Lecture 13: Mid-term Review

In this lecture we review the take-home portion of the mid-term, and go on to examine some strategies for decision-making under uncertainty.

(Text of this lecture will appear following the mid-term.)

Assumptions

The mid-term take-home was the closest we have so far come to a real analysis. One thing that shows we are getting close to reality is that, although the text of the question ties down the working assumptions fairly tightly, there are still several possible solutions, none of them necessarily wrong, corresponding to different interpretations.

For this reason, it is important in any analysis to make your assumptions explicit and state them clearly.

One point on which individual solutions differed was on whether we should include the cost of buying extrusions externally at 5 cents each when internal production falls short of requirements. The answer to this is a definite ``No''; our calculations already include a 5c benefit for not having to buy externally, so when we do need to buy externally, we just remove this benefit.

A second point occurs in the evaluation of the `sequence of E1s' case, where we have to decide when to buy the second E1. The text states that we should buy it at the end of the year before we run out of internal production capacity; however, we are still free to decide when we actually switch the machine on -- and hence when we start selling its output and paying its maintenance costs. It would be plausible to suggest that we shouldn't switch it on till internal capacity runs out. A further level of sophistication would be to switch it on only when the external price is high enough that it's profitable to use the machine to produce for the external market.

While each of these strategies can be modelled, it is advisable to do a simple order-of-magnitude check to see whether the resulting change in the final figures is worth the labour of building a more sophisticated model. In this case, the question of when we switch on the second E1 does not affect the final situation -- the sequence option remains the worst of the three.

The text suggests that we look at a ten-year study period. To know how much confidence we should place in the conclusions of the study, it would be useful to know how sensitive the results are to the length of the study period. From the figures you need to calculate the ten-year results, it is very easy also to plot comparisons for shorter study periods. These plots show that the relative merits of the three strategies are about the same for any study period beyond seven years.

Solution Method

It is excessively laborious to do all the calculations for this study by hand. Spread-sheet programs are faster and easier. The most general solution method may be to write a simple program in C, or a symbolic language such as Maple. This allows easy graphical output, and all the assumptions of interest can be represented as particular choices for the value of variables. It is then straightforward to investigate the sensitivity of the solution to each assumption.

Clarity of Presentation

This is the most important part of the exercise. The conclusions of your analysis should be easy to read and interpret. This begins with purely ergonomic considerations: does the reader have to search through 12 pages to compare the merits of the twenty-seven options, or are they presented in a single table? Are the results all presented to the same number of decimal places, so that the profile made by the left-hand digits of the results reflects their magnitude?

It's worth remembering that, in industry, your memos and reports may be the only aspect of your work visible to management. Managers put a high value on their time, and appreciate reports that can be interpreted quickly and easily.

The simplest way of presenting the results of the study is as a table of recommendations:

		5%	10%	15%
		Growth Rate
External Price	$0.03	E1	E1	E1
	$0.035	E2	E2	E2
	$0.040	E2	E2	E2

This shows clearly what strategy should be followed in each case, and indicates that the most important missing information is the external price; the growth rate doesn't identify any particular strategy. However, it is also desirable to provide the reader with some additional information: the relative sizes of the advantage gained by each strategy, and the numerical values of the present values. The first of these goals can be met by a bar chart, the second by a table of numerical values. Either present worth or IRR can be used; the advantage of present worth is that an incremental comparison of alternatives is not necessary.

The report to management should also highlight any facts of particular interest that emerge from the figures. One such fact emerges when we look at the present worth for an E2 machine, assuming that all the output is sold on the external market at $0.04/unit. This turns out to be positive, assuming the machine lasts at least seven years. In other words, there is an investment opportunity here to realise a high rate of return on a new activity. Further analysis may be needed to see if the opportunity can be realised, but you will certainly get credit for noticing it.

As mentioned, the report should recommend further investigation of the external price for extrusions. This recommendation can be made more focused by setting an aspiration level for the external price: by plotting the present worths of the E1 and E2 options as a function of the external price, you can determine that E2 forges ahead as soon as the external price crosses $0.034; this conclusion is stable under all values of growth rate considered.

Decision-making Under Uncertainty

(This topic is discussed in Fleischer, pp. 374-384). If you want to recommend a particular strategy before doing the market study (or following a market study whose results leave some uncertainty in the external price), you are in the realm of decision-making under risk or decision-making under uncertainty. The former applies when you can assign probabilities to each external price range, the latter when you have no information on probabilities. For the present, we'll assume the latter case applies; we'll discuss risk later in the semester.

We first note that the sequence-of-E1's strategy always performs less well than a single E1; in analyst's jargon, the sequence strategy is dominated by the E1 strategy. So we can eliminate it from further consideration.

To decide between E1 and E2, we may follow one of several decision-making procedures:

The Minimax Strategy

This strategy is based on the pessimistic assumption that things will turn out for the worst, so you should make the choice that will make the best of the available worst outcomes. (If we are discussing losses, the principle says that we should minimize the maximum loss; if we're discussing gains, we should maximize the minimum gain.)

		5%			10%			15%			Max
		Growth Rate
		$0.03	$0.035	$0.04	$0.03	$0.035	$0.04	$0.03	$0.035	$0.04
Strategy	E1	10	25	39	27	37	48	35	43	52	10
Strategy	E2	-40	38	117	-8	63	134	30	91	153	-40

From this table, we see that if we do E1, the worst that can happen is that we make $10,000 whereas if we do E2, the worst that can happen is that we lose $40,000. So, choosing the better of these options, we elect to do E1.

The Maximax Principle

This principle takes the contrary viewpoint, that we should plan to take maximum advantage of the best that can happen. Thus, we compare the best that can happen under each strategy, given that everything turns out as we'd like, and go for the maximum advantage:

		5%			10%			15%			Max
		Growth Rate
		$0.03	$0.035	$0.04	$0.03	$0.035	$0.04	$0.03	$0.035	$0.04
Strategy	E1	10	25	39	27	37	48	35	43	52	52
Strategy	E2	-40	38	117	-8	63	134	30	91	153	153

Thus in this case, we would choose E2, since it offers a maximum payoff of $153,000, in preference to the $52,000 payoff from E1.

The Principle of Minimax Regret

This principle, also known as the Savage Principle, is based on the psychologically plausible premise that, if you make a plan based on the assumption that A will happen, and then B happens instead, you will regret losing the benefits you could have had if you'd been smart enough to guess right. So you adopt the strategy of minimizing the regret you might otherwise be obliged to feel.

To do this, you first construct a regret matrix:

		5%			10%			15%			Max
		Growth Rate
		$0.03	$0.035	$0.04	$0.03	$0.035	$0.04	$0.03	$0.035	$0.04
Strategy	E1	0	13	78	0	26	86	0	48	101	101
Strategy	E2	50	0	0	35	0	0	5	0	0	50

This matrix is constructed by asking, for each strategy and each outcome, ``What would I lose if I chose this strategy and this outcome came up?''. We then examine the maximum loss for each strategy -- $101,000 for E1 and $50,000 for E2 -- and on this basis decide to minimize our potential regret by choosing E2.

One drawback with this scheme is that the outcome can be changed by adding to the regret matrix another possible strategy, itself less desirable than either existing alternative. (See Fleischer for discussion).

The Laplace Principle

This is sometimes known as the Principle of Insufficient Reason; it states that, in the absence of any information about the probabilities of different futures, we should assume they are all equally probable. It is then possible to calculate the expected value of the present worth of each strategy:

Expected value of E1 = (Sum{Present Worth of E1 in the event that j})/(Number of possible futures)

On the basis of this principle, the expected value of E1 is $35,100, while the expected value of E2 is $64,200; so we should choose E2.

A drawback with this principle is that we can change the results by adding possibilities: for example, we could consider that the external price for extrusions could be $0.03, $0.031, $0.032, $0.035 or $0.04. Applying the Laplace principle to present worths calculated for these possibilities would increase the expected value of E1.

What's the Point?

It is reasonable to ask, why are we bothering to discuss these principles with fancy names, when at the end of the day we're just going to chose either E1 or E2? There doesn't seem to be any compelling reason to choose one principle over another, and the principles don't consistently point to any one solution, so why not just toss a coin?

One reason for discussing these principles is that it provides a vocabulary in which we can talk about decision-making. Without the knowledge that different principles exist, decision-makers can get stuck in unprofitable disagreements, arising because each person is implicitly taking a different principle as the only possible basis for decision.

A second reason for articulating these principles arises when we look at a more complex situtation, with four or five different available strategies interacting in complicated ways. Intuition becomes less reliable here, and it is useful to have a decision method that can be applied automatically. Each of the strategies we've discussed can be expressed as an algorithm and applied to problems of arbitrary size, and each strategy has at least some claim to be rational.

John Jones
Fri Jan 2 10:38:23 PDT 2008