Sahidul Islam
University of kalyani, India
E-mail: sahidul.math@gmail.com
Wasim Akram Mandal
University of kalyani, India
E-mail: wasim0018@gmail.com
Submission: 02/09/2016
Accept: 13/10/2016
ABSTRACT
In this paper, an Inventory model
with unit production cost, time depended holding cost, with-out shortages is
formulated and solved. We have considered here a single objective inventory
model. In most real world situation, the objective and constraint function of
the decision makers are imprecise in nature, hence the coefficients, indices,
the objective function and constraint goals are imposed here in fuzzy
environment. Geometric programming provides
a powerful tool for solving a variety of imprecise optimization problem. Here
we have used nearest interval approximation method to convert a triangular
fuzzy number to an interval number then transform this interval number to a
parametric interval-valued functional form and solve the parametric problem by
geometric programming technique. Here two necessary theorems have been derived. Numerical
example is given to illustrate the model through this
Fuzzy Parametric
Geometric-Programming (FPGP) method.
Keywords: Inventory model, Fuzzy number,
Space constraint, Geometric Programming, Interval-valued function
1. INTRODUCTION
An inventory deals with decision
that minimize the total average cost or maximize the total average profit. For
this purpose the task is to construct a mathematical model of the real life
Inventory system, such a mathematical model is based on various assumptions and
approximations. In ordinary inventory model it consider all parameter like
set-up cost, holding cost, interest cost
a fixed. But in real life situation it will have some little
fluctuations. So consideration of fuzzy variables is more realistic.
Geometric Programming (GP) method is
an effective method used to solve a non-linear programming problem like
structural problem. It has certain advantages over the other optimization
methods. Here, the advantage is that it is usually much simpler to work with
the dual than the primal one. Solving a non-linear programming problem by GP
method with degree of difficulty (DD) plays essential role. (It is defined as
DD = total number of terms in objective function and constraints – total number
of decision variables – 1). Since late 1960’s, Geometric Programming (GP) used
in various field (like OR, Engineering science etc.). Geometric Programming
(GP) is one of the effective methods to solve a particular type of Non linear
programming problem. The theory of Geometric Programming (GP) first emerged in
1961 by Duffin and Zener. The first publication on GP was published by Duffin
and Zener on (1967). There are many references on applications and methods of
GP in the survey paper by Ecker. They describe GP with positive or zero degree
of difficulty. But there may be some problems on GP with negative degree of
difficulty. Sinha et al. proposed it theoretically. Abot-El-Ata and his group
applied modified form of GP in inventory modelsPark and Wang studied shortages
and partial backlogging of items. Friedman (1978) presented continuous time
inventory model with time varying demand. Ritchie (1984) studied in inventory
model with linear increasing demand. Goswami, Chaudhuri (1991) discussed an
inventory model with shortage. Gen et. Al. (1997) considered classical inventory
model with Triangular fuzzy number. Yao and Lee (1998) considered an economic
production quantity model in the fuzzy sense. Kumar, 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 and 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). Kotb, Fergancy (2011), presented
Multi-item EOQ model with both demand-depended unit cost and varying Lead time via Geometric Programming. Dey and Roy (2015) presented Optimum shape design of structural model with
imprecise coefficient by parametric geometric programming.
In this paper we first considered crisp inventory model.
There after it transformed to fuzzy inventory mode and developed. Here two
necessary theorems have been derived. At
last it made an example and solved it by Parametric Geometric-Programming
Technique.
2. MATHEMATICAL MODEL
An Inventory model is developed
under the following notations and assumptions.
2.1.
Notations
I(t): Inventory level at any
time, t≥0.
D: Demand per unit time, which
is constant.
T: Cycle of length.
S: Set-up cost per unit time.
H: Holding cost per unit item,
which is time depended.
P: Unit demand and set-up cost
dependent production cost.
q: Production quantity per
batch.
f(D,S): Unit production cost per
cycle.
TAC(D,S,q): Total average cost
per unit time.
w0: Space area per
unit quantity.
W: Total storage space area.
2.2.
Assumptions
a) The inventory system involves
only one item.
b) The replenishment occur
instantaneously at infinite rate.
c) The lead time is negligible.
d)
Demand rate is constant.
e)
The unit production cost is continuous function of demand and Set-up cost and
take
the following form:
P = θ, θ,x (>0).
f)
Holding cost is time depended, as “at”.
2.3.
Crisp model
Figure1:
Inventory Model
The differential equation describing
I(t) as follows
, 0≤t≤T (2.3.1)
With the boundary condition I(0) = q,
I(T) = 0.
The solution of (2.3.1) is obtained as
I(t) = q – Dt
(2.3.2)
Also there are T= q/D.
Here inventory holding cost = H = . (2.3.3)
Total inventory related cost per cycle =
set-up cost + holding cost + production cost
= S
+ +Pq (2.3.4)
So total average cost per cycle is given
by
TAC(D,S,q)
+ + θ. (2.3.5)
And storage area = w0q.
So the inventory model can be written as
Min TAC(D,S,q) + + θ (2.3.6)
subject to w0q ≤ W, D,S,q > 0.
2.4.
Fuzzy model
When the objective and constraint goals, coefficients and exponents
become fuzzy sets and fuzzy Numbers respectively, the crisp model (2.3.6)
written to be a fuzzy model, as
TAC(D,S,q) =
+ +
subject to q ,
D,S,q > 0.
(2.4.1)
3. GEOMETRIC PROGRAMMING (GP) PROBLEM
3.1.
Primal program
Primal
Geometric Programming (PGP) problem is:
Minimize
(t) =
(3.1)
subject to (t) =
(r=1,2,…,l)
tj> 0, (j=1,2,….,m).
Where
C0k(>0) (k=1,2,…..,T0), Crk(>0) and αrkj
(k=1,2,….,1+Tr-1,…..,Tr;r = 0,1,2,……,l; j = 1,2,….,m) are
real numbers. It is constrained polynomial PGP problem. The number of term each
polynomial constrained functions varies and it is denoted by Tr for
each r = 0,1,2,…,l. Let T = T0+
T1+ T2+…. Tl be the total number of terms in
the primal program. The Degree of Difficulty is (DD) = T – (m+1).
3.2.
Dual program:
Dual programming (DP) problem of (3.1)
is:
Maximize d(δ) =
(3.2)
Subject to = 1 (Normality
condition)
= 0, (Orthogonal conditions)
δrk
> 0, (k=1,2,……,Tr) (Positivity constant)
·
Case-1
: For T0 ≥ M+1, the dual
program presents a system of linear equations for the dual variables, where the
number of linear equations is either less than or equal to dual variables. More or unique solutions exist for the dual
vectors.
·
Case-2 : For
T0 < M+1, the dual program presents a system of linear equations
for the dual variables, where 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 one
can get an approximate solution vector
for the system using either the Latest Square(SQ) or Max-Min(MN) method.
These
are applied to solve such a system of linear equations. Ones optimal dual variable vector are known, the corresponding values of the
primal variable vector x is found from the following relations:
= if i
and = if i (k=1,2, …….., ).
3.3.
Solution procedure of crisp
model by Geometric Programming (G.P) technique:
Here the primal problem is
Min TAC(D,S,q)
= + + θ (3.1.1)
subject to w0q ≤ W, D ,S, q > 0.
Corresponding
dual form of (3.1.1) is given by
Max d() =
subject to = 1
(3.1.2)
= 0
+ (1-x)= 0
+ 2= 0
0.
From (3.1.2) we get .
Putting the values
in (3.1.2) we get the optimal solution of dual problem. The values of D, S, q
is obtained by using the primal dual relation as follows.
From primal dual relation we get
= (,
= (,
= (,
= 1.
The optimal solution of the
model through the parametric approach is
given by
(
and S*
,
D* ,
q*
.
4. FUZZY NUMBER AND ITS NEAREST INTERVAL APPROXIMATION:
4.1.
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-cut of The α-cut of , is defined by Aα={x:
μA(x)≥α.}
Figure2:
Trapezoidal fuzzy number of with α-cut.
Aα
is a non-empty bounded closed interval in X and it can be denoted by Aα
= [AL(α), AR(α)]. Where AL(α) and AR(α)
are the lower and upper bounds of the closed interval respectively.
Figure
2 shows a fuzzy number with α-cuts Aα1 = [AL(α1),
AR(α1)], Aα2 = [AL(α2),
AR(α2)]. It Seen that if α2 ≥ α1
then AL(α2) ≥ AL(α1) and AR(α1)
≥ AR(α2).
4.2.
Interval number
An
interval number A is defined by an ordered pair of real numbers as follows A =
[ where andare the left and 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.
4.3.
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 is a fuzzy number. We have to find a closed
interval , which is the nearest to with
respect to 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 to a given fuzzy number is [. In the centre and half–width
form the interval number of is defined as .
4.4.
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 strictly
monotone, continuous function and its inverse exits. Let be the inverse of g(s), then
s.
5. GEOMETRIC PROGRAMMING WITH FUZZY COEFFICIENT:
When
all coefficients of Eq. (6) are triangular fuzzy number, then the geometric
programming problem is of the form
Min
(x)
(5.1)
subject to (x) (1≤i≤n)
x>0.
Its objective functions
(x) =
and constraints of the form
(x) = (0≤i≤n)
are all posynomials of x in
which coefficients and indexes
are fuzzy numbers.
Where = ( and = (. Using nearest interval
approximation method, we transform all triangular fuzzy number into interval
number i.e. [and [. The geometric programming
problem with imprecise parameters is of the following form
Min
(x) (5.2)
subject
to (x) 1
(1≤i≤n)
x>0,
Its objective function is
(x) = ,
and constraints of the form is
(x) = , (1≤i≤n).
Where denote the interval counterparts i.e. ∈ [ and∈ [
for all i and k. Using parametric
interval-valued functional form, the problem (5.2) reduces to
Min (x,s) = (5.3)
Subject to
(x,s) =
xj > 0 for i = 1,2,…….n, j = 1,2,……..m.
This
is a parametric geometric programming problem. We get different solutions of
this problem for different value of the parameter s.
the dual programming of (5.3) is as follows:
Max d(δ,s)
= (5.4)
Subject to
= 1,
= 0,
> 0.
5.1.
Theorem 5.1.
If x is a feasible vector for the
constraints PGP and if δ is a feasible vector for the corresponding DP, then (x,s) ≥ d(δ,s) (Primal- Dual
Inequality).
5.2.
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 (5.1.1)
Again
(x,s) can be written as
(x,s) = .
Now
applying A.M.-.G.M inequality, we get
Or
(5.1.2)
Using
1, [as
(x,s) 1]
We
have
1
(5.1.3)
Multiplying
(5.1.1) and (5.1.3) we get
(5.1.4)
Using
orthogonal condition the inequality (5.1.4) becomes
= d(δ,s) (5.1.5)
i.e.,
(x,s) ≥ d(δ,s) . (Proof).
5.3.
Theorem 5.2.
δ is a feasible vector for the dual
programming (DP) problem , then d(δ,1) ≥
d(δ,0).
5.4.
Proof:
We
have , for
all k, (k=1,2,…….,).
Or
Or
Or
Or
Or
i.e.,
d(δ,1) ≥ d(δ,0). (proof)
5.5.
Solution procedure of fuzzy
model by Geometric Programming (G.P) technique:
When =
(), =
(), =
() and =
() are triangular fuzzy number .then the
fuzzy model is
TAC(D,S,q) =
+ + (5.3.1)
subject to
q , D, S, q > 0.
Using
nearest interval approximation method, the interval number corresponding
triangular number = () is [] = []. Similarly interval number
corresponding and are [] = [], [] = [] and [] = [] respectively. The problem
(5.3.1) reduces to
Min TAC(D,S,q) = + + (5.3.2)
subject to w0q ≤ [],D, S, q > 0.
Which
is equivalent to
Min TAC(D,S,q) =
+ + (5.3.3)
subject to q , D, S, q > 0,
where ∈
[], ∈
[], ∈
[] and ∈
[].
According
to section 4.4, the fuzzy model (5.2.3) reduces to a parametric programming by
replacing
where s ∈
[0,1].
The model
takes the reduces form as follows
Min TAC(D,S,q) =
+ +
(5.3.4)
subject to q
, D,S,q > 0
Corresponding
dual form of (5.3.4) is given by
Max d() =
subject to = 1 (5.3.5)
= 0
+ (1-x)= 0
+ 2= 0
≥ 0.
From
(5.3.5) we get .
Putting
the values in (5.3.5) we get the optimal solution of dual problem. The values
of D, S, q is obtained by using the primal dual relation as follows:
From
primal dual relation we get
= (,
= (,
= (,
= 1.
The
optimal solution of the model through
the parametric approach is given by
(
and
S* = ,
D* = ,
q* = .
6. NUMERICAL EXAMPLE AND SOLUTION:
A
manufacturing company produces a machine. It is given that the inventory
carrying cost of the machine is $15 per unit per year. The production cost of
the machine varies inversely with the demand and set-up cost. From the past
experience, the production cost of the machine is 120 where D is the demand rate and S is set-up
cost. Storage space area per unit time () and total storage space area
(W) are 100 sq. ft. and 2000 sq. ft. respectively. Determine the demand rate
(D), set-up cost (S), production quantity (q), and optimum total average cost
(TAC) of the production system.
Then the input value of the
model (2.3.6) is
Table-1(Input values)
a |
H |
x |
θ |
|
W |
7 |
15 |
1.75 |
120 |
100 |
2000 |
Then the model is of the form
Min TAC(D,S,q) = + +
subject to 100q ≤ 2000, D, S, q
> 0. (6.1)
Table 2: Optimal solution of (2.3.6) for crisp model
Crisp model |
S* |
D* |
q* |
TAC*(S*,D*,q*)$ |
G.P |
0.684 |
4048 |
20 |
140.517 |
N.L.P |
0.685 |
4047 |
20 |
140.685 |
When the input data
of inventory model is taken as triangular fuzzy number i.e.. = (5,7,9), =
(13,15,17), = (116,120,124) and = (1800,2000,2200). Using
nearest interval approximation method, we get the corresponding interval number
and interval-valued function i.e.
≈ [6,8], ∈ [6,8],
≈ [14,16], ∈ [14,16],
≈ [118,122], ∈ [118,122],
≈ [1900,2100], ∈ [1900,2100], where
s ∈
[0,1].
The optimal solution
of the fuzzy model by interval-valued parametric geometric programming is
presented in Table
Table 3: Optimal Solution for Fuzzy Inventory Model
s |
S* |
D* |
q* |
TAC*(S*,D*,q*)$ |
0.0 |
0.820 |
2983.86 |
21.00 |
119.801 |
0.1 |
0.786 |
3175.16 |
20.79 |
123.060 |
0.2 |
0.753 |
3378.71 |
20.58 |
126.396 |
0.3 |
0.722 |
3595.32 |
20.38 |
129.924 |
0.4 |
0.693 |
3825.81 |
20.18 |
133.735 |
0.5 |
0.664 |
4071.08 |
19.97 |
137.533 |
0.6 |
0.637 |
4332.07 |
19.78 |
141.519 |
0.7 |
0.610 |
4609.80 |
19.58 |
145.473 |
0.8 |
0.585 |
4905.33 |
19.38 |
149.793 |
0.9 |
0.561 |
5219.81 |
19.19 |
154.196 |
1.0 |
0.538 |
5554.45 |
19.00 |
158.767 |
For s=0, the lower
bound of the interval value of the parameter is used to find the optimal
solution. For s=1, the upper bound of interval value of the parameter is used
for the optimal solution. These results yield the lower and upper bounds of the
optimal solution. The main advantage of the proposed technique is that one can
get the intermediate optimal result using proper value s.
Here we have given a
rough graph, which shown how change the value of TAC*(S*,D*,q*) for difference values of s.
Figure 3: Change of the value of objective function
for change of s, by Fuzzy Geometric Programming Technique.
7. SENSITIVITY ANALYSIS:
Effect, for increment the parameter “s”.
(1)
For
increasing of “s”, set-up cost S* is decreasing.
(2) For increasing of “s”, demand rate D* is
increasing.
(3) For increasing of “s , Production
quantity q* is decreasing.
(4)
For
increasing of “s”, Total average cost TAC*(S*,D*,q*)
is increasing.
8. CONCLUSION:
In this paper, we have proposed a real
life inventory problem in a fuzzy environment and presented solution along with
sensitivity analysis approach. The inventory model developed with unit
production cost, time depended holding cost, with-out shortages. This model has
been developed for single item.
In
this paper, we first create a crisp model then it transformed to fuzzy model
and solved by parametric Geometric-Programming technique.
Here decision maker may obtain the optimal results according to his expectation
.In fuzzy we have
considered triangular fuzzy number(T.F.N) In future, the other type of
membership functions such as piecewise linear hyperbolic, L-R fuzzy number, Trapezoidal Fuzzy Number (TrFN), Parabolic flat Fuzzy
Number (PfFN), Parabolic Fuzzy Number (pFN), pentagonal fuzzy number etc can be
considered to construct the membership function and then model can be easily
solved.
ACKNOWLEDGEMENTS
The authors are thankful to
University of Kalyani for providing financial assistance through DST-PURSE
Programme. The authors are grateful to the reviewers for their comments and
suggestions.
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