Proceedings of 5th International Symposium on Intelligent Manufacturing Systems, May 29-31, 2006: 1382-1401 Sakarya University, Department of Industrial Engineering

Supplier Selection with Genetic Algorithm and Fuzzy AHP

Cemalettin Kubat (*), Baris Yuce (*)

*Department of Industrial Engineering, Sakarya Univ ersty, Sakarya, Turkey

Abstract

Nowadays, within new important strategies for production price and quality,

supplier plays a key role in the corporate competition. Because of this reason, supplier

selection must be considerate for all corporate. Supplier selection may include a multi

criteria problem which includes both qualitative and quantitative factors for example

purchase cost, quality level, supplier risk etc... Selecting best supplier is necessary to

make a trade off between tangible and intangible factors. In this work we suggested to

integrate Analytic Hierarchy Process (AHP), Fuzzy AHP and Genetic Algorithm (GA) to

determine best suppliers. Fuzzy set will be utilized linguistic factor to organize criteria and

sub criteria weight, with pair wise compare with fuzzy AHP; it will be utilized to organize all

factors and which assigned weighting for related factor. Finally, a hypothetical supplier

selection problem will be solved by proposed (GA) algorithm.

Keywords: Fuzzy Logic; Analytic Hierarchy Process; Genetic Algorithms

1. Introduction

Supplier selection decisions are an important tool for achieving corporate

competition. In most industries the cost of raw materials and component parts constitutes

the main cost of a product, such that in some cases it can account for up to 70% [1]. Since

different factories supply under different supplier parameter, such as total cost, service

level, quality rate, on time delivery etc… selection of supplier and allocation of demand

really become a hard problem. Such that manner, corporate purchasing department play a

key and vital role. Several conflicting factors affect supplier selection conditions. Also they

affect supplier performance. Because of these circumstances, our model has to translate

a Multi criteria decision making problem [2]. Stamm and Golhar [3], Ellram [4] and Roa

and Kiser [5] identified, respectively, 13, 18, and 60 criteria for supplier selection. Since

supplier selection problem a multiple criteria problem, it requires to trade off between

tangible and intangible factors to find best supplier.

In this paper, firstly, we use fuzzy AHP for uncertain weight which are linguistic

expressions; then, determined each of suppliers’ weight by using identified factors; finally,

determined best supplier by using Genetic Algorithm for each of part order.

We organized this paper as follows. The section 2 gives literature knowledge about

supplier selection. In section 3 defined Fuzzy AHP and determine criteria weight by

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entropy .In section 4 defined Genetic Algorithm In section 5 defined problem descriptions.

In section 6 generate a model with Fuzzy AHP and GA In section 7 give an example

model for Multi Source Supplier Selection. In section 7.1 define a numerical example

and solution. In section 8 concluded about supplier selection problem. In section 9 defined

references.

2. Literature review

The literature in this area discusses both single sourcing and multiple sourcing.

Based on this idea there are two kind of supplier selection process and two kind of aim

such as;

1. Single Sourcing: In single sourcing model, the constraints are not considered by

purchasing officer. That mean, all supplier can satisfy the buyer’s requirement of

demand, quality, delivery time and the specs. At that point, the purchase officer

makes only one decision, which one is the best supplier.

2. Multiple Sourcing: In that type process, there is some limitation such as, supplier

cost, supplier capacity, supplier quality or the buyer needs to purchase some part

of demand from one supplier and the other etc… At these conditions the officer,

make two decisions, the first one ,which is the best supplier and second one, how

many percent of order should be respond by selected supplier?

The vast majority of the decision models applied to the supplier selection are linear

weighting models, fuzzy AHP approach and mathematical programming model.

2.1. Fuzzy AHP Approach Model

Fuzzy set theory has proven advantages within uncertain and imprecise condition. Fuzzy

set theory resembles as human reasoning in the use of approximate information and use

linguistic expressions for solving uncertainty. In supplier selection processes, suppliers’

weights are given fuzzy number; we can use fuzzy set rules for fuzzy weights turn to

certain number.

Generally , in supplier selection problem ,some researcher use fuzzy set theory with AHP

such as Felix T. S. Chan and Niraj Kurman [6]or fuzzy with multi objective problem such

as A. Amid ,S.H. Ghodsypour and C. O’Brien.[2]

2.2Linear weighting models

In linear weighting models, weights are given to the criteria, the biggest weight indicating

the highest importance rating on the criteria are multiplied by their weights and summed in

order to obtain a single score for each supplier [7].

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2.3. Mathematical programming models

In mathematical programming, decision maker has one or more than objective. Decision

maker wants to be satisfied from all objectives. Some times, we can not have any efficient

model for supplier selection because of the intractable factor. So we need to use heuristic

method or another alternative usage is using mathematical method with AHP.

2.4. Analytical Hierarchical Process (AHP):

AHP is developed by Saaty[8] . It is a well proven multi-attribute decision making

methodology, especially powerful for those complex problems with asset of highly

interrelated factors [9-10]. With a pool of potential options, AHP make pair comparison

and helps to determine which alternative is the better than the other criteria. If there is not

any constraint for problem, AHP is enough for making decision. such as, single source

which is mentioned in this article. In Fig.3 we can see the hierarchic structure of the

supplier selection’s factors. If the value for alternative I and j are respectively iw and jw ,

the preference of alternative is

i to j is equal to iw / jw .Hence the pairwise comparison matrix is;

1w / 1w 1w / 2w ……. 1w / nw ,

2w / 1w 2w / 2w ……. 2w / nw ,

nw / 1w nw / 2w ……. nw / nw ,

As this matrix is consistent the weight of each element is its relative normalized amount

[11]:

Weight of ith element =

∑=

n

ii

i

w

w

1

The priority of alternative i to j for negative criteria, such as cost, is equal to jw / iw , then

the pairwise comparison matrix is;

As this matrix is also consistent, the weights of elements are the normalized amount of

any columns, which is equal to the inverse normalized amount of the alternatives:

Weight of ith element (for negative criteria) =

∑=

n

i i

i

w

w

1

1

1

[12]

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3. Fuzzy AHP

Fuzzy set theory has proven advantages within fuzzy , imprecise and uncertain

manner and looks like as human reasoning in its use of approximate information and

uncertainty to generate decisions. Fuzzy set theory implements classes and grouping of

data with boundaries that are not sharply defined (i.e. fuzzy). In conventional, AHP, the

pairwise comparison is established using a nine-point scale which converts the human

preferences between available alternatives as equally, moderately, strongly, very strongly

or extremely preferred. Even though the discrete scale of AHP has the advantages of

simplicity and ease of use, it is not sufficient to take into account the uncertainty

associated with the mapping of one’s perception to a number [13]. The linguistic

assessment of human feelings and judgements are vague and it is not reasonable to

represent it in terms of precise numbers. It feels more confident to give interval

judgements than fixed value judgements. Hence, triangular fuzzy numbers are used to

decide the priority of one decision variable over other. Synthetic extent analysis method is

used to decide the final priority weights based on triangular fuzzy numbers and so-called

as fuzzy extended AHP (FEAHP) [6].The FEAHP is the fuzzy extension of AHP to

efficiently handle the fuzziness of the data involved in the decision of best global supplier.

It is easier to understand and it can effectively handle both qualitative and quantitative

data in the multi-attribute decision making problems. We used in this article triangular

fuzzy number shown as

fig 1.A Fuzzy triangulars membership function

A fuzzy set [14,15] is characterized by a membership function, which assigns to each

object a grade of membership ranging between 0 and 1. In this set the general terms

such as “large”, “medium”, and “small” each will be used to capture a range of numerical

values.

Aµ

A

)( ylA )( yrA

1a 2a 3a

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A fuzzy number is a special fuzzy set, such that;

}),(,{ RxxxM M ∈= µ where the x value is lies on the R 1 i.e. ∞≤≤∞− x and )(xMµ is

continious from R1 to close interval[0,1]. And )(xMµ is defined each fuzzy numbers

membership function ,which is shown as eq.1 [6].

≤≤−−≤≤−−

=otherwise

axaaaxa

axaaaax

xM

0

)/()(

)/()(

)( 32233

21121

µ Eq.1.

3.1 Calculation Each of AHP’s value by Fuzzy AHP Mo del

If object set is denoted by P={ }nppp ........, 21 and the objective set is denoted by

Q={ }nqqq ........, 21 then according to extended concept analysis [16] each object is taken

and extend analysis for each objective Oi is performed, respectively. Hence the m extent

analysis values for each object are obtained with thefollowing signs:

oim

oioi AAA ,...., 21 i=1,2, ....,n, oikA k=(1,2,...m) are triangular fuzzy numbers. The value of

fuzzy synthetic extent with respect to the ith object is defined as eq.2.;

1

1 1 1

−

= = =∑ ∑∑

⊗=m

k

n

i

m

k

oik

oik

i AAF eq.2.

The value of ∑=

m

k

oikA

1

can be found by performing the fuzzy addition operation of m extent

analysis values from a particular matrix shown as eq.3.;

∑=

m

k

oikA

1

=

∑∑∑

===

m

kk

m

kk

m

kk aaa

13

12

11 ,, eq.3

And the value of ∑∑= =

n

i

m

k

oikA

1 1

is shown as eq.4.

∑∑= =

n

i

m

k

oikA

1 1

=

∑ ∑∑

= ==

n

i

n

ik

n

ikk aaa

1 13

121 ,,

eq.4.

Because of eq.2 , we shold tranformed eq.4 such eq.5.

∑∑∑===

n

ik

n

ik

n

ik aaa

11

12

13

1,

1,

1 eq.5.

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the degree of possibility of A1 ={ } { }2322212131211 ,,,, aaaAaaa =≥ is defined as

[ ]))(),(min(sup)( 221 1xxAAV AA

yx

µµ≥

=≥ when a pair (x,y) exists such that yx ≥ and

1)()( 21== yx AA µµ then we have )( 21 AAV ≥ =1 .eq.6

Since A 1 and A 2 convex number. If 2211 aa ≥ then )( 21 AAV ≥ =1 and )( 12 AAV ≥ then

)( 12 AAV ≥ = )()(121 dAAhgt Aµ=∩ where d is the ordinate of the highest intersection

point. D between 1Aµ and 2Aµ . When A1 A 1 = ( )131211 ,, aaa and ( )2322212 ,, aaaA = so we

compute the degree of possibility such as;

)( 12 AAV ≥ = )( 21 AAhgt ∩ =)()( 11122322

2311

aaaa

aa

−−−−

eq.7

For the comprasion of A1 and A 2 , both the value of )( 21 AAV ≥ and )( 12 AAV ≥ are

required.The degree possibility for a convex fuzzy number to be greater than j convex

fuzzy numbers A i (i = 1, 2, . . . , j) can be defined by

V =≥ ),....,( 21 jAAAA [ ]).......()()( 21 jAAandAAandAAV ≥≥≥ =

)min( iAA ≥ , i = 1, 2, . . . , k. eq.8.

if m(P i ) = min )( ji FFV ≥ , eq.9

for j = 1, 2, . . . , n; j ≠ i. then the weight vector is given by W P = TnPmPmPm ))(),...(),(( 21

where ),...2,1( niPi == are n elements. After normalizing W P , we get the normalized

weight vectors W = TnPmPmPm ))(),...(),(( 21 where W is a non-fuzzy number and this

gives the priority weights of one alternative over other.[6]

4. Genetic Algorithm

Genetic Algorithm (GA) is one of modern heuristic technique, which has been

widely adopted by many researchers in solving various problems. GA was developed by

John Holland in 1960. A GA works with a population of individual strings (chromosomes),

each representing a possible solution to a given problem. Each chromosome (individual)

is assigned a fitness value according to the result of the fitness (or objective) function.

Such highly fit chromosomes will survive more frequently than other in the population, and

they are given more opportunities to reproduce and the offspring (child) share features

taken from their parents. It is a heuristic optimization algorithm, which imitate the natural

genetic evolution’s mechanism. If there is big solution space for problem, we can use GA

for solving the problem. Genetic algorithm is a search method in solution space. It work

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such as; firstly determine an initial solution pool, each solution is named chromosome.

And each chromosome is formed by genes which are one of problems attributes.

Generally, the initial pool generated randomly [17].

5. Problem description

In this article, we discussed about more factors which affected decision makers’ stability.

In production configuration, the finished product is usually composed of many parts. Each

of parts can be provided by different suppliers, which can be different location. Another

problem is tangible and intangible factors and all these factors have conflict

circumstances. Hence these condition, our processes follow these steps;

1. By the fuzzy AHP, we determine factors weight from linguistic expression and

Determined each supplier weight and final criteria weight by AHP;

2. Searched the best supplier and determined order quantity of that supplier with

GA.

We can see that proposed methodology of problem in fig.2. step by step.

Fig2. The algorithm of the proposed method

Supplier Evoulation

Data collect

Determine weight of each criterias and subcriterias

Fuzzy set theory

Determined final criteras for each supplier by

Fuzzy AHP

Determine the best

supplier and quantity

Genetic Algorithm

Create initial pool’s population size

• Set crosover rate • Set mutation rate • Set terminatination

rule; generation size, time etc...

Generation i

Stop

Reproduct

Evaluate

Crossover

Mutation

Genaration i+1

Stop

No

Yes

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6. Generate a model with Fuzzy AHP and GA

Firstly; we must determine which criteria affect to our supplier selection process.

We determine main and sub criteria in Fig 3.

Fig.3. Hierarchy for Supplier Selection

We can see some criteria at figure3. which have sub criteria. Firstly, we should determine

each criterion’s weight, use pairwise comparison between each of criteria by Fuzzy AHP.

We have linguistic variable for each criteria. And we must transform each of them to the

numeric value .Our Scales is shown on table.1. For our model it shown as (Fuzzy number)

on table 1.2.3.4.5.6 then We determined each criteria’s and sub criteria’s final weight

Finally the used each crisp value for determined best suppliers and its assignment rate by

using A[6].

We presume that our purchase order is inspect the all candidate supplier and after

evaluation, they give score to them and each of score is constituted each linguistic value’s

membership function parameters. It shown on table1. We signified Saaty’s 1-9 scales in

fuzzy set.

Supplier Profile (C4)

Service Performance

(C3)

Quality (C2)

Cost (C1)

Supplier Evoulation and Selection

Risk Factor (C5)

C21 C22

C23

C31 C32

C33 C34

C41

C43

C51 C52

C53

Suppler 1 Suppler 2 Suppler 3 Suppler 4 Suppler 5 Suppler 6

C42

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Table .1 Each of membership functions’ parameter for Saaty’s scale

Linguistic Expressions a 1 a 2 a 3

Equal 1 1 2

Equal -Moderate 1 2 3

Moderate 2 3 4

Moderate- Fairly Strong 3 4 5

Fairly Strong 4 5 6

Fairly Strong- Very Strong 5 6 7

Very Strong 6 7 8

Very Strong- Absolute 7 8 9

Absolute 8 9 9

Table.2 Fuzzy number for each main criterion.

Criteria MC 1 MC 2 MC 3 MC 4 MC 5 Weight

MC 1 (1,1,1) (2,3,4) (2,3,4) (3,4,5) (3,4,5) 0,416

MC 2 (1/4,1/3,1/2) (1,1,1) (2,3,4) (3,4,5) (3,4,5) 0.341

MC 3 (1/4,1/3,1/2) (1/5,1/4,1/3) (1,1,1) (2,3,4) (2,3,4) 0.162

MC 4 (1/5,1/4,1/3) (1/5,1/4,1/3) (1/4,1/3,1/2) (1,1,1) (2,3,4) 0.017

MC 5 (1/5,1/4,1/3) (1/5,1/,4,1/3) (1/4,1/3,1/2) (1/4,1/3,1/2) (1,1,1) 0.064

The each main criteria are defined in fuzzy set respectively ; F 1 , F 2 , F 3 , F 4 , F 5

F 1 =(11,15, 19) ⊗ ( 1/53.347,1/ 41.927, 1/31.4)= (0.206, 0.357, 0.605)

F 2 =(9.25, 12.34 , 15.5) ⊗ ( 1/53.347,1/ 41.927, 1/31.4)= (0.1734, 0.294, 0.4936)

F 3 =(5.45, 7.58, 10.01) ⊗ ( 1/53.347,1/ 41.927, 1/31.4)= (0.102, 0.18, 0.318)

F 4 =(3.65, 4.84, 6.167) ⊗ ( 1/53.347,1/ 41.927, 1/31.4)= (0.068,0.1154, 0.1964)

F 5 =(1.9, 2.167, 2.67) ⊗ ( 1/53.347,1/ 41.927, 1/31.4)= (0.0356, 0.051, 0.085)

The degree of possibility of F i over F k k ≠ i can be determined in eg.6 -8.

)( 21 FFV ≥ =1 , )( 31 FFV ≥ =1 , )( 41 FFV ≥ = 1 , )( 51 FFV ≥ =1;

)( 12 FFV ≥ =

−−−−

)206.0357.0()4936.0294.0(

4936.0206.0=0.82 ; )( 32 FFV ≥ =1 )( 42 FFV ≥ =1 ;

)( 52 FFV ≥ =1 , )( 13 FFV ≥ =0.39 ; )( 23 FFV ≥ =0.56 ; )( 43 FFV ≥ =1 ; )( 53 FFV ≥ =1

)( 14 FFV ≥ =0.041 ; )( 24 FFV ≥ =0.07255; )( 34 FFV ≥ =0.594 ; )( 54 FFV ≥ =1

)( 15 FFV ≥ = 0.65; )( 25 FFV ≥ = 0.57; )( 35 FFV ≥ =0.151 ; )( 45 FFV ≥ =0.509

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M(C 1 )= min { )( 21 FFV ≥ , )( 31 FFV ≥ , )( 41 FFV ≥ , )( 51 FFV ≥ }=min{1,1,1,1}=1 the same way

M(C 2 )= 0.82 ; M(C 3 )=0.39; M(C 4 )=0.041 ; M(C 5 )=0.151

Hence the weight vektor of main criteria W C ={1,0.82,0.39,0.041,0.151} T . Now we must

normalised each of criteria, because we want to show total of that criteria =1. So if Normalised

them; each of ourr criteria’s weight is respectively; W C ={0,416 , 0.341 , 0.162 , 0.017,

0.064} T found; Used the same way, we determined each other sub criteria’s weight , which is

shown as table 3,4,5,6

Table.3 Fuzzy number of Quality factor’s Subcriteria.

SubCriteria SC 21 SC 22 SC 23 Weight

SC 21 (1,1,1) (4,5,6) (1/4,1/3,1/2) 0.512

SC 22 (1/6,1/5,1/4) (1,1,1) (1,2,3) 0.150

SC 23 (2,3,4) (1/3,1/2,1) (1,1,1) 0.338

Table.4 Fuzzy number of Service Performance factor’s Subcritearia.

SubCriteria SC 31 SC 32 SC 33 SC 34

Weight

SC 31 (1,1,1) (1/4,1/3,1/2) (4,5,6) (2,3,4) 0.520

SC 32 (2,3,4) (1,1,1) (1/3,1/2,1) (1,2,3) 0.198

SC 33 (1/6,1/5,1/4) (1,2,3) (1,1,1) (1/3,1/2,1) 0.125

SC 34 (1/4,1/3,1/2) (1/3,1/2,1) (1,2,3) (1,1,1) 0.157

Table.5 Fuzzy number of Supplier Profile factor’s Subcritearia.

SubCriteria SC 41 SC 42 SC 43 Weight

SC 41 (1,1,1) (2,3,4) (1,1,2) 0.686

SC 42 (1/4,1/3,1/2) (1,1,1) (2,3,4) 0.157

SC 43 (1/2,1,1) (1/4,1/3,1/2) (1,1,1) 0.157

Table.6 Fuzzy number of Risk factor’s Subcritearia.

SubCriteria SC 51 SC 52 SC 53

Weight

SC 51 (1,1,1) (4,5,6) (1/4,1/3,1/2) 0.512

SC 52 (1/6,1/5,1/4) (1,1,1) (1,2,3) 0.150

SC 53 (2,3,4) (1/3,1/2,1) (1,1,1) 0.338

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7. An Example Model for Multi Source Supplier Selec tion

To demonstrate the ability of this proposed article considering the product part

change , it is applied an example idea which is affected by an some researcher , who are

H.S Wang and Z.H.Che.[19].Our product tree is seem in figure 4.(Each of part composed

1 subpart.)

Fig.4. Showed R’s product tree.

We interested in X half product, which is composed five purchasing material in fig.5.

Fig.5. Shows X half product’s product tree

.

In table 7. we can see that X’s parts and whose appropriate suppliers are shown.

Table7.Each parts and Suppliers

A B C D E

Supplier 1,2,3 Supplier 1,2,4 Supplier 5 Supplier3,5 Supplier4,5

R

W X

X

A B C D E

T Y Z

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Table8.Supplier’s Quantitative Information

Sup. Cost ($)

(*)

Technical

Level(Fuzzy)

Scale (*)

Defect

Rate(%)

(*)

Reliability

Rate (%)

(**)

Flexibility

(Rate)(%)

(**)

On time delivery

(Rate)(%) (**)

Response

(Rate)(%)

(**)

S1 A 5 Moderate 0.05 0.85 0.60 0.90 0.70

B 3 Strong 0.02 0.90 0.50 0.85 0.60

S2 A 3.5 Fairly Strong 0.02 0.95 0.50 0.90 0.90

B 2.5 Strong 0.03 0.95 0.40 0.90 0.80

S3 A 4 Strong 0.03 0.90 0.60 0.80 0.80

D 6 Very strong 0.06 0.75 0.70 0.87 0.80

S4 B 4 Moderate 0.04 0.90 0.80 0.85 0.85

E 5 Fairly Strong 0.03 0.90 0.40 0.90 0.80

S5 C 5 Absolute 0.02 0.95 0.30 0.98 0.825

D 5 Strong 0.03 0.95 0.30 0.90 0.65

E 6 Fairly Strong 0.04 0.85 0.45 0.80 0.75

Sup. Cominac.

Status

(Fuzzy) (**)

Financial

Status

(Fuzzy) (**)

Supplier

Capacity

(Part)

Supplier

Experien.

(Year)(**)

Geograph.

Status

(Fuzzy) (*)

Machine Status

(Fuzzy) (*)

Worker

Status

(Fuzzy) (*)

S1 A Absolute Moderate 800 5 Very

strong

Absolute Moderate

B Strong Strong 500 5 Moderate Fairly Strong-

Very Strong

Strong

S2 A Strong Very strong 400 10 Strong Moderate Strong

B Strong Very strong 1000 7 Moderate Fairly Strong-

Very Strong

Very strong

S3 A Very strong Strong 1000 6 Fairly

Strong

Fairly Strong Very strong

D Strong Moderate 1200 3 Very

strong

Moderate Strong

S4 B Very strong Very Strong 400 2 Very

Strong

Strong Moderate

E Moderate Moderate 600 1 Strong Moderate Strong

S5 C Very Strong Absolute 1500 1 Absolute Absolute Absolute

D Moderate Moderate 500 10 Moderate Moderate Absolute

E Moderate Absolute 1000 4 Fairly

Strong-

Very

strong

Fairly Strong-

Very Strong

Absolute

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(*)

∑=

n

i i

i

w

w

1

1

1 , (**)

∑=

n

i

i

w

w

1

.

Table 9. Each supplier weight for priority weight of Part A

PartA MC1 MC2 MC3

Criter.

Weight

Main

Criter.

Weight

SC21 SC22 SC23 Criter.

Weight

Main

Criter.

Weight

SC31 SC32 SC33 SC34 Criter.

Weight

Main

Criter.

Weight

0,416 0.512 0.150 0.338 0.341 0.520 0.198 0.125 0.157 0.162

Sup1 0.473 0.196 0.510 0.460 0.170 0.387 0.132 0.470 0.080 0.360 0.431 0.373 0.060

Sup2 0.262 0.108 0.290 0.330 0.420 0.340 0.116 0.135 0.612 0.430 0.524 0.327 0.053

Sup3 0.265 0.110 0.200 0.210 0.410 0.273 0.093 0.395 0.308 0.210 0.045 0.300 0.048

PartA MC4 MC5 Total

Criter.

Weight

SC41 SC42

SC43 Criter.

Weight

Main

Criter.

SC51 SC52 SC53 Criter.

Weight

Main

Criter.

0.686 0.157 0.157 0.017 0.512 0.150 0.338 0.064

Sup1 0.510 0.460 0.170 0.448 0.076 0.250 0.540 0.453 0.362 0.023 0.420

Sup2 0.290 0.330 0.420 0.317 0.054 0.410 0.351 0.178 0.323 0.020 0.340

Sup3 0.200 0.210 0.410 0.235 0.004 0.340 0.109 0.369 0.315 0.020 0.140

Table 10. Each supplier’s final weight for each of part units Priority weight for Part B

PartB MC1 MC2 MC3

Criter.

Weight

Main

Criter.

Weight

SC21 SC22 SC23 Criter.

Weight

Main

Criter.

Weight

SC31 SC32 SC33 SC34 Criter.

Weight

Main

Criter.

Weight

0,416 0.512 0.150 0.338 0.341 0.520 0.198 0.125 0.157 0.162

Sup1 0.258 0.107 0.326 0.512 0.375 0.370 0.126 0.651 0.182 0.600 0.431 0.517 0.083

Sup2 0.425 0.176 0.124 0.312 0.264 0.200 0.068 0.135 0.465 0.082 0.524 0.240 0.039

Sup4 0.317 0.132 0.450 0.176 0.361 0.379 0.130 0.214 0.353 0.318 0.045 0.288 0.047

PartB MC4 MC5 Total

Criter.

Weight

SC41 SC42 SC43 Criter.

Weight

Main

Criter.

SC51 SC52 SC53 Criter.

Weight

Main

Criter.

0.686 0.157 0.157 0.017 0.512 0.150 0.338 0.064

Sup1 0.333 0.540 0.178 0.341 0.058 0.345 0.178 0.258 0.290 0.018 0.499

Sup2 0.167 0.124 0.800 0.260 0.042 0.340 0.369 0.654 0.450 0.029 0.317

Sup4 0.500 0.336 0.022 0.400 0.068 0.315 0.453 0.088 0.259 0.016 0.331

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Table 11. Each supplier’s final weight for each of part units Priority weight for Part C PartC MC1 MC2 MC3

Criter.

Weight

Main

Criter.

Weight

SC21 SC22 SC23 Criter.

Weight

Main

Criter.

Weight

SC31 SC32 SC33 SC34 Criter.

Weight

Main

Criter.

Weight

0,416 0.512 0.150 0.338 0.341 0.520 0.198 0.125 0.157 0.162

Sup5 1 0.416 1 1 1 1 0.341 1 1 1 1 1 0.162

PartC MC4 MC5 Total

Criter.

Weight

SC41 SC42

SC43 Criter.

Weight

Main

Criter.

SC51 SC52 SC53 Criter.

Weight

Main

Criter.

0.686 0.157 0.157 0.017 0.512 0.150 0.338 0.064

Sup5 1 1 1 1 0.017 1 1 1 1 0.064 1

Table 12. Each supplier’s final weight for each of part units Priority weight for Part D

PartD MC1 MC2 MC3

Criter.

Weight

Main

Criter.

Weight

SC21 SC22 SC23 Criter.

Weight

Main

Criter.

Weight

SC31 SC32 SC33 SC34 Criter.

Weight

Main

Criter.

Weight

0,416 0.512 0.150 0.338 0.341 0.520 0.198 0.125 0.157 0.162

Sup3 0.480 0.200 0.278 0.651 0.500 0.409 0.140 0.470 0.513 0.623 0.578 0.514 0.080

Sup5 0.520 0.216 0.722 0.349 0.500 0.591 0.201 0.530 0.487 0.377 0.422 0.469 0.076

PartD MC4 MC5 Total

Criter.

Weight

SC41 SC42 SC43 Criter.

Weight

Main

Criter.

SC51 SC52 SC53 Criter.

Weight

Main

Criter.

0.686 0.157 0.157 0.017 0.512 0.150 0.338 0.064

Sup3 0.081 0.700 0.500 0.234 0.004 0.478 0.460 0.450 0.465 0.030 0.454

Sup5 0.919 0.300 0.500 0.756 0.013 0.522 0.540 0.550 0.535 0.034 0.546

1396

Table 13 . Each supplier’s final weight for each of part units Priority weight for Part E

The Final weight of each criteria and priorty of them seem on table 9-13.

In this article we proposed to compute supplier selection following that, we solve an

example problem for supplier selection;

In our model we have capacity, demand, on time delivery, quality constraints and in our

model we think about wide size Bill Off Material (BOM), so our problem turn a hard

problem we must solve it by genetic algorithm. Our fitness function is equation (10);

In this article , We proposed that a chromosome had composed as fig.6.The first gene

takes only number 1, 2 ,3; the second is 1,2,4 , the third is take only 5 , the fourth is 3, 5

and the fifth is take only 4, 5 because others do not product the other production we can

see in table.7.

We use GA’s the fitness function in Eq.10.and in our model, the crossing over is shown

such as in fig.6 A,B,C,D,E is respectively purchase 1,2,3, 1,2,4 , 5 , 3,5 and 4,5 an

example of our chromosome construct is shown as fig.6 each of arrow define the each

products shows alternative supplier .If each gene include a supplier we shows its number

other we write 0.And all time the biggest supplier is written each parts first member. Fig.6 Crossing over process

PartE MC1 MC2 MC3

Criter. Weight

Main Criter. Weight

SC21 SC22 SC23 Criter. Weight

Main Criter. Weight

SC31 SC32 SC33 SC34 Criter. Weight

Main Criter. Weight

0,416 0.512 0.150 0.338 0.341 0.520 0.198 0.125 0.157 0.162

Sup4 0.462 0.192 0.211 0.578 0.278 0.289 0.099 0.623 0.700 0.465 0.651 0.622 0.061

Sup5 0.538 0.270 0.789 0.422 0.722 0.711 0.242 0.377 0.300 0.535 0.349 0.378 0.101

PartE MC4 MC5 Total

Criter.

Weight

SC41 SC42 SC43 Criter.

Weight

Main

Criter.

SC51 SC52 SC53 Criter.

Weight

Main

Criter.

0.686 0.157 0.157 0.017 0.512 0.150 0.338 0.064

Sup4 0.213 0.815 0.074 0.285 0.004 0.154 0.789 0.215 0.750 0.048 0.404

Parent chromosomes Child chromosomes

Crossover point

0 4 5 3 5 4 0 1

0 2 1

4 5 0 5 5 1 4 2 3 1 2

4 5 0 5 5 1 4 2 0 2 1

0 4 5 3 5 4 0 1 3 1 2

1397

Fitness (Objective) Function: Identify the acceptable solution of the part supplier combination which maximizes the total

supplier’s gain and determine both best supplier and best quantities under the capacity ,

demand , Quality, Delivery constraints.

iS set of suppliers offering item i

jK set of items offered by supplier j

ijw final weight of part i for supplier j

ijq defective rate of part i offered by supplier j

iQ the buyer’s maximum acceptable defective rate of item i

ijt on-time delivery rate of part i offered by supplier j

ijT the buyer’s minimum acceptable on-time delivery rate of item i

ijC maximum supply capacity of part i offered by supplier j

ijD total demand of item i

ijX units of part i to purchase from supplier j

ijN order rate of part i to supplier j

Maximize ∑∑= =

n

i

m

ijijijij NXw

1

eq.10

Constraints:

Capacity Constraint As supplier j can provide up to C ij units of item i and its order quantity

X ij should be equal or less than capacity such as;

iKi Ki

ijijji SjCNXj j

∈≤∑ ∑∈ ∈

Demand Constraint:

The total order of each part should be equal i. buyer’s demands;

ijSj

jiSj

ij NDXii

∑∑∈∈

= jKi ∈

Quality Constraint:

Since Q i is the buyer’s maximum acceptable defective rate of item i and q ij is the

defective rate of supplier j, the quality constraint can be shown as [7].

1398

∑ ∑∈ ∈

≤j iSj Sj

jijiijijij DQNXq

Delivery Constraint: Since T i is the buyer’s minimum acceptable on- time delivery rate of

item i and ijt

is the on time delivery rate of supplier j, the delivery constraint can be shown

as[7].

∑ ∑∈ ∈

−≤−j iSj Sj

jijiijij DTXt )1()1(

∑∈

=ĐSj

ijijij XXN / jKi ∈ , iSj ∈

We want to validate the model at hand, because of it a numeric examples is designed

and performed by concrete data[7]. Suppose that five suppliers and five units are allowed

for our computation. In model, we used each data at table.15

7.1. A Numerical example

In this problem the purchase officer wishes to purchase for each part’s best supplier and

allocate order quantities to them; for instance if demand of A, B, C, D, E is1500 units, the

minimum acceptable on time rate of A, B, C, D, E, are respectively 0.85, 0.75, 0.90, 0.85,

0.80 and maximum acceptable defect rate of A, B, C, D, E, are respectively 0.05, 0.02,

0.03, 0.05, 0.04. For each part of X’s supplier seem on the table 14. We use each supplier

weights from table 17. and we use Genetic algorithm for best supplier and allocate best

order quantities to them.

Our fitness function is;

Max Z= 0.42 N11 X11 + 0,34 12N X12 + 0,140 13N X 13 + 0,499 21N X 21 + 0,317 22N X 22 + 0,331

24N X 24 + 35N X 35 + 0,454 43N X 43 + 0,546 45N X 45 + 0,404 54N X 54 +0,596 55N X 55

N11 X11 + 12N X 12 + 13N X13 =1500

21N X 21 + 22N X 22 + 24N X 24 =1500

35N X 35 =1500

43N X 43 + 45N X 45 =1500

54N X 54 + 55N X 55 =1500

X11 + X12 + X13 ≤ 800N11 1312 N1000 N400 ++

X 21 + X 22 + X 24 ≤ N400 N1000 500N 242221 ++

X 51 ≤ 35 1500N

1399

X 42 + X 45 ≤ 4543 N5001200N +

X 54 + X 55 ≤ 5554 N1000600N +

0.05 N11 X11 +0.02 12N X12 +0.03 13N X13 ≤ 75

0.02 21N X 21 +0.03 22N X 22 +0.04 24N X 24 ≤ 30

0.02 35N X 35 ≤ 45

0.06 43N X 43 +0.03 45N X 45 ≤ 75

0.03 54N X 54 +0.04 55N X 55 ≤ 75

0.15 N11 X11 +0.05 12N X12 +0.10 13N X13 ≤ 225

0.10 21N X 21 +0.05 22N X 22 +0.10 24N X 24 ≤ 375

0.05 35N X 35 ≤ 150

0.25 43N X 43 +0.05 45N X 45 ≤ 225

0.10 54N X 54 +0.15 55N X 55 ≤ 300

N j1 = X j1 / ∑∈ ĐSj

jX 1 (j=1,2,3) ; N j2 = X j2 / ∑∈ ĐSj

jX 2 (j=1,2,4) ; N j3 = X j3 / ∑∈ ĐSj

jX 3 (j=5) ;

N j4 = X j4 / ∑∈ ĐSj

jX 4 (j=2,5) ; N j5 = X j5 / ∑∈ ĐSj

jX 5 (j =4,5)

X11 , X12 ,X13 , X 21 , X 22 , X 24 ,X 35 ,X 43 , X 45 , X 54 , X 55 ≥ 0

0 ≤ N11 , 12N , 13N , 21N , 22N , 24N , 35N , 43N , 45N , 54N , 55N ≤ 1

The solution of GA is determined by using a Genetic Program , and our mutation rate is

0.1 the rossing over rate is 0.9 and population size = 40 in 8minutes and after 267140

iterations we determined follow result and our genetic code, which is shown on fig.7.

X11: 799,990; X12: 397,890; X13: 302,122;X21: 498,792; X22: 778,477; X24: 222,730;

X35: 1499,977; X43: 1106,914; X45: 393,080;X54: 500,592; X55: 999,404; N11 =0.534,

12N =0.265, 13N =0.201, 21N =0.332, 22N =0.519, 24N =0.149, 35N =1, 43N =0.738,

45N =0.262; 54N =0.333; 55N =0.667

fig.7.Solution Chromosomes 2 1 3 2 5 4 1 2 3 2 1

1400

8. Conclusion In this article, we discussed an integrated model for supplier selection with using GA and

Fuzzy AHP techniques. Our aimed determined the best supplier and whose optimal

appointment of quantities. We used Fuzzy AHP for determining linguistic expression and

fuzzy weight to transformed crisp weights of each supplier criteria. By using these crisp

weights, we determined the best supplier and optimal order quantity of them with using

GA. We suggested an example, which involve an example product and on that example,

we develop a GA and Fuzzy AHP model for multi factor supplier selection’s problems. By

Using Fuzzy AHP, we transformed each linguistic weight to crisp value and made up a

mathematical model, which called GA’s objective function, for determining best supplier

and order quantity. We use Autofit3.0 program for solving GA model. The Result is shown

in fig 7. We believe that this model a new approach to determining supplier selection.

There are a lot of model for determining best supplier and more of them appropriate for

single source supplier selection problem. Because globalism, increasing business cost,

quality order and etc. factors are directing administrator to decrease their cost and search

different and suitable supplier from different location. And they also consider alternative

supplier each of product’s part, so this problem turn a multi source problem and a complex

structure .Our model is a different approach to this type of problems.

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