Supply Chain Model with Imperfect Production Process and ... · Singh and Singh (2008) discussed the supply chain model with different conditions. In the above models, the demand

37Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost

Supply Chain Model with ImperfectProduction Process and Stochastic

Demand Under Chance and ImpreciseConstraints with Variable Holding Cost

S R Singh* and Urvashi**

This paper develops a mathematical model for a single period multi-product manmanufacturing system of stochastically imperfect items with continuous stochasticdemand under budget and shortage constraints. Here the constraints are of threetypes: (1) both stochastic, (2) one stochastic and other imprecise (fuzzy), and(3) both imprecise. The stochastic constraints have been represented by chanceconstraints and fuzzy constraints in the form of possibility/necessity constraints.Shortages are allowed for the retailer’s model. The model is illustrated throughnumerical examples.

IntroductionSupply chain management is a cross-functional approach to managing themovement of raw materials into an organization and the movement of finishedgoods out of the organization toward the end-consumer. As corporations strive tofocus on core competencies and become more flexible, they have reduced theirownership of raw material sources and distribution channels. These functions areincreasingly being outsourced to other corporations that can perform the activitiesbetter or more cost-effectively. The effect has been to increase the number ofcompanies involved in satisfying the consumer demand, while reducing managementcontrol of daily logistics operations. Less control and more supply chain partnersled to the creation of the concept of supply chain management. The purpose ofsupply chain management is to improve trust and collaboration among the supplychain partners, thus improving the inventory visibility and velocity. Several modelshave been proposed for understanding the activities required to manage materialmovements across organizational and functional boundaries. Studies on supplychain management have emphasized the importance of a long-term strategic

Keywords: Inventory, Stochastic demand, Imperfect production, Chance constraint, and Possibility/necessity constraint

© 2009 IUP. All Rights Reserved.

* Reader, Department of Mathematics, D N College, Meerut, India. E-mail: [email protected]** Lecturer, Department of Mathematics, D N College, Meerut, India. E-mail: [email protected]

The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200938

relationship between the producer and the retailer. With the recent advances incommunication and information technologies, the integration of these functionsis a common phenomenon. Moreover, due to limited resources, increasingcompetition and market globalization, enterprises are forced to develop supplychain that can respond quickly to customer needs with minimum stock andmaximum service level. Recently Yang and Wee (2003), Hans et al. (2006) andSingh and Singh (2008) discussed the supply chain model with different conditions.

In the above models, the demand rate is considered to be a deterministicconstant or time dependent or stock dependent. In some realistic situation likenewsboy problem, the demand is uncertain in a stochastic sense. Hadley and Whitin(1963) first extended the classical Economic Order Quantity (EOQ) model tothe stochastic model. Traditional stochastic models have been considered primarilywith demand and lead time uncertainties. Nahmias (1978) and Silver (1981)discussed that type of models. Fitzsimmoms and Sullivan (1978) investigated analgorithm using probabilistic goals on the concept of chance constraint due toCharnes and Cooper (1959), where the goal can be stated in terms of probabilityof satisfying the aspiration level. Further, Ben-Daya and Raouf (1993), andKalpakam and Sapna (1995 and 1996) discussed the inventory models withstochastic lead time. The models discussed above are inventory models that dealwith perfect quality of items. This is not a realistic situation. Salameh and Jaber(2000) considered imperfect quality in stochastic environment.

In general, the Economic Production Quantity (EPQ) models are formulatedwith constant holding cost. In real life, it may not be so. This paper presents aninventory model with stochastic environment and variable holding cost. Theholding cost is a linear function of time mans; as time increases, the holding costwill also increase. Some products are kept in storage for long time, the moresophisticated the storage facilities and services needed, the higher the holdingcost. Goh (1992), Giri et al. (1996), and Shao et al. (2000), etc., investigatedinventory models with variable holding cost.

In general, the EPQ models are formulated under crisp resource constraintswhich may not be possible in real life. For example, at the beginning of abusiness, it may be launched with some capital. But during the period ofbusiness, it may happen to meet the unexpected increase in demand. Theproduction rate may have to be stepped up, and in doing so, the organizationwould have to invest some more capital. This augmented amount is normallyfuzzy in nature for a new company, for which past data are not available.Regarding an old company, it is possible to have the past data for the variationin budget, which may be represented by a probability distribution. Hence, theresource constraints become stochastic or imprecise in nature. During recentyears, there are some EPQ models. Chang (1999), Hsieh and Chen (1999),


Chen and Hsieh (2000), Hsieh (2000), Sourveloudis et al. (2000), and Leeand Yao (2004), investigated the models in fuzzy environment. The ChanceConstrained Programming (CCP) technique is one which is used to solveproblems involving chance constraints, i.e., constraints having randomparameters. The CCP was originally developed by Charnes and Cooper (1959)and has, in recent years, been extended in several directions for variousapplications. Liu and Iwamura (1998) solved CCP with fuzzy parameters.

Earlier investigations took one parameter or a resource as fuzzy and solvedusing fuzzy set theory and extension principle. They did not consider fuzzy andstochastic parameters and/or both fuzzy and stochastic constraints in a singlemodel. In these models, resource constraints were not imposed in a possibility/necessity sense. Such a complex model is investigated in this paper.We also discussed the model with variable holding cost which is realistic in nature.Numerical example is given to show the concavity of the model, and sensitivityanalysis is also given.

2. Assumptions and Notations2.1 Assumptions

• The inventory system is an imperfect production system and involvesmultiple items.

• This is a single period inventory model.

• Production rate is finite and constant.

• Total demand over the period of cycle is stochastic and uniform over time.

• Percentage of imperfectness is stochastic.

• Shortages are permitted and fully backlogged.

• Screening costs for all items are same.

The inventory system involves n items and for ith item (i= 1, 2, …, n), thefollowing notations are used:

(1) Ai’s, Bi’s and Ri’s are constants in the density function fi (xi), where:

elsewhere00 iiiiiii RxxBAxf

(2) bi’s and di’s are constants in the density function gi(ei), where

elsewhere00 iiiii bedeg


(3) B, B are maximum budget (total production cost and screening cost)which are considered as stochastic and fuzzy respectively.

(4) C1i is holding cost per unit item per unit time for producer.

(5) Cpi is production cost per unit; piC being the fuzzy value of Cpi.

(6) Csi is shortage cost per unit item; siC being the fuzzy value of Csi.

(7) ei is rate of imperfect units.

(8) EAPm (Q1, Q2, …, Qn) is total expected average profit of the producer.

(9) E(B), E(SM) is expectation of the random variables B, SM respectively.

(10) EPmi is the Expected profit for ith item of producer.

(11) fi(xi) is probability density function of the demand xi (0 < xi < ).

(12) gi(ei) is probability density function for the rate of defective units ei.

(13) Ki is selling price per unit item of imperfect quality.

(14) Li is salvage value per unit.

(15) Pi is production rate per unit time.

(16) Pii is production rate of good (perfect) units which satisfies the relation

Pii = (1– ei)Pi

(17) qi(t) is on hand inventory at time t 0.

(18) Qi is total production.

(19) Qii is total production of good units which satisfies the relationQii = (1– ei)Qi.

(20) Qsi is the shortage amount.

(21) ri’s are probabilities, i = 1, ..., k.

(22) SC is screening cost per unit item.

(23) Si is selling price per unit item of good quality.

(24) SM,

MS are maximum shortage costs allowed which are considered asstochastic and fuzzy respectively.

(25) t1i is the production time.

(26) t2i is the time after which shortages occur in the retailer’s model.

(27) Ti is fixed duration of the cycle.

(28) xi is total demand over time period (0, Ti), which is stochastic.


(29) (ri) is a real number, where

ir

i trdtt

, being the standard normal

density function, i = 1, ..., k.

(30) 1 is possibility/necessity of the budget constraint; same notation isbeing used in different models with different values in general.

(31) 2 is possibility/necessity of the shortage constraint, same notation isbeing used in different models with different values in general.

(32) (B), (SM) are standard deviations of the random variables B, SM

respectively.

(33) C2i is holding cost per unit item per unit time for producer.

(34) Clsi is lost sale cost per unit item.

3. Mathematical Model3.1 Producer’s Model

ii

iiii

i ttTx

Petqdt

tdq101 , ...(1)

iii

ii

i TttTx

tqdt

tdq 1, …(2)

With boundary conditions qi(0) = 0, qi(Ti) = 0

Solutions of the above equations are given below

it

i

iiii tte

Tx

Ptq 1011

,

…(3)

iii

i

i

tTiii

i TttTx

TetP

tqi

11 ,

…(4)

where Pii = (1– ei)Pi

Since shortages do not occur, we must have qi(Ti) 0

01

i

i

i

tTiii

Tx

TetP i

xi Piit1i …(5)


Now by the notations Qi = Pit1i and Qii = Piit1i,

Expected holding cost for non defective units of ith item

1

0 0 0

1

1

11ii iQ tt

i

iii dte

Tx

PtC

.C.H

iiiiii

T

t i

i

i

tTiii deegdxxfdttC

Tx

TetPi

i

i

1

11

1

0 0

111

111 1ii iiQ T

i

iiii

t

i

iiii

eTtP

te

Tx

P

C.C.H

iiiiiiiii

i deegdxxftTT

x

1

1

0 02

211 12

1 11ii iiQ t

it

i

i

iii

etetTx

P

iiiiii

tTtTii

i

iii deegdxxfeetTTtP iiii

211 111

...(6)

Expected holding cost for defective units of ith item is given as:

i

t

iiiii deegPtetCi

1

0 0

1

1

.C.H

1

0

31

0

2

1 32 iiii

iiiii

i

iii deeg

PQe

deegPQe

C …(7)

where Qi = Pit1i

Salvage value of the system is given as:

1

00

1

iii

Q

iiiiiii deegdxxfxQLi

.V.S …(8)

Screening cost of the system is given as:


S.C. = SCPit1i= SCQi ...(9)

Production cost of the system is given as:

P.C. = CpiPit1i = CpiQi ...(10)

Revenue from the sales of perfect units

1

0 0

iii

Q

iiiii deegdxxfxSii

.R.S …(11)

Revenue from the sales of imperfect units

1

0

iiiiii deegeQk.R.S …(12)

Expected Profit for ith item= EPmi

= Revenue from the sales of perfect units + revenuefrom the sales of imperfect units – holding costfor non defective units – holding cost for– defective units – screening cost – production cost

…(13)

Hence, for all items total expected average profit is given as:

n

imi

inm EP

TQQQEAP

121

1,, , …(14)

3.2 Retailer’s Model3.2.1 Case 1When shortages do not occur:

The inventory level qi(t) is governed by the differential equations, i = 1, 2, ..., n.

kT

tTx

tqdt

tdq i

i

ii

i 0, …(15)

With boundary condition 0

kT

q ii

Solution of the above equation is:

1t

kT

i

ii

i

eTxtq

…(16)


Expected holding cost for the system is given as:

1

0 0 02 iii

Q

iii

kT

ii deegdxxfdttqtCii

i

.C.H …(17)

3.2.2 Case 2When shortages occur:

The governing differential equations are:

ii

ii

i ttTx

tqdt

tdq20 , …(18)

kTtt

Txe

dttdq i

ii

it

i

2,

…(19)

With boundary condition qi(t2i) = 0

Solution of the above equations is given as:

itt

i

ii tte

Tx

tq i2012 ,

…(20)

kT

tteeTx

tq ii

tt

i

ii

i 2

2 ,

...(21)

Expected holding cost for the system is given as:

iiiiii

Q

t

ii deegdxxfdttqtCii

i

1

0 0

2

2

.C.H …(22)

Expected Shortage cost for the ith item = CsiQsi

iii

Q

kT

t

iiiisi deegdxxftqCii

i

i

1

0 2

.C.S …(23)

where iii

Q

kT

t

iiiisi deegdxxftqQii

i

i

1

0 2


Expected lost sale cost for the ith item is:

iii

Q

kT

t

iiit

i

iLsi deegdxxfe

Tx

Cii

i

i

1

0 2

1L.S.C.

…(24)

Sales Revenue for the ith item is:

iii

Q

Q

iiiiiiiiii deegdxxfQdxxfxSii

ii

1

0 0

S.R. …(25)

Expected Profit for ith item of the retailer = EPri …(26)

= Sales Revenue – holding cost– shortage cost – lost sale cost

Hence, for all items total expected average profit is given as:

EAPr(Q1, Q2, …,Qn, t2i) =

n

iri

i

EPT1

1…(27)

Now, the total expected average profit for the whole supply chain system

EAP(Q1, Q2, …, Qn, t2i) =

n

imiri

i

EPEPT1

1…(28)

Max EAP(Q1, Q2, …, Qn, t2i)

Subject to:

n

iiCpi BQSC

1

and

n

iMsisi SQC

1…(29)

We now study the general problem by considering particular density functionsfor the demand ‘x’and for ei, the percentage of defective units in item Qi.

We consider the density functions for demand as linear, i.e.,

elsewhere00 iiiiiii RxxBAxf

...(30)


where Ai’s and Bi’s are constants.

From the property of probability density function, i.e.,

1dxxf

Simplifying we get the following conditions

12

2

iiii

RBRA

We consider the density functions for ei’s as:

elsewhere00 iii

ii

bedeg ...(31)

where bidi = 1, i = 1, 2, 3, .., n

Under these considerations from Equation (29), we have

EAP(Q1, Q2, …, Qn, t2i)

n

ii

iii

iiii

iii

i

i bQB

QRB

RAb

bSTd

1

4322

111222

i

ipiCi

iii

iii d

QCSb

QBb

AQL 432 11

2411

6

22

22

21

21

k

T

ii

i

ik

T

i

i

ii

ek

Tk

TTk

TeT

C

4

33

2

1124

116 i

iii

ii bQB

bQA

iiiiitk

T

iit

i

si bRBRA

eetkTe

TC

ii

i

321 32

333

ii

Liii

iii t

kT

CbQB

bQA

si 34

33

2

1112

116


22

2

3214

3

21

22211

12 iiii

i

i

i

iii

ii bQkb

PQ

PQC

bQB

332

1 113

113

1ii

PQ

iiiii

iii bQ

ePPPAb

QAC

i

i

2

2

2

233

2

311

116

i

i

i

iPQ

iiiiPQ

iii

i

ii

i

ii ePAPAePQATb

bTQA

443

33

114

118

116 i

iii

iiii

ii bBP

bQBP

bQB

43

44

2

2

2

2

1112

1116 i

i

iii

ii

iiPQ

iiPQ

ii bT

QBb

TPQBePPePQ i

i

i

i

43

23

2 112

113 i

i

PQ

iiii

i

PQ

iii bTePQB

bTePQB i

i

i

i

13

111

211 3

4

43 ii

i T

i

iiiPQ

i

iii eT

bQAe

TbPB

1211

1211

611 434332

iii

ii

iiiiii bQBPT

bQAbQA

311

16111

811 34443

i

ii

iiiT

i

iii bPT

bQBeT

bQB i

ii

PQ

iiiiiiPQ

iii

i

ii QPePPAPQA

ePQA

PQA i

i

i

i

2224


i

PQ

iiiiii

i

PQ

ii

ii

iii

TePQAbb

bPT

eQAPT

bQA i

i

i

i

4

32

3

244 23

11232

11

i

i

i

i

i

PQ

i

i

PQ

ii

i

PQ

ii

TQ

TP

TeP

TePQ

TePA i

i

i

i

i

i

3

2

5

2

5

2

45

2

22

3

2

3

2

2

244 264

11

iiiPQ

iiiPQ

iii

i

iii PQBePQBeQPBPQBb i

i

i

i

i

PQ

iii

i

PQ

iii

i

PQ

ii

ii

iiiii

TePQB

TeQPB

TeQB

TPQBQPB i

i

i

i

i

i

5

2

4

3

3

3

2

5

2

2 63102

i

PQ

ii

i

ii

i

PQ

iii

i

PQ

iii

i

ii

TePB

TQB

TePQB

TeQPB

TQB i

i

i

ii

i

6

3

6

3

5

2

4

2

3

3 2223

34 11

32

iii b

PB …(32)

Here,

iiiiitk

T

iit

isi b

RBRAeet

kT

eT

Q ii

i

3211 32

333

43

32

1112

116 i

iii

ii bQB

bQA

Hence the problem, given by Equation (29), is reduced to:


Subject to:

n

iicpi BQSC

1


and

niQSQC i

n

iMsisi ,,,,, 3210

1

…(33)

Now we shall consider different cases for the constraints, i.e., (a) both theconstraints as stochastic, (b) one constraint as stochastic and another constraintas fuzzy, (c) both the constraints as fuzzy.

4. Constraint4.1 Probabilistic Constraint

• When limitations on total production cost and screening cost isprobabilistic then,

Prob 10 111

rrBQSCn

iiCpi ,

• When limitation on shortages becomes probabilistic, the constraintbecomes,

Prob 10 221

rrSQCn

iMsisi ,

Fuzzy (Possibility/Necessity) constraints:

If Msipi SCCB ˆˆ,ˆ,ˆ are imprecise in nature, the above constraints are of thefollowing form:

n

iMsisii

n

iCpi SQCandBQSC

11

ˆˆˆˆ

(Here wavy cap^ denotes fuzzyfication of the parameters.)

Hence, Equation (33) is reduced to:


Subject to:

niQSQCandBQSC i

n

iMsisii

n

iCpi ,,,,,,ˆˆˆˆ 3210

11

...(34)

5. Chance Constrained Programming (CCP) TechniqueA stochastic nonlinear programing problem with some linear chance constraintscan be expressed as:


Max Z(x1, x2, …, xn) …(35)

Subject to:

Prob i

n

jijij rbxa

1 …(36)

xj 0, j = 1, 2, …, n, ri(0, 1), i = 1, 2, …, k …(37)

where aij, bi are normal random variables and ri are specified probabilities.For simplicity, we assume that the decision variables xj are deterministic. Here, weconsider the case: where only bi’s are normally distributed random variables withknown means and variances.

Let E(bi) and Var(bi) denote the mean and variance of the normal randomvariable bi.

The constraints in Equation (18) can be written as:

Prob

i

i

n

j ijij

i

ii rbVar

bEx

bVar

bEb

1

… (38)

where i

ii

bVarbEb

is a standard normal variation. Considering (ri), where

ir

i tr dt t

, , being the standard normal density function, we have

i

i

n

j ijijr

bVar

bExa

1 which can be written as:

n

j iiiiij rbVarbExa1

…(39)

Thus the probabilistic nonlinear programming problem stated in Equations(35 to 39) is equivalent to the following deterministic nonlinear programmingproblem:

Max Z(x1, x2, …, xn) …(40)

Subject to:

kirnjxrbVarbExa ij

n

j iiijij ,,,,,,,,, 21102101


6. Possibility/Necessity Programming Technique6.1 Possibility/Necessity in Fuzzy EnvironmentIf A

�

and B�

be two fuzzy subsets of real numbers with membership functions �A

and �Brespectively, then taking degree of uncertainty as the semantics of fuzzy

number, according to Liu and Iwamura (1998), and Dubois and Prade (1980

and 1983):

Pos (A�

* B�

) = sup {min (�A(x), �

B(y)), x, y , (x * y)} …(41)

where the abbreviation Pos represents possibility and * is any one of the relations>, <, =, , . On the other hand, necessity measure of an event A

�

* B�

is a dual ofpossibility measure. The grade of necessity of an event is the grade of impossibilityof the opposite event and is defined as:

Nes (A�

* B�

) = 1 – Pos (A�

*—––

B�

) …(42)

where the abbreviation Nes represents necessity measure and A�

*—––

B�

represents

complement of the event A�

* B�

. Also necessity measures satisfy the condition

Min (Nes (A�

* B�

), Nes (A�

*—––

B�

)) = 0

The relationship between possibility and necessity measures also satisfies thefollowing conditions (cf. Dubois and Prade, 1978):

Pos (A�

* B�

) Nes (A�

* B�

), Nes (A�

* B�

> 0) Pos (A�

* B�

) = 1and

Pos (A�

* B�

) < 1 Nes (A�

* B�

) = 0.

If A�

, B�

and C�

= f(A�

, B�

) where f: be a binary operation, then membership

function �C

of C~

is defined as �C(v) = sup {min (�

A(x), �

B(y)), x, y and

v = f(x, y) v }

6.2 Triangular Fuzzy Number

In particular, if A�

be a Triangular Fuzzy Number (TFN) (Figure 1) then �A(x) is

defined as follows:

�A

(x) =

otherwise0

for

for

3223

3

2112

1

axaaaxa

axaaaax


where a1, a2 and a3 are real numbers. Also this type of triangular fuzzy number isdenoted as:

A�

= (a1, a2, a3),

6.2.1 Cut of a Fuzzy Number

An cut of a fuzzy number, A�

, is defined as a crisp set:

A = {x: �A(x) x, }, where [0, 1]

6.3 Imprecise Constraints

Let us consider the constraint A�

B�

. This can be represented in the necessity and

possibility sense as: Nes (A�

B�

) and Pos (A�

B�

) . (Nes (A�

B�

) . Pos (A�

B�

))

estimates that an event “A�

B�

” will occur with the minimum (maximum) chance at

least g (say by DM). Hence, the said constraint can be represented as Nes (A�

B�

) >

(Pos (A�

B�

) > ). Let A�

= (a1, a2, a3) and B�

= (b1, b2, b3) be two TFNs. Then forthese fuzzy numbers, following Inuiguchi et al. (1994), and Wang and Shu (2005),Lemmas 1 and 2 can be derived.

Lemma 1

Nes (A�

B�

) > iff

1

2312

13

bbaaab

.

Figure 1: Membership Function of TFN A�

I

0 a1 a2

a3

�A (x)


Proof

Let us have Nes (A�

B�

) > . From Figure 2, it is clear that

Figure 2: Two TFN A�

and B�

and Pos (A�

B�

)

I

0 b1a1 b2 a2 a3b3

�B

(y) �A

(x)

1

Pos (A�

B�

) =

31

13222312

131

22

for0

for

for1

ba

abbabbaa

abba

,

Hence, Nes (A�

B�

) > (1 – Pos (A�

B�

)) > .

Therefore, Nes (A�

B�

) > iff 1= 13222312

13 1 abbabbaa

ab

,, .

Lemma 2

Pos (A�

B�

) > iff 13222312

13 1 babaaabb

ba

,, .

Proof

Let us consider Pos (A�

B�

) > (Figure 3).

Pos (A�

B�

) =

31

13222312

132

22

for0

for

for1

ba

abbaaabb

baba

,

Pos (A�

B�

)> iff 13222312

13 1 babaaabb

ba

,, .


7. Different Models7.1 Both Constraints are Stochastic7.1.1 ModelIn this case our problem is:

Max EAP (Q1, Q2, …, Qn, t2i)

Subject to:

Prob 11

rBQSCn

iiCpi

Prob ,21

rSQCn

iMsisi

Qi > 0, i =1, 2, 3, …, n, 0 < r1 < 1,

0 < r2 < 1 …(43)

Hence, according to chance constraint programming technique, the optimizingproblem in Equation (35) is restated as:

Max EAP(Q1, Q2, …, Qn, t2i) ...(44)

Subject to:

011

rBVarBEQSCn

iiCpi …(45)

021

rSVarSEQC M

n

iMsisi …(46)

Qi > 0, Qsi > 0, i = 1, 2, 3, …, n, 0 < r1 < 1, 0 < r2 < 1

Figure 3: Two TFN A�

and B�

and Pos (A�

B�

)

I

0 b1 a1 a2 b2 a3b3

�A

(x) �B

(y)

2


7.2 One Constraint is Stochastic and the Other is FuzzyIn this case there are four sub cases. We consider the model under different casesof constraints.

7.2.1 ModelBudget constraint as stochastic and shortage constraint necessity type. In thiscase, the problem is:

Max EAP1 (Q1, Q2, …, Qn, t2i)

Subject to:

Prob 11

rBQSCn

iiCpi

,

Nes 21

n

iMsisi SQC ˆˆ

Equivalent crisp representation of the above problem is given by:

Max EAP1(Q1, Q2, …, Qn, t2i)

Subject to:

011

rBVarBEQSC i

n

iCpi

2

12312

11

1

n

isisisiMM

n

iMsisi

QCCSS

SQC

i.e., Max EAP1(Q1, Q2, …, Qn, t2i) ...(47)

Subject to:

01

1

n

iiCpi rBVarBEQSC

and

n

iMMsisisi SSQCC

122122232 11 …(48)

Qi > 0, Qsi > 0, i =1, 2, 3, …, n, 0 < r1 < 1, 0 < 2 < 1


7.2.2 Model

Budget constraint as stochastic and shortage constraint is of possibility type.

Max EAP1(Q1, Q2, …, Qn, t2i) …(49)

Subject to:

31

22222 11 12 M

n

iMsisisi SSQCC

…(50)

constraint Equation (38)

Qi > 0, Qsi > 0, i =1, 2, 3, …, n, 0 < r1< 1, 0 < 2 < 1

7.2.3 ModelBudget constraint is of necessity type and shortage constraint is stochastic.

Max EAP1(Q1, Q2, …, Qn, t2i) … (51)

Subject to:

21111

2131 11 BBQSCSC i

n

iCpiCpi

…(52)


where EAP1 = EAP with CpiCpiCpi SCSCSC 2131 1

Qi > 0; i =1, 2, 3, …, n, 0 < r2 < 1, 0 < 1 < 1

7.2.4 Model

Budget constraint is possibility type and shortage constraint is stochastic.

Max EAP1(Q1, Q2, …, Qn, t2i) …(53)

Subject to:

n

iiCpiCpi BBQSCSC

131211121 11 …(54)



Qi > 0; i =1, 2, 3, …, n, 0 < r2 < 1, 0 < 1 < 1

7.3 Both Constraints are Fuzzy

Here also we have four sub-cases.


7.3.1 Model

Budget constraint is of Nes type and shortage constraint is also Nes type.

Max EAP1(Q1, Q2, …, Qn, t2i) …(55)

Subject to:

constraint Equations (40) and (44)


and 1222 1 sisisi CCC , Qi > 0; i = 1, 2, 3, …, n, 0 < 2 < 1, 0 < 1 < 1

7.3.2 ModelBudget constraint is of Nes type and shortage constraint is Pos type.

Max EAP1(Q1, Q2, …, Qn, t2i) …(56)

Subject to:



and 2232 1 sisisi CCC , Qi > 0; i =1, 2, 3, …, n, 0 < 2 < 1, 0 < 1 < 1

7.3.3 ModelBudget constraint is Pos type and shortage constraint is of Nes type.

Max EAP1(Q1, Q2, …, Qn, t2i) …(57)

Subject to:



and 2232 1 sisisi CCC , Qi > 0; i =1, 2, 3, …, n, 0 < 2 < 1, 0 < 1 < 1

7.3.4 ModelBudget constraint is of Pos type and shortage constraint is Pos type.

Max EAP1(Q1, Q2, …,Qn, t2i) …(58)

Subject to:



and 2232 1 sisisi CCC , Qi > 0; i = 1, 2, 3, …, n, 0 < 2 < 1, 0 < 1 < 1


8. Numerical ExampleNow, the crisp model is illustrated numerically.

The common input parameters, which are used in the crisp model, are given as:

= 0.04, k = 1, = 0.05, =0.001.

The common input parameters for the ith items are given in Table 1.

1. 5.61980 6.04850 10,278.1000 31,378.200 131,079,000

2. 2.83250 3.08100 2,282.6600 7,039.200 132,640,000

3. 1.89261 2.06491 977.6900 3,021.910 132,902,000

4. 1.42010 1.55260 540.1100 1,671.070 132,988,000

5. 1.13750 1.23890 341.9920 1,058.690 133,027,000

6. 0.94830 1.03760 235.8190 730.274 133,048,000

7. 0.81307 0.88995 172.3860 533.966 133,061,000

8. 0.71160 0.77910 131.4880 407.357 133,069,000

9. 0.63260 0.69280 103.5890 320.969 133,075,000

10. 0.56940 0.62370 83.7126 259.409 133,079,000

Table 2: Variation in the Number of Deliveries

k t21 t22 Qs1 Qs2 EAP

Figure 4: Variation in the Number of Deliveries

140,000,000

120,000,000

100,000,000

80,000,000

60,000,000

40,000,000

20,000,000

0

No. of Deliveries

Exp

ecte

d A

vera

ge P

rofi

t

EAP k

1 2 3 4 5 6 7 8 9 10

The variation in the number of deliveries and product quantity are presented inTables 2 and 3 and Figures 4 to 7.

Table 1: Common Input Parameters

I 0.035 0.0050 0.65 12 18 45 6 800 6 25 0.04 0.2 0.90 4 10

II 0.030 0.0045 0.90 13 19 50 7 900 7 20 0.05 0.3 0.95 6 12

Item Ai Bi siC1i Li Pi Ti Ri Ki di bi CPi C2i CSi CLSi


Tabl

e 3:

Var

iati

on in

the

Pro

duct

ion

Qua

ntit

y

1,1

00

6.0

48

53

1,3

78

.20

13

1,0

79

,00

01

,10

06

.04

85

31

,37

8.2

08

33

,84

2,0

00

1,1

00

6.0

48

53

1,3

78

.20

1,5

75

,10

0,0

00

1,1

10

6.2

21

32

8,1

86

.00

64

6,3

36

,00

01

,11

06

.22

13

28

,18

6.0

01

,34

7,6

70

,00

01

,11

06

.22

13

28

,18

6.0

02

,08

8,9

30

,00

0

1,1

20

6.3

87

62

4,2

20

.60

1,1

91

,32

0,0

00

1,1

20

6.3

87

62

4,2

20

.60

1,8

92

,65

0,0

00

1,1

20

6.3

87

62

4,2

20

.60

2,6

33

,91

0,0

00

1,1

30

6.5

47

71

9,4

60

.10

1,7

66

,85

0,0

00

1,1

30

6.5

47

71

9,4

60

.10

2,4

68

,18

0,0

00

1,1

30

6.5

47

71

9,4

60

.10

3,2

09

,44

0,0

00

1,1

40

6.7

02

31

3,8

82

.90

2,3

73

,75

0,0

00

1,1

40

6.7

02

31

3,8

82

.90

3,0

75

,08

0,0

00

1,1

40

6.7

02

31

3,8

82

.90

3,8

16

,34

0,0

00

1,1

50

6.8

52

07

,46

6.8

23

,01

2,8

60

,00

01

,15

06

.85

20

7,4

66

.82

3,7

14

,19

0,0

00

1,1

50

6.8

52

07

4,6

6.8

24

,45

5,4

50

,00

0

Q1=

1000

, t 2

1=

5.61

98,

Qs1=

1027

8.1

Q1=

1010

, t 21

=5.

772,

Qs1=

6750

.99

Q1=

1020

, t 2

1=

5.91

83,

Qs1=

2636

.78

Q2

t 22Q

s2E

AP

Q2

t 22Q

s2E

AP

Q2

t 22Q

s2E

AP

Figure 5: Variation in 2nd Items Quantitywhen Q1 = 1,000

3,500,000,000

3,000,000,000

2,500,000,000

2,000,000,000

1,500,000,000

1,000,000,000

500,000,000

01,100 1,110 1,120 1,130 1,140 1,150

Q2

EAP

Q2 EAP

Figure 6: Variation in 2nd Items Quantitywhen Q1= 1,010

3,500,000,000

3,000,000,000

2,500,000,000

2,000,000,000

1,500,000,000

1,000,000,000

500,000,000

01,100 1,110 1,120 1,130 1,140 1,150

Q2

EAP

Q2 EAP

4,000,000,000


ConclusionIn real world, defective products cannot be avoided in some production processes.It may be possible and reasonable to discuss the model with defective products.This paper highlights the situation of imperfect production process with fuzzyand stochastic constraint. Our model is realistic due to market situations. Numericalillustration is given to show the optimality of the model. From the sensitivityanalysis, we can say that as the number of deliveries increase, the profit alsoincreases. The profit is highly sensitive towards the total number of units produced.Numerical is shown for the two types of items. We investigated the model withvariable holding cost whose function is dependent on time. Further, the modelcan be developed with a variable rate of deterioration and inflation.

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Figure 7: Variation in 2nd Item Quantity when Q1 = 1,020

5,000,000,000

4,500,000,000

4,000,000,000

3,500,000,000

3,000,000,000

2,500,000,000

2,000,000,000

1,500,000,000

1,100 1,110 1,120 1,130 1,140 1,150

Q2

EAP

1,000,000,000

500,000,000

0

Q2 EAP


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Reference # 61J-2009-03-03-01

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