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Title: Logistics Decision Analysis Methods


1
Logistics Decision Analysis Methods
  • Analytic Hierarchy Process
  • Presented by Tsan-hwan Lin
  • E-mail percy_at_ccms.nkfust.edu.tw

2
Motivation - 1
  • In our complex world system, we are forced to
    cope with more problems than we have the
    resources to handle.
  • What we need is not a more complicated way of
    thinking but a framework that will enable us to
    think of complex problems in a simple way.
  • The AHP provides such a framework that enables us
    to make effective decisions on complex issues by
    simplifying and expediting our natural
    decision-making processes.

3
Motivation - 2
  • Humans are not often logical creatures.
  • Most of the time we base our judgments on hazy
    impressions of reality and then use logic to
    defend our conclusions.
  • The AHP organizes feelings, intuition, and logic
    in a structured approach to decision making.

4
Motivation - 3
  • There are two fundamental approaches to solving
    problems the deductive approach(???)and the
    inductive (???or systems) approach.
  • Basically, the deductive approach focuses on the
    parts whereas the systems approach concentrates
    on the workings of the whole.
  • The AHP combines these two approaches into one
    integrated, logic framework.

5
Introduction - 1
  • The analytic hierarchy process (AHP) was
    developed by Thomas L. Saaty.
  • Saaty, T.L., The Analytic Hierarchy Process, New
    York McGraw-Hill, 1980
  • The AHP is designed to solve complex problems
    involving multiple criteria.
  • An advantage of the AHP is that it is designed to
    handle situations in which the subjective
    judgments of individuals constitute an important
    part of the decision process.

6
Introduction - 2
  • Basically the AHP is a method of (1) breaking
    down a complex, unstructured situation into its
    component parts (2) arranging these parts, or
    variables into a hierarchic order (3) assigning
    numerical values to subjective judgments on the
    relative importance of each variable and (4)
    synthesizing the judgments to determine which
    variables have the highest priority and should be
    acted upon to influence the outcome of the
    situation.

7
Introduction - 3
  • The process requires the decision maker to
    provide judgments about the relative importance
    of each criterion and then specify a preference
    for each decision alternative on each criterion.
  • The output of the AHP is a prioritized ranking
    indicating the overall preference for each of the
    decision alternatives.

8
Major Steps of AHP
  • 1) To develop a graphical representation of the
    problem in terms of the overall goal, the
    criteria, and the decision alternatives. (i.e.,
    the hierarchy of the problem)
  • 2) To specify his/her judgments about the
    relative importance of each criterion in terms of
    its contribution to the achievement of the
    overall goal.
  • 3) To indicate a preference or priority for each
    decision alternative in terms of how it
    contributes to each criterion.
  • 4) Given the information on relative importance
    and preferences, a mathematical process is used
    to synthesize the information (including
    consistency checking) and provide a priority
    ranking of all alternatives in terms of their
    overall preference.

9
Constructing Hierarchies
  • Hierarchies are a fundamental mind tool
  • Classification of hierarchies
  • Construction of hierarchies

10
Establishing Priorities
  • The need for priorities
  • Setting priorities
  • Synthesis
  • Consistency
  • Interdependence

11
Advantages of the AHP
The AHP provides a single, easily understood,
flexible model for a wide range of unstructured
problems
The AHP enables people to refine their definition
of a problem and to improve their judgment and
understanding through repetition
The AHP integrates deductive and systems
approaches in solving complex problems
The AHP does not insist on consensus but
synthesizes a representative outcome from diverse
judgments
The AHP can deal with the interdependence of
elements in a system and does not insist on
linear thinking
The AHP reflects the natural tendency of the mind
to sort elements of a system into different
levels and to group like elements in each level
The AHP takes into consideration the relative
priorities of factors in a system and enables
people to select the best alternative based on
their goals
The AHP provides a scale for measuring
intangibles and a method for establishing
priorities
The AHP leads to an overall estimate of the
desirability of each alternative
The AHP tracks the logical consistency of
judgments used in determining priorities
12
Q A
13
Hierarchy Development
  • The first step in the AHP is to develop a
    graphical representation of the problem in terms
    of the overall goal, the criteria, and the
    decision alternatives.

14
Pairwise Comparisons
  • Pairwise comparisons are fundamental building
    blocks of the AHP.
  • The AHP employs an underlying scale with values
    from 1 to 9 to rate the relative preferences for
    two items.

15
Pairwise Comparison Matrix
  • Element Ci,j of the matrix is the measure of
    preference of the item in row i when compared to
    the item in column j.
  • AHP assigns a 1 to all elements on the diagonal
    of the pairwise comparison matrix.
  • When we compare any alternative against itself
    (on the criterion) the judgment must be that they
    are equally preferred.
  • AHP obtains the preference rating of Cj,i by
    computing the reciprocal (inverse) of Ci,j (the
    transpose position).
  • The preference value of 2 is interpreted as
    indicating that alternative i is twice as
    preferable as alternative j. Thus, it follows
    that alternative j must be one-half as preferable
    as alternative i.
  • According above rules, the number of entries
    actually filled in by decision makers is (n2
    n)/2, where n is the number of elements to be
    compared.

16
Preference Scale - 1
17
Preference Scale - 2
  • Research and experience have confirmed the
    nine-unit scale as a reasonable basis for
    discriminating between the preferences for two
    items.
  • Even numbers (2, 4, 6, 8) are intermediate values
    for the scale.
  • A value of 1 is reserved for the case where the
    two items are judged to be equally preferred.

18
Synthesis
  • The procedure to estimate the relative priority
    for each decision alternative in terms of the
    criterion is referred to as synthesization(????).
  • Once the matrix of pairwise comparisons has been
    developed, priority(?????????)of each of the
    elements (priority of each alternative on
    specific criterion priority of each criterion on
    overall goal) being compared can be calculated.
  • The exact mathematical procedure required to
    perform synthesization involves the computation
    of eigenvalues and eigenvectors, which is beyond
    the scope of this text.

19
Procedure for Synthesizing Judgments
  • The following three-step procedure provides a
    good approximation of the synthesized priorities.
  • Step 1 Sum the values in each column of the
    pairwise comparison matrix.
  • Step 2 Divide each element in the pairwise
    matrix by its column total.
  • The resulting matrix is referred to as the
    normalized pairwise comparison matrix.
  • Step 3 Compute the average of the elements in
    each row of the normalized matrix.
  • These averages provide an estimate of the
    relative priorities of the elements being
    compared.
  • Example

20
Example Synthesizing Procedure - 0
  • Step 0 Prepare pairwise comparison matrix

Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
21
Example Synthesizing Procedure - 1
  • Step 1 Sum the values in each column.

Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
Column totals 13/8 19/6 15
22
Example Synthesizing Procedure - 2
  • Step 2 Divide each element of the matrix by its
    column total.
  • All columns in the normalized pairwise comparison
    matrix now have a sum of 1.

Comfort Car A Car B Car C
Car A 8/13 12/19 8/15
Car B 4/13 6/19 6/15
Car C 1/13 1/19 1/15
23
Example Synthesizing Procedure - 3
  • Step 3 Average the elements in each row.
  • The values in the normalized pairwise comparison
    matrix have been converted to decimal form.
  • The result is usually represented as the
    (relative) priority vector.

Comfort Car A Car B Car C Row Avg.
Car A 0.615 0.632 0.533 0.593
Car B 0.308 0.316 0.400 0.341
Car C 0.077 0.053 0.067 0.066
Total 1.000
24
Consistency - 1
  • An important consideration in terms of the
    quality of the ultimate decision relates to the
    consistency of judgments that the decision maker
    demonstrated during the series of pairwise
    comparisons.
  • It should be realized perfect consistency is very
    difficult to achieve and that some lack of
    consistency is expected to exist in almost any
    set of pairwise comparisons.
  • Example

25
Consistency - 2
  • To handle the consistency question, the AHP
    provides a method for measuring the degree of
    consistency among the pairwise judgments provided
    by the decision maker.
  • If the degree of consistency is acceptable, the
    decision process can continue.
  • If the degree of consistency is unacceptable, the
    decision maker should reconsider and possibly
    revise the pairwise comparison judgments before
    proceeding with the analysis.

26
Consistency Ratio
  • The AHP provides a measure of the consistency of
    pairwise comparison judgments by computing a
    consistency ratio(?????).
  • The ratio is designed in such a way that values
    of the ratio exceeding 0.10 are indicative of
    inconsistent judgments.
  • Although the exact mathematical computation of
    the consistency ratio is beyond the scope of this
    text, an approximation of the ratio can be
    obtained.

27
Procedure Estimating Consistency Ratio - 1
  • Step 1 Multiply each value in the first column
    of the pairwise comparison matrix by the relative
    priority of the first item considered. Same
    procedures for other items. Sum the values
    across the rows to obtain a vector of values
    labeled weighted sum.
  • Step 2 Divide the elements of the vector of
    weighted sums obtained in Step 1 by the
    corresponding priority value.
  • Step 3 Compute the average of the values
    computed in step 2. This average is denoted as
    lmax.

28
Procedure Estimating Consistency Ratio - 2
  • Step 4 Compute the consistency index (CI)
  • Where n is the number of items being compared
  • Step 5 Compute the consistency ratio (CR)
  • Where RI is the random index, which is the
    consistency index of a randomly generated
    pairwise comparison matrix. It can be shown that
    RI depends on the number of elements being
    compared and takes on the following values.
  • Example

29
Random Index
  • Random index (RI) is the consistency index of a
    randomly generated pairwise comparison matrix.
  • RI depends on the number of elements being
    compared (i.e., size of pairwise comparison
    matrix) and takes on the following values

n 1 2 3 4 5 6 7 8 9 10
RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49
30
Example Inconsistency
  • Preferences If, A ? B (2) B ? C (6)
  • Then, A ? C (should be 12) (actually 8)
  • Inconsistency

Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
31
Example Consistency Checking - 1
  • Step 1 Multiply each value in the first column
    of the pairwise comparison matrix by the relative
    priority of the first item considered. Same
    procedures for other items. Sum the values
    across the rows to obtain a vector of values
    labeled weighted sum.

32
Example Consistency Checking - 2
  • Step 2 Divide the elements of the vector of
    weighted sums by the corresponding priority value.

Step 3 Compute the average of the values
computed in step 2 (lmax).
33
Example Consistency Checking - 3
  • Step 4 Compute the consistency index (CI).

Step 5 Compute the consistency ratio (CR).
  • The degree of consistency exhibited in the
    pairwise comparison matrix for comfort is
    acceptable.

34
Development of Priority Ranking
  • The overall priority for each decision
    alternative is obtained by summing the product of
    the criterion priority (i.e., weight) (with
    respect to the overall goal) times the priority
    (i.e., preference) of the decision alternative
    with respect to that criterion.
  • Ranking these priority values, we will have AHP
    ranking of the decision alternatives.
  • Example

35
Example Priority Ranking 0A
  • Step 0A Other pairwise comparison matrices

Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
Price Car A Car B Car C
Car A 1 1/3 ¼
Car B 3 1 ½
Car C 4 2 1
Criterion Price MPG Comfort Style
Price 1 3 2 2
MPG 1/3 1 1/4 1/4
Comfort 1/2 4 1 1/2
Style 1/2 4 2 1
MPG Car A Car B Car C
Car A 1 1/4 1/6
Car B 4 1 1/3
Car C 6 3 1
Style Car A Car B Car C
Car A 1 1/3 4
Car B 3 1 7
Car C 1/4 1/7 1
36
Example Priority Ranking 0B
  • Step 0B Calculate priority vector for each
    matrix.

Price MPG Comfort Style
Car A Car B Car C
Criterion
Price MPG Comfort Style
37
Example Priority Ranking 1
  • Step 1 Sum the product of the criterion priority
    (with respect to the overall goal) times the
    priority of the decision alternative with respect
    to that criterion.

Step 2 Rank the priority values.
Alternative Priority
Car B 0.421
Car C 0.314
Car A 0.265
Total 1.000
38
Hierarchies A Tool of the Mind
  • Hierarchies are a fundamental tool of the human
    mind.
  • They involve identifying the elements of a
    problem, grouping the elements into homogeneous
    sets, and arranging these sets in different
    levels.
  • Complex systems can best be understood by
    breaking them down into their constituent
    elements, structuring the elements
    hierarchically, and then composing, or
    synthesizing, judgments on the relative
    importance of the elements at each level of the
    hierarchy into a set of overall priorities.

39
Classifying Hierarchies
  • Hierarchies can be divided into two kinds
    structural and functional.
  • In structural hierarchies, complex systems are
    structured into their constituent parts in
    descending order according to structural
    properties (such as size, shape, color, or age).
  • Structural hierarchies relate closely to the way
    our brains analyze complexity by breaking down
    the objects perceived by our senses into
    clusters, subclusters, and still smaller
    clusters. (more descriptive)
  • Functional hierarchies decompose complex systems
    into their constituent parts according to their
    essential relationships.
  • Functional hierarchies help people to steer a
    system toward a desired goal like conflict
    resolution, efficient performance, or overall
    happiness. (more normative)
  • For the purposes of the study, functional
    hierarchies are the only link that need be
    considered.

40
Hierarchy
  • Each set of elements in a functional hierarchy
    occupies a level of the hierarchy.
  • The top level, called the focus, consists of only
    one element the broad, overall objective.
  • Subsequent levels may each have several elements,
    although their number is usually small between
    five and nine.
  • Because the elements in one level are to be
    compared with one another against a criterion in
    the next higher level, the elements in each level
    must be of the same order of magnitude.
    (Homogeneity)
  • To avoid making large errors, we must carry out
    clustering process. By forming hierarchically
    arranged clusters of like elements, we can
    efficiently complete the process of comparing the
    simple with the very complex.
  • Because a hierarchy represents a model of how the
    brain analyzes complexity, the hierarchy must be
    flexible enough to deal with that complexity.

41
Types of Functional Hierarchy
  • Some functional hierarchies are complete, that
    is, all the elements in one level share every
    property in the nest higher level.
  • Some are incomplete in that some elements in a
    level do not share properties.

42
Constructing Hierarchies - 1
  • Ones approach to constructing a hierarchy
    depends on the kind of decision to be made.
  • If it is a matter of choosing among alternatives,
    we could start from the bottom by listing the
    alternatives.
  • (decision alternatives gt criteria gt overall
    goal)
  • Once we construct the hierarchy, we can always
    alter parts of it later to accommodate new
    criteria that we may think of or that we did not
    consider important when we first designed it.
  • (AHP is flexible and time-adaptable)
  • Sometimes the criteria themselves must be
    examined in details, so a level of subcriteria
    should be inserted between those of the criteria
    and the alternatives.

43
Constructing Hierarchies - 2
  • If one is unable to compare the elements of a
    level in terms of the elements of the next higher
    level, one must ask in what terms they can be
    compared and then seek an intermediate level that
    should amount to a breakdown of the elements of
    the next higher level.
  • The basic principle in structuring a hierarchy is
    to see if one can answer the question Can you
    compare the elements in a lower level in terms of
    some all all the elements in the next higher
    level?
  • The depth of detail (in level construction)
    depends on how much knowledge one has about the
    problem and how much can be gained by using that
    knowledge without unnecessarily tiring the mind.
  • The analytic aspects of the AHP serve as a
    stimulus to create new dimensions for the
    hierarchy. It is a process for inducing
    cognitive awareness. A logically constructed
    hierarchy is a by-product of the entire AHP
    approach.

44
Constructing Hierarchies II - 1
  • When constructing hierarchies one must include
    enough relevant detail to depict the problem as
    thoroughly as possible.
  • Consider environment surrounding the problem.
  • Identify the issues or attributes that you feel
    contribute to the solution.
  • Identify the participants associated with the
    problem.
  • Arranging the goals, attributes, issues, and
    stakeholders in a hierarchy serves two purposes
  • It provides an overall view of the complex
    relationships inherent in the situation.
  • It permits the decision maker to assess whether
    he or she is comparing issues of the same order
    of magnitude in weight or impact on the
    solution.(???????????????????????????????????????
    ??)

45
Constructing Hierarchies II - 2
  • The elements should be clustered into homogeneous
    groups of five to nine so they can be
    meaningfully compared to elements in the next
    higher level.
  • The only restriction on the hierarchic
    arrangement of elements is that any element in
    one level must be capable of being related to
    some elements in the next higher level, which
    serves as a criterion for assessing the relative
    impact of elements in the level below.
  • Elements that are of less immediate interest can
    be represented in general terms at the higher
    levels of the hierarchy and elements critical to
    the problem at hand can be developed in greater
    depth and specificity.
  • It is often useful to construct two hierarchies,
    one for benefits and one for costs to decide on
    the best alternative, particularly in the case of
    yes-no decisions.

46
Constructing Hierarchies II - 3
  • Specifically, the AHP can be used for the
    following kinds of decision problems
  • Choosing the best alternatives
  • Generating a set of alternatives
  • Setting priorities
  • Measuring performance
  • Resolving conflicts
  • Allocating resources (Benefit/Cost Analysis)
  • Making group decisions
  • Predicting outcomes and assessing risks
  • Designing a system
  • Ensuring system reliability
  • Determining requirements
  • Optimizing
  • Planning
  • Clearly the design of an analytic hierarchy is
    more art than science. But structuring a
    hierarchy does require substantial knowledge
    about the system or problem in question.

47
Need for Priorities - 1
  • The analytical hierarchy process deals with both
    (inductive and deductive) approaches
    simultaneously.
  • Systems thinking (inductive approach) is
    addressed by structuring ideas hierarchically,
    and causal thinking (deductive approach) is
    developed through paired comparison of the
    elements in the hierarchy and through synthesis.
  • Systems theorists point out that complex
    relationships can always be analyzed by taking
    pairs of elements and relating them through their
    attributes. The object is to find from many
    things those that have a necessary connection.
  • The object of the system approach (,which
    complemented the causal approach) is to find the
    subsystems or dimensions in which the parts are
    connected.

48
Need for Priorities - 2
  • The judgment applied in making paired comparisons
    combine logical thinking with feeling developed
    from informed experience.
  • The mathematical process described (in priority
    development) explains how subjective judgments
    can be quantified and converted into a set of
    priorities on which decisions can be based.

49
Setting Priorities - 1
  • The first step in establishing the priorities of
    elements in a decision problem is to make
    pairwise comparisons, that is, to compare the
    elements in pairs against a given criterion.
  • The (pairwise comparison) matrix is a simple,
    well-established tool that offers a framework for
    1 testing consistency, 2 obtaining additional
    information through making all possible
    comparisons, and 3 analyzing the sensitivity of
    overall priorities to changes in judgment.

50
Setting Priorities - 2
  • To begin the pairwise comparison, start at the
    top of the hierarchy to select the criterion (or,
    goal, property, attribute) C, that will be used
    for making the first comparison. Then, from the
    level immediately below, take the elements to be
    compared A1, A2, A3, and so on.
  • To compare elements, ask How much more strongly
    does this element (or activity) possess (or
    contribute to, dominate, influence, satisfy, or
    benefit) the property than does the element with
    which it is being compared?
  • The phrasing must reflect the proper relationship
    between the elements in one level with the
    property in the next higher level.
  • To fill in the matrix of pairwise comparisons, we
    use numbers to represent the relative importance
    of one element over another with respect to the
    property.

51
Synthesis II
  • To obtain the set of overall priorities for a
    decision problem, we have to pull together or
    synthesize the judgments made in the pairwise
    comparisons, that is, we have to do weighting and
    adding to give us a single number to indicate the
    priority of each element.
  • The procedure is described earlier.

52
Consistency II - 1
  • In decision making problems, it may be important
    to know how good our consistency is, because we
    may not want the decision to be based on
    judgments that have such low consistency that
    they appear to be random.
  • How damaging is inconsistency?
  • Usually we cannot be so certain of our judgments
    that we would insist on forcing consistency in
    the pairwise comparison matrix (except diagonal
    ones).
  • As long as there is enough consistency to
    maintain coherence among the objects of our
    experience, the consistency need not be perfect.
  • When we integrate new experiences into our
    consciousness, previous relationships may change
    and some consistency is lost.
  • It is useful to remember that most new ideas that
    affect our lives tend to cause us to rearrange
    some of our preferences, thus making us
    inconsistent with our previous commitments.

53
Consistency II - 2
  • The AHP measure the overall consistency of
    judgments by means of a consistency ratio.
  • The procedure for determining consistency ratios
    is described earlier.
  • Greater inconsistency indicates lack of
    information or lack of understanding.
  • One way to improve consistency when it turns out
    to be unsatisfactory is to rank the activities by
    a simple order based on the weights obtained in
    the first run of the problem.
  • A second pairwise comparison matrix is then
    developed with this knowledge of ranking in mind.
  • The consistency should generally be
    better.(??????????)

54
Backup Materials
55
Interdependence
  • So far we have considered how to establish the
    priority of elements in a hierarchy and how to
    obtain the set of overall priorities when the
    elements of each level are independent.
  • However, often the elements are interdependent,
    that is, there are overlapping areas or
    commonalities among elements.
  • There are two principal kinds of interdependence
    among elements of a hierarchy level
  • Additive interdependence
  • Synergistic interdependence

56
Additive Interdependence
  • In additive interdependence(??????), each element
    contributes a share that is uniquely its own and
    also contributes indirectly by overlapping or
    interacting with other elements.
  • The total impact can be estimated by 1
    examining the impacts of the independent and the
    overlapping shares and then 2 combining the
    impacts.
  • In practice, most people prefer to ignore the
    rather complex mathematical adjustment for
    additive interdependence and simply rely on their
    own judgment (putting higher priority on those
    elements having more impacts).

BACK
57
Synergistic Interdependence - 1
  • In synergistic interdependence(??????), the
    impact of the interaction of the elements is
    greater than the sum of the impacts of the
    elements, with due consideration given to their
    overlap.
  • This type of interdependence occurs more
    frequently than additive interdependence and
    amounts to creating a new entity for each
    interaction.
  • Much of the problem of synergistic
    interdependence arises from the fuzziness of
    words and even the underlying ideas they
    represent.
  • The qualities that emerge cannot be captured by a
    mathematical process (such as Venn diagrams).
    What we have instead is the overlap of elements
    with other elements to produce an element with
    new priorities that are not discernible in its
    parent parts.

58
Synergistic Interdependence - 2
  • With synergistic interdependence, one needs to
    introduce (for evaluation) additional criteria
    (new elements) that reveal the nature of the
    interaction.
  • The overlapping elements should be separated from
    its constituent parts. Its impact is added to
    theirs at the end to obtain their overall impact.
    Synergy of interaction is also captured at the
    upper levels when clusters are compared according
    to their importance
  • Note that if we increase the elements being
    compared by one more element and attempt to
    preserve the consistency of their earlier
    ranking, we must be careful how we make
    comparisons with the new element.
  • Once we compare one of the previous elements with
    a new one, all other relationships should be
    automatically set otherwise there would be
    inconsistency and the rank order might be changed.

59
Synergistic Interdependence - 3
  • The AHP provides a simple and direct means for
    measuring interdependence in a hierarchy.
  • The basic idea is that wherever there is
    interdependence, each criterion becomes an
    objective and all the criteria are compared
    according to their contributions to that
    criterion.
  • This generates a set of dependence priorities
    indicating the relative dependence of each
    criterion on all the criteria.
  • These priorities are then weighted by the
    independence priority of each related criterion
    obtained from the hierarchy and the results are
    summed over each row, thus yielding the
    interdependence weights.

60
Synergistic Interdependence - 4
  • Note that prioritization from the top of the
    hierarchy downward includes less and less synergy
    as we move from the larger more interactive
    clusters to the small and more independent ones.
  • Interdependence can be treated in two ways.
  • Either the hierarchy is structured in a way that
    identifies independent elements or dependence is
    allowed for by evaluating in separate matrices
    the impact of all the elements on each of them
    with respect to the criterion being considered.

BACK
61
Advantages of the AHP
Unity
Process Repetition
Complexity
Interdependence
Judgment and Consensus
AHP
Tradeoffs
Hierarchic Structuring
Synthesis
Measurement
Consistency
62
Research Issues
  • Hierarchy construction
  • Method to deal with interdependence
  • Fuzziness in relationships among elements?
  • Priority setting
  • Scale vs. other scaling methods
  • How to make subjective judgment more objective
  • Application
  • Performance measurement via AHP vs. DEA
  • Network vs. hierarchic structure
  • How to deal with situation when subjective
    judgment depends on relative weight of the
    criterion based?
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