Fuzzy Mixture Inventory Model Involving Fuzzy Lead Time Demand and Fuzzy Total Demand.

S. Kumar^* and B. Mukherjee

Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India

.*Corresponding Author E-mail: manshashiva@gmail.com

ABSTRACT:

This paper is Fuzzy mixture inventory model involving fuzzy lead time demand and fuzzy total demand. The model is inventory model system ,Expected annual total cost in crisp sense calculated but lead time crashing cost ,holding cost ,stock out cost ,setup cost is assume as fuzzy random variable and total expected annual cost in fuzzy sense also calculated . Expected annual total cost is represented by Trapezoidal fuzzy number. We have used the Graded mean integration of defuzzification. Numerical example has been shown to compare the both result in crisp and fuzzy sense.

KEYWORDS: Lead Time Crashing Cost; Demand; Trapezoidal Fuzzy Number; Expectated Annual Total Cost.

INTRODUCTION:

Ouyang and Wu (1998) [1] viewed that the demand of lead time can be any known and free cumulative distribution. Moon & Choi(1998) [2] extended Owang et al model (1996) considering the reorder point to be another decision variable. Liao and Shyu[3] considered lead time as a variable and controlled it by paying extra crashing cost . This model has been extendented by Ben-Dayu and Raouf, Ouyang [4] mixture inventory model involving variable lead time with backorders and lost sales by further considering the fuzziness of lead time demand and annual average demand .Chang et al have proposed the mixed inventory model involving variable lead time with backorder , lost sales and fuzzify the total demand to be the triangular fuzzy number and derive the fuzzy total cost. Then we further fuzzify the total demand to be trapezoidal fuzzy total cost in fuzzy sense. By the centriod method of defuzzification we derive the estimate of total cost in fuzzy sense. A numerical example is provided to illustrate the result.

CONCLUSIONS:

For mixed inventory model with lead time, has applied the fuzzy sets to deal with uncertain annual average demand , while the lead-time demand is treated as an ordinary (crisp) random variable with unknown form of probability distribution we also consider a mixture inventory model and address the issue of lead time reduction in the fuzzy environments.

The annual average demand D is a crisp value and the random lead time demand X is normally distributed, we first fuzzify.we find out the solution of the model with all fuzzy components are the same as of under all fuzzy parameters with non fuzzy parameter .we derive total expected annual cost in the fuzzy sense. Example are exhibited that in the fuzzy circumstances the optimal batch size has more raise when we use crisp number than when we employ trapezoidal fuzzy number, we have deuced that while utilizing trapezoidal fuzzy number the optimal total annually cost is more impressed and has more change in compare the we position we use corresponding crisp case as well as triangular fuzzy number.

REFERENCES:

[1] L.Y. Ouyang, and K.S. Wu, A Minimax Distribution Free Procedure for Mixed Inventory Model with Variable Lead Time, International Journal of Production Economics, 56, 511-516 (1998).

[2] I. Moon, and S. Choi, A Note on Lead Time and Distributional Assumptions in Continuous Review Inventory Models, Computers & Operations Research, 25, 1007-1012 (1998)

[3] C.J. Liao, C.H. Shyu, An analytical determination of lead time with normal demand, International Journal of Operations and Production Management 11, 72–78 (1991).

[4] M. Ben-Daya, Inventory models involving lead time as decision variable, Journal of the Operational Research Society 45, 579-582, (1994).

[5] L.Y. Ouyang, N.C. Yeh, K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society 47, 829–832 (1996).

[6] L.Y. Ouyang, N.C. Yeh and K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society 47, 829-832, (1996).

[7] H.M. Lee, J.S. Yao, Economic production quantity for fuzzy demand quantity and fuzzy production quantity, European Journal of Operational Research 109, 203–211 (1998).

[8] C.J. Liao, C.H. Shyu, An analytical determination of lead time with normal demand, International Journal of Operations and Production Management 11, 72–78 (1991).

[9] D.C. Lin, J.S. Yao, Fuzzy economic production for production inventory, Fuzzy Sets and Systems 111, 465–495 (2000).

[10] L.Y. Ouyang, N.C. Yeh, K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society 47, 829–832 (1996).

Received on 19.01.2014 Accepted on 29.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 134-139