Logistics Decision Analysis Methods Analytic Hierarchy Process Presented by Tsan-hwan Lin E-mail: percy@ccms.nkfust.edu.tw

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.

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.

Motivation - 3 There are two fundamental approaches to solving problems: the deductive approach（演繹法）and the inductive (歸納法；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.

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.

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.

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.

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.

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

Establishing Priorities The need for priorities Setting priorities Synthesis Consistency Interdependence

Advantages of the AHP AHP Unity Process Repetition Tradeoffs Synthesis 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 Unity Process Repetition Tradeoffs Synthesis Consistency Measurement Interdependence Complexity AHP Judgment and Consensus Hierarchic Structuring 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

Q & A

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. Overall Goal: Criteria: Decision Alternatives: Select the Best Car Price MPG Comfort Style Car A Car B Car C

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.

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.

Preference Scale - 1

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.

Synthesis The procedure to estimate the relative priority for each decision alternative in terms of the criterion is referred to as synthesization（綜合；合成）. Once the matrix of pairwise comparisons has been developed, priority（優先次序；相對重要性）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.

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:

Example: Synthesizing Procedure - 0 Step 0: Prepare pairwise comparison matrix Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6

Example: Synthesizing Procedure - 1 Step 1: Sum the values in each column. Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6 Column totals 13/8 19/6 15

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 8/13 12/19 8/15 4/13 6/19 6/15 1/13 1/19 1/15

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. 0.615 0.632 0.533 0.593 0.308 0.316 0.400 0.341 0.077 0.053 0.067 0.066 Total 1.000

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:

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.

Consistency Ratio The AHP provides a measure of the consistency of pairwise comparison judgments by computing a consistency ratio（一致性比率）. 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.

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.

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:

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 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49

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 1 2 8 1/2 6 1/8 1/6

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.”

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).

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.

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:

Example: Priority Ranking – 0A Step 0A: Other pairwise comparison matrices Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6 Price Car A Car B Car C 1 1/3 ¼ 3 ½ 4 2 Criterion Price MPG Comfort Style 1 3 2 1/3 1/4 1/2 4 MPG Car A Car B Car C 1 1/4 1/6 4 1/3 6 3 Style Car A Car B Car C 1 1/3 4 3 7 1/4 1/7

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

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

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.

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.

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.

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.

Constructing Hierarchies - 1 One’s 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 => criteria => 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.

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.

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.（我們無法直接比較蘋果與橘子；卻可以根據它們的甜度、營養、價格來決定誰是比較好的水果。）

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.

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.

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.

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.

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.

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.

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.

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.

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.（由於已有先入為主看法）

Backup Materials

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

Additive Interdependence In additive interdependence（累加性依賴性）, 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

Synergistic Interdependence - 1 In synergistic interdependence（綜效性依賴性）, 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.

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.

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.

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

Advantages of the AHP AHP Unity Process Repetition Complexity Interdependence Judgment and Consensus AHP Tradeoffs Hierarchic Structuring Synthesis Measurement Consistency

Research Issues Hierarchy construction Priority setting Application 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?