International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
156
An Intuitionistic Fuzzy Inventory Backorder Problem using
Triangular Intuitionistic Fuzzy Numbers
Mahuya Deb
1, Prabjot Kaur
21
Department of Commerce, Gauhati University, Assam, India
2Department of Mathematics, Birla Insitute of Technology, Mesra. Jharkhand, India
Abstract-- One of the worst stresses for a business owner is not having inventory available for product sales. Stock outs incur a cost called shortage cost which comprises of delays, labour time, wastage, lost production , loss of profit from lost sales, loss of profit due to loss of good will. In inventory models with shortages, the general assumption is that the unmet demand is either completely lost or completely backlogged. However, it is quite possible that while some customers leave, others are willing to wait till fulfillment of their demand. Inventory backorder is a situation in which a customer’s order cannot be filled when presented, and for which the customer is prepared to wait for some time .This quantity is proportional to the average customer waiting time, so it directly measures customer dissatisfaction due to delivery delays. The percentage of items backordered and the number of backorder days are important measures of the quality of a company's customer service and the effectiveness of its inventory management .In this paper economic order quantity (EOQ)model for inventory system with backorder is proposed. The model is constructed taking a few parameters as triangular intuitionistic fuzzy numbers.An illustrative example is used to explain the proposed approach. The results obtained is compared to that of a fuzzy model which reveals that intuitionistic fuzzy quantity and cost are similar to that of a fuzzy model justifying the fact that intuitionistic fuzzy sets with a non-membership degree have higher describing possibilitiesand that modal operators can be defined over IFS.
Keywords: intuitionisticfuzzy EOQ, backorder, triangular intuitionistic fuzzy Numbers
I. INTRODUCTION
Business environment being very dynamic today survival and growth have become the buzz words. Due to the uncertainty in demand estimations, organizations face a problem of stock outs ie a situation when demand exceeds the supply of their goods and services. This is so because stock outs cost time and money. Inventory backorder is a situation in which a customer’s order cannot be filled when presented, and for which the customer is prepared to wait for some time.
However backorder situations can signal poor inventory management or purchasing behavior and after considering the tangible and intangible costs of backorders, sometimes it is just cheaper in the longer run to make more product and carry more inventory. When shortage costs are accounted for, the basic EOQ model becomes slightly more general and the basic optimising model becomes a special one. [5]. In such a scenario inventory analysis has attained limelight considering the investments involved in maintaining and managing inventories. Classical inventory models assume deterministic parameters. However in the real case there are many uncertainities that should be considered.[11].What is usually observed in practice is that the demand quantities, inventory costs and order costs are not usually crisp values or real numbers. They inhibit vagueness or uncertainitywhich becomes difficult to be described with some probability distribution . Hence in situations like this it is more appropriate that these parameters are expressed in fuzzy numbers. Fuzzy set theory was introduced by Zadeh[15] in 1965. The application of fuzzy set theory to inventory problems has been proposed by Park in [11].In 1987, Park [11] used fuzzy set concepts to treat the inventory problem with fuzzy inventory cost under arithmetic operations of Extension Principle. In the past, several researchers have applied the fuzzy sets theory to deal with the inventory problems. Li et al. [8] established a fuzzy economic order quantity model based on generalized defuzzyfying approach. They used the extension of the lagrangean method to solve the inequality constraints. Yao and Lee [14] developed a fuzzy backorder inventory model. They applied the Extension Principle to find the optimal back order quantity. Chen and Hsieh [7] discussed a fuzzy backorder inventory model wherein the Second Function Principle is applied to the fuzzy arithmetical operations on generalized trapezoidal fuzzy numbers. Purnomoet al
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
157
Sen and Malakar [13] considered an EOQ model with shortages considering the various parameters as triangular, trapezoidal and parabolic fuzzy numbers. As an extension of the fuzzy set, the concept of intuitionistic fuzzy set (IFS) was introduced by Atanassov[2]. It is characterized by two functions expressing the degree of membership and the degree of non-membership, respectively. IFS is a very suitable tool to describe the imprecise or uncertain decision information and deal with the uncertainty and vagueness in decision making. Banerjee and Roy [3]generalized the application of the intuitionistic fuzzy optimization in the constrained multi objective stochastic inventory model. Nagoorgani [10] also used triangular intuitionistic fuzzy number in solving a linear programming problem. Chakraborty et al. [6] gave the solution for the basic EOQ model using intuitionistic fuzzy optimization technique wherein the various parameters including shortage cost are first treated as fuzzy numbers .Mahapatra[9]gave a multi objective inventory model of deteriorating items with some constraints in an intuitionistic fuzzy environment.Taking a novel approach this paper proposes to construct the inventory backorder model taking the parameters as triangular intuitionistic fuzzy numbers. The signed distance function defined by [4] is used for defuzzification.The organisation of the paper is as follows: In section 2 the preliminaries on fuzzy and Intuitionistic fuzzy sets is detailed. Section 3 deals in establishment of the Inventory Model in the Intuitionistic fuzzy environment.. Section 4 deals with a numerical to illustrate the result. The last Section 5 provides the conclusion for this study.
II. DEFINITIONS AND PRELIMINARIES ON FUZZY AND INTUITIONISTIC FUZZY SETS
2.1 Fuzzy Numbers
A fuzzy subset
A
of the real line R with membership function]
1
,
0
[
:
)
(
x
R
A
is called a fuzzy number ifi.
A
is normal, (i.e.) there exist an elementx
0such that
A(
x
0)
1
ii .
A
is fuzzy convex, ie]
1
,
0
[
,
,
),
(
)
(
]
)
1
(
[
1
2
1
2 1 2
Ax
x
Ax
Ax
x
x
R
iii .
A(
x
)
is upper continuous, andiv. supp
A
is bounded, where supp}
0
)
(
:
{
x
R
x
A
A2.2 Triangular Fuzzy Number:
A fuzzy number
A
of the universe of discourse U may be characterized by a triangular distribution function parameterized by a triplet(
a
1,
a
2,
a
3)
.
The membership function of the fuzzy number
A
is defined as
3 3 2
2 3
2 1
1 2
1
1
,
0
,
,
,
0
)
(
a
x
a
x
a
a
x
x
a
a
x
a
a
a
a
x
a
x
x
A
2.3 Arithmetic operations of fuzzy numbers
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
158
0 ) 1 , 2 , 31 (( 0 ) 3 , 2 , 1 ( ) ( ) 1 / 3 , 2 / 2 , 3 / 1 ( ) 3 , 2 , 1 ( ) 3 , 2 , 1 ( ) 1 1 , 2 1 , 3 1 ( 1 1 ) ( ) 1 3 , 2 2 , 31 1 ( ) 3 , 2 , 1 ( ) 3 , 2 , 1 ( is A from B of n subtractio then the ) 1 , 2 , 3 ( ) ( ) 3 3 , 2 2 , 1 1 ( ) 3 , 2 , 1 ( ) 3 , 2 , 1 ( ) ( ) 3 3 , 2 2 , 1 1 ( ) 3 , 2 , 1 ( ) 3 , 2 , 1 ( ) ( if a a a A if a a a A then R If v b a b a b a b b b a a a B A b b b B B iv Ib a b a b a b b b a a a B A b b b B iii b a b a b a b b b a a a B A ii b a b a b a b b b a a a B A i2.4 Intuitionistic Fuzzy Set: Let a set X be fixed. An IFS
A
~
in X is an object having the form}
:
)
(
),
(
,
{
~
~~
x
x
x
X
x
A
A
A
, where the]
1
,
0
[
:
)
(
~
x
X
A
and ~(
x
)
:
X
[
0
,
1
]
A
define thedegree of membership and degree of non membership respectively of the elememt
x
X
to the setA
~
, which is a subset of the set X , for every element of1
)
(
)
(
0
,
~
~
X
x
x
x
A
A2.5Intuitionistic Fuzzy Number
Intuitionistic Fuzzy Number: An IFN
A
~
is defined as follows:(i)an intuitionistic fuzzy subset of the real line
ii) normal, i.e., there is any
x
0
R
such that)
0
)
(
(
1
)
(
~~
x
so
x
A
A
(iii)a convex set for the membership function ~
(
x
)
A
i.e.]
1
,
0
[
,
,
))
(
),
(
min(
)
)
1
(
(
1 2 ~ 1 ~ 2 1 2~
Ax
x
Ax
Ax
x
x
R
(iv) a concave set for the non-membership function ~
(
x
)
A
ie]
1
,
0
[
,
,
))
(
),
(
max(
)
)
1
(
(
1 2 ~ 1 ~ 2 1 2~
Ax
x
Ax
Ax
x
x
R
2.6 A triangular intuitionistic fuzzy number
A
~
(
a
1,
a
2,
a
3;
w
~a,
u
a~)
is a subset of intuitionistic fuzzy set set on the set of realInternational Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
159
otherwise
a
x
a
a
a
a
x
a
x
a
a
a
x
a
x
and
otherwise
a
x
a
a
a
x
a
a
x
a
a
a
a
x
x
AA
1
,
,
)
(
,
0
,
)
(
,
)
(
)
(
3 22 3
2
2 1
2 1
2
~ 3
2 2 3
3
2 1
1 2
1
~
2.7 Arithmetic Operations of Triangular Intuitionistic Fuzzy Number
If
~
A
(
a
1,
a
2,
a
3)(
a
1
,
a
2,
a
3
)
andB
~
(
b
1,
b
2,
b
3)(
b
1
,
b
2,
b
3
)
are two TIFNs, then1. Addition of two TIFN is
)
,
,
)(
,
,
(
~
~
3 3 2 2 1 1 3 3 2 2 1
1
b
a
b
a
b
a
b
a
b
a
b
a
B
A
is also TIFN2. Subtraction of two TIFN
)
,
,
)(
,
,
(
~
~
1 3 2 2 3 1 1 3 2 2 3
1
b
a
b
a
b
a
b
a
b
a
b
a
B
A
is also TIFN3. Multiplication of two TIFN
)
,
,
)(
,
,
(
~
~
3 3 2 2 1 1 3 3 2 2 1
1
b
a
b
a
b
a
b
a
b
a
b
a
B
A
is also a TIFN4. If TIFN
A
~
(
a
1,
a
2,
a
3)(
a
1
,
a
2,
a
3
)
and y=ka (with k>0) then~
y
k
A
~
5. Division of two TIFN is
)
,
,
)(
,
,
(
~
~
1 3
2 2
3 1
1 3
2 2
3 1
b
a
b
a
b
a
b
a
b
a
b
a
B
A
is also a TIFN
2.8 Defuzzification for Triangular Fuzzy Number
The defuzzification value for a triangular fuzzy number(a1,a2,a3) is given by
4
2
2 31
a
a
a
A
(1)
2.9 Defuzzification for Triangular Intuitionistic Fuzzy Number
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
160
8
2
2
]
)}
)(
1
(
{
}
)
)(
1
(
{
)}
(
{
}
)
(
{
[
4
1
]
)
(
)
(
)
(
)
(
[
4
1
)
ˆ
,
ˆ
(
3 2 1 3 2 1
1
0 2 3 2
1
0 2 2 1
1
0 3 3 2
1
0 1 2 1
1
0 1
0 1
0 1
0
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
L
L
L
L
O
A
D
s
III. MODEL FORMULATION IN CRISP ENVIRONMENT
3.1 Assumptions
i. Only a single order is produced at the beginning of each cycle and the entire lot is delivered in one batch. ii. Cs is the shortage cost per unit quantity per unit time
& Co is the ordering cost per order, known and
constant.
iii. Q is the lot-size per cycle whereas S1 is the initial inventory level after fulfilling the back-logged quantity of previous cycle and Q-S1 is the maximum shortage level.
iv. T is the cycle length or scheduling period where as t1is the period with no shortage
3.2 Notations
The following notations are used for developing the model
Ch: carrying cost per unit quantity per unit time
Cs: shortage cost per unit quantity per unit time
Co: set up cost per order T: cycle length or scheduling period
D: total demand TC: total cost
S1: initial inventory level after fulfilling the back logged
quantity of previous cycle Q: lot size per cycle
:
h
C
Fuzzy carrying cost per unit quantity per unit time:
s
C
Fuzzy shortage cost per unit quantity per unit timeh
C
~
: Triangular Intuitionistic Fuzzy carrying osts
C
~
: Triangular Intuitionistic Fuzzy backorder costC
T
: Fuzzy total costF (q): defuzzified total cost
F (q*): Minimum defuzzified total cost
C
T
~
: intuitionistic fuzzy total cost)
(q
F
:defuzzified total cost)
(
q
F
: Minimum defuzzified total cost(intuitionistic)Q : Order Quantity
*
Q
: Optimal order quantity:
Q
Optimal order quantity(intuitionistic fuzzy)Regarding the cycle length or scheduling period of the inventory system, it is assumed that the cycle length or scheduling period T is constant
Here T is constant i.e., inventory is to be replenished after every time period T. As 𝑡1is the no shortage period,
D
S
t
Dt
S
1 1
1 1
Now, inventory carrying cost during period [0 , t1] is
2
2
2 1 1
1
C
t
C
S
S
C
h hInternational Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
[image:6.612.114.537.151.359.2]161
Fig 1. Inventory model with shortages[1]
Again shortage cost during the interval [𝑡1, T] is given
by
]
[.
)
(
2
1
)
)(
(
2
1
1 1
2 1
1 1
D
S
Q
t
T
S
Q
C
D
t
T
S
Q
C
C
s s s
Therefore the total average cost of the system is given by
T
S
Q
C
D
D
S
C
TC
h s/
]
)
(
2
1
2
[
1 22
1
Since the set up cost
C
0& time period T are constant,the average set up cost
T
C
0also being constant is excluded
in the cost expression.
∵ T is constant, Q = DT is also constant. Hence the above expression i.e., the expression for average cost is a function of single variable 𝑆1. So, we can minimise the above expression with respect to 𝑆1.
Therefore differentiating the total cost w.r.t S1 is given
by
s h
s h
s h
s h
s h
s
s h
s
C
C
DT
C
C
C
C
Q
C
C
C
C
C
DT
C
C
C
Q
C
S
min 1
3.3 Fuzzy ModelWith Shortages
Let
C
h
(
C
h1,
C
h2,
C
h3)
)
,
,
(
s1 s2 s3s
C
C
C
C
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
162
]
2
)
(
2
,
2
)
(
2
,
2
)
(
2
[
)
,
,
(
2
)
(
2
)
,
,
(
]
)
(
2
1
2
[
1
3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1 3 2 1 2 1 2 1 3 2 1 2 1 2 1 s h s h s h s s s h h h s hC
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
C
C
D
S
Q
D
S
C
C
C
S
Q
C
D
D
S
C
T
C
T
Usingthe signed distance for defuzzification of triangular fuzzy number we get,
)]
2
)
(
2
(
)
2
)
(
2
(
2
)
2
)
(
2
[(
4
1
)
(
3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 11 s h s h s
h
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
q
F
Differentiating F(q) w.r.t S1 we get
3 3 2 2 1 1 3 2 1 3 3 2 2 1 1 3 2 1 * 1
)
(
2
)
2
(
)
(
2
)
2
(
s h s h s h s s s s h s h s h s s sC
C
C
C
C
C
DT
C
C
C
C
C
C
C
C
C
Q
C
C
C
S
Also at S1 =S1* we get that
0
)
(
1 1 2
S
S
F
Thus TC is minimum at S1 =S1* and the minimum cost is given by
)]
2
)
(
2
*
(
)
2
)
(
2
(
2
)
2
)
(
2
*
[(
)
(
3 2 1 * 2 1 3 2 2 1 * 2 * 1 2 1 2 1 * 2 1 1 * s h s h s hC
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
q
F
3.4 Intuitionistic Fuzzy Model With Shortages
Let
C
h
(
C
h1,
C
h2,
C
h3)(
C
h1
,
C
h2,
C
h3
)
)
,
,
)(
,
,
(
1 2 3 1
2 3
s s s s s ss
C
C
C
C
C
C
C
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
163
)
2
)
(
2
,
2
)
(
2
,
2
)
(
2
(
)
2
)
(
2
,
2
)
(
2
,
2
)
(
2
(
)
,
,
)(
,
,
(
2
)
(
2
)
,
,
)(
,
,
(
]
)
(
~
2
1
2
~
[
1
~
3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1 3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1 3 2 1 3 2 1 2 1 2 1 3 2 1 3 2 1 2 1 2 1
s h s h s h s h s h s h s s s s s s h h h h h h s hC
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
C
C
C
C
C
D
S
Q
D
S
C
C
C
C
C
C
S
Q
C
D
D
S
C
T
C
T
Using the signed distance for defuzzification of triangular intuitionistic fuzzy number we get,
)]
2
)
(
2
(
)
2
)
(
2
(
2
)
2
)
(
2
(
)
2
)
(
2
(
)
2
)
(
2
(
2
)
2
)
(
2
[(
8
1
)
(
3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1 3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1
s h s h s h s h s h s hC
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
q
F
Therefore the optimal shortage quantity and the optimal cost is given by
Q
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
S
s h s h s h s h s h s s s s s)
(
)
(
)
(
)
(
4
)
(
)
4
(
*
3 3 1 1 3 3 2 2 1 1 3 1 3 2 1 1
)]
2
)
(
2
(
)
2
)
(
2
(
2
)
2
)
(
2
*
(
)
2
*)
(
2
*
(
)
2
)
*
(
2
*
(
2
)
2
*)
(
2
*
[(
8
1
)
(
3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1 3 2 1 2 1 3 2 2 1 2 1 2 1 2 1 2 1 1
s h s h s h s h s h s hC
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
C
D
S
Q
D
S
C
q
F
IV. NUMERICAL SOLUTION
A commodity is to be supplied at a constant rate of 25 units per day. A penalty cost will be charged at a rate of Rs 10 per day, if it is late for missing the scheduled delivery date. The cost of carrying the commodity in inventory is Rs 16 per unit per month. The production process is such that each month (30 days) a batch of items is started and is available for delivery any time after the end of the month. Find the optimal level of inventory at the beginning of each month.
Solution:
D=25units per day
Ch=Rs 16/30 =0.53 per unit per day
Cs=Rs10 per unit per day
The optimal inventory level is given by
S1=712.25 units
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
164
Fuzzy Case Intuitionistic Fuzzy Case
99
.
380
)
(
81
.
708
)
11
,
9
,
8
(
)
56
.
0
,
54
.
0
,
51
.
0
(
25
1
q
F
S
C
C
day
per
units
D
s h
70
.
380
)
(
28
.
708
1
)
12
,
9
,
6
)(
11
,
9
,
8
(
~
)
58
.
0
,
54
.
0
,
49
.
0
)(
56
.
0
,
54
.
0
,
51
.
0
(
~
25
q
F
S
s
C
h
C
day
per
units
D
Table 1:
Sensitivity analysis when associated costs are TFN and TIFN
Fuzzy Intuitionistic Fuzzy
Cs=(8,9,11),Ch=(0.51,0.54,0.56) Cs=(8,9,11)(6,9,12),Ch=(0.51,0.54,0.56)(0.49,0.54,0.58)
sl no D S F(q) S F(q)
1 15 425.29 228.59 424.97 228.42
2 20 567.05 304.79 566.62 304.56
3 25 708.81 380.99 708.28 380.70
4 30 850.57 457.18 849.94 456.84
5 35 992.34 533.38 991.59 532.98
6 40 1134.10 609.58 1133.25 609.12
7 45 1275.86 685.78 1274.90 685.26
V. CONCLUSION
Analysis of the problem under fuzzy and intuitionistic environment has shown that the shortage quantity obtained under the intuitionistic environment is closer to the fuzzy quantity. Therefore, when membership function is not always accurately defined due to the lack of personal error, an intuitionistic fuzzy set may help in solving the problem. A comprehensive sensitivity analysis has been performed to illustrate the impact of demand on the ordering policy. The relative size of Ch and Cshas an influence on shortages.
For large Ch relative to Cs the effect on Q is considerable.
If on the other hand,Ch is small relative to Cs, minor
changes in quantity and cost can be expected. However for very large shortage cost, the backorder model is the basic EOQ model without shortages.
REFERENCES
[1]. Anderson, Sweeney, Williams, “ An Introduction to Management
Science.”Cengage Learning .India Edition
[2]. Atanassov, T.K (1986), “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20,87-96.
[3]. Banerjee,S.andRoy,K.T(2010), “Solution of single and multiobjective stochastic inventory models with fuzzy cost components by intutionistic fuzzy optimization techniques”, Advances in Operations Research 2010.
[4]. Bharati,S.K and Singh, S.R, ( 2015)A note on solving a fully intuitionistic fuzzy linear programming problem based on sign distance, International Journal of Computer Application (0975-8887) Vol 119-No23
[5]. Buffa .S Elsewood, Dyer S James “Management Science / Operations Research”. A Wiley /Hamilton Publication
[6]. Chakraborty Susovan, Pal Madhumangal, Nayak Kr Prasun (2011) Intuitionistic Fuzzy Optimization technique for the solution of an EOQ model, Fifteenth International Conference on IFS, Burgas,NIFS 172,52-64.
[7]. Chen .H.C, Hsieh .C.H (1999) Optimization of Fuzzy Simple Inventory Models”,1999 IEEE Intrnational Fuzzy Systems Conference Proceedings, August 22-25,1999,Seoul, Korea
[8]. Li .Q, Zhang Q,Shen .H,(2006) Fuzzy Inventory Model with Backorder under Function Principle 1-4244-0318-9/06/$20.00@2006IEEE
[9]. Mahapatra N.K (2012) Multi-objective Inventory Model of Deteriorating Items with Some Constraints in an Intuitionistic Fuzzy Environment , International Journal of Physical and Social Sciences ,vol 2(9).
[10].Nagoorgani ,A., and.Ponnalagu, K. (2012) A New Approach on Solving Intuitionistic Fuzzy Linear Programming Problem”, Applied Mathematical Sciences, Vol. 6, no. 70, 3467 – 3474
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 8, August 2016)
165
[12].Purnomo D H , Wee .M.H, Chiu.Y ,(2012),Fuzzy Economic OrderQuantity Model with Partial Backorder , International Conference on Management , Behavioral sciences and Economics Issues (ICMBSE’2012) Penang, Malaysia
[13].Sen, N. and Malakar, S. A Fuzzy Inventory Model with Shortages Using Different Fuzzy Numbers(2015) American Journal of Mathematics and Statistics 5(5): 238-248
[14].Yao S.J , Lee M.H, (2000) Fuzzy inventory with or without backorder for fuzzy order quantity with trapezoidal fuzzy number,Fuzzy Sets and Systems, vol.105,pp 311-337,16 March 2000 [15].Zadeh, L.A. (1965) Fuzzy Sets, Information and Control, vol8,
![Fig 1. Inventory model with shortages[1]](https://thumb-us.123doks.com/thumbv2/123dok_us/8690515.877025/6.612.114.537.151.359/fig-inventory-model-with-shortages.webp)
