Title: New Frontiers in Fuzzy MCDM for Promoting ValueCreated Business Competitiveness in EEra
1New Frontiers in Fuzzy MCDM for Promoting
Value-CreatedBusiness Competitiveness in E-Era
- Gwo-Hshiung Tzeng
- Distinguished Chair Professor
- Department of Business Administration
- Kainan University
- International Symposium on Management
Engineering, March 11, 2006 - URL http//www.knu.edu.tw/letures
- E-mail ghtzeng_at_mail.knu.edu.tw
ghtzeng_at_cc.nctu.edu.tw - Tel886-3-341-2500 ext.1101
- Fax886-3-341-2456
2New Frontiers of Fuzzy Multiple Attribute
Decision Making
- Chapters of this Book (with Jih-Jeng Huang)
- Fuzzy analytic hierarchy process (FAHP)
- Fuzzy analytic network process (FANP) .
- Fuzzy simple additive weighting (FSAW)
- Fuzzy TOPSIS
- ELECTRE
- PROMETHEE
3New Frontiers of Fuzzy Multiple Attribute
Decision Making
- Fuzzy integral
- Grey relation model
- Rough sets
- Applications
- Structural models
- Interpretive structural modeling (ISM)
- DEMATEL
- Fuzzy cognition maps (FCM)
4Agenda
- Profile of Multiple Criterion Decision Making
- Historical Development of Multiple Objective
Decision Making - Historical Development of Multiple Attribute
Decision Making - Multiple Criterion Decision Making Methods
- Structural Model
- Conclusions
5New Thinking Frame for Social Science Research
Fig.1 Overview of Social Science Research with
MCDM
6Concepts of Systems for Research Methods
7Multiple Attribute Utility Theory with Weights
Access for MCDM
- Weightings
- AHP / ANP Fuzzy
- Entropy Measure
- Fuzzy Integral
- MADM Methods
- SAW
- TOPSIS, VIKOR
- PROMETHEE
- ELECTRE
- Grey Relation Analysis
- Additive Types MAUT
- Non-additive Types MAUT
- Fuzzy Integral
8Data Processing / Statistical and Multivariate
Analysis (1)
Fig.2 Data Process for Knowledge Discovery
9Data Processing / Statistical and Multivariate
Analysis (2)
Fig.3 Data Mining for Intelligent Computation in
Knowledge Economy Era
10Concepts of Smile Curve for RD, Production, and
Marketing
Goal
Aspects
Criteria/Attributes
How restructures for building the relation
structures in MCDM problems? ISM, DEMATEL, Fuzzy
Cognitive Map, AHP?AHP, ANP, Fuzzy Integral
Innovation/Creativity
Customer needs (Multi-attribute)
After 1980
Knowledge-Based Marketing Knowledge-Based Tech
nology Knowledge-Based
RD
Marketing
Before 1980
Production
Value-added
Value-added
Value-added
11Business Competitiveness in E-Era
E-Era
Information/Internet Service Providers
Society
Min negative environment impacts Min ecologicl
impacts
Information platform and Information Flow
...
ERP
Customers
Min price Max quality Max level of
service
Enterprise
Max profit SPiQi-costs(MPWT)
Max competitivity
MRP
...
...
Global Distribution
Distribution in Global
Customers
Suppliers)
Money Flow
For Satisfying Customer Needs
Logistics (Physical Distribution)
DRP (Distribution Requirements Planning
??????
12Data Processing / Statistical and Multivariate
Analysis (3)
Fig.4 Multivariate Statistical Data Analysis
13Data Processing / Statistical and Multivariate
Analysis (4)
Fig.5 Multivariate Statistical Data Analysis
(contd)
14Fig.6 Development of Forecasting Model
15MCDM Overview (1)
- Problems for MCDM are common occurrences in
everyday life corresponding to many fields. MCDM
means the process of determining the best
feasible solution according to the established
criteria (representing different effects). - Practical problems are often characterized by
several noncommensurable and conflicting
(competing) criteria, and there may be no
solution satisfying all criteria simultaneously. - Fuzzy Sets Theory - Zadeh (1965)
- Decision-Making in a Fuzzy Environment Bellman
and Zadeh (1970), Management Science, 17(4)
141-164. - Toward a Generalized Theory of Uncertainty
(GTU) an OutlineZadeh (2005), Information
Science, 172(1)1-40.
16MCDM Overview (2)
- Hwang and Yoon (1981) suggest that the MCDM
problems can classify into two categories - Multiple Objective Decision Making (MODM)
- Multiple Attribute Decision Making (MADM).
- MODM usually be fitted in planning/designing
aspects, which is to achieve the optimal goals by
considering the various interactions within the
given constrains. - MADM usually be applied in evaluating/choosing
facet, which associated with a limited number of
predetermined alternatives.
17Basic Concept of MCDM
- Data sets Crisp Sets/ Fuzzy Sets/ Rough Sets/
Grey Hazy Sets - Fuzzy Number and Linguistic Variables
- Fuzzy Relation / Grey Relation
- Fuzzy Logic Fuzzy Reasoning
- Fuzzy Measures and Fuzzy Integral Possibility
Measures and Necessity Measures Belief Measures
and Plausibility Measures - Fuzzy Synthetic and Assessment.
18Treating Business Problems for Promoting
Competitiveness Could Be Solved by Multiple
Criteria Decision Making
- To develop a new business operations and
management systems, there are always some
objectives to meet. The cost? The schedule? The
performance? - So many objectives and alternatives, to choose a
better alternative is more important for the
engineers and tax payer. - The cost/benefit analysis is a common method for
decision making. What is benefit ? - Cost can be quantified. Time can be quantified by
dollar, sometimes, not always. Engineers
experience and skill? Political consideration?
(from U.S. or French?)
19Treating Business Problems for Promoting
Competitiveness Could Be Solved by Multiple
Criteria Decision Making (Cont.)
- Linguistic variable always exits in criteria.
- A faster, more powerful, stealthier
next-generation air fighter? - Is faster enough?
- faster more powerful?
- faster more powerful stealthier?
- 112 ?
20Fig.7 Historical Development of Multiple
Objective Decision Making
21Some military affairs could be solved by
Multiple Objective Decision Making
- The desire is infinite, the resource is limited.
- Are 8 submarines enough for us? Why not 12? Where
the money , pilots and logistic equipment come
from? - A major weapon system needs budget over NT 1
billion and spans several years. All projects
inherently involve multiple criteria and
uncertainty/vagueness. - The military budget can not be decided until the
Bill signed by the President. (FMS will be
influenced) - A programs first year budget doesnt mean which
will survive next year.
22Treating Business Problems for Promoting
Competitiveness Could Be Solved by Multiple
Criteria Decision Making (Cont.)
- Conflict of resource allocation always exits.
- Budget and human resource are more important
recently. How to allocate these resource? - The assessment of educational efficiency.
- Chung Cheng Institute of Technology
- National Defense management college
- National Defense Medical Center
23Multiple Objective Programming
- In almost all multiobjective programming (MOP)
problems can mathematically be represented as - MOP problems can be solved by using
- Weighting Method
- e-constraint
- SWT (Surrogate Worth Trade-off Method)
(2)
24Compromise solution(1)
Compromise solution method originally proposed by
Yu and Zeleney in 1972.
25Compromise solution(2)
26Compromise solution(3)
- Thinking (normalization)
- Problem
27Fuzzy Goal Programming (1)
- In fuzzy goal programming problems, we can refer
to the concept of TOPSIS for MODM with compromise
solution (Lai et al. 1994) to define the
membership function of fuzzy goal as follows
(3)
28Fuzzy Goal Programming (2)
- We can transfer Eq.(2) to expression method as
Eq.(4)
(4)
29Fuzzy Goal Programming (3)
- we also can employ max-min method to transfer
Eqs.(4) as Eq.(5)
(5)
30Fuzzy Goal and Fuzzy Constraint Programming (1)
- In fuzzy goal and fuzzy constraint programming
problems, it can mathematically be represented as
(6)
31Fuzzy Goal and Fuzzy Constraint Programming (2)
- First, we can define the membership function of
fuzzy goal as follows
(7)
32Fuzzy Goal and Fuzzy Constraint Programming (3)
- Second, we can also define the membership
function of fuzzy constraints as follows
(8)
33Fuzzy Goal and Fuzzy Constraint Programming (4)
- We can transfer Eq.(6) to expression method as
follows
(9)
34Fuzzy Goal and Fuzzy Constraint Programming (5)
- we also can employ max-min method to transfer
Eqs.(9) as follows
(10)
35Fuzzy Multiple Objective Linear Programming (1)
- The fuzzy multiple objectives linear programming
(FMOLP) usually has the following formulation
(11)
36Fuzzy Multiple Objective Linear Programming (2)
- The FMOLP problem (10) can be solved by
transferring it into a crisp MOLP shown as (12)
(12)
37Fuzzy Multiple Objective Linear Programming (3)
- Problem (11) can be solved by fuzzy algorithm
interactively. - For details, see Zimmermann (1978), Lee and Li
(1993). - For applications and extensions along this line
see Sakawa (1993), Sakawa et al. (1995), Shibano
et al. (1996), Shih et al. (1996), Ida and Gen
(1997), and Shih and Lee (1999) and quoted there
in.
38Fuzzy Multiple Objective Linear Programming (4)
- In almost all fuzzy multiobjective programming
(FMOP) problems can mathematically be represented
as
(12)
39Two Phase Approach for Solving FMOLP Problem (1)
- With two phase approach for solving FMOLP
problems, the mathematical programming is
described as follows - where represents binary relation and
defined as follows
-
means or
(13)
40Two Phase Approach for Solving FMOLP Problem (2)
- In first phase, we consider crisp multiple
objective linear programming problems, which
mathematical programming formula as follows
(14)
41Two Phase Approach for Solving FMOLP Problem (3)
- According to Zimmermann (1978), there are two
important relation between and - (1) Optimal level of and , that is
- (2) Having trade-off relation between and
- Then the mathematical programming formula become
as follows
(15)
42Two Phase Approach for Solving FMOLP Problem (4)
43Two Phase Approach for Solving FMOLP Problem (5)
- How to find the optimal solution, using iteration
procedure is proved a good approach, when
, then stop it. Furthermore, in second
phase, only to find , such that
44Two Phase Approach for Solving FMOLP Problem (6)
- Lee and Li (1993) proposed algorithm for solving
FMOLP problems as follows - Step 1. Setting tolerable error t, step width
eand initial a-cut (a1.0 ), iterative frequency
t1 - Step 2. Putting aa-te , solve c-LP problem,
then obtainedßand x - Step 3. If a-ß?t, let ?min(a, ß), go to
step 4 - otherwise, go back step 2. If width e is too
large, - let ee/2 and t1, go back step 2
- Step 4. Obtained ?,a,ß, and x end.
45Two Phase Approach for Solving FMOLP Problem (7)
- Furthermore, using first phase a,ß and refer to
algorithm of Lee and Li (1993), solving c-LP2
problems as following mathematical programming,
for more description refer to Ida and Gen (1997).
(16)
46Bi-Level Programming
- Single objective bi-level programming
- Multi-objective bi-level programming
47Multistage Decision Making with Multiple Criteria
(1)
- Yu and Seiford (1981) proposed a general
framework for multicriteria finite stage problems
as follows
48Multistage Decision Making with Multiple Criteria
(2)
- the decision variable is
with each - . The state variables
are generated by - where and that is a set
which specifies the set of alternatives when the
state is reach. - The sequence generated serially by
is a path in the state space.
(17)
49Multistage Decision Making with Multiple Criteria
(3)
- The familiar constraints in mathematical
programming such as
50Goal Programming with Achievement Functions (1)
- Goal programming (GP) is an analytical approach
devised to address decision-making problems where
targets have been assigned to all the attributes
and where the decision-maker is interested in
minimizing the non-achievement of the
corresponding goals (Romero, 2002) - Initially conceived as an application of single
objective linear programming by Charnes and
Cooper (1955, 1961), and then GP gained
popularity in the 1960s and 70s from the works of
Ijiri (1965), Lee (1972), and Ignizio (1976).
51Goal Programming with Achievement Functions (2)
- GP is ideal for criteria with respect to which
target values of achievement are of significance
(Steurer,1986). - Goal programming is distinguished from linear
programming by - (1)The conceptualization of objectives as
goals - (2)The assignment of priorities and/or weights
to the achievement of the goals - (3)The presence of deviational variables and
to measure overachievement and
underachievement from target or (threshold)
levels - (4)The minimization of weighted-sums of
deviational variables to find solutions that best
satisfy the goals.
52Goal Programming with Achievement Functions (3)
- Tamiz and others (1995) show that around 65 of
GP applications reported in the literature use
lexicographic achievement functions, 21 weighted
achievement functions and the rest other types of
achievement functions, such as a MINMAX structure
in which the maximum deviation is minimized.
53Goal Programming with Achievement Functions (4)
- Weighted GP (WGP) Model
- The mathematical programming of a WGP model is
the following (Ignizio 1976)
(18)
54Goal Programming with Achievement Functions (5)
- where
- if is unwanted, otherwise
- if is unwanted, otherwise
. - The parameters and are the weights
reflecting preferential and normalizing purposes
attached to achievement of the i-th goal.
55Goal Programming with Achievement Functions (6)
- Lexicographic GP (LGP) Model
- The achievement function of LGP model is made up
of an ordered vector whose dimension coincides
with the Q number of priority levels established
in the model. - Each component in this vector represents the
unwanted deviation variables of the goals placed
in the corresponding priority level.
56Goal Programming with Achievement Functions (7)
- The mathematical programming of a LGP model is
the following (Ignizio 1976) - where represents the index set of goals
placed in the r-th priority level.
(19)
57Goal Programming with Achievement Functions (8)
- Lexicographic achievement functions imply a
non-compensatory structure of preferences. - In other words, there are no finite trade-offs
among goals placed in different priority levels
(Romero 1991)
58Goal Programming with Achievement Functions (9)
- MINMAX GP (MGP) Model
- The achievement function of a MGP model seeks for
the minimization of the maximum deviation from
any single goal. - If we represent by D this maximum deviation, the
mathematical programming of a LGP model is the
following (Flavell 1976)
(20)
59Goal Programming with Achievement Functions (10)
- MGP implies the optimization of a utility
function where the maximum deviation is
minimized. - MGP provides the most balanced solution among the
achievement of different goals. - In other words, it is the solution of maximum
equity among the achievement of the different
goals. - That is, preferentially MGP solutions represent
the opposite pole with respect to the WGP
solution (Tamiz and others 1998)
60De Novo Programming Method (1)
- Dealing with a multiple criteria optimization
problem, we usually confront a situation that is
almost impossible to optimize all criteria in a
given system. This property is so-called
trade-offs. - Zeleny (1981,1986) suggested that trade-offs are
properties of inadequately designed system and
thus can be eliminated through designing better,
preferably optimal system.
61De Novo Programming Method (2)
- Zeleny (1995) proposed the concept of optimal
portfolio of resources which is design of system
resources in the sense of integration, i.e. the
levels of individual resources are not determined
separately, so that there are no trade-offs in a
new designed system. - Zeleny developed a De Novo programming for
designing optimal system by reshaping the
feasible set.
62De Novo Programming Method (3)
- Zeleny suggested an optimum-path ratio to
contract the budget to available budget along the
optimal path. - Shi (1995) discussed different budgets from
different point of views and define six type
optimum-path ratios to find alternatives for
optimal system design. - No matter what optimum-path ratio is used, it
only can provide a certain path to locate a
solution in the decision space of the new system.
63De Novo Programming Method (4)
- A multicriteria problem can be described as
follows (Yu, 1985) - where and are
matrices, - ,and
.
(21)
64De Novo Programming Method (5)
Graph Example max f1 profit max f2
quality Reshaping the feasible set in order to
include missing g alternative
Fig.8 Given design with natural quality profit
trade-offs
65De Novo Programming Method (6)
- Let the k-th row of C be denoted by
- , so
that , is the kth criteria or objective
function (k1,,q). - the ideal point of Eq.(21) is
, where for
k1,,q. - If there exists
, such that - , then
the called the ideal solution.
66De Novo Programming Method (7)
- When the purpose is to design an optimal system
rather than optimize a given system, it is of
interest to consider following problem - where ,
and
present the unit prices of resources and total
available budget respectively. We can call this
kind of problem as a multi-criteria optimal
system design (MOSD) problem.
(22)
67De Novo Programming Method (8)
- The synthetic solution for MOSD problem
- If we consider each objective function
separately, then Eq.(20) can be written as
follows
(23)
68De Novo Programming Method (9)
- Problem (21) is a continuous Knapsack problem,
and the solution is - where
69De Novo Programming Method (10)
- By the definition of ideal point of the ordinary
system, , if the number of criteria is less than
that of variables, we can individually solve the
problem and obtain synthetic solutions as
follows - Shi (1995) further defined the synthetic optimal
solution as follows, is the optimal
solution of Eq.(20).
70De Novo Programming Method (11)
- A simple production problem involving two
products suits and dresses, in quantities
and , each of them consuming five different
resources (unit market prices of resources are
given). The data are summarized as following
71De Novo Programming Method (12)
- The costs of the given resources portfolio
- Unit costs of producing one unit of each of the
two products - Expected profit margins (price-cost) are
72De Novo Programming Method (13)
- Maximizing total value of function f1
- Maximizing total quality index f2
73De Novo Programming Method (14)
- Maximizing levels of two products can be
calculated by mathematical programming - Maximum f1 in profit
- Maximum f2 in total quality index
74De Novo Programming Method (15)
- Minimizing the total cost by considering the
following constraints - Maximum f1 in profit
- Maximum f2 in total quality index
- Cost of the newly designed system
75De Novo Programming Method (16)
- The new portfolio of resources proposed by the
consultant is as following
76De Novo Programming Method (17)
- 8.2 A meta-optimal of MOSD problem
- Zeleny (1986) proposed this method to locate a
solution is to solve a meta-optimal problem,
which mathematic programming is shown as - Let an optimal solution of (21) is denoted by
. - Eq.(22) means that minimizing the needed budget
under the constraints of attaining the ideal
point of objective function.
(24)
77De Novo Programming Method (18)
- 8.3 A flexible-constraint meta-optimal of
- MOSD problem
- Shi (1995) proposed a flexible-constraint
meta-optimal problem, which replaces the equality
constrains in Eq.(22) with the inequality
constraints
(25)
78De Novo Programming Method (19)
- Let an optimal solution of above problem is
denoted by . Problem (23) is to minimize
the needed budget with the constraints that the
objective functions must be better than the ideal
point. - Comparing problem (22) with problem (23), we can
notice that the feasible region of problem (23)
is larger than that of (22). Hence, there should
be no doubt that the optimal solution of (23) is
better or equal to that of (22).
79De Novo Programming Method (20)
- All the above methods can locate a solution
however, we still need to check whether the
needed budgets are less than the total available
budget. If so, then we can say it is a solution
for a MOSD problem. - A synthetic-optimal budget, meta-optimal budget,
and flexible-constraint meta-optimal budget, are
defined as follows, respectively.
80De Novo Programming Method (21)
- Shi (1995) gave a theorem for stating the
relations among these budgets, the theorem states
that
81DEA Methods for Assessment of Efficiency (1)
- DEA measures efficiency by estimating an
empirical production function which represents
the highest values of outputs that could be
generated by relevant inputs, as obtained from
observed input-output vectors for the analyzed
DMU. - The inefficiency of a DMU is then measured the
distance from the point representing its input
and output values to the corresponding reference
point.
82DEA Methods for Assessment of Efficiency (2)
- Common notation used in the follow-up is
summarized below. - Indices
- k DMUs, k1,,n
- i inputs, i1,,r
- j outputs, j1,,s
- Data
- - the value of i-th input for the k-th
DMU - - the value of j-th output for the k-th
DMU - - a small positive number called
- non-Archimedean quantity
83DEA Methods for Assessment of Efficiency (3)
- Variables
- - slacks corresponding to input i, output j
- respectively
- - virtual multipliers for input i, output j
- respectively
- - weight of in the facet for the
- evaluated DMU
- - relative efficiency of .
84DEA Methods for Assessment of Efficiency (4)
- The CCR Model
- This model proposed by Charnes A., Cooper W.W.
and Rhodes E. (1978), according to their model,
for each DMUk solve
(26)
85DEA Methods for Assessment of Efficiency (5)
- The dual program solves for each as
(27)
86DEA Methods for Assessment of Efficiency (6)
(28)
87DEA Methods for Assessment of Efficiency (7)
- The corresponding primal has a slightly different
objective from Eq.(26)
(29)
88DEA Methods for Assessment of Efficiency (8)
- Fuzzy DEA Model
- The value of output identifying by triangular
fuzzy number, the mathematics programming of
fuzzy DEA based on CCR model described as
(30)
89DEA Methods for Assessment of Efficiency (9)
- This optimal value of objective function
expressed by triangular fuzzy number, conducting
the fuzzy objective value as follows for with
90DEA Methods for Assessment of Efficiency (10)
- On the other hand, we conduct the fuzzy objective
value as follows for with
91DEA Methods for Assessment of Efficiency (11)
- The mathematics programming of DEA with fuzzy
output based on CCR model can then described as
(31-1)
92DEA Methods for Assessment of Efficiency (12)
(31-2)
93DEA Methods for Assessment of Efficiency (13)
- Because this programming is one kind of multiple
objective linear programming, using max-min
concept can then transfer as follows formulation
(32-1)
94DEA Methods for Assessment of Efficiency (14)
(32-2)
- for more detail referring to Tanino (1995) and
Tanaka (2001)
95Fig.9 Development of Multiple Attribute Decision
Making
96Some military affairs could be solved by
Multiple Attribute Decision Making
- According to the Government procurement law
Article 18 Invitation to Tender - The tendering procedures for procurement include
open tendering procedures, selective tendering
procedures, and limited tendering procedures. - The procurement personnel should base on the
consideration of public interest, procurement
efficiency or professional judgment to make an
appropriate procurement decision and not contrary
to the provisions of this Law. - How to judge all criteria and make a right
decision?
97Basic Concept for MADM (1)
- MADM basically comprise two phases (Dubois and
Prade 1980) - Phase 1 to aggregate the performance score with
respect to each alternative/strategy - Phase 2 to rank all alternatives/strategies
according to their synthetic value (or utility
value) from Phase 1. - The hierarchical process of MADM
- Step1. Defining the nature of problem
- Step 2. Building a hierarchy system for
evaluating (Fig. 9)
98Basic Concept for MADM (2)
- Step 3. Selecting the appropriate evaluating
method - Step 4. Determining the relative weights and
performance score of each attribute with respect
to each alternative, both which data may be in
crisp and/or fuzzy - Step 5. Calculating the synthetic utility values,
which are the aggregation value of relative
weights and performance scores corresponding to
alternatives - Step 6. Outranking the alternatives refer to
their synthetic utility values from Step. 5
99Fig.10 Hierarchy System for Multiple
Attribute Decision-Making
100AHP (1)
- AHP developed by Saaty (1977)
- The procedure for AHP
- Step 1. Set up the hierarchy system by
decomposing the problem into a hierarchy of
interrelated elements. - Step 2. Generate input data consisting of
pairwise comparison matrix to find the
comparative weight among the attribute of the
decision elements. - Step 3. Synthesize the individual subjective
judgment and estimate the relative weight. - Step 4. Determine the aggregating relative
weights of the decision elements to arrive at a
set of ratings for the decision
alternatives/strategies.
101AHP (2)
- If we wish to compare a set of n criteria
pairwise according to their relative importance
(weights), then denote the criteria by
C1,C2,,Cn and their weights by w1,w2,,wn. If
w (w1,w2,,wn)T is given, the pairwise
comparisons may be represented by matrix A of the
following formulation - (A-?maxI) w 0
(33)
- Eq.(33) denotes that A is the positive reciprocal
matrix of pairwise comparison values derived by
intuitive judgment for ranking order.
102AHP (3)
- In order to derive the priority eigenvector, we
must find the eigenvector w with respective ?max
which satisfies Aw ?maxw. - Saaty suggested the consistency index (C.I.
(?maxn)/(n-1)) to test the consistency of the
intuitive judgment. - In general, a value of C.I. is less than 0.1 is
satisfactory.
103Entropy measure (1)
- Entropy measure originally introduced by Shannon
Weaver (1949), defined entropy measure - where k is positive constant, and pj
satisfied - Entropy in information theory is a measure of
uncertainty of certain message in evaluating
system.
(34)
(35)
104Entropy measure (2)
- Procedure (Hwang Yoon, 1981)
- 1. Let the decision matrix D of m alternatives
and n attributes (criteria) be - 2. The project outcomes of attribute j can be
defined as
(36)
(37)
105Entropy measure (3)
- 3. The entropy value Ej of the set of project
outcomes of attribute j is -
- where and guarantees that
- 4. Calculate the degree of diversification dj
(38)
(39)
106Entropy measure (4)
- 5. If the DM has no reason to prefer one
criterion over another, the principle of
insufficient reason suggested that each one
should be equally preferred. Then the best weight
set he can expect, instead of the equal weight,
is -
- 6. If the DM has a prior, subjective weight ?j,
then this can be adapted with the help of wj
information. The new weight wj0 is
(40)
(41)
107Simple Additive Weighting method (1)
- SAW method is a probably the best known and very
widely used method of MADM. - To each of attributes in SAW, the DM assigns
importance weights which become the coefficients
of the variables. - To reflect the DMs marginal worth assessments
within attributes, the DM also make a numerical
scaling of intra-attribute value. - The DM can then obtain a total score for each
alternative simply by multiplying the scale
rating for each attribute value by the importance
weight assigned to the attribute and then summing
these products over all attributes.
108Simple Additive Weighting method (2)
- Suppose the DM assigns a set of importance weight
to the attributes, - .Then the most
preferred alternative, A, is selected such that - Usually, the weights are normalized so that
109TOPSIS (1)
- TOPSIS (Technique for Order Preference by
Similarity to Ideal Solution) developed by
Hwang and Yoon (1981). - Basic principle The chosen alternative should
have the shortest distance from the Positive
Ideal Solution (PIS) and the farthest distance
from the Negative Ideal Solution (NIS).
110TOPSIS (2)
- Generally, the global criteria method measures
the distance by using Minkowskis Lp-metric. - when p increase, distance dp decrease, i.e.
(42)
111TOPSIS (3)
- p1 implies are equal weights for all these
deviations - p2 implies that these deviations are weighted
proportionately with the largest deviation having
the largest weight - for p8, the largest deviation completely
dominates the distance determination.
(43)
112TOPSIS (4)
- Considering the incommensurability nature among
objectives or criteria, Yu and Zeleny (1975)
normalized the distance family of Eq.(42) to
remove the effects of the incommensurability by
using the reference point.
(44)
113TOPSIS (5)
- Lai et al., (1994) extended the concept of TOPSIS
by Hwang and Yoon (1981) to develop a methodology
for solving MODM problems. - where is the PIS , is the NIS, and
p1,2,, 8. - The value chosen for p reflects the way of
achieving a compromise by minimizing the weighted
sum of the deviations of criteria from their
respective reference points.
(45)
114TOPSIS (6)
- With the concept of optimal compromise solution,
the best alternative or decisions are those that
have - the shortest distance from PIS
- the farthest distance from NIS.
- Sometimes the compromise solution based on PIS is
not identical to that which is based on NIS they
divided the objectives by which characteristic
as follows
115TOPSIS (7)
- they obtained the distance functions as follows
- where
(46)
(47)
116TOPSIS (8)
Fig.11 TOPSIS method for two dimensional case
117TOPSIS (9)
- They transfer problem of Eq.(45) into the
following bi-objective problem with two
commensurable objectives to obtain the compromise
solution
(48)
where
118TOPSIS (10)
- They further utilized the membership functions of
fuzzy sets theory representing these two
objective functions as follows
(49)
119TOPSIS (11)
(50)
where
120TOPSIS (12)
- They also resolved (48) by using the max-min
operation proposed by Bellman and Zadeh (1970)
and applied by Zimmermann (1978). - The satisfying decision, x , be obtained by
solving the following problem
(51)
121TOPSIS (13)
- Let and generally the
value of are selected by subjective from DM
in practice. - We will have the following equivalent model of
(11) that if giving the same values of - where is the satisfactory level for both
criteria of the shortest distance from PIS and
the farthest distance from the NIS.
(52)
122TOPSIS (14)
- Evaluating procedure (Hwang Yoon, 1981)
- Step 1. Construct the normalized decision matrix
- Step 2. Construct the weighted normalized
decision matrix - Step 3. Determin the ideal solution and
negative-ideal solution - Step 4. Calculate the separation measure
- Step 5. Calculate the relative closeness to the
ideal solution - Step 6. Rank the preference order.
123VIKOR (1)
- Considering the relative importance of
alternatives in using compromise solution method
for MCDM problems, the Minkowskis Lp-metric can
be expressed as follows - where fij is the value of i-th criterion
function for the j-th alternative, fi and fi-
are the best value and the worst value of i-th
criterion function, respectively
(53)
124VIKOR (2)
- The VIKOR developed by S. Opricovic (1998), which
method is based on the aggregated function Lp,j
,which has following steps - 1. Determination of the best value fi and
the worst value - fi- of all criterion functions, that is,
for criterions - i1,,n, we have
- 2. Compute the values Sj and Rj for j1,,J
which defined as follows
(54)
125VIKOR (3)
(55)
- 3. Compute the values Qj for for j1,,J which
defined as - where
- 4. Rank the alternatives, sorting by the values
S,R and Q. The results are three ranking lists. - 5. For given criteria weights, the alternative
which is the best ranked by the measure Q
if the following two conditions are satisfied
126VIKOR (4)
- C1. Acceptable advantage
, where - is the alternative with second position
in the ranking list - by Q, and for J
is the number of - alternatives. Specifically,
- C2. Acceptable stability in decision making
the - alternative has to the best ranked
by S or by R, or - both, as well. This compromise solution
is stable within - decision-making process, which could be
voting by - majority rule when vgt0.5 is needed or
by consensus - v?0.5 or with veto vlt0.5 . v is the
weight of decision - making strategy with the majority of
criteria.
127ELECTRE (1)
- Benayoun et al.(1966) were originally used the
concept of outranking relation to introduce the
ELECTRE (Elimination et Choice Translating
Reality) method. - ELECTRE models based on the nature of problem
statement (to find kernel solution or to rank
order of alternatives), the degree of
significance of criteria which take in account
(true or pseudo), and the preferential
information (weights, concordance index,
discordance index, veto effect).
128ELECTRE(2)
- ELECTRE I model firstly developed by Roy (1968)
to find the kernel solution under situation of
true criteria and restricted outranking relation
be given, this method cannot derive the ranking
of alternatives. - Then Roy proposed ELECTRE IS method to reform the
drawback of kernel consistency for ELECTRE I
method. - Roy and Bertier (1973) developed ELECTRE II to
find the partial outranking of possible
alternatives for situation of true criteria
given.
129ELECTRE (3)
- Moscarola and Roy (1977) developed ELECTRE A
method to solve some specific problems in the
banking sector. - Roy (1977 1978) developed ELECTRE III which
extending the crisp outranking relations for
modeling decision makers preferences to fuzzy
condition. - Roy and Bouyssou (1983) proposed ELECTRE IV to
simplify procedure of ELECTRE III model.
130ELECTRE (4)
- ELECTRE IV method use pseudo-criteria as in
ELECTRE III. The basic difference between the two
methods is that no weights for the criteria are
introduced in ELECTRE IV in case of which
parameter is difficult measure by objectively in
practice. - The exploiting ranking procedure used in ELECTRE
III is generally by following steps (Vincke,1992)
- Step 1. Construct a complete preorder by
descending distillation procedure. - Step 2. Construct a complete preorder by
ascending distillation procedure. - Step 3. Construct the partial preorder as the
final result.
131ELECTRE (5)
Fig.12 General structure of ELECTRE III
132PROMETHEE (1)
- PROMETHEE (Preference Ranking Organization Method
for Enrichment Evaluations ) proposed by Brans,
Mareschal and Vincke (1984 ) for solving
multi-criteria decision-making problems as
follows - where is a set of
possible actions (or alternatives) and
is a set of considered
criteria, represents performance of
action with respect to the j-th criterion.
(56)
133PROMETHEE (2)
- If for a given pair of alternatives a and b have
- for and
at least one inequality is strict, then a
dominates b. - PROMETHEE methods belong to the outranking
methods consisting in enriching the dominance
order. They include three phases - 1. Construction of generalized criteria
- 2. Determination of an outranking relation on A
- 3. Evaluation of this relation in order to given
an answer.
134PROMETHEE (3)
Table 1 Generalized criteria (Brans et al. 1984)
135PROMETHEE (4)
- Given the performance matrix
136PROMETHEE (5)
(57)
(58)
137PROMETHEE (6)
- In PROMETHEE methods, the higher the leaving flow
and the lower entering flow, the better the
alternative. The leaving and entering flow induce
respectively the following preorder on
alternatives on A
(59)
(60)
138PROMETHEE (7)
139PROMETHEE (8)
- PROMETHEE II
- PROMETHEE III associates to each action a, an
interval and define a complete
interval order as follows
140PROMETHEE (9)
- PROMETHEE IV extends PROMETHEE II to the case of
a continuous set of actions A, such a set arises
when the actions are, for instance, percentages,
dimensions of a product, compositions of an
alloy, investments, and so on. - Besides, the leaving flow, the entering flow, and
the net flow for continuous set A are defined as
follows
141Fuzzy Measures (1)
142Fuzzy Measures (2)
- General fuzzy measure
- Basic idea
143Fuzzy Integral (1)
- Let h be a measurable set function defined on the
fuzzy measurable space ,and suppose that
, then the
fuzzy integral of fuzzy measure with
respect to can be defined as follows
(Ishii Sugeno,1985)
(1)
144Fuzzy Integral (2)
- where
- In addition, if and
then - is
not necessary.
145Concepts of Smile Curve for RD, Production, and
Marketing
Goal
Aspects
Criteria/Attributes
How restructures for building the relation
structures in MCDM problems? (ISM, DEMATEL, Fuzzy
cognitive map,)
146Building Criteria/Features Structure Relations
for Evaluation-Weightings
- 1.Methods for Building Criteria/Features
Structure Structure Model - - Linear Structure Model Path Analysis
(1900s), Cause Relation (1960s), Linear Structure
Relation (1990s) - - - ISM (Interpretive Structure Modeling)
- - - DEMATEL (decision-making trial and
evaluation laboratory)
147- 2. Kinds of Criteria/Features Structure Relations
for Evaluation-Weightings - (1) Independent Relations in Criteria/FeaturesAHP
(Analytic Hierarchy Process) - (2) Dependent Relations in Criteria/Features
- - Feedback relationsANP (Analytic Network
Professor) - - Interdependent relationsFuzzy Integral
148ISM (Interpretive Structure Modeling)
149DEMATEL (decision-making trial and evaluation
laboratory)
150FCM (Fuzzy Cognition Maps )
- Given a 4-tuple ( ) where
denotes the set of n objects, denotes the
connection matrix which is composed of the
weights between objects, is the state matrix,
where is the initial matrix and is the
state matrix at certain iteration t, and f is a
threshold function, which indicates the weighting
relationship between and .
151- Several formulas have been used as threshold
functions such as
(Hard line function)
(Hyperbolic-tangent function)
(Logistic function)
152- The influence of the specific criterion to other
criteria can be calculated using the following
updating equation - ,
- where denotes the identity matrix.
,
The vector-matrix multiplication operation to
derive successive FCM states is iterated until it
converges to a fixed point situation or a limit
state cycle.
153Conclusions
- Social science research have to face the real
world situations. - MCDM methods with intelligent techniques and
habitual domain expansion make people to cope
with problems correctly and fast. - Evolutionary soft computation with genetic
algorithms are powerful and widely used
stochastic optimization techniques which will be
the main stream for solving MCDM problems. - The more knowledge you possess, the more
advantage and success will be.