal SK, Papachristos S (2007) Economic ordering policy for an item with imperfect quality subject to the in-house inspection. Int J Syst Sci 38(6):473–482

    Article  MATH  Google Scholar 

  26. Mohan S, Mohan G, Chandrasekhar A (2008) Multi-item, economic order quantity model with permissible delay in payments and a budget constraint. Eur J Ind Eng 2(4):446–460

    Article  Google Scholar 

  27. Maddah B, Jaber MY (2008) Economic order quantity for items with imperfect quality: revisited. Int J Prod Econ 112(2):808–815

    Article  Google Scholar 

  28. Khan M, Jaber MY, Wahab MIM (2010) Economic order quantity model for items with imperfect quality with learning in inspection. Int J Prod Econ 124(1):87–96

    Article  Google Scholar 

  29. Roy MD, Sana SS, Chaudhuri K (2011) An economic order quantity model of imperfect quality items with partial backlogging. Int J Syst Sci 42(8):1409–1419

    Article  MathSciNet  MATH  Google Scholar 

  30. Pal B, Sana SS, Chaudhuri K (2012) Multi-item EOQ model while demand is sales price and price break sensitive. Econ Model 29(6):2283–2288

    Article  Google Scholar 

  31. Khalilpourazari S, Pasandideh SHR (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

    Google Scholar 

  32. Ross SM (1996) Stochastic processes, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  33. Inuiguchi M, Ramık J (2000) Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst 111(1):3–28

    Article  MathSciNet  MATH  Google Scholar 

  34. Mula J, Poler R, Garcia JP (2006) MRP with flexible constraints: a fuzzy mathematical programming approach. Fuzzy Sets Syst 157(1):74–97

    Article  MathSciNet  MATH  Google Scholar 

  35. Torabi SA, Hassini E (2008) An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets Syst 159(2):193–214

    Article  MathSciNet  MATH  Google Scholar 

  36. Pishvaee MS, Razmi J, Torabi SA (2012) Robust possibilistic programming for socially responsible supply chain network design: a new approach. Fuzzy Sets Syst 206:1–20

    Article  MathSciNet  MATH  Google Scholar 

  37. Heilpern S (1992) The expected value of a fuzzy number. Fuzzy Sets Syst 47(1):81–86

    Article  MathSciNet  MATH  Google Scholar 

  38. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24(3):279–300

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu B, Iwamura K (1998) Chance constrained programming with fuzzy parameters. Fuzzy Sets Syst 94(2):227–237

    Article  MathSciNet  MATH  Google Scholar 

  40. Gen M (1997) Genetic algorithm and engineering design. Wiley, New York

    Google Scholar 

  41. Mirjalili S (2016) Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput Appl 27(4):1053–1073

    Article  Google Scholar 

  42. Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133

    Article  Google Scholar 

  43. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  44. Mirjalili S, Mirjalili SM, Hatamlou A (2016) Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 27(2):495–513

    Article  Google Scholar 

  45. Mirjalili S (2015) Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowl-Based Syst 89:228–249

    Article  Google Scholar 

  46. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67

    Article  Google Scholar 

  47. Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water Cycle algorithm–a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110:151–166

    Article  Google Scholar 

  48. Zhang Y, Song S, Zhang H, Wu C, Yin W (2012) A hybrid genetic algorithm for two-stage multi-item inventory system with stochastic demand. Neural Comput Appl 21(6):1087–1098

    Article  Google Scholar 

  49. Memari A, Ahmad R, Rahim ARA, Hassan A (2017) Optimizing a just-in-time logistics network problem under fuzzy supply and demand: two parameter-tuned metaheuristics algorithms. Neural Comput Appl. https://doi.org/10.1007/s00521-017-2920-0

    Article  Google Scholar 

  50. Mortazavi A, Khamseh AA, Naderi B (2015) A novel chaotic imperialist competitive algorithm for production and air transportation scheduling problems. Neural Comput Appl 26(7):1709–1723

    Article  Google Scholar 

  51. Mousavi SM, Tavana M, Alikar N, Zandieh M (2017) A tuned hybrid intelligent fruit fly optimization algorithm for fuzzy rule generation and classification. Neural Comput Appl. https://doi.org/10.1007/s00521-017-3115-4

    Article  Google Scholar 

  52. Dalfard VM, Kaveh M, Nosratian NE (2013) Two meta-heuristic algorithms for two-echelon location-routing problem with vehicle fleet capacity and maximum route length constraints. Neural Comput Appl 23(7–8):2341–2349

    Article  Google Scholar 

  53. Abdelaziz AY, Ali ES, Elazim SA (2016) Implementation of flower pollination algorithm for solving economic load dispatch and combined economic emission dispatch problems in power systems. Energy 101:506–518

    Article  Google Scholar 

  54. Ali ES, Elazim SA, Abdelaziz AY (2016) Ant lion optimization algorithm for renewable distributed generations. Energy 116:445–458

    Article  Google Scholar 

  55. Ali ES, Elazim SA (2016) Mine blast algorithm for environmental economic load dispatch with valve loading effect. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2650-8

    Article  Google Scholar 

  56. Byrd RH, Gilbert JC, Nocedal J (2000) A trust region method based on interior point techniques for nonlinear programming. Math Program 89(1):149–185

    Article  MathSciNet  MATH  Google Scholar 

  57. Byrd RH, Hribar ME, Nocedal J (1999) An interior point algorithm for large-scale nonlinear programming. SIAM J Optim 9(4):877–900

    Article  MathSciNet  MATH  Google Scholar 

  58. Waltz RA, Morales JL, Nocedal J, Orban D (2006) An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math Program 107(3):391–408

    Article  MathSciNet  MATH  Google Scholar 

  59. Khalilpourazari S, Mohammadi M (2016) Optimization of closed-loop supply chain network design: a Water Cycle Algorithm approach. In: 2016 12th international conference on industrial engineering (ICIE). IEEE, pp 41–45

  60. Sadollah A, Eskandar H, Lee HM, Yoo DG, Kim JH (2016) Water Cycle algorithm: a detailed standard code. SoftwareX 5:37–43

    Article  Google Scholar 

  61. Sadollah A, Eskandar H, Bahreininejad A, Kim JH (2015) Water Cycle algorithm with evaporation rate for solving constrained and unconstrained optimization problems. Appl Soft Comput 30:58–71

    Article  Google Scholar 

  62. Watkins WA, Schevill WE (1979) Aerial observation of feeding behavior in four baleen whales: Eubalaena glacialis, Balaenoptera borealis, Megaptera novaeangliae, and Balaenoptera physalus. J Mammal 60(1):155–163

    Article  Google Scholar 

  63. Roy RA (1990) Primer on the Taguchi method. Society of Manufacturing Engineers, New York

    MATH  Google Scholar 

  64. Taguchi G, Chowdhury S, Wu Y (2005) Taguchi’s quality engineering handbook. Wiley, Hoboken

    MATH  Google Scholar 

  65. Khalilpourazari S, Khalilpourazary S (2016) Optimization of production time in the multi-pass milling process via a Robust Grey Wolf Optimizer. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2644-6

    Article  Google Scholar 

  66. Khalilpourazari S, Pasandideh SHR, Niaki STA (2016) Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA, and WCA. Appl Soft Comput 49:770–791

    Article  Google Scholar 

  67. Kayvanfar V, Teymourian E (2014) Hybrid intelligent water drops algorithm to unrelated parallel machines scheduling problem: a just-in-time approach. Int J Prod Res 52(19):5857–5879

    Article  Google Scholar 

  68. Pishvaee MS, Rabbani M, Torabi SA (2011) A robust optimization approach to closed-loop supply chain network design under uncertainty. Appl Math Model 35(2):637–649

    Article  MathSciNet  MATH  Google Scholar 

  69. Vahdani B, Veysmoradi D, Shekari N, Mousavi SM (2016) Multi-objective, multi-period location-routing model to distribute relief after earthquake by considering emergency roadway repair. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2696-7

    Article  Google Scholar 

  70. Talaei M, Moghaddam BF, Pishvaee MS, Bozorgi-Amiri A, Gholamnejad S (2016) A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: a numerical illustration in electronics industry. J Clean Prod 113:662–673

    Article  Google Scholar 

  71. Khalilpourazari S, Khalilpourazary S (2017) A lexicographic weighted Tchebycheff approach for multi-constrained multi-objective optimization of the surface grinding process. Eng Optim 49(5):878–895

    Article  MathSciNet  Google Scholar 

  72. Khalilpourazari S, Khamseh AA (2017) Bi-objective emergency blood supply chain network design in earthquake considering earthquake magnitude: a comprehensive study with real world application. Ann Oper Res. https://doi.org/10.1007/s10479-017-2588-y

    Article  Google Scholar 

  73. Khalilpourazari S, Khalilpourazary S (2017) An efficient hybrid algorithm based on Water Cycle and Moth-Flame optimization algorithms for solving numerical and constrained engineering optimization problems. Soft Comput. https://doi.org/10.1007/s00500-017-2894-y

    Article  Google Scholar 

  74. Mohammadi M, Khalilpourazari S (2017) Minimizing makespan in a single machine scheduling problem with deteriorating jobs and learning effects. In: Proceedings of the 6th international conference on software and computer applications. ACM, pp 310–315

  75. Khalilpourazari S, Khalilpourazary S (2017) A Robust Stochastic Fractal Search approach for optimization of the surface grinding process. Swarm Evol Comput. https://doi.org/10.1016/j.swevo.2017.07.008

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

    Soheyl Khalilpourazari, Seyed Hamid Reza Pasandideh & Ali Ghodratnama

Authors
  1. Soheyl Khalilpourazari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seyed Hamid Reza Pasandideh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali Ghodratnama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seyed Hamid Reza Pasandideh.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khalilpourazari, S., Pasandideh, S.H.R. & Ghodratnama, A. Robust possibilistic programming for multi-item EOQ model with defective supply batches: Whale Optimization and Water Cycle Algorithms. Neural Comput & Applic 31, 6587–6614 (2019). https://doi.org/10.1007/s00521-018-3492-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-018-3492-3

Keywords

Search

Navigation