American Journal of Oper ations Research, 2011, 1, 155-159
doi:10.4236/ajor.2011.13017 Published Online September 2011 (http://www.SciRP.org/journal/ajor)
Multi-Item Fuzzy Inventory Model Involving Three
Constraints: A Karush-Kuhn-Tucker Conditions Approach
R. Kasthuri, P. Vasanthi, S. Ranganayaki*, C. V. Seshaiah
Department of Mathematics, Sri Ramakrishna Engineering College, Coimbatore, India
Received July 7, 2011; revised July 22, 2011; accepted August 22, 2011
Abstract
In this paper, a multi-item inventory model with storage space, number of orders and production cost as con-
straints are developed in both crisp and fuzzy environment. In most of the real world situations the cost pa-
rameters, the objective functions and constraints of the decision makers are imprecise in nature. This model
is solved with shortages and the unit cost dependent demand is assumed. Hence the cost parameters are im-
posed here in fuzzy environment. This model has been solved by Kuhn-Tucker conditions method. The re-
sults for the model without shortages are obtained as a particular case. The model is illustrated with numeri-
cal example.
Keywords: Multi-Item Inventory Model, Membership Function, Karush-Kuhn-Tucker Condition
1. Introduction
The literal meaning of inventory is the stock of goods for
future use (production/sales). The control of inventories
of physical goods is a problem common to all enterprises
in any sector of an economy. The basic objective of in-
ventory control is to reduce investment in inventories
and ensuring that production process does not suffer at
the same time.
In general the classical inventory problems are de-
signed by considering that the demand rate of an item is
constant and deterministic and that the unit price of an
item is considered to be constant and independent in na-
ture. But in practical situation, unit price and demand
rate of an item may be related to each other. When the
demand of an item is high, an item is produced in large
numbers and fixed costs of production are spread over a
large number of items. Hence the unit cost of the item
decreases. .i.e., the unit price of an item inversely relates
to the demand of that item. So demand rate of an item
may be considered as a decision variable.
Zadeh [1] first gave the concept of fuzzy set theory.
Later on, Bellman and Zadeh [2] used the fuzzy set the-
ory to the decision-making problem. Zimmerman [3]
gave the concept to solve multi objective linear pro-
gramming problem. Fuzzy set theory has made an entry
into the inventory control systems. Sommer [4] applied
the fuzzy concept to an inventory and production sched-
uling problem. Park [5] examined the EOQ formula in
the fuzzy set theoretic perspective associating the fuzzi-
ness with the cost data. Hence we may impose ware-
house space, cost parameters, number of orders, produc-
tion cost etc, in fuzzy environment.
The Kuhn-Tucker conditions [6] are necessary condi-
tions for identifying stationary points of a non linear
constrained problem subject to inequality constraints.
The development of this method is based on the La-
grangean method.
These conditions are also sufficient if the objective
function and the solution space satisfy the conditions in
the following Table 1.
The conditions for establishing the sufficiency of the
Kuhn-Tucker conditions [7] are summarized in the fol-
lowing Table 2.
Kuhn-Tucker conditions, also known as Karush-Kuhn-
Tucker (KKT) conditions was first developed by W. Ka-
rush in 1939 as part of his M.S. thesis at the University
of Chicago. The same conditions were developed inde-
pendently in 1951 by W. Kuhn and A. Tucker.
In this paper, a multi-item, multi-objective inventory
problem with shortages along with three constraints such
as limited storage space, number of orders and produc-
tion cost has been formulated. The unit cost is considered
here in fuzzy environment. The problem has been solved
by KKT conditions method. This model is illustrated by
numerical example [8-11].

156 R. KASTHURI ET AL.
Table 1. The objective function and the solution space.
Required conditions
Sense of optimization
Objective function Solution space
Maximization Concave Convex Set
Minimization Convex Convex Set
Table 2. The sufficiency of the Kuhn-Tucker conditions.
Problem Kuhn-Tucker conditions
1) Max z = f(X)
subject to hi(X) ≤ 0
X ≥ 0, 1, 2,,im
 
 
1
0
0,0,1,2,,
0,1,2,,
m
i
i
i
jj
ii
i
i
fX hX
xx
hXhX im
im












2) Min z = f(X)
subject to hi(X) ≥ 0
X ≥ 0,
1, 2,,im
 
 
1
0
0,0,1,2,,
0,1,2,,
m
i
i
i
jj
ii
i
i
fX hX
xx
hXhX im
im












2. Assumptions and Notations
A multi-item, multi-objective inventory model is devel-
oped under the following notations and assumptions.
2.1. Notations
n = number of items
t = number of orders
W = Floor (or) shelf-space available
B = Total investment cost for replenishment
For ith item: (i = 1, 2, , n) 
Di = Di (pi) demand rate [function of unit cost price]
Qi = lot size (decision variable)
Mi = Shortage level (decision variable)
Si = Set-up cost per cycle
Hi = Inventory holding cost per unit item
mi = Shortage cost per unit item
pi = price per unit item (decision variable)
wi = storage space per item
TC(p, Q, M) = expected annual total cost
2.2. Assumptions
1) replenishment is instantaneous
2) lead time is zero
3) demand is related to the unit price as
i
i
i
ii
i
A
DA
pi
p




where Ai (>0) and βi (0 < βi < 1) are constants and real
numbers selected to provide the best fit of the estimated
price function. Ai > 0 is an obvious condition since both
Di and pi must be non-negative.
3. Formulation of Inventory Model with
Shortages
Let the amount of stock for the ith item (i = 1, 2, , n)
be Ri at time t = 0. In the interval (0, Ti(= t1i + t2i)), the
inventory level gradually decreases to meet demands. By
this process the inventory level reaches zero level at time
t1i and then shortages are allowed to occur in the interval
(t1i, Ti). The cycle then repeats itself (Figure 1).

The differential equation for the instantaneous inven-
tory qi(t) at time t in (0, Ti) is given by


1
1
for 0
for
i
i
ii
dq tDt
dt
DttT
i
i
t




(1)
with the initial conditions qi(0) = Ri(= Qi – Mi), qi(Ti) =
–Mi, qi(t1i) = 0.
For each period a fixed amount of shortage is allowed
and there is a penalty cost mi per items of unsatisfied
demand per unit time.
From (1)



1
11
for 0
.for
iii i
ii i
qtRDtt t
Dtttt T
 
i


So Dit1i = Ri, Mi = Dit2i, Qi = DiTi
Holding cost =


12
0
d2
i
t
iii
ii
i
HQ M
i
H
qt tT
Q



Shortage cost =

1
2
d2
i
i
T
ii
ii
i t
mM
mqtt T
Q

i
Production cost = piQi

22
The total costProduction costSet up cost
Holding costShortage cost
22
ii ii
ii iiii
ii
QM mM
pQ SHTT
QQ




 
The total average cost of the ith item is
 
22
,, 22
iii
iii i
ii iii
ii
HQ M
SD mM
TCpQMp DQQ

 
i
Q


1
22
,,
22
ii
ii
iii iiii
i
ii iii
ii
AS
TCpQMA pp
Q
HQ MmM
QQ
 



(2)
for 1,2, 3,,in

.

R. KASTHURI ET AL.
157
Figure 1. Inventory level of the ith item.
There are some restrictions on available resources in
inventory problems that cannot be ignored to derive the
optimal total cost.
1) There is a limitation on the available warehouse
floor space where the items are to be stored i.e.
1
n
ii
i
wQ W


;
2) Investment amount on total production cost cannot
be infinite, it may have an upper limit on the maximum
investment. i.e. ;
1
n
ii
i
pQB



3) An upper limit on the number of orders that can be
made in a time cycle on the system (i.e.)
1
n
i
ii
Dt
Q


.
The problem is to find price per unit item, the lot size,
the shortage amount so as to minimize the total average
cost function (2) subject to the total space and total pro-
duction cost restrictions.
It may be written as
Min TCi(pi, Qi, Mi) for all 1,2,3,,in

.
Subject to the inequality constraints
1
n
ii
i
wQW



1
n
ii
i
pQB



1
n
i
ii
Dt
Q



4. Fuzzy Inventory Model with Shortages
When
s
i
p are fuzzy decision variables, the above crisp
model under fuzzy environment reduces to


1
1
22
,,
22
ii
n
ii
ii i
ii
ii iii
ii
AS
MinTCpQMA pp
Q
HQ MmM t
QQ
 














subject to the constraints
1
n
ii
i
wQ W



1
n
ii
i
pQB



11
1
i
n
iii
i
pAQ
 


t
[Here cap ‘~’ denotes the fuzzification of the parame-
ters.]
The above fuzzy non-linear programming can be
solved using Kuhn-Tucker conditions.
4.1. Membership Function
The membership function for the fuzzy variable pi is de-
fined as follows

1,
,
0,
i
i
ii
ii
i
iL
Li
pL
LL
iL
pL
Up
XLp
UL
pU

i
iL
U


















Here ULi and LLi are upper limit and lower limit of pi
respectively.

158 R. KASTHURI ET AL.
4.2. Fuzzy Inventory Model without Shortages
When
s
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
A
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
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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
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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-
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doi:10.1080/03081077608547470
[4] G. Sommer, “Fuzzy Inventory Scheduling,” In: G. Lasker,
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[5] K. S. Park, “Fuzzy Set Theoretic Interpretation of Eco-
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[6] H. A. Taha, “Operations Research: An Introduction,” Pren-
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Research (Methods & Solutions),” Sultan Chand Co., New
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[8] E. A. Silver and R. Peterson, “Decision Systems for In-
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[9] H. Tanaka, T. Okuda and K. Asai, “On Fuzzy Mathe-

R. KASTHURI ET AL.
159
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[10] F. E. Raymond, “Quantity and Economic in Manufac-
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