FIGURE 1 Decision Matrix
The "decision matrix" is a useful tool to help make choices between complex alternatives. This particular heuristic can serve a different purpose at each stage in our work. First, it can help to an analyze the problem, the task, or the objective by having them broken down into a number of requirements. Once the requirements have been determined, the heuristic helps sorts them in their relative importance or weights.
FIGURE 1 Decision Matrix
The same heuristic can be used as a "sub-matrix" for combining the various benefits that constitute a complex requirement and for noting their relative contribution towards meeting this requirement. Finally, the heuristic provides a framework for evaluating various courses of action, or choices, in order to select:
More important, enumerating and putting the requirements into this framework makes people think about the requirements and the alternatives at a detail level. The matrix, of course, does not make decisions, it simply lines up the information for the exercise of judgment by the person responsible.
It serves as a two-dimensional checklist which forces one to consider, one at a time, the major factors leading to a decision. For instance, they must choose between the best combination for the given resources, or the least resources for a given combination. Trying for both the best combination and the least resources is like trying to get both ends of a seesaw to go up at the same tune. No technique of can achieve this.
Whether studying a wafer-thin integrated circuit or a new aircraft, design must start by identifying and evaluating what are the important attributes of the objective (customer wants, the desired benefits etc.). These are listed along the top of the matrix shown in Figure 1.
The letters and their subscripts are simply addresses on the matrix. They are used for discussion to identify location. They do come in handy in understanding all sorts of arrays presented in this fashion.
In each locations labeled B1 to Bn write each of the desired attributes. Using only those which differ among the various alternatives being considered. If all the alternatives are equal with respect to an attribute, obviously it would not be meaningful evaluating the alternatives.
Under objective, at W1 to Wn, is put a number, the weighting, showing the relative importance of the attribute with respect to the others. It should be emphasized that the sum of the weightings must equal unity. The weighting cannot be add indiscriminately without adversely changing the others.
If we are very familiar with the product and the requirements, the weights may be assigned directly. Otherwise, we can start by dividing 1 equally among the requirements, and then varying the weights as the relative importance of each requirement is compared with the others, keeping the sum always equal to 1.
The e in the cells represents the "normalized utility" of the alternative with respect to the requirement heading the column. "Just how well does this alternative satisfy the requirement heading this column." If we refer to e12 (e one,two), we mean how well does alternate C1 meet the requirement B2.
We cannot compare alternatives with respect to different attributes and simply add them up. We must consider each alternative with respect to one attribute at a time in terms of a normalized utility scale.
FIGURE 2 Decision Matrix for Wave Guide
Figure 2 shows the required attributes of a complex wave-guide such as: smoothness of surface finish, economy of production, dimensional accuracy, and lightness of weight. The customer, the program manager, and the systems and design engineers have arrived at the weights of 0.2, 0.2, 0.5 and 0.1 respectively, adding up to 1. Now the normalized utilities must be entered into the matrix.
FIGURE 3 Limits for establishing a standard scale.
Figure 3 shows actual values of the parameters are converted into normalized utilities.
Fully as important as finding the commensurable utilities is the standard scale. The normalized utility scale extends between two agreed upon values. These values establish upper and lower limits for the parameter and their utility.
Simply put, the limits set at the least acceptable measurable value of the parameter, and the best practical value (Fig. 4). The direct measure of performance can be related to utility by first identifying the values at which the performance, (a) begins to improve the product and, (b) beyond which it cease to improve the product. These are the performance values that correspond to the minimum and maximum values on the normalized utility scale.
FIGURE 4 Developing normalized utilities for surface
finish as roughness values.
In our example, does 8 microinches have a full normalized utility of 90; and 64 micro inches, 70? This is where the decision maker uses all the information, judgment, wisdom, and insight he can muster. The lower limit of the raw data is define as the "least acceptable", and the upper limit is the "most that can be effectively used", or the "best available", whichever is less.
It is importance that these leasts and mosts really be of equivalent utility among the requirements. The best surface finish must contribute no more and no less than the best price, the lightest weight, or the greatest dimensional accuracy. (See Fig. 4)
The linear function in Fig. 3, used to transform the range of 64 to 8 microinches measures into the 70 to 90 range of utility, is called a normalizing function. All it does is convert variations along one scale to identical variations on a standard scale. In this case, we decided that the effectiveness of the interior finish varied inversely as the surface roughness, but at the same rate.
When Daniel Bernoulli wrote in 1738 that the utility of money to a man declined in proportion to how much money he already had, and that this decline was not uniform but logarithmic, he was stating the law of marginal utility. For our purpose it simply means that the line on the graph which transforms raw data into commensurable units may not always be a straight line or a pat mathematical function.
FIGURE 5 Differnces in cost normalized into economy
of function.
The fact that economy of production may decrease at a different rate than the increase in price is shown by the curvature of the graph in Fig. 5. This curve not only normalizes the "dollar-cost of production" into the 70 to 90 normalized utility scale, but it also adjusts the dollar figures to take into account the "utility" of money to the customer.
It turns out, however, that the concept of utility goes beyond money. The utility of dimensional accuracy is not uniformly proportional to the actual change in tolerances (Fig. 6), and the value to the user of decreased weight is not uniformly proportional to the change in weight (Fig. 7).
FIGURE 6 Normalized utility for tolerances into
dimensional accuracy.
What has been done in Figs. 4, 5, and 6 is to normalize, as well as adjust for utility, thus providing commensurable units which truly reflect both magnitude and utility. In Fig. 3, we simply normalized because the change in surface finish and utility is proportional.
FIGURE 7 Normalized utility for "lightness of weight"
By multiplying the normalized utility en in each cell by the weighting factor Wn heading its column we get that the contribution that the alternatives performance with respect to that requirement makes to the overall value (utility) of that alternative. All that remains is to sum the utilities along the row and find a number expressing the relative utility of the of that alternative with respect to the others. This relative utility includes a balance consideration all the requirements.
In the example Fig. 2, it is seen that electroforming provides the highest expected value, for this particular application. And the reasoning is visible. Others can also see the reasoning that went into this numerical rating. They also see that fabrication could have been the best choice, had cost been more important than dimensional accuracy.
This example illustrates another valuable property of this type of matrix. It is not only useful in ordering our alternatives in a rational manner, but it makes our reasons for the choice visible. persuading others that the preferred alternative should be implemented becomes much easier.
FIGURE 8 Alternate designs
FIGURE 9 Matrix Form
FIGURE 10 Adding weighting factors
FIGURE 11 Adding utility factors
FIGURE 12 Computing weighted utility
FIGURE 13 Finished decision matrix.
The order of the "SUM" gives the order of preference according to the listed criteria.