2 METHODOLOGIES

2.1 Fuzzy Analytic Network Process (Fanp)

Because analytic network process (ANP) can consider the interrelationships among elements in a problem setting, the use of the ANP instead of analytic hierarchy process (AHP) has increased substantially in recent years. To consider the fuzziness and vagueness in the decision making process, fuzzy set theory can be incorporated into the ANP, so called FANP. An example of the procedures for the FANP is as follows (Kang, Lee and Yang, 2010; Lee, Wang and Lin, 2010):

1. Decompose the problem into a network.

2. Prepare a questionnaire based on the constructed network, and ask experts to fill out the questionnaire. The questionnaire will be prepared based on pairwise comparison with Saaty’s nine point scales (Saaty, 1980). Experts are asked to fill out the questionnaire.

Consistency index and consistency ratio for each comparison matrix are calculated to examine the consistency of each expert’s judgment (Saaty, 1980). If the consistency test is not passed, the original values in the pairwise comparison matrix must be revised by the expert.

3. Aggregate the results of the experts’

questionnaires. The scores of pairwise comparison are transformed into linguistic variables by the transformation concept.

According to Buckley (1985) the fuzzy positive reciprocal matrix can be defined as:

[ ]

ij ^k

k a~

~ =

Α (1)

Α~k : a positive reciprocal matrix of decision maker k;

a~ij : relative importance between decision elements i and j;

j i aij = ,1∀ =

~ and a a i j n

ji

ij 1~ , , 1,2, ,

~ _{= ∀ = KK}

If there are k experts P1, P2…., Pk, every pairwise comparison between two criteria has k positive reciprocal triangular fuzzy numbers. Employ geometric average approach to aggregate multiple experts’ responses, and the aggregate fuzzy positive reciprocal matrix is:

[ ]

a~ij

~^*= ^*

Α (2)

where a~ij^*₌

(

a~ij¹_⊗a~ij²_⊗..._⊗a~ij^k

)

^1/k

4. Defuzzy the synthetic triangular fuzzy numbers

( )

* , ,

ij ij ij ij

a% = x y z into crisp numbers. For instance, the center of gravity (COG) method can be applied.

(

^{x y z}

)

_{i j n}

aij ^{ij ij ij} 3 , , 1,2, ,

* + + ∀ = KK

= (3)

5. Form pairwise comparison matrices using the defuzzificated values, and apply software, such as Super Decisions or Excel, to form an unweighted supermatrix. Next, form a weighted supermatrix to ensure column stochastic.

6. Calculate the limit supermatrix by taking the weighted supermatrix to 2q +1 powers so that the supermatrix converges into a stable supermatrix.

Obtain the priority weights of the alternatives from the limit supermatrix.

2.2 Goal Programming (GP)

Goal programming (GP) is useful in dealing with multi-criteria decision problems where the goals cannot simultaneously be optimized, and decision makers can consider several objectives together in finding a set of acceptable solutions and to obtain an optimal compromise (Lee, Kang and Chang, 2009).

The purpose of GP is to minimize the deviations between the achievement of goals and their aspiration levels (Chang, 2007). GP has been applied in various studies. For example, an integrated AHP and preemptive goal programming methodology is developed by Wang, Huang and Dismukes (2004) to select the best set of multiple suppliers to satisfy capacity constraint.

The achievement function of GP is (Chang, 2007; Lee et al., 2009):

Min ( )

1

−

=

++

∑ i

n

i wi di d (4)

s.t. f_i(X)- + =∑

=

−

+ m

j ij ij

i

i d g S B

d

1 ( ), i=1,2,...,n (5) 0

, ⁻≥

+ i

i d

d ,i=1,2,...,n (6) ),

( ) (B U x

Sij ∈ i i=1,2,...,n (7) F

X∈ (F is a feasible set) (8) where di is the deviation from the target value gi; wi represents the weight attached to the deviation;

) ) ( , 0

max( _{i i}

i f X g

d⁺= − and

)) ( , 0

max( g f X

d_i⁻= _i− _i are, respectively, over- and under-achievements of the ith goal; Sij(B) An Evaluation Model for Green and Low-Carbon Suppliers

605

represents a function of binary serial number; and )

(x

Ui is the function of resources limitations.

Based on the fuzzy theory, the highest possible value of membership function is 1 for something that is more/higher the better in the aspiration levels (Charnes and Cooper, 1961). To achieve the maximization of g_ijS_ij(B), the flexible membership function goal with aspiration level 1 (i.e., the highest possible value of membership function) is (Chang, 2007):

) 1 (

min max

min − + =

−

− _{+ −}

i i ij

ij d d

g g

g B S

g (9)

where

g

_maxand

g

_min are, respectively, the upper and lower bound of the right-hand side (i.e., aspiration levels) of equation (5).

For a simpler calculation, the fractional form of equation (9) is:

L 1 ) 1 L (

1

min− + =

− _i⁺ _i⁻

i ij

ij i

d d g B S

g (10)

where Li=gmax−gmin.

For something that is less/lower the better in the aspiration levels, the similar idea of maximization of

) (B S

g_{ij ij} can be used to achieve the minimization of )

(B S

gij ij . The flexible membership function goal with the aspiration level 1 (i.e., the lowest possible value of membership function) is (Chang, 2007):

) 1 (

min max

max − + =

−

− _{+ −}

i i ij

ij d d

g g

B S g

g (11)

where gmaxand gmin are, respectively, the upper and lower bound of the right-hand side (i.e., aspiration levels) of equation (5).

The fractional form of equation (11) can be converted into a polynomial form:

1 )

L ( 1 L

1

max− _{ij ij} − _i⁺ + _i⁻ =

i i

d d B S g

g (12)

3 AN INTEGRATED MODEL FOR FANP AND GP MODEL

The steps of the proposed FANP and GP model are summarized as follows:

Step 1. Define the green and low-carbon supplier evaluation problem, and construct an evaluation network with criteria, detailed

criteria and alternatives.

Step 2. Prepare and distribute a questionnaire. A questionnaire with five linguistic terms, as shown in Table 1, is prepared based on the constructed network.

Table 1: Triangular fuzzy numbers.

Linguistic

variable Fuzzy number

Membership function of fuzzy

number Extremely

strong 9~ _(9,9,9)

Intermediate 8~ _(7,8,9)

Very strong 7~ _(6,7,8)

Intermediate 6~ _(5,6,7)

Strong 5~ _(4,5,6)

Intermediate 4~ _(3,4,5)

Moderately

strong 3~ _(2,3,4)

Intermediate 2~ _(1,2,3)

Equally strong 1~ _(1,1,1)

Step 3. Prepare pairwise comparison matrix. With pairwise comparison of criteria with respect to the overall objective, we can obtain a matrix (Α~_1k) for expert k:

1 2

1 12 1

2 2

12

1

1 2

C C C C C

C 1

C 1 1

1 C 1

1 1

C

1

1 1 1

C

i j m

k mk

k mk

k i ijk

j ijk

mk mk

m

a a

a a

a a

a a

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

A

L L

% L L L L %

%

L L L L

%

M M L L L L

% M M M M % L L

M M M % L L

M L L L L L L

L L L L

% %

(13)

where m is the number of criteria (C).

Step 4. Aggregate experts’ opinions and build an aggregated fuzzy pairwise comparison matrix. Geometric average approach is employed to aggregate experts’ responses and to obtain a synthetic triangular fuzzy number (Lee, 2009; Lee et al., 2009):

(

ij ij ijk

)

^k

ij a a a

a ~1 ~2 ~ ¹

~ = ⊗ ⊗_KK⊗ (14)

where a~ =ijk

(

lijk,tijk,uijk

)

The fuzzy aggregated pairwise comparison matrix is:

ICINCO 2012 - 9th International Conference on Informatics in Control, Automation and Robotics

606

12 1 12 2

1

1 2

1

1 1

1 1

1 1

1

1 1 1

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

=⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

% L L L L %

L L L L %

%

M M L L L L

M M M % L L

%

M M M % L L

L L L L L L

L L L L

% %

j j

ij

ij

j j

a a

a a

a a

a a

A

(15)

where a~ =ij

(

lij,tij,uij

)

Step 5. Calculate crisp relative importance weights (priority vectors) for factors by adopting the center of gravity.

Step 6. The consistency test (Saaty, 1980) is performed by calculating the consistency index (CI) and consistency ratio (CR). If the consistency test is not passed, the expert will be asked to re-do the part of the questionnaire.

max

1

= −

− CI n

n

λ (16)

RI

CR=CI (17)

where λmax is the largest eigenvalue of A1, n is the number of items being compared in the matrix, and RI is random index defined by Saaty (1980). If CR is less than 0.1, the threshold for consistency, the expert’s judgment is consistent. If the consistency test is not passed, the expert will be asked to re-do the part of the questionnaire.

Step 7. Calculate the weights of sub-criteria, the interdependence among sub-criteria with respect to the same upper-level criterion, and the performance of suppliers with respect to each sub-criterion using a similar procedure from Step 3 to Step 6.

Step 8. Form an unweighted supermatrix. The local priority vectors calculated from Step 5 and 7 are entered in the appropriate columns of a matrix, known as an unweighted supermatrix, as follows.

21 22

32 33

43

Goal Criteria Sub-criteria Alternatives I

Goal Criteria W

W W

Sub-criteria

W I

Alternatives w

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

S (18)

where w21 is a vector that represents the impact of the goal on the criteria,W₃₂ is a matrix that represents the impact of criteria on sub-criteria, W₂₂ indicates the

interdependency of the criteria, W₄₃ is a matrix that represents the impact of criteria on each of the alternatives, W₃₃ indicates the interdependency of the sub-criteria, and I is the identity matrix (Saaty 1996).

Step 9. Transform the unweighted supermatrix into a weighted supermatrix (Saaty, 1996; Lee, Chen and Tong, 2008).

Step 10. Calculate the limit supermatrix. The weighted supermatrix is raised to powers to obtain the limit supermatrix.

Step 11. Rank the suppliers. The priority weights of the suppliers can be found in the alternative-to-goal block, i.e. block (4,1), in the limit supermatrix.

Step 12. Construct a GP model for the green and low-carbon supplier selection and order allocation problem. Set the GP model based on the results from Step 11 to maximize satisfaction:

Max Z0=g1×G1+g2×G2+ +.... gn×Gn (19) Step 13. Formulate the GP model by adopting

equations (21) to (27) to minimize the aspiration level of i^th objective. It is as follows:

1 1 2 2

1

Min (^{n i} _{i i i}( _{i i} ))

i i

Z g d d L d d

L

+ − + −

=

=∑ + + + ⁽²⁰⁾

s.t. f_i(X)-

∑

=

−

++ = ^m

j ij ij i

i d g S B

d

1

)

( =1,2,...,n (21)

) (X

fi -d_i⁺₁+d_i1⁻=g_i^maxz_i+g_i^min(1−z_i) i=1,2,...,n (22)

= +

−

−

+ ^{min +}₂ ⁻₂

max (1 ))

1(

i i i i i i i

d d z g z L g

) 1(max min

i i i

g or L g

n

i=1,2,..., (23)

0 , , , ,

, ⁻ ₁^{+ −}₁ ⁺₂ ⁻₂≥

+

i i i i i

i d d d d d

d ^i=1,2,...,n (24)

B

X∈ (B is a feasible set) (25)

{ }^0,1

i∈

z (26)

4 A CASE STUDY

A case study is used to examine the practicality of An Evaluation Model for Green and Low-Carbon Suppliers

607

the proposed FANP with GP model. A committee of experts in the IC industry is formed to define the problem of supplier selection. A questionnaire is constructed and is targeted on the experts in the IC design company. Based on the collected opinions of the experts and the proposed model, the performance results of the suppliers can be generated. The five criteria and their respective sub-criteria are listed in Table 2.

Table 2: Criteria and sub-criteria of FANP.

Criteria Sub-criteria C1

Purchasing management

C11 Low pollution C12 Material label C13 Recycling C2

Process management

C21 Modularization C22 Process control C23 Technology level

C24 Process improvement capability

C3

Quality control

C31 Environmental regulation fulfilment C32 Product quality control

C33 Capability of handling abnormal products

C34 Delivery quality and date C35 Quality certification C4

Business management

C41 Internal education and training C42 Green R&D design capability C43 Pollution control

C44 Regulation of harmful material control C5

Cost control

C51 Production cost C52 Businesscost C53 Purchase cost

在文檔中 行政院國家科學委員會專題研究計畫 成果報告 (頁 87-90)