# Multiattribute utility theory

Engineers are always making design decisions, whether it's the design of a thermal fin or the location of a new manufacturing plant. Poor decisions could result in the loss of money, resources, and time. Therefore, it is important that engineers make logical and well reasoned decisions.

However, the decision process can prove to be quite complicated, especially when trade offs need to be made, such as between strength and weight of a beam. Given the complexity of technology and systems, when there are dozens of attributes, there are can be hundreds of alternatives to choose from, which can lead to a seemingly infinite number of possible combinations. So, how does one choose the best combination?

The purpose for using utility theory in decision making is to create a mathematical model to aid the process. It gives the decision maker the ability to quantify the desirability of certain alternatives. Utility theory is for design scenarios where uncertainty and risk are considered. The end result of using this method is a function which represents the designer's preferences, given a certain set of design attributes

The validity of this method is questioned by Barzilai, based on the reasoning that mathematical operations on utility functions are incorrect. Barzilai's arguments are not the consensus view in any research community, and the interested reader is encouraged to consult his reference for more details.

# Axioms

Decision theory has a set of axioms, which are meant to eliminate inconsistencies and suboptimal choices when it comes to trade-offs and uncertainty. They provide the basis for good decision making, and structures the problem so determining the utility function is more straightforward. The first three axioms are for determining a value function to allow ranking of alternatives. The second three axioms structure a preference function.

• Completeness of complete order: X is preferred/less preferred/equally preferred to Y
• Transitivity: if $X \succ Y$ and $Y \succ Z$, then $X \succ Z$
• Monotonicity: more of an attribute is preferred, or less of an attribute is preferred
• Probabilities exist and can be quantified
• Monotonicity of Probability: decision maker prefers a larger probability of a good outcome than a smaller probability of a good outcome
• Substitution Independence: preferences are linear with respect to probability

However, it is important to remember that the axioms are just guidelines, not rules, and are meant to guide decision-making. There may be cases where the axioms do not accurately represent a particular problem. But in general, if the decision-maker's stated preferences violate the axioms, it is more likely that the decision maker is inconsistent.

# Building the Utility Function

To determine the utility value, or the desirability, of the design, there are six steps.

1. Identify significant design attributes and generate alternative designs
2. Verify relevant attribute conditions or bounds
3. Use the lottery (described below) to determine the designer's preference
4. Evaluate Single Attribute Utility (SAU) function and trade-off preferences
5. Combine SAUs into Multi-Attribute Utility function (MAU)
6. Select alternative with the highest MAU value by ranking the alternatives

## Selection of Attributes

Attributes are selected so that the designer's preference will be reflected in the attribute features. The range of the attribute must be selected so that it it is useful, manageable, and should indicate the expected performance of the design. When choosing the attributes, they must be:

1. Complete, such that important aspects are reflected in the design formulation
2. Operational, so that design decision analysis can be meaningfully implemented
3. Non-redundant, so there's no double counting
4. Minimal, for simplicity

## Selection of Attribute Bounds

The upper and lower bounds of an attribute are chosen by the designer. It is possible to use mathematical optimization techniques to choose the limits, however there is no rule as to the size of the range. The range of the attribute can change the weight of the scaling factors, when using the multiattribute utility model.

## Lottery

The lottery is the step in the process where the designer's preferences are determined. In this step, the designer needs to make a decision between two choices. The first choice is the having the probability p of the most preferred alternative or 1-p of the least preferred alternative. The second choice is the absolute certainty of a particular alternative, or the certainty value, between the most and least preferred. The goal of the lottery is to determine the probability p where the decision maker is indifferent between the two choices. The indifference between the two choices is called certainty equivalence.

For example, let's say a decision maker had the choice between getting a million dollars, or a p chance of getting 5 million dollars and 1-p chance of getting 0 dollars. If the decision maker was indifferent between a 75% chance of getting 5 million and 25% chance of getting nothing, and getting 1 million, then the certainty equivalence occurs when p = .75, and the 1 million is the certainty value.

## Development of Single Attribute Utility

SAU functions are obtained by using a set of lottery questions based on certainty equivalence. They are monotonic functions, where the best outcome is set at 1, and the worst at 0. SAU functions are then developed to describe the designer's compromise between the best and worst alternatives based on the lottery questions.

An analytical function is typically used for preference description, and exponential functions are usually used to describe its shape. The general form is u(x) = aebx + c, where a and c are parameters which guarantee the utility is normalized between 0 and 1, and b is the risk coefficient which shows degree of risk attitude, reflecting rate at which risk attitude changes with different attribute level.

The three data points used to determine the unknown coefficients are obtained from the equation u(x) = pu(xo) + (1 − p)u(x * ), where x is the certainty value, xo is the best alternative, and x * is the worst alternative. Given that the utility is scaled between 1 and 0, u(xo) = 1, u(x * ) = 0, so u(x) = p.

Using the example from the previous section, the three points used to determine the unknown values of a, b, and c are the following:

u(5 million) = 1

u(0) = 0

u(1 million) = 0.75

## Example of Single Attribute Utility

For an entry level engineering job, let's say that a decision maker is trying to determine their preference for jobs based on salary. The upper end is a salary of $75,000, which is assigned to a utility value of 1. The lower end is a salary of$40,000, which is assigned to a utility value of 0.

Let's also say that for a salary of \$50,000, the decision maker's indifference probability is 0.75. Using the the three points u(0) = 40,000, u(1) = 75,000 and u(0.75) = 50,000, the exponential form u(x) = aebx + c can be solved, which is the single attribute utility function.

Note: In a real engineering problem, the upper and lower bounds can be determined through optimization methods.

## Development of Multi-Attribute Utility

When certain independence conditions are met, a mathematical combination of all the SAU functions, with scaling constants, results in the MAU function, which is the overall utility function with all attributes considered. Scaling constants reflect designer's preference on the attributes, which is based on scaling constant lottery questions and preference independence questions.

The form of the MAU function depends upon the particular independence conditions fulfilled by the different SAU functions. Keeney and Raiffa are a good reference for details beyond what is presented here. The general form for the condition that each attribute is mutually utility independent, meaning that each attribute satisfies the utility independence condition with every subset of the attribute set, then the following equation holds true.

$u(x) = \sum_{i=1}^n k_iu_i(x_i) + k\sum_{i=1,j>i}^n k_ik_ju_i(x_i)u_j(x_j)$

$+ k^2\sum_{i=1, j>i, l>j}^n k_ik_jk_lu_i(x_i)u_j(x_j)u_l(x_l)$

+ ... + kn − 1k1k2...knu1(x1)u2(x2)...un(xn)

where u is normalized between 0 and 1, ui(xi) is a normalized single attribute utility function, k_i is the scaling constant for the single attribute utility function, and K is a scaling constant that is the solution to $1 + K = \prod_{i=1}^n (1 + Kk_i)$. By factoring the equation and applying the conditions of preferential independence, utility independence, and additive independence, this general form can be simplified to the additive and multiplicative forms.

The multiplicative formulation is used if and only if preferential independence and utility independence conditions are satisfied. It is in the form

$U(x) = \frac{1}{K}({\prod[Kk_iU_i(x_i)+1]}-1)$

where U(x) is scaled from 0 to 1, xi is the performance level of attribute i, Ui(xi) is the SAU for attribute i, ki is the single attribute scaling constant, and K is the normalizing constant which scales U(x) from 0 to 1. The value of K is derived from

$1+K = \prod(1 + Kk_i)$

The advantage of an additive formulation is its simplicity, but assumptions can be restrictive. It is only valid when preferential, utility and additive independence conditions are satisfied. It takes the form

$U(x) = \sum k_iU_i(x_i)$

The values of ki can be solved by generating a system of n equations, the equations being the SAUs, with n unknown ki's. It would require another set of utility functions where the utility value is the same. However, this can by difficult to solve. Therefore, finding the ki's is typically determined through a second set of lottery questions.

For determining the scaling constants, two types of questions can be used. Question 1: For what probability p are you indifferent between:

• the resulting lottery giving a p chance at the best alternative and a 1 - p chance at the worst, and
• the result of an set of alternatives between the two.

The decision maker's answer comes out to the value of the scaling factor.

Question 2: Select a level for attribute 1, x1', and a level for attribute 2, x2', so that for any fixed levels of all other attributes, you are indifferent between:

• the result of getting x1' and the worst alternative 2 option and
• the result of getting x2' and the worst alternative 1 option.

The utilities can be then expressed as $k_iu_i(x_i^') = k_ju_j(x_j^')$. From here, one of the k's can be solved.

For more information regarding the derivation of the multiplicative and additive formulations as well as determining the scaling constants, see the Keeney and Raiffa reference.