Journal of Mathematical Modelling and Algorithms

, Volume 10, Issue 4, pp 341–356

# A Multi-Objective Production Inventory Model with Backorder for Fuzzy Random Demand Under Flexibility and Reliability

Article

## Abstract

In this paper, an Economic Production Quantity (EPQ) model is developed with flexibility and reliability consideration of production process in an imprecise and uncertain mixed environment. The model has incorporated fuzzy random demand, an imprecise production preparation time and shortage. Here, the setup cost and the reliability of the production process along with the backorder replenishment time and production run period are the decision variables. Due to fuzzy-randomness of the demand, expected average demand is a fuzzy quantity and also imprecise preparation time is represented by fuzzy number. Therefore, both are first transformed to a corresponding interval number and then using the interval arithmetic, the single objective function for expected profit over the time cycle is changed to respective multi-objective functions. Due to highly nonlinearity of the expected profit functions it is optimized using a multi-objective genetic algorithm (MOGA). The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.

### Keywords

Fuzzy random variable Imprecise preparation time Reliability Flexibility Interval arithmetic

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### References

1. 1.
Bag, S., Chakraborty, D., Roy, A.R.: A production inventory model with fuzzy random demand and with flexibility and reliability considerations. Comput. Ind. Eng. 56, 411–416 (2009)
2. 2.
Bector, R.C., Chandra, S.: Fuzzy Mathematical Programming and Fuzzy Matrix Games. Springer, New York (2005)
3. 3.
Bhandari, R.M., Sharma, P.K.: The economic production lot size model with variable cost function. Opsearch 36, 137–150 (1999)Google Scholar
4. 4.
Cheng, T.C.E.: An economic production quantity model with flexibility and reliability considerations. Eur. J. Oper. Res. 39, 174–179 (1989a)
5. 5.
Cheng, T.C.E.: An economic order quantity model with demand-dependent unit cost. Eur. J. Oper. Res. 40, 252–256 (1989b)
6. 6.
Cheng, T.C.E.: An economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Trans. 23, 23–28 (1991)
7. 7.
Chung, K., Hou, K.: An optimal production run time with imperfect production processes and allowable shortages. Comput. Oper. Res. 30, 483–490 (2003)
8. 8.
Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001)
9. 9.
Deb, K., Goel, T.: Controlled elitist non-dominated sorting genetic algorithms for better convergence. In: Proceedings of the First International Conference on Evolutionary Multi-criterion Optimization, Zurich, pp. 67–81 (2001)Google Scholar
10. 10.
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)
11. 11.
Dubois, D., Prade, H.: Operations of fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)
12. 12.
Grzegorzewski, P.: Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst. 130, 321–330 (2002)
13. 13.
Hansen, E., Walster, G.: Global Optimization Using Interval Analysis. Marcel Dekker Inc., New York (2004)
14. 14.
Hax, A.C., Canada, D.: Production and Inventory Management. Prentice-Hall, New Jersey (1984)Google Scholar
15. 15.
Holland, H.J.: Adaptation in Natural and Artifcial Systems. University of Michigan (1975)Google Scholar
16. 16.
Islam, S., Roy, T.K.: A fuzzy EPQ model with flexibility and reliability consideration and demand dependent unit production cost under a space constraint: a fuzzy geometric programming approach. Appl. Math. Comput. 176, 531–544 (2006)
17. 17.
Karmakar, S., Mahato, S.K., Bhunia, A.K.: Interval oriented multi-section techniques for global optimization. J. Comput. Appl. Math. 224, 476–491 (2009)
18. 18.
Knowles, J., Corne, D.: Approximating the non-dominated front using the Pareto archived evolution strategy. Evol. Comput. 8, 149–172 (2000)
19. 19.
Khouja, M.: The economic production lot size model under volume flexibility. J. Comput. Oper. Res. 22, 515–523 (1995)
20. 20.
Kwakernaak, H.: Fuzzy random variables: definition and theorems. Inform. Sci. 15, 1–29 (1978)
21. 21.
Leung, K.F.: A generalized geometric programming solution to an economic production quantity model with flexibility and reliability considerations. Eur. J. Oper. Res. 176, 240–251 (2007)
22. 22.
Maiti, M.K., Maiti, M.: Production policy of damageable items with variable cost function in an imperfect production process via genetic algorithm. Math. Comput. Model. 42, 977–990 (2005)
23. 23.
Mahapatra, N.K., Maiti, M.: Inventory model for breakable items with uncertain setup time. Tamsui Oxf. J. Manag. Sci. 20, 83–102 (2004)
24. 24.
Pal, P., Bhunia, A.K., Goyal, S.K.: On optimal partially integrated production and marketing policy with variable demand under flexibility and reliability considerations via Genetic Algorithm. Appl. Math. Comput. 188, 525–537 (2007)
25. 25.
Panda, D., Maiti, M.: Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: a geometric programming approach. Math. Comput. Model. 49, 1733–1749 (2009)
26. 26.
Porteus, E.L.: Investing in reduced set-ups in the EOQ model. Manag. Sci. 31, 998–1010 (1985)
27. 27.
Porteus, E.L.: Optimal lot sizing, process quality improvement and set-up cost reduction. Oper. Res. 34, 137–144 (1986)
28. 28.
Rosenblatt, M.J., Lee, H.L.: Economic production cycle with imperfect production processes. IIE Trans. 18, 47–55 (1986)Google Scholar
29. 29.
Roy, M.D., Sana, S.S., Chaudhuri, K.: An economic order quantity model of imperfect quality items with partial backlogging. Int. J. Syst. Sci. 42, 1409–1419 (2011a)
30. 30.
Roy, M.D., Sana, S.S., Chaudhuri, K.: An optimal shipment strategy for imperfect items in a stock-out situation. Math. Comput. Model. 54, 2528–2543 (2011b)
31. 31.
Sana, S.S.: A production-inventory model in an imperfect production process. Eur. J. Oper. Res. 200, 451–464 (2010a)
32. 32.
Sana, S.S.: An economic production lot size model in an imperfect production system. Eur. J. Oper. Res. 201, 158–170 (2010b)
33. 33.
Sana, S.S.: A production-inventory model of imperfect quality products in a three-layer supply chain. Decis. Support Syst. 50, 539–547 (2011)
34. 34.
Sana, S.S., Chaudhuri, K.: On a volume flexible production policy for a deteriorating item with time-dependent demand and shortages. Adv. Model. Optim. 6, 57–74 (2004a)
35. 35.
Sana, S.S., Chaudhuri, K.: On a volume flexible production policy for a deteriorating item with stock-dependent demand rate. Nonlinear Phenom. Complex Syst. 7, 61–68 (2004b)Google Scholar
36. 36.
Sana, S. S., Chaudhuri, K.: An EMQ model in an imperfect production process. Int. J. Syst. Sci. 41, 635–646 (2010)
37. 37.
Sana, S.S., Goyal, S.K., Chaudhuri, K.: An imperfect production process in a volume flexible inventory model. Int. J. Prod. Econ. 105, 548–559 (2007a)
38. 38.
Sana, S.S., Goyal, S.K., Chaudhuri, K.: On a volume flexible inventory model for items with an imperfect production system. Int. J. Oper. Res. 2, 64–80 (2007b)
39. 39.
Sarkar, B., Sana, S.S., Chaudhuri, K.: An imperfect production process for time varying demand with inflation and time value of money - An EMQ model. Expert Syst. Appl. 38, 13543–13548 (2011)Google Scholar
40. 40.
Silver, E.A.: Establishing the order quantity when the amount received is uncertain. INFOR 14, 32–39 (1976)Google Scholar
41. 41.
Tersine, K.D.: Principles of Inventory and Materials Management. North-Holland, New York (1982)Google Scholar
42. 42.
Van Beek, P., Putten, C.: OR contributions to flexibility improvement in production/inventory systems. Eur. J. Oper. Res. 31, 52–60 (1987)
43. 43.
Wright, C.M., Mehrez, A.: An overview of representative research of the relationships between quality and inventory. Omega 26, 29–47 (1998)
44. 44.
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
45. 45.
Zitzler, E., Thiele, L.: An evolutionary algorithm for multi-objective optimization: the strength Pareto approach, Technical report no. 43. Zurich, Computer engineering and networks laboratory Switzerland (1998)Google Scholar