Logistics Decision Analysis Methods

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Logistics Decision Analysis Methods

• Analytic Hierarchy Process
• Presented by Tsan-hwan Lin
• E-mail percy_at_ccms.nkfust.edu.tw

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Constructing Hierarchies

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

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Establishing Priorities

• The need for priorities
• Setting priorities
• Synthesis
• Consistency
• Interdependence

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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
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Q A
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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.

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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.

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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.

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Preference Scale - 1
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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.

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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.

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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

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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
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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
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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
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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
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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

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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.

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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.

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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.

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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

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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
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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
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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.

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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).
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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.

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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

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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
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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
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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
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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.

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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.

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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.

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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.

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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.

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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.

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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.(???????????????????????????????????????
  ??)

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.(??????????)

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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

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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.

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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.

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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?