The Abacus Multi-model Decision Support System™ (MDSS™) is a software package that integrates three common and one proprietary decision models into a single unified system for increased effectiveness in making critical long-range decisions.  The four models are (1) Decision Trees, (2) the Multi-Attribute Utility Model (MAUM), (3) the Abacus Priority Rank Order Model™ (PROM™), and (4) Probability Calibration (PC).  A user-friendly Window interface allows a Decision Maker to navigate his expanding decision tree while using the PROM™ and/or MAUM models to determine utility values.  Special nodes allow specification of events with probabilities depicting situations that are out of the Decision Maker′s control.  The Probability Calibration model aids in refining probability estimates.  A "rollback" algorithm determines the best alternative.  In addition, a sensitivity analysis process shows the Decision Maker which part of the tree is the most effective to expand.  The MDSS™ Configuration Management Module™ (CMM™) allows a number of individuals to participate in a group decision-making process.

Decision Trees

A Decision Tree is a hierarchical structure for aiding an individual who must select one of a number of mutually exclusive alternatives available to him sometime in the future.  For example, a manager may wish to use a Decision Tree to help him decide which company to award a contract to.  In the figure on the right, three alternatives are considered: Company X, Y or Z.  The "decision node" is depicted by a square box with the alternatives shown as branches.  A "utility value" must be determined for each branch.  This is accomplished by direct assignment or by using one of the available decision models.  The best alternative is the branch with the highest value.

The values placed on the branches of any decision node in the tree are assumed to be preliminary estimates.  By using the available models, these estimates are refined and hopefully approach a more accurate value representing the Decision Maker′s utilities for the choices.  One method of refinement is tree expansion.  One of the currently ending ("leaf") branches is given a new node and branches of its own.  The node can be one of two possible types: (1) decision node or (2) event node.  A new decision node behaves like the box shown in the figure.  An event node, however, represents possible situations that are out of the Decision Maker′s control.  Consequently, every branch emanating from an event node must be assigned a probability of occurrence in addition to a utility value.  The value of an event node is the expected value of its branches: the sum of the product of each utility value and the corresponding branch probability.  Generally, the more the tree is expanded, the more accurate the values of the main decision node become and the more valuable the tree is to aiding the Decision Maker in making a choice.

The best alternative is determined by using a "rollback" procedure that starts from the leaf nodes and calculates the utility value of every internal node in the tree.  The recommended choice, then, is the major alternative with the highest calculated utility value.

MAUM Analysis

Regardless of how much the tree is expanded, there is still the problem of assigning reasonable utility values to all unexpanded leaf branches.  Rather than personally estimating values and placing them directly into the tree, the Decision Maker can use another decision model to help make a determination.  The Multi-Attribute Utility Model (MAUM) can be used to fix values on any subset of the branches of any node in the tree.

To use the MAUM, the Decision Maker must generate a number of attributes that are relevant to the decision node being analyzed.  Then, a utility value must be assigned to each alternative for each attribute.  A utility matrix is usually used to structure the values. Second, the Decision Maker must assign relative weights to the set of attributes.  These weights, which must sum to 100%, represent the importance of each attribute to the decision.  After value and weight assignments, the matrices are normalized and an expected value is calculated for each alternative by summing the product of the normalized value with the corresponding weight.  If desired, MDSS will insert the final values into the decision tree at the proper nodes and become the starting point for the tree rollback.

PROM™ Analysis

As an alternative to MAUM, the Abacus Priority Rank Order Model™ (PROM™) can be used to determine the values of any subset of branches at any node.  This approach also requires attributes but it is not necessary to assign fixed values or weights.  Instead, every alternative is ranked with the others in a list for each attribute.  Using the ranks, a set of utility values emerges for each of the selected alternatives.  These values can be sent back to the Decision Tree and inserted into their proper nodes.

Sensitivity Analysis

One of the problems in using a Decision Tree is determining which node to expand next.  The Decision Maker does not want to waste time expanding a node if it will have a small effect in determining the best decision.  Sensitivity Analysis can help find the most important parts of the tree.  MDSS™ asks the following question for every leaf branch: "How much does the branch value have to change before there is a shift in the best initial decision?"  The node that has the highest calculated value is the most "sensitive" and should be the one to expand.

Where do the numbers on the tree come from?

The Abacus integrated model approach consists of building a decision tree while using utilities and probabilities to determine branch values with MAUM and/or PROM™ used to determine values of the leaf branches in the expanding tree.  Generally, the larger the decision tree, the better the final recommendation provided that the assigned values represent strong insight by the Decision Maker.

Utility Values

The utility number assigned at a leaf branch represents the Decision Maker′s determination of the value or "worth to him" of the entire path back to the root node.  At each decision node on the path, he asks, "Suppose I did this?" and at each event node, he asks, "Suppose that this happened?"  At the end, he asks, "What would this final situation be worth to me?" and assigns a number between 0 and 100.

MAUM and PROM™

The MAUM and PROM™ models aid in assigning utility values to the branches of a decision node.  With MAUM, real numbers like cost, warrantee time, average life time, etc can be placed into the model and utility values are produced that can be put into the decision tree.  It is not necessary for the Decision Maker to provide estimates out of his head.  With PROM™, even real numbers are not necessary.  It uses rank-order comparisons to produce utility values.

Using the output of the MAUM and PROM™ models as input to the decision tree works only when the branches of the decision node are directly related to the alternatives used in the model.  For example, suppose that a decision node in a company decision tree has branches, (1) "Hire Mr. A", (2) "Hire Mr. B", and (3) "Hire Mr. C" and suppose that if Mr. B is hired, the company must buy him a car.  The MAUM or PROM™ models can be used to pick the best car with attributes such as automobile cost, reliability, looks, etc. but the final numbers cannot be transferred back into the decision tree because they do not describe the worth of hiring Mr. B.  However, if the model contained attributes such as personal ability, reliability, timeliness, etc., the output numbers would have meaning in the tree.  For intermediate situations, a feature has been provided that allows the output of the models to modify the current tree estimates within a limited range.

Estimating Probability Values

The probability numbers on the branches represent the Decision Maker′s determination of how likely it is for a particular event to occur given the entire preceding path back to the root node.  The sum of the probabilities emanating from any given event node must be 100% with each one greater than zero.  The Decision Maker must first estimate these numbers to the best of his ability.  Then, the MDSS™ Probability Calibration model can be used to refine the estimates.  Based on a sample of previously estimated probabilities for events that have actually occurred, a newly estimated probability can be adjusted given the past performance of the Decision Maker.

User interaction sequence for the MDSS™ Decision Tree Module demonstration

Configuration Management Module (CMM)