International Journal of Fuzzy Mathematics and Systems.
ISSN 2248-9940 Volume 4, Number 2 (2014), pp. 193-201
© Research India Publications
http://www.ripublication.com
On Optimal Total Cost and Optimal Order Quantity for
Fuzzy Inventory Model without Shortage
1
1
D. Stephen Dinagar and 2J. Rajesh kannan
PG. and Research Department of Mathematics, T.B.M.L College, Porayar, INDIA.
2
Department of Basic Engineering, S.S.P College, Puthur, INDIA
1
Email: dsdina@rediffmail.com 2Email: raja.npr16@gmail.com
Abstract
In this paper, we develop fuzzy optimal total cost and fuzzy optimal order
quantity for the proposed inventory model. Holding cost, Ordering cost and
Total demands are taken as hexagonal fuzzy numbers. An inventory model
without shortage has been considered in a fuzzy environment. Distinct fuzzy
arithmetic operations on hexagonal fuzzy numbers are proposed. Numerical
examples are presented to illustrate the process of obtaining the fuzzy optimal
order quantity and the fuzzy minimal optimal total inventory cost.
Keywords: Hexagonal fuzzy numbers, Fuzzy inventory model, Fuzzy optimal
total cost, Fuzzy optimal order quantity,
Introduction
In 1915, the first inventory model was developed by F. Harris[3] later in 1965, first
time the concept of fuzzy sets was introduced by Lofti A. Zadeh [11]. In 1970 L. A.
Zadeh and R. E. Bellman proposed a mathematical model decision making in a fuzzy
environment. The economic lot size models have been studied extensively. Since
Harris [3] & Wilson [9] presented the EOQ model serves useful approximation to
many real life problems. Urgeletti [8] treated, EOQ model in fuzzy sense and used
triangular fuzzy numbers. Chen and Wang [1] used trapezoidal fuzzy numbers to
fuzzify the order cost, inventory cost and back order cost in the total cost of inventory
model without back order. Yao et al. [10] considered the fuzzified problems for the
inventory with or with or without backorder models. Jain [4] worked on the decision
making in the presence of fuzzy variables. Kacpryzk et al. [5] discussed some longterm inventory policy making through the fuzzy-decision making models.
Rajarajesvari et.al [6] proposed the hexagonal fuzzy number without any restrictions
of parameters. Stephen and Rajesh [7] have developed the fuzzy inventory model with
194
D. Stephen Dinagar and J. Rajesh kannan
allowable shortage using hexagonal fuzzy numbers. We have modified the definition
of the number by including conditions for the convexity of the number and few more
results are also included in the work.
In this paper, the optimal total cost and optimal order quantity for the inventory
model in fuzzy environment have been studied. An inventory model considering
holding cost, ordering cost and total demand are all in terms of hexagonal fuzzy
numbers. The new arithmetic operations are defined and applied the fuzzy total cost
and optimal order quantity. We apply fuzzy set theoretic approach. An algorithm is
developed to find the fuzzy optimal order quantity and also minimizing the fuzzy total
cost. Sensitivity analysis is carried out through the numerical examples.
In this article, in section 2, some basic definitions and new arithmetic operations
on hexagonal fuzzy numbers are presented. In section 3, we describe in brief notions
and assumption used in the developed model. In section 4, the fuzzy mathematical
models and algorithm. In section 5, numerical examples are given to illustrate the
model and sensitivity analysis has been mode for different changes in the parameter
values. In section 6, the concluding remarks are given.
2. Definitions and preliminaries
Definition 2.1: Fuzzy set
A fuzzy set A in a universe of discourse x is defined as the following set of pairs
A  {( x,  A ( x)) : x  X } . Here  A : X  [0,1] is a mapping called the membership
value of x  X in a fuzzy set A .
Definition 2.2: convex fuzzy set
A fuzzy set A  {( x,  A ( x))}  X is called convex fuzzy set if all A x are convex sets
i.e. for every element x1  A and x2  A for every   [0,1]  x1  (1   ) x2  A
  [0,1]. Otherwise the fuzzy set is called non convex fuzzy set.
Definition 2.3: Hexagonal Fuzzy Number
A fuzzy number on A h is a Hexagonal fuzzy number denoted by
(a1  a2  a3  a4  a5  a6 ) are
Ah  (a1, a2 , a3 , a4 , a5 , a6 ) where
real
numbers
Satisfying a2  a1  a3  a2 and a5  a4  a6  a5 and its membership function  Ah ( x) is
given as;
On Optimal Total Cost and Optimal Order Quantity
195
 x  a1

 1  x  a1  a  x  a
1
2
 2  a2  a1 

 1 1  x  a2 
  
a2  x  a3
2
2
a

a

3
2



 Ah ( x) 1,
a3  x  a4

1  1  x  a4  a  x  a
4
5
 2  a5  a4 

 1  a6  x 
 a5  x  a6
 
2
a

a

6
5


0,
x >a6
Remark: 2.3.1
1. The Hexagonal fuzzy numbers Ah becomes trapezoidal fuzzy numbers if
a2  a1  a3  a2 and a5  a4  a6  a5
2. The Hexagonal fuzzy numbers Ah becomes non-convex fuzzy numbers if
a2  a1  a3  a2 and a5  a4  a6  a5 .
Fig .1.Hexagonal fuzzy number
Definition 2.4: Equality of two Hexagonal fuzzy numbers
Two Hexagonal fuzzy numbers A  (a1 , a2 , a3 , a4 , a5 , a6 ) and B  (b1 , b2 , b3 , b4 , b5 , b6 ) are
said to be equal i.e. A  B if and only if a  b , a  b , a  b , a  b , a  b , a  b .
1
1
2
2
3
3
4
4
5
5
6
6
Definition 2.5: Symmetric Hexagonal fuzzy number
A fuzzy number A  (a1, a2,a3, a4, a5,a6 ) is said to be a symmetric Hexagonal fuzzy
number, if a3  a1  a6  a4 . Otherwise the fuzzy number is called non symmetric
fuzzy number.
196
D. Stephen Dinagar and J. Rajesh kannan
Fig.2 Symmetric Hexagonal fuzzy number
Definition 2.6:
We define a ranking function R : f ( R )  R which maps each fuzzy numbers to the
real line f ( R ) represents the set of all hexagonal fuzzy numbers. If R be any linear
 a  a  a3  a4  a5  a6 
ranking function, then. R( A )   1 2

6


Definition 2.7: Equivalent hexagonal fuzzy numbers
A fuzzy number A is said to be equivalent to a fuzzy number B if its value of the
Ranking function are the same. i.e. A  B if R ( A )  R ( B )
Definition 2.8: (New Arithmetic Operations)
The new arithmetic operations between hexagonal fuzzy numbers are proposed given
below. Let us consider A1  (a1 , a2 , a3 , a4 , a5 , a6 ) and A2  (b1 , b2 , b3 , b4 , b5 , b6 ) be two
hexagonal fuzzy numbers. Then,
The addition of A1 and A2 is
A () A  (a  b , a  b , a  b , a  b , a  b , a  b )
1
2
1
1
2
2
3
3
4
4
5
5
6
6
The subtraction of A1 and A2 is
A1 () A2  (a1  b6 , a2  b5 , a3  b4 , a4  b3 , a5  b2 , a6  b1 )
The multiplication of A and A is
1
2
a
a
a
a
a
a

A1 () A2   1  b , 2  b , 3  b , 4  b , 5  b , 6  b 
6
6
6
6
6
6

Where  b  (b1  b2  b3  b4  b5  b6 )
The division of A and A is
1
2
 6 a 6a 6a 6 a 6 a 6a 
A1 () A2   1 , 2 , 3 , 4 , 5 , 6 
 b b b b b b 
if  b  0 Where  b  (b1  b2  b3  b4  b5  b6 )
On Optimal Total Cost and Optimal Order Quantity
If k  0 is a scalar kA is defined as
(ka1 , ka2 , ka3 , ka4 , ka5 , ka6 ), if k > 0
kA  
(ka6 , ka5 , ka4 , ka3 , ka2 , ka1 ), if k < 0
A  a1 , a2 , a3 , a4 , a5 , a6  a1 , a2 , a3 , a4 , a5 , a6

197

Where a1 , a2 , a3 , a4 , a5 , a6 are non zero positive real numbers.
Case II
Multiplication of A1 and A2 is
A1 () A2   a1  b1 , a2  b2 , a3  b3 , a4  b4 , a5  b5 , a6  b6 
Division of A and A is
1
2
a a a a a a 
A1 () A 2   1 , 2 , 3 , 4 , 5 , 6  where bi  0, i  1, 2,3, 4, 5, 6
 b6 b5 b4 b3 b2 b1 
3. Notations and Assumptions
3.1 Notations
We define the following symbols
C : Fuzzy holding cost per unit quantity per unit time
S : Fuzzy Setup cost (or) ordering cost per order
: Length of the plan

D : Fuzzy Total demand over the planning time period [0, 1]
Q : Fuzzy Order quantity per cycle
Tc : Fuzzy total cost for the period [0, 1]
( )∗: Minimum Fuzzy optimal total cost for [0, 1]
∗
: Fuzzy Optimal order quantity
3.2 Assumptions
In this paper the following assumptions are considered as
Total demand is fuzzy nature
Time plan is constant
Holding cost, Ordering cost are fuzzy in nature
Shortages are not allowed.
4. Fuzzy mathematic model – I
In this model, the new arithmetic operation as in case - I have been utilized also fuzzy
demand over the planning time period [0,1], fuzzy holding cost permit quantity per
unit time and fuzzy set up cost or ordering cost are taken in terms of hexagonal fuzzy
numbers. Now we fuzzifying total cost is given by [2]
198
D. Stephen Dinagar and J. Rajesh kannan
  SD

CTQ
Tc 

2
Q
Our aim is to apply the hexagonal fuzzy number fuzzy total cost and obtain the
fuzzy optimal order quantity by using the simple calculus technique.
Suppose
C  (C1 , C2 , C3 , C4 , C5 , C6 ) S  (S1 , S2 , S3 , S4 , S5 , S6 )
D  ( D , D , D , D , D , D ) are hexagonal fuzzy numbers
1
2
3
4
5
6
  SD

 CTQ
Tc  


Q 
 2
By using new arithmetic operations and simplifying we get,
 D TQ
 D TQ
 D TQ
 D TQ
 D TQ
 D 
TQ
3
5
1
2
4
C


,
C


,
C


,
C


,
C


,
C  6 a 

1
a
2
a
3
a
4
a
5
a
 2 6Q
 2 6Q
 2 6Q
 2 6Q
 2 6 6Q

Tc  2 6Q


 T  F Q

 c
(4.1)
*

The fuzzy optimal order quantity Q which is minimize the total inventory cost
Tc  F (Q ) is obtained as the solution of the first order fuzzy differential equation
d 
(Tc )  0 and it is found as
dQ
 
Q *  (Q1 , Q2 , Q3 , Q4 , Q5 , Q6 ), using defintion 2.8 (vi)
Where
 a  (C1  C2  C3  C4  C5  C6 )
 b  (S1  S 2  S3  S 4  S5  S6 )
 c  (Q1  Q2  Q3  Q4  Q5  Q6 )
d 2 F (Q )
 0 this show that F (Q ) is minimum at Q  Q *
Also Q  Q * we have
2

dQ
and from (4.1)
 D TQ
  D TQ
  D TQ
  D TQ
  D TQ
  D 
* TQ
1b
1a
2b
2a
3b
3a
4a
4a
5b
5a

FQ   ,
 ,
 ,
 ,
 , 6 b  6 a
 12 c 12 c 12 c 12 c 12 c 12 c 
 
4.1 Fuzzy mathematical model – II
Suppose
C  (C1 , C2 , C3 , C4 , C5 , C6 ) S  (S1 , S2 , S3 , S4 , S5 , S6 )
D  ( D , D , D , D , D , D ) are hexagonal fuzzy numbers
1
2
3
4
5
6
On Optimal Total Cost and Optimal Order Quantity
199
  SD

 CTQ
Tc  


Q 
 2
By using arithmetic operations and simplifying we get,
 CTQ S D C TQ S D C TQ S D C TQ S D C TQ S D C TQ S D 
Tc   1  1 1 , 2  2 2 , 3  3 3 , 4  4 4 , 5  5 5 , 6  6 6 
2
2
2
2
2
Q
Q
Q
Q
Q
Q 
 2
Tc  F Q
(4.1.1)
*

The fuzzy optimal order quantity Q which is minimize the total inventory cost
Tc  F (Q ) is obtained as the solution of the first order fuzzy differential equation
d 
(Tc )  0 and it is found as
dQ
 
Q * 
2 S1 D1 2 S 2 D2 2 S 3 D3 2 S 4 D4 2 S 5 D5 2 S 6 D6
,
,
,
,
,
TC6
TC5
TC 4
TC3
TC 2
TC1
Q *  (Q1 , Q2 , Q3 , Q4 , Q5 , Q6 ), using defintion 2.8 (vi)
d 2 F (Q )
*


 0 this show that F (Q ) is minimum at Q  Q *
Also Q  Q we have
2

dQ
and from (4.1.1)
TCQ S D TCQ S D TCQ S D TCQ S D TCQ S D TCQ S D 
F Q*   1 1  1 1 , 2 2  2 2 , 3 3  3 3 , 4 4  4 4 , 5 5  5 5 , 6 6  6 6 
Q6
2
Q5
2
Q4
2
Q3
2
Q2
2
Q1 
 2
 
4.3 Algorithm for finding fuzzy total cost and fuzzy optimal order quantity.
Step 1
Calculate the model fuzzy total cost for the fuzzy values of C , S , D and T
Step 2
Now determine fuzzy total cost using new arithmetic operations fuzzy holding cost,
fuzzy ordering cost, fuzzy shortage cost and fuzzy demand taken in terms of
hexagonal fuzzy numbers.
Step 3
Find the fuzzy optimal order quantity which can be obtain by putting the first
derivative of F (Q ) equal to zero and second derivate is positive at Q  Q *
5. Numerical examples:
Fuzzy
model
–
I
Let
C  (6, 7,11,13,17,18) , S  (14,15,19, 21, 25, 26) ,
D  (250,350,500,550,650, 700), T = 6days
then
Q *  (11.78,13.94,16.66,17.48,19.00,19.72) F (Q * )
200
D. Stephen Dinagar and J. Rajesh kannan
=(728.40,927.88,1208.40,1298.78,1475.23,1562.01) .
Sensitivity analysis
S.
)
Demand ( D
No
.
1 (200,300,450,
500,600,650)
2 (225,325,475,
525,625,675)
3 (250,350,500,
550,650,700)
4 (275,375,525,
575,675,725)
5 (300,400,550,
600,700,750)
S  (14,15,19, 21, 25, 26)
C  (6,7,11,13,17,19)
S  (14,15,18, 22, 25, 26)
C  (6,7,10,14,17,18)
Q *
F (Q * )
Q *
F (Q * )
(10.54,12.90,15.81,
16.66,18.25,19.00)
(11.18,13.43,16.24,
17.07,18.64,19.36)
(11.78,13.94,16.66,
17.48,19.00,19.72)
(12.36,14.43,17.07,
17.87,19.36,20.06)
(12.90,14.90,17.48,
18.25,19.72,20.41)
(637.06,850.83,1148.80,12
43.81,1429.86,1521.26)
(683.96,890.06,1178.88,12
71.31,1452.94,1541.41)
(728.40,927.88,1208.40,12
98.78,1475.23,1562.01)
(771.20,964.36,1237.35,13
25.47,1497.75,1582.56)
(811.68,999.45,1265.97,13
51.57,1520.26,1602.98)
(10.54,12.90,15.81,
16.66,18.25,19.00)
(11.18,13.43,16.24,
17.07,18.64,19.36)
(11.78,13.94,16.66,
17.48,19.00,19.72)
(12.36,14.43,17.07,
17.87,19.36,20.06)
(12.90,14.90,17.48,
18.25,19.72,20.41)
(637.06,850.83,1148.80,12
43.81,1429.86,1521.26)
(683.96,890.06,1178.88,12
71.31,1452.94,1541.41)
(728.40,927.88,1208.40,12
98.78,1475.23,1562.01)
(771.20,964.36,1237.35,13
25.47,1497.75,1582.56)
(811.68,999.45,1265.97,13
51.57,1520.26,1602.98)
Fuzzy model – II
Let C  (7,9,11,13,15,17) , S  (15,17,19, 21, 23, 25) ,
D  (250, 350, 450,550, 650, 750), T = 6days
then
Q *  (8.05,10.14,15.60,18.70, 27.81, 31.79)
F (Q* )  (254.99, 401.72,1022.82,1469.68,3020.87,3977.52)
Sensitivity analysis
S.
)
Demand ( D
No
.
1 (200,300,450,
500,600,650)
2 (225,325,475,
525,625,675)
3 (250,350,500,
550,650,700)
4 (275,375,525,
575,675,725)
5 (300,400,550,
600,700,750)
S  (15,17,19, 21, 23, 25)
C  (7,9,11,13,15,17)
S  (15,17,18, 22, 23, 25)
C  (7, 9,10,14,15,17)
Q *
F (Q * )
Q *
F (Q * )
(7.20,9.39,14.80,17
.83,26.72,30.64)
(7.63,9.77,15.21,18
.27,27.27,31.22)
(8.05,10.14,15.60,1
8.70,27.81,31.79)
(8.44,10.50,15.99,1
9.12,28.34,32.36)
(8.81,10.84,16.36,1
9.54,28.86,32.91)
(220.98,365.6,967.92,1404
.82,2942.16,4001.78)
(238.23,383.93,995.9,1437
.38,2990.05,3986.01)
(254.99,401.72,1022.82,14
69.68,3020.87,3977.52)
(270.89,418.98,1049.37,15
00.83,3052.48,3980.85)
(286.2,435.54,1074.68,153
2.23,3086.25,3990.53)
(7.20,9.39,14.80,17
.83,26.72,30.64)
(7.63,9.77,15.21,18
.27,27.27,31.22)
(8.05,10.14,15.60,1
8.70,27.81,31.79)
(8.44,10.50,15.99,1
9.12,28.34,32.36)
(8.81,10.84,16.36,1
9.54,28.86,32.91)
(220.98,365.6,967.92,1404
.82,2942.16,4001.78)
(238.23,383.93,995.9,1437
.38,2990.05,3986.01)
(254.99,401.72,1022.82,14
69.68,3020.87,3977.52)
(270.89,418.98,1049.37,15
00.83,3052.48,3980.85)
(286.2,435.54,1074.68,153
2.23,3086.25,3990.53)
From table – I, it is observed that,
 The fuzzy optimal order quantity is closer to crisp optimal order quantity
 The fuzzy total cost is closer to crisp total cost
 For different values of S and C , changing only middle two spreads the fuzzy
optimal order quantity remains fixed. The same is true for fuzzy total cost.
From table – II, it is observed that,
On Optimal Total Cost and Optimal Order Quantity


201
The fuzzy optimal order quantity increases and this fuzzy total cost is increase
is linear.
For the different values of S and C changing only middle two spreads, the
fuzzy optimal order quantity remains fixed, but the fuzzy total cost is not fixed
on the middle two spreads are changed.
Conclusion
In this paper we have studied the optimal order quantity and the optimal total cost
under the fuzzy environment with the aid of hexagonal fuzzy numbers. The new
‘fuzzy arithmetic’ operations in fuzzy model-1 are proposed and applied in the
proposed notions. By using hexagonal fuzzy number ranking method and the
corresponding change have been observed. From the table, it is observed that the
fuzzy optimal order quantity and fuzzy optimal total cost are very close to the
classical model than the fuzzy model -2. Thus it can be concluded that the proposed
method is more effective and efficient for the computational purpose.
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D. Stephen Dinagar and J. Rajesh kannan