a fuzzy mathematical programming model for a supply network...
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A Fuzzy Mathematical Programming Model for a Supply
Network Design of Raw Materials under Uncertainty – A
Case Study
M. Amirkhan, A. Norang* & R. Tavakkoli-Moghaddam
Mohammad Amirkan, M.Sc., Department of Industrial Engineering, Imam Hossein University, Tehran, Iran. m.amirkhan.ie@gmail.com
Ahmad Norang*, Assistance Professor, Department of Industrial Engineering, Imam Hossein University, Tehran, Iran. a.norang@gmail.com Reza Tavakkoli-Moghaddam, Professor, Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.
tavakoli@ut.ac.ir
Keywords 1 ABSTRACT
With respect to environmental and business conditions, the issue of
designing a supply chain network has attracted lots of attention of
many researchers during the recent years. On the other hand, the
presence of uncertainties in these networks led many researchers to
work on identification of uncertainties and considering them in their
models followed by developing the appropriate decision making
approaches. In this paper, using fuzzy set theory, a bi-objective
mixed-integer linear programming model for supply network of raw
materials of a food industries complex is proposed and with respect
to this complex. In addition to cost factor, two effective and
important factors (i.e., quality and time) are considered in this
model. Furthermore, both parameters and constraints are
considered in fuzzy. To solve the presented model, a two-phase fuzzy
approach is used, Such that the results indicate the total costs of
complex, the quantity of purchasing, the raw materials transported
between facilities, the inventory of raw materials, backorder or
surplus delivery of raw materials to complex and suppliers used in
each period for the complex and intermediaries.
© 2014 IUST Publication, IJIEPM. Vol. 25, No. 2, All Rights Reserved
**
Corresponding author. Ahmad Norang Email: a.norang@gmail.com
August 2014, Volume 25, Number 2
pppp.. 221177--223355
hhttttpp::////IIJJIIEEPPMM..iiuusstt..aacc..iirr//
IInntteerrnnaattiioonnaall JJoouurrnnaall ooff IInndduussttrriiaall EEnnggiinneeeerriinngg && PPrroodduuccttiioonn MMaannaaggeemmeenntt ((22001144))
Network design,
Supply chain, Uncertainty,
Fuzzy programming
**
m.amirkhan.ie@gmail.com
a.norang@gmail.com
tavakoli@ut.ac.ir
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%
-
][
-
hhttttpp::////IIJJIIEEPPMM..iiuusstt..aacc..iirr//
ISSN: 2008-4870
-
-
][
ABCTCO
JIT
AHP-
1 Activity-based Costing 2 Total Cost of Ownership 3 Just in Time 4 Analytic Hierarchy Process
-
PSt Ta
Op
SPr
MPr
SS
Sup
Pro
Dis
Cr
U
SP
MP
MRM
SRM
SOb
MOb
L
NL
F
IL
CDS
UC
TG
UN
TW
SS
VFL
PQ
TQ
TM
DD
In
Ro
OC
CLC
B
RO
VP
SEx
Hu
MHu
(IA)Y-IA
N-IA
[ ] St, Op, MPr, SS, Dis, Cr, SP, SOb, NLP, F/ DS, UC/ FL, TM/ C, CLC/ Hu/ Y-IA
[ ] Ta, MPr, Pro, Dis, U, MP, MOb, NLP, F/ DS, UC/ PQ, TQ, In/ C,CLC, B, RO/ Hu/ N-IA
[ ] St, Op, SPr, Sup, Pro, Dis, Cr, SP, SRM, SOb, LP, F/ DS, UC, UN/ PQ, TQ, FL/ C/ MHu/ N-IA
[ ] Ta, MPr, Pro, Dis, Cr, MP, MOb, LP, F/ UC, TG/ PQ, TQ/ C, CLC/ Ex/ N-IA
[ ] St, Op, SPr, SS, Sup, Pro, Dis, Cr, SP, MRM, SOb, LP, F/ DS, UC, UN, SS/ FL, PQ, TQ, DD/ C/ Hu/ Y-IA
[ ] St, SPr, Dis, U, SP, SOb, LP, F/ DS, UC, UN/ FL, TQ/ C/ Hu/ N-IA
[ ] Ta, MPr, Sup, Pro, Dis, U, MP, MRM, SOb, LP, F/ DS, UC/ PQ, TQ, In/ C/ Ex/ Y-IA
[ ] Ta, MPr, Pro, Dis, U, MP, MOb, LP, F/ DS, UC/ PQ, TQ, In/ C, CLC/ Hu/ Y-IA
[ ] Ta, Op, MPr, Dis, Cr, MP, MOb, LP, F/ UC/ TQ/ C, CLC/ MHu/ N-IA
[ ] St, Ta, MPr, Sup, Pro, Dis, Cr, SP, MRM, SOb, LP, F/ DS, UC/ FL, TQ/ C/ Ex/ N-IA
[ ] Ta, SPr, Dis, Cr, MP, SOb, LP, F/ DS, UC/ In, TQ/ C/ Ex/ N-IA
[ ] St, Ta, SPr, SS, Dis, Cr, SP, MOb, LP, F/ DS, UC/ FL, TQ, TM/ C, CLC/ Ex/ N-IA
[ ] St, Ta, Op, MPr, Pro, Dis, Cr, SP, SOb, LP, F/ UC, DS/ FL, PQ, TQ/ C/ Ex/ Y-IA
[ ] Ta, MPr, Sup, Pro, Dis, U, MP, MRM, MOb, LP, F/ DS, UC, TG/ PQ, TQ, In/ C, VP/ Hu/ Y-IA
[ ] St, Ta, SPr, Pro, Dis, Cr, SP, SOb, LP, IL/ DS/ FL, PQ, TQ/ C, CLC/ Hu, Ex/ N-IA
[ ] St, Ta, SPr, Pro, Dis, U, MP, MOb, LP, IL/ DC, UC/ FL, PQ, TQ/ C, CLC/ Hu, Ex/ N-IA
[ ] St, Ta, SPr, Dis, Cr, SP, SOb, NLP, F/ DS, TG/ FL, TQ, TM, RO/ C, CLC/ Hu/ Y-IA
[ ] Ta, MPr, Sup, Pro, Dis, U, MP, MRM, MOb, LP, F/ DS, UC, TG/ PQ, TQ, In/ C, VP, CLC/ Hu, Ex/ Y-IA
[ ] St, Ta, MPr, Pro, Dis, U, SP, MOb, NLP, IL/ FL, PQ, TQ/ C, CLC/ Ex/ Y-IA
[ ] St, Ta, MPr, Dis, U, SP, MOb, LP, F/ DS, UC/ FL, TQ/ C, CLC/ MHu/ N-IA
[ ] St, Ta, SPr, Sup, Pro, Dis, U, SP, SOb, LP, F/ DS, UC, UN, SS/ FL, DD, PQ, TQ, In/ RO/ Hu, Ex N-IA
[ ] Ta, Op, MPr, Dis, Cr, MP, MOb, LP, F/ UC/ TQ/ C, CLC/ MHu/ N-IA
St, Ta, MPr, Sup, U, MP, MRM, MOb, LP, F / UC, TG / TQ, TM, In / C, CLC, VP / Hu, Ex / Y-IA
1کننده تأمين
2کننده تأمين
3کننده تأمين
4کننده تأمين
1واسطه
2واسطه
کارخانه
”“”“
-
-
JIT
AHP-
1-
2-
3-
4-
Jj = 1, … , J
Kk = 1, … , K
T, … , T1= t,
Pp = 1, … , P
J K= S
1S S
1J J
1slc
11jslcj
11jslcj
12kslck
1olc
11( )jolc t-
jt12
( )kolc tk
t1
ulc
11( )pjc tp
jt 12
( )pkc tpk
t
11( )pjaulc t
pjt12
( )pkaulc t
pk t1
( )ph tpt
1( )pcshor tp
t1
( )pcsur tp
t2kslck
2jkslcj
k2kolck
2( )jkolc t-
jtk2kulck
2( )pjkc tp
jkt 2
( )pjkaulc t
pj t
k 2
( )pkh tpt
k
1( )pkcshor tp
kt
1( )pkcsur tp
kt1sRs
2jkRj
k
1pbp
2pkbpk
1( )pde tpt
2( )pkde tptk
1 ( )pw t
pt
2 ( )pkw t
ptk
1 ( )pww t
pt
2 ( )pkww t
ptk
1( )prn tp
t
2( )pkrn tpk
t
1( )pInfI tpt
2( )pkInfI tpt
k
1( )pMaxI tpt
2( )pkMaxI tpt
k
1
pf
p
2pkf
pk
pjqp-
j
pkqpk
ptqp
jsl
j
tsl
1 ( )pjx tptj
2 ( )pjkx tptj
k
1 ( )pky tptk
)(1 tI ppt
)(2 tI pkpt
k
)(1 tsurpp
t
)(1 tshorpp
t
)(2 tsurpkpk
t
)(2 tshorpkp
kt
1
jz-
j
1
ku-
k
2
jkz-
jk
)(1 tm jt-
j
)(1 tnkt
k
)(2 tm jkt
jk
Min TCO =
[11 1.j j
j
slc z +12 1.k k
k
slc u + 11 1( ). ( )j j
t j
olc t m t +12 1( ). ( )k k
t k
olc t n t +1111 1( ( ) ( )). ( )pj pj pj
p t j
c t aulc t x t +
1212 1( ( ) ( )). ( )pk pk pk
p t k
c t aulc t y t +1 1( ). ( )p p
t p
h t I t ] + [ 2 2.jk jk
k j
slc z +2 2( ). ( )jk jk
k j t
olc t m t +
22 2( ( ) ( )). ( )pjk pjk pjk
k p t j
c t aulc t x t +2 2( ). ( )pk pk
k t p
h t I t ] [1 1( ). ( )p p
p t
csur t sur t +1 1( ). ( )p p
p t
cshor t shor t
+2 2( ). ( )pk pk
p k t
csur t sur t +2 2( ). ( )pk pk
p k t
cshor t shor t ]
Max TVP = 1 1 1.( ( ) ( ))s ps ps
s p t p t
R x t y t + 2 2 ( )jk pjk
j k p t
R x t
1 ( )pj
j
x t + 1 ( )pk
k
y t - 1
( )pde t = )(1 tsurp - )(1 tshorp tp,
)(1 tsurp )(1 tvp . 1 ( )pw t .1
( )pde t tp,
)(1 tshorp (1- )(1 tvp ). 1 ( )pww t .1
( )pde t tp,
1pb + )1(1
psur - )1(1
pshor + (1
(1)pde -1
(1)prn ) = )1(1
pI p
)1(1 tI p + )(1 tsurp - )(1 tshorp + (1
( )pde t -1
( )prn t ) = )(1 tI p , 1p t T
1 ( )pInf I t )(1 tI p 1
( )pMaxI t tp,
2 ( )pjkl
j
x t - 2
( )pkde t = )(2 tsurpk - )(2 tshorpk ktp ,,
)(2 tsurpk )(2 tvpk . 2 ( )pkw t .2
( )pkde t ktp ,,
)(2 tshorpk (1- )(2 tvpk ). 2 ( )pkww t .2
( )pkde t ktp ,,
2
pkb + )1(2
pksur - )1(2
pkshor + (2
(1)pkde -2
(1)pkrn ) = )1(2
pkI kp,
)1(2 tI pk + )(2 tsurpk - )(2 tshorpk + (2
( )pkde t -2
( )pkrn t ) = )(2 tI pk , , 1p k t T
2
( )pkInfI t )(2 tI pk 2
( )pkMaxI t ktp ,,
1 ( )pky t )(2 tI pk ktp ,,
1
1 ( )pj
j s
x t
+
1
1 ( )pk
k s
y t
1
pf . ( 1 ( )pj
j s
x t
+ 1 ( )pk
k s
y t
) tp,
1
2 ( )pjk
j J
x t
2
pkf .2 ( )pjk
j
x t , ,p k t
1 ( )pjx t (1
( )p
t
de
) . )(1 tm j tjp ,,
1 ( )pky t (1
( )p
t
de
) . )(1 tnk tkp ,,
2 ( )pjkx t (2
( )pk
t
de
) . )(2 tm jk tkjp ,,,
)(1 tI p 1
1( )
t
p
t
de
,p t T T
11 1. ( )pjpj
j
q x t + 12 1. ( )pkpk
k
q y t 1 1.( ( ) ( ))pj pkp
j k
Tq x t y t tp,
11 1. ( )j pj
p j
sl x t +12 1. ( )k pk
p k
sl y t 1 1.( ( ) ( ))pj pk
p j p k
Tsl x t y t t
t
jj tmz )(11 j
11 )( jj ztm tj,
t
kk tnu )(11 k
11 )( kk utn ,k t
t
jkjk tmz )(22 kj,
22 )( jkjk ztm tkj ,,
1 ( )pjx t , 2 ( )pjkx t , 1 ( )pky t , )(1 tI p , )(2 tI pk , )(1 tsurp , )(1 tshorp , )(2 tsurpk , )(2 tshorpk 0
1
jz , 2
jkz , 1
ku , )(1 tm j , )(2 tm jk , )(1 tnk , 1( )pV t , )(2 tvpk {0 , 1}
.
-
tjk1 ( )jm t)(1 tnk
)(1 txpjl)(1 typkl
t
jk)(2 tm jk
)(2 txpjkl
1
jz
j1 ( ) 0jtm t
1
jz
j
1
1
1
. 1,...,
1,...,
0
n
j j
j
n
ij i
j
n
ij ij
j
Minimize z c x
s t a b i l
a x b i l m
x
][
1
1
1
. + (1- ) 1,...,
1,...,
0
n
j j
j
n
ij i i
j
n
ij ij
j
Minimize z c x
s t a b t i l
a x b i l m
x
][
1
1
1
( )3
. ( ) ( )+( + )(1- ) 1,...,3 3 3
( ) ( ) 3 3
j j
ij ij i i i i
ij ij i i
nc c
j j
j
na a b b t t
ij j i i
j
na a b b
ij j i
j
d dMinimize z c x
d d d d d ds t a x b t i l
d d d da x b
1,...,
0
i l m
x
jcdjcdjc
Rj cj rj
j j jd c r j j jd R c
jcu
1
Min TCO = [11 11
11 1( ).
3
j jslc slc
j j
j
d d
slc z
+ 12 12
12 1( ).3
k kslc slck k
k
d dslc u
+
11 1111 1( ( ) ). ( )
3
j jolc olc
j j
t j
d dolc t m t
+
12 1212 1( ( ) ). ( )
3
k kolc olck k
t k
d dolc t n t
+
11 11 11 1111 11 1( ( ) ( ) ). ( )
3 3
pj pj pj pjc c aulc aulc
pj pj pj
p t j
d d d dc t aulc t x t
+
12 12 12 1212 12 1( ( ) ( ) ). ( )
3 3
pk pk pk pkc c aulc aulc
pk pk pk
p t k
d d d dc t aulc t y t
+
1 11 1( ( ) ). ( )
3
p ph h
p p
t p
d dh t I t
] +
[2 2
2 2( ).3
jk jkslc slc
jk jk
k j
d dslc z
+
2 22 2( ( ) ). ( )
3
jk jkolc olc
jk jk
k j t
d dolc t m t
+
2 2 2 22 2 2( ( ) ( ) ). ( )
3 3
pjk pjk pjk pjkc c aulc aulc
pjk pjk pjk
k p t j
d d d dc t aulc t x t
+
2 22 2( ( ) ). ( )
3
pk pkh h
pk pk
k t p
d dh t I t
] +
[1 1
1 1( ( ) ). ( )3
p pcsur csur
p p
p t
d dcsur t sur t
+
1 11 1( ( ) ). ( )
3
p pcshor cshor
p p
p t
d dcshor t shor t
+
2 22 2( ( ) ). ( )
3
pk pkcsur csur
pk pk
p k t
d dcsur t sur t
+
2 22 2( ( ) ). ( )
3
pk pkcshor cshor
pk pk
p k t
d dcshor t shor t
]
Max TVP = 1 1 1.( ( ) ( ))s ps ps
s p t p t
R x t y t + 2 2. ( )jk pjk
j k p t
R x t
1 ( )pj
j
x t + 1 ( )pk
k
y t - 1 1
1( ( ) )3
p pde de
p
d dde t
= )(1 tsurp - )(1 tshorp ,p t
)(1 tsurp )(1 tvp . 1 ( )pw t .1 1
1( ( ) )3
p pde de
p
d dde t
,p t
)(1 tshorp (1- )(1 tvp ). 1 ( )pww t .1 1
1( ( ) )3
p pde de
p
d dde t
,p t
1
pb + )1(1
psur - 1(1)pshor + (1 1
1( (1) )3
p pde de
p
d dde
-
1 11( (1) )
3
p prn rn
p
d drn
) = )1(1
pI p
)1(1 tI p + )(1 tsurp - )(1 tshorp + (1 1
1( ( ) )3
p pde de
p
d dde t
-
1 11( ( ) )
3
p prn rn
p
d drn t
) = )(1 tI p
, 1p t T
1 1
8 818( ( ) ) ( ).(1 )
3 3
p pInfI InfI t tp
d d d dInfI t t
)(1 tI p ,p t
)(1 tI p 1 1
8 818( ( ) ) ( ).(1 )
3 3
p pMaxI MaxI t tp
d d d dMaxI t t
,p t
2 ( )pjkl
j
x t - 2 2
2( ( ) )3
pk pkde de
pk
d dde t
= )(2 tsurpk - )(2 tshorpk , ,p t k
)(2 tsurpk )(2 tvpk .2 ( )pkw t .
2 22( ( ) )
3
pk pkde de
pk
d dde t
, ,p t k
)(2 tshorpk (1- )(2 tvpk ).2 ( )pkww t .
2 22( ( ) )
3
pk pkde de
pk
d dde t
, ,p t k
2
pkb + )1(2
pksur - )1(2
pkshor + (2 2
2( (1) )3
pk pkde de
pk
d dde
-
2 22( (1) )
3
pk pkrn rn
pk
d drn
) = )1(2
pkI ,p k
)1(2 tI pk+ )(2 tsurpk
- )(2 tshorpk+(
2 22( ( ) )
3
pk pkde de
pk
d dde t
-
2 22( ( ) )
3
pk pkrn rn
pk
d drn t
)= )(2 tI pk
, , 1p k t T
2 2
14 14214( ( ) ) ( ).(1 )
3 3
pk pkInfI InfI t tpk
d d d dInfI t t
)(2 tI pk , ,p t k
)(2 tI pk 2 2
14 14214( ( ) ) ( ).(1 )
3 3
pk pkMaxI MaxI t tpk
d d d dMaxI t t
, ,p t k
1 ( )pky t )(2 tI pk , ,p t k
1
1 ( )pj
j s
x t
+
1
1 ( )pk
k s
y t
1 11( )
3
p pf fp
d df
( 1 ( )pj
j s
x t
+ 1 ( )pk
k s
y t
) 16 16 .(1 )16( )
3
t td dt
,p t
1
2 ( )pjk
j J
x t
2 22( ).
3
pk pkf f
pk
d df
17 172
17 .(1 )( ) ( )3
t tpjk
j
d dx t t
, ,p k t
1 ( )pjx t (1 1
1( ( ) )3
p pd d
p
t
d dd
) . )(1 tm j , ,p j t
1 ( )pky t (1 1
1( ( ) )3
p pd d
p
t
d dd
) . )(1 tnk , ,p k t
2 ( )pjkx t (2 2
2( ( ) )3
pk pkd d
pk
t
d dd
) . )(2 tm jk , , ,p j k t
)(1 tI p 1 11
1( ( ) )3
p p
td d
p
t
d dd
,p t T T
11 1111 1( ). ( )
3
pj pjq q
pj pj
j
d dq x t
+
12 1212 1( ). ( )
3
pk pkq q
pk pk
k
d dq y t
1 1( ).( ( ) ( ))
3
p pTq Tq
p pj pk
j k
d dTq x t y t
,p t
11 1111 1( ). ( )
3
j jsl sl
j pj
p j
d dsl x t
+
12 1212 1( ). ( )
3
k ksl slk pk
p k
d dsl y t
1 1( ).( ( ) ( ))
3
Tsl Tslpj pk
p j p k
d dTsl x t y t
t
t
jj tmz )(11 j
11 )( jj ztm ,j t
t
kk tnu )(11 k
11 )( kk utn ,k t
t
jkjk tmz )(22 ,j k
22 )( jkjk ztm , ,j k t
1 ( )pjx t , 2 ( )pjkx t , 1 ( )pky t , )(1 tI p , )(2 tI pk , )(1 tsurp , )(1 tshorp , )(2 tsurpk , )(2 tshorpk 0
1
jz , 2
jkz , 1
ku , )(1 tm j , )(2 tm jk , )(1 tnk , 1( )
pV t , )(2 tvpk {0 , 1}
][min-
max
min-max][
][
dd
αα-PISα
α-NISα
2 2
-Nis2 22 2 2 2
2 2
2 2
1 if Z
( ) if Z
0 if Z
Pis
NisPis
Pis Nis
Nis
Z
Z Zx Z Z
Z Z
Z
1 1
-Pis1 11 1 1 1
1 1
1 1
1 if Z
( ) if Z
0 if Z
Pis
NisNis
Nis Pis
Nis
Z
Z Zx Z Z
Z Z
Z
(x)hµh
][
Max λ(x) = γλ0 + (1 – γ) ( )h h
h
x
s.t. λ0 µh(x) , h = 1, 2
x F(x) , λ0 and λ [0 , 1]
F(x)
hθγh
}(x)hµ{h=min0λ
γhθ
-
γαhθ
-
AHP
2θ1θγα
-
GAMS 22.9.2
α2θ 1θ
1PisZ
1NisZ
2NisZ
2PisZ
1Z 2Z 1 2
α2θ1θ
1Z 2Z 0 1 2
12α
1 2 1Z 2Z 0 1 2
γα2θ1θ
γ
γ
γγ
γ
2θ1θαγ
2θ1θ
1 Balanced Solution
AHP
Degraeve, Z., Roodhooft, F., Doveren, B.V, “The Use
of Total Cost of Ownership for Strategic Procurement:
a Company-Wide Management Information System”,
Journal of the Operational Research Society, Vol. 56,
2005, pp. 51-59.
[3]
Jang, Y., Jang, S., Chang, B., Park, J., “A Combined
Model of Network Design and Production
/Distribution Planning for a Supply network”,
Computer & Industrial Engineering, Vol. 43, 2002,
pp. 263-281.
[4]
Croom, S., Romano, P., Giannakis. M., “Supply
Chain Management: an Analytical Framework for
Critical Literature Review”. European Journal of
Purchasing & Supply Management, Vol. 6, 2000, pp.
67-83.
[5]
Peidro, D., Mula, J., Polar, R., Lario, F.C.,
“Quantitative Models for Supply Chain Planning
Under Uncertainty: a Review”, International Journal
of Advance Manufacture Technology, Vol. 43, 2009,
pp. 400–420.
[6]
Eskigun, E., Uzsoy R., Preckel, P.V., Beaujon, G.,
Krishnan, S., Tew, J.D., “Outbound Supply Chain
Network Design with Mode Selection, Lead Times
and Capacitated Vehicle Distribution Centers”,
European Journal of Operational Research, Vol. 165,
No. 1, 2005, pp. 182-206.
[7]
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