Applied Mathematical Sciences, Vol. 7, 2013, no. 102, 5085 - 5094
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2013.37362
Multi Item Inventory Model with Shortages under
Limited Storage Space and Set up Cost Constraints
via Karush Kuhn Tucker Conditions Approach
P. Vasanthi and C. V. Seshaiah
Department of Mathematics
Sri Ramakrishna Engineering College
Coimbatore, TamilNadu, India
Copyright © 2013 P. Vasanthi and C. V. Seshaiah. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
A multi-item inventory model with shortages and demand dependent on
unit cost has been formulated along with storage space and set up cost constraints.
In most of the real world situations the cost parameters are imprecise in nature.
Hence the cost parameters are imposed here in fuzzy environment. This model is
solved by Kuhn-Tucker conditions method. The model is illustrated with
numerical example. The model is solved for without shortages as a special case.
Keywords: Inventory, KKT conditions, demand dependent on unit cost, Fuzzy
unit cost
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P. Vasanthi and C. V. Seshaiah
1. Introduction
The inventory control is the function of directing the movement of goods
through the entire manufacturing cycle from the requisitioning of raw materials to
the inventory of finishing goods in an orderly manner to meet the objectives of
maximum outcomes service with minimum investment and efficient plant
operation. Stock level of various constraints such as limited warehouse space,
limited budget available for inventory, degree of management attention towards
individual items in the inventory and customer service level.
Many researchers have studied the inventory model with limited storage
space and investment. Teng and Yang[8] proposed a deterministic inventory lot
size models with time-varying demand and cost under generalized holding cost.
Abou-El-Ata and Kotb[1] applied geometric programming approach to solve
some inventory models with variable inventory costs. Shawky and Abou-ElAta[7] solved a constrained production lot size model with trade policy by
geometric programming and Lagrangean methods. Other related inventory models
were written by Cheng[2], Jang and Klein[3] and Mandal et al[5]. Recently
Kotb[4] discussed multi item EOQ model with limited storage space and set up
cost constraints with varying holding cost.
Zadeh[9] first gave the concept of fuzzy set theory. Zimmerman[10] gave
the concept to solve multi objective linear programming problem. Fuzzy set
theory has made an entry into the inventory control system. T.K. Roy,
N.K.Mandal and M.Maiti[6] solved a fuzzy nonlinear programming under fuzzy
environment.
This paper determines the total cost of an inventory model with shortages
under storage space and set up cost constraints which has been solved using
Karush Kuhn Tucker conditions. The price per unit item is fuzzified for better
result.
Multi item inventory model with shortages
5087
The Kuhn Tucker conditions are necessary conditions for identifying
stationary points of a non-linear constrained problem subject to inequality
constraints. These conditions are given in table 1.1
Table 1.1
Sense
of Required conditions
optimization
Objective function
Solution space
Maximization
Concave
Convex Set
Minimization
Convex
Convex Set
The conditions for establishing the sufficiency of the Kuhn-Tucker
conditions are summarized in the following table 1.2
Table 1.2
Problem
1.Max z = f(X)
Kuhn-Tucker conditions
subject to hi(X)≤ 0
m
∂
∂ i
f ( X ) − ∑ λi
h (X ) = 0
∂x j
∂x j
i =1
X ≥ 0, i = 1,2,…..m
λi hi ( X ) = 0, hi ( X ) ≤ 0, i = 1, 2,....m
λi ≥ 0, i = 1, 2,......m
2.Min z = f(X)
subject to hi(X) ≥ 0
m
∂
∂ i
f ( X ) − ∑ λi
h (X ) = 0
∂x j
∂x j
i =1
X ≥ 0, i= 1,2,…..m
λi hi ( X ) = 0, hi ( X ) ≥ 0, i = 1, 2,....m
λi ≥ 0, i = 1, 2,......m
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P. Vasanthi and C. V. Seshaiah
2. Assumptions & Notations
The multi item model is developed under the following notations and
assumptions.
Notations:
n = number of different items carried in inventory
W = Floor (or) shelf-space available
K= Limitations on the total set up cost
For ith item: (i= 1,2,… n)
Di = Annual demand rate
Qi = lot size (decision variable)
Mi = shortage level (decision variable)
Si = Ordering cost or set up cost
Hi = Unit holding (inventory carrying) cost per item
pi= unit purchase (production) cost (decision variable)
mi = shortage cost per unit item
wi= storage space per item
TC(D, Q, M) = Average annual total cost
Assumptions:
(i)
Replenishment is instantaneous
(ii)
Lead time is zero
(iii)
Demand is related to the unit price as
Di = Ai pi − βi where Ai (>0) and βi (0< βi<1) are constants and real
numbers selected to provide the best fit of the estimated proceed
function. Ai>0 is an obvious condition since both Di and pi must be
non-negative.
Multi item inventory model with shortages
5089
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 at time t1i and then
shortages are allowed to occur in the interval (t1i, Ti). The cycle then repeats itself.
The differential equation for the instantaneous inventory qi(t) at time t in
(0,Ti) is given by
dqi (t )
= − Di ,
dt
= -Di,
for 0≤t≤t1i
for t1i≤t≤Ti -------------------------(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 item of unsatisfied demand per unit time.
From (1)
qi(t) = Ri – Dit for 0 ≤ t ≤ t1i
= Di(t1i –t)for t1i≤ t ≤ Ti
So Dit1i= Ri , Mi = Dit2i , Qi = DiTi
t1i
Holding cost = H i ∫ qi ( t )dt =
0
Ti
Shortage cost = mi ∫ qi ( t )dt =
t1i
H i ( Qi − M i )
2Qi
2
mi M i 2
Ti
2Qi
Production cost = piQi
Total cost = Production cost + Set up cost + Holding cost + Shortages cost
= pi Qi + Si +
H i (Qi − M i )2 Ti mi M i 2Ti
+
2Qi
2Qi
The total average cost of the ith item is
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P. Vasanthi and C. V. Seshaiah
TC ( pi ,Qi , M i ) = pi Di +
Si Di H i (Qi − M i )2 mi M i 2
+
+
Qi
2Qi
2Qi
TC ( pi ,Qi , M i ) = Ai pi1− βi +
---------------------- (2)
Ai Si pi − βi H i (Qi − M i )2 mi M i 2
+
+
---------------------- (3)
Qi
2Qi
2Qi
For i = 1, 2….n
There are some restrictions on available resources in inventory problems
that cannot be ignored to derive the optimal total cost.
(i)
There is a limitation on the available warehouse floor space where the
items are to be stored.
n
∑wQ ≤W
i =1
(ii)
i
i
Investment amount on total set up cost is limited.
n
n
Di
Si ≤ K
∑
i =1 Qi
(or)
∑Ap
i =1
i
i
− βi
Qi −1Si ≤ K
The problem is to find price per unit item, the lot size, shortage amount so
as to minimize the total average cost function subject to total space and setup cost
restrictions.
Min TC (pi,Qi,Mi) for all i = 1,2,….n subject to the inequality constraints
n
∑wQ ≤W
i =1
i
i
n
∑Ap
i =1
i
i
− βi
Qi −1Si ≤ K
4. Fuzzification of the Cost Parameter
Here the unit cost price pi is imposed under fuzzy environment. The
membership function for the fuzzy variable pi is defined as follows.
Multi item inventory model with shortages
5091
⎧
⎫
1, pi ≤ LLi
⎪
⎪
⎪ U Li − pi
⎪
, LLi ≤ pi ≤ U Li ⎬
μ pi ( X ) = ⎨
⎪U Li − LLi
⎪
⎪
⎪
0, pi ≥ U Li
⎩
⎭
Here ULi and LLi are upper limit and lower limit of pi respectively.
5. Inventory Model without Shortages
The above inventory model without shortages reduces to
⎡
Ai Si pi − βi H i Qi ⎤
1− β i
Min TC(pi,Qi) = ∑ ⎢ Ai pi +
+
⎥
2 ⎦
Qi
i =1 ⎣
------------ (4)
n
Subject to the constraints
n
∑wQ ≤W
i =1
i
i
n
∑Ap
i =1
i
i
− βi
Qi −1Si ≤ K
6. Numerical Example
To solve the above nonlinear programming using Kuhn-Tucker conditions,
the following values are assumed and solved for a single item.
n = 1, A1=100, K=$200, S1=$100, H1=$1, w1=2sq ft, W=150 sqft,
m1=$1 and $10≤p1≤$20
Here Kuhn-Tucker conditions are used as trial and error method by taking
different value for β until an optimum result is obtained.
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P. Vasanthi and C. V. Seshaiah
Table 6.1
βi
pi
µpi
Qi
Di
Mi
value
Average
Total cost
0.88
10.18
0.982
72.05
12.98
36.03
168.14
0.89
12.41
0.759
65.21
10.63
32.60
164.52
0.90
15.40
0.460
58.43
8.54
29.22
160.67
0.91
19.56
0.044
51.70
6.68
25.85
157.55
From the above table it follows that pi=10.18 have the maximum
membership value 0.982.
Hence the required optimum solution is p1 = $10.18, Q1 = 72.05, M1 =
36.03, Minimum expected Total cost = $ 168.14
Optimum solution for the model without shortages are given in the
following table 6.2
Table 6.2
βi
pi
µpi value
Qi
Di
Average
Total cost
0.88
18.56
0.144
39.51
7.65
181.10
0.87
15.38
0.462
43.51
9.28
185.73
0.85
10.98
0.902
51.60
13.05
194.33
7. Conclusion
In this paper we have proposed a concept of the optimal solution of the
inventory problem with fuzzy cost price per unit item under storage space and set
up cost restrictions with unit price per item, lot size and shortage level as decision
variables. The model is solved for only a single item. From the above solution it is
Multi item inventory model with shortages
5093
obvious that the unit cost increases, but the lot size and demand value decreases as
the β value increases and hence the total average cost also decreases as β value
increases. This model can be extended to solve for more than one item with other
restrictions like budget limitations, lot size etc.
References
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Varying Holding Cost under two restrictions; A Geometric Programming
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[2] T.C.E. Cheng, Economic Order Quantity Model with Demand Dependent Unit
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5094
P. Vasanthi and C. V. Seshaiah
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Received: July 2, 2013