Title: Logistics Decision Analysis Methods
1Logistics Decision Analysis Methods
- Analytic Hierarchy Process
- Presented by Tsan-hwan Lin
- E-mail percy_at_ccms.nkfust.edu.tw
2Motivation - 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.
3Motivation - 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.
4Motivation - 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.
5Introduction - 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.
6Introduction - 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.
7Introduction - 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.
8Major 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.
9Constructing Hierarchies
- Hierarchies are a fundamental mind tool
- Classification of hierarchies
- Construction of hierarchies
10Establishing Priorities
- The need for priorities
- Setting priorities
- Synthesis
- Consistency
- Interdependence
11Advantages 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
12Q A
13Hierarchy 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.
14Pairwise 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.
15Pairwise 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.
16Preference Scale - 1
17Preference 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.
18Synthesis
- 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.
19Procedure 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
20Example 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
21Example 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
22Example 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
23Example 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
24Consistency - 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
25Consistency - 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.
26Consistency 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.
27Procedure 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.
28Procedure 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
29Random 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
30Example 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
31Example 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.
32Example 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).
33Example 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.
34Development 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
35Example 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
36Example 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
37Example 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
38Hierarchies 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.
39Classifying 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.
40Hierarchy
- 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.
41Types 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.
42Constructing 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.
43Constructing 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.
44Constructing 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.(???????????????????????????????????????
??)
45Constructing 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.
46Constructing 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.
47Need 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.
48Need 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.
49Setting 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.
50Setting 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.
51Synthesis 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.
52Consistency 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.
53Consistency 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.(??????????)
54Backup Materials
55Interdependence
- 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
56Additive 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
57Synergistic 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.
58Synergistic 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.
59Synergistic 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.
60Synergistic 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
61Advantages of the AHP
Unity
Process Repetition
Complexity
Interdependence
Judgment and Consensus
AHP
Tradeoffs
Hierarchic Structuring
Synthesis
Measurement
Consistency
62Research 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?