Decision Making

1. Decision Making A scientific approach

2. Why study decisions? • Morris (1963, p.11) argues that management science is about making decisions. Specifically; • “… improvement will occur if the methods of science are applied to the decisions which managers must take …” • This is where the ‘money’ is.

3. Decision Making From 164 (how much do you remember?): • Decision Tables: Minimax and Regret • Decision Trees – structuring a decision problem • Utility Functions – quantifying preferences • Maximising Expected Utility – decision criteria under uncertainty • Value of Information – how much are data worth? • All Prescriptive approach.

4. What is a decision? • A decision is the choice of a course of action. • An action has formal meaning, and relates to the term ‘strategy’. It could be to employ candidate ‘A’ for the job. Another action would be to employ candidate ‘B’.

5. Enumerate the Actions … • A decision is only required where there are two or more possible actions. • The actions should be ‘specific’, so employ ‘A’ is an action whereas employ ‘someone else’ is insufficient to allow useful analysis. • The first step is to identify the possible actions that are available. Indeed this is where ‘added value’ comes in. • We denote the actions d1, d2, …

6. Exhaustive and Exclusive • A list is said to be exhaustive where it covers all possibilities. • Items in a list are said to be exclusive where they are such that only one will be amenable to being chosen. • A list is exhaustive and exclusive where all possible actions are covered, and the decision maker has to choose exactly one.

7. Starter Garlic Bread Garlic Mushrooms Green Salad Main Calzone Pizza Steak Sitting in a restaurant with friends, you have to choose what to eat. Identify a mutually exclusive and exhaustive list of ‘choices’, that is actions, given the menu. Menu Decision

8. Aside … • Thus, the choice of what to eat is the selection from a finite list. • In this case we will ignore the multivariate nature of utility – that is the multiple criteria aspect. • Additionally, we will not complicate things by consider the larger collection of possibilities (sharing a bit of each with a companion.)

9. Uncertainty • If we knew in advance how ‘tasty’ or ‘useful’ each dish was then the decision wouldn’t be hard – maximise ‘tastiness.’ • However, it might be, for example, that the quality of the calzone depends on the chef on duty – something about which we don’t know. • The state of not knowing is called being uncertain. And it is this ‘not knowing’ or uncertainty which complicates our decision greatly.

10. Notation • The uncertain quantity or event may be described by q. • Where the uncertain event relates to a discrete number of possibilities then it is usual to consider the sequence q1,…,qn. • Thus in the case of the restaurant there may be two chefs. The event that the first chef is on duty is q1 whereas the second one is q2. • The collection of uncertain events should be exclusive and exhaustive.

11. Consequences • A consequence is determined by choice of action, together with the realisation of some uncertain event. • Thus it can be totally described by the combination of a decision di and an (a priori) uncertain event. • That is Cij = (di,qj) denotes a consequence.

12. Example • A manufacturer has to decide whether to inspect a product or ship it without inspection. • Inspection has a non-zero cost associated with it. • Inspection will catch poor product, allowing replacement before shipping. • If the product is OK, the customer will be happy. • If the product is substandard, the customer will return it and the manufacturer will have to ship a replacement. • Identify di, qj, and the Cij.

13. Answers

14. Job of decision maker • The decision maker has to choose a ‘row’ on the table. • That is, which consequences would you be happy with? • Note that q1 and q2 are not necessarily equally likely – the probability of each could be quantified in advance, though. (In the example let P(Dodgy)=0.2.)

15. Value of the Consequence • In general, it may not be possible to look at the table of consequences and trade the rows easily against each other. • A way to assist in the decision making process is to assign values to each of the consequences. These values are termed utilities. • The decision maker then has to ‘maximise’ this value somehow.

16. Utility Defined • A number of different definitions of utility exist, and some of these were explored in earlier years. • Here, the key feature of interest is that the ‘best’ consequence has highest utility and ‘worst’ consequence has lowest utility. • Lindley (1985) restricts utility to [0,1]. • In the example, let C21=1.0, C22=0.0, C11=0.9 and C12=0.5.

17. Decision Table

18. Combining u and p • The decision maker is in charge of d. That is, they can choose any row they wish. • Thus, we should summarise each row in turn, then decide between the summaries. • The summary advocated by many is to take the weighted sum of the utilities. This is formally termed expected utility.

19. Expected Utility • Expected utility may be calculated as; • Probability may depend on decision but usually not! • This is identical to expected value seen before;

20. Summary • Thus, when faced with a decision, the prescribed procedure is as follows; • List all courses of action, that is all possible decisions that can be taken; d1,…,dm. • Identify uncertain events that can impact upon the consequences associated with the course of action. That is, identify q1,…,qn. • Quantify the probabilities of these. • Assign utilities to the consequences. • Calculate the expected utility for each decision.

21. Exercise • Calculate the expected utility for each decision in the example.

22. Decision Rule • Choose the decision that maximises expected utility. • In this case, that is to inspect. • However, note that there is not much different between the utilities, and changing the error rate from 20% downward, makes the abandoning of inspection attractive. (Try error rate of 10%).

23. Exercise 1 Which decision is better? Why do you not need probabilities?

24. Exercise 2 Calculate expected utilities in terms of p. Thus, explain which decision is better for p<0.5