I NTRODUCTION TO THE ANP AND ITS SUPERMATRICES
In this introduction we review the ANP process and the SuperDecisions software and show some applications. Applications may be simple, consisting of a single network, or complex, consisting of a main network and two or more layers of sub-networks. Each network and sub-network is created in its own window.
In practice an application consists of:
• A Simple Network – HamburgerModel- All the clusters and their nodes are in a single window. An example of a simple network would be a “market share” application such as the Hamburger model. The simple network itself is the decision network because it contains the cluster of nodes that serve as the alternatives of the decision. In a market share application they are the competitors for whom market share is being predicted; for example, McDonald’s, Burger King and Wendy’s.
• A Two-level Network - Car Purchase BCR-There is a top-level network with Merit nodes such as Benefits, Opportunities, Costs and Risks, each of which has a sub-network. The alternatives cluster is in each of the sub-networks. The sub-networks are the decision networks because they contain the alternatives.
• A Complex Network- National Missile Defense Model -There is a main network of Merits nodes (Benefits, Opportunities, Costs and Risks), each having an attached sub-network that contains among others nodes that will serve as control criteria. The nodes selected to serve as control criteria, the high priority nodes in the network, have decision networks containing the alternatives attached to them. In practice this is the most complex system we work with, though there is no theoretical limitation on the number of levels of sub-networks.
A model is contained in a network system that physically is a single file. All the networks and sub-networks are in the same file. The file has the extension .sdmod; for example, the Hamburger model is in the file named Hamburger.sdmod. If you have the SuperDecisions software installed a model file can be launched by double-clicking on it within Windows Explorer. Sometimes the files for the Encyclicon applications have been saved in an old .mod format or a zipped format with a .mod.gz extension to reduce their sizeSuch a zipped file cannot be launched by double-clicking. Use the File Open command in SuperDecisions to open the file.
There are four sample models that we will discuss in the introduction: Bridge, Hamburger, Car Purchase BCR, and National Missile Defense. All can be found in the software under the SuperDecisions Help>Sample Models>Introductory Models command. The Bridge example is a simple network used to illustrate the idea of feedback. The Hamburger example is a simple network used to estimate market share of fast food restaurants. The Car Purchase BCR model is a two-level network with Benefits, Costs and Risks nodes in the top-level network, and the alternatives in the sub-networks attached to them. They range from the simplest example of feedback to a complex multi-level network structure. All of these examples are included in the sample models of the SuperDecisions software. To load a sample model use the Help, Sample Models command in the software and select the one you want. Sample models are located in the c:/program files/super decisions/samples directory.
Bridge is the simplest example in a single network and we use it to demonstrate the supermatrix idea. Hamburger is used to estimate the market share of three fast-food hamburger places. The model consists of a single network containing the factors that consumers consider when choosing a fast-food restaurant. Some of its clusters are inner dependent with the nodes in them being compared with respect to other
nodes in the same cluster. It has cluster comparisons as well as node comparisons. We use this model to explain more complicated supermatrices, inner and outer dependence and the motivation behind doing cluster comparisons. It is a simple network model in a single window.
Car Purchase BCR is a model for selecting the best kind of car to buy: Japanese, European or American, by taking into consideration the benefits, costs and risks of each type of car. It is a complex two-layer model with three sub-networks. The top-level network contains a benefits node, a costs node and a risks node, each of which has a sub-network where the alternatives are located. Judgments in a sub-network are made from the perspective of its controlling node in the network above.
National Missile Defense is the most complex kind of application with a top-level control model in which the priorities of the BOCR have been obtained by rating them against the US’s national objectives. It has control criteria sub-networks under that and finally decision networks containing the alternatives at the bottom.
D EMONSTRATION OF TH
E S IMPLEST T YPE O
F F EEDBACK
N ETWORK, THE B RIDGE M ODEL
Bridge is a decision problem to pick the best of two bridge designs. It is a simple network of one level that contains only two clusters, with two nodes in each cluster, and links between the nodes. A network is structured of clusters, nodes and links. We use this model to show how feedback arises in a network decision structure and how the pairwise comparison questions are formulated when there is feedback. Here the clusters are outer dependent, that is, nodes in a cluster are compared only with respect to nodes in the other cluster.
Load the Bridge model by selecting Help, Sample Models from the main menu and selecting
The Bridge model, a simple network model, is shown in Figure 1. Clusters may be re-sized by left-clicking on the small button at the lower right hand corner and dragging. To select a cluster left-click on the title bar. A cluster is selected when its title bar is highlighted. Left-click on the title bar of a cluster window and drag to move it to a different location.
Figure 1. The Bridge Model: a Simple Network.
The decision in this model is to select the best bridge. The objectives are to have a safe bridge and an aesthetically pleasing bridge. If one were doing the model from the top down as in a hierarchy, there would be a goal with Aesthetics and Safety as the criteria and Bridge A and Bridge B as the alternatives. One would compare Aesthetics to Safety, Safety would likely be perceived as extremely more important, so the safest bridge would be the "best" choice.
But in a feedback network one compares the bridges for preference with respect to Aesthetics and to Safety, and one also compares the prevalence of Aesthetics versus Safety for each bridge. The net result of this is that priorities are obtained for all four nodes in the system. Suppose the safest bridge, B, is unattractive, and the nicer looking bridge, A, is very safe, though not as safe as B. The priorities of the criteria depend on the bridges available and since both are quite safe, the priority of safety in the feedback system ends up less than it would be in a hierarchy where one compares Safety to Aesthetics in the abstract and apart from any specific bridge. It makes common sense that if both bridges are very safe, one should pick the better-looking bridge, even though one bridge is far safer than the other. The Analytic Network Process through feedback guides us to the best choice in a way that matches our common sense.
The Aesthetics node in the Objectives cluster is linked to Bridge A and to Bridge B, and because there is a link between nodes, a link appears from the Objectives cluster to the Alternatives cluster. Because at least one node in the Objectives cluster is linked to nodes in the Alternatives cluster, a link appears automatically from the Objectives cluster to the Alternatives cluster. Aesthetics is the parent node and Bridge A will be compared to Bridge B with respect to it. The node Safety is also linked to Bridge A and Bridge B, and they will be compared for preference with respect to safety.
To turn on the “show connections” mode, as shown in Figure 2 click on the star-shaped icon . When you place your cursor over a node when this icon is depressed, the nodes connected from it will be outlined in red. Try this by placing the cursor over the Bridge A node and the Aesthetics and Safety nodes will be outlined in red.
When a node has had the comparisons marked as completed for nodes within a cluster that are connected to it, the cluster window of these nodes will also be outlined in red. Both bridges are also connected to the nodes in the Objectives cluster, so holding the cursor over the Bridge A node will show Aesthetics and Safety outlined in red, and the Objectives cluster being outlined in red indicates that the comparison of these nodes with respect to Bridge A is complete.
Figure 2. The Aesthetics Node is connected to Bridge A and Bridge B.
F EEDBACK L INKS
It is easy for those who have used the Analytic Hierarchy Process to understand how to pairwise compare Bridge A and Bridge B with respect to Aesthetics. Bridge A would be highly preferred. But what may be new is the idea that criteria may be compared with respect to an alternative. What does that mean? When comparing, for example, Aesthetics and Safety with respect to Bridge A, the question is: What is more a more pronounced or prevalent characteristic of Bridge A, its aesthetics or its safety? Bridge A is beautiful and that is what we like best about it. Its safety, though quite adequate, is nothing notable. So we strongly prefer its aesthetics to its safety.
For Bridge B what is its more preferable characteristic, aesthetics or safety? Since it is quite ugly, the answer is that the Safety of Bridge B is extremely preferable to its Aesthetics. These kinds of preference questions and answers, both directions, help us establish our true priorities for all the elements in the problem.
T HE S UPERMATRIX
The comparison process will be covered in the next demonstration. Here we will show the various computations involving the supermatrix. To show the three different supermatrices, select the Computations command from the menu shown in Figure 3.
Figure 3. The Computations Menu.
T HE U N-WEIGHTED,W EIGHTED AND L IMIT SUPERMATRICES
There are three supermatrices associated with each network: the Unweighted Supermatrix, the Weighted Supermatrix and the Limit Supermatrix. Supermatrices are arranged with the clusters in alphabetical order across the top and down the left side, and with the elements within each cluster in alphabetical order across the top and down the left side. To change the ordering in a supermatrix, you need only re-name the clusters and/or the elements, so their alphabetical order will be the order you want. Changing names after building a model and making comparisons is permitted and will correctly preserve any judgments that have been made.
The unweighted supermatrix is composed of column vectors that are the priorities obtained by comparing nodes is a cluster with respect to a parent node. The column for a given node contains all the priority vectors in the system with that node as a parent of the comparison. A parent node may have children in many different clusters, so the priority vectors are stacked on top of each other in the parent node’s column. Each priority vector sums to 1.0, so the numbers in the column of a given node may sum to more than 1, though the sum will always be an integer (or zero if that node is not connected to any other nodes in the entire system)..
Each column has to sum to 1 for a supermatrix to converge to the limit supermatrix. This is done by weighting all the numbers in each “component” of the supermatrix by the cluster priorities. If the clusters have not been pairwise compared, their priorities are assumed to be equal. The [A,B] component of a supermatrix, for example, is the submatrix consisting of the priority vectors from pairwise comparing elements in the B cluster linked from parents in the A cluster. sets with the parent in the A cluster and the children in the B cluster.
When you multiply an unweighted supermatrix component by a constant, you multiply all the numbers in the component by the same constant.
The effect of doing this is that all the columns become stochastic, that is, they sum to 1.0, and the supermatrix will now converge.
The unweighted supermatrix contains the local priorities derived from the pairwise comparisons throughout the network as shown in Figure 4. For example, the priorities of the elements Aesthetics and Safety, with respect to Bridge A are shown in the two bottom cells of the first column, 0.857143 and 0.142857. This may be interpreted with the statement, "The Aesthetics characteristic of Bridge A is between strongly and very strongly, or 6 times, more its dominant preferred characteristic than its Safety aspect." All the local priority information can be read directly from the unweighted Supermatrix.
Figure 4. The Unweighted Supermatrix for the Bridge Model.
Definition of Component: A component in a supermatrix is the block defined by a cluster name at the left and a cluster name at the top. For example, the (Alternatives, Alternatives) component in Figure 4 is composed of the block of four zeros in the upper left-hand corner shown in the screen clip below.
Detail of (Alternatives, Alternatives) Component from Figure 4