HelpSmith – SuperDecisions Help v2.7
Creative Decisions Foundation
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SuperDecisions Software SuperDecisions Help SuperDecisions Manual
Copyright © 201 5 Creative Decisions Foundation
The SuperDecisions software implements the Analytic Network Process (ANP) for decision making. The ANP is a framework for handling very complex oftentimes messy problems that are not reducible to quantitative analysis because of the intangible considerations. It allows you to decompose a problem systematically and incorporate judgments on intangible factors alongside tangible factors.
Software Versions The Analytic Network Process for decision making was created by Thomas L. Saaty. The SuperDecisions
software was designed by William J. Adams We currently have distributions for:
1) Windows 2000 to Windows 10
2) Mac OS X (version 10.4 and above). PPC and Intel processors are both supported.
The goal and criteria are one comparison group with the goal as the parent and the criteria as the children. The criteria will be pairwise compared with respect to the Goal for importance.
Each criterion connected to the cars forms a comparison group with that criterion as the parent and the cars as children. The cars will be pairwise compared with respect to the criterion for preference. There are
5 comparison groups for this model. Below is a figure showing the hierarchy under construction; thus far the goal is connected to the criteria
and the first criterion, Prestige, is connected to the cars, so at this point two comparison groups have been established.
Below the figures are shown as they are typically presented in the AHP theory. The second figure is a "complete" hierarchy, that is, every node in a level is connected to every node in the level below. Not all hierarchies have this same complete form.
A Hierarchy under Construction
The Finished Complete Hierarchy
A Hierarchy in the SuperDecisions Software
In the SuperDecisions software a decision model is made up of clusters, nodes and links. Below is a screenshot of the car choice hierarchy as it appears in the software.
Clusters are groupings of nodes which are logically related factors of the decision. Connections are made among nodes to establish comparison groups and when nodes are connected links automatically appear between their clusters. Though there are no levels the clusters may be arranged to look like a hierarchy by dragging and dropping them to stack them.
In a hierarchy the links go only downward: from the goal node to the criterion nodes and from each criterion node to the alternative nodes.
Note: Numbers are sometimes used to preface the cluster and node names because in the supermatrix (discussed a few lines below) they are in alphabetical order and if you want to control the order, numbering is the best way to do it.
SuperDecisions Hierarchical Model Screenshot
This model can be found in Help>Sample Models>Tutorial_Models with the name: Tutorial_1_Acura_Relative_Model.sdmod
Showing Children of a Parent Node
The arrow from one cluster to another is merely an indicator that some parent node or nodes in the cluster at the base of the arrow, the "from" cluster, are linked to some node or nodes in the cluster at the point of the arrow, the "to" cluster, but it does not specifically indicate which nodes are connected. The parent node or nodes are in the "from" cluster and their respective groups of children are in the "to" cluster.
Turn on the "show connections" mode as shown in the screenshot below. Holding the cursor over a node will cause its children nodes to be outlined in red. If the entire cluster window of the children nodes is also outlined in red it means the pairwise comparisons for that family have been finished and marked as completed.
Screenclip Showing Children Nodes
The Fundamental Scale of the AHP and ANP
The pairwise comparison judgments used in the AHP pairwise comparison matrix are defined as shown in the Fundamental Scale of the AHP below.
THE FUNDAMENTAL SCALE OF THE AHP
Intensity of importance
Definition
Explanation
1 Equal importance
Two elements contribute equally to the objective
3 Moderate importance
Experience and judgment slightly favor one element over another
5 Strong importance
Experience and judgment strongly favor one element over another
7 Very strong importance
An activity is favored very strongly over another
9 Absolute importance
The evidence favoring one activity over another is of the highest possible order of affirmation
Used to express intermediate values
For comparing elements that are very close Rational numbers
Decimals
Ratios arising from the scale
Use these ratios to complete the matrix if
above that may be greater
consistency were to be forced based on an
than 9
initial set of n numerical values
Reciprocals
If element i has one of the
If the judgment is k in the (i, j) position in matrix
above nonzero numbers
A , then the judgment 1/ k must be entered in
assigned to it when compared
the inverse position (j, i).
with element j , then j has the reciprocal value when compared with i
To compare n elements in pairs construct an n x n pairwise comparison matrix A of judgments expressing dominance. For each pair choose the smaller element serves as the unit and the judgment that expresses how many times more is the dominant element .Reciprocal positions in the matrix are inverses, that is, a ij = 1/a ji .
The Pairwise Comparison Matrix
The goal is the parent node of the criteria and they comprise one of the comparison groups in this model. The criteria will be pairwise compared with respect to the goal. The pairwise comparison judgments are made using the Fundamental Scale of the AHP and the judgments are arranged in a matrix, the pairwise comparison matrix.
The numbers in the cells in an AHP matrix, by convention, indicate the dominance of the row element over the column element; a cell is named by its position (Row, Column) with the row element first then the column element. In the AHP pairwise comparison matrix below the (Price, MPG) cell has a judgment of 3 in it, meaning Price is 3 times as important as Miles Per Gallon (MPG). So logically this means MPG has to be 1/3 as important as Price so 1/3 is automatically entered in the (MPG, Price) cell.
Only the judgments in the unshaded area need to be made and entered because the inverse of a judgment, for example, (Price, MPG) is automatically entered in its transpose cell (MPG, Price). The diagonal elements are always 1, because an element equals itself in importance. Matrices with this property are called inverse matrices.
Only judgments in the unshaded area need to be made and entered. There will be 6 judgments required for these 4 elements. If the number of elements is n the number of judgments is n(n-1)/2 to do the complete set of judgments. It is possible to make less than this number of judgments and obtain a rough estimate, but there must be a minimum of n -1 judgments.
AHP Pairwise Comparison Matrix
GOAL Prestige Price
Deriving Priorities from the AHP Pairwise Comparison Matrix
Priorities for the criteria are obtained by calculating the principal eigenvector of the above matrix. A short computational way to obtain this vector is to raise the matrix to powers. Fast convergence is obtained by successively squaring the matrix. The row sums are calculated and normalized. The computation is stopped when the difference between these sums in two consecutive calculations of the power is smaller than a prescribed value.
The eigenvector of the above matrix to four significant decimals is:
Criteria Priorities
Prestige 0.0986
Price 0.425 MPG
The Mathematics behind the Pairwise Comparison Matrix
The priorities of an AHP pairwise comparison matrix are obtained by solving for the principal eigenvector of the matrix. The mathematical equation for the principal eigenvector w and principal eigenvalue λ max of a
matrix A is given below. It says that if a matrix A times a vector w equals a constant (λ max is a constant)
times the same vector, that vector is an eigenvector of the matrix. Matrices have have more than one eigenvector; the principal eigenvector which is associated with the principal eigenvalue λ max
(that is, the
largest eigenvalue) of A is the solution vector used for an AHP pairwise comparison matrix.
Aw = λ max w
Perron's Theorem shows that for a matrix of positive entries, there exists a largest real eigenvector and its eigenvector has positive entries. This is an important theorem that supports the use of the eigenvector solution in AHP theory to obtain priorities from a pairwise comparison matrix. The book, Fundamentals of Decision Making and Priority Theory by Thomas L. Saaty, gives more details about the mathematics of the pairwise comparison matrix.
For more on the mathematics of the AHP click here .
The Pairwise Comparison Matrix View
There are five modes for making assessments: judgments drawn for the Fundamental Scale of the AHP are used in the graphical, verbal, matrix, and questionnaire modes and direct data is used in the Direct mode. The matrix mode is shown in the screenclip below. Only the judgments in the unshaded cells of the AHP Matrix above need to be entered so the cells shown in the software are limited to these.
The (Prestige, Price) cell has a red 4 in it with the arrow pointing up to Price, indicating Price is 4 times more important than Prestige (the value 1/4 is used in the computations.) And 4 is used for the (Price, Prestige) cell though it is not displayed. Prestige does not even show up as a column heading because the entire column is greyed out in the AHP Matrix view. However, Prestige does appear as a row heading so it is involved in as many comparisons as any other node.
The priorities for the criteria are the results shown in the panel at the right below and are entered in the supermatrix in the column of the parent node of the comparison, the goal node in this case. Interpret numbers in red as fractions, and numbers in blue as whole numbers; for example the red 4 in the (Prestige, Price) cell represents the 1/4 in the AHP matrix above. The arrow points to the dominant factor for each cell.
The Inconsistency of 0.077 is given above the derived priorities in the Results panel below. The inconsistency should be less than 0.10, that is, 10%.
Matrix Comparison Mode as Shown in the SuperDecisions Software
Note that the priorities in the Results panel are the same as those given mathematically above for the AHP comparison matrix. The inconsistency is also given. In this instance it is .07684 which is satisfactory. A rule of thumb is that the inconsistency should be less than 0.10.
The Supermatrix
The AHP uses a data structure called a supermatrix that contains priorities from the comparison groups with the priorities of the children from a comparison group of children and parent appearing in the column of the parent. The name supermatrix is used for this matrix not because it is specially wonderful, but because it is made up of column vectors of priorities, each of which was obtained from a matrix! So in a way it is a matrix of matrices.
There are three supermatrices: 1) The unweighted supermatrix contains all the pairwise comparison results. 2) The weighted supermatrix, weighted by the importance of clusters, is important only in network models. The weighted supermatrix is the same as the unweighted supermatrix for hierarchies. In the weighted supermatrix all columns must sum to zero.
3) The limit supermatrix is the final version of the supermatrix obtained by raising the weighted supermatrix to powers (with modifications depending on the model structure).
In this example, the priorities of the criteria are arranged beneath the goal in the first column and the priorities of the cars are arranged beneath each criterion. The initial supermatrix of the derived priorities is called the unweighted supermatrix. In the ANP component blocks of the supermatrix are multiplied by constants so that the columns will sum to 1. This is the weighted supermatrix.
The Unweighted Supermatrix
The simple way to get the answer for a hierarchy is to multiply the priority of each element in the hierarchy (derived through pairwise comparisons) by the weight of its parent element and sum the bottom level priorities of the alternatives to get the answers. However, the solution may also be obtained using a supermatrix.
The Unweighted Supermatrix for the Car Hierarchy
The Limit Supermatrix
The SuperDecisions software uses a special algorithm to remember and display additional priorities in the Limit isupermatrix that appeared in successive powers of the matrix and give useful informatioin. The final overall priorities for the alternatives, in raw unnormalized form, appear in the column beneath the goal. The priorities for the criteria in the goal column, when normalized, are the original priorities derived by pairwise comparison.
The weighted supermatrix is raised to powers until it converges to the limit supermatrix which contains the final results, the priorities for the alternatives, as well as the overall priorities for all the other elements in the model. It happens that the weighted supermatrix is the same as the unweighted supermatrix for an AHP hierarchy, so raise the matrix above to powers.
A hierarchy is a special kind of network with a goal cluster from which an arrow only goes out, called a source, and an alternatives cluster with arrows only coming in, called a sink. This type of model is actually more difficult when it comes to finding the results than a more typical network model with connections going every direction. To do the computations yourself, using Microsoft Excel for example, raise the unweighted supermatrix to powers until all the cells go to zero, then back up to the previous power to find the final non-zero priorities for the alternatives. All the rest of the cells will be zero.
Thomas Saaty uses a different method in his books that is more theoretically correct. With the method above it is simply an observation that at the power when the matrix goes to zero, there must have been priorities in it in the previous step, so back up and use those. But with the Saaty method each alternative is connected to itself, resulting in an unweighted supermatrix containing an identity matrix of 1's in the (alternative, alternative) block of cells. This matrix will not go to zero, instead it reaches a steady state containing only the priorities of the alternatives with all the other cells zero as it is raised to powers.
SuperDecisions Limit Matrix with Final Priorities
Getting the Answer from the Limit Supermatrix
The result we are seeking is the priorities of the alternatives, the Raw numbers shown in the third column are directly from the limit supermatrix: Acura TL 0.172133, Toyota Camry 0.100103, and Honda Civic 0.227764. To convert them to the priorities in the Normals column, they are normalized to 1, which means sum the raw numbers and divide each by this sum. The Normals priorities add up to 1. The priorities in the Ideals column are obtained by dividing each Raw number by the largest, 0.172133, resulting in the "Ideal" alternative having a value of 1.
The synthesis command does the work for you, extracting the raw numbers from the limit supermatrix and idealizing and normalizing them. From the perspective of the person who made the judgments in this model, the Honda Civic is the best choice. This is, of course, a subjective outcome.
Synthesizing to show that the Honda Civic is the Best Choice
Sensitivity Analysis
To perform sensitivity analysis one asks what the decision would be if the priorities of the criteria were different. Below is a screenshot of the dynamic sensitivity barchart showing that if the priority of the Prestige criterion increases considerably from its original .04 value, the more prestigious car, the Acura TL, represented by the red bar, becomes the best choice. Note: To get to this screen select the Computations>New Sensitivity command, the Horz Barchart tab, set the "Node for sensitivity" to Prestige and click and drag the Parameter button to the right.
Sensitivity Barchart increasing Priority of Prestige
Changing from AHP Hierarchical Thinking to ANP Network Thinking
In the Analytic Network Process (ANP), which is a generalization of the AHP, the decision elements are organized in a network of clusters with links between the elements going in either direction or both directions. The simplest type of network is a feedback model between criteria and alternatives. To illustrate what this means we will convert the hierarchical car model to a network model.
A hierarchy is a network too, but a special kind of network with a goal cluster from which all the arrows lead away, and a sink cluster (the alternatives) that all the arrows lead into. Links go only downward in a hierarchy. In a typical network one has neither sinks nor sources; and the links can go in any direction. A network can more faithfully represent the relative world we really live in. One does not buy a car by determining in the abstract the importance of the criteria before going shopping and looking at a few cars. The available cars determine how important the criteria are. And when new cars are added to those being considered, the importance of the criteria may change.
A Network in the SuperDecisions Software for the Car Choice
In this network each car is linked to all the criteria and they are evaluated for their importance in each car resulting in what one might call profiles of the cars with the priorities of the criteria being evaluated in terms of the available cars - this is feedback.
Step 1. Compare the alternatives with respect to each criterion (as is done in AHP models). Step 2. Compare the criteria with respect to each of the cars to get a profile of the importance of the criteria in that car. Step 3. Synthesize to get the overall priorities for the cars, and, incidentally, overall priorities for the criteria. Note that the overall criteria priorities very much depend on the cars that have been included for consideration. They are specific for this group of cars.
The main difference between this ANP model and the AHP model is that the importance of the criteria has been derived from the available alternatives (using feedback) not established top down in an abstract way from the goal.
Deriving the Priorities of the Criteria through Feedback
A typical pairwise comparison question to determine priorities of the criteria from the alternatives, known as feedback, instead of from their importance to the goal, would be: “For the Acura TL, which do you like better, its prestige or its price?”, “Its prestige or its MPG?” etc. The result of these questions for the Acura TL is a profile of the importance of the criteria for the Acura. Such a profile is developed for each car, then the profiles are combined in the limit supermatrix to give overall priorities for the criteria.
Profile of Criteria for the Acura TL
Priorities of the criteria for the Acura TL
Similar profiles can be obtained for the Toyota Camry and Honda Civic as shown below: Toyota Camry
Honda Civic
The Unweighted Supermatrix for the Car Network
The Limit Supermatrix for the Car Network
The Final Synthesized Result for the ANP Model Notice that the best car is no longer the Honda Civic as determined in the AHP model, but is now the more
prestigious Acura TL. This is because the criterion of Prestige is more important in the ANP model than we estimated in the AHP model by evaluating the importance of the criteria directly with respect to the goal. In the ANP where the priorities of the criteria are determined by feedback, getting information about the criteria from the cars rather than from a goal, we find out it is more important than we knew.
Use the Synthesis command to get the final results, either the Syn shortcut or Computations>Synthesize from the menu.
Showing Priorities of all the Nodes in the Model
In contrast to the solution using an AHP model where the Prestige node had the lowest priority at 0.09 when pairwise comparing with respect to the goal, here the Prestige node has the highest priority in the criteria cluster at 0.317. The Limiting Priorities of all the nodes in the model sum to 1.0 and are shown below in the right hand column. The limiting priorities are normalized for each cluster to sum to 1.0 shown in the left hand column.
Limiting Priorities of all Nodes in Model
An Example of a Larger Network Model
To build a network model, considers first what the decision is about. What is the goal? Then try to determine factors that seem to play a part in the decision and what the alternatives will be. The factors are arranged in some logical way into groups. In the software the term for the logical groupings is clusters and the term for a factor is node.
The Hamburger network model shown below is for estimating relative market share of three hamburger restaurant chains. It has factors customers might think about in choosing one of the restaurants represented by nodes in the model and the nodes are logically grouped into 4 clusters.
Hint: To load one of the sample model files, click on the Help command on the main menu, then click on the name of the model you want. This model is named Hamburger.sdmod.
1 Alternatives
1 McDonald's 2 Burger King 3 Wendy's
2 Advertising
1 Creativity 2 Promotion 3 Frequency
3 Quality of Food
1 Nutrition 2 Taste 3 Portion
4 Other
1 Price 2 Location 3 Service 4 Speed 5 Cleanliness 6 Menu Item 7 Take-out 8 Reputation
Below is the model as it appears in the SuperDecisions software. It is still referred to as a simple network model as all the clusters and nodes are in the same window. This is one of the sample introductory models. To load it go to: Help>Samples>1_Introductory_Models>Hamburger.sdmod
SuperDecisions Simple Network Model
This model can be loaded from Samples>Tutorial Models>hamburger.sdmod. Links are made among the nodes to indicate influence. To show which nodes are connected from a given,
click on the "show connections" icon to turn on the "show connections" mode. Mouse over any node to see the nodes to which it connects outlined in red.
Showing Creativity Node linked to Nodes in many Clusters
For example, hold the cursor over the Creativity node to see it is linked to nodes in three clusters; its own 2 Advertising cluster (inner dependence), the 4 Other cluster and the 1 Alternatives cluster. Thus there are three comparison groups involving the Creativity node. Similar views of the children of other nodes can be seen by moving the cursor from node to node in the model. A parent node can have children in many different clusters and be involved in many sets of pairwise comparisons as the parent.
Comparison Groups and Priorities
A comparison group consists of a parent node and its children nodes. The children nodes of a comparison group must all be in the same cluster, though a parent node can have children in several different clusters. The children nodes in each cluster are pairwise compared with respect to the parent node. The results are priorities for the children nodes. Priorities sum to 1.000 for each group of children nodes and are combined throughout the model to give an overall answer for the alternatives of the decision.
Final Synthesized Results
For this model, the results are a set of priorities that reflected the relative market share of the restaurants quite accurately at the time the model was done. The Raw values are obtained directly from the Limit Supermatrix, normalized to give the Normals column, and divided by the largest of them to give the Ideals column.
Synthesis Results
An Example of a Hierarchical Model
The SuperDecisions software can be used to create both hierarchical and network models. In the Analytic Hierarchy Process a decision model is structured as a hierarchy with a goal node at the top, criteria influencing the goal in the level below (there may also be several additional levels of sub-criteria), and the alternatives of the decision in the bottom level. Here is a graphic of a hierarchical decision model to choose the best car. The criteria will be pairwise compared for importance to establish their priorities with respect to the goal. The cars will be pairwise compared for preference to establish their priorities with respect to each criterion. The results of all these comparisons will be combined to give the best car overall; that is, the car with the highest priority.
This is an example of a decision model to choose the best car. The goal is connected to the criteria that will be employed in choosing a car and they form a comparison group with the goal being the parent and the criteria being the children. The criteria will be pairwise compared with respect to the Goal for importance.
Each of the criteria is connected to the three cars. Thus altogether there are five sets of pairwise comparisons in this model: the criteria for importance with respect to the goal and the cars with respect to each of the 4 criteria for preference. Below is a figure showing the hierarchy under construction; thus far the goal is connected to the criteria and the first criterion, Prestige, is connected to the cars, so two comparison groups have been defined.
The Finished Hierarchy
A Hierarchy in the SuperDecisions Software
In the SuperDecisions software a decision model is made up of clusters, nodes and links. Below is a screenshot of the car choice hierarchy as it appears in the software.
Clusters are groupings of nodes which are logically related factors of the decision. Connections are made among nodes to establish comparison groups and when nodes are connected links automatically appear between their clusters. Though there are no levels the clusters may be arranged to look like a hierarchy by dragging and dropping them to stack them.
A Hierarchy under Construction
In a hierarchy the links go only down, from the goal node to the criterion nodes and from each criterion node to the alternative nodes.
Note: Numbers are sometimes used to preface the cluster and node names because in the supermatrix they are in alphabetical order and if you want to control the order, numbering clusters and nodes is the best way to do it.
SuperDecisions Hierarchical Model Screenshot
Go to Help>Sample Models>Tutorial_Models to load this car choice model: Tutorial_1_Acura_Relative_Model.sdmod
The arrow from one cluster to another is merely an indicator that some parent node or nodes in the cluster at the base of the arrow, the "from" cluster, are linked to some node or nodes in the cluster at the point of the arrow, the "to" cluster, but it does not specifically indicate which nodes are connected. The parent node or nodes are in the "from" cluster and their respective groups of children are in the "to" cluster.
Turn on the "show connections" mode as shown in the screenshot below. Holding the cursor over a node will cause its children nodes to be outlined in red. If the entire cluster window of the children nodes is also outlined in red it means the pairwise comparisons for that family have been finished and marked as completed.
Pop-up Descriptions Click on the ? icon to select it and depress it to turn on the "show descriptions" mode. Then holding the cursor over a cluster or node for a few seconds will cause its description to pop up, if one was entered when they were created.
How to show which children nodes are connected from a parent node
In this hierarchical model the Goal node is the parent of the criteria nodes, and each criterion node is the parent of the alternatives. Criteria are compared using for importance to the goal; the cars are compared for preference with respect to each of the criterion nodes.
Thus there are 4 sets of comparisons to be made for this model.
Deriving Priorities from the AHP Pairwise Comparison Matrix
AHP Pairwise Comparison Matrix
GOAL Prestige Price
Priorities for the criteria are obtained by calculating the principal eigenvector of the above matrix. A short computational way to obtain this vector is to raise the matrix to powers. Faster convergence can be obtained by successively squaring the matrix. The row sums are calculated and normalized at each iteration. The computation is stopped when the difference between these sums in two consecutive calculations of the power is smaller than a prescribed value.
The principal eigenvector of the above matrix to four significant decimals is:
Criteria Priorities
Prestige 0.0986
Price 0.425 MPG
What does this result mean, in everyday language? It says the most important thing in buying a car is the price (presumably one would prefer a less expensive car - though if a Russian billionaire were making the decision he might prefer the most expensive car, still price is a big factor) at 42.5% and the second most important thing is Comfort at 30.77%. The least important thing about a car is its prestige.
Remember, this is top-down thinking, the kind of thinking where your judgments reflect what you believe to be the politically correct proper answer. It is not necessarily the way you think when you go to the dealer showroom and see a few cars. Somehow one's opinion about the importance of prestige may take a dramatic swing upwards. People continue to buy those fancy cars, and the ANP is a more realistic decision process that captures this reasoning because it allows feedback; that is, you must see and evaluate a few alternatives before you can actually determine the importance of the criteria. This is called feedback.
The ANP can integrate the top-down view with a bottom-up view. Sensitivity Analysis To perform sensitivity analysis one asks what the decision would be if the priorities of the criteria were
different. Below is a screenshot of the dynamic sensitivity barchart showing that if the priority of the Prestige criterion increases considerably from its original .04 value, the more prestigious car, the Acura TL, represented by the red bar, becomes the best choice. Note: To get to this screen select the Computations>New Sensitivity command, the Horz Barchart tab, set the "Node for sensitivity" to Prestige and click and drag the Parameter button to the right.
Sensitivity Barchart showing Results of increasing Priority of Prestige
To create an ANP decision model first consider the decision problem to come up with a collection of factors that seem to represent the issues in the decision, define the purpose of the decision carefully, and determine some possible alternatives of choice. In the software the term for a factor is node and the nodes are logically arranged into groups called clusters.
What is a Model?
The term model is sometimes used as a verb meaning to create a framework in some way that represents something in the real world. Models range from things like statues created by artists to miniatures of buildings that architects use to a mental understanding of a situation. It is the latter meaning that we have in mind when talking about a decision model; the model starts with a conception of what the decision is about, what the alternatives are and what factors should be taken into account in someone's mind. The mental model is then built using the software resulting in a computer file. Information about the mental model the decision maker has in mind is transferred into the software, including the factors and alternatives of the problem, and their structure, how they are grouped and linked together. SuperDecisions models are files with the extension .sdmod.
After the software implementation of the mental model of the decision we usually refer to it as "the model"; it is a representation of how we see the decision: what the decision is about, the factors that come into play, and the alternatives of the decision. The decision maker(s) then make assessments using judgments or data about the elements of the problem and the software combines these judgments to prioritize the alternatives, or rank them. This is the end result, what we were looking for when we conceptualized and created the decision model.
Market Share Models as Validation Exercises
Models to estimate the market share of products or companies involve both tangibles and intangibles and have been found to be a good means of validating the ANP decision process. One may ask, "Why try to estimate market share when the incomes and other market measures of most public companies are widely available on the internet?" Early on when the ANP was first being applied, it was important to find real-life situations that had measurable outcomes against which the model results could be checked to validate the ANP as a decision making process. The success of a market share model does depend on the familiarity of the person who is designing the model and making the judgments with the product.
The Hamburger Market Share Model
The Hamburger model was created for the purpose of estimating the relative market share of three fast food hamburger chains. The factors become the nodes in the model and they are grouped into clusters of nodes that are similar in some way. They become the nodes in the model and include tangible things such as price and soft intangible things such as creativity of advertising, location of restaurants, and other things customers might think about in deciding where to eat, and they are logically grouped into the 4 clusters listed below.
Hint: Sample models are under the Help command on the main menu. This sample model can be loaded by double-clicking on it from Help>Sample Models>Introductory Models>Hamburger.sdmod
1 Alternatives
1 McDonald's 2 Burger King 3 Wendy's
2 Advertising
1 Creativity 2 Promotion 3 Frequency
3 Quality of Food
1 Nutrition 2 Taste 3 Portion
4 Other
1 Price 2 Location 3 Service 4 Speed 5 Cleanliness 6 Menu Item 7 Take-out
8 Reputation Below is a screenshot of the hamburger model. This is referred to as a simple network because all the
clusters are in the same window.
SuperDecisions Network Model for to Estimate Market Share of Hamburger Places
Links are made among the nodes to indicate influence. It is not possible from this view of the model to determine precisely which nodes are connected. An arrow from one cluster to another is a general indicator that some node(s) in the cluster at its base are connected to some node(s) in the cluster at its point.
To show which nodes are connected from a given, click on the "show connections" icon, the fan-shaped icon next to the ? help icon, to turn on the "show connections" mode. With this mode turned on mousing over any node in the model will show the nodes it is connected to outlined in red.
For example, the Creativity node is linked to nodes in three clusters; its own 2 Advertising cluster, the 4 Other cluster and the 1 Alternatives cluster.
Showing Which Nodes are Linked from the Creativity Node
Comparison Groups
A comparison group consists of a parent node and its children nodes that are in the same cluster. The Creativity node in the screenshot above is the parent in three different comparison groups as it is connected to children nodes in three clusters: 1 Alternatives, 4 Other and 2 Advertising clusters.
The children nodes are pairwise compared with respect to the parent node for dominance. Dominance may be expressed in terms of importance, preference or likelihood. The results are priorities for the children nodes. Priorities sum to 1.000 for each group of children nodes. Priorities are combined and synthesized for all the nodes throughout the model to give the answer, the priorities of the alternatives of the decision.
For this model, these priorities reflect the relative market share of the restaurants. For example, 1 McDonald's, 2 Burger King and 3 Wendy's must be pairwise compared with respect to 1
Creativity to establish priorities for them and thus rank them for creativity in advertising. See the section on pairwise comparing for more details.
The pairwise comparison judgments used in the AHP are defined in the Fundamental Scale of the AHP shown below. Elements may be pairwise compared with respect to importance, preference or likelihood. Most comparisons can be broadly categorized as one of these three types.
THE FUNDAMENTAL SCALE OF THE AHP
Intensity of importance
Definition
Explanation
1 Equal importance
Two elements contribute equally to the objective
3 Moderate importance
Experience and judgment slightly favor one element over another
5 Strong importance
Experience and judgment strongly favor one element over another
7 Very strong importance
An activity is favored very strongly over another
9 Absolute importance
The evidence favoring one activity over another is of the highest possible order of affirmation
Used to express intermediate values
For comparing elements that are very close Rational numbers
Decimals
Ratios arising from the scale
Use these ratios to complete the matrix if
above that may be greater
consistency were to be forced based on an
initial set of n numerical values Reciprocals
than 9
If element i has one of the
If the judgment is k in the (i, j) position in matrix
above nonzero numbers
A , then the judgment 1/ k must be entered in the
assigned to it when compared
inverse position (j, i).
with element j , then j has the reciprocal value when compared with i
To compare n elements in pairs construct an n x n pairwise comparison matrix A of judgments expressing dominance. For each pair choose the smaller element serves as the unit and the judgment that expresses how many times more is the dominant element .Reciprocal positions in the matrix are inverses, that is, a ij = 1/a ji .
A network is composed of clusters, nodes and links among the nodes in a window. Hierarchies are special cases of networks in which the links point from the goal to the criteria to the alternatives. In general networks the links may go in any direction, and there is no goal.
Node Comparison Groups and Network Links
A comparison group is comprised of a parent node and its children nodes in a given cluster that will be compared with respect to it. The children nodes must all be in the same cluster, though a parent node may be the parent of several groups and have sets of children in different clusters. A parent node may even have children in its own cluster, and thus be involved in several comparison sets.
A node may serve as a parent in one comparison group and a child in another. An arrow will appear from one cluster to another when there is at least one link from a node in the first cluster to a
node in the second cluster. Most commonly a node in the first cluster will be connected to several nodes in the second cluster. When a node is linked to a node or nodes in its own cluster, the arrow becomes a loop on that cluster.
The arrows that indicate links automatically appear whenever nodes are linked. Clusters cannot be linked in any other way.
Cluster Comparisons
Cluster comparisons arise in networks when 2 or more clusters are connected by arrows from a given cluster. They must be pairwise compared for impact on the given cluster. Select the Cluster tab in the comparisons mode to perform the pairwise comparisons. To see the results of all the cluster comparisons select Computations>Cluster matrix. Each column gives the relative impact of the clusters connected from the column heading cluster.
Mathematical Note: Each value in the cluster matrix is multiplied times all the entries in the corresponding component of the unweighted supermatrix to yield the weighted supermatrix in which all the columns sum to 1. The software performs this computation automatically.
Complex Models
A model in which all the clusters and nodes are in a single window is a simple network model. Nodes in a simple network can have subnetworks attached to them, and the resulting model is called a complex model. Complex models may have any number of cascading levels, but usually are limited to three for a BOCR (Benefits, Opportunities, Costs and Risks) model. When a subnetwork is created for a node a blank window appears and a simple network of clusters, nodes and connections is built there.
2-Level Complex Model
An example of a 2-level BOCR (benefits, opportunities, costs, and risks) model is shown below for the sample introductory model Car_BOCR.sdmod. This is a model to choose whether to buy an American, Japanese or European car. There are 4 subnets, one for each of the BOCR nodes (known as merit nodes). The priority vectors for the cars obtained in the subnets are combined in the top level window using a formula. The additive(negative) formula is being used here: bB+oO-cC-rR. The b,o,c and r stand for the priorities of the Merit Nodes in the top level window, while the B,O,C, and R stand for the synthesized vectors of priorities of the cars in the respective subnets.
Top Level of Two-level BOCR Network
The judgments in a subnet are made with respect to the controlling node, the 1Benefits node in the top level window, so all the judgments are with respect to which of the nodes in a pairwise comparison has the more positive benefit.
Subnet under 1Benefits node Judgments in this subnetwork are made with respect to benefits
Subnet under 2Opportunities node Judgments made with respect to opportunities. Opportunities are possibilities that may be available in the future as opposed to benefits which are things that are known and for which the benefits can be estimated with confidence.
Judgments in the costs and risks subnets are made posing the question as to which is the more costly or risky. The highest priority alternative should be the most costly and most risky in order for the formula to work correctly.
Subnet under 3Costs node Judgments made with respect to costs (in the present). Judgments are made so the most costly is given the higher priority or preference. Synthesizing in a costs subnet should result in the most costly alternative having the highest priority.
Subnet under 4Risks node Judgments made with respect to risks (possible things that might happen in the future). Judgments are made so the most risky has the highest priority when the priorities for the alternatives are synthesized.
Outer Dependence
Outer Dependence is when a cluster is connected to a cluster other than itself by an arrow. Such a connection means at least one node in the first cluster is the parent for a comparison group of children nodes in the other cluster.
Inner Dependence
Inner Dependence is when a cluster is connected to itself, which occurs when the parent node and the children nodes forming a comparison group are in the same cluster. A loop will appear on the cluster in this case.
The Design Command
To build a model you must create clusters, nodes within the clusters and connections among the nodes. When the software starts the blank window shown below will appear.
There are two ways to begin building a model: •
Use the Design menu command •
Use the File>New command Using the Design menu command directly begin creating clusters, nodes and connections in the blank
window. •
Design>Cluster to create, edit and modify cluster appearance Shortcut: Press N (Capital N. Hold down the Shift key and press letter n) •
Design>Node to create, edit and modify node appearance Shortcut: Press n (Press the letter n) •
Design>Node connexions from to make node connections.
The Design Command
The File>New command opens the template window which has a number of possibilities:
• Select the template you want to use
• Full Template - creates a complex BOCR model with three levels: the top Level contains Benefits, Opportunities, Costs and Risks nodes, (the BOCR nodes); a second level of control hierarchies; and a third level of decision subnets containing the alternatives.
• Small Template - creates a two level complex model with BOCR nodes in the first level attached directly to decision subnets (containing the alternatives) in the second level.
• Open File opens a browser for loading a pre-existing model •
Simple Network does nothing except display the error message below. In the opening screen you must create a network of clusters, nodes and connections using the Design menu commands.
• File>Recent Files shows a list of recently used models.
To build a model you must create clusters, nodes within the clusters and make connections among the nodes.
Creating a Cluster in a New Model
From the main menu: •
Design>Cluster brings up the new cluster dialog box where you can to create, edit and modify cluster appearance
• Keyboard Shortcut Press <Shift> <n> (i.e. capital N) with the cursor located anywhere on the background of the main screen to bring up the New Cluster Dialog Box. Upon saving, the new cluster window will be located where the cursor was when you pressed the keys.
• Mouse Shortcut Press <Shift> and <Left-Mouse-Button> to bring up the New Cluster Dialog Box. Upon saving the new cluster window will be located where the cursor was when you implemented the shortcut.
In the New Cluster Dialog box shown below type the name of the cluster in the Name field. Press <Tab> or click in the next field with the mouse to move to the next field. Type a description of the cluster in the Description field, or leave it blank.
For a complete description of setting Fonts, Icons, Pictures and Colors click here .
Click the Create Another button at the bottom to save and begin creating the next cluster immediately. Click Save to save the current cluster and return to the main screen.
The name of the previous cluster remains selected in the Name field when you create the next cluster immediately so you can begin typing the new name. Pressing <Enter> before typing anything will result in the software trying to save a new cluster with the old name and you will get the warning shown below.
Click the Save button to stop creating clusters and return to the main window.
Pop-ups Menu Icon (the ? menu icon)