A hybrid dynamic MADM model for problem-improvement in economics and business.
Peng, Kua-Hsin ; Tzeng, Gwo-Hshiung
Introduction
Decision making (DM) regarding single criterion problems is highly
intuitive, being a simple matter of choosing the alternative with the
highest preference rating. However, when DM evaluates alternatives
involving multiple criteria, numerous problems, including criteria
weights, preferences or influence dependence, and conflicts among
criteria, complicate the problems and require sophisticated solutions
(Tzeng, Huang 2011). Bernoulli proposed the expected-utility principle
in 1738 (Luce, Raiffa 1957). Additionally, von Neumann and Morgenstern
(1944) presented an expected utility criterion that typically aggregates
these elements in ranking possible actions to decide the optimal
selection. In the early 1970s, multiple criteria decision making (MCDM)
was introduced as a promising and important field of study, then,
research on MCDM has been increasing extremely (Carlsson, Fullr 1996;
Wallenius et al. 2008). Multiple criteria decision making (MCDM) is a
scientific analytical method for evaluating a set of alternatives by
considering multiple criteria to determine a priority ranking and
improvement for alternative implementation (Tsaur et al. 1997; Wang, Lee
2009; Chang et al. 2012). MCDM methods generally aim to help
decision-makers make better decisions by selecting the best from among
multiple feasible alternatives under the presence of multiple choice
criteria and diverse criteria priorities (Jankowski 1995; Mollaghasemi,
Pet-Edwards 1997). Additionally, MCDM methods attempt to improve
decision quality, through clearer, more reasonable and more efficient
decision processes.
Hwang and Yoon (1981) classified MCDM problems into two main
categories, namely multiple attribute decision making (MADM) and
multiple objective decision making (MODM) (Fig. 1), based on the
different purposes and data types. The former mainly involve the
evaluation/improvement/selection facets/dimensions, which are usually
associated with a limited number of predetermined alternatives and
discrete preference ratings. The latter category exist particularly in
the areas of design/planning, and generally involve attempting to
optimize goals by considering the various interactions within the given
constrains, so that both decision and objective spaces are changeable in
new research concepts. However, this study proposed that the traditional
MCDM ignores some important new concepts and limitations/defects for
solving the real-world problems. First, conventional MADM assumes
independent criteria with a hierarchical structure. However,
relationships among criteria or dimensions are usually interdependent
for real-world problems, and in some cases feedback effects exist. The
criteria in practical MADM problems are generally interactive, and thus
some interdependent models have been proposed (such as DANP
(DEMATEL-based ANP), etc.). Second, conventional MADM only obtains
relatively good solutions from existing alternatives, but also avoids
"choosing the best among inferior
choices/options/alternatives", i.e. avoids "Pick the best
apple from a barrel of rotten apples", it should be replaced by the
aspiration levels. Third, conventional MADM merely allows the selection
and ranking of alternatives or strategies, but these alternative methods
shift the focus from how to conduct "ranking" or
"selection" of the most preferable alternatives to how to
"improve" them. Fourth, information fusion/aggregation, such
as fuzzy integral, a non-additive/super-additive model, has been
developed for performance aggregation. Therefore, a Hybrid Dynamic
Multiple Attribute Decision Making (HDMADM) method is needed to overcome
the defects of the conventional MADM method and solve the complications
of dynamic problems in the real world (Campanella, Ribeiro 2011). This
study presented two categories of HDMADM. The first category used the
basic concept of ANP (Saaty 1996) with DEMATEL (call DANP, DEMATEL-based
ANP) to yield influential weights of dimensions/criteria, and combined
influential weights with the additive types of VIKOR. The second
category also used DANP to yield influential weights of
dimensions/criteria, but combined the influential weights of the DANP
with non-additive/super-additive types of fuzzy integral to assess and
improve complex practical problems. Finally, this study presented two
empirical cases to demonstrate the ability of the HDMADM method to
overcome the defects of the conventional MADM method. The remainder of
this paper is organized as follows. Section 1 reviews the MCDM method.
Section 2 then introduce the HDMADM method. Subsequently, Section 3
presents some empirical cases to demonstrate the effectiveness of the
HDMADM method. Finally, the last section presents conclusions.
[FIGURE 1 OMITTED]
1. MCDM method
For studies wishing to know how to develop improvement strategies
to achieve the goal or aspiration level, for example pursuing higher
performance, competitiveness and satisfactory service, an important
question is which research methods are most suitable and practical for
solving real world problems. Based on the above thinking, the first part
of the study attempts to figure out how many attributes or criteria
should consider. On the other hand, the study must collect adequate data
that reflect the behaviors of attributes or criteria. Additionally, the
study should build a set of possible alternatives or strategies to
guarantee that the goal or aspiration level is achieved using MCDM
methods. Then, the next step is to select appropriate MCDM methods that
help decision-makers to evaluate, improve and choose possible
alternatives or strategies.
Multiple criteria decision making (MCDM) is a scientific analytical
method for evaluating a set of alternatives based on multiple criteria
(Campanella, Ribeiro 2011; Tsaur et al. 1997; Wang, Lee 2009; Loban
1997). MCDM techniques have been used in recent years to solve a wide
variety of problems (Chen, Liao 2004; Hung, Chiang 2008; Ou Yang et al.
2008), such as supplier selection (Deng, Chan 2011); performance
evaluation of higher education (Wu et al. 2012); improving
airline's service quality (Kuo 2011); evaluating website quality
(Chou, Cheng 2012); product design and selection (Liu 2011); evaluating
hot spring hotels service quality (Tseng 2011); prioritizing sustainable
electricity production technologies (Streimikiene et al. 2012), etc.
Additionally, most MCDM problems in the real world thus occur in hybrid
situations, which include goals, aspects (or dimensions), attributes (or
criteria), and possible alternatives (or strategies). Furthermore, most
real-world decision problems are dynamic, however, the traditional MCDM
model is unable to capture this dynamicity (Campanella, Ribeiro 2011)
and hybrid situation, thus, should develop a suitable HDMADM method to
solve complication dynamic problems in the real world.
2. Methodology for solving the real world problems
This section is divided into five parts: the first part describes
the concept of HDMADM method, the second part presents DEMATEL method,
the third part presents DANP, the fourth part presents VIKOR method, and
the last part describes fuzzy integral for focusing on how to aggregate
the performance in non-additive/super-additive situations to suit the
real world problems.
2.1. Hybrid Dynamic Multiple Attribute Decision Making (HDMADM)
This study proposed the DEMATEL technique and combines a DANP with
additive types of VIKOR and non-additive types of fuzzy integral to
address the problems of conventional MADM method. The DEMATEL technique
is used to build an influential network relations map (INRM), then for
obtaining the influential weights of each criterion, DANP use the basic
concept of ANP (Saaty 1996) and taking the transpose of normalized
total-influence matrix [T.sub.c] (denoting
([T.sup.[alpha].sub.c])') by dimensions to get the un-weighted
super-matrix [W.sub.c] (i.e. [W.sub.c] = ([T.sup.[alpha].sub.c])')
and taking the normalized total-influence matrix [T.sub.D] (obtaining
[T.sup.[alpha].sub.D]) multiplying the un-weighted super-matrix
[W.sub.c] to obtain the weighted super-matrix [W.sup.[alpha].sub.c](i.e.
[W.sup.[alpha].sub.c] = [T.sup.[alpha].sub.D][W.sub.c]). According to
the weighted super-matrix [W.sup.[alpha].sub.c], it multiplies by itself
multiple times to obtain limit super-matrix [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII](Appendix B). Then, the VIKOR method or Fuzzy
Integral with influential weights (DANP) is used to integrate the
performance gaps. Finally, it is possible to determine how to improve
performance and reduce the gaps to achieve the aspiration level based on
INRM. The processes of HDMADM are illustrated as Fig. 2.
[FIGURE 2 OMITTED]
2.2. DEMATEL method
DEMATEL is an analytical technique for building a structural model.
DEMATEL is mainly used to clarify and solve complex problems. DEMATEL
uses matrix and related mathematical theories (Boolean operation) to
calculate the cause and effect relationships involved in each element.
This technique is widely used to solve various complex problems, and
particularly to understand complex problem structures and provide
practical problem-solving methods. The DEMATEL technique involves five
steps (see Appendix A). The first step is to confirm that the system has
n elements and develop the evaluation scale, using a pair-wise of
dimensions to perform the comparison, and also using the measuring scale
0, 1, 2, 3, 4, where (0) represents no influence whatsoever, (1)
represents low influence, (2) represents medium influence, (3)
represents high influence, and (4) represents extremely high influence.
The second step calculates the initial matrix to directly obtain the
influential matrix (Lin, Tzeng 2009; Chen et al. 2010). The third step
normalizes the matrix such that at least one column or row, but not all,
sums to one. The fourth step then obtains the total influence matrix.
Finally, the fifth step builds the influential network relation map
(INRM).
2.3. Finding the influential weights using DANP
This study not only uses the DEMATEL technique to build the
interactive relationship among the various dimensions/criteria, but also
seeks the most accurate influential weights. This study found that ANP
can serve this purpose. This study used the basic concept of ANP (Saaty
1996), which eliminates the limitations of Analytic Hierarchy Process
(AHP) and is applied to solve nonlinear and complex network relations
(Saaty 1996). DANP is intended to solve interdependence and feedback
problems of criteria in influential weights. This study thus applies the
characteristics of influential weights based on basic concept of ANP and
combines them with DEMATEL (call DANP, DEMATEL-based ANP) to solve these
kinds of problems (see Appendix B). This approach yields more practical
results in real world problem.
2.4. VIKOR Method
The VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje)
method (see Appendix C) was developed for multicriteria optimization of
complex systems. It introduces the multicriteria ranking index based on
the particular measure of "closeness" to the "ideal"
solution (Opricovic 1998). VIKOR uses the class distance function (Yu
1973) based on the concept of the positive-ideal (or in this study adopt
the Aspiration level) solution and negative-ideal (or in this study
adopt the Worst level) solution and orders the results. For normalized
class distance function it is better to be near the positive-ideal
points (the aspiration level) and far from the negative-ideal point (the
worst value) for normalized class distance function (Lee et al. 2009; Ho
et al. 2011). Opricovic and Tzeng (2004) proposed the compromise ranking
method (VIKOR) as a suitable technique for implementation within MCDM
(Tzeng et al. 2002a, b, 2005; Opricovic, Tzeng 2002, 2003, 2007). VIKOR
comprises the following steps: The first step is to check the best and
worst values of the assessment criteria. The second step is to calculate
the mean group utility based on the sum of all individual-criterion
regrets (i.e. average overall performance gaps, and those for each
dimension, and for each criterion; and strategies for reducing these
gaps), and calculate the maximal regret for an individual-criterion for
improvement priority, both overall and for each dimension. The third
step is to obtain the comprehensive/integrating indicators and sort the
results provided to the decision-maker to implement improvement
strategies and reduce competitiveness gaps in both overall performance
and individual dimensions of performance.
2.5. The [lambda] fuzzy measure and fuzzy integral
In order to overcome non-additive problem, Sugeno (1974) introduced
the concept of fuzzy measure and fuzzy integral (see Appendix D). This
study presents that used DANP to yield influential weights of
dimensions/criteria, then combined the influential weights of the DANP
with non-additive types of fuzzy integral to integrate the performance
gaps and improve complex practical problems.
3. An empirical case
This section comprises two parts: the first part describes an
empirical case involving Taiwan to explore strategies for improving
tourism destination competitiveness (TDC) based on a HDMADM model using
DEMATEL, DANP and VIKOR; the second part presents an empirical case
involving a Taiwanese company for supplier evaluation and improvement
based on a fuzzy integral-based hybrid MADM model that addresses the
dependence/relationships among the various criteria and the non-additive
gap-weighted analysis.
3.1. Tourism destination competitiveness (TDC) of Taiwan
The following presents an empirical case involving Taiwan to
explore strategies for improving tourism destination competitiveness
(TDC) using a HDMADM model. This study identifies three dimensions of
expert cognition and opinion, and also identifies the relationship
between the degrees of the impact, which is compared with other
dimensions, as listed in Table 1. According to the total influential
prominence ([r.sub.i] + [d.sub.i]), "Regulatory framework
([D.sub.1])" is the highest total influential prominence among
other factors that means the most important influencing factors;
additionally, "Human cultural and natural resources
([D.sub.3])" is the factors with the weakest total influential
prominence among other factors. According to the influential relation
([r.sub.i] - [d.sub.i]), "Regulatory framework ([D.sub.1])"
represents the highest degree of impact relationship and directly
affects other factors. Otherwise, "Business environment and
infrastructure ([D.sub.2])" is more vulnerable to influence than
other dimensions.
Table 2 lists the relationship between the direct or indirect
impacts and compares them with other criteria. "Prioritization of
Travel and Tourism ([C.sub.5])" is the most important criterion
among those considered; additionally, "Safety and security
([C.sub.3])" is the criterion with the smallest impact on other
criteria. Furthermore, Table 2 shows that "Policy rules and
regulations ([C.sub.1])" has the strongest relationship among all
the criteria. Otherwise, "Tourism infrastructure ([C.sub.8])"
is the most vulnerable criteria to outside influences.
This study builds the assessment model using DEMATEL, which is
combined with the DANP (DEMATEL-based ANP) model to obtain the
influential weights of each criterion, as listed in Table 2.
Furthermore, the influential weights combine with the VIKOR in
weightings to assess the priority of problem-solving improvement based
on the competitiveness gaps identified by VIKOR and the influential
relation map.
A real case involving Taiwan is used to assess the total
competitiveness using the VIKOR method, as listed in Table 3. The scores
of each criterion and the total average gap ([S.sub.k]) of Taiwan are
obtained, using the relative influential weights from DANP to multiply
the gap ([r.sub.kj]). Consequently, this study obtains the total
competitiveness gap of Taiwan.
Additionally, to improve the human cultural and natural resources
([D.sub.3]) dimension, this study finds that the criterion of
"Natural resources ([C.sub.13])" is the maximal performance
gap. Furthermore, the criterion of "Human resources
([C.sub.11])" is the most important and influential criterion, and
thus can be considered the critical criterion for improving natural
resources. Thus, "Human resources ([C.sub.11])" can be
considered the critical criterion for improving the regulatory
framework. Additionally, the comprehensive indicator ([R.sub.k]) can be
obtained, which value of v can determine by the expert that is defined
as v = 0.5 in this paper. This study identifies the comprehensive
indicator ([R.sub.k]) as 0.602, indicating that the Taiwanese government
must improve the gap of TDC. Furthermore, the government can identify
the problem-solving strategy according to the DEMATEL technique combined
with DANP and VIKOR (called the hybrid MCDM model).
The DEMATEL technique (Fig. 3) can obtain valuable cues for making
accurate decisions. This system structure model reveals that Taiwan
suffers a significant gap in the "Human cultural and natural
resources ([D.sub.3])" dimensions, making it necessary focus on the
[FIGURE 3 OMITTED]
"Regulatory framework ([D.sub.1])" dimensions for
improving the TDC of Taiwan. Furthermore, for improving the regulatory
framework ([D.sub.1]) dimension, this study finds that the criterion of
"Health and hygiene ([C.sub.4])" prioritizes reducing the
maximal competitiveness gap. Fig. 3 reveals that the criteria of
"Policy rules and regulations ([C.sub.1])",
"Prioritization of Travel & Tourism ([C.sub.5])" and
Environmental sustainability ([C.sub.2]) are the most important and
influential criteria because they are most closely related to other
criteria in the ([D.sub.1]) dimension. Additionally, for improving the
human cultural and natural resources ([D.sub.3]) dimension, this study
finds that the criterion of "Natural resources ([C.sub.13])"
is the maximal performance gap. Furthermore, the criteria of "Human
resources ([C.sub.11])" is the most important and influential
criteria, and thus can be considered the critical criteria for improving
natural resources. Thus, the criteria of "Human resources
([C.sub.11])" can be considered the critical criterion for
improving the regulatory framework.
Consequently, Fig. 3 shows valuable cues for making accurate
decisions. The influential network relations map provides an initial
tool for demonstrating that the degrees of influence differ among
dimensions and criteria. This study utilizes the most important and
influential criteria as critical factors to improve the maximal gap of
competitiveness.
3.2. Supplier evaluation and improvement involving a Taiwanese
company
The supplier selection four dimensions and 11 criteria are
developed based on literature review and discussions with the managers
of the case company. Following the DEMATEL method, the influential
network-relationship can be visualized by drawing an influential
network-relationship map (INRM) of the four dimensions and their
subsystems, as shown in Fig. 4 (the contents summarized from
Tzengs' research group (Liou et al. 2012)).
This real case study used the DANP (DEMATEL-based ANP) model to
obtain the influential weights of each criterion, as listed in Table 4.
[FIGURE 4 OMITTED]
This real case study utilizes fuzzy integrals to aggregate the
weighted gaps. Because the criteria within the same dimension have
interdependent relationships, their weighted gaps should be integrated
rather than being treated as individual values. Similarly, the
integrated weighted gaps of the four dimensions should be further
considered with their final synthesized values. Through a questionnaire
survey conducted by managers of the case study company, the fuzzy
integral 1 values, which range from -1 to positive infinity [infinity]
(i.e. -1 [less than or equal to] [lambda] < [infinity]), that
represent the properties of substitution or multiplication between
criteria are obtained. Substitutive effects exist among attributes of
risk, and a multiplicative effect exists among compatibility, quality,
and cost. Table 5 lists the l values and the fuzzy measures g(*).The
integrated weighted gaps of each potential supplier are then calculated
as shown in Table 6 (the contents summarized from Tzengs' research
group (Liou et al. 2012)).
In the case study, the proposed fuzzy integral-based model
addresses this problem, and the results reveal a different priority:
[A.sub.3] > [A.sub.2] > [A.sub.1] > [A.sub.4] > [A.sub.5]
(Table 6). Obviously, [A.sub.3] is the best service provider considering
both the criterion weights and performance interdependence. This
non-additive model should be more reasonable than previous additive
models because if network relationships exist between criteria, the
performances should have the same effect (the contents summarized from
Tzengs' research group (Liou et al. 2012)).
Conclusions
This study proposed some important new concepts and limit
limitations/defects of traditional MADM. Additionally, this study
presented empirical cases to demonstrate that the HDMADM method could
overcome the defects of the conventional MADM method. First, the
traditional model assumes that the criteria are independent and
hierarchical in structure; however, real-world problems frequently
involve interdependent criteria. This study presented a HDMADM method
that applies the characteristics of influential weights ANP and combines
them with DEMATEL (call DANP, DEMATEL-based ANP) to solve
interdependence and feedback problems of criteria. Second, the VIKOR
method set the best [f.sup.*.sub.j] values as the aspiration level and
the worst [f.sup.-.sub.j] values as the tolerable level for all
criterion functions, j = 1, 2, ..., n to avoid "Choosing the best
among a range of inferior choices/options/alternatives" (i.e. this
study avoids picking the best from a barrel of rotten apples). Third,
the HDMADM method shifts the concept from the "ranking" or
"selection" of the most preferable alternatives to the
"improvement" of their performances or competitiveness to
achieve the aspiration level based on influential network relation map
(INRM) using the DEMATEL technique. Finally, this study describes an
empirical case involving supplier evaluation and improvement of
Taiwanese company based on a novel fuzzy integral-based hybrid MADM
model that addresses the dependence/relationships among the various
criteria and non-additive gap-weighted analysis.
Appendix A. DEMATEL technique
The DEMATEL technique is used to construct the
interactions/interrelationship between criteria to build an influential
relation map. The method is divided into three steps:
Step 1: Find the average influence matrix A
The first step is to calculate initial matrix, using pair of degree
of interaction/interrelationship to obtain directly influence matrix A =
[[[a.sub.ij]].sub.n x n], where [a.sub.ij] represents the degree of
effect on i factor effects j factor (Lin, Tzeng 2009; Chen et al. 2010).
A = [[[a.sub.ij]].sub.n x n] = [[1/H [H.summation over (h = 1)]
[a.sup.h.sub.ij]].sub.n x n], (1)
where h is the [h.sup.th] expert and h = 1, 2, ..., H.
Step 2: Calculate the normalized influence matrix X
When the elements of i have a direct effect on the elements of j,
then [a.sub.ij] [not equal to] 0, otherwise [a.sub.ij] = 0. The second
step is to normalize the matrix. It can be obtained from Eqs. (2) and
(3). Its diagonal is 0, and maximum sum of row or column is 1, but not
all.
X = sA, (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
Step 3: Compute the total influence matrix T
The total-influence matrix T can be obtained through Eq. (4), in
which I denotes the identity matrix.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Explanation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where X = [[[x.sup.ij.sub.c]].sub.n x n], 0 [less than or equal to]
< 1, 0 < [[summation].sup.n.sub.j = 1] [x.sup.ij.sub.c] [less than
or equal to] 1 and 0 < [[summation].sup.n.sub.i = 1] [x.sup.ij.sub.c]
[less than or equal to] 1, and at least one row or column of the
summation, but not all, equals one; then, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] can be guaranteed.
To sum of each row and column of the total influence matrix T =
[[[t.sub.ij]].sub.n x n], in these results the sum of all rows (vector r
= [[[[summation].sup.n.sub.j = 1][t.sub.ij]].sub.n x 1] =
[[[r.sub.i]].sub.n x 1] = ([r.sub.1], ..., [r.sub.i], ...,
[r.sub.n])') and the sum of all columns(vector d =
[[[[summation].sup.n.sub.i = 1][t.sub.ij]].sub.n x 1] =
[[[d.sub.j]].sub.n x 1] = ([d.sub.1], ..., [d.sub.j], ..., [d.sub.n])
can be obtained. If [r.sub.i] represents the sum of all rows of the
total-influence matrix T, meaning directly or/and indirectly affects to
other criteria; [d.sub.j] represents the sum of all columns of the
total-influence matrix T, meaning is affected by other criteria.
[r.sub.i] represents the factor which will affect other factors,
[d.sub.j] represents the factor that is affected by other factors.
According to the definition, when i = j, then [r.sub.i] + [d.sub.j]
presents the degree of relationship between the factors, meaning
"prominence"; [r.sub.i] - [d.sub.j] presents the degree of
effect and effected for the factors, meaning "relation" (Tzeng
et al. 2007) in dynamic influence.
Appendix B. To find the weights by DANP model
DANP can be divided into following steps:
Step 1: Develop the structure of the question
The questions are clearly described then break them down to level
structure.
Step 2: Develop Unweighted Supermatrix
Firstly, each level with total degree of effect that obtains from
the total-influence matrix t of DEMATEL as shown in Eq. (5).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Normalize [T.sub.c] with total-influence will be obtained
[T.sup.[alpha].sub.c] that shows in Eq. (6).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Normalize [T.sup.[alpha]11.sub.c] will be obtained by Eqs (7) and
(8), according to the same fashion will be obtained
[T.sup.[alpha]nn.sub.c].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
And then, total-influence matrix is normalized into Supermatrix
according to the group in relying relationship to obtain Unweighted
Supermatrix as show in Eq. (9).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
In addition, we will be obtained matrix [W.sup.11] and [W.sup.12]
by Eq. (10). If blank or 0 shown in the matrix means the group or
criteria is independent, according to the same fashion will be obtained
matrix [W.sup.nn].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Step 3: Obtain Weight Supermatrix
Let each dimension of total-influence matrix [T.sub.D] as (11) be
normalized with total degree of influence to obtain
[T.sup.[alpha].sub.D], the result as Eq. (12).
[d.sub.i] = [[summation].sup.n.sub.j = 1][t.sup.ij.sub.D], i = 1,
2, ..., n;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
Then, drive the normalized [T.sup.[alpha].sub.D] into Unweight
Supermatrix W to obtain Weight Supermatrix [W.sup.[alpha]], the result
as shown in Eq. (13):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Step 4: Obtain limit supermatrix
According to the weighted super-matrix [W.sup.[alpha]], it
multiplies by itself multiple times to obtain limit supermatrix based on
basic concept of Markov Chain. Then, the DANP influential weights of
each criterion can be obtained by [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII], where z represents any number for power.
Appendix C. VIKOR method
VIKOR method can be divided into follow steps:
Step 1: Check the best value [f.sup.*.sub.j] and the worse value
[f.sup.-.sub.j]
There [f.sup.*.sub.j] represents the positive-ideal point, that
means the expert gives the scores of the best value (aspired levels) in
each criterion and [f.sup.-.sub.j] represents the negative-ideal point,
that means the expert gives the scores of the worst values in each
criterion. We use Eqs. (14) and (15) to obtain the results.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
We set the best [f.sup.*.sub.j] values to be the aspiration level
and the worst [f.sup.-.sub.j] values as the tolerable level for all
criterion functions, j = 1,2, ..., n. In this study, we modify the
traditional approach (suppose the jth function denotes benefits:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and shift the
concept from the "ranking" or "selection" of the
most preferable alternatives to the "improvement" of their
performances to achieve the aspiration level for each dimension and
criterion. Therefore, the [f.sup.*.sub.j] and [f.sup.-.sub.j] values can
be set by decision makers, so that [f.sup.*.sub.j] is the aspiration
level and [f.sup.-.sub.j] is the worst value. For example, in
questionnaires we can use performance scores ranging from 0 to 10 (from
very dissatisfied or very bad [left arrow] 0,1,2, ...,9,10 [right arrow]
very satisfied or very good) expressed natural language, wherein the
aspiration level can be set at 10 and the worst value at zero. In this
study, we set [f.sup.*.sub.j] = 10 as the aspiration level and
[f.sup.-.sub.j] = 0 as the worst value, which differs from the
traditional approach. This allows us to avoid "choosing the best
among inferior options/alternatives (i.e. avoid picking the best apple
from among a barrel of rotten apples)".
Step 2: Calculate the mean of group utility [S.sub.k] and maximal
regret [Q.sub.k]
There [S.sub.k] represents the ratios of distance to the
positive-ideal, it means the synthesized gap for all criteria; [w.sub.j]
represents the influential weights of the criteria from DANP; [r.sub.kj]
represents the average gap-ratios (regret) of normalized distance to the
aspired level point, and [Q.sub.k] represents the maximal gap-ratios
(regret) of normalized distance to the aspired level in all criteria, it
means the maximal gap in j criteria for prior improvement. Those values
can be computed respectively by Eqs. (16) and (17).
[S.sub.k] = [n.summation over (j = 1)][w.sub.j][r.sub.kj] =
[n.summation over (j = 1)][w.sub.j] ([absolute value of [f.sup.*.sub.j]
- [f.sub.kj]])/([absolute value of [f.sup.*.sub.] - [f.sup.-.sub.j]]);
(16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
Step 3: Obtain the comprehensive indicator [R.sub.k] and sorting
results
The values can be computed respectively by Eq. (18).
R = v([S.sub.k] - [S.sup.*])/([S.sup.-] - [S.sup.*]) + (1 -
v)([Q.sub.k] - [Q.sup.*])/([Q.sup.-] - [Q.sup.*]). (18)
Those values derived from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII] or setting [S.sup.*] = 0 (the aspired level), [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] or setting [S.sup.-] = 1 (the
worst situation); [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or
setting [Q.sup.*] = 0 (the aspired level), and [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII] or setting [Q.sup.-] = 1 (the worst
situation). Therefore, when [S.sup.*] = 0 and [S.sup.-] = 1, and
[Q.sup.*] = 0 and [Q.sup.-] = 1, we can re-write the Eq. (43) as
[R.sub.k] = v[S.sub.k] + (1 - v)[Q.sub.k]. Weight v = 1 represents only
to be consider the average gap (average regret) weight and weight v = 0
represents only to be consider the max gap to be prior improvement. It
can provide the decision-makers by experts. Generally v = 0.5 (the
majority of criteria), it could be adjusted depends on the situation.
Appendix D. The [lambda] fuzzy measure and fuzzy integral
Let [g.sub.[lambda]] denote a [lambda] fuzzy measure which is
defined on a power set P(x), for the finite set X = {[x.sub.1],
[x.sub.2], ..., [x.sub.n]}. The fuzzy measure has the following property
(Tzeng, Huang 2011):
[for all]A, B [member of] P(X), A [intersection] B = [empty set],
[g.sub.[lambda]](A [union] B) = [g.sub.[lambda]](A) +
[g.sub.[lambda]](B) + [lambda][g.sub.[lambda]](A)[g.sub.[lambda]] (B)
for -1 < [lambda] < [infinity]. (19)
The density of the fuzzy measure [g.sub.i] = [g.sub.[lambda]]
({[x.sub.i]}) can be obtained from questionnaire responses (thus
[g.sub.[lambda]]({[x.sub.i]}) = u([x.sup.*.sub.i], x 0/i)). Assume the
existence of a single product, for which all criteria are perfect, and
where the product equals 1. Now assume that for this product only one
criterion [x.sup.*.sub.i] is completely perfect, while besides
[x.sup.*.sub.i] all other criteria x 0/i are inferior. The question
becomes the attractiveness of the product in this situation. The local
weights ([w.sub.1], [w.sub.2], ..., [w.sub.n]) can be obtained using
DANP. Next, the fuzzy measure weights are set to:
([g.sub.[lambda]]({[x.sub.1]}), [g.sub.[lambda]]({[x.sub.2]}), ...,
[g.sub.[lambda]]({[x.sub.n]})) = q([w.sub.1], [w.sub.2], ..., [w.sub.n])
= ([w.sub.1]q, [w.sub.2]q, ..., [w.sub.n]q), (20)
where q denotes the adjusted weight coefficient.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
Based on the above properties, one of the three following
situations is sustained for a specific case involving attributes,
[x.sub.1] and [x.sub.2].
a. If l > 0, then [g.sub.[lambda]] (A [union] B) >
[g.sub.[lambda]] (A) + [g.sub.[lambda]](B) which implies that [x.sub.1]
and [x.sub.2] have a multiplicative effect in {A, B};
b. If l = 0, then [g.sub.[lambda]](A [union] B) =
[g.sub.[lambda]](A) + [g.sub.[lambda]](B) which implies that [x.sub.1]
and [x.sub.2] have an additive effect in {A,B};
c. If l < 0, then [g.sub.[lambda]](A [union] B) <
[g.sub.[lambda]](A) + [g.sub.[lambda]](B) which means that [x.sub.1] and
[x.sub.2] have a substitutive effect in {A,B}.
In this model, the performance values are replaced by the gaps
which equal aspiration levels minus the evaluated values with respect to
each criterion. Let h denote a measurable set function (gap function)
defined on the fuzzy measurable space, and supposing that h([x.sub.1])
[greater than or equal to] h([x.sub.2]) [greater than or equal to]...
[greater than or equal to] h([x.sub.n]), then the fuzzy integral of
fuzzy measure g(*) with respect to h(*) can be defined as follows
(Ishii, Sugeno 1985), as shown in Fig. D1.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
[FIGURE D1 OMITTED]
The fuzzy integral defined in Eq. (22) is called the Choquet
integral (Sugeno 1974; Ishii, Sugeno 1985; Sugeno et al. 1998; Chen et
al. 2000, 2001a; Chiou, Tzeng 2002, 2003; Chiou et al. 2005; Liou, Tzeng
2007; Chu et al. 2007; Larbani et al. 2011). Using the fuzzy integral to
formulate the original data can not only extract fewer and more
representative factors to describe the system, but can also consider the
interactions between attributes. This study used [integral]h dg =
[a.sub.in] as the integrated weighted gaps of cluster [C.sub.n] at
alternative i.
Caption: Fig. 1. Basic concepts on overview of social science
research with MCDM (Liou, Tzeng 2012)
Caption: Fig. 2. Model procedures of Hybrid Dynamic Multiple
Attribute Decision Making (HDMADM) (Chiu et al. 2013)
Caption: Fig. 3. The influential network relations map of each
dimension and criteria
Caption: Fig. 4. Influential network-relationship map within
systems Source: Liou et al. (2012).
Caption: Fig. D1. Concept of fuzzy integral
doi: 10.3846/20294913.2013.837114
Received 13 August 2012; accepted 13 April 2013
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Kua-Hsin PENG (a), Gwo-Hshiung TZENG (b)
(a) Institute of Leisure and Recreation Management, Kainan
University, No. 1, Kainan Road, Luchu, Taoyuan 338, Taiwan
(b) Graduate Institute of Urban Planning, College of Public
Affairs, National Taipei University, No. 151, University Road, San Shia
237, Taiwan
(b) Institute of Management of Technology, National Chiao-Tung
University, No. 1001, Ta-Hsueh Road, Hsin-Chu 300, Taiwan
Corresponding author Gwo-Hshiung Tzeng E-mail:
ghtzeng@mail.ntpu.edu.tw; ghtzeng@cc.nctu.edu.tw
Kua-Hsin PENG received her PhD degree from the Department of
Management Science of National Chiao Tung University, Taiwan in 2011.
Currently, she is an Assistant Professor in the Department of Leisure
and Recreation Management at Kainan University. She has published in
numerous journals, including Tourism Management, Current Issues in
Tourism, Journal of the Operational Research Society, Quality &
Quantity, Expert Systems with Applications, Journal of Grey System, etc.
Her research interest is applying hybrid MCDM method to solve some
problems in tourism and leisure management, accreditation performance,
etc.
Gwo-Hshiung TZENG. In 1967, he received the Bachelor's degree
in Business Management from the Tatung Institute of Technology (now
Tatung University), Taiwan; in 1971, he received the Master's
degree in Urban Planning from Chung Hsing University (Now Taipei
University), Taiwan; and in 1977, he received the PhD degree course in
management science from Osaka University, Osaka, Japan. He was an
Associate Professor at Chiao Tung University, Taiwan (1977-1981), a
Research Associate at Argonne National Laboratory (July 1981-January
1982), a Visiting Professor in the Department of Civil Engineering at
the University of Maryland, College Park (August 1989-August 1990), a
Visiting Professor in the Department of Engineering and Economic System,
Energy Modeling Forum at Stanford University (August 1997-August 1998),
a Professor at Chaio Tung University (1981-2003), and a Chair Professor
at Chiao Tung University. His current research interests include
statistics, multivariate analysis, network, routing and scheduling,
multiple criteria decision making, fuzzy theory, hierarchical structure
analysis for applying to technology management, energy, environment,
transportation systems, transportation investment, logistics, location,
urban planning, tourism, technology management, electronic commerce,
global supply chain, etc.
Table 1. Total influential matrix of T and the sum of the effects
on the dimensions
Dimensions [D.sub.1] [D.sub.2]
[D.sub.1] Regulatory framework 0.305 0.825
[D.sub.2] Business environment and 0.321 0.237
infrastructure
[D.sub.3] Human cultural and 0.290 0.435
natural resources
Dimensions [D.sub.3] [r.sub.i]
[D.sub.1] Regulatory framework 0.782 1.912
[D.sub.2] Business environment and 0.332 0.891
infrastructure
[D.sub.3] Human cultural and 0.208 0.932
natural resources
Dimensions [d.sub.i] [r.sub.i] +
[d.sub.i]
[D.sub.1] Regulatory framework 0.916 2.828
[D.sub.2] Business environment and 1.497 2.388
infrastructure
[D.sub.3] Human cultural and 1.322 2.254
natural resources
Dimensions [r.sub.i] -
[d.sub.i]
[D.sub.1] Regulatory framework 0.996
[D.sub.2] Business environment and -0.606
infrastructure
[D.sub.3] Human cultural and -0.389
natural resources
Note : [1/[n.sup.2]] [n.summation over (i = 1)]
[n.summation over (j = 1)] [[absolute value of ([t.sup.p.sub.ij] -
[t.sup.p - 1.sub.ij]/[t.sup.p.ub.ij]] x 100% = 3.11% < 5%, i.e.
significant confidence is 96.89%, where p = 10 denotes the
number of experts and [t.sup.p.sub.ij] is the average influence of i
criterion on j; and n denotes number of dimensions, here n = 3 and
n x n matrix.
Table 2. The sum of influences, weights and rankings of each
criterion
Dimensions/Criteria [r.sub.i] [d.sub.i]
[D.sub.1] Regulatory framework
[C.sub.1] Policy rules and regulations 1.750 0.882
[C.sub.2] Environmental sustainability 0.865 0.933
[C.sub.3] Safety and security 0.716 0.846
[C.sub.4] Health and hygiene 0.764 0.886
[C.sub.5] Prioritization of Travel 1.857 1.192
and Tourism
[D.sub.2] Business environment
and infrastructure
[C.sub.6] Air transport infrastructure 0.726 0.935
[C.sub.7] Ground transport 0.735 0.936
[C.sub.8] Tourism infrastructure 0.754 1.020
[C.sub.9] ICT infrastructure 0.734 0.884
[C.sub.10] Price competitiveness 0.690 1.014
[D.sub.3] Human cultural and
natural resources
[C.sub.11] Human resources 1.103 0.778
[C.sub.12] Affinity for 0.729 0.930
travel & tourism
[C.sub.13] Natural resources 0.884 0.896
[C.sub.14] Culture resources 0.803 0.977
Dimensions/Criteria [r.sub.i] + [r.sub.i] -
[d.sub.i] [d.sub.i]
[D.sub.1] Regulatory framework
[C.sub.1] Policy rules and regulations 2.633 0.868
[C.sub.2] Environmental sustainability 1.798 -0.068
[C.sub.3] Safety and security 1.562 -0.131
[C.sub.4] Health and hygiene 1.651 -0.122
[C.sub.5] Prioritization of Travel 3.048 0.665
and Tourism
[D.sub.2] Business environment
and infrastructure
[C.sub.6] Air transport infrastructure 1.661 -0.209
[C.sub.7] Ground transport 1.670 -0.201
[C.sub.8] Tourism infrastructure 1.774 -0.266
[C.sub.9] ICT infrastructure 1.618 -0.150
[C.sub.10] Price competitiveness 1.704 -0.325
[D.sub.3] Human cultural and
natural resources
[C.sub.11] Human resources 1.881 0.325
[C.sub.12] Affinity for 1.659 -0.202
travel & tourism
[C.sub.13] Natural resources 1.780 -0.013
[C.sub.14] Culture resources 1.781 -0.174
Degree of
Dimensions/Criteria importance Ranking
(Global
weight)
[D.sub.1] Regulatory framework 0.2866 3
[C.sub.1] Policy rules and regulations 0.0544 3
[C.sub.2] Environmental sustainability 0.0546 2
[C.sub.3] Safety and security 0.0500 5
[C.sub.4] Health and hygiene 0.0537 4
[C.sub.5] Prioritization of Travel 0.0739 1
and Tourism
[D.sub.2] Business environment 0.3803 1
and infrastructure
[C.sub.6] Air transport infrastructure 0.0744 3
[C.sub.7] Ground transport 0.0739 4
[C.sub.8] Tourism infrastructure 0.0809 1
[C.sub.9] ICT infrastructure 0.0717 5
[C.sub.10] Price competitiveness 0.0794 2
[D.sub.3] Human cultural and 0.3332 2
natural resources
[C.sub.11] Human resources 0.0769 4
[C.sub.12] Affinity for 0.0837 3
travel & tourism
[C.sub.13] Natural resources 0.0841 2
[C.sub.14] Culture resources 0.0885 1
Table 3. The performance evaluation of the case study by VIKOR
Dimensions / Criteria Local Global
weights weights
(by DANP)
[D.sub.1] Regulatory framework 0.2866(3)
[C.sub.1] Policy rules and 0.1898 0.0544(3)
regulations
[C.sub.2] Environmental 0.1905 0.0546(2)
sustainability
[C.sub.3] Safety and security 0.1745 0.0500(5)
[C.sub.4] Health and hygiene 0.1874 0.0537(4)
[C.sub.5] Prioritization of travel 0.2579 0.0739(1)
and tourism
[D.sub.2] Business environment 0.3803(1)
and infrastructure
[C.sub.6] Air transport 0.1956 0.0744(3)
infrastructure
[C.sub.7] Ground transport 0.1943 0.0739(4)
[C.sub.8] Tourism infrastructure 0.2127 0.0809(1)
[C.sub.9] ICT infrastructure 0.1885 0.0717(5)
[C.sub.10] Price competitiveness 0.2088 0.0794(2)
[D.sub.3] Human cultural and 0.3332(2)
natural resources
[C.sub.11] Human resources 0.2308 0.0769(4)
[C.sub.12] Affinity for travel 0.2512 0.0837(3)
and tourism
[C.sub.13] Natural resources 0.2524 0.0841(2)
[C.sub.14] Culture resources 0.2656 0.0885(1)
Total performances
Total gap ([S.sub.k])
Dimensions / Criteria Case study of Taiwan
Score Gap
([r.sub.kj])
[D.sub.1] Regulatory framework 4.40 0.433
[C.sub.1] Policy rules and 4.80 0.367
regulations
[C.sub.2] Environmental 4.20 0.467
sustainability
[C.sub.3] Safety and security 5.50 0.250
[C.sub.4] Health and hygiene 3.30 0.617
[C.sub.5] Prioritization of travel 4.20 0.467
and tourism
[D.sub.2] Business environment 4.90 0.357
and infrastructure
[C.sub.6] Air transport 3.80 0.533
infrastructure
[C.sub.7] Ground transport 5.70 0.217
[C.sub.8] Tourism infrastructure 4.40 0.433
[C.sub.9] ICT infrastructure 5.30 0.283
[C.sub.10] Price competitiveness 5.10 0.317
[D.sub.3] Human cultural and 3.90 0.517
natural resources
[C.sub.11] Human resources 5.70 0.217
[C.sub.12] Affinity for travel 4.60 0.400
and tourism
[C.sub.13] Natural resources 2.40 0.767
[C.sub.14] Culture resources 2.90 0.683
Total performances 4.40 --
Total gap ([S.sub.k]) -- 0.437
Table 4. Influential weights of system factors
Dimensions Local Rankings Criteria
Weights
[D.sub.1] 0.306 1 [C.sub.11] Relationship
Compatibility [C.sub.12] Flexibility
[C.sub.13] Information
sharing
[D.sub.2] 0.231 3 [C.sub.21] Knowledge
Quality skill
[C.sub.22] Customers'
satisfactions
[C.sub.23] On time rate
[D.sub.3] Cost 0.204 4 [C.sub.31] Cost saving
[C.sub.32] Flexibility
in billing
[D.sub.4] Risk 0.259 2 [C.sub.41] Labor union
[C.sub.42] Loss of
management control
[C.sub.43] Information
security
Dimensions Local Rankings Local Rankings Global
Weights Weights Weights
[D.sub.1] 0.306 1 0.367 1 0.112
Compatibility 0.310 3 0.095
0.324 2 0.099
[D.sub.2] 0.231 3 0.281 3 0.065
Quality
0.379 1 0.088
0.340 2 0.079
[D.sub.3] Cost 0.204 4 0.506 1 0.103
0.494 2 0.101
[D.sub.4] Risk 0.259 2 0.327 2 0.085
0.351 1 0.091
0.322 3 0.083
Source: Liou et al. (2012).
Table 5. Fuzzy measure g(x) of each parameter and parameter
combination
Fuzzy Measure g(*)
Supplier Selection (evaluating systems) [lambda] = -0.597,
q = 1.358
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[D.sub.1]) = ({[D.sub.1] ({[D.sub.1]
0.415 [D.sub.2]}) = [D.sub.2],
0.651 [D.sub.3]}) =
0.821
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[D.sub.2]}) = ({[D.sub.1] ({[D.sub.1]
0.314 [D.sub.3]}) = [D.sub.2],
0.624 [D.sub.4]}) =
0.866
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[D.sub.3]}) = ({[D.sub.1] ({[D.sub.1]
0.277 [D.sub.4]}) = [D.sub.3],
0.680 [D.sub.4]}) =
0.844
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[D.sub.4]}) = ({[D.sub.2], ({[D.sub.2],
0.352 [D.sub.3]}) = [D.sub.3],
0.539 [D.sub.4]}) =
0.778
[g.sub.[lambda]]
({[D.sub.2],
[D.sub.4]}) =
0.600
[g.sub.[lambda]]
({[D.sub.3],
[D.sub.4]}) =
0.571
[g.sub.[lambda]] [g.sub.[lambda]]
({[D.sub.1]) = ({[D.sub.1]
0.415 [D.sub.2],
[D.sub.3],
[D.sub.4]}) =1
Compatibility ([D.sub.1]) [lambda] = 0.358, q = 0.900
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.11]}) = ({[C.sub.11], ({[C.sub.11],
0.330 [C.sub.12]}) = [C.sub.12],
0.642 [C.sub.13]}) = 1
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.12]}) = ({[C.sub.12]
0.279 [C.sub.13]}) =
0.656
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.13]}) = ({[C.sub.12]
0.291 [C.sub.13]}) =
0.599
Fuzzy Measure g(*)
Quality ([D.sub.2]) [lambda] = 3.902, q = 0.539
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.21]}) = ({[C.sub.21] ({[C.sub.21],
0.151 [C.sub.22]}) = [C.sub.22],
0.476 [C.sub.23]}) = 1
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.22]}) = ({[C.sub.21]
0.204 [C.sub.23]}) =
0.443
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.23]}) = ({[C.sub.22]
0.183 [C.sub.23]}) =
0.533
Cost ([D.sub.3]) [lambda] = 1.268, q = 0.798
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.31]}) = ({[C.sub.31]
0.403 [C.sub.32]}) = 1
[g.sub.[lambda]]
({[C.sub.33]}) =
0.395
Risk ([D.sub.4]) [lambda] = -0.073, q = 1.025
[g.sub.[lambda]] [g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.41]}) = ([C.sub.41] ({[C.sub.41],
0.336 [C.sub.42]}) = [C.sub.42],
0.687 [C.sub.43]}) = 1
[g.sub.[lambda]] [g.sub.[lambda]]
([C.sub.42]}) = ([C.sub.41]
0.360 [C.sub.43]}) =
0.657
[g.sub.[lambda]] [g.sub.[lambda]]
({[C.sub.43]}) = ([C.sub.42]
0.330 [C.sub.43]}) =
0.681
Source: Liou et al. (2012).
Table 6. Gap ratio values of potential suppliers by Fuzzy Integral
Criteria Weights Alternatives
Local
[A.sub.1] [A.sub.2]
Compatibility ([D.sub.1]) 0.306 0.240 0.179
Relationship ([C.sub.11]) 0.367 0.264 0.208
Flexibility ([C.sub.12]) 0.310 0.214 0.211
Information sharing ([C.sub.13]) 0.324 0.242 0.175
Quality ([D.sub.2]) 0.231 0.286 0.224
Knowledge skills ([C.sub.21]) 0.281 0.280 0.221
Customer satisfaction ([C.sub.22]) 0.379 0.286 0.255
On time rate ([C.sub.23]) 0.340 0.302 0.213
Cost ([D.sub.3]) 0.204 0.242 0.300
Cost saving ([C.sub.31]) 0.506 0.246 0.333
Flexibility in 0.494 0.239 0.278
billing ([C.sub.32])
Risk ([D.sub.4]) 0.259 0.252 0.245
Labor unions ([C.sub.41]) 0.327 0.257 0.292
Loss of management 0.351 0.255 0.208
control ([C.sub.42])
Information security ([C.sub.43]) 0.322 0.242 0.235
Total gap -- 0.359 0.350
(rank) (3) (2)
Criteria Weights
Local
[A.sub.3] [A.sub.4]
Compatibility ([D.sub.1]) 0.306 0.197 0.182
Relationship ([C.sub.11]) 0.367 0.199 0.198
Flexibility ([C.sub.12]) 0.310 0.198 0.176
Information sharing ([C.sub.13]) 0.324 0.194 0.173
Quality ([D.sub.2]) 0.231 0.227 0.227
Knowledge skills ([C.sub.21]) 0.281 0.275 0.224
Customer satisfaction ([C.sub.22]) 0.379 0.227 0.265
On time rate ([C.sub.23]) 0.340 0.213 0.214
Cost ([D.sub.3]) 0.204 0.327 0.339
Cost saving ([C.sub.31]) 0.506 0.313 0.324
Flexibility in 0.494 0.348 0.362
billing ([C.sub.32])
Risk ([D.sub.4]) 0.259 0.227 0.249
Labor unions ([C.sub.41]) 0.327 0.214 0.219
Loss of management 0.351 0.218 0.248
control ([C.sub.42])
Information security ([C.sub.43]) 0.322 0.249 0.278
Total gap -- 0.345 0.361
(rank) (1) (4)
Criteria Weights
Local
[A.sub.5]
Compatibility ([D.sub.1]) 0.306 0.263
Relationship ([C.sub.11]) 0.367 0.268
Flexibility ([C.sub.12]) 0.310 0.264
Information sharing ([C.sub.13]) 0.324 0.258
Quality ([D.sub.2]) 0.231 0.214
Knowledge skills ([C.sub.21]) 0.281 0.214
Customer satisfaction ([C.sub.22]) 0.379 0.203
On time rate ([C.sub.23]) 0.340 0.246
Cost ([D.sub.3]) 0.204 0.268
Cost saving ([C.sub.31]) 0.506 0.267
Flexibility in 0.494 0.269
billing ([C.sub.32])
Risk ([D.sub.4]) 0.259 0.277
Labor unions ([C.sub.41]) 0.327 0.275
Loss of management 0.351 0.288
control ([C.sub.42])
Information security ([C.sub.43]) 0.322 0.268
Total gap -- 0.376
(rank) (5)
Note: For example Alternative [A.sub.1], [D.sub.1]: (0.264-0.242) x
0.330) + (0.242-0.214) x 0.656) + (0.214 x 1) = 0.240,
total ratio gap: (0.286-0.252) x 0.314) + (0.252-0.242) x 0.600) +
(0.242-0.240) x 0.778) + (0.240 x 1) = 0.359
(non-additive). Source: Liou et al. (2012).
