158 R. KASTHURI ET AL.

4.2. Fuzzy Inventory Model without Shortages

When

i

p are fuzzy decision variables, the above crisp

model without shortages under fuzzy environment re-

duces to

1

1

Min ,2

ii

n

iii i

ii i

ii

SH

TCpQA pp

Q

Q

subject to the constraints

1

n

ii

i

wQW

1

n

ii

i

pQB

1

1

i

n

iii

i

pAQ t

5. Numerical Example

To solve the above non-linear programming using Kuhn-

Tucker conditions, the following values are assumed.

n = 1, t = 3 A1 = 100, S1 = $100, H1 = $1,

w1 = 2 sq. ft, W = 150 sq. ft, B = $1200,

m1 = $1 and $10 ≤ p1 ≤ $20

By the method of Kuhn-Tucker conditions, consider

the four cases

1) λ1 = 0, λ2 = 0

2) λ1 ≠ 0, λ2 = 0

3) λ1 = 0, λ2 ≠ 0

4) λ1 ≠ 0, λ2 ≠ 0

Here Kuhn-Tucker conditions are used as trial and er-

ror method by taking different values for β1 until an op-

timum result is obtained.

Optimal solutions for the fuzzy model with shortages

β1 p1 µP1

value Q1 D1 M1 Expected

Total cost

0.88 10.179 0.982 72.04612.978 36.023168.133

0.89 12.406 0.759 65.22010.85 32.609164.484

0.90 15.405 0.4595 58.4288.533 29.214160.664

0.91 19.561 0.0439 51.6936.681 25.847156.533

From the above table it follows that 10.179 has the

maximum membership value 0.982.

Hence the required optimum solution is

p1 = 10.179, Q1 = 72.046, M1 = 36.023

Minimum expected Total cost = $168.133.

Optimal solutions for the fuzzy model without short-

ages

β1 p1 µP1

value Q1 D1 Expected

Total cost

0.89 10.788 0.921 75.000 12.042 183.460

0.85 11.178 0.882 50.693 12.850 194.327

0.92 16.00 0.400 75.000 07.802 172.733

From the above table it follows that 10.788 has the

maximum membership value 0.921.

Hence the required optimum solution is

p1 = 10.788, Q1 = 75

Minimum expected Total cost = $183.460.

6. Conclusions

In this paper we have proposed a concept of the optimal

solution of the inventory problem with fuzzy cost price

per unit item. Fuzzy set theoretic approach of solving an

inventory control problem is realistic as there is nothing

like fully rigid in the world. By solving the above fuzzy

inventory model using Kuhn-Tucker condition method

we have the values of imprecise variable for decision

making. The above discussed model can be developed

with many limitations, such as their inventory level,

Warehouse space and budget limitations, etc.

7. References

[1] L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol.

8, No. 3, 1965, pp. 338-353.

doi:10.1016/S0019-9958(65)90241-X

[2] R. E. Bellman and L. A. Zadeh, “Decision-Making in a

Fuzzy Environment,” Management Science, Vol. 17, No.

4, 1970, pp. B141-B164. doi:10.1287/mnsc.17.4.B141

[3] H. J. Zimmermann, “Description and Optimization of

Fuzzy Systems,” International Journal of General Sys-

tems, Vol. 2, No. 4, 1976, pp. 209-215.

doi:10.1080/03081077608547470

[4] G. Sommer, “Fuzzy Inventory Scheduling,” In: G. Lasker,

Ed., Applied Systems and Cybernetics, Vol. 6, Academic

Press, New York, 1981.

[5] K. S. Park, “Fuzzy Set Theoretic Interpretation of Eco-

nomic Order Quantity,” IEEE Transactions on Systems,

Man, and Cybernetics, Vol. 17, No. 6, 1987, pp. 1082-1084.

[6] H. A. Taha, “Operations Research: An Introduction,” Pren-

tice-Hall of India, Delhi, 2005, pp. 725-728.

[7] P. K. Gupta and M. Mohan, “Problems in Operations

Research (Methods & Solutions),” Sultan Chand Co., New

Delhi, 2003, pp. 609-610.

[8] E. A. Silver and R. Peterson, “Decision Systems for In-

ventory Management and Production Planning,” John

Wiley, New York, 1985.

[9] H. Tanaka, T. Okuda and K. Asai, “On Fuzzy Mathe-

Copyright © 2011 SciRes. AJOR