1. INTRODUCTION
An
inventory management deals with the decision that minimizes total averages cost
or maximizes total average profit. In an ordinary inventory model are
considered all parameters like shortage cost, carrying cost etc. as a fixed,
but in a real life situation there some small fluctuations. Therefore,
consideration of fuzzy number is more realistic and interesting.
The study of inventory model where
demand rates vary with time is the last decades. Datta and Pal investigated an
inventory system with power demand pattern and deterioration. Park and Wang
studied shortages and partial backlogging of items. Friedman (1978) presented
continuous time inventory model with time varying demand.
Ritchie (1984) studied an inventory
model with linear increasing demand. Goswami and Chaudhuri (1991) discussed an
inventory model with shortages. Gen et. al. (1997) considered classical
inventory model with triangular fuzzy number. Yao and Lee (1998) considered an
economic production quantity model in fuzzy sense. De, Kundu and Goswami (2003)
presented an economic production quantity inventory model involving fuzzy
demand rate.
Syde and Aziz (2007) applied sign
distance method to fuzzy inventory model without shortage. D.Datta and Pravin
Kumar published several paper of fuzzy inventory with or without shortage.
Islam, Roy (2006) presented a fuzzy EPQ model with flexibility and reliability
consideration and demand depended unit production cost under a space
constraint.
A solution method of posynomial
geometric programming with interval exponents and coefficients was developed by
Liu (2008). Kotba. M. Kotb, Halla. Fergancy (2011) presented Multi-item EOQ
model with both demand-depended unit cost and varying lead time via geometric
programming.
Jana, Das and Maiti (2014) presented
multi-item partial backlogging inventory models over random planning horizon in
random fuzzy environment. Samir Dey and Tapan Kumar Roy (2015) presented
optimum shape design of structural model with imprecise coefficient by
parametric geometric programming.
A signomial optimization problem
often provides much more accurate mathematical representation of real-world
nonlinear optimization problems. Initially Passy and Wilde (1967), and Blau and
Wilde (1969) generalized some of the prototype concepts and theorems in order
to treat signomial programs (SP).
In other work that general type of
signomial programming (SP) has been done by Charnes et. al. (1988), who
proposed methods for approximating signomial programs with prototype geometric
programs. Islam and Roy (2005) proposed EOQ model with shortages under fully
backlogging and constant demand is formulated and solved. Here the model is
solved by fuzzy signomial geometric programming (FSGP) technique. Fuzzy
signomial geometric programming (FSGP) technique provides a powerful technique
for solving many non-linear problems.
2. FUZZY NUMBER AND ITS NEAREST INTERVAL APPROXIMATION
2.1.
Fuzzy
number
A real number described as fuzzy subset on the real line
whose membership function
has the following characteristics with
=
Where
is continuous and strictly increasing and
is continuous and strictly decreasing.
α- level set: The α- level of a fuzzy number is
defined as a crisp set where A(α) = [x: μA(x)≥α, xϵX] where αϵ [0,1]. A(α) is a non-empty bounded closed interval
contained in X and it can be denoted by Aα = [AL(α), AR(α)]. AL(α) and AR(α)
are the lower and the upper bounds of the closed interval, respectively.
2.2.
Interval
number
An interval number A is defined by
an ordered pair of real numbers as follows A = [ where
and
are the left and the right bounds of interval
A, respectively. The interval A, is also defined by center (
) and half-width (
) as follows
A
= ( = {x:
where
=
is the center and
=
is
the half-width of A.
2.3.
Nearest
interval approximation
Here we want to approximate a fuzzy
number by a crisp model. Suppose and
are two fuzzy numbers with α-cuts are [AL(α),
AR(α)] and [BL(α), BR(α)], respectively. Then the distance between
and
is
d() =
.
Given We have to find a closed interval
, which is closest to
with respect to some metric. We can do it,
since each interval is also a fuzzy number with constant α-cut for all α ∈ [0, 1]. Hence (
,
. Now we have to minimize
d(
with respect to .
In order to minimize d(, it is sufficient to minimize
the function D(
,
= (
)). The first partial
derivatives are
and
Solving and
we get
CL =
and CR =
.
Again,
since (D(
,
)) =2 > 0,
(D(
,
)) =2 > 0 and
H(,
) =
(D(
,
)).
(D(
,
)) –
= 4 > 0.
So,
D(,
) i.e. d(
is global minimum. Therefore, the interval Cd(
= [
] is the nearest interval
approximation of fuzzy number
with respect to the metric d.
Let
= (a1, a2, a3) be a triangular fuzzy number.
The α-cut interval of
is defined as
Aα
= [,
] where
= a1+α(a2 - a1) and
= a3 - α(a3 – a2). By nearest interval
approximation method the lower limit of the interval is
CL
= =
=
and the upper limit of the
interval is
CR
= =
=
.
Therefore, the interval number
corresponding is [
In the centre and half –width form the
interval number of
is defined as
.
2.4.
Parametric
Interval-valued function
Let [m, n] be an interval, where m
> 0, n > 0. From analytical geometry point of view, any real number can
be represented on a line. Similarly, we can express an interval by a function.
The parametric interval-valued function for the interval [m, n] can be taken as
g(s) = for s ∈ [0, 1], which is a strictly
monotone, continuous function and its inverse exits. Let
be the inverse of g(s), then
s.
3. DETERMINISTIC EOQ MODEL
In many real-life situations
shortages occur in an EOQ model. When Shortages occurs, costs are incurred. The
purpose of this section is to discuss the deterministic EOQ model in crisp
environment. The notations to be used are:
Tac(Q,S): Total average cost of the EOQ model.
Q: Order quantity.
Maximum shortage that occurs
under an ordering policy
: Carrying cost per item per
unit time.
: Shortages cost per item per
unit time.
: Ordering cost per order.
D: Demand rate per unit time.
0 t
Figure
2: EOQ model
Variables of the EOQ model are Q, S
and are constant parameters.
Thus,
Total
carrying cost = ,
Total
shortages cost = ,
So
total cost =
And
total average cost Tac(Q,S) =
= ,
.
i.e.,
problem is
Minimize Tac(Q,S)
(3.1)
subject to Q, S
4. FUZZY EOQ MODEL
In the inventory model we take the
parameters ,
and
are fuzzy numbers.
Then from (3.1) we have
Minimize (Q,S) =
+
+
(4.1)
subject to
Q, S
5. UNCONSTRAINED FUZZY SIGNOMIAL GP PROBLEM
A problem without any restrictions
is called unconstrained problem. I.e., a problem of the form
Minimize
(5.1)
Subject to
j
1, 2,……,m,
is called unconstrained problem.
Primal problem:
A primal fuzzy signomial GP
programming problem is of the form
Minimize
(5.2)
Subject to
j
1, 2,……,m.
Where
Here are real numbers and coefficient
are fuzzy triangular, as
.
Using nearest interval approximation
method, transformed all triangular fuzzy number into interval number i.e., [. Then the fuzzy signomial
geometric programming problem is of the following form
Min
(5.3)
Subject
to
j
1, 2,……,m.
Where denotes the interval counter parts i.e.,
for all i. Using parametric interval-valued
functional form, the problem () reduces to
Min
(5.4)
Subject
to
j
1, 2,……,m.
This
is a parametric geometric programming (PGP) problem.
Dual signomial GP problem:
Dual GP problem of the given primal GP problem
is
Maximize
(5.5)
Subject to
,
,
Case I: n>m+1, (i.e. DD >0) so
the DP presents a system of linear equations for the dual variables. Here the
number of linear equations is less than the number of dual variables. More
solutions of dual variable vector exist. In order to find an optimal solution
of DP, we need to use some algorithmic methods.
Case II: n< m+1, (i.e. DD <0) so
the DP presents a system of linear equations for the dual variables. Here the
number of linear equations is greater than the number of dual variables. In
this case generally no solution vector exists for the dual variables. However,
using Least Square (LS) or Min-Max (MM) method one can get an approximate
solution for this system.
Furthermore the primal-dual relation is
.
(5.6)
Note: A
Weak Duality theorem would say that
For any
primal-feasible x and dual-feasible but this is not true of the pseudo-dual fuzzy
signomial GP problem.
Corollary:
When the value
of is 1, then a fuzzy signomial geometric
programming (FSGP) problem transform to ordinary geometric programming problem.
Theorem 1: When is 1, then
(x, s) ≥
(δ, s) (Primal- Dual
Inequality).
Proof
The expression for(x, s) can be written as
(x, s) =
.
Here the weights are
and positive terms are
, ……… ,
.
Now applying
A.M.-.G.M inequality, we get
(
)
Or
[
Or
Or
=
i.e., (x, s) ≥
(δ,s) .
Ex. 1: Minimize (Q,S) =
+
+
subject to Q, S
With input values
Table-1
(Input data)
|
|
|
|
D |
|
(16, 20, 24) |
(40, 50, 60) |
(40, 50, 60) |
10 |
Using nearest approximation method
Then the problem is
Min. Tac(Q,S,s)+
+
Sub. Q, S
i.e., Min. Tac(Q,S,s)+
Sub. Q, S
This is primal problem and corresponding
dual problem is
Subject to
Solving above
equations, we have
,
,
,
i.e., (5.7)
Taking
log on both side of (5.7) and then partially differentiating with respect to and using the conditions of finding optimal
solution we get this equation
From primal-dual
relation
Solving
above relations with difference values of weight, we get the list of values in
table-2
Table -2: optimal solution
|
|
Optimal values objectives |
||||
|
s |
1 |
Optimal dual variables |
Optimal primal variables |
|
|
|
0.1 |
0.9 |
|
|
87.464 |
87.464 |
|
0.3 |
0.7 |
|
|
92.929 |
92.929 |
|
0.5 |
0.5 |
|
|
98.736 |
98.736 |
|
0.7 |
0.3 |
|
|
104.906 |
104.906 |
|
0.9 |
0.1 |
|
|
111.462 |
111.462 |
6. FUZZY MODIFIED SIGNOMIAL GEOMETRIC PROGRAMMING PROBLEM
(FMSGP)
6.1.
Primal
problem:
A primal modified signomial GP
programming problem is of the form
Minimize
(6.1)
Subject to
j
1, 2,……,m.
Where
Using nearest interval approximation
method, transformed all triangular fuzzy number into interval number i.e., [. Then the fuzzy signomial
geometric programming problem is of the following form
Minimize
(6.2)
Subject to
j
1, 2,……,m.
Where
Where
denotes the interval counter parts i.e.,
for all i. Using parametric interval-valued
functional form, the problem () reduces to
Minimize
(6.3)
Subject to
j
1, 2,……,m.
Where
This
is a parametric geometric programming (PGP) problem.
Dual
signomial GP problem:
Dual
GP problem of the given primal GP problem is
Maximize
(6.4)
Subject to
,
,
Case
I: nknm+n, (i.e. DD>0) So the DP
presents a system of linear equations for the dual variables. Here the number
of linear equations is less than the number of dual variables. More solutions
of dual variable vector exist. In order to find an optimal solution of DP, we
need to use some algorithmic methods.
Case
II: nk<nm+n,
(i.e. DD <0) So the DP presents a system of linear equations for the dual
variables. Here the number of linear equations is greater than the number of
dual variables. In this case generally no solution vector exists for the dual
variables. However, using Least Square (LS) or Min-Max (MM) method one can get
an approximate solution for this system.
Furthermore
the primal-dual relation is
,
s
(6.5)
Note 2: A Weak Duality theorem would say that
For any primal-feasible x and
dual-feasible but this is not true of the pseudo-dual fuzzy
modified signomial GP problem.
Corollary 2: When the values of is 1, then a fuzzy modified signomial
geometric programming (FMSGP) problem transform to ordinary modified geometric
programming problem.
Theorem
2: When is 1, then
(
) ≥ n
(Primal- Dual Inequality).
Proof.
The
expression for(
) can be
written as
(
) =
.
Here the weights are and positive
terms are
, ……… ,
.
Now
applying A.M.-.G.M inequality, we get
Or
Or
[
Or
=
i.e., (
) ≥ n
.
Ex.2:
Minimize (
,
) =
subject
to ,
With input
values
Table:
3 (Input data)
|
i |
|
|
|
|
|
i=1 |
(16, 20, 24) |
(40, 50, 60) |
(40, 50, 60) |
10 |
|
i=2 |
(6, 10, 14) |
(105, 125, 145) |
(21, 25, 29) |
15 |
Using
nearest approximation method
Then the
problem is
Min. Tac(Q,S,s)+
+
+
+
Sub. ,
i.e.,
Min .Tac(Q,S,s)+
Sub. ,
This is
primal problem and corresponding dual problem is
.
Subject to
Solving
above equations, we have
,
,
And
,
,
i.e.,
(6.6)
Taking
log on both side of (6.6) and then partially differentiating with respect to and
respectively and using the
conditions of finding optimal solution we get;
And
From
primal-dual relation
Solving
above relations with difference values of weight we get the list of values in
table-4.
Table 4: optimal solution
|
|
Optimal
values objectives |
||||
|
s |
1 |
Optimal
dual variables |
Optimal
primal variables |
|
|
|
0.1 |
0.9 |
|
|
8187.095 |
195.347 |
|
0.3 |
0.7 |
|
|
9205.062 |
206.989 |
|
0.5 |
0.5 |
|
|
10349.600 |
219.326 |
|
0.7 |
0.3 |
|
|
11636.450 |
232.372 |
|
0.9 |
0.1 |
|
|
13083.300 |
246.191 |
7. CONCLUSION
In
this paper, a fuzzy EOQ model with shortages under fully backlogging and
constant demand is formulated and solved. Here the model is solved by fuzzy
signomial geometric programming (FSGP) technique. For fuzzy coefficient we used
only triangular fuzzy number (TrFN). In future other types of fuzzy numbers
would be used. The methodology proposed in this paper may also be applicable to
other EOQ models.
Our
approach provide here a simple EOQ model, but in the future it should be used
many complex EOQ models. For future research of uncertainty in economic order
quantity (EOQ) model, by using different type of fuzzy numbers such as
pentagonal, hexagonal fuzzy numbers of generalized fuzzy numbers be
analytically more challenging and interesting. Inflation plays an important
role in present day-to-day life, but we have neglected it. Therefore,
consideration of inflation problem would be more realistic.
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