Asia Pacific Management Review (2004) 9(5), 783-800
A Use of Analytic Network Process
for Supply Chain Management
Tomokatsu Nakagawa* and Kazuyuki Sekitani**
Abstract
Strategic decision analysis on Supply Chain Management (SCM) involves tangible and intangible factors and it needs to be evaluated by interdependent multiple criteria. Analytic Hierarchy Process (AHP) is a practical and popular decision-making tool for SCM. Interdependency
among criteria becomes a source of difficulty in directly applying AHP to various SCM decision problems. Hence, the managerial issue is examined by Analytic Network Process (ANP),
the general form of AHP. This study addresses a new use of ANP on SCM strategic decision
analysis such as a supplier selection and improvement of supply chain performance.
Keywords: E-business, Analytic network process, Supply chain management
1. Introduction
Recent advances in information technologies/Information systems
(IT/IS) generate economic and strategic changes in many aspects of modern
business, e.g. manufacturing, logistics, finance and marketing. In the competitive business environment, IT/IS accelerates reorganization of a series of
decision making processes where a product or service is manufactured from
raw materials (the origin of products/service) to the end-products/service.
Such a computer-based reproduction process is considered as a supply chain
(SC) in this study. In discussing the SC management, we need to mention a
fact that companies are interested in supply chain management (SCM) [35],
because they face to increases and varieties of customer demands, competition in global environment, decreases in governmental regulations and increases in environmentally conscious business practices.
SCM involves coordinating and managing all the activities from raw
materials procurement to the delivery of the final product to customers by
the efficient use of IT/IS. This framework is depicted in Figure 1. The aim of
SCM is to globally optimize material and information flows in SC by horizontal integration between companies within SC and vertical integration of
*
Graduate school of Science and Technology, Shizuoka University, 3-5-1. Jouhoku, Hamamatsu, Shizuoka, 432-8561, Japan. E-mail: [email protected]
**
Dept. Systems Engineering, Shizuoka University, 3-5-1. Jouhoku, Hamamatsu, Shizuoka,
432-8561, Japan. E-mail: [email protected]
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Tomokatsu Nakagawa et al.
Information flow
Supplier Supplier Purchasing Materials Production Physical Marketing Customer Customer
management
Distribution &Sales
Product
Product flow
CUSTOMER RELATIONSHIP MANAGEMENT
Supply Chain Business Process
CUSTOMER
SERVICE MANAGEMENT
ORDER
FULFILLMENT
MANUFACTURING FLOW MANAGEMENT
PROCUREMENT
PRODUCT DEVELOPMENT
RETURNS
CHANNEL
Figure 1 A Framework for Supply Chain Management [8]
existing business processes in each company. In order to attain the aim of
SCM, managers within SC need to make strategic decisions for supplier selection, buying strategies, capital equipment purchasing, supplier performance evaluation, a long-term partnerships between buyers and suppliers, effective purchasing and distribution, etc. These are usually ambiguous and
unstructured problems that include both tangible and intangible factors under
complicated criteria with interdependence relationships (e.g., [17] and [24]).
A desirable methodology for such managerial issues is to allow for the
synthesis of these factors and to help managers to structure the decisionmaking problem.
Analytic Hierarchy Process (AHP) and its extension, Analytic Network
Process (ANP) are one of such systematic approaches that can deal with both
quantitative and qualitative factors under multiple criteria [27, 28]. As
widely known, AHP and ANP are practical tools of multiple criteria decision
analysis. A lot of applications and case-studies using AHP and ANP are reported in various fields of business-world and industries [39, 41]. Especially,
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Asia Pacific Management Review (2004) 9(5), 783-800
AHP in SCM has been a popular approach for supplier selection [2, 7, 14,
16, 24, 25, 26, 38, 40], the design of supply chain networks [5, 6, 18, 19],
supplier performance evaluation [12, 15]. ANP is also applied to a same type
of problems on SCM as AHP [1, 30, 31], because ANP allows for the network structure modeling including all AHP models. The network modeling
capability adds ANP with new applications to SCM such as the strategic
decision analysis of a long-term partnership within SC [22, 23]. AHP, the
origin of ANP, sometimes provides an irrational ranking (see [10, 11, 36] for
the details) and it is called rank reversal phenomenon. By exploiting the
network modeling effectively, ANP may mitigate the possibility of rank reversal phenomenon [32].
This paper presents a new issue of using ANP for SCM decision problems. This issue may be specified as follows: The final results obtained from
ANP do not have any response to varying the essential part of input-data in
ANP. Hence, ANP provides the final results that are not affected by a critical
factor for the overall decision-makings. This irrationality is different from
the conventional issue of AHP, rank reversal phenomenon. This paper shows
that the irrational final result is caused by certain network structures of the
ANP model. Such network structures of the ANP model often appear in applications to SCM.
The remainder of the paper is organized as follows: Section 2 describes
a brief review of AHP/ANP and the importance of ANP on the supplier selection problem. Section 3 shows the irrationality of ANP by illustrating an
example of supplier selection [31] and discusses mathematically why such a
problem occurs. Moreover the potential irrationality of the ANP analysis for
SCM [1] is reported. Finally, the last section discusses how to handle the
irrationality related to ANP/SCM.
2. Research Background
2.1 AHP and ANP
ANP is a more general form of AHP. Both AHP and ANP provide the
overall weights of alternatives under the multiple criteria. AHP decomposes
a decisional problem into several decision levels in such a way that they
form a hierarchy. Therefore, in AHP, the top element of the hierarchy is the
overall goal for a decision-model. Alternatives are listed in the bottom level
of the hierarchy. Each element in the hierarchy is supposed to be independent, and a relative ratio scale of measurement is derived from pairwise comparisons of the elements in a level of the hierarchy with respect to an ele-
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Tomokatsu Nakagawa et al.
ment of the preceding level. However, in many cases, there is interdependence among criteria and alternatives. ANP does not require such independence among elements and hence, it can be used as an effective tool in the
decisional cases with element independence [28]. For example, ANP allows
for elements and a subset of elements, say ‘cluster’, to ‘control’ and be ‘controlled’ by the varying levels, clusters of decision criteria or alternatives.
Interdependencies are represented as directed arcs among levels, clusters of
decision criteria and alternatives. For example, a looped arc exists if a pair of
mutually controlling clusters/elements is in the same level. ANP models a
decision-making, using a network structure. Since a hierarchical structure is
one of network structures, ANP includes AHP completely.
In ANP, the relative importance or strength of the impacts on a given
element is determined, similar to AHP, by using pairwise comparisons with
a scale of 1-9. However, ANP must evaluate interdependencies within levels
of clusters and mutually dependent elements in a cluster. To complete this
evaluation, Saaty [28] has developed a square matrix ‘supermatrix’ whose
size is the number of all elements in the network. The supermatrix forming
requires a series of steps as follows:
1.
completing pairwise comparisons of the elements on their controlling elements;
2.
taking the results relative importance weights (eigenvectors) and
placing them in submatrices within the supermatrix;
3.
adjusting the supermatrix so that it is ‘column stochastic’ (i.e. the
summation of the values in the columns of the supermatrix sum to
one);
The final analysis of ANP is to derive the overall weight of each element. These overall weights are usually calculated by raising the supermatrix to a sufficiently large power until the weights have converged and remained stable. If the weights have not converged and remained unstable,
Saaty and Vargas [29] developed the following three steps:
1.
the supermatrix is decomposed into network portions and hierarchical portions
2.
the sub-supermatrix corresponding to the network portions is
raised to a sufficiently large power
3.
a cross-product of the converged network data supermatrix and
the hierarchical network data.
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Asia Pacific Management Review (2004) 9(5), 783-800
2.2 Application of AHP and ANP to SCM
The SC consists of all links from suppliers to customers of a product and
SCM involves various types of strategic decision problems [3]. Goffin et al.
[13] have stated that supplier management is one of the key issues of SCM,
because a cost of raw materials and component constitutes the main cost of a
product. Moreover, SCM requires suppliers to manufacture and deliver to a
company and such requirement needs the precise quantity and quality of
material at a prescribed time. Thus the performance of suppliers becomes a
key of SCM's success or failure.
A number of empirical articles on supplier selection have appeared [4,
38]. Based on empirical data collected from 170 purchasing managers,
members of the National Association of Purchasing Management, Dickson
[9] identified quality, cost, and delivery performance history as the three
most important criteria in vendor selection. Recently, this fact is confirmed
by the similar questionnaire survey of Vonderembse and Tracey [37]. Therefore, buyer-supplier relationships based on only the price factor has not been
appropriate in SCM. In addition, the supplier selection decision must include
strategic and operational criteria as well as tangible and intangible criteria in
the analysis [31]. Thus, it is easily found that the supplier selection is complicated by many criteria. This is a typical multiple criteria decision-making
problem [38].
As stated previously, AHP and ANP have been used for the supplier selection. Since Narasimhan [25] proposed the use of AHP for the supplier
selection, Nydick and Hill [26] and Barbarosoglu and Yazgaç [2], Khurrum
and Faizul [16] and Mohanty and Deshmukh [24] have developed various
decision hierarchy models for the supplier selection and improved the AHP
models. Ghodsypour and O'Brien [14] and Çebi and Bayraktar [7] combined AHP and mathematical programming models to deal with multiple
suppliers selection problems. Korpela, Lehmusvaara and Tuominen [17] and
Massella and Rangone [21] developed supplier selection systems based on
AHP. The application of AHP to the supplier selection problem has an analytical advantage that may be summarized as follows: AHP deals with subjective judgments of a buyer in supplier choice. Concretely, the buyer is only
required to give verbal, qualitative statements regarding the relative importance of one criterion versus another criterion and similarly regarding the
relative preference for one supplier versus another on a criterion. This merit
is shared with ANP applied to the supplier selection problem. However, the
hierarchy structure imposes independence of criteria on AHP and ANP is
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Tomokatsu Nakagawa et al.
free from the restriction.
The survey paper by Boer, Labro and Morlacchi [4] has reported that the
supplier selection process is decomposed into four stages, initial problem
definition, formulation of criteria, qualification of potential suppliers and
final choice among the qualified suppliers. They state that AHP is useful for
two stages, the qualification and the final choice. They conclude from [20]
that Interpretive Structure Modeling (ISM) is a tool of the criteria formulation stage because ISM aids the buyer by separating dependent criteria from
independent criteria. Therefore, the combination of ISM and AHP may be
effective. However, ANP does not need the aid of ISM because it can deal
with dependent criteria. Hence, it is effective to some parts of the criteria
formulation stage.
By exploiting the analytical advantage of ANP, Sarkis and Talluri [31]
and Agarwal and Shanker [1] propose the use of ANP for the supplier selection. Additionally, ANP is applied to strategic analysis for organizational
supply chain relationship [22] and a hub-facility location problem [30].
3. A Drawback of ANP/SCM Decision Problems
In [31], the ANP decision hierarchy for the supplier selection problem
includes a network port consisting of one element and three clusters, the
overall goal, the planning horizon cluster, the organizational factors cluster
and the strategic performance metrics cluster. The planning horizon cluster
has two elements, short and long terms, the organizational factors cluster has
Goal
Strategic Supplier Selection
Organizational Factors
Strategic Performance Metrics
Culture
Technology
Relationship
Short Term
Cost
Quality
Time
Flexibility
Long Term
Figure 2 Network Potions of Decision Hierarchy for Supplier Selection [31]
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Asia Pacific Management Review (2004) 9(5), 783-800
Table 1 Initial Supermatrix for Network Portion of Decision Hierarchy [31]
Planning
Horizon
short
long
term
term
goal
goal
Strategic Performance Metrics
cost
quality
Time
Organizational factors
flex
culture
technology
relationship
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.667
0.500
0.500
0.500
0.667
0.500
0.500
0.000
0.000
0.000
0.333
0.500
0.500
0.500
0.333
0.500
0.500
cost
0.140
0.127
0.250
0.000
0.249
0.528
0.286
0.000
0.000
0.000
quality
0.340
0.312
0.250
0.169
0.000
0.140
0.143
0.000
0.000
0.000
short
term
long
term
time
0.281
0.280
0.250
0.443
0.594
0.000
0.571
0.000
0.000
0.000
flexibility
0.239
0.280
0.250
0.387
0.157
0.332
0.000
0.000
0.000
0.000
culture
0.090
0.163
0.169
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.728
0.540
0.443
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.181
0.297
0.387
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Techno
-logy
Relationship
three elements, culture, technology and relationship and the strategic performance metrics cluster has four elements, cost, quality, time, flexibility.
The network port is graphically and numerically summarized in Figure 2 and
Table 1, respectively.
As stated in Section 2.2, ANP normalizes each column of Table 1 such that
the column-sum is one and generates the following supermatrix:
⎡0.000
⎢0.000
⎢
⎢0.000
⎢
⎢0.070
⎢0.170
M=⎢
⎢0.140
⎢0.120
⎢
⎢0.045
⎢0.364
⎢
⎢⎣0.091
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎤
0.000 0.000 0.333 0.250 0.250 0.250 0.667 0.500 0.500⎥⎥
0.000 0.000 0.167 0.250 0.250 0.250 0.333 0.500 0.500⎥
⎥
0.064 0.125 0.000 0.125 0.264 0.143 0.000 0.000 0.000⎥
0.156 0.125 0.085 0.000 0.070 0.076 0.000 0.000 0.000⎥
⎥.
0.140 0.125 0.222 0.297 0.000 0.285 0.000 0.000 0.000⎥
0.140 0.125 0.193 0.078 0.166 0.000 0.000 0.000 0.000⎥
⎥
0.082 0.084 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎥
0.270 0.222 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎥
⎥
0.148 0.194 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎥⎦
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((1)
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Tomokatsu Nakagawa et al.
To discuss the mathematical properties of the overall weights of the elements within the network portion, the supermatrix M is represented by
⎡0 0 T ⎤ , where `T‘ is the transpose operator for vector or matrix, 0 is a zero
⎢
⎥
⎣x Y ⎦
vector,
⎡0.000
⎢0.000
⎢
⎢0.064
⎢
⎢0.156
Y = ⎢0.140
⎢
⎢0.140
⎢0.082
⎢
⎢0.270
⎢
⎣0.148
0.000 0.333 0.250 0.250 0.250 0.667 0.500 0.500⎤
0.000 0.167 0.250 0.250 0.250 0.333 0.500 0.500⎥⎥
0.125 0.000 0.125 0.264 0.143 0.000 0.000 0.000⎥
⎥
0.125 0.085 0.000 0.070 0.076 0.000 0.000 0.000⎥
0.125 0.222 0.297 0.000 0.285 0.000 0.000 0.000⎥ .
⎥
0.125 0.193 0.078 0.166 0.000 0.000 0.000 0.000⎥
0.084 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎥
⎥
0.222 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎥
⎥
0.194 0.000 0.000 0.000 0.000 0.000 0.000 0.000⎦
(2)
( 2)
and
T
x = [0.000 0.000 0.070 0.170 0.140 0.120 0.045 0.364 0.091] .
(3)
Note that Y is irreducible and primitive. (For an irreducible and primitive
matrix, see [28, 33])
Sarkis et al. [31] report that the supermatrix M is numerically converged after raising M to the 64th power. This convergence of M on the numerical calculation is theoretically guaranteed by the following lemmas:
Lemma 1 Let x and 0 be an n-dimensional vector and an n-dimensional
zero vector, respectively. Let Y be a matrix of order n and suppose
⎡0 0T ⎤ , then for any natural number k
M=⎢
⎥
⎣x Y ⎦
⎡ 0
0T ⎤ .
M k = ⎢ k −1
k⎥
⎣Y x Y ⎦
(4)
Proof: We will prove (4) by induction of k. When k = 2, we have
⎡0 0 T ⎤ ⎡ 0 0 T ⎤ ⎡ 0 0 T ⎤
.
M2 = ⎢
⎥⋅⎢
⎥=⎢
2⎥
x
Y
x
Y
Yx
Y
⎣
⎦ ⎣
⎦ ⎣
⎦
790
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Asia Pacific Management Review (2004) 9(5), 783-800
0T ⎤ , then we have
Suppose that M k = ⎡⎢ 0
k −1
k⎥
⎣Y x Y ⎦
⎡ 0
0 T ⎤ ⎡0 0 T ⎤ ⎡ 0
0T ⎤ .
M k +1 = M k ⋅ M = ⎢ k −1
⋅
=
⎥ ⎢ k
k⎥ ⎢
k +1 ⎥
⎣ Y x Y ⎦ ⎣x Y ⎦ ⎣Y x Y ⎦
(6)
From (5) and (6) the induction is completed.
Lemma 2 Let e be an n-dimensional vector whose element is all one. Let x
be an n-dimensional nonnegative vector such that eTx=1 and let Y be an
irreducible and primitive nonnegative matrix of order n such that eTY=eT.
T
Consider M = ⎡0 0 ⎤ and let u be a principal eigenvector of Y such that
⎢
⎥
⎣x Y ⎦
eTu=1, then
⎡0 0 T ⎤
lim M k = ⎢
T⎥ .
k →∞
⎣⎢u ue ⎥⎦
(7)
Proof: Since the matrix Y is irreducible and primitive, it follows from
eTY=eT that lim k →∞ Yk converges to a Y∞ and each column of Y∞ is the
same vector v. Hence,
lim Y k = Y ∞ = ve T .
k →∞
(8)
It follows from Lemma 1, (8) and eTx = 1 that
0T ⎤
0
⎡ 0
0T ⎤ ⎡
=
lim M k = lim ⎢ k −1
⎢
⎥
⎥
k →∞
k →∞ Y
Y k −1 x lim Y k ⎥
x Y k ⎦ ⎢⎣ lim
⎣
k →∞
k →∞
⎦
⎡ 0
0T ⎤ ⎡ 0
=⎢ T
=⎢
T ⎥
⎣ ve x ve ⎦ ⎣ v
0T ⎤
⎥.
ve T ⎦
It follows from Theorem 2 of [33] that v=u, and hence,
⎡0 0T ⎤
.
lim M k = ⎢
T⎥
k →∞
u
ue
⎣
⎦
According to the definition of the overall weight by Saaty [28] and Sarkis et
al. [30], the overall weights vector for all elements within the planning hori-
791
Tomokatsu Nakagawa et al.
zon cluster, the organizational factors cluster and the strategic performance
metrics cluster corresponds to the first column of M∞. In fact, Sarkis et al.
[31] report that the overall weights for cost, quality, time, flexibility, culture,
technology and relationship are:
[cost quality time flexibility
= [0.094 0.081 0.126 0.099
culture technology relationship] (9)
0.033 0.099 0.068].
The normalized principal eigenvector u of (2) is
[0.213
T
0.187 0.094 0.081 0.126 0.099 0.033 0.099 0.068] .
(10)
This means that [u3, . . . ,u9] of (10) coincides with the overall weight vector
(9) and hence, Lemma 2 is valid for the supermatrix of (1).
Lemma 2 shows that the convergence of M depends entirely upon only
the mutual evaluation values Y among criteria, not the evaluation values x
from the goal to criteria. This fact is summarized as follows:
T
Theorem 3 Suppose the same condition of M = ⎡⎢0 0 ⎤⎥ as Lemma 2. For
⎣x Y ⎦
every n-dimensional nonnegative vector z such that eTz=1,
⎡ 0
⎡ 0
0T ⎤
0T ⎤ ⎡ 0 0T ⎤ ,
=
lim ⎢ k −1
lim
⎥
⎢
⎥=⎢
⎥
k →∞ Y
x Y k ⎦ k →∞ ⎣Y k −1 z Y k ⎦ ⎣u ue T ⎦
⎣
(11)
where u is a principal eigenvector of Y such that eTu=1.
T
T
⎡
⎤
⎡
⎤
Proof: By applying Lemma 2 to ⎢0 0 ⎥ and ⎢0 0 ⎥ , respectively, it
⎣z Y ⎦
⎣x Y ⎦
follows that
⎡ 0
⎡ 0
0T ⎤ ⎡0 0T ⎤
0T ⎤ ⎡ 0 0T ⎤ .
and
=
lim ⎢ k −1
lim
⎥ ⎢
⎥
⎢
⎥=⎢
⎥
k →∞ Y
k →∞ Y k −1 x
z Y k ⎦ ⎣u ue T ⎦
Y k ⎦ ⎣u ue T ⎦
⎣
⎣
This proves the assertion.
Theorem 3 implies that overall weight vector of (9) is invariant to varying the evaluation values from the goal, i.e., x. Hence, x of (3) does not affect the overall weights of all criteria and alternatives. This is a serious issue
for ANP because the judgments of the decision maker must be reflected in
overall weights of some criteria or alternatives and the evaluation values
from the top level element, goal, should affect overall weights of all elements
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Asia Pacific Management Review (2004) 9(5), 783-800
Supply Chain Performance Index
Lead Time
Market sensitiveness (MS)
Cost
Process integration
Customer
Responsiveness
(CR)
New product
Introduction
(NPI)
Service Level
Information-driven
CPB
CSD
Delivery
Speed
(DS)
EDI
CSS
MOI
DA
Figure 3 Decision Hierarchy for Supply Chain Improvement [1]
except the goal. This issue is called “nonresponse phenomenon.” Theorem 3
implies that the non-response phenomenon is not caused by the choice of the
primitive matrix Y.
The nonresponse phenomenon arises from SCM decision problems [31].
In [1], criteria of the SC performance are formulated as four levels hierarchy
whose bottom level includes network structures, so-called ‘inner dependence’ by Saaty [28]. The criteria hierarchy is shown in Figure 3.
Without loss of generality, the network structure discussed here is restricted within the third level element MS and its lower cluster consisting of
DS, NPI and CR. The supermatrix corresponding to this network structure
can be specified as follows:
⎡0.000
⎢ 0.691
M=⎢
⎢ 0.091
⎢
⎣0.218
0.000 0.000 0.000⎤
0.000 0.800 0.857⎥⎥ .
0.250 0.000 0.143⎥
⎥
0.750 0.200 0.000⎦
(12)
From Lemma 2, it follows that the overall weights of DS, NPI and CR are
[0.456
T
0.168 0.376 ] .
793
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Tomokatsu Nakagawa et al.
Theorem 3 implies that the overall weights of (13) are determined by
only the mutual evaluation among DR, NPI and CR and they are independent of varying the evaluation from MS to DS, NPI and CR. Furthermore,
evaluation values among the top level, goal, the second one and MS within
the decision hierarchy of Figure 3 do not affect the overall weights of (13)
by the following theorem:
Theorem 4 Let ej be a j-dimensional vector whose component is all one. Let
Y be an irreducible and primitive matrix of order n such that enTY = enT and
let X and Q be an (n × m) nonnegative matrix with enTX = emT and an (m × l)
nonnegative matrix with emTQ = elT, respectively. Let p be an l-dimensional
nonnegative vector such that elTp=1. Consider a nonnegative matrix
⎡0 0 0 0 ⎤
⎢p 0 0 0 ⎥
⎥ with order (1+l+m+n) and let u be a principal eigenvecM=⎢
⎢0 Q 0 0 ⎥
⎢
⎥
⎣ 0 0 Y X⎦
tor of Y such that enTu = 1, then
0
⎡0
⎢0
0
lim M = ⎢
⎢0
k →∞
0
⎢
T
⎣u ue l
k
0
0
0
ue Tm
0 ⎤
0 ⎥⎥ .
0 ⎥
⎥
ue Tn ⎦
(14)
Proof: By induction of k, we have
0
0
0
0 ⎤
⎡
⎢
0
0
0
0 ⎥⎥
Mk = ⎢
⎢
0
0
0
0 ⎥
⎢ k −3
k −2
k −1
k⎥
⎣Y XQp Y XQ Y X Y ⎦
(15)
for every k ≥ 4. Since the matrix Y is irreducible and primitive, it follows
from eTY = eT that lim k →∞ Yk converges to a Y∞ and each column of Y∞ is a
principal eigenvector u of Y. That is,
lim k →∞ Yk = uenT.
This implies from enTX = emT, emTQ = elT and elTp = 1 that
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Asia Pacific Management Review (2004) 9(5), 783-800
0
0
0
⎡
⎢
0
0
0
⎢
lim M k = ⎢
0
0
0
k →∞
⎢
k −3
k −2
k −1
lim
Y
XQp
lim
Y
XQ
lim
Y
X
⎢⎣k →∞
k →∞
k →∞
0
0
0
0 ⎤
⎡
⎢
0
0
0
0 ⎥⎥
=⎢
⎢
0
0
0
0 ⎥
⎢ T
T
T
T⎥
ue
XQp
ue
XQ
ue
X
ue
n
n
n⎦
⎣ n
0
⎡0
⎢0
0
=⎢
⎢0
0
⎢
T
u
ue
l
⎣
0
0
0
ue Tm
⎤
0 ⎥⎥
0 ⎥
⎥
lim Y k ⎥
k →∞
⎦
0
0 ⎤
0 ⎥⎥
.
0 ⎥
⎥
ue Tn ⎦
Theorem 4 means that the overall weights of the elements within the bottom
level are not affected by varying p, Q or X. Hence, all judgments of the decision maker in all levels except the bottom one are vain.
The last part of this section discusses a relationship between the princiT
pal eigenvector of the supermatrix M = ⎡⎢0 0 ⎤⎥ and its overall weight
⎣x Y ⎦
vector. Saaty [28] and Sekitani et al. [33] have shown the equivalence between the principal eigenvector of the irreducible supermatrix and its overall
weight vector. However, their results cannot be directly applied to the supermatrix M because M is reducible. The relationship between the principal
eigenvector of Y and that of M is as follows:
Theorem 5 Suppose the same condition of M as Lemma 2 and let u be a
principal eigenvector of Y such that eTu=1, then the (n + 1)-dimensional
vector [0, uT]T is a principal eigenvector of M.
Proof: Since eT x =1 and eTY = eT, it follows that eTM = eT, and hence, the
supermatrix M has the largest eigenvalue that is 1. Let [v0, vT]T be an eigenvector of M corresponding to the largest eigenvalue 1, then
⎡ 0 0 T ⎤ ⎡v 0 ⎤ ⎡v 0 ⎤
⎢
⎥⎢ ⎥ = ⎢ ⎥ .
⎣x Y ⎦ ⎣ v ⎦ ⎣ v ⎦
795
Tomokatsu Nakagawa et al.
This means that v0 = 0 and xv0 + Yv = Yv = v. Since the matrix Y is primitive and irreducible, v is a principal eigenvector of Y, that is v = u. Therefore, the supermatrix M has a principal eigenvector [0, uT]T.
From Theorem 5, the first component of the principal eigenvector of the
T
⎡
⎤
supermatrix M = ⎢0 0 ⎥ is 0, which means the nonresponse phenome⎣x Y ⎦
non occurs in the overall weights.
4. Concluding Comments
In the context of application of ANP to SCM decision analysis, this
study documents that nonresponse phenomenon occurs in the overall evaluation process. The nonresponse phenomenon is inherent in an ANP network
structure that is referred to as “sinarchy” by Saaty [28]. The sinarchy is a
hierarchy within a feedback cycle between the last two (bottom or sink) levels as shown in Figure 4. As stated in Theorem 3, the nonresponse phenomenon is a state of the overall weights that is not affected by any judgments of the decision-maker within any levels except last two levels. A
sinarchy structure in the SCM decision problem appears in the supplier selection [31] and the SC performance evaluation [1], and the nonresponse
phenomenon arises in both the applications. These facts mean that overall
Goal
Figure 4 Structure of Sinarchy
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Asia Pacific Management Review (2004) 9(5), 783-800
weights within the sinarchy must not be derived from raising the supermatrix
M to the sufficiently large power.
As stated in Theorem 5, overall weights by the eigenvalue method for
T
the supermatrix M = ⎡⎢0 0 ⎤⎥ is equivalent to the method of raising M to the
⎢⎣x
Y ⎥⎦
infinity power. Therefore, the nonresponse phenomenon cannot be solved by
a direct application of the eigenvalue method to M. For the reducible supermatrix Sekitani and Takahashi [33] has developed the analytic tool of overall
weights of all elements with the ANP network. They considers that mi is the
number of positive values in the ith row of M for i = 2, . . ., n + 1 and let m1
⎡m1
= 1. By generating N = ⎢⎢
⎢0
⎣
0T ⎤
⎥ , −1 ⎡1 0T ⎤ is called the averaged
O
⎥
⎥ N ⋅⎢
⎣⎢x Y ⎦⎥
m n +1 ⎥
⎦
supermatrix (see [34] for the average principle of AHP). They propose overall weights derived from the principal eigenvector of the averaged supermaT⎤
⎡
trix N −1 ⋅ ⎢1 0 ⎥ . Since the first component of the principal eigenvector of
⎢⎣x
Y ⎥⎦
the averaged supermatrix is positive, the nonresponse phenomenon may be
avoided by the method of Sekitani and Takahashi [33]. However, their
method may be more difficult and complex for the ANP users who are faced
with the real world problems. This shortcoming needs to be further investigated as an extension of this study.
This study deals with an irreducible and primitive matrix Y in order to
simplify the mathematical discussion and it is easy to extend all lemmas and
theorems of section 3 into the case of an irreducible matrix Y with more than
one cycle index.
Acknowledgment
This research is partially supported by the Japan Society for the Promotion
of Science under grant No. 15510123.
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