Published in FRUAA-2011, NIT Durgapur
Application of Uncertain Programming to an EOQ
Model for Imperfect Quality Items
Arindum Mukhopadhyay
Adrijit Goswami
Department of Mathematics
Indian Institute of Technology Kharagpur
Kharagpur, India
E-mail - [email protected]
Department of Mathematics
Indian Institute of Technology Kharagpur
Kharagpur, India
E-mail - [email protected]
Abstract—Uncertainty theory has got quite attention after it
was pioneered by Baoding Liu in 2007. Uncertainty is the newest
area among the concepts to express imprecise quantities which
can be expressed by the ambiguity of human language. Such
an unpredictable value can be neither accurately described by
probability theory nor fuzzy logic. In inventory models, costs are
very prominent part of the systems which determines the profit
of the organization. There are some external factors by which
there is always a chance to fluctuation in costs .Thus, costs can
be considered as another form of human uncertainty because it is
the organization’s own decision that order will be placed or not;
and if orders are placed then how much? In an inventory system,
there are alaways some imperfect items exist from lot size due
to imperfect production process, natural disasters, damages, or
many other reasons. Adopting the Uncertainty theory methodology for inventory models used by Rong, the expected value model
and chance-constrained programming model of economic order
quantity for imperfect items with Uncertain costs are developed
when the all type of the costs are assumed to be uncertain
variables. Applying the result from the Liu’s uncertainty theory
the models can be transformed into a deterministic form, and
finally we get the solution by 99-method. The effectiveness of the
proposed models can be explained by numerical example. The
paper concludes with Conclusion and scope of future work.
Index Terms—Uncertain Variable; Imperfect quality; Economic Order Quantity; Expected Value Model; Chanceconstrained Programming
I. I NTRODUCTION
The world we live in is not deterministic, namely it is not the
one in which past events fully determine future ones. So one
has to think the nature of world full of uncertainty and many
theories are developed to cope with uncertainty theoretically
and more precisely, Mathematically. An error in a scientific
measurement means the inevitable uncertainty that is present
in all measurements and they cannot be eliminated. A suitable
approach to deal with uncertainty in decision-making should
take into consideration the human subjectivity, rather than
employing only objective probability measures. This kind of
uncertainty of human behavior led to the development of a new
area in decision -analysis called Fuzzy Logic. The concept of
a fuzzy set has been coined by Zadeh [7] to deal with the
representation of classes whose boundaries are ill-defined, or
flexible, by means of characteristic functions taking values in
the interval [0, 1].Fuzzy methodologies are mainly useful for
approximate reasoning, mainly for the systems in which there
is some vagueness and imprecision to deal with the linguistic
variables; but these kind of model assumes that there is a possibility of occurrence of some value and a suitable membership
function is derived for that. Although it is quite useful for
decision models with incomplete information but it is a matter
of fact that there is a requirement of a lot of information to develop fuzzy model; mainly the possible values and a quantifier
to support that value. Also the fuzzy logic was not self-dual.
So there was a lot of scope to develop a theory which should
account for the almost completely unknown information. Liu
in 2007 developed Uncertainty theory which is quite able to
deal with such a high level subjective uncertainty. According
to Liu[1]-” uncertainty theory attempts to model uncertain
phenomena that themselves might be relatively invariant but
the real quantities cannot be exactly observed (e.g., oil ?eld
reserve), or they are essentially sets but their boundaries are
not sharply described”. For example-A completely unknown
number, an Arbitrary constant any word from any dictionary,
warm etc can be explained better by uncertain variable of
the uncertain theory than any other theories developed so far.
The uncertainty theory concerns an incomplete or imperfect
knowledge of something which is necessary to solve the
problem. Peng and Iwamura [3] gave a sufficient and necessary condition of uncertainty distribution. Gao [4] provided
some mathematical properties of uncertain measure. You [9]
proved some convergence theorems of uncertain sequences.
In addition, Liu and Ha [10] proved some expected value
formulas for functions of uncertain variables. As an application
of uncertainty theory, Liu [2] proposed a spectrum of uncertain
programming that is a type of mathematical programming
involving uncertain variables. A brief comparisons between
Probability, Fuzzy and Uncertainty theory are given in TABLE
I
The classical economic order quantity (EOQ) and economic
production quantity (EPQ) models were two traditional inventory model which were popular among researchers and
management professionals for their simplicity. But it is a fact
that they seem to be based on few unrealistic assumptions.
One of them is assuming that all the quantity considered is
of perfect quality. In reality, the production process is not
always free of defects.A fraction of the items produced always
comes with defects. The main key of a successful business is
TABLE I
A BRIEF COMPARISION BETWEEN PROBABILITY ,FUZZY AND
UNCERTAINTY THEORY
PROBABILITY
THEORY
FUZZY
THEORY
UNCERTAINTY
THEORY
Kolmogorov
Zadeh
Liu
Based on
Probability measure
Based on
Possibility Measure
Based on
Uncertain Measure
Self-Dual
Not Self-Dual
Self-Dual
Deals with the
values based on
observed results
Deals with known but
imprecise values with
un-sharp boundaries
Deals with unpredictable
values with imprecise
boundaries.
to provide the customer his demand within shortest possible
time, with the best quality, and all at a competitive price.
That will be easier if the price of production can be reduced
by taking advantage of some discount and the production
process involves 100 % screening, so that there is no chance
of presence of defective items in the product. During last
decades there have been a lot-of work done in the area of
EOQ and EPQ of Imperfect quality items. Rosenblatt and Lee
[12] discussed an EPQ model where they assumed that the
defective items could be reworked instantaneously at a cost
and found that the presence of defective products motivates
smaller lot sizes. Shwaller [13] presented a procedure and
assumed that imperfect quality items are present in a known
proportions and considered fixed and variable inspection costs
are applied for finding and removing the item. Zhang and
Gherchak [14] considered a joint lot sizing and inspection
policy here a random proportion of lot size were defective.
They assumed that defective items are not reworkable and
thus assumed the concept of replacement of them by the
good quality items Salameh and Jaber [15] assumed that the
defective items could be sold at a discounted price in a single
batch by the end of the 100 % screening process and found
that the economic lot size quantity tends to increase as the
average percentage of imperfect quality items increases. Goyal
et al. [16] made some modifications in the model of Jaber et
al. to calculate actual cost and actual order quantity. Chang
[17] considered inventory problem for items received with
imperfect quality, where, upon the arrival of order lot, 100%
screening process is performed and the items of imperfect
quality are sold as a single batch at a discounted price, prior
to receiving the next shipment. He assumed defective rate as a
fuzzy number. He also presented a model with fuzzy defective
rate and fuzzy demand. Papachristos et al[18] pointed out that
the sufficient conditions for prevent shortages given in Jaber
et al. may not really prevent their occurrence and considering
the timing of withdrawing the imperfect quality items from
stock, they clarified a point not clearly stated in Salameh and
Jaber. Wee et al [19] develops a optimal inventory model
for items with imperfects quantity and shortage backorder.
They allow 100% screening of items which is greater than
the demand rate.Chung [20] et al considered an inventory
model with imperfect quality items under the condition of two
warehouses for storing items. Jaber et al. [21] incorporated
the concept of entropy cost in the extension of the inventory
model with imperfect quality and they supposed that there two
different types of holding costs for the items perfect quality
and imperfect quality. Jaber et al.[22] also considered the
assumptions of learning curve and assumed that percentage
of defective lot size reduces according to the learning curve.
The paper is organized as follows: Section I contains
introduction and brief literature review. In section II some
preliminary concepts of Uncertainty theory are discussed.
Section III contains mathematical model and incorporation of
Uncertain Programming. Section IV contains two examples
to describe the proposed models and Section V contains
conclusion and scope of future work.
II. PRELIMINARIES
A. Basic Definitions
Let Γ be a nonempty set, and let A be a σ-algebra over
Γ. Each Λ ∈A is called an event . In order to provide an
axiomatic definition of uncertain measure, it is necessary to
assign to each event Λ; a number M{Λ}which indicates the
level that Λ will occur. In order to ensure that the number
M{Λ}has uncertain mathematical properties, Liu [1] proposed
the following four axioms:
Axiom 1 (Normality) M{Γ }= 1.
Axiom 2 (Monotonicity) M{Λ1 } ≤ M{Λ2 } wheneverΛ1 ⊆
Λ2 .
Axiom 3 (Self-Duality) M{Λ} + M{Λc } = 1 for any
event Λ .
Axiom 4 (Countable Subadditivity) For every countable
sequence of events{Λi }, we have
M{
∞
∪
i=1
Λi } ≤
∞
∑
M {Λi }
(1)
i=1
Definition 1 (Liu [1]) The set function M is called an
uncertain measure if it satisfies the normality, monotonicity,
self-duality, and countable subadditivity axioms.
Definition 2 (Liu [1]) An uncertain variable is measurable
function ξ from an uncertainty space (Γ, A,M) to the set of
real numbers, i.e., for any Borel set B of real numbers, the
set
{ξ ∈ B} = {γ ∈ Γ|ξ{γ} ∈ B}
(2)
is an event.
Definition 3 (Liu [1]) The uncertainty distribution Φ : R
[0, 1] of an uncertain variable ξ is defined by
Φ(x) = M {ξ ≤ x}
(3)
Definition 4 (Liu [1]) Let ξ be an uncertain variable. Then
the expected value of ξ is defined by
∫ ∞
∫ 0
E[ξ] =
M {ξ ≥ r}dr −
M {ξ ≤ r}dr
(4)
0
−∞
TABLE II
THE 99-TABLE OF ξ
TABLE IV
THE 99-TABLE OF ξi
α
0.01
0.02
.......
0.99
α
0.01
0.02
.......
0.99
x
x1
x2
.......
x99
xi
xi1
xi2
.......
xi99
TABLE V
THE 99-TABLE OF f (ξ1 , ξ2 , ξ3 , ...., ξn )
TABLE III
THE 99-TABLE OF f (ξ)
α
f (x)
0.01
f (x1 )
0.02
.......
f (x2 )
.......
α
0.99
f (x1 , x2 , ..., xn )
f (x99 )
Definition 5 (Liu [1]) Let ξ be an uncertain variable, and
α ∈ (0, 1]. Then
ξinf (α) = inf {r|M {ξ ≤ r} ≥ α}
(5)
is called the α-pessimistic value to ξ.
B. Useful Results
Theorem 1 (Liu [1]) Let ξ be an uncertain variable with
uncertainty distribution Φ. If the expected value exists, then
∫ 1
E[ξ] =
ϕ−1 (α)dα
(6)
0
Theorem 2 (Liu [1]) Let ξ be an uncertain variable with
uncertainty distribution Φ. Then its α-pessimistic value is
ξinf (α) = ϕ−1 (α)
(7)
Theorem 3 (Liu [1]) Let ξ1 , ξ2 , ξ3 , ...., ξn be independent uncertain variables with uncertainty distributions
Φ1 , Φ2 , Φ3 , ...., Φn respectively. If f is a strictly increasing
function. Then
ξ = f (ξ1 , ξ2 , ξ3 , ...., ξn )
(8)
(9)
is an uncertain variable with inverse uncertainty distribution
−1
−1
−1
ϕ−1 (α) = f (ϕ−1
1 (α), ϕ2 (α), ϕ3 (α), ...., ϕn (α))
(10)
99-method 1 (Liu [1]) An uncertain variable ξ with uncertainty distribution Φ is represented by a 99-table as Table 2.
Where 0.01, 0.02, ... . , 0.99 in the first row are the values
of uncertainty distribution Φ, and quantities in the second row
are the corresponding values of x1 , x2 , x3 , ...., xn .
It can be observed that, the 99-table is a discrete representation of uncertainty distribution Φ. Then for any strictly
increasing function f(x), the uncertain variable f (ξ) has a 99table as Table 3.
99-method 2 (Liu [1]) Let us assume ξ1 , ξ2 , ξ3 , ...., ξn
are uncertain variables, and each ξi is represented by
a 99-table as Table 4. Then for any strictly increasing function f (x1 , x2 , x3 , ...., xn ) the uncertain variable
f (ξ1 , ξ2 , ξ3 , ...., ξn ) has a 99-table as Table 5.
0.01
.......
0.99
1
2
n
f (x11 , x21 , ...., xn
1 ) ....... f (x99 , x99 , ...., x99 )
III. M ATHEMATICAL M ODEL
Economic order quantity model and its variants mainly
deal with making two decisions; one of them is regarding
suitable amount of order quantity and another is suitable time
to reorder. The optimal decision is one which either minimize
total cost or maximize total profit.
A. Brief review of the crisp model for imperfect quality items
We are reusing a modified and simplified form of Salameh
and Jaber’s[15] model . Notations: Following notations are
used in the Crisp model- λ : demand rate;
s : screening rate; s>λ
h : holding cost per unit, per unit time;
a : ordering cost ;
p : purchasing cost per unit;
w : screening cost per unit;
Assumptions:
Following assumptions are considered in the model1) Replenishment is instantaneous; Lead time is zero.
2) The time horizon is infinite.
3) Demand is constant.
4) The lot size y is a decision variable in the model.
5) The defective percentage p is a fixed constant for a lot
size.
6) The screening process and demand proceeds simultaneously, but the screening rate is greater than demand rate,
s>λ
7) A single product is considered.
8) Inventory level reaches to zero at the end of each cycle.
The total cost is given by the following expression:
[ 2
]
y (1 − p)2
py 2
T C(y) = a + cy + wy + h
+
(11)
2λ
s
The total annual average cost is, thus given by:
T C(y)
T C(y)
=
T
y(1 − p)/λ
[
[
]]
a
y(1 − p)2
py
λ
=
+c+w+h
+
y
2λ
s
(1 − p)
T AC(y) =
(12)
Differentiating with respect to y we get,
[
[
]]
−a
(1 − p)2
p
λ
′
T AC (y) =
+h
+
T AC”(y)
y2
2λ
s (1 − p)
2a λ
= 3
y (1 − p)
TABLE VI
THE 99-TABLE OF ξi
α
0.01
0.02
.......
0.99
xi
xi1
xi2
.......
xi99
uncertainty distribution Ψ−1 (α) where Ψ−1 (α)
[
]]
[ −1
y(1 − p)2
py
λ
Φ1 (α)
−1
−1
−1
+ Φ2 (α) + Φ3 (α) + Φ4 (α)
+
=
y
2λ
s
(1 − p)
Above inverse uncertainty distribution will be helpful in calculating expected value model and chance constrained model
stated below.
TABLE VII
THE 99-TABLE OF f (ξ1 , ξ2 , ξ3 , ξ4 ; y)
α
[
f (x1 , x2 , x3 , x4 , y)
[
x41
0.01 .......
x1
1
y
+ x21 + x31 +
y(1−p)2
2λ
]]
+
py
s
0.99
[
......
λ
(1−p)
[
x499
x1
99
y
+ x299 + x399 +
y(1−p)2
2λ
]]
+
(Since 0 ≤ p < 1, y > 0, λ > 0, a > 0 )
So, T AC(y) is convex in y. Thus, optimum EOQ is given by
the equation T AC ′ (y) = 0
And the optimum EOQ is:
√
2aλs
∗
y =
(13)
h[s(1 − p)2 + 2λp]
B. Modification of the model in Uncertain environment
Rong has considered a traditional EMQ model with backorder and developed two uncertain programming models; One
as Expected value model and another as Chance Constrained
programming model. We, in our paper Follow the methodology of his Paper but with more simplicity and elaborately and
each step we show that how it is converted into corresponding
crisp model’s result ; which according to our opinion will
enhance the understanding and comparability of Uncertain
Programming with traditional Mathematical Programming.
Let us consider all types of cost as uncertain variables.
Let
ξ1 Ordering cost; an uncertain variable.
ξ2 Purchase cost per unit; an uncertain variable.
ξ3 Screening cost per unit; an uncertain variable.
ξ4 Holding cost per unit per unit time; an uncertain variable.
Φi Uncertainty distribution of ξi ;(i = 1, 2, 3, 4)
If each ξi (i = 1, 2, 3, 4) is represented by a 99-table as
Table 6, then the f (ξ1 , ξ2 , ξ3 , ξ4 ; y) has a 99-table as Table 7.
The total cost per period includes ordering cost, purchase
cost, screening cost and. holding cost .Let ξi ;(i = 1, 2, 3, 4) be
independent uncertain variables with uncertainty distribution
Φi ;(i = 1, 2, 3, 4). Ifξi ;(i = 1, 2, 3, 4) are ordering cost,
purchase cost per unit, screening cost per unit and. holding
cost per unit per unit time, and the total cost per unit time is
represent by
f (ξ1 , ξ2 , ξ3 , ξ4 ; y)
[
]]
[
y(1 − p)2
py
λ
ξ1
+ ξ2 + ξ3 + ξ4
+
=
y
2λ
s
(1 − p)
Then the total cost per unit time being the function of
uncertain variables is itself a uncertain variable with inverse
py
s
λ
(1−p)
C. Expected Value Model
The cost function is itself a uncertain variable, minimization
of which is meaningless. Thus, one has to convert the uncertain
cost function to its expected value so that some mathematical
programming method can be applied to obtain a optimum
order quantity for which total cost per unit time is minimum.
Let as assume that expectations of the variables ξi exist
and so that of f (ξ1 , ξ2 , ξ3 , ξ4 ; y) also exists. Now we can
develop expected value model as follows:
min E[f (ξ1 , ξ2 , ξ3 , ξ4 ; y)]
subject to:
y>0
(3.7)
Here y is a decision variable .One has to find the
optimal value of y; let it be y∗such that the value
E[f (ξ1 , ξ2 , ξ3 , ξ4 ; y∗)] is minimum.
Using the theorem 1 we
∫ 1 get ,
E[f (ξ1 , ξ2 , ξ3 , ξ4 ; y)] = 0 Ψ−1 (α)dα
∫ 1 [ −1
Φ1 (α)
λ
−1
+ Φ−1
=
2 (α) + Φ3 (α) +
(1 − p) 0
y
]]
[
y(1 − p)2
py
Φ−1
(α)
+
4
2λ
s
=
∫ 1
∫ 1
λ
λ
Φ−1
[Φ−1 (α) + Φ−1
(α)dα
+
3 (α)]dα +
(1 − p)y 0 1
(1 − p) 0 2
]
[
pyλ
y(1 − p)
+
Φ−1
4 (α)
2
(1 − p)s
which is a function of y only ; [say e(y)]
So we have to obtain the value of y ∗ such that e(y ∗ )
is minimum.
That is the transformed crisp expected value model; defined
as-
Minimize e(y)
Subject to y > 0
c1
y
Assume that the pessimistic value criterion holds. If we want
to minimize the pessimistic value of the objective function
subject to some chance constraints, the corresponding chanceconstrained programming problem is as follows,
Minimize f
Subject to:M f (ξ1 , ξ2 , ξ3 , ξ4 ; y) ≤ f ≥ α
Where α is a predetermined confidence level and min f is
α− pessimistic return. ξinf (α) = Ψ−1 (α)
]]
[ −1
[
Φ
1
=
(α)
y
y(1 − p)2
−1
−1
−1
+Φ
(α) + Φ
(α) + Φ
(α)
2
3
4
py
λ
s
(1 − p)
+
2λ
=
−1
(α)
Φ
1
y
[
−1
−1
−1
(α)
(α) + Φ
(α) + Φ
+Φ
4
3
2
y(1 − p)2
]]
py
λ
s
(1 − p)
+
2λ
Let us suppose ξi ; (i = 1, 2, 3, 4) be linear uncertain variables.
Then, ξi ∼ L(ai , bi ) where ai , bi for (i = 1, 2, 3, 4) are real
numbers. are real numbers. So inverse uncertainty distribution is
given by :
Φ−1
i (α) = (1 − α)ai + αbi ;for (i = 1, 2, 3, 4)
Thus, ∫
1
(ai +bi )
E[ξi ] = 0 Φ−1
i (α)d(α) =
2
Now, if we apply the above result to the expected value model
developed in section C; we get
as
] cost
[ of the model ]]
[ expected total
E[f (ξ1 , ξ2 , ξ3 , ξ4 ; y)] =
λ
1−p
e1
y
y(1−p)
pyλ
+ (1−p)s
2
+e2 +e3 +e4
i)
where ei = (ai +b
2
Which is a convex function; since,
2λ
E ′′ [f (ξ1 , ξ2 , ξ3 , ξ4 ; y)] = (1−p)y
3 > 0∀y > 0
′
Thus E [f (ξ1 , ξ2 , ξ3 , ξ4 ; y)] = 0 gives the optimal solution of the
proposed expected value model.
Thus the solution is
√
y∗ =
2(a1 + b1 )λs
(a4 + b4 )[s(1 − p)2 + 2λp]
√
2c1 λs
c4 [s(1 − p)2 + 2λp]
y∗ =
√
∗
y =
2((1 − α)a1 + αb1 )λs
((1 − α)a4 + αb4 )[s(1 − p)2 + 2λp]
(15)
For√
perfect quality items;p = 0
1) p = 0 ; y ∗ =
Let ξi be normal uncertain variables.
ie.ξi ∼ N (ei , σi ) where ei , σi for (i = 1, 2, 3, 4) are real numbers.
Now
(
)
Φ−1
i (α) = ei + σi
α
1−α
ln
√
2e1 λ
y∗ =
;
e4
which is the traditional EOQ.
Crisp case, for p = 0,α = 0.5;
For solving the chance constrained programming model, like first
example we substitute Φ−1
i (α) = ci
Which gives
√
2c1 λs
c4 [s(1 − p)2 + 2λp]
∗
y =
√
∗
2e1 +
√
σ1 π3 ln
(
α
1−α
)
λs
Special cases : which is the crisp EOQ for imperfect quality items.
2(a1 +b1 )λs
(a4 +b4 )
2) Crisp Case a1 = b1 ,a4 = b4 and p = 0 y ∗ =
[Traditional EOQ]
Using chance constrained programming model we have to
minimize
[
−1
−1
−1
+Φ
(α) + Φ
(α) + Φ
(α)
2
3
4
3
π
2e1 λs
So optimal solution is :y ∗ =
; [Crisp EOQ with
e4 [s(1−p)2 +2λp]
imperfect
quality
items]
Crisp
case,
for
p
=
0,α
=
0.5;
√
y =
√
√
Thus
∫1
E[ξi ] = 0 Φ−1
i (α)dα = ei ;for (i = 1, 2, 3, 4)
(14)
Special Cases :
y
λ
(1−p)
B. Example 2
A. Example 1
−1
Φ
(α)
1
+
2b1 λ
y =
b4
Thus,it gives traditional EOQ model.
IV. E XAMPLES
=
+ c2 + c3 + c4
py
s
subject to y > 0
Which essentially gives :
∗
Subject to:
y>0
[
y(1−p)2
2λ
For crisp
√ case; α = 1
2b1 λs
y∗ =
; [Traditional EOQ]
b4 [s(1−p)2 +2λp]
So we obtain a crisp chance constraining problem as follows:
Minimize
[
]]
[
[
D. Chance-Constrained Programming model
y(1 − p)2
2λ
√
1) Further; p =0 gives
2a1 λ
a4
;
]]
py
λ
s
(1 − p)
+
Subject to:
y>0
Now for ξi ∼ L(ai , bi ) where ai , bi for (i = 1, 2, 3, 4) are real
numbers.
So inverse uncertainty distribution is given by :
Φ−1
for (i = 1, 2, 3, 4)
i (α) = (1 − α)ai + αbi = ci (say);
Now we have to minimize:
Which is nothing but Basic EOQ model.
V. C ONCLUSION
We have applied Uncertain programming approach to develop an
Uncertain EOQ model for imperfect quality items. The Uncertainty
theory has been incorporated by considering the Costs as Uncertain
variables. We develop Expected value model and Chance constrained
Programming model as two approaches of Uncertain models independently and we have shown that above Uncertain models can be
reduced to corresponding crisp model and traditional EOQ model
subjected to certain conditions. Finally two Numerical examples are
given for each of these models. In our working paper we are in the
process of modifying this model to one with shortages, time varying
demand and Entropy cost.
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