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Title: New Frontiers in Fuzzy MCDM for Promoting ValueCreated Business Competitiveness in EEra


1
New 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

2
New 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

3
New 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)

4
Agenda
  • 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

5
New Thinking Frame for Social Science Research
Fig.1 Overview of Social Science Research with
MCDM
6
Concepts of Systems for Research Methods
7
Multiple 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

8
Data Processing / Statistical and Multivariate
Analysis (1)
Fig.2 Data Process for Knowledge Discovery
9
Data Processing / Statistical and Multivariate
Analysis (2)
Fig.3 Data Mining for Intelligent Computation in
Knowledge Economy Era
10
Concepts 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
11
Business 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
??????
12
Data Processing / Statistical and Multivariate
Analysis (3)
Fig.4 Multivariate Statistical Data Analysis
13
Data Processing / Statistical and Multivariate
Analysis (4)
Fig.5 Multivariate Statistical Data Analysis
(contd)
14
Fig.6 Development of Forecasting Model
15
MCDM 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.

16
MCDM 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.

17
Basic 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.

18
Treating 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?)

19
Treating 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 ?

20
Fig.7 Historical Development of Multiple
Objective Decision Making
21
Some 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.

22
Treating 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

23
Multiple 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)
24
Compromise solution(1)
Compromise solution method originally proposed by
Yu and Zeleney in 1972.
25
Compromise solution(2)
  • Basic thinking
  • Problem

26
Compromise solution(3)
  • Thinking (normalization)
  • Problem

27
Fuzzy 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)
28
Fuzzy Goal Programming (2)
  • We can transfer Eq.(2) to expression method as
    Eq.(4)

(4)
29
Fuzzy Goal Programming (3)
  • we also can employ max-min method to transfer
    Eqs.(4) as Eq.(5)

(5)
30
Fuzzy Goal and Fuzzy Constraint Programming (1)
  • In fuzzy goal and fuzzy constraint programming
    problems, it can mathematically be represented as

(6)
31
Fuzzy Goal and Fuzzy Constraint Programming (2)
  • First, we can define the membership function of
    fuzzy goal as follows

(7)
32
Fuzzy Goal and Fuzzy Constraint Programming (3)
  • Second, we can also define the membership
    function of fuzzy constraints as follows

(8)
33
Fuzzy Goal and Fuzzy Constraint Programming (4)
  • We can transfer Eq.(6) to expression method as
    follows

(9)
34
Fuzzy Goal and Fuzzy Constraint Programming (5)
  • we also can employ max-min method to transfer
    Eqs.(9) as follows

(10)
35
Fuzzy Multiple Objective Linear Programming (1)
  • The fuzzy multiple objectives linear programming
    (FMOLP) usually has the following formulation

(11)
36
Fuzzy Multiple Objective Linear Programming (2)
  • The FMOLP problem (10) can be solved by
    transferring it into a crisp MOLP shown as (12)

(12)
37
Fuzzy 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.

38
Fuzzy Multiple Objective Linear Programming (4)
  • In almost all fuzzy multiobjective programming
    (FMOP) problems can mathematically be represented
    as

(12)
39
Two 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)
40
Two 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)
41
Two 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)
42
Two Phase Approach for Solving FMOLP Problem (4)
  • where

43
Two 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

44
Two 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.

45
Two 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)
46
Bi-Level Programming
  • Single objective bi-level programming
  • Multi-objective bi-level programming

47
Multistage Decision Making with Multiple Criteria
(1)
  • Yu and Seiford (1981) proposed a general
    framework for multicriteria finite stage problems
    as follows

48
Multistage 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)
49
Multistage Decision Making with Multiple Criteria
(3)
  • The familiar constraints in mathematical
    programming such as

50
Goal 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).

51
Goal 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.

52
Goal 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.

53
Goal Programming with Achievement Functions (4)
  • Weighted GP (WGP) Model
  • The mathematical programming of a WGP model is
    the following (Ignizio 1976)

(18)
54
Goal 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.

55
Goal 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.

56
Goal 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)
57
Goal 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)

58
Goal 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)
59
Goal 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)

60
De 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.

61
De 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.

62
De 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.

63
De Novo Programming Method (4)
  • A multicriteria problem can be described as
    follows (Yu, 1985)
  • where and are
    matrices,
  • ,and
    .

(21)
64
De 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
65
De 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.

66
De 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)
67
De 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)
68
De Novo Programming Method (9)
  • Problem (21) is a continuous Knapsack problem,
    and the solution is
  • where

69
De 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).

70
De 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

71
De 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

72
De Novo Programming Method (13)
  • Maximizing total value of function f1
  • Maximizing total quality index f2

73
De 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

74
De 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

75
De Novo Programming Method (16)
  • The new portfolio of resources proposed by the
    consultant is as following

76
De 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)
77
De 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)
78
De 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).

79
De 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.

80
De Novo Programming Method (21)
  • Shi (1995) gave a theorem for stating the
    relations among these budgets, the theorem states
    that

81
DEA 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.

82
DEA 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

83
DEA 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 .

84
DEA 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)
85
DEA Methods for Assessment of Efficiency (5)
  • The dual program solves for each as

(27)
86
DEA Methods for Assessment of Efficiency (6)
  • The BCC Model

(28)
87
DEA Methods for Assessment of Efficiency (7)
  • The corresponding primal has a slightly different
    objective from Eq.(26)

(29)
88
DEA 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)
89
DEA 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

90
DEA Methods for Assessment of Efficiency (10)
  • On the other hand, we conduct the fuzzy objective
    value as follows for with

91
DEA Methods for Assessment of Efficiency (11)
  • The mathematics programming of DEA with fuzzy
    output based on CCR model can then described as

(31-1)
92
DEA Methods for Assessment of Efficiency (12)
(31-2)
93
DEA 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)
94
DEA Methods for Assessment of Efficiency (14)
(32-2)
  • for more detail referring to Tanino (1995) and
    Tanaka (2001)

95
Fig.9 Development of Multiple Attribute Decision
Making
96
Some 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?

97
Basic 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)

98
Basic 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

99
Fig.10 Hierarchy System for Multiple
Attribute Decision-Making
100
AHP (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.

101
AHP (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.

102
AHP (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.

103
Entropy 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)
104
Entropy 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)
105
Entropy 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)
106
Entropy 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)
107
Simple 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.

108
Simple 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

109
TOPSIS (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).

110
TOPSIS (2)
  • Generally, the global criteria method measures
    the distance by using Minkowskis Lp-metric.
  • when p increase, distance dp decrease, i.e.

(42)
111
TOPSIS (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)
112
TOPSIS (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)
113
TOPSIS (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)
114
TOPSIS (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

115
TOPSIS (7)
  • they obtained the distance functions as follows
  • where

(46)
(47)
116
TOPSIS (8)
Fig.11 TOPSIS method for two dimensional case
117
TOPSIS (9)
  • They transfer problem of Eq.(45) into the
    following bi-objective problem with two
    commensurable objectives to obtain the compromise
    solution

(48)
where
118
TOPSIS (10)
  • They further utilized the membership functions of
    fuzzy sets theory representing these two
    objective functions as follows

(49)
119
TOPSIS (11)
(50)
where
120
TOPSIS (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)
121
TOPSIS (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)
122
TOPSIS (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.

123
VIKOR (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)
124
VIKOR (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)
125
VIKOR (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

126
VIKOR (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.

127
ELECTRE (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).

128
ELECTRE(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.

129
ELECTRE (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.

130
ELECTRE (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.

131
ELECTRE (5)
Fig.12 General structure of ELECTRE III
132
PROMETHEE (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)
133
PROMETHEE (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.

134
PROMETHEE (3)
Table 1 Generalized criteria (Brans et al. 1984)
135
PROMETHEE (4)
  • Given the performance matrix

136
PROMETHEE (5)
  • Idea

(57)
(58)
137
PROMETHEE (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)
138
PROMETHEE (7)
  • PROMETHEE I

139
PROMETHEE (8)
  • PROMETHEE II
  • PROMETHEE III associates to each action a, an
    interval and define a complete
    interval order as follows

140
PROMETHEE (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

141
Fuzzy Measures (1)
142
Fuzzy Measures (2)
  • General fuzzy measure
  • Basic idea

143
Fuzzy 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)
144
Fuzzy Integral (2)
  • where
  • In addition, if and
    then
  • is
    not necessary.

145
Concepts 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,)
146
Building 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

148
ISM (Interpretive Structure Modeling)
 
 
149
DEMATEL (decision-making trial and evaluation
laboratory)
  • Digraph of the DEMATEL

150
FCM (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.
153
Conclusions
  • 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.
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