* Corresponding Author. Email address: [email protected]; Tel: +91 9153222799 1
A Fuzzy E.O.Q Model With Unit Production Cost, Time Depended
Holding Cost, With-Out Shortages Under A Space Constraint: A
Fuzzy Geometric Programing (FGP) Approach
Sahidul Islam1, Wasim Akram Mandal2*
(1) Department of mathematics, University of kalyani, kalyani, W.B, India
(2) Beldanga D.H.Sr.Madrasah, Beldanga-742189, Murshidabad, WB, India
Copyright 2017 Β© Sahidul Islam and Wasim Akram Mandal. 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
In this paper, an Economic Order quantity (E.O.Q) model with unit production cost, time depended
holding cost, with-out shortages is formulated and solved. In most real world situation, the objective
and constraint function of the decision makers are uncertainty in nature so the coefficients, indices the
objective function and constraint goals are imposed here in fuzzy environment The problem is then solved
using both Fuzzy Max-Min Geometric-Programming technique and Fuzzy parametric Geometric-
Programming. Sensitivity analysis is also presented here.
Keywords: E.O.Q model, Fuzzy set, Max-Min operator, Geometric Programming, Parametric Geometric
Programming Technique.
2010 Mathematics Subject Classification: 90B05, 90C70.
1 Introduction
An inventory deals with decision that minimize the cost function or maximize the profit function. For
this purpose the task is to construct a suitable mathematical model of the real life Inventory system, such a
mathematical model is based on various assumption and approximation.
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Volume 2017, Issue 1, Year 2017 Article ID ojids-00009, 14 Pages
doi:10.5899/2017/ojids-00009
Research Article
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In ordinary inventory model it considers all parameter like set-up cost, carrying cost, interest cost,
shortages etc as a fixed. But in real life situation it will have some little fluctuations. So consideration of
fuzzy variables is more realistic.
Since late 1960βs, 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 (NLP) problem. The theory of Geometric Programming (GP) first emerged in 1961 by
Duffin and Zener and it further developed by Duffin. Duffin R J, Peterson E L, Zener C M studied
Geometric Programming-Theory and Application. Friedman (1978) developed continuous time inventory
model with time varying demand. Ritchie (1984) studied in inventory model with linear increasing
demand. Goswami, Chaudhuri (1991) presented an inventory model with shortage. Gen et. Al. (1997)
proposed classical inventory model with Triangular fuzzy number. Yao and Lee (1998) presented an
economic production quantity (EPQ) model in the fuzzy sense. Sujit Kumar De, P.K.Kundu and
A.Goswami (2003) discussed an economic production quantity (EPQ) inventory model involving fuzzy
demand rate. J.K.Syde and L.A.Aziz (2007) applied sign distance method to fuzzy inventory model
without shortages. D.Datta and Pravin Kumar published several papers of fuzzy inventory with or without
shortage. S. Islam, T.K. Roy (2006) presented a fuzzy EPQ model with flexibility and reliability
consideration and demand depended unit Production cost under a space constraint
S. Islam, T.K. Roy (2010), studied Multi-Objective Geometric-Programming Problem and its Application.
A.M Kotb, Halaa, Fergancy (2011) presented Multi-item EOQ model with both demand-depended unit
cost and varying Lead time via Geometric Programming approach. Samir Dey and Tapan Kumar Roy
(2015) proposed Optimum shape design of structural model with imprecise coefficient by parametric
geometric programming.
In this paper we consider crisp inventory model, there after it transformed to fuzzy inventory mode and
developed by geometric programming approach. First we solved the model by Fuzzy Max-Min Geometric-
Programming technique and then it solved by Fuzzy parametric Geometric -Programming technique. At
last it made an example and solved it by both 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, tβ₯0.
D: Demand per unit time, which is constant.
T: Cycle of length of the given inventory.
S: Set-up cost per unit time.
H: Holding cost per item per unit time, 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 of the inventory.
2.2. Assumptions
a) The inventory system involves only one item.
b) The replenishment occurs instantaneously at infinite rate.
c) The lead time is negligible.
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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= ΞΈπ·βπ₯πβ1, ΞΈ,xβ β (>0).
f) Holding cost time depended as at.
2.3. Crisp model
q
D
0 T
Figure 1: E.O.Q Model
The differential equation describing I(t) as follows ππΌ(π‘)
ππ‘= βπ· , 0β€tβ€T (2.1)
With the boundary condition I(0) = q and I(T) = 0.
The solution of (2.1) is obtained as
I(t) = q β Dt (2.2)
Also there are
T = q/D.
Now holding cost = Hβ« ππ‘. πΌ(π‘)ππ‘π
0 =
ππ»π3
6π·2 (2.3)
Total inventory related cost per cycle = set-up cost + holding cost + production cost
= S + ππ»π3
6π·2 +pq (2.4)
i.e., total average cost per cycle is given by
TAC(D,S,q) = ππ·
π +
ππ»π3
6π·2 + ΞΈπ·1βπ₯πβ1 (2.5)
And storage area = w0q.
So the inventory model can be written as,
Min TAC(D,S,q) = ππ·
π +
ππ»π2
6π· + ΞΈπ·1βπ₯πβ1 (2.6)
subject to w0q β€ W, D,S,q > 0.
2.4. Fuzzy model
When the objective constraint goals and coefficients become fuzzy sets and fuzzy numbers, then the crisp
model (2.6) written to be a fuzzy model, as
πποΏ½οΏ½ TAC(D,S,q) = ππ·
π +
οΏ½οΏ½οΏ½οΏ½π2
6π· + οΏ½οΏ½π·1βπ₯πβ1
subject to π€0q β² W , D,S,q > 0. (2.7)
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3 Some basic concept & definition
3.1. Pre-requisite mathematics
Fuzzy sets first introduced by Zadeh [18] in 1965 as a mathematical way of representing vagueness in
every life.
Definition 3.1. A fuzzy set οΏ½οΏ½ on the given universal set X is a set of order pairs, οΏ½οΏ½={(x,ποΏ½οΏ½(x): xΟ΅X} where
ποΏ½οΏ½ (x)β[0,1] is called a membership function.
Definition 3.2. The Ξ±-cut of οΏ½οΏ½, is defined by AΞ± ={x: ποΏ½οΏ½(x) β₯ Ξ±}
ΞΌA(x)
1
πΌ2 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦
πΌ1 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.
0 AL(Ξ±1) AL(Ξ±2) β¦β¦β¦β¦β¦.. AR(Ξ±2) AR(Ξ±1) x
β¦β¦β¦β¦β¦ core
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..
Ξ-cut
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦
support
Figure 2: Trapezoidal fuzzy number of οΏ½οΏ½ with Ξ±-cuts.
AΞ± is a non-empty bounded closed interval in X and it can be denoted by AΞ± = [AL(Ξ±), AR(Ξ±)]. Where AL(Ξ±)
and AR(Ξ±) are the lower and upper bounds of the closed interval respectively. Figure 2 shows a fuzzy
number οΏ½οΏ½ with Ξ±-cuts AΞ±1 = [AL(πΌ1), AR(πΌ1)], AΞ±2 = [AL(πΌ2), AR(πΌ2)]. It Seen that if πΌ2β₯ πΌ1 then AL(πΌ2)
β₯ AL(πΌ1) and AR(πΌ1) β₯ AR (πΌ2).
Definition 3.3. A is normal if there exists xΟ΅X such that ΞΌA(x) =1.
3.2. Mathematical analysis
Consider a non-linear programming (NLP) as follows,
Min g0(x)
Subject to gi(x) β€ 1 (1β€iβ€n), (3.8)
x>0.
Its objective and constraints of the form
gi(x) = β πΆπππππ=1 β π₯π
πΌπππππ=1 (0β€iβ€n)
xj> 0, (J=1,2,β¦.,m)
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Here πππ(>0), (k=1,2,β¦β¦.,T0) and πΌ ikj be any real numbers. When the objective and constraint goals,
coefficients and exponents become fuzzy sets and fuzzy numbers respectively, then we transform (3.8) into
a fuzzy geometric programming as follows,
πποΏ½οΏ½ g0(x)
subject to gi(x) β² 1 (1β€iβ€n) (3.9)
x>0,
Its objective and constraints of the form
gi(x) = β οΏ½οΏ½πππππ=1 β π₯π
οΏ½οΏ½πππππ=1 (0β€iβ€n)
are all posynomials of x in which coefficients οΏ½οΏ½ππ and indexes οΏ½οΏ½πππ are fuzzy numbers.
3.2.1. Some definitions and theorems
Definition 3.4. For n-th parabolic flat fuzzy number (a1,a2,a3,a4)PfFN containing the coefficients
οΏ½οΏ½ππ (0β€iβ€n; 1β€kβ€Ti), the membership function of οΏ½οΏ½ππ is
ΞΌcik (cik) =
{
1 β (
a2βcik
a2βa1)n for a1 β€ cik β€ a2
1 for a1 β€ cik β€ a2
1 β (cikβa3
a4βa3)n for a3 β€ cik β€ a4
0 for otherwise.
(3.10)
Similarly, we can determine the membership function of the indexes οΏ½οΏ½πππ (0β€iβ€n; 1β€kβ€Ti; 1β€iβ€m).
Note:
(a) when n=1, οΏ½οΏ½ππ become Trapezodial Fuzzy Number (TrFN),
(b) when n=1, and a3=a4, οΏ½οΏ½ππ become Triangular Fuzzy Number (TFN),
(c) when n=2, οΏ½οΏ½ππ become Parabolic flat Fuzzy Number (PfFN),
(d) when n=2, and a3=a4, οΏ½οΏ½ππ become Parabolic Fuzzy Number (pFN),
Definition 3.5. Here Ξ΄-cut of οΏ½οΏ½ππ (0β€iβ€n; 1β€kβ€Ti) is given by
ΞΌcikβ1(Ξ΄) = [ ΞΌcikL
β1(Ξ΄), ΞΌcikRβ1(Ξ΄)] = [a1 + β1 β Ξ΄
n(a2 β a1), a4 - β1 β Ξ΄
n(a4 β a3)]. (3.11)
Similarly, we can determine the Ξ΄-cut of οΏ½οΏ½πππ (0β€iβ€n; 1β€kβ€Ti; 1β€iβ€m).
Proposition 3.1. When the coefficients and indexes of the fuzzy geometric programming problem are
taken as fuzzy numbers, then
πποΏ½οΏ½ β οΏ½οΏ½πππππ=1 β π₯π
οΏ½οΏ½πππππ=1
subject to β οΏ½οΏ½πππππ=1 β π₯π
οΏ½οΏ½πππππ=1 β²1 (1β€iβ€n), (3.12)
xj > 0,
using Ξ΄-cut of fuzzy numbers coefficients and indexes, the above problem is reduces to the following form
πποΏ½οΏ½ β [ πππππΏβ1(πΏ), πππππ
β1(πΏ)]π0π=1 β π₯π
[ποΏ½οΏ½ππππΏβ1(πΏ),ποΏ½οΏ½ππππ
β1(πΏ)]ππ=1
subject to β [ πππππΏβ1(πΏ), πππππ
β1(πΏ)]πππ=1 β π₯π
[ποΏ½οΏ½ππππΏβ1(πΏ),ποΏ½οΏ½ππππ
β1(πΏ)]ππ=1 β² 1 (1β€iβ€n),
xj > 0,
Which is equivalent to
πποΏ½οΏ½ β πππππΏβ1(πΏ)
πππ=1 β π₯π
ποΏ½οΏ½ππππβ1
(πΏππ=1 )
subject to β πππππΏβ1(πΏ)
πππ=1 β π₯π
ποΏ½οΏ½ππππβ1
(πΏππ=1 ) β€ 1 (1β€iβ€n) (3.13)
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where
ποΏ½οΏ½ππππβ1(πΏ) = {
ποΏ½οΏ½ππππΏβ1(πΏ) π€βππ οΏ½οΏ½ππππΏ > 0,
ποΏ½οΏ½ππππ β1(πΏ) π€βππ οΏ½οΏ½ππππΏ < 0,
(1β€iβ€n)
Definition 3.6. For any x β βm and feasible index di β β (β is the real number set), if
gi(x,πΏ) = β πππππΏβ1(πΏ)
πππ=1 β π₯π
ποΏ½οΏ½ππππβ1
(πΏππ=1 ) β€ 1 (1β€iβ€n),
then the linear membership function are given by
π0(g0(x,πΏ)) = {
1 ππ g0(x, πΏ) β€ π§0,
(π§0+π0βg0(x,πΏ)
π0) ππ π§0 β€ g0(x, πΏ)
0 ππ g0(x, πΏ) β₯ π§0 + π0,
β€ π§0 + π0, (3.14)
πi(gi(x,πΏ)) = {
1 ππ gi(x, πΏ) β€ π§0,
(1+ππβi(x,πΏ)
π0) ππ 1 β€ gi(x, πΏ)
0 ππ gi(x, πΏ) β₯ 1 + ππ,
β€ 1 + ππ , (3.15)
Based on Zimmerman, first finding πΏ-cut of the fuzzy numbers in coefficients and indexes then we built
membership functions of both objective and constraints goals and using max-min operator the above
problem (3.13) reduced to a fuzzy Non-Linear Programming (FNLP) problem
Max π
subject to ππ(β πππππΏβ1(πΏ)
πππ=1 β π₯π
ποΏ½οΏ½ππππβ1
ππ=1 (πΏ)) β₯ π (1β€iβ€n), (3.16)
x > 0, π, πΏ β [0,1],
which is equivalent to a geometric programming problem with parameters π, πΏ variation
Min πβ1
subject to ππ(β πππππΏβ1(πΏ)
πππ=1 β π₯π
ποΏ½οΏ½ππππβ1
ππ=1 (πΏ)) β₯ π (1β€iβ€n), (3.17)
x > 0, π, πΏ β [0,1],
Theorem 3.1. Let the membership function πi(gi(x,πΏ)), ππππ(cik), ποΏ½οΏ½πππ(πΌikj) be all continuous and strictly
monotone. Then (4.1b.4.4) is equivalent with following form
Min πβ1
subject to β ποΏ½οΏ½πππΏ
β1(πΏ)πππ=1
β π₯πποΏ½οΏ½ππππ
β1
(πΏ)ππ=1
ππβ1(πΏ)
β€ 1,
x > 0, π, πΏ β [0,1], (0β€iβ€n, 1β€jβ€m).
Proof. Pls. see reference [13] S.Islam, T.K. Roy (2006).
Corollary 3.1. Let the membership function πi(gi(x,πΏ)), ππππ(cik), ποΏ½οΏ½πππ(πΌikj) be all continuous and strictly
monotone and the problem reduce to
Min πβ1
subject to β ποΏ½οΏ½πππΏ
β1(πΏ)πππ=1
β π₯πποΏ½οΏ½ππππ
β1
(πΏ)ππ=1
ππβ1(πΏ)
β€ 1, (3.18)
x > 0, π, πΏ β [0,1], (0β€iβ€n, 1β€jβ€m).
which is a classical posynomial geometric programming (GP) with parameters πΎ, πΏ. Its dual form is
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Max d(π) = (πβ1
π00)π00 β β (
ποΏ½οΏ½ππβ1(πΏ)/ππ
β1(π)
πππ)πππ
πππ=1
ππ=0 (3.19)
subject to π00 = 1,
π00 = β π0ππ0π=1
(Ξ(Ξ΄))ππ = 0, π, πΏ β [0,1],
π β₯ 0
Where πππ = πππ(Ξ΄,π)
and Ξ(Ξ΄) =
(
οΏ½οΏ½011β1(πΏ)β¦ οΏ½οΏ½01π
β1(πΏ)β¦ οΏ½οΏ½01πβ1(πΏ)
β¦ β¦ β¦οΏ½οΏ½0π½01
β1(πΏ)β¦ οΏ½οΏ½0π½01β1(πΏ)β¦ οΏ½οΏ½0π½01
β1(πΏ)β¦ β¦ β¦
οΏ½οΏ½π11β1(πΏ)β¦ οΏ½οΏ½π1π
β1(πΏ)β¦ οΏ½οΏ½π1πβ1(πΏ)
β¦ β¦ β¦οΏ½οΏ½ππ½π1
β1(πΏ)β¦ οΏ½οΏ½ππ½π1β1(πΏ)β¦ οΏ½οΏ½ππ½π1
β1(πΏ))
4 Solution procedure of fuzzy model
4.1. Fuzzy MAX-MIN Geometric Programming Technique on EOQ Model
When coefficient and exponents are taken as a triangular fuzzy number i.e., in general οΏ½οΏ½ = (π1, π2, π3).
Then the πΏ-cut of the fuzzy number π, is given by
π(πΏ) = [π1 + πΏ(π2 β π1), π3 β πΏ(π3 β π2)], πΏ β [0,1].
Taking the membership function as in (3.14) and (3.15) turn the problem (2.7) into (3.18) and Obtained,
Min πβ1
subject to βππ·πβ1β(π»1+πΏ(π»2βπ»1))ππ2π·β1/6β(π1+πΏ(π2βπ1))π·1βπ₯πβ1
(β(π§0+π0β1)+π0π) β€ 1 (4.20)
(π€0
1+πΏ(π€02βπ€0
1))π
π+π1π β€ 1
D,S,q > 0, πΎ, πΏ β [0,1].
The dual form of (4.c1.1) is given by
Max d(π) =
(πβ1
π00)π00
(
1
π01
(π§0+π0β1)βπ0π)
π01
(
(π»1+πΏ(π»2βπ»1))π
6π02
(π§0+π0β1)βπ0π)
π02
(
(π1+πΏ(π2βπ1)
π03
(π§0+π0β1)βπ0π)
π03
(
(π€01+πΏ(π€0
2βπ€01)
π11
π+π1π)
π11
subject to
π00 = 1,
π01+π02+π03+π11 = π00, (4.21)
π01 β π₯π03 = 0
π01 β π02 + (1 β π₯)π03 = 0
β π01 + 2π02 +π11= 0
π01, π02, π03, π11 β₯ 0.
From (4.21) we get π01 =1
4βπ₯, π02 =
2βπ₯
4βπ₯, π03 =
1
4βπ₯, π11 =
2π₯β3
4βπ₯
Putting the value of the objective function of the problem (4.c1.2), we get
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d(π) = πβ1 (4βπ₯
(π§0+π0β1)βπ0π)
1
4βπ₯(
(π»1+πΏ(π»2βπ»1))π(4βπ₯)
6(2βπ₯)
(π§0+π0β1)βπ0π)
2βπ₯
4βπ₯
Γ ((π1+πΏ(π2βπ1)(4βπ₯)
(π§0+π0β1)βπ0π)
1
4βπ₯((π€0
1+πΏ(π€02βπ€0
1)(4βπ₯)/(2π₯β3)
π+π1π)
2π₯β3
4βπ₯ (2π₯β3
4βπ₯)2π₯β3
4βπ₯
We can obtained π by the aid of d(π) = πβ1. Then the above equation is reduces to the following form
(4βπ₯
(π§0+π0β1)βπ0π)
1
4βπ₯(
(π»1+πΏ(π»2βπ»1))π(4βπ₯)
6(2βπ₯)
(π§0+π0β1)βπ0π)
2βπ₯
4βπ₯
Γ ((π1+πΏ(π2βπ1)(4βπ₯)
(π§0+π0β1)βπ0π)
1
4βπ₯((π€0
1+πΏ(π€02βπ€0
1)(4βπ₯)/(2π₯β3)
π+π1π)
2π₯β3
4βπ₯ (2π₯β3
4βπ₯)2π₯β3
4βπ₯ =1 (4.22)
Solving the above equation of πΎ for given πΏ β [0,1] by Newton-Raphson method, we obtain the value of
πβ. Putting the value of πβ, we obtained the values of the dual objective function.
Again from the between primal-dual relation, we get
βππ·πβ1
(β(π§0+π0β1)+π0πβ)
= π01
β
π00β =
1
4βπ₯,
β(π»1+πΏ(π»2βπ»1))ππ2π·β1/6
(β(π§0+π0β1)+π0πβ)
= π02
β
π00β =
2βπ₯
4βπ₯,
β(π1+πΏ(π2βπ1))π·1βπ₯πβ1
(β(π§0+π0β1)+π0πβ)
= π03
β
π00β =
1
4βπ₯,
(π€01+πΏ(π€0
2βπ€01))π
π+π1πβ =
π11β
π11β =1. (4.23)
Solving (4.23) we have
S* =6π1
βπ2β((π§0+π0β1)βπ0π
β)2
π(π»1+πΏ(π»2βπ»1)),
D* = π(π»1+πΏ(π»2βπ»1))π2
6π2β((π§0+π0β1)βπ0π
β),
q* = π+π1π
β
(π€01+πΏ(π€0
2βπ€01)) .
4.2. Fuzzy Parametric Geometric Programming Technique on EOQ Model
Taking οΏ½οΏ½ = π»1 + πΏ(π»2 βπ»1), οΏ½οΏ½ =π1 + πΏ(π2 β π1) , π€0 = π€01 + πΏ(π€0
2 βπ€01) , πππ οΏ½οΏ½ =π1 +
πΏ(π2 βπ1) where Ξ±β [0, 1] in (2.7). The model takes the reduces form as follows
Min TAC(D,S,q) = ππ·
π +
π(π»1+πΏ(π»2βπ»1))π2
6π· + (π1 + πΏ(π2 β π1)))π·1βπ₯ πβ1 (4.24)
subject to (π€01 + πΏ(π€0
2 βπ€01)) q β€ (π1 + πΏ(π2 βπ1)) , D,S,q > 0.
Applying geometric programming GP technique the dual programming of the problem (4.24) is
Max π(π) = (1
π1) π1(
π(π»1+πΏ(π»2βπ»1))
6π2)π2(
π1+πΏ(π2βπ1)
π3)π3(
π€01+πΏ(π€0
2βπ€01)
(π1+πΏ(π2βπ1)))π01π01
π01
subject to π1 +π2 +π3 = 1,
π1 β π3 = 0,
π1 β π2 + (1-x)π3= 0,
β π1 + 2π2 +π01= 0,
π1, π2, π3, π01 β₯ 0.
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i.e., we get π1 =1
4βπ₯, π2 =
2βπ₯
4βπ₯, π3 =
1
4βπ₯, πππ π01 =
2π₯β3
4βπ₯.
Putting the values in (4.24) 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 the primal dual relation we have, ππ·
π = π1
β Γ πβ(π),
π(π»1+πΏ(π»2βπ»1))π2
6π· = π2
β Γ πβ(π),
(π1 + πΏ(π2 β π1))π·1βπ₯πβ1 = π3β Γ πβ(π),
(π€01+πΏ(π€0
2βπ€01)) π
(π1+πΏ(π2βπ1)) = 1.
The optimal solution of the given model through the parametric approach is given by
πβ(π) = (4 β π₯)1
4βπ₯ (π(π»1+πΏ(π»2βπ»1))(4βx)
(2βπ₯)6)
2βπ₯
4βπ₯((π1 + πΏ(π2 β π1))(4 β π₯))
1
4βπ₯ Γ
((π€0
1+πΏ(π€02βπ€0
1)) (4βπ₯)
(π1+πΏ(π2βπ1))(2π₯β3))2π₯β3
4βπ₯ (2π₯β3
4βπ₯)2π₯β3
4βπ₯ . (4.25)
and
S* =6π1
βπ2βπβ(π)2
π(π»1+πΏ(π»2βπ»1)),
D* = π(π»1+πΏ(π»2βπ»1))π2
6π2βπβ(π)
,
q* = (π1+πΏ(π2βπ1))
(π€01+πΏ(π€0
2βπ€01)) .
5 Numerical example and solution:
A manufacturing company produces a item. It is given that the inventory holding cost of the item is $15
per unit per year. The production cost of the item varies inversely with the demand and set-up cost. From
the past experience, the production cost of the item is 120π·β3πβ1 where D is the demand rate and S is set-
up cost. Storage space area per unit item (π€0) 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.
Input values of the model (2.6) are
Table 1
a H x ΞΈ π€0 W
7 15 1.75 120 100 2000
Then the model is of the following form
Min TAC(D,S,q) = ππ·
π +
105π2
6π· + 120π·β0.75πβ1
subject to 100q β€ 2000, D,S,q > 0. (5.26)
the crisp solution of the given model is in table 2.
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Table 2: (Optimal solution of (2.6) for crisp model) 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
For fuzzy Geometric-Programming (FGP) method, lets we consider z0 = 15.5, fuzzy objective goal d0 = 1
and Total storage space area tolerance d1 = 100, also taking H =(14,16,18), ΞΈ = (116,120,124), w0
=(96,100,104) (as a fuzzy triangular number), πΏ =0.5, then from (4.c1.3) we get π = 0.007.
For Fuzzy Parametric Geometric βProgramming (FGP) Technique taking Ξ± = 0.5, H =(14,16,18), ΞΈ =
(116,120,124), w0 = (96,100,104) and W = (2000,2200,2400).
Then corresponding solution is in table 3.
Table 3: (Optimal solution of (2.7) for fuzzy model) Table-3( Optimal solution of (2.4.1) for fuzzy model)
fuzzy model S* D* q* TAC*(S*,D*,q*)
F.G.P(MAX-MIN) 0.677 4236 20.415 142.561
F.G.P(PARAMETRIC) 0..662 4718 21.428 147.780
6 Sensitivity analysis
6.1. Sensitivity test of fuzzy E.O.Q problem
We now examine to sensitivity analysis of the optimal solution of the given problem for changes of Ξ±,
keeping the other parameters unchanged. The initial data is given from the above numerical example.
Value of Ξ± % ππ πβππππ F.G.P(MAX-MIN) F.G.P(PARAMETRIC)
0.1 -80 144.939 142.077
0.2 -60 144.295 142.983
0.3 -40 143.849 143.570
0.4 -20 143.108 144.349
0.5 0 142.561 144.830
0.6 +20 141.934 145.257
0.7 +40 141.484 145.542
0.8 +60 140.802 145.775
0.9 +80 140.271 145.989
Here we have given a rough graph, which shown how change the value of TAC*(S*,D*,q*) for different
values of Ξ±.
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146 _
145 _
144 _
TAC*(S
*,D
*,q
*) 143 _
142 _
141 _
| | | | | | | | | |
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ξ±
Figure 3: Change of the value of objective function for change of Ξ± by Fuzzy Max-Min Geometric Programming
Technique.
146 _
145 _
TAC(πβ,π·β, πβ) 144 _
143 _
142 _
141 _
| | | | | | | | | |
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Ξ±
Figure 4: Change of the value of objective function for change of Ξ± by Fuzzy Parametric Geometric Programming
Technique.
6.2. Outcome of sensitivity analysis
Effect, for increment parameters-
1) Fig.3. shows that as Ξ± changes increasingly the total average cost of the given problem decreases.
2) Fig.4. shows that as Ξ± changes increasingly the total average cost of the given problem increases.
7 Conclusion
In this paper, we have proposed a real life inventory problem in crisp and fuzzy environment and
presented solution along with sensitivity analysis approach. The inventory model is developed with unit
production cost, time depended holding cost, with-out shortages. This model has been developed for a
single item.
In this paper, we first create a model then it transformed as a fuzzy model. At last we give a real example
and solved it various methods. In fuzzy we have considered triangular fuzzy number (T.F.N) and solved by
fuzzy Max-Min Geometric-Programming and fuzzy Parametric Geometric-Programming Technique. In
future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number,
Oxford Journal of Intelligent Decision and Data Science 2017 No. 1 (2017) 1-14 12
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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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