### Problems

Problems do **NOT** involve the allocation of resources; they generally involve physical systems governed by the laws of nature and are removed from human values and subjectivity. A few examples of forms problems may manifest in:

- Given the state of a system at one point in time, determine the state of the system at another point in time
- Given elements of the state of a system at a point in time, determine other elements of its state at the same time
- Determine the logical conclusions that can be drawn from a set of data

Problems have answers that depend only on the problem statement, data, boundary conditions, and the laws of nature and mathematics.

Examples:

- Given the current position and velocity vectors of a spacecraft, determine its velocity 30 seconds from now
- Determine the strain on the front axle of a Ford F-150 given these conditions…
- Given these lens measurements, can the refractive index of the glass be determined?

### Decisions

Decisions involve resource allocation, and the decision that represents the best decision depends on a subjective set of preferences. In making a decision, one has to consider concepts such as **options**, **expectations**, and **values** – these are defined further down.

Examples:

- Due to budget cuts, a manager has to fire one of his employees. Should the manager fire employee X or employee Y?
- Should NASA complete a manned mission to the moon again prior to a manned mission to Mars?
- Should the aircraft fuselage be built with Carbon Composites or Aluminum?

**Values- **An expression of human needs, wants, and desires. A scalar measure that enables the rank ordering of all feasible choices of control variables.

**Expectations-** Range of possible outcomes of a decision paired with their probabilities of occurrence. The point of modeling is to determine the expectation for each option. They are more related to physical outcomes that value.

** ** IMPORTANT:** **Expectations are NOT the same as values!!!

**Options-** A set of control variables. Different courses of action we can decide to take.

### What is the Preferred Decision?

The preferred decision is the option whose expectation has the highest value.

### Transitive vs. Intransitive

An example of a transitive preference ordering is:

A>B>C

An example of an intransitive preference ordering is

A>B>C>A

This is derived from the transitive property of math which states that if A>B and B>C, then A>C. In the intransitive case, we see that if A>B and B>C, it is still possible that C>A.

### Irrational

An individual who does not make decisions that reflect his/her stated preference. An individual who makes an intransitive preference ordering is considered irrational.

### Arrows Impossibility Theorem

A group of fully rational individuals does NOT imply the group will exhibit transitive preferences, and, in general, the group will NOT exhibit transitive preferences. Group utility is likely NOT transitive.

### Utility

An abstract generalized concept to quantify value. Utility may take the form of money, happiness, or a combination of multiple attributes depending on the context.

### Expected Utility Theory

The utility of a lottery is the sum of the utilities of all possible outcomes of the lottery weighted by their probabilities of occurrence. This theory provides a measure of utility under uncertainty and risk.

**6 Axioms of vN-M Utility (von Neumann – Morgenstern Utility)**

- All outcomes of a vN-M lottery can be ordered in terms of the decision maker’s preferences, and that ordering is transitive (remember, this condition is necessary in order that rational decision making be possible).

- Any compound lottery, that is, any lottery that has as an outcome another lottery, can be reduced to a simple lottery that has among its outcomes all the outcomes of the compound lottery with the associated probabilities of their occurrence (this condition equivalences compound and simple lotteries).

- If the outcomes of a lottery. A,, A2,. . ., A^, are ordered from most desired to least desired, then there exists a number p , 0 < p < 1, such that one is indifferent between an outcome A,, 1 < i < r, and the lottery A1 with probability p and A, with probability p – 1 (this assures the continuity of preferences between outcomes At and A^).

- For any lottery such as that given in axiom 3, with p specified, there exists a certainty outcome S, that can be substituted for A,, and the preferences of the decision maker will remain unchanged (this axiom provides that any lottery can be reduced to an equivalent lottery that contains only the outcomes Ai and A^).

- The decision maker’s preferences and indifferences among lotteries is transitive (this assures that rational preferences exist among lotteries).

- Given two lotteries, each with only two outcomes, and which differ only in terms of the probabilities of the outcomes (that is, they have identical outcomes), the lottery in which the probability of the more desired outcome is higher is the preferred lottery (this is a statement of preference).

### Exogenous Variables

Variable that affect the system, but engineers can’t control. Examples include the weather, geopolitics, the future availability of resources etc.

### References

Hazelrigg, George A. “A framework for decision-based engineering design.”*Journal of mechanical design* 120.4 (1998): 653-658. link