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),
39Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200940
(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.
41Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
(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)
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200942
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:
43Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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)
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200944
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
45Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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)
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200946
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
47Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200948
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:
Max EAP(Q1, Q2, …, Qn, t2i)
Subject to:
n
iicpi BQSC
1
49Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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:
Max EAP(Q1, Q2, …, Qn, t2i)
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:
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200950
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
51Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200952
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)
53Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
,, .
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200954
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
55Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200956
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)
constraint Equation (38)
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)
constraint Equation (38)
where EAP1 = EAP with CpiCpiCpi SCSCSC 1121 1
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.
57Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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)
where EAP1 = EAP with CpiCpiCpi SCSCSC 2131 1
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:
constraint Equations (42) and (44)
where EAP1 = EAP with CpiCpiCpi SCSCSC 2131 1
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:
constraint Equations (40) and (46)
where EAP1 = EAP with CpiCpiCpi SCSCSC 1121 1
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:
constraint Equations (40) and (46)
where EAP1 = EAP with CpiCpiCpi SCSCSC 1121 1
and 2232 1 sisisi CCC , Qi > 0; i = 1, 2, 3, …, n, 0 < 2 < 1, 0 < 1 < 1
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200958
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
59Supply Chain Model with Imperfect Production Process and Stochastic Demand UnderChance and Imprecise Constraints with Variable Holding Cost
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
The IUP Journal of Computational Mathematics, Vol. II, No. 1, 200960
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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5,000,000,000
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4,000,000,000
3,500,000,000
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