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# Computing optimal choice

In this step we show you a simple example of the multi-attribute utility approach developed to support complex decisions with multiple objectives.

Multi-Attribute Decision Analysis tells us how to make a decision when the question at hand is complex and has multiple consequences. It was not devised to describe how actual people make decisions. Rather, it was developed to tell us how decisions should be made to ensure they are optimal in the sense that they are consistent with the underlying preferences. It posits that every problem, however complicated it may seem, can be divided into smaller parts, which are easier to solve. Once we solve these smaller problems we can combine what we learned into a final decision.

This approach is based on multiattribute utility concept. To use it, the decision-maker must go through three basic steps. We talk about them below using an example of buying a house, which seems to be a decision complex enough to apply this approach.

Step 1: What are the important attributes

First, we need to divide the big problem into smaller ones. For this, we decide what are the things about possible options that contribute to these options’ attractiveness. These things are called attributes. For the house buying example, important attributes could be, for instance, the time needed to travel to work, the spaciousness of the house, and how nice the neighbourhood is. The table below gives you three houses to choose from along with their attributes.

Step 2: What are the utilities of the attributes values

Second, we need to determine how attribute values translate into utilities. This is done for each attribute separately. It is typically assumed that the utilities used are cardinal, which means that we need to rank the attribute values and decide how much one value is better than the other. Without going into complicated details of how this can be done and what it exactly means, we can notice several things that can happen in the house example. For the work-travel time, we can think that 30 minutes is twice as good as 15 minutes travel, but 35 and 30 minutes are virtually the same value and have the same utility. For the area, we might decide that the 180 m2 house is the best because house A is too small and house B is too big. The neighbourhood attribute is the most subjective one, and we cannot measure it in metres or seconds, which we will later translate into utilities. Say that our preferences are reflected by the following numbers:

Step 3: How important is each attribute

Before we combine the information on attributes’ utilities into options’ utilities, we have to determine how important each attribute is. For this, we need to decide, for example, how willing we are to resign from getting to work fast in exchange for a better space. Once again, without going into details of how this can be done and what it exactly means, we can, for instance, decide that getting fast to work is worth twice as much, and assign twice higher weight to the travel time attribute. We choose the weights so they sum up to 100%. We could, for example, come up with the following solution:

### Making a decision

Once we complete the steps listed above, the utility of each option can be computed if we make some additional assumptions. It is the simplest to assume that the final utility is a sum of the utilities of the attributes weighted by how important each attribute is. For example, the utility of HOUSE A is given by:

Utility of travel time from house A x importance of travel time + utility of house A area x importance of area + utility of the house A neighbourhood x importance of the neighbourhood

Or in numbers: 10 x 0,60 + 4 x 0,30 + 6 x 0,10 = 7,8.

Similarly, the utility of HOUSE B is: 5 x 0,60 + 6 x 0,30 + 10 x 0,10 = 5,8.

And for HOUSE C it is: 5 x 0,60 + 10 x 0,30 + 8 x 0,10 = 6,8

Given the computed utilities, we should choose HOUSE A.

### Evaluating the decision

The last step of the procedure of Multi-Attribute Decision Analysis is evaluating whether we are happy with the reached decision. The outcome of the analysis can differ from our intuitive choice. This can happen because some more emotional attributes are not well represented in the attributes. For example, a certain house just may “feel good” to live in, despite some attributes being far from optimal. This opens up the door for reevaluating what is important in the given situation. It also shows us that ultimately, it is the intuition that we rely on when making a decision

If you are interested in a simple and non formal description of an example of using this procedure you can read this short paper.