DOI 10.1007/s10845-011-0563-z
Decision-making for the best selection of suppliers by using
minor ANP
Toshimasa Ozaki · Mei-Chen Lo · Eizo Kinoshita · Gwo-Hshiung Tzeng
Received: 31 July 2010 / Accepted: 20 June 2011 / Published online: 7 August 2011 © Springer Science+Business Media, LLC 2011
Abstract There are many studies on the application exam-ples and the definitions of the super-matrix in the Analytic Network Process (ANP). The ANP applies network struc-tures with the alternatives and the criteria to clarify a com-plex decision-making. It is important to obtain the consent from both the criteria and the alternatives because the eval-uation is relative and reciprocal. In addition, there are very few studies on the interaction between the criteria and the alternatives. There is no realistic interpretation of the max-imum eigenvalue in the ANP and the negative elements of the criteria matrix. Based on our study, the evaluation with only the alternatives consists of the missing values or the non-square matrix, it is not easy to make. The ANP that con-tains these irregular alternatives is defined as “Minor ANP”. Then, we describe these methods of making the priority of T. Ozaki (
B
)Faculty of Commerce, Nagoyagakuin University, 1-25 Atsuta-Nishimachi, Atsuta, Nagoya, Aichi, Japan e-mail: [email protected]
M.-C. Lo
Department of Business Management, National United University, 1, Lienda, Miaoli 36003, Taiwan
e-mail: [email protected] E. Kinoshita
Faculty of Urban Science, Meijo University, Kani, Gifu, Japan
e-mail: [email protected] G.-H. Tzeng
Institute of Project Management, Kainan University, Taoyuan, Taiwan
G.-H. Tzeng
Institute of Management of Technology, Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected]
only the alternative’s matrix values by using the ANP. This study has proven that the missing values in the alternative matrix can be replaced by zeros such that the criteria matrix can be derived without using the inverse alternative matrix. Then, the priority of alternative selection can be obtained through the calculation of eigenvector. An empirical case of the best selection of supplier illustrates our proposed meth-ods for applying to the manufacturing industry. The result reveals the methods can be well adopted in the real world. Keywords Decision analysis· Analytic network process (ANP)· Multiple criteria decision making (MCDM) · Non-square matrix
Introduction
There are many studies on the application examples and the definitions of the super-matrix in the Analytic Network Pro-cess (ANP) (Saaty 1996). The ANP completely relies on human side to assist in solving social decision problems. The ANP applies network structures with the alternatives and the criteria to clarify a complex decision-making. It is important to obtain the consent from both the criteria and the alter-natives because the evaluation is relative and reciprocal. In addition, there are very few studies on the interaction between the criteria and the alternatives. Either, there is no realistic interpretation of the maximum eigenvalue in the ANP and the negative elements of the criteria matrix.Ozaki et al.(2009) proved the missing value in the alternative matrix which can be replaced as “zero” to be calculated without considering of criteria direct effect, instead of deriving the criteria matrix from inversed alternative matrix. Then, the priority of alter-native selection can be obtained through the calculation of eigenvector.
The way of considering both criteria matrix and alternative matrix can be named as “Major ANP” or “standard ANP”. As for the proposed “Minor ANP” which can be named as the “irregular ANP” with only gives a consideration on the alternative matrix. This study focuses on the “Minor ANP”, and describes the types of the decision-making through the meaning of the eigenvalue, and the negative elements of the criteria matrix. And the experiments numeric calculations via case study of suppliers’ selection are applied.
Since selecting the supplier that provides the best design and manufacturing expertise can make the product more competitive and cost effective. A supplier should offer more than just “parts that meet spec.” Choosing the right supplier involves much more than scanning price lists in manufacture industry. The choice will depend on a wide range of fac-tors such as value for money, quality, reliability and service. To weigh up the importance of these different factors will depend on the business’s priorities and strategy as well as the preference of decision makers. Besides, selecting a sup-plier is both a quantitative and qualitative process. The way of calculation on strategically choosing suppliers can help understand when some missing values occurred. Addition-ally, it can also help the potential customers to weigh up their purchasing decisions in products.
The reminder of this paper is organized as follows. Next Section, features of “Minor ANP” are reviewed and dis-cussed. Then, the evaluation value of the non-square matrix is developed. After Section, an empirical case of experimen-tal numbers for the best selection of suppliers is illustrated to show and discuss our proposed methods for applying to the manufacturing industry. Finally, conclusion is presented in last Section.
Minor ANP—features of the method
The existing methods such asHarker(1987a) andNishizawa
(2005, 2007) are those using the characteristics of mathe-matical or AHP calculationSaaty(1980). However, there is still a problem which exist an unclear condition when both methods are used simultaneously. For example, the miss-ing values in alternative matrix can be nearly presumed by Harker method, by means of a combination with Nishizawa method to presume the missing values. Obviously, the result is hardly to find the way out on calculation for its eigenvec-tor. From Nishizawa method, the comparison values of two pairs or more are necessary to presume the missing values, relatively; the result also has raised a problem to reveal the method is appropriate and affected others at the same time.
According to the issue from Nishizawa method, the prob-lem solving is demonstrated by using the proposed method. As a token cases are shown in Table 1 where the miss-ing values are arbitrarily generated. The decision makers
Table 1 Case of alternative selection Criteria C1 C2 Alternative\weight w1(0.8) w2(0.2) Value D1 A1 0.7 0.6 0.68 (a1) A2 0.3 0.4 0.32 (a2) Alternative\weight w1(0.5) w2(0.5) Value D2 A1 0.1 0.7 0.40 (b1) A3 0.9 0.3 0.60 (b2)
Alternative\weight w1(0.3) w2(0.7) Value
D3 A2 0.2 0.7 0.55 (c1)
A3 0.8 0.3 0.45 (c2)
Table 2 Alternative selection by decision makers
Alternative\decision maker D1 D2 D3
A1 a1 b1
A2 a2 c1
A3 b2 c2
Di,(i = 1, 2, 3) give selection to alternatives A1, A2, and A3
upon given criteria (C1, C2)by subjective weights (w1, w2)
for priority measuring via calculation of performance value. According to Table1, the matrix of alternative selection by decision makers, as Table2, is formed and the missing values have appeared with the sign of “”.
Priority level of alternatives matrix with missing value The priority calculation by using the ANP via the alterna-tive’s matrix, it is a beginning of the “Minor ANP” as part of the ANP characteristics. The missing values also were mentioned byHarker(1987b) which based upon the graph-theoretic structure of the pairwise comparison matrix and the gradient of the right Perron vector. The alternatives matrix
Ureplaced the missing values with “zero” to be matrix
UA H P are given as follows.
U= ⎡ ⎣aa12 b c1 1 b2 c2 ⎤ ⎦ ⇒ UA H P = ⎡ ⎣aa12 b01 0c1 0 b2 c2 ⎤ ⎦ (1) In this case, Harker method can not obtain the priority through eigenvector calculation. Due to Harker method (Harker & Vargas 1987), the eigenvector x= (x1, x2, x3, x4)
of the alternative matrix is the imperfect evaluation matrix which adds the numbers of missing values as “one” to the diagonal. Nevertheless, the eigenvalue λ is not able to be
calculated. Sugiura & Kinoshita (2005, 2007) proposed a solution to calculate eigenvector by transforming the matrix
UA H P as Eq. (2). UA H P = ⎡ ⎣aa12/a/a11 aa21/a/a22 cb11/c/b22 b2/b1 c2/c1 c2/c2 ⎤ ⎦ (2)
where the calculation Eq. (3) can be written as:
UA H P = ⎡ ⎣ww12/w/w11 ww21/w/w22 ww21/w/w33 w3/w1 w3/w2 w3/w3 ⎤ ⎦ = ⎡ ⎣aa12/a/a11 aa21/a/a22 cb11/c/b22 b2/b1 c2/c1 c2/c2 ⎤ ⎦ = ⎡ ⎣1/A1 A1 CB 1/B 1/C 1 ⎤ ⎦ (3)
This study refers to the methods from Harker & Vargas
(1987);Sugiura & Kinoshita(2005) andOzaki et al.(2009) which use the matrix inverse of the matrix UA H P to define
the criteria matrix W by replacing the missing values’ posi-tion as “zero”. And the eigenvectors to the eigenvalue k of the ANP are assumed to be x, z.
0 W U 0 x z = k x z (4) The eigenvector z= (z1, z2, z3) of the above matrix as the
Eq. (4) which calculates the equivalent of the eigenvector of
UA H P. From this study, the eigenvector can be obtained only
by way of deriving from the alternatives matrix U. Calculating the eigenvectors in AHP and ANP
Since the matrix UA H P is represented by A = a1/a2, B =
b1/b2 and C = c1/c2. The maximum eigenvalue and the
eigenvector are assumed to beα and y (y1, y2, y3= 1).
y1= B{AC (1 − α) − B} B(1 − α) − AC and y2= C{B (1 − α) − AC} AC(1 − α) − B
are the eigenvector of UA H P. On the other hand, the
maxi-mum eigenvalue and the eigenvector in UW based on Eq. (4) are assumed to be k and z (z1, z2, z3= 1)
z1= B(B + AC) 1− k2− AC (B + AC) 1− k2− B z3 z2= C(B + AC) 1− k2− B (B + AC) 1− k2− AC z3
are obtained. When equation z1and equation z2are rewritten
by 1−1−k12 = α − 1, then z1= y1and z2= y2are obtained. Therefore, the eigenvector UW becomes equivalent with the eigenvector of UA H P.
Presuming the missing values in the imperfect alternative matrix
A presumption method of the missing values is introduced by four dimensions’ matrix in an imperfect AHP (as Eq.5). The imperfect evaluation matrix UA H Pis defined as follows.
UA H P = ⎡ ⎢ ⎢ ⎣ 1 0 a b 0 1 c d a−1 c−1 1 0 b−1 d−1 0 1 ⎤ ⎥ ⎥ ⎦ (5)
In this Section, we propose a new presumption method of the missing values, which can be distinguishable from the methods ofHarker & Vargas(1987).
As Eq. (5), the missing values presumption to the criteria matrix W s obtained from the missing value matrix. And the eigenvector of UWAN Pin the ANP matrix is calculated, as
Eq. (6). W = 1 a2d2+abcd+b2c2 ⎡ ⎢ ⎢ ⎣ abcd 0 ab2c2 a2bd2 0 abcd a2cd2 b2c2d ad2 b2c abcd 0 bc2 a2d 0 abcd ⎤ ⎥ ⎥ ⎦ UW = k ⎡ ⎢ ⎢ ⎣
1/ (bc+ad) ab abc abd
cd 1/ (bc+ad) acd bcd d b 1/ (bc+ad) bd c a ac 1/ (bc+ad) ⎤ ⎥ ⎥ ⎦ (6)
where, k is constant when the eigenvector z(z1, z2, z3, z4) in
the matrix UW is assumed to be z4 = 1. Then, the missing
value z1/z2can be calculated as
√
abc−1d−1, and missing value z3/z4to be solved as
√
a−1bc−1d.
On the other hand, when Eq. (5) is described as max-imum eigenvalue λH and eigenvector y (y1, y2, y3, y4)
by the matrix of Harker, (λH − 2)y1 = ay3 + by4 and
(λH− 2)y2= cy3+ dy4are obtained from row one and row
two of the matrix. Simultaneously, y1/y2=
√
abc−1d−1and y3/y4 =
√
a−1bc−1d are obtained as the missing value by the geometric mean calculation..
As a result, the estimations by this proposed method reveals the same values as the Harker method (Harker & Vargas 1987). Therefore, the validity of this method can be proved and its effective calculations.
Evaluation in the non-square matrix
The square matrix of the alternatives is described in the pre-ceding Section, and the non-square matrix as Eq. (7) of the alternatives describe in this Section.
U= ⎡ ⎣aa12 bb12 a3 b3 ⎤ ⎦ (7)
Kinoshita & Sugiura(2008) applied the idea of “Dominant AHP” (Kinoshita & Nakanishi 1999;Kinoshita et al. 2002;
Kinoshita & Sugiura 2008) with non-square matrix to the evaluation. ⎡ ⎣a21/a1b2/b11 a3/a1b3/b1 ⎤ ⎦ .a1/(a1+b1) b1/(a1+b1) = ⎡ ⎣(a(a12+b+b21)/(a)/(a11+b+b11)) (a3+b3)/(a1+b1) ⎤ ⎦ as of ⎡ ⎣(a1+b1)/ (ai+bi) (a2+b2)/ (ai+bi) (a3+b3)/ (ai+bi) ⎤ ⎦ (8)
This study draws an institution process of the problem-solving where the alternative matrix and criteria matrix are regularized. And Kinoshita method adopts a computational approach and considers the weights from each criterion and alternative.
While,Nishizawa(2007) explained that the evaluation value of the regularized super-matrix S was corresponding to Ki-noshita’s solution. The evaluation value of the non-square matrix was calculated with the same way as normalized weights as the Saaty type ANP. Besides, Nishizawa’s pro-posal provides a possible solution which can be obtained from this problem solving if a suitable criteria matrix is given. The proposed method
We take the matrix inverse of the alternative matrix U, and then define the left inverse U’ as U’U=I. Hence, the eigen-vectors in the ANP calculation to the eigenvalue k is assumed to be x and z. 0 U U 0 x z = k x z (9) Then, we propose the method by assuming an eigenvector z of UUto be the evaluation value in Eq. (10).
U= ⎡ ⎢ ⎢ ⎢ ⎣ a2b3− a3b2+ b2 a1b2− a2b1 −a1b3+ a3b1− b1 a1b2− a2b1 1 −a2+ a2b3− a3b2 a b − a b a1− a1b3+ a3b1 a b − a b 1 ⎤ ⎥ ⎥ ⎥ ⎦ (10)
Additionally, we assume the eigenvector z and the homoge-neous simultahomoge-neous linear equations(UW − k2I)z = 0.
UWz= ⎡ ⎣aa12 bb12 a3 b3 ⎤ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ a2b3− a3b2+ b2 a1b2− a2b1 −a1b3+ a3b1− b1 a1b2− a2b1 1 −a2+ a2b3− a3b2 a1b2− a2b1 a1− a1b3+ a3b1 a1b2− a2b1 1 ⎤ ⎥ ⎥ ⎥ ⎦ = k2z as of (a2b3− a3b2+ b2) ai + (a2b3− a3b2− a2) bi a1b2− a2b1 z1 +(a3b1− a1b3− b1) ai + (a1− a1b3+ a3b1) bi a1b2− a2b1 z2 +(ai + bi)z3 = k2zi = 0 (i = 1, · · · , 3)
where k = 0. The homogeneous simultaneous linear equa-tions z1 = aa1+b1 3+b3−1z3 ∼= a1+b1 a3+b3z3and z2 = a2+b2 a3+b3−1z3 ∼= a2+b2 a3+b3z3are obtained. z3= (a3+ b3)/ (ai+ bi) is substi-tuted, z1 = (a1 + b1)/(ai + bi) and z2 = (a2 + b2)/ (ai + bi), z2 = (a2 + b2)/ (ai+ bi) are con-verted.
An empirical case of experimental numbers for the best selection of suppliers
Supplier or vendor selections are complex due to the fact that various criteria must be considered in decisions making process (Spekman 1988). The analysis of criteria for selec-tion and measuring the performance of suppliers has been the focus of many studies. Therefore, we take the most effective empirical study on high-tech companies of suppliers’ selec-tion, and focus its famous sector on wafer foundry manu-facturing in Taiwan. The considerations on choosing right suppliers as of first tier supplier and second tier supplier could be important to ensure the production line stability. Therefore, the aspects of demand side, supplier selection may concern several criteria, such as supplier status, deliv-ery capability, processes, technical status, supplier culture, financial/commercial, support. From the aspects of supply side, there are a number of key characteristics that should look for when identifying and short listing possible suppli-ers. Good supplier should be able to demonstrate that they can offer the benefits, such as reliability, quality, value for money, strong service and clear communication, financial security, a partnership approach, etc.
Decision making on selecting suppliers
Several methods have been proposed in the literature for sin-gle sourcing supplier selection. Although, selecting a sup-plier is both a quantitative and qualitative process. A strategic
approach (Spekman 1988) to choosing suppliers can also help to understand how the own potential customers weigh up their purchasing decisions. The most effective suppliers are those who offer products or services that match—or exceed—the needs of the business. So when we are looking for suppliers, it is best to be sure of our business needs and what you want to achieve by buying, rather than simply paying for what sup-pliers want to sell you. For example, if we want to cut down the time it takes us to serve our customers, suppliers that offer us faster delivery will rate higher than those that compete on price alone.
The supplier selection process occurs after the buyer or sourc-ing team has determined the decision criteria on which to base the decision and gathered data on each potential sup-plier. Supplier selection is the point at which the buyer or sourcing team decides how much volume to place with each specific supplier(s). The criteria to the supplier selection pro-cess are driven primarily by the value of the purchase to the organization and the risk of acquiring the purchase in the mar-ketplace. The higher the value and risk of the purchase, the more likely that cross-functional sourcing teams will make the supplier selection decision. If the logic behind the selec-tion process is flawed or something missing to be included into the decision process, these errors can be costly for both the buying and selling organizations in terms of unacceptable supplier performance and a negative impact on operational and/or strategic capabilities.
The criteria of selecting supplier
The criteria of selecting supplier (Nydick & Hill 1992) who will meet production needs are to make manufacturing easier, obtain higher quality, increase reliability, lower cost, improve performance, increase life, lower maintenance, lower parts count, reduce size and weight, a more saleable product, increase energy efficiency, add intelligence (Kahraman et al. 2003;Ghodsypour & O’Brien 2001).
As for how many suppliers should a company maintain, it is well worth examining how many suppliers really need. In the best case of choosing the right supplier as well as buying from a carefully targeted group, it could have a number of
Supplier selection Control suppliers (C1) Competitive advantages (C2)
c11Quality of the product or service provided
c12Competitive prices
c13 On-time, cost-effective delivery of products c14 Technology infrastructure of the supplier
c21Demographics c22Financial solvency c23Experience
c24Environmentally conscious policies
A1 A2 A3 Criteria Alternative Aspects Goal
Fig. 1 Hierarchy Structure
Table 3 Supplier selection data
Alternative\criteria C1 C2 Total Ratio
A1 36 16 52 0.3662
A2 12 32 44 0.3099
A3 42 4 46 0.3239
Total 90 52 142 1
benefits (Ghodsypour & O’Brien 1998) as the concerns of criteria.
It will be easier to control suppliers (C1) and business will
become more important to them, may be able to make deals that give an extra competitive advantage (C2).
Key factors (Choi & Hartley 1996) when selecting a sup-plier/service provider, there are considered as:
1. Quality of the product or service provided (c11)
2. Competitive prices (c12)
3. On-time, cost-effective delivery of products (c13)
4. Technology infrastructure of the supplier (c14)
5. Demographics (c21)
6. Financial solvency (c22)
7. Experience (c23)
8. Environmentally conscious policies (c24)
Figure1has summarized above statement as of the hierar-chy structure for the discussion on missing value while the evaluation work to be proceeding.
Kinoshita’s method
According to (Kinoshita & Nakanishi 1999; Kinoshita et al. 2002; Kinoshita & Sugiura 2008) which evaluation in non-square matrix method have applied ideas on “Dominant AHP” with non-square matrix evaluation. The weights of the criteria are obtained from the ratios of the evaluation from the dominant alternatives. When two or more dominant alter-natives and two or more criteria are weighted, the evaluation value also can be obtained by the concurrent convergence method (CCM) (Sugiura & Kinoshita 2005).
Table3provides a simple case that Kinoshita method sum up the values to calculate the ratios from two criteria(C1, C2)
and three alternatives (A1, A2, A3), i.e. A1 = 52/142 =
0.3662, A2 = 44/142 = 0.3099 and A3 = 46/142 =
0.3239. This study draws an institution process of the prob-lem solving where the alternative matrix and criteria matrix are regularized. Kinoshita method adopts a computational approach and considers the weights from each criterion and alternative.
Nishizawa’s method
Nishizawa explained that the evaluation value of the regular-ized super-matrix S was corresponding to Kinoshita’s solu-tion and perform the criteria matrix W by using the trans-posed matrix from alternative matrix U. However, the eval-uation value of the non-square matrix has been calculated as same the normalized weights as the Saaty type ANP.
S = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0.6923 0.2727 0.9130 0 0 0.3077 0.7273 0.0870 0.4 0.3077 0 0 0 0.1333 0.6154 0 0 0 0.4667 0.0769 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⇒ 0.3662 0.3099 0.3239 (11)
Therefore, as Eq. (11) the Nishizawa’s proposal provides a possible solution which can be obtained from this problem solving if a suitable criteria matrix is given.
The proposed method
We take the matrix inverse of the alternative matrix U, and then define the left inverse Uas UU=I. Hence, the
eigen-vectors in the ANP calculation to the eigenvalue k is assumed to be x and z.
Then, we applied the numbers from Table3to calculate
UU as follow. UU= ⎡ ⎣3612 1632 42 4 ⎤ ⎦−1.3167 0.5333 1 −1.3625 0.5875 1 = ⎡ ⎣−69.2−59.4 2825.6.2 5244 −60.75 24.75 46 ⎤ ⎦ (12)
The Eq. (12) yields (0.3688, 0.3121, 0.3191) as an eigen-vector when the minimum eigenvalue is k = 0. The result reveals that this eigenvector calculation is extremely close to Kinoshita’s solution, i.e.(0.3662, 0.3099, 0.3239).
UU= ⎡ ⎣10.1391 1.1556 1.1304.9778 1.0182 0.9565 1 1 1 ⎤ ⎦ (13)
In Eq. (13), the elements of UUlof the third row are assumed to be “one”. Therefore, the column in the center of this vec-tor shows the highest value, and the right column reveals the same result as Kinoshita’s normalized solution.
Validity of this method in the non-square matrix
The study clarifies the meaning of minimum eigenvalue we have proposed. Then, numbers of the token cases (as Table4) of the evaluation matrix U are generated. Directly, the inverse matrix U’ is defined to be criteria matrix W, and then the eigenvector of UW to be calculated.
It is a minimum eigenvalue when the eigenvector stabi-lizes to be calculated and proved through several cases (as Table4). The eigenvector of UW are obtained according to the follows Eq. (14).
z1= a1+ b1 a3+ b3− 1 z3∼= a1+ b1 a3+ b3 z3 and z2= a2+ b2 a3+ b3− 1 z3∼= a2+ b2 a3+ b3 z3 (14)
when a minimum eigenvalue is assumed to be zero. The eigenvector in this method yields the same solution as Kinoshita’s when assuming z3 = (a3+ b3)/
(ai + bi).
Moreover, the correlation coefficient of our cases by this method and Kinoshita’s solutions is 0.999318, obviously, they are almost the same. Therefore, it seems to be no obsta-cle, even if the left inverse is assumed to be criteria matrix W. It is clarified that the eigenvector to the minimum eigenvalue of UW yields the evaluation of the non-square matrix.
For the purpose of showing the difference from those men-tioned methods. We simply utilized our proposed method to have the comparison with Harker method and Nishizawa method which shows as Table4. And the advantages of the proposed method have its calculation approach in difference from the Harker method which is an analytical technique for requesting it by the simultaneous equations. On the other hand, our method demonstrates the matrix with the missing value which is caught with the decision making problem as an uncertain evaluation procession. And it is shown that we need mutual evaluation with entirely opposite to the evalua-tion in order to make the proper decision in these samples.
Discussion and Implications
Missing values presumption by ANP in an imperfect evaluation matrix
Even if the eigenvector in the imperfect matrix is obtained, the missing values cannot be always presumed. This problem has been ignored up to now. From our proposed model, the solution in the ANP means “an agreement” with the alter-native matrix U and the criteria matrix W. The situation of “Negotiation breakdown” might be occurred, when the max-imum eigenvalue and the eigenvector do not exist. Therefore,
Table 4 Examples of calculations
Case U UW UW Column Eigenvalue Eigenvector
1st 2nd 3rd Max.λ1 Mid.λ2 Min.λ3
1 31 38 −73.29 −72.08 69 0.7563 0.7667 0.7582 1.00001 0.1919 0.7704 0.7667 14 2 −17.23 −15.71 16 0.1778 0.1671 0.1758 0.99849 0.7594 0.1633 0.1778 48 43 −96.89 −94.02 91 1 1 1 1.50E-03 1 1 1 2 20 13 −57.65 3.92 33 0.6360 0.6471 0.6346 None None 0.6351 0.6470 13 43 −99.52 7.65 56 1.0980 1.2633 1.0769 0.99650 1.0841 1.0980 34 18 −90.63 6.05 52 1 1 1 0.00354 1 1 3 49 16 32.93 −113.37 65 1.0641 1.0317 1.0156 1.01535 0.9884 −2.1702 1.0324 31 24 27.01 −94.93 55 0.8730 0.8639 0.8594 0.99999 0.8516 −0.0378 0.8733 30 34 30.94 −109.88 64 1 1 1 –1.54E-02 1 1 1 4 7 13 145.86 −60.54 20 0.3197 0.3175 0.3125 1.03056 0.3228 −0.1162 0.3174 26 43 499.78 −207.86 69 1.0952 1.0900 1.0781 1.00000 1.1026 0.0522 1.0953 28 36 456.32 −190.70 64 1 1 1 –3.07E-02 1 1 1 5 21 42 −187.06 200.81 63 1.2533 1.2600 1.2353 1.10364 1.2547 0.0000 1.2605 9 34 −128.36 138.06 43 0.8600 0.8663 0.8431 1.10364 0.8619 −0.3137 0.8598 34 17 −149.26 159.38 51 1 1 1 -0.10414 1 1 1
Eigenvalue—Maximum (Upper)λ1, Middleλ2, Minimum (Lower)λ3thenλ1≥ λ2≥ λ3 Remark
(1) whole agreement with value positive and more than 1
(2) negotiation breakdown happens when any eigenvector cannot be obtained (3) partial agreement to be with value negative or close to 0
this method provides the ways of judgement and presuming the existence of the missing value, also might check the valid-ity of the Harker method.
Meaning of eigenvalue and Validity of Kinoshita’s method Kinoshita method adopts a computational approach and considers the weights from each criterion and alternative. According to this method, the solution of the super-matrix of the ANP is an eigenvector to the maximum eigenvalue which becomes an index of an agreement. That is, if this value is large, it means that an agreement becomes near complete agreement. The minimum eigenvalue is used in Table4for the evaluation. It indicates a minimum unit for agreement. The agreement seems to collapse extremely easily when a minimum eigenvalue is close to zero. However, the correla-tion coefficient with the evaluacorrela-tion values of this method is 0.999318 with the same result as Kinoshita’s solutions, and naturally think that both solutions are a nearly same. Negative evaluation values in the criteria matrix
From our proposed method, the elements of the criteria matrix might become negative. Therefore, the elements of Ul adjust to a positive values is an option. That means the nega-tive elements does not affect directly to the eigenvector cal-culation. For example, if all elements become a positive for
the criteria matrix i.e. that adds 2 and 10 in the Eq. (14). The result shows the eigenvector z does not change anyway. Then, the eigenvector z+2that adds 2 and the eigenvector z+10that adds 10 are obtained. Therefore, the relationship between the eigenvector z+2and the eigenvector z+10become the same
z+2 = z+10 = (0.3662, 0.3099, 0.3239). While, the
eigen-vector z = (0.3688, 0.3121, 0.3191) and z ∼= z+2 = z+10
exist little differences in the eigenvectors. An effective con-version regime could be found out, the criteria matrix might be changed by converting “the standard” of the evaluation. Classification type of the decision-making
From this study, we figure out the fact that exists an agree-ment between the alternative matrix and the criteria matrix when the maximum eigenvalue exists in the “Major ANP”. Whereas, the maximum eigenvalue and the eigenvector do not exist, then, there are no agreement between the alter-native matrix and the criteria matrix. This situation named as “Negotiation breakdown” when any eigenvector cannot be obtained. The “whole agreement” happens when exists positive eigenvalue and more than 1. “partial agreement” are with value negative or close to 0. This study states that there is a minimum unit in an agreement between the criteria and alternatives. “Minor ANP” becomes a tool to give a clas-sification on the cases of selection decisions that accepts a realistic type on decision-making process. And these three
types of decision-making enable to reflect a realistic human behavior in decision-making.
A choice with missing values towards to the supplier selection
It is important to have a choice of sources via the supplier selection. Buying from only one supplier can be dangerous— where do you go if they let you down, or even go out of busi-ness? Equally, while exclusivity may spur some suppliers to offer a better service, others may simply become complacent and drop their standards. But the judgments of supplier selec-tion still exist quite unknown with subjective or objective way of choosing. Missing parts of consideration from decision makers may cause both tangible and intangible loss. Also, it may count on the missing parts as an unknown message which still push decision makers ahead to move business for-ward by reducing to a minimum risk (as of giving the missing values a reasonable assumptions and predicting the possible way to go through).
Conclusion
This study describes “Minor ANP” which the method of mak-ing the priority by usmak-ing irregular ANP which utilize the calculation with only the alternative matrix, i.e. with the miss-ing value and non-square matrix. Also, this paper shows that there are three types of decision makings of “Whole agree-ment”, “Negotiation breakdown”, and “Partial agreement”. Besides, this proposed method examined the validity of Har-ker method or Nishizawa method, and clarified the meaning without presumption.
With the case of supplier selection when there are some limitations in suppliers’ capacity, quality, etc. In other words, no one supplier can satisfy the buyer’s total requirements and the buyer needs to purchase some part of his/her demand from one supplier and the other part from another supplier to com-pensate for the shortage of capacity or low quality of the first supplier. And prior to the strategic decision may exist fuzzy or uncertainty and lack sufficient information to access the decision making process.
This method suggests that the elements of the criteria matrix need not necessarily be always positive. These con-clusions are the results of approaching the restoration func-tion that exists inside the ANP by the interacfunc-tion between the alternative matrix and the criteria matrix. Finally, “Minor ANP” introduces user-friendliness and reflects the behavior of human decision-making. “Minor ANP” also allows for the intuitive recognition and a range of missing values to be possibly included into decision procedures. Furthermore, we expect to advance the research of “Minor ANP” and benefit its simplified calculation in the real world.
Acknowledgments This work has been supported partly by National Science Council- Taipei (NSC 099-2811-H-424-001) with 3-year Post Doctoral Research Program.
References
Choi, T.Y., & Hartley, J.L. (1996). An exploration of supplier selec-tion practices across the supply Chain. Journal of Operaselec-tions Management, 14(4), 333–343.
Ghodsypour, S.H., & O’Brien, C. (1998). A decision support system for supplier selection using an integrated analytic hierarchy pro-cess and linear programming. International Journal of Production Economics, 56-57, 199–212.
Ghodsypour, S.H., & O’Brien, C. (2001). The total cost of logis-tics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of Production Economics, 73(1), 15–27.
Harker, P.T. (1987). Incomplete pairwise comparisons in the analytic hierarch process. Mathematical Modelling, 9(11), 837–848. Harker, P.T. (1987). Derivatives of the Perron root of a positive
reciprocal matrix: With application to the analytic hierarchy pro-cess. Applied Mathematics and Computation, 22(2/3), 217–232. Harker, P.T., & Vargas, L.G. (1987). The theory of ratio scale estimation: Saaty’s analytic hierarchy process. Management Sci-ence, 33(11), 1383–1403.
Kahraman, C., Cebeci, U., & Ulukan, Z. (2003). Multi-criteria sup-plier selection using fuzzy AHP. Logistics Information Manage-ment, 16(6), 382–394.
Kinoshita, E., & Nakanishi, M. (1999). Proposal of new AHP model in light of dominant relationship among alternatives. Journal of the Operations Research Society of Japan, 42(2), 180–197. Kinoshita, E., Sekitani, K., & Shi, J. (2002). Mathematical properties
of dominant AHP and concurrent convergence method. Journal of the Operations Research Society of Japan, 45(2), 198–213. Kinoshita, E., & Sugiura, S. (2008). A comparison of dominant AHP
and similar dominant models. Journal of Research Institute of Meijo University, 7, 113–116.
Nishizawa, K. (2005). Estimation of unknown comparisons in incom-plete AHP. Report of the Research Institute of Industrial Technol-ogy, Nihon University, 77, 1–9.
Nishizawa, K. (2007). Simple AHP based on three-level evaluation and two-cluster ANP for the decision makers. Journal of Japanese Symposium on The Analytic Hierachy Process of the Operations Research Society of Japan, 1, 97–104.
Nydick, R.L., & Hill, R.P. (1992). Using the analytic hierarchy pro-cess to structure the supplier selection procedure. International Journal of Purchasing and Materials Management, 28(2), 31–36. Ozaki, T., Sugiura, S., Kinoshita, E., Tzeng, G.H. (2009). Dissolution of the dilemma or circulation problem using the Analytic Net-work Process. Proceedings of the 10t hInternational Symposium on the Analytic Hierarchy Process 2009, Pittsburgh.
Saaty, T.L. (1980). The analytic hierarchy process. New York: McGraw-Hill.
Saaty, T.L. (1996). Decision making with dependence and feedback: The analytic network process. Pittsburgh: RWS Publications. Spekman, R.E. (1988). Strategic supplier selection: Understanding
long-term Buyer relationships. Business Horizons, 31(4), 75–81. Sugiura, S., & Kinoshita, E. (2005). Solution of circular logic by con-current convergence method of evaluating value. Urban Science Studies of Meijo University, 10, 115–121.
Sugiura, S., & Kinoshita, E. (2007). A dissolution of circular logic on social choice. Journal of Japanese Symposium on the Ana-lytic Hierarchy Process of the Operations Research Society of Japan, 1, 75–83.
