Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem.

*(English)*Zbl 1211.90022Summary: Ppossibility and necessity representations of fuzzy inequality constraints are presented and then crisp versions of the constraints are derived. Here analogous to chance constraints, real-life necessity and possibility constraints in the context of two warehouse multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic two warehouse multi-item production-inventory model with fuzzy constraints has been formulated for a finite period of time and solved for optimal production with the objective of having maximum profit. The rate of production is unknown, assumed to be a function of time and considered as a control variable. Also the present system produces some defective units alongwith the perfect ones and the rate of produced defective units is stochastic in nature. Demand of the good units is stock dependent and known and the defective units are sold at a reduced price. The space required per unit item and available storage space are assumed to be imprecise. The inequality of budget constraints is also imprecise. The space and budget constraints are expressed as necessity and/or possibility types. The model is reduced to an equivalent deterministic model using fuzzy relations and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn-Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and pictorial forms.

##### MSC:

90B05 | Inventory, storage, reservoirs |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

##### Keywords:

two warehouse; fuzzy-stochastic inventory; necessity and possibility; optimal production; optimal control
PDF
BibTeX
Cite

\textit{K. Maity}, Appl. Math. Modelling 35, No. 3, 1252--1263 (2011; Zbl 1211.90022)

Full Text:
DOI

##### References:

[1] | Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy set syst., 1, 3-28, (1978) · Zbl 0377.04002 |

[2] | Dubois, D.; Prade, H., Possibility theory, (1988), Academic Press New York, London · Zbl 0645.68108 |

[3] | Maiti, M.K.; Maiti, M., Fuzzy inventory model with two warehouses under possibility constraints, Fuzzy set syst., 157, 52-73, (2006) · Zbl 1085.90004 |

[4] | Dubois, D.; Prade, H., The Mean value of a fuzzy number, Fuzzy set. syst., 24, 79-300, (1987) · Zbl 0634.94026 |

[5] | Liu, B.; Iwamura, K.B., Chance constraint programming with fuzzy parameters, Fuzzy set. syst., 94, 27-237, (1998) |

[6] | Maity, K.; Maiti, M., Possibility and necessity constraints and their defuzzification – a multi-item production-inventory scenario via optimal control theory, Eur. J. oper. res., 177, 882-896, (2007) · Zbl 1109.90035 |

[7] | Urban, T.L., Inventory models with demand rate dependent on stock and shortage levels, Int. J. prod. econ., 22, 85-93, (1995) |

[8] | Levin, R.I.; Mclaughlim, C.P.; Lamone, R.P.; Kottas, J.F., Production/operations management: contemporary policy for managing operating systems, (1972), McGraw-Hill New York |

[9] | Salameh, M.K.; Jaber, M.Y., Economic production quantity model for items with imperfect quality, Int. J. prod. econ., 64, 59-64, (2000) |

[10] | Goyal, S.K.; Cardenas-Barron, L.E., Note on economic production quantity model for items with imperfect quality a practical approach, Int. J. prod. econ., 77, 85-87, (2002) |

[11] | Wee, H.M.; Yu, J.; Chen, M.C., Optimal inventory model for items with imperfect quality and shortage back ordering, Omega, 35, 7-11, (2007) |

[12] | Maity, A.K.; Maity, K.; Maiti, M., A production-recycling-inventory system with imprecise holding costs, Appl. math. model., 32, 2241-2253, (2008) · Zbl 1156.90303 |

[13] | Pakkala, T.P.M.; Achary, K.K., Discrete time inventory model for deteriorating items with two warehouses, Opsearch, 29, 90-103, (1992) · Zbl 0775.90142 |

[14] | Bhunia, A.K.; Maiti, M., A two warehouses inventory model for deteriorating items with a linear trend in demand and shortages, J. oper. res. soc., 49, 287-292, (1997) · Zbl 1111.90308 |

[15] | Mandal, S.; Maity, K.; Mondal, S.; Maiti, M., Optimal production inventory policy for defective items with fuzzy time period, Appl. math. model., 34, 810-822, (2010) · Zbl 1185.90015 |

[16] | Roy, A.; Maity, K.; Kar, S.; Maiti, M., A production-inventory model with remanufacturing for defective and usable items in fuzzy-environment, Comput. ind. eng., 56, 87-96, (2009) |

[17] | Xionga, G.; Heloa, P., An application of cost-effective fuzzy inventory controller to counteract demand fluctuation caused by bullwhip effect, Int. J. prod. res., 44, 5261-5277, (2006) · Zbl 1128.90317 |

[18] | Inuiguchi, M.; Sakawa, M.; Kume, Y., The usefulness of possibility programming in production planning problems, Int. J. prod. econ., 33, 45-52, (1994) |

[19] | Sana, S.; Goyal, S.K.; Chaudhuri, K.S., A production-inventory model for a deteriorating item with trended demand and shortages, Eur. J. oper. res., 157, 357-371, (2004) · Zbl 1103.90335 |

[20] | Kuo, Y., Optimal adaptive control policy for joint machine maintenance and product quality control, Eur. J. oper. res., 171, 586-597, (2006) · Zbl 1090.90053 |

[21] | Pontryagin, L.S.; Boltyanski, V.G., The mathematical theory of optimal process, (1962), Inter Science New York |

[22] | Wang, J.; Shu, Y.F., Fuzzy decision modelling for supply chain management, Fuzzy set syst., 150, 107-127, (2005) · Zbl 1075.90532 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.