A Fuzzy Mathematical Programming Model for a Supply...

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. [email protected]

Ahmad Norang*, Assistance Professor, Department of Industrial Engineering, Imam Hossein University, Tehran, Iran. [email protected] Reza Tavakkoli-Moghaddam, Professor, Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.

[email protected]

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: [email protected]

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

http://ihu.academia.edu/

mailto:[email protected]

http://ihu.academia.edu/



**

[email protected]

[email protected]

[email protected]

-

%

-

][

-

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




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]

Chen, C., Lee, W., “Multi-Objective Optimization of

Multi-Echelon Supply Chain Networks with Uncertain

Product Demands and Prices”, Computers and

Chemical Engineering, Vol. 28, 2004, pp. 1131–1144.

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