solution of stochastic inventory models with chance-constraints by

137

Int. Jour. of Business & Inf. Tech. Vol-1 No. 1 June 2011

Solution of Stochastic Inventory Models with

Chance-Constraints by Intuitionistic Fuzzy

Optimization Technique S. Banerjee, T.K. Roy

Department of Mathematics, Bengal Engineering and Science University,

Shibpur, Howrah-711103, West Bengal, India

Abstract: Paknejad et al.’s model is analyzed in this paper

where the lead time demand follows normal distribution

under deterministic, stochastic and mixed environment,

respectively and thus the model becomes more realistic.

Expected annual cost is measured, with varying defective

rate. After that the Item wise multi objective models with

chance-constraints for both exponential and uniform lead

time demand are taken and the results are compared

numerically both in fuzzy optimization and intuitionistic

fuzzy optimization techniques. Objective of this paper is to

establish that intuitionistic fuzzy optimizaion method is

better than usual fuzzy optimization technique as expected

annual cost of this inventory model is more minimized in

case of intuitionistic fuzzy optimization method. Necessary

graphical presentations are also given besides numerical

illustrations.

Key-Words: Fuzzy optimizations, intuitionistic fuzzy

optimization, lead time demand, chance-constraint, multi-

objective stochastic model.

.

1. Introduction

Today most of the real-world decision-making

problems in economic, technical and environmental

ones are multi-dimensional and multi-objective. It is

significant to realize that multiple-objectives are often

non-commensurable and conflict with each other in

optimization problem. An objective within exact target

value is termed as fuzzy goal. So a multi-objective

model with fuzzy objectives is more realistic than

deterministic of it.

In conventional inventory models, uncertainties are

treated as randomness and are handled by appealing to

probability theory. However, in certain situations

uncertainties are due to fuzziness and these cases the

fuzzy set theory, originally introduced by Zadeh [23] is

applicable. In decision making process, first, Bellman

and Zadeh[6] introduced fuzzy set theory, Tanaka et. al.

[42] applied concept of fuzzy sets to decision making

problems to consider the objectives as fuzzy goals over

the α-cuts of a fuzzy constraints. Zimmerman [44, 45]

showed the classical algorithms can be used in few

inventory models. Li, Kabadi and Nair [28] discussed

fuzzy models for single-period inventory problem in

2002. Abou-El-Ata, Fergany and Wakeel[1] considered

in 2003, a probabilistic multi-item inventory model

With varying order cost. A single-period inventory

model with fuzzy demand isanalyzed by Kao and Hsu

[7]. Fergany, H.A. and El-Wakeel[8] considered a

probabilistic single-item inventory problem with

varying order cost under two linear constraints. A

survey of literature on continuously deteriorating

inventory models is discussed by F. Raafat[10]. Hala

and EI-Saadani[11] analyzed a constrained single

period stochastic uniform inventory model with

continuous distributions of demand and varying holding

cost. some inventory problems with fuzzy shortage cost

is discussed by Hideki Katagiri and Hiroaki [13]. I.

Moon and S. Choi[14] implemented A note on lead

time and distributional assumptions in continuous

review inventory models. Lai and Hwang [24,25]

elaborately discussed fuzzy mathematical programming

and fuzzy multiple objective decision making in their

two renowned contributions. Liang-Yuh and Hung-Chi

[26] analyzed A minimax distribution free procedure

for mixed inventory models invoving variable lead time

with fuzzy lost sales. Mahapatra and Roy [29]

discussed fuzzy multi-objective mathematical

programming on reliability optimization model. Hariga

and Ben-Daya [32] considered some stochastic

inventory models with deterministic variable lead time.

A fuzzy EOQ model with demand-dependent unit cost

under limited storage capacity is implemented by Roy

and Maiti [41]. Zheng [43] discussed optimal control

policy for stochastic inventory systems with Markovian

discount opportunities. Jaggi and Arneja [15]

considered stochastic integrated vendor-buyer model

with unstable lead-time and set up cost. Sadi-Nezhad ,

Nahavandi and Nazemi [40] analyzed periodic and

continuous inventory models in the presence of fuzzy

costs. Halim , Giri and Choudhuri [12] discussed fuzzy

production planning models for an unreliable

production system with fuzzy production rate and

stochastic and fuzzy demand. Banerjee and Roy [2]

presented a probabilistic inventory model with fuzzy

cost components and fuzzy random variable.

Intuitionistic Fuzzy Set (IFS) was introduced by K.

Atanassov [17] and seems to be applicable to real world

problems. The concept of IFS can be viewed as an

alternative approach to define a fuzzy set in case where

available information is not sufficient for the definition

of an imprecise concept by means of a conventional

fuzzy set. Thus it is expected that, IFS can be used to

simulate human decision-making process and any

138


activitities requiring human expertise and knowledge

that are inevitably imprecise or totally reliable. Here the

degree of rejection and satisfaction are considered so

that the sum of both values is always less than unity

[17]. Atanossov also analyzed Intuitionistic fuzzy sets

in a more explicit way. Atanassov[18] discussed an

Open problems in intuitionistic fuzzy sets theory. An

Interval valued intuitionistic fuzzy sets was analyzed by

Atanassov and Gargov[18]. Atanassov and

Kreinovich[19] implemented Intuitionistic fuzzy

interpretation of interval data. The temporal

intuitionistic fuzzy sets are discussed also by

Atanossov[20]. Intuitionistic fuzzy soft sets are

considered by Maji Biswas and Roy[31]. Nikolova,

Nikolov, Cornelis and Deschrijver[33] presented a

Survey of the research on intuitionistic fuzzy sets.

Rough intuitionistic fuzzy sets are analyzed by Rizvi,

Naqvi and Nadeem[39]. Angelov [35] implemented the

Optimization in an intuitionistic fuzzy environment. He

[36, 37] also contributed in his another two important

papers, based on Intuitionistic fuzzy optimization.

Pramanik and Roy [38] solved a vector optimization

problem using an Intuitionistic Fuzzy goal

programming. A transportation model is solved by Jana

and Roy [16] using multi-objective intuitionistic fuzzy

linear programming. Fidanova and Atanassov [9]

discussed generalized net models and intuitionistic

fuzzy estimation of the process of ant colony

optimization. Banerjee and Roy [3] analyzed a

probabilistic fixed order interval system by general

fuzzy programming technique and intuitionistic fuzzy

optimization technique. Banerjee and Roy [4] also

solved a stochastic inventory model with fuzzy cost

components by fuzzy geometric and intuitionistic fuzzy

geometric programming technique. Mahapatra and

Mahapatra [30] presented the redundancy optimization

by intuitionistic fuzzy multi-objective programming.

Banerjee and Roy [5] discussed the solution of a

constrained stochastic model by fuzzy geometric and

intuitionistic fuzzy geometric programming.

Paknejad et.al. [34] presented a quality adjusted

lot-sizing model with stochastic demand and constant

lead time and studied the benefits of lower setup cost in

the model. We note that the previous literature focuses

on the issue of setup cost reduction in which

information about lead-time demand, whether constant

or stochastic, is assumed completely known. . Liang-

Yuh and Hung-Chi[27] modifies Paknejad et al.‟s

inventory model by relaxing the assumption that the

stochastic demand during lead time follows a specific

probability distribution and by considering that the

unsatisfied demands are partially backordered. Also,

instead of having a stockout cost in the objective

function, a service level constraint is employed.

As a single objective stochastic inventory problem,

Paknejad et al.‟s model[34] is analyzed in this paper

under deterministic, stochastic and mixed environment,

respectively, where the lead time demand follows

normal distribution. Expected annual cost is measured,

with varying defective rate, in deterministic

environment. After that an Item wise multi objective

models with chance-constraints for both exponential

and uniform lead time demand are taken and the results

are compared numerically both in fuzzy optimization

and intuitionistic fuzzy optimization techniques.

Objective of this paper is to establish that intuitionistic

fuzzy optimizaion method is better than usual fuzzy

optimization technique as expected annual cost of this

inventory model is more minimized in case of

intuitionistic fuzzy optimization method. Necessary

graphical presentations are also given besides numerical

illustrations.

2. Mathematical Model

Paknejad et.al. [26] presented a quality adjusted

lot-sizing model with stochastic demand and constant

lead time and studied the benefits of lower setup cost in

the model. We note that the previous literature focuses

on the issue of setup cost reduction in which

information about lead-time demand, whether constant

or stochastic, is assumed completely known. This paper

considers Paknejad et al.‟s model along with the

notations and some assumptions that will be taken into

account throughout the paper. Each lot contains a

random number of defectives following binomial

distribution. After the arrival purchaser examines the

entire lot. An order of size Q is placed as soon as the

inventory position reaches the reorder point s. the

shortages are allowed and completely backordered.

Lead-time is constant and probability distribution of

lead-time demand is known.

Now, we use the following notations:

D = expected demand per year

Q = lot size

s = reorder point

K = setup cost

θ = defective rate in a lot of size Q, 0 ≤ θ ≤ 1

h = nondefective holding cost per unit per year

h = defective holding cost per unit per year

π = shortage cost per unit short

= cost of inspecting a single item in each lot

μ = expected demand during lead time

)(sb = the expected demand short at the end of the

cycle

s

dxxfsxsb )()()(

Where, )(xf is the density function of lead-time

demand. ),( sQEC = Expected annual cost given that

a lot size Q is ordered.

139


3. Single Objective Stochastic Inventory

Model SOSIM)

Thus a quality adjusted lot-sizing model is formed

as:

),( sQMinEC = Setup cost + non-defective item

holding cost + stockout cost + defective item holding

cost + inspecting cost

=

1)1(

)1(

)(

)))1((2

1(

)1(

DvQh

Q

sbD

QshQ

DK

(3.1)

0, sQ

It is the stochastic model, which minimizes the

expected annual cost.

4. Multi Item Stochastic Inventory Model

(MISIM)

This model is analyzed here firstly in deterministic

environment and after that it is discussed in stochastic

and mixed environment also.

4.1 Model with deterministic budget and storage

To solve the problem in equation (3.1) as a MISIM,

it can be reformulated as:

),......,,,......,( 11 nn ssQQMinEC =

n

i 1

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQf

1

1

.,.......,2,10, nisQ ii

Here, For the i-th item (i =1, 2, ………….,n),

pi = price per unit item ,

fi = floor space available per unit,

n = number of item,

F = available floor space,

B = total budget.

4.2 Model with stochastic budget and storage

),......,,,......,( 11 nn ssQQMinEC =

n

i 1

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQf

1

1

ˆˆ

ˆˆ

.,.......,2,10, nisQ ii

( Here, „‟ indicates the randomization of the

parameters.)

4.3 Model with stochastic budget and fuzzy

storage

),......,,,......,( 11 nn ssQQMinEC =

n

i 1

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQfPoss

1

1

1

ˆˆ

~~

nisQ ii ,.......,2,10, , 0< 1 < 1.

(Here, „‟ indicates the fuzzification of the parameters

and „Poss‟ indicates the impreciseness of the constraint)

140


5 .Multi-Objective Stochastic Inventory

Model (MOSIM)

In reality, a managerial problem of a responsible

organization involves several conflicting objectives to

be achieved simultaneously subject to a system of

restrictions (constraints) that refers to a situation on

which the DM has no control. For this purpose a latest

tool is linear or non-linear programming problem with

multiple conflicting objectives. So the following model

which is an item wise multi objective may be

considered:

),( iii sQMinEC

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

.,.......,2,10, nisQ ii

5.1 Model with deterministic budget and storage

),( iii sQMinEC

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQf

1

1

.,.......,2,10, nisQ ii

5.2 Model with stochastic budget and storage

),( iii sQMinEC

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQf

1

1

ˆˆ

ˆˆ

.,.......,2,10, nisQ ii

( Here, „‟ indicates the randomization of the

parameters.)

5.3 Model with stochastic budget and fuzzy

storage

),( iii sQMinEC

i

ii

iii

ii

iii

iiiiii

ii

ii

vDQh

Q

sbD

QshQ

KD

1)1(

)1(

)(

)))1((2

1(

)1(

Subject to

n

i

ii

n

i

ii

BQp

FQfPoss

1

1

1

ˆˆ

~~

.,.......,2,10, nisQ ii

( Here, „‟ indicates the fuzzification of the

parameters and „Poss‟ indicates the impreciseness of

the constraint)

6. Fuzzy Non-linear Programming (FNLP)

Technique to Solve Multi-Objective

141


Non-Linear Programming Problem

(MONLP)

A Multi-Objective Non-Linear Programming

(MONLP) or Vector Minimization problem (VMP)

may be taken in the following form:

T

k xfxfxfxM ))(,),........(),(()inf( 21

Subject to

},......,2,1)(:{ mforjbororxgRxXx jj

n

….(6.1)

and ),....,2,1( niuxl ii

Zimmermann [36] showed that fuzzy programming

technique can be used to solve the multi-objective

programming problem.

To solve MONLP problem, following steps are used:

STEP 1: Solve the MONLP of equation (6.1) as a

single objective non-linear programming problem using

only one objective at a time and ignoring the others,

these solutions are known as ideal solution.

STEP 2: From the result of step1, determine the

corresponding values for every objective at each

solution derived. With the values of all objectives at

each ideal solution, pay-off matrix can be formulated as

follows:

)(........)()( 21 xfxfxf k

kx

x

x

....

2

1

)(....)()(

................

)(....)()(

)(....)()(

*

21

22*

2

2

1

11

2

1*

1

k

k

kk

k

k

xfxfxf

xfxfxf

xfxfxf

Here kxxx ,......,, 21

are the ideal solutions of the

objective functions )(),.......,(),( 21 xfxfxf k

respectively.

So

)}(),.......(),(max{ 21 krrrr xfxfxfU

and

)}(),.......(),(min{ 21 krrrr xfxfxfL

[Lr and Ur be lower and upper bounds of the thr

objective functions )(xf r ),.....,2,1 kr ]

STEP 3: Using aspiration level of each objective of the

MONLP of equation (6.1) may be written as follows:

Find x so as to satisfy

rr Lxf ~

)( ),........,2,1( kr

Xx

Here objective functions of equation (6.1) are

considered as fuzzy constraints. These type of fuzzy

constraints can be quantified by eliciting a

corresponding membership function:

0)(( xf rr or 0 if rr Uxf )(

)(( xf rr if rrr UxfL )(

),........,2,1( kr

= 1 if rr Lxf )(

….(6.2)

Having elicited the membership functions (as in

equation (6.2)) )(( xf rr for r = 1, 2, …… , k,

introduce a general aggregation function

))).((....,)),.......(()),((()( 2211~ xfxfxfGx kkD

So a fuzzy multi-objective decision making problem

can be defined as

Max )(~ xD

subject to Xx

….(6.3)

Here we adopt the fuzzy decision as:

Fuzzy decision based on minimum operator (like

Zimmermann‟s approach [44]. In this case equation

(6.3) is known as FNLPM.

Then the problem of equation (6.3), using the

membership function as in equation (6.2), according to

min-operator is reduced to:

Max

….(6.4)

Subject to )(( xf ii for

),........,2,1 ki

Xx ]1,0[

STEP 4: Solve the equation (6.4) to get optimal

solution.

7. Mathematical Analysis

A stochastic non-linear programming problem is

considered as:

Min f0(X)

Subject to

fj(X) cj (j=1, 2, ………..,m)

X 0.

142


i.e Min f0(X)

….(7.1.1)

Subject to

fj(X) 0 (j=1, 2, ………..,m)

X 0.

Where, fj(X) = fj(X) - cj

Here X is a vector of N random variables y1, y2,

……….,yn and it includes the decision variables x1, x2,

……….,xn.

Expanding the objective function f0(X) about the mean

value iy of iy and neglecting the higher order term:

ii

N

i i

yyXy

fXfXf

1

0

00 )()( = )(X

(say) ….(7.1.2)

If yi (i=1, 2, ……. ,n) follow normal distribution then

so does )(X . The mean and variance of )(X are

given by:

= )(X

….(7.1.3)

2

2

1

0

iy

N

i i

Xy

f

….(7.1.4)

When some of the parameters of the constraints are

random in nature then the constraints will be

probabilistic and thus, the constraints can be written as:

jj rfP )0( (j=1,2 , ………,m)

….(7.1.5)

Then in the light of the theoretical convention

given above, equivalent deterministic constraints are:

0)(

2/1

2

1

iy

N

i i

j

jjj Xy

frf (j=1,2 ,

………,m) ….(7.1.6)

where, )( jj r is the value of the standard normal

variate corresponding to the probability rj.

When some of the parameters of the constraints are

fuzzy then the constraints will be imprecise and thus,

we are to consider the following theorem:

Theorem

Let X : R be a normal fuzzy variable

with parameters (a, b). For a chosen confidence level ,

1 if [Poss(X = x)] then, x [ XL , X

U ]

Where, XL = a - b log , X

U = a + b

log .

Proof: From definition, X(x) = [ X-1

(x)], x R.

Now, [Poss(X = x)]

X(x)

when, X N~

(a, b)

X(x) = exp(-((x-a)/b)2), - < X < .

Therefore,

log

2

b

ax

loglog

b

ax

loglog baxba

If the fuzzy constraint is of the form:

Poss j

n

iijij

bxA

1

11

~~ , j

= 1, 2,

………..,J

….(7.1.7)

Then, we define J normal fuzzy variables as follows:

ij

Y~

n

iijij

bxA1

11

~~, j

= 1, 2,

………..,J

where 1

~jiA and jib

~are mutually min-related normal

fuzzy variables and

jiY

~ ),(

~ji

Yji

Y dmN .

So, the fuzzy constraint (7.1.7) changes to:

Poss jjiY 0~

, j .

Hence, from the above Theorem, we have J pairs of

equivalent crisp constraints as follows:

0log

jjiY

jiY dm ,

0log

jjiY

jiY dm , j

….(7.1.8)

8. Formulation of Intuitionistic Fuzzy

Optimization [IFO]

When the degree of rejection (non-membership) is

defined simultaneously with degree of acceptance

(membership) of the objectives and when both of these

degrees are not complementary to each other, then IF

sets can be used as a more general tool for describing

uncertainty.

To maximize the degree of acceptance of IF

objectives and constraints and to minimize the degree

of rejection of IF objectives and constraints, we can

write:

143


nK,1,2,......i ,),(min

nK,1,2,......i ,),( max

i

i

RXX

RXX

Subject to

0

1)()(

)()(

,0)(

X

XX

XX

X

ii

ii

i

Where )(Xi denotes the degree of membership

function of )(X to the thi IF sets and )(Xi denotes

the degree of non-membership (rejection) of )(X from

the thi IF sets.

9. An Intuitionistic Fuzzy Approach for

Solving MOIP with Linear Membership

and Non-Membership Functions

To define the membership function of MOIM

problem, let acc

kL and acc

kU be the lower and upper

bounds of the thk objective function. These values are

determined as follows: Calculate the individual

minimum value of each objective function as a single

objective IP subject to the given set of constraints. Let **

2

*

1 ,......, kXXX be the respective optimal solution

for the k different objective and evaluate each objective

function at all these k optimal solution. It is assumed

here that at least two of these solutions are different for

which the thk objective function has different bounded

values. For each objective, find lower bound (minimum

value) acc

kL and the upper bound (maximum value)

acc

kU . But in intuitionistic fuzzy optimization (IFO),

the degree of rejection (non-membership) and degree of

acceptance (membership) are considered so that the

sum of both values is less than one. To define

membership function of MOIM problem, letrej

kL and

rej

kU be the lower and upper bound of the objective

function )(XZ k where acc

kL rej

kL rej

kU acc

kU . These values are defined as follows:

The linear membership function for the objective

)(XZ k is defined as:

)( if 0

)(L if )(

)( Zif 1

))(( k

k

acc

kk

acc

kk

acc

acc

k

acc

k

k

acc

k

acc

k

kk

UXZ

UXZLU

XZU

LX

XZ

…. (9.1)

)( if 0

)(L if )(

)( Zif 1

))(( k

k

rej

kk

rej

kk

rej

rej

k

rej

k

rej

kk

rej

k

kk

LXZ

UXZLU

LXZ

UX

XZ

…. (9.2)

μk,νk)

μk ( )(XZk ) 1

1

νk ( )(XZk

Lkacc

Lkrej

Ukacc

=Ukrej

Zk(X)

Figure-1: Membership and non-membership functions

of the objective goal

Lemma: In case of minimization problem, the lower

bound for non-membership function (rejection)) is

always greater than that of the membership function

(acceptance).

Now, we take new lower and upper bound for the non-

membership function as follows:

)(acc

k

acc

k

acc

k

rej

k LUtLL where 10 t

)(acc

k

acc

k

acc

k

rej

k LUtUU for 0t

Following the fuzzy decision of Bellman-Zadeh [6]

together with linear membership function and non-

membership functions of (9.1) and (9.2), an

intuitionistic fuzzy optimization model of MOIM

problem can be written as:

K,1,2,......k ,),(min

K,1,2,......k ,),( max

k

k

RXX

RXX

....(9.3)

Subject to

144


0

1)()(

)()(

,0)(

X

XX

XX

X

kk

kk

k

The problem of equation (9.3) can be reduced following

Angelov [35] to the following form:

Max

….(9.4)

Subject to

)()(acc

k

acc

k

acc

kk LUUXZ

)()(rej

k

rej

k

rej

kk LULXZ

0

1

0

X

Then the solution of the MOIM problem is summarized

in the following steps:

Step 1. Pick the first objective function and solve it as a

single objective IP subject to the constraint, continue

the process K-times for K different objective functions.

If all the solutions (i.e.

n),1,2,......j ;,.....,2,1(......**

2

*

1 miXXX k

same, then one of them is the optimal compromise

solution and go to step 6. Otherwise go to step 2.

However, this rarely happens due to the conflicting

objective functions.

Then the intuitionistic fuzzy goals take the form

)(XZ k kk XL *)(~ .,.......,2,1 Kk ,

Step 2. To build membership function, goals and

tolerances should be determined at first. Using the ideal

solutions, obtained in step 1, we find the values of all

the objective functions at each ideal solution and

construct pay off matrix as follows:

)(............)()(

........................

........................

........................

)(............)()(

)(............)()(

**

2

*

1

*

2

*

22

*

21

*

1

*

12

*

11

kkkk

k

k

XZXZXZ

XZXZXZ

XZXZXZ

Step 3. From Step 2, we find the upper and lower

bounds of each objective for the degree of acceptance

and rejection corresponding to the set of solutions as

follows:

acc

kU = ))(max(*

rk XZ and acc

kL =

))(min(*

rk XZ

kr 1 kr 1

For linear membership functions,

)(acc

k

acc

k

acc

k

rej

k LUtLL where 10 t

)(acc

k

acc

k

acc

k

rej

k LUtUU for 0t

Step 4. Construct the fuzzy programming problem of

equation (9.3) and find its equivalent LP problem of

equation (9.4).

Step 5. Solve equation (9.4) by using appropriate

mathematical programming algorithm to get an optimal

solution and evaluate the K objective functions at these

optimal compromise solutions

.

Step 6. STOP.

10. Few Stochastic Models

CASE 1: Demand follows Uniform distribution

We assume that lead time demand for the

period for the thi item is a random variable which

follows uniform distribution and if the decision maker

feels that demand values for item I below ia or above

ib are highly unlikely and values between ia and ib

are equally likely, then the probability density function

)(xf i are given by:

)(xf i =

otherwise 0

a if 1

i i

ii

bxab

for

.,.....,2,1 ni

So, )(2

)()(

2

ii

ii

iiab

sbsb

for .,.....,2,1 ni

….(10.1)

Where, )( ii sb are the expected number of shortages

per cycle and all these values of )( ii sb affect all the

desired models.

CASE 2: Demand follows Exponential distribution

We assume that lead-time demand for the

period for the thi item is a random variable that follows

145


exponential distribution. Then the probability density

function )(xf i are given by:

)(xf i = )( x

iie ,

x>0 for .,.....,2,1 ni

= 0, otherwise

So, )(

)(i

s

ii

iiesb

for .,.....,2,1 ni

….(10.2)


per cycle and all these values of )( ii sb affect all the

desired models.

CASE 3: Demand follows Normal distribution

We assume that lead-time demand for the

period for the ith item is a random variable, which

follows normal distribution. Then the probability

density function fi(x) are given by:

)2/)(exp(2

1)(

22

ii

i

i xxf

,

x

)2/)(exp(2

1

))(1(2

)()(

22

iii

i

ii

i

ii

ii

s

sssb

for .,.....,2,1 ni ….(10.3)


per cycle and all these values of )( ii sb affects all the

desired models and )( ix represents area under

standard normal curve from -∞ to ix .

11. Numericals

A. Solution of the model (3.1)

We consider the following data:

D=2750; K=10; h=0.25; v =0.02; π =1; h =0.15.

[All the cost related parameters are measured in „$‟]

Here, the lead time demand follows Normal distribution

with mean (μ) =20 and standard deviation (σ) =2.

The expected demand short at the end of the cycle takes

up the value as:

)2/)(exp(2

1

))(1(2

)()(

22

s

sssb

In the Table-1, a study of expected annual cost EC (Q,

s) with lot size Q and reorder point s is given for

different defective rate θ. We conclude from the

following Table-1 as well as graph-1 that, the order

quantity as well as the expected annual cost increase as

θ increases.

θ EC* (Q, s) Q

*

0.1 186.3471 489.2484

0.2 202.8538 513.7929

0.3 223.2740 543.9329

0.4 249.3887 581.9080

0.5 284.3121 631.4677

0.6 334.0731 699.4850

0.7 412.2208 800.3515

0.8 557.5926 971.4324

0.9 952.6872 1361.609

Graph-1: B. Solution of the model (4.1), (4.2) and

(4.3). In this case our objective is to analyze the

expected annual cost of the Multi-item model in

deterministic, stochastic and mixed environment

respectively when the lead-time demand follows normal

distribution and thus the model becomes more practical

and realistic.

0

200

400

600

800

1000

1200

1400

θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

EC* (Q, s) 186.347 202.854 223.274 249.389 284.312 334.073 412.221 557.593 952.687

Q* 489.248 513.793 543.933 581.908 631.468 699.485 800.352 971.432 1361.61

1 2 3 4 5 6 7 8 9

146


To solve (4.1), we use the methods described in the

section 7 and the following data are considered:

D1=2750; K1=10; h1=0.25; θ1=0.3; μ1= 20; σ1= 2;

v1=0.02; π1=2; 1h =0.15; p1=2.

D2=2700; K2=14; h2 =0.65; θ2=0.5; μ2 = 10; σ2= 1;

v2=0.01; π2=1; 2h =0.35; p2=4.

B=40000, F=50000, f1=2, f2=3.

For model (4.2) the following data are taken:

D1=2750; K1=10; h1=0.25; θ1=0.3; μ1= 20; σ1= 2;

v1=0.02; π1=2; 1h =0.15; D2=2700; K2=14; h2 =0.65;

θ2=0.5; μ2 = 10; σ2= 1; v2=0.01; π2=1; 2h =0.35; 1p̂

=(3, 0.03), 1f̂ =(2,0.02), r=0.95, 2p̂ =(2, 0.02) 2f̂ =(3,

0.03) B̂ =(40000, 40) F̂ =(50000,50).

For model (4.3) the following data are taken:

D1=2750; K1=10; h1=0.25; θ1=0.3; μ1= 20; σ1= 2;

v1=0.02; π1=2; 1h =0.15; D2=2700; K2=14; h2 =0.65;

θ2=0.5; μ2 = 10; σ2= 1; v2=0.01; π2=1; 2h =0.35; 1p̂

=(1, 0.01), 1

~f =(1,0.01), r=0.95, 2p̂ =(2, 0.02) 2

~f =(3,

0.03) B̂ =(50000, 60) F~

=(60000,60), 1=0.5.


Thus, we have the results obtained below:

Table-2

Models

under

Normal

Distribution

EC($) Q1 Q2 s1 s2

Determinist

ic

Environmen

t

1331.2

1

473.2

6

297.1

2

41.2

2

61.3

2

Stochastic

Environmen

t

1294.7

6

488.1

4

797.0

6 4.34 6.58

Mixed

Environmen

t

912.19 388.4

0

297.8

4

18.4

2

11.6

5

C. Solution of the model (5.1), (5.2), (5.3)

In case of intuitionistic fuzzy optimization not only the

membership function is maximized but also, the non-

membership function is minimized (as it is described

in section 8 and 9), whereas in the fuzzy optimization

technique only the memberships function is

maximized (as in section 6). So, our objective is to

establish the fact that IFO makes the betterment of

the solution than FO.

To solve MOSIM of section 5.1, we use the methods

described in the sections 6, 7, 8 and 9 and the following

data are considered:

Case1. The lead time demand follows uniform

distribution and thus )( ii sb , the expected demand

short at the end of the cycle takes up the value

according to (10.1).

We consider two different set of data as:

D1=2700; K1=12; h1=0.55; θ1=0.6; μ1=(a1+b1)/2; v1

=0.03; π1=1; h 1=0.25; a1=20; b1=70; μ1=(a1+b1)/2,

p1=3, f1=2.

D2=2750; K2=10; h2=0.25; θ2=0.8; μ2=(a2+b2)/2; v2

=0.02; π2=2; h 2=0.15; a2=10; b2=50; μ2=(a2+b2)/2,

p2=2, f2=3, B=40000, F=50000.


The pay-off matrix is:

EC1 EC2

5142.5743421.852

5697.7992213.512

We take, U1acc

=852.3421; L1acc

=512.2213;

U2acc

=799.5697; L2acc

=574.5142;

U1rej

=852.3421; L1rej

=530; U2rej

=799.5697;

L2rej

=598.

Table-3: Comparison of solutions of FO and IFO in

deterministic environment

(UNIFORM LEAD-TIME DEMAND)

MET

HODS

Q1 Q2 s1 S2 EC1

($)

EC2

($)

α*

β*

Fuzzy

Optim

izatio

n

42

3.

7

10

66.

6

71.

78

4

49.

45

3

522.

348

7

591.

768

3

0.

7

9

--

Intuiti

onistic

Fuzzy

optimi

zation

97

0.

5

11

54.

4

64.

43

2

51.

65

4

518.

321

9

567.

650

8

0.

8

9

0.

04

5

147


Graph-2: The above result shows that Intuitionistic

fuzzy optimization [IFO] obtains more optimized

values of EC1 and EC2 than fuzzy optimization [FO].

Also solution obtained by IFO (518.3219, 567.6508) is

closer to the ideal solution (512.2213, 574.5142) than

the solution obtained by FO.

Case2. The lead time demand follows exponential

distribution and thus )( ii sb , the expected demand

short at the end of the cycle takes up the value

according to (10.2).

We consider two different set of data as:

D1=2700; K1=8; h1=1; θ1=0.4; μ1=1/λ1; v1 =0.03; π1=1;

h 1=0.25; λ1=1, p1=3, f1=2.

D2=2750; K2=10; h2=1;θ2=0.7; v2 =0.02; π2=1.1; h

2=0.15; μ2=1/λ2; λ2=1.1, p2=2, f2=3, B=40000,

F=50000.


The pay-off matrix is :

EC1 EC2

4329.3718945.562

3498.5273421.365

We take, U1acc

=562.8945; L1acc

=365.3421;

U2acc

=527.3498; L2acc

=371.4329;

U1rej

=562.8945; L1rej

=390; U2rej

=527.3498;

L2rej

=415.


deterministic environment

(EXPONENTIAL LEAD-TIME DEMAND)

Graph-3: The above result shows that Intuitionistic

fuzzy optimization [IFO] obtains more optimized

values of EC1 and EC2 than fuzzy optimization [FO].

Also solution obtained by IFO (378.0912, 432.6754) is

closer to the ideal solution (365.3421, 371.4329) than

the solution obtained by FO.

To solve MOSIM of section 5.2 with uniform lead-time

demand, using equation (10.1), we use the methods



D1=2700; K1=12; h1=0.55; θ1=0.6; μ1=(a1+b1)/2; v1

=0.03; π1=1; h 1=0.25; a1=20; b1=70; μ1=(a1+b1)/2, 1p̂

=(3, 0.03), 1f̂ =(2,0.02), r=0.95.

D2=2750; K2=10; h2=0.25; θ2=0.8; μ2=(a2+b2)/2; v2

=0.02; π2=2; h 2=0.15; a2=10; b2=50; μ2=(a2+b2)/2,

2p̂ =(2, 0.02) 2f̂ =(3, 0.03) B̂ =(40000, 40) F̂

=(50000,50).


460

480

500

520

540

560

580

600

FO 522.3487 591.7683

IDEAL 512.2213 574.5142

IFO 518.3219 567.6508

EC1 EC2

0

50

100

150

200

250

300

350

400

450

FO 386.7813 441.7865

IDEAL 365.3421 371.4329

IFO 378.0912 432.6754

EC1 EC2

MET

HODS

Q1 Q2 s1 S2 EC1

($)

EC2

($)

Α*

β*

Fuzzy

Optim

izatio

n

34

2.0

9

94

6.2

5

1.

9

2

2.

6

7

386.

781

3

441.

786

5

0.4

12

5

--

Intuiti

onistic

Fuzzy

optimi

zation

21

2.9

7

94

1.7

8

3.

5

6

2.

2

6

378.

091

2

432.

675

4

0.7

23

4

0.

11

2

148



stochastic environment


MET

HODS

Q1 Q2 s1 S2 EC1

($)

EC2

($)

α*

β*

Fuzzy

Optim

izatio

n

41

5.

9

10

41.

8

67.

09

1

45

.8

7

533.

987

6

574.

675

4

0.

78

--

Intuiti

onistic

Fuzzy

optimi

zation

87

1.

9

11

29.

6

63.

98

7

49

.3

4

527.

912

6

567.

987

6

0.

86

2

0.

03

2

Again, to solve the model of section 5.2 with

exponential lead-time demand, using equation (10.2),

we use the methods described in the sections 6, 7, 8 and

9 and consider the following data:

D1=2700; K1=8; h1=1; θ1=0.4; μ1=1/λ1; v1 =0.03; π1=1;

h 1=0.25; λ1=1, 1p̂ =(3, 0.03), 1f̂ =(2,0.02), r=0.95.

2p̂ =(2, 0.02) 2f̂ =(3, 0.03) B̂ =(40000, 40) F̂

=(50000,50), D2=2750; K2=10; h2=1;θ2=0.7; v2 =0.02;

π2=1.1; h 2=0.15; μ2=1/λ2; λ2=1.1.



stochastic environment (EXPONENTIAL LEAD-TIME

DEMAND)

MET

HODS

Q1 Q2 s1 S

2

EC1

($)

EC2

($)

α*

β*

Fuzzy

Optim

izatio

n

32

0.4

0

93

3.0

5

1.

7

5

3.

1

3

382.

093

2

432.

986

4

0.4

43

1

--

Intuiti

onistic

Fuzzy

optimi

zation

33

2.9

6

94

2.9

8

3.

0

1

2.

6

5

377.

543

0

420.

743

1

0.7

79

8

0.0

31

4

To solve MOSIM of section 5.3 with uniform lead-time

demand, using equation (10.2), we use the methods



D1=2700; K1=12; h1=0.55; θ1=0.6; μ1=(a1+b1)/2; v1

=0.03; π1=1; h 1=0.25; a1=20; b1=70; μ1=(a1+b1)/2, 1p̂

=(1, 0.01), 1

~f =(1,0.01), r=0.95

D2=2750; K2=10; h2=0.25; θ2=0.8; μ2=(a2+b2)/2; v2

=0.02; π2=2; h 2=0.15; a2=10; b2=50; μ2=(a2+b2)/2,

2p̂ =(2, 0.02) 2

~f =(3, 0.03) B̂ =(50000, 60) F

~

=(60000,60), 1=0.5.





METH

ODS

Q1 Q2 s1 S2 EC

1

($)

EC

2

($)

α*

β*

Fuzzy

Optimi

zation

42

4.

8

10

72.

9

68

.9

1

44

.2

2

53

5.0

9

57

3.8

7

0.7

69

1

--

Intuiti

onistic

Fuzzy

optimi

zation

77

8.

3

11

37.

9

65

.0

8

46

.9

7

52

8.4

8

56

3.9

8

0.8

13

2

0.

03

1

To solve MOSIM of section 5.3 with exponential lead-

time demand we use the methods described in the

sections 6, 7, 8 and 9 and the following data are

considered:

D1=2700; K1=8; h1=1; θ1=0.4; μ1=1/λ1; v1 =0.03; π1=1;

h 1=0.25; λ1=1, D2=2750; K2=10; h2=1;θ2=0.7; v2

=0.02; π2=1.1; h 2=0.15; μ2=1/λ2; λ2=1.1, 1p̂ =(1,

0.01), 1

~f =(1,0.01), r=0.95, 2p̂ =(2, 0.02) 2

~f =(3, 0.03)

B̂ =(50000, 60) F~

=(60000,60), 1=0.5.




(EXPONENTIAL LEAD-TIME DEMAND)

METH

ODS

Q1 Q2 s1 s2 EC

1

($)

EC

2

($)

α*

β*

Fuzzy

Optimi

zation

32

7.8

4

94

2.9

8

2.

2

1

3.

4

3

39

1.7

6

43

6.0

9

0.6

75

4

--

Intuiti

onistic

Fuzzy

optimi

zation

33

4.0

3

95

1.4

2

2.

8

7

2.

9

1

38

0.3

4

42

0.4

3

0.7

16

5

0.0

21

5

The above results also show that Intuitionistic fuzzy

optimization [IFO] obtains more optimized values of

EC1 and EC2 than fuzzy optimization [FO].

12. Conclusion

Objective of this paper is to analyze the Paknejad

et al.‟s model in stochastic or mixed environment and

149


thus this model becomes more practical and realistic.

Item wise multi objective models for both exponential

and uniform lead time demand are taken and the results

are compared numerically both in fuzzy optimization

and intuitionistic fuzzy optimization techniques. From

Our numerical as well as graphical presentations, it is

clear that intuitionistic fuzzy optimization obtains better

results than fuzzy optimization. In case of intuitionistic

fuzzy optimization not only the membership function is

maximized but also, the non-membership function is

minimized. That is why intuitionistic fuzzy

optimization always improves the solution than fuzzy

optimization. Thus expected annual cost is more

minimized in case of intuitionistic fuzzy optimization

than the usual fuzzy optimization technique. This model

can also be extended taking lead-time demand as fuzzy

random variables.

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