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