Index
TABLE OF CONTENTS:

1.	Introduction

2.	Terms

3.	Program Start-Up Window

4.	Select Matrix Type & Matrix Set Up

5.	Relative Value Matrix

6.  	Multiplication Matrix

7.	Weighting of Criteria

8.	Override of Criteria Weights

9.	Consistency Ratio

10.	Sensitivity Analysis

11	Reordering the Matrix

12.	Matrix Output

13.	Saving Data

14.	Importing to Powerpoint


Introduction
1.  Introduction:

a.	The Decision Matrix (DECMAT) is a tool decision-makers can use to assist them in solving problems with multiple, and often competing, decision criteria.  Some of these criteria may be more important than others.  This DECMAT program provides a structured approach, first to establish criteria weights and then to apply these weights within the decision matrix.  In this way, decision-makers have a relatively objective tool to use for their decisions.

b.	This program was developed for use at the Combined Arms and Services Staff School (CAS3) at the U.S. Army Command and General Staff College, Ft. Leavenworth, Kansas.  The program author is a 1997 graduate of CAS3.  This program is considered freeware and is intended for dissemination throughout the U.S. Army.

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Terms
2.  Terms:

a.	Objective:  The desired end state or goal of the decision.

b.	Evaluation Criteria (or States of Nature): Aspects of the situation over which the decision-maker has no control.

c.	Courses of Action (or Strategies): Alternative choices available to the decision-maker.

d.	Raw Data (Payoffs): The dependent variable.  Results of the interaction between the criteria and the courses of action (i.e., the cost of a course of action given an evaluation criterion).

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Program Start-up
3.  Program Start-Up Window:

a.	Upon executing the DECMAT Program, one finds an empty, unlabeled 2x2 DECMAT in the program start-up window.  This program has the capability to work up to a 10x15 matrix.

b.  This program always calculates values based on lower values being better, both within the matrix and for the total column.

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Matrix Type and Matrix Setup
4.  Select Matrix Type and Matrix Set Up:

a.	Select the matrix type by dragging down the Matrix menu and highlighting either the Multiplication Matrix or the Relative Value Matrix.  An alternate method to select the matrix type is to select the circled M or R on the tool bar to select a Multiplication or Relative Value Matrix, respectively.  The default selection is a "Multiplication Matrix."

b.  Evaluation Criteria are shown along the top of the matrix.  Click on the + or - button with the horizontal arrow (located on the tool bar) to add or delete Evaluation Criteria columns.  Add the criteria names to title boxes at the top of the matrix; the title boxes accept two lines of entry.  By convention, Evaluation Criteria are shown in order of weight, descending from left to right in the matrix.  See Weighting of Criteria for further explanation.

c.  Courses of Action are shown along the left side of the matrix.  Click on the + or - button with the vertical arrow (located on the tool bar) to add or delete Courses of Action rows.  Add the Courses of Action names to the boxes on the left side of the matrix; the Course of Action name boxes accept two lines of entry.

d.  Add a matrix title to the box at the top of the matrix.

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Relative Value Matrix
5. 	Relative Value Matrix:

a.	This matrix type is the easier of the two methods to use; however, this method is also less accurate since it does not account for the magnitude of difference between raw data values.

b.	To use the relative value matrix:

(1)	For a given evaluation criterion, rank the courses of action based on the payoff raw data.

(2) 	Assign 1 to the best course of action for a given evaluation criterion and continue rank ordering the rest of the courses of action within that evaluation criterion.

(3) 	If two or more courses of action have the same raw data value within a given evaluation criterion, average their rankings and assign that average to each of these courses of action.

EXAMPLE:

Two courses of action are tied for the 2d and 3d ranking.

(2+3) / 2 = 2.5

Assign 2.5 to each course of action.

(4)	Input the relative values to the DECMAT in the appropriate raw data box corresponding to that combination of Course of Action and Evaluation Criterion.

c.	Computation:  The relative value method computes the total for each Course of Action by adding the products of each evaluation criterions relative value times the evaluation criterions weight along a Course of Action row (see Weighting of Criteria), as follows:

(RV1)(W1) + (RV2)(W2) + . . . + (RVn)(Wn) = TotalREL VAL

Where:	RVn = Relative Value for the Raw Data of the

nth Evaluation Criterion

Wn = Weight of the nth Evaluation Criterion

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Multiplication Matrix
6.	Multiplication Matrix:

a.	This method is the more accurate of the two methods since it uses the actual raw data payoffs for each course of action in its computation.  In this way, the multiplication method can account for the magnitude of difference among an evaluation criterions raw data for the Courses of Action.

EXAMPLE:

In a case where the raw data for a criterions three courses of action is 3, 21, and 22, the method uses the values of 3, 21, and 22 to determine its solution rather than assigning the less precise relative values of 1, 2, and 3.  Notice how the middle value would have an advantage in the relative value matrix by appearing to be closer in value to the best value than it actually is.

b.	To use the multiplication matrix:

(1)	Input the raw data directly to the DECMAT in the appropriate raw data box corresponding to that combination of Course of Action and Evaluation Criterion.

(2)	Min and Max Tabs.  Below each Evaluation Criterion column is a tab for Min and Max.  The default is that Min is selected.  The matrix in the DECMAT program is designed to analyze on the basis that lower values are better.  To properly use this matrix:

·	For each Evaluation Criterion, determine whether the value for that Evaluation Criterion is better as a higher number or as a lower number.

·	If the Evaluation Criterion is better as a lower value, leave Min selected below that Evaluation Criterion and analyze the next Evaluation Criterion.

·	If the Evaluation Criterion is better as a higher value, select the Max tab appearing below that Evaluation Criterion.  The program will automatically invert this value in the matrix during its computation so that the larger values are expressed numerically as smaller values.  The final printout will show these values as inverted fractions.

EXAMPLE:

For an automobile purchase, you are considering Fuel Economy as an Evaluation Criterion, measured in miles per gallon (MPG).  In this case, higher values are better.  By selecting the Max tab below the Evaluation Criterion, you will have the program invert these values.  The values of 15 MPG and 20 MPG become 1/15 MPG and 1/20 MPG, respectively, in the matrix as the program calculates the total.  The final print out will show 1/15 and 1/20 for these values.

(3)	Computation.  The multiplication method computes the total for each Course of Action by multiplying each evaluation criterions raw data value taken to the exponential power of the evaluation criterions weight (see Weighting of Criteria), as follows:

(RD1)(W1) . (RD2)(W2) . . . . . (RDn)(Wn) = TotalMULTIPLICATION

Where:	RDn = Raw Data of the nth Evaluation Criterion

Wn = Weight of the nth Evaluation Criterion as

the exponent

(4)  Limitations on the use of the Multiplication Matrix.  One cannot use the multiplication method, and therefore must use the relative value method, if either of the following conditions exists:

·	Evaluation Criterion which cannot be expressed with a numerical value.  Do not assign relative values for use in a multiplication matrix, as this introduces inconsistencies derived from using a hybrid matrix.

·	Raw Data with a zero value.  Zero multiplied by anything will always equal zero for the total, no matter how large other criteria values may be.  For this situation, one may redefine the evaluation criteria in a way that results in values other than zero.  If this cannot be done, one must use the Relative Value Matrix.  Do not simply add 1 (or any other value) to the raw data as this will introduce subjective inaccuracies to the matrix.

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Weighting of Criteria
7.	Weighting of Criteria:

a.	Some criteria will often be more important than other criteria.  This program uses the technique of Pairwise Comparison to lend objectivity to what otherwise would be a subjective ranking of many criteria simultaneously -- something the human mind has difficulty doing.  In short, the program translates a verbal comparison of the relative importance of the Evaluation Criteria into numerical values and then uses a mathematical model to determine an appropriate weight that accurately reflects the logic.

b.	To determine criteria weights using the Pairwise Comparison techniques:

(1)	Step 1:  Rank order the evaluation criteria in general importance as you see them at this point.

EXAMPLE:

1	Evaluation Criterion #1

2	Evaluation Criterion #2

:	:

:	:

n	Evaluation Criterion #n

(2)  Step 2:  Pull down the Matrix menu and select Criteria Weights.  The window for determining Pairwise Comparison will appear.

(3)  Step 3:  Review the factors for criteria comparison at the bottom right of the screen:

1	Equal

2	Slightly Favored

3	Favored

4	Strongly Favored

(4)  Step 4:  Input the numerical importance factor to the chart by comparing each of the Evaluation Criteria with each of the other criteria.  The example below shows the process by which one can determine the pairwise comparison:

EXAMPLE:

·	Which Evaluation Criterion is more important ... Evaluation Criterion #1 or Evaluation Criterion #2?

Answer:  Evaluation Criterion #1

·	By what importance factor?

Answer:  3 (Favored)

·	Enter 3 in the box where Evaluation Criteria #1 along the left intersects Evaluation Criterion #2 along the top.

·	Continue to evaluate all pairs until the chart is complete.

(5)  Step 5:  Conduct a logic check.  All pairwise comparison importance factors should increase in value or be equal in value as you move from left to right along a row of the chart.

EXAMPLE:

EC#2	EC#3	EC#4

EC#1	  3      --->	  3     --->	  4

EC#2	  	  1     --->	  3

EC#3	  	  3

If the values do not increase or remain the same as you move from left to right in the chart, one of two conditions exists:

(a)  Evaluation Criteria are originally rank ordered out of sequence.  To solve, change the rank order of the Evaluation Criteria and rework the Pairwise Comparison.  Change the values of the columns in the DECMAT to show the Evaluation Criteria in the same order.

(b)	The logic of the pairwise comparison is incorrect.  To solve, review the pairwise comparison factors.

Conduct a similar check within a given Evaluation Criterion.  The values should decrease or remain the same as you move from top to bottom in the chart.

(6)  Step 6:  To print the Pairwise Comparison chart, select "Print."

NOTE:  To return to the DECMAT without sending the calculated weights to the matrix, select "Cancel."

(7)  Step 7:  Select the Send button.  The computer calculates the weights for each of the Evaluation Criteria, imports the values to the DECMAT, and returns you to the DECMAT window.

c.  The mathematical model which determines the criteria weighting is based on the Eigenvector method as described by Ching-Lai Hwang and Kwangsun Yoon in their monograph Lecture Notes in Economics and Mathematical Systems (186: Multiple Attribute Decision Making:  Methods and Applications) (Springer-Verlag, Berlin, 1981).  The methodology for solving the Eigenvalues for a specific Eigenvector is described by Anthony J. Pettofrezzo in his book, Matrices and Transformations  (Dover Publications, Inc., New York, 1966).

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Override of Criteria Weights
8.  Override of Criteria Weights.

a.  To override one or more of the Pairwise-Comparison-calculated weights, select Criteria Weights from the Matrix menu.  Understand, however, that overriding the Pairwise Comparison weights will introduce an element of subjectivity to the matrix.

b.	The "Weight of Evaluation Criteria" screen will appear listing the Evaluation Criteria and their weights.

c.	To enter a different weight, delete the Pairwise-Comparison-calculated weight and insert the new value for the weight.

d.	Select Print to print the revised weights.

e.	Select Cancel to return to the DECMAT.  The original weights will remain unchanged.

f.	Select Override Off to return to the Pairwise Comparison screen.  The original weights will remain unchanged.

g.	Select Send Values to input the revised values to the DECMAT and to return to the DECMAT screen.

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Consistency Ratio
9.  Consistency Ratio:

a.  After you have determined the criteria weights, the program determines the Consistency Ratio.  The Consistency Ratio is a numerical value measuring how well the Pairwise Comparison values maintain a logical series of relationships.  The program uses another mathematical model, based on the least squares method, to measure how well the logic fits.

b.  The program shows the Consistency Ratio as a percentage below the right side of the DECMAT.

c.  For this particular mathematical model, a Consistency ratio of 95% or more means the logic of the Pairwise Comparison is acceptable enough to use the resultant weights in the DECMAT.

·	NOTE:  CAS3s previous version of the DECMAT program used a different mathematical model to model the logic, resulting in an acceptable Consistency Ratio threshold of 90%.  The value of 90% in the older model is equivalent to 95% in the new version.

d.  Should you input the Pairwise Comparison so that the resultant Consistency Ratio is below 95%, an error box will appear alerting you of this fact.  Re-evaluate your Pairwise Comparison for logic errors.

e.  With a large number of Evaluation Criteria, the program model may not catch a single, obvious error in the Pairwise Comparison logic.  This limitation to the model makes Step 5 of the Pairwise Comparison very important.

f.   The mathematical model which determines the Consistency Ratio is based  on the method described by Cliff T. Ragsdale in his book, Spreadsheet Modeling and Decision Analysis: A Practical Introduction to Management Science, 2d Edition (South-Western College Publishing, Stamford, Connecticut, 1997).

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Sensitivity Analysis
10.	Sensitivity Analysis:

a.	Sensitivity analysis identifies the degree to which the DECMAT results are subject to change with only small changes in the evaluation criteria weights.  A solution which is not sensitive to changes in weights provides the decision-maker with confidence that he or she has a valid solution.

b.  Together with the Consistency Ratio, Sensitivity Analysis is a measure of the DECMATs subjectivity.  A solution that is "sensitive" to changes in weights is a red flag for the decision-maker to re-evaluate the relationships among the Evaluation Criteria.  With a sensitive solution, the decision-maker must review the pairwise relationships of the criteria to see if the criteria relationships really reflect how the decision-maker feels.  In other words, the decision-maker must develop confidence in the criteria weighting.

c.  The program conducts sensitivity analysis by changing each Evaluation Criterion weight independently, within the range of plus or minus three points, and recalculates the matrix to determine whether the solution changes.  The program does not re-calculate when the weights are taken below the value of 1.00.  The program resets the criterion weight to its original value before proceeding on to analyze the sensitivity of the next Evaluation Criterion.

d.  EXAMPLE:  For an Evaluation Criterion weight of 2.38, the program:

·	Sets the weight incrementally lower (e.g., 2.37, 2.36, etc.), recalculates, and determines the weight where the solution may change.  NOTE:  In this example, the program does not set the value lower than 1.00 and therefore does not complete the calculation within the full range of -3.

·	Sets the weight incrementally higher (e.g., 2.39, 2.40, etc.), recalculates, and determines the weight where the solution may change.  Stops calculating at the weight of 5.58 to complete the range of +3.

·	Resets the weight to the original value of 2.38 and moves to analyze the next Evaluation Criterion.

e.  To conduct a Sensitivity Analysis, drag down on the "Matrix" menu and select "Sensitivity Analysis.

f.  The Sensitivity Analysis screen will appear with the completed results.

(1)  To print the Sensitivity Analysis, select "Print."

(2)  To return to the DECMAT screen, select "OK."

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Reordering the Matrix
11.	Reordering The Matrix:

a.  By convention, show the Evaluation Criteria in decreasing order of weight from left to right in the matrix.

b.  To reorder a Evaluation Criterion, place the cursor along the top of the matrix above the Evaluation Criterion desired until a hand appears.  Click and drag the hand to the desired location for the Evaluation Criterion and release the mouse button.  The Evaluation Criterion will be "dropped" at that location, reordering the Evaluation Criteria as a column.

c.  Reorder the Courses of Action in similar manner.

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Matrix Output
12.	Matrix Output:

a.	The DECMAT program calculates the totals of each Course of Action and shows the values in the far right column of the matrix.  The program re-calculates the totals whenever a value is changed.

b.	The type of matrix selected (Relative Value or Multiplication) is shown at the bottom right of the matrix.

c.  	The Consistency Ratio also appears at the bottom right of the matrix.

d.  	The program circles the recommended solution (the smallest total value).

e.	Should the total values be extremely large or small such that the number, the program sets the total values to the same order of magnitude (power of 10) and shows this power of 10 at the bottom right of the matrix.  An exception to this occurs when the difference in the numbers is greater than 108.

f.  	You may print the DECMAT, showing all comments at the matrixs bottom right, by selecting the printer button along the tool bar or by dragging down on the "File" menu and selecting "Print."

g.	You may print the Pairwise Comparison chart by dragging down on the "Matrix" menu and selecting "Criteria Weights."  When the Pairwise Comparison screen appears, review the values to ensure they are correct and select "Print."

h.	You may print the Sensitivity Analysis by dragging down on the "Matrix" menu and selecting "Sensitivity Matrix."  When the Sensitivity Matrix screen appears, select "Print."

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Saving Data
13.  Saving Data:  To save your file:

a.	Select the disc button on the tool bar or drag down on the "File" menu and select "Save."

b.	Name the file as a "*.dec" file.

c.	Select the correct directory and select "OK."

d.	To resave a file, select the disc button on the tool bar or drag down and select Resave.  Follow the same procedure as above.

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Importing to PowerPoint
14.  Importing to PowerPoint:

a.	There is no direct method to import the matrix to PowerPoint.

b.	To depict the decision matrix in a PowerPoint presentation:

(1)  Open PowerPoint concurrently with the Decmat program.

(2)  Print Screen in the Decmat program.

(3)  Select Paste in PowerPoint.

(4)  The Decmat screen will appear in a bit map format in PowerPoint.

c. 	As the bit map format is memory intensive, one may:

(1)  Crop out the tool bar portion of the Decmat screen using the Crop Picture function of PowerPoint.

(2)  Resize the matrix to fit using the object corner.

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Aspects of the situation over which the decision-maker has no control


A verbal comparison of the relative importance of the Evaluation Criteria into numerical values