Unknown Label Type continuous Decision Tree
The management of a company that I shall call Stygian Chemical Industries, Ltd., must decide whether to build a small plant or a large one to manufacture a new product with an expected market life of ten years. The decision hinges on what size the market for the product will be.
Possibly demand will be high during the initial two years but, if many initial users find the product unsatisfactory, will fall to a low level thereafter. Or high initial demand might indicate the possibility of a sustained high-volume market. If demand is high and the company does not expand within the first two years, competitive products will surely be introduced.
If the company builds a big plant, it must live with it whatever the size of market demand. If it builds a small plant, management has the option of expanding the plant in two years in the event that demand is high during the introductory period; while in the event that demand is low during the introductory period, the company will maintain operations in the small plant and make a tidy profit on the low volume.
Management is uncertain what to do. The company grew rapidly during the 1950's; it kept pace with the chemical industry generally. The new product, if the market turns out to be large, offers the present management a chance to push the company into a new period of profitable growth. The development department, particularly the development project engineer, is pushing to build the large-scale plant to exploit the first major product development the department has produced in some years.
The chairman, a principal stockholder, is wary of the possibility of large unneeded plant capacity. He favors a smaller plant commitment, but recognizes that later expansion to meet high-volume demand would require more investment and be less efficient to operate. The chairman also recognizes that unless the company moves promptly to fill the demand which develops, competitors will be tempted to move in with equivalent products.
The Stygian Chemical problem, oversimplified as it is, illustrates the uncertainties and issues that business management must resolve in making investment decisions. (I use the term "investment" in a broad sense, referring to outlays not only for new plants and equipment but also for large, risky orders, special marketing facilities, research programs, and other purposes.) These decisions are growing more important at the same time that they are increasing in complexity. Countless executives want to make them better—but how?
In this article I shall present one recently developed concept called the "decision tree," which has tremendous potential as a decision-making tool. The decision tree can clarify for management, as can no other analytical tool that I know of, the choices, risks, objectives, monetary gains, and information needs involved in an investment problem. We shall be hearing a great deal about decision trees in the years ahead. Although a novelty to most businessmen today, they will surely be in common management parlance before many more years have passed.
Later in this article we shall return to the problem facing Stygian Chemical and see how management can proceed to solve it by using decision trees. First, however, a simpler example will illustrate some characteristics of the decision-tree approach.
Displaying Alternatives
Let us suppose it is a rather overcast Saturday morning, and you have 75 people coming for cocktails in the afternoon. You have a pleasant garden and your house is not too large; so if the weather permits, you would like to set up the refreshments in the garden and have the party there. It would be more pleasant, and your guests would be more comfortable. On the other hand, if you set up the party for the garden and after all the guests are assembled it begins to rain, the refreshments will be ruined, your guests will get damp, and you will heartily wish you had decided to have the party in the house. (We could complicate this problem by considering the possibility of a partial commitment to one course or another and opportunities to adjust estimates of the weather as the day goes on, but the simple problem is all we need.)
This particular decision can be represented in the form of a "payoff" table:
Much more complex decision questions can be portrayed in payoff table form. However, particularly for complex investment decisions, a different representation of the information pertinent to the problem—the decision tree—is useful to show the routes by which the various possible outcomes are achieved. Pierre Massé, Commissioner General of the National Agency for Productivity and Equipment Planning in France, notes:
"The decision problem is not posed in terms of an isolated decision (because today's decision depends on the one we shall make tomorrow) nor yet in terms of a sequence of decisions (because under uncertainty, decisions taken in the future will be influenced by what we have learned in the meanwhile). The problem is posed in terms of a tree of decisions."1
Exhibit I illustrates a decision tree for the cocktail party problem. This tree is a different way of displaying the same information shown in the payoff table. However, as later examples will show, in complex decisions the decision tree is frequently a much more lucid means of presenting the relevant information than is a payoff table.
Exhibit I. Decision Tree for Cocktail Party
The tree is made up of a series of nodes and branches. At the first node on the left, the host has the choice of having the party inside or outside. Each branch represents an alternative course of action or decision. At the end of each branch or alternative course is another node representing a chance event—whether or not it will rain. Each subsequent alternative course to the right represents an alternative outcome of this chance event. Associated with each complete alternative course through the tree is a payoff, shown at the end of the rightmost or terminal branch of the course.
When I am drawing decision trees, I like to indicate the action or decision forks with square nodes and the chance-event forks with round ones. Other symbols may be used instead, such as single-line and double-line branches, special letters, or colors. It does not matter so much which method of distinguishing you use so long as you do employ one or another. A decision tree of any size will always combine (a) action choices with (b) different possible events or results of action which are partially affected by chance or other uncontrollable circumstances.
Decision-event chains
The previous example, though involving only a single stage of decision, illustrates the elementary principles on which larger, more complex decision trees are built. Let us take a slightly more complicated situation:
You are trying to decide whether to approve a development budget for an improved product. You are urged to do so on the grounds that the development, if successful, will give you a competitive edge, but if you do not develop the product, your competitor may—and may seriously damage your market share. You sketch out a decision tree that looks something like the one in Exhibit II.
Exhibit II. Decision Tree with Chains of Actions and Events
Your initial decision is shown at the left. Following a decision to proceed with the project, if development is successful, is a second stage of decision at Point A. Assuming no important change in the situation between now and the time of Point A, you decide now what alternatives will be important to you at that time. At the right of the tree are the outcomes of different sequences of decisions and events. These outcomes, too, are based on your present information. In effect you say, "If what I know now is true then, this is what will happen."
Of course, you do not try to identify all the events that can happen or all the decisions you will have to make on a subject under analysis. In the decision tree you lay out only those decisions and events or results that are important to you and have consequences you wish to compare. (For more illustrations, see the Appendix.)
For readers interested in further examples of decision-tree structure, I shall describe in this appendix two representative situations with which I am familiar and show the trees that might be drawn to analyze management's decision-making alternatives. We shall not concern ourselves here with costs, yields, probabilities, or expected values.
New Facility
The choice of alternatives in building a plant depends upon market forecasts. The alternative chosen will, in turn, affect the market outcome. For example, the military products division of a diversified firm, after some period of low profits due to intense competition, has won a contract to produce a new type of military engine suitable for Army transport vehicles. The division has a contract to build productive capacity and to produce at a specified contract level over a period of three years.
Figure A illustrates the situation. The dotted line shows the contract rate. The solid line shows the proposed buildup of production for the military. Some other possibilities are portrayed by dashed lines. The company is not sure whether the contract will be continued at a relatively high rate after the third year, as shown by Line A, or whether the military will turn to another newer development, as indicated by Line B. The company has no guarantee of compensation after the third year. There is also the possibility, indicated by Line C, of a large additional commercial market for the product, this possibility being somewhat dependent on the cost at which the product can be made and sold.
If this commercial market could be tapped, it would represent a major new business for the company and a substantial improvement in the profitability of the division and its importance to the company.
Management wants to explore three ways of producing the product as follows:
1. It might subcontract all fabrication and set up a simple assembly with limited need for investment in plant and equipment; the costs would tend to be relatively high and the company's investment and profit opportunity would be limited, but the company assets which are at risk would also be limited.
2. It might undertake the major part of the fabrication itself but use general-purpose machine tools in a plant of general-purpose construction. The division would have a chance to retain more of the most profitable operations itself, exploiting some technical developments it has made (on the basis of which it got the contract). While the cost of production would still be relatively high, the nature of the investment in plant and equipment would be such that it could probably be turned to other uses or liquidated if the business disappeared.
3. The company could build a highly mechanized plant with specialized fabrication and assembly equipment, entailing the largest investment but yielding a substantially lower unit manufacturing cost if manufacturing volume were adequate. Following this plan would improve the chances for a continuation of the military contract and penetration into the commercial market and would improve the profitability of whatever business might be obtained in these markets. Failure to sustain either the military or the commercial market, however, would cause substantial financial loss.
Either of the first two alternatives would be better adapted to low-volume production than would the third.
Some major uncertainties are: the cost-volume relationships under the alternative manufacturing methods; the size and structure of the future market—this depends in part on cost, but the degree and extent of dependence are unknown; and the possibilities of competitive developments which would render the product competitively or technologically obsolete.
How would this situation be shown in decision-tree form? (Before going further you might want to draw a tree for the problem yourself.) Figure B shows my version of a tree. Note that in this case the chance alternatives are somewhat influenced by the decision made. A decision, for example, to build a more efficient plant will open possibilities for an expanded market.
Plant Modernization
A company management is faced with a decision on a proposal by its engineering staff which, after three years of study, wants to install a computer-based control system in the company's major plant. The expected cost of the control system is some $30 million. The claimed advantages of the system will be a reduction in labor cost and an improved product yield. These benefits depend on the level of product throughput, which is likely to rise over the next decade. It is thought that the installation program will take about two years and will cost a substantial amount over and above the cost of equipment. The engineers calculate that the automation project will yield a 20% return on investment, after taxes; the projection is based on a ten-year forecast of product demand by the market research department, and an assumption of an eight-year life for the process control system.
What would this investment yield? Will actual product sales be higher or lower than forecast? Will the process work? Will it achieve the economies expected? Will competitors follow if the company is successful? Are they going to mechanize anyway? Will new products or processes make the basic plant obsolete before the investment can be recovered? Will the controls last eight years? Will something better come along sooner?
The initial decision alternatives are (a) to install the proposed control system, (b) postpone action until trends in the market and/or competition become clearer, or (c) initiate more investigation or an independent evaluation. Each alternative will be followed by resolution of some uncertain aspect, in part dependent on the action taken. This resolution will lead in turn to a new decision. The dotted lines at the right of Figure C indicate that the decision tree continues indefinitely, though the decision alternatives do tend to become repetitive. In the case of postponement or further study, the decisions are to install, postpone, or restudy; in the case of installation, the decisions are to continue operation or abandon.
An immediate decision is often one of a sequence. It may be one of a number of sequences. The impact of the present decision in narrowing down future alternatives and the effect of future alternatives in affecting the value of the present choice must both be considered.
Adding Financial Data
Now we can return to the problems faced by the Stygian Chemical management. A decision tree characterizing the investment problem as outlined in the introduction is shown in Exhibit III. At Decision #1 the company must decide between a large and a small plant. This is all that must be decided now. But if the company chooses to build a small plant and then finds demand high during the initial period, it can in two years—at Decision #2—choose to expand its plant.
Exhibit III. Decisions and Events for Stygian Chemical Industries, Ltd.
But let us go beyond a bare outline of alternatives. In making decisions, executives must take account of the probabilities, costs, and returns which appear likely. On the basis of the data now available to them, and assuming no important change in the company's situation, they reason as follows:
- Marketing estimates indicate a 60% chance of a large market in the long run and a 40% chance of a low demand, developing initially as follows:
- Therefore, the chance that demand initially will be high is 70% (60 + 10). If demand is high initially, the company estimates that the chance it will continue at a high level is 86% (60 ÷ 70). Comparing 86% to 60%, it is apparent that a high initial level of sales changes the estimated chance of high sales in the subsequent periods. Similarly, if sales in the initial period are low, the chances are 100% (30 ÷ 30) that sales in the subsequent periods will be low. Thus the level of sales in the initial period is expected to be a rather accurate indicator of the level of sales in the subsequent periods.
- Estimates of annual income are made under the assumption of each alternative outcome:
1. A large plant with high volume would yield $1,000,000 annually in cash flow.
2. A large plant with low volume would yield only $100,000 because of high fixed costs and inefficiencies.
3. A small plant with low demand would be economical and would yield annual cash income of $400,000.
4. A small plant, during an initial period of high demand, would yield $450,000 per year, but this would drop to $300,000 yearly in the long run because of competition. (The market would be larger than under Alternative 3, but would be divided up among more competitors.)
5. If the small plant were expanded to meet sustained high demand, it would yield $700,000 cash flow annually, and so would be less efficient than a large plant built initially.
6. If the small plant were expanded but high demand were not sustained, estimated annual cash flow would be $50,000.
- It is estimated further that a large plant would cost $3 million to put into operation, a small plant would cost $1.3 million, and the expansion of the small plant would cost an additional $2.2 million.
When the foregoing data are incorporated, we have the decision tree shown in Exhibit IV. Bear in mind that nothing is shown here which Stygian Chemical's executives did not know before; no numbers have been pulled out of hats. However, we are beginning to see dramatic evidence of the value of decision trees in laying out what management knows in a way that enables more systematic analysis and leads to better decisions. To sum up the requirements of making a decision tree, management must:
1. Identify the points of decision and alternatives available at each point.
2. Identify the points of uncertainty and the type or range of alternative outcomes at each point.
3. Estimate the values needed to make the analysis, especially the probabilities of different events or results of action and the costs and gains of various events and actions.
4. Analyze the alternative values to choose a course.
Exhibit IV. Decision Tree with Financial Data
Choosing Course of Action
We are now ready for the next step in the analysis—to compare the consequences of different courses of action. A decision tree does not give management the answer to an investment problem; rather, it helps management determine which alternative at any particular choice point will yield the greatest expected monetary gain, given the information and alternatives pertinent to the decision.
Of course, the gains must be viewed with the risks. At Stygian Chemical, as at many corporations, managers have different points of view toward risk; hence they will draw different conclusions in the circumstances described by the decision tree shown in Exhibit IV. The many people participating in a decision—those supplying capital, ideas, data, or decisions, and having different values at risk—will see the uncertainty surrounding the decision in different ways. Unless these differences are recognized and dealt with, those who must make the decision, pay for it, supply data and analyses to it, and live with it will judge the issue, relevance of data, need for analysis, and criterion of success in different and conflicting ways.
For example, company stockholders may treat a particular investment as one of a series of possibilities, some of which will work out, others of which will fail. A major investment may pose risks to a middle manager—to his job and career—no matter what decision is made. Another participant may have a lot to gain from success, but little to lose from failure of the project. The nature of the risk—as each individual sees it—will affect not only the assumptions he is willing to make but also the strategy he will follow in dealing with the risk.
The existence of multiple, unstated, and conflicting objectives will certainly contribute to the "politics" of Stygian Chemical's decision, and one can be certain that the political element exists whenever the lives and ambitions of people are affected. Here, as in similar cases, it is not a bad exercise to think through who the parties to an investment decision are and to try to make these assessments:
- What is at risk? Is it profit or equity value, survival of the business, maintenance of a job, opportunity for a major career?
- Who is bearing the risk? The stockholder is usually bearing risk in one form. Management, employees, the community—all may be bearing different risks.
- What is the character of the risk that each person bears? Is it, in his terms, unique, once-in-a-lifetime, sequential, insurable? Does it affect the economy, the industry, the company, or a portion of the company?
Considerations such as the foregoing will surely enter into top management's thinking, and the decision tree in Exhibit IV will not eliminate them. But the tree will show management what decision today will contribute most to its long-term goals. The tool for this next step in the analysis is the concept of "rollback."
"Rollback" concept
Here is how rollback works in the situation described. At the time of making Decision #1 (see Exhibit IV), management does not have to make Decision #2 and does not even know if it will have the occasion to do so. But if it were to have the option at Decision #2, the company would expand the plant, in view of its current knowledge. The analysis is shown in Exhibit V. (I shall ignore for the moment the question of discounting future profits; that is introduced later.) We see that the total expected value of the expansion alternative is $160,000 greater than the no-expansion alternative, over the eight-year life remaining. Hence that is the alternative management would choose if faced with Decision #2 with its existing information (and thinking only of monetary gain as a standard of choice).
Exhibit V. Analysis of Possible Decision #2 (Using Maximum Expected Total Cash Flow as Criterion)
Readers may wonder why we started with Decision #2 when today's problem is Decision #1. The reason is the following: We need to be able to put a monetary value on Decision #2 in order to "roll back" to Decision #1 and compare the gain from taking the lower branch ("Build Small Plant") with the gain from taking the upper branch ("Build Big Plant"). Let us call that monetary value for Decision #2 its position value. The position value of a decision is the expected value of the preferred branch (in this case, the plant-expansion fork). The expected value is simply a kind of average of the results you would expect if you were to repeat the situation over and over—getting a $5,600 thousand yield 86% of the time and a $400 thousand yield 14% of the time.
Stated in another way, it is worth $2,672 thousand to Stygian Chemical to get to the position where it can make Decision #2. The question is: Given this value and the other data shown in Exhibit IV, what now appears to be the best action at Decision #1?
Turn now to Exhibit VI. At the right of the branches in the top half we see the yields for various events if a big plant is built (these are simply the figures in Exhibit IV multiplied out). In the bottom half we see the small plant figures, including Decision #2 position value plus the yield for the two years prior to Decision #2. If we reduce all these yields by their probabilities, we get the following comparison:
Build big plant: ($10 × .60) + ($2.8 × .10) + ($1 × .30) – $3 = $3,600 thousand
Build small plant: ($3.6 × .70) + ($4 × .30) – $1.3 = $2,400 thousand
Exhibit VI. Cash Flow Analysis for Decision #1
The choice which maximizes expected total cash yield at Decision #1, therefore, is to build the big plant initially.
Accounting for Time
What about taking differences in the time of future earnings into account? The time between successive decision stages on a decision tree may be substantial. At any stage, we may have to weigh differences in immediate cost or revenue against differences in value at the next stage. Whatever standard of choice is applied, we can put the two alternatives on a comparable basis if we discount the value assigned to the next stage by an appropriate percentage. The discount percentage is, in effect, an allowance for the cost of capital and is similar to the use of a discount rate in the present value or discounted cash flow techniques already well known to businessmen.
When decision trees are used, the discounting procedure can be applied one stage at a time. Both cash flows and position values are discounted.
For simplicity, let us assume that a discount rate of 10% per year for all stages is decided on by Stygian Chemical's management. Applying the rollback principle, we again begin with Decision #2. Taking the same figures used in previous exhibits and discounting the cash flows at 10%, we get the data shown in Part A of Exhibit VII. Note particularly that these are the present values as of the time Decision #2 is made.
Exhibit VII. Analysis of Decision #2 with Discounting Note: For simplicity, the first year cash flow is not discounted, the second year cash flow is discounted one year, and so on.
Now we want to go through the same procedure used in Exhibit V when we obtained expected values, only this time using the discounted yield figures and obtaining a discounted expected value. The results are shown in Part B of Exhibit VII. Since the discounted expected value of the no-expansion alternative is higher, that figure becomes the position value of Decision #2 this time.
Having done this, we go back to work through Decision #1 again, repeating the same analytical procedure as before only with discounting. The calculations are shown in Exhibit VIII. Note that the Decision #2 position value is treated at the time of Decision #1 as if it were a lump sum received at the end of the two years.
Exhibit VIII. Analysis of Decision #1
The large-plant alternative is again the preferred one on the basis of discounted expected cash flow. But the margin of difference over the small-plant alternative ($290 thousand) is smaller than it was without discounting.
Uncertainty Alternatives
In illustrating the decision-tree concept, I have treated uncertainty alternatives as if they were discrete, well-defined possibilities. For my examples I have made use of uncertain situations depending basically on a single variable, such as the level of demand or the success or failure of a development project. I have sought to avoid unnecessary complication while putting emphasis on the key interrelationships among the present decision, future choices, and the intervening uncertainties.
In many cases, the uncertain elements do take the form of discrete, single-variable alternatives. In others, however, the possibilities for cash flow during a stage may range through a whole spectrum and may depend on a number of independent or partially related variables subject to chance influences—cost, demand, yield, economic climate, and so forth. In these cases, we have found that the range of variability or the likelihood of the cash flow falling in a given range during a stage can be calculated readily from knowledge of the key variables and the uncertainties surrounding them. Then the range of cash-flow possibilities during the stage can be broken down into two, three, or more "subsets," which can be used as discrete chance alternatives.
Conclusion
Peter F. Drucker has succinctly expressed the relation between present planning and future events: "Long-range planning does not deal with future decisions. It deals with the futurity of present decisions."2 Today's decision should be made in light of the anticipated effect it and the outcome of uncertain events will have on future values and decisions. Since today's decision sets the stage for tomorrow's decision, today's decision must balance economy with flexibility; it must balance the need to capitalize on profit opportunities that may exist with the capacity to react to future circumstances and needs.
The unique feature of the decision tree is that it allows management to combine analytical techniques such as discounted cash flow and present value methods with a clear portrayal of the impact of future decision alternatives and events. Using the decision tree, management can consider various courses of action with greater ease and clarity. The interactions between present decision alternatives, uncertain events, and future choices and their results become more visible.
Of course, there are many practical aspects of decision trees in addition to those that could be covered in the space of just one article. When these other aspects are discussed in subsequent articles,3 the whole range of possible gains for management will be seen in greater detail.
Surely the decision-tree concept does not offer final answers to managements making investment decisions in the face of uncertainty. We have not reached that stage, and perhaps we never will. Nevertheless, the concept is valuable for illustrating the structure of investment decisions, and it can likewise provide excellent help in the evaluation of capital investment opportunities.
1. Optimal Investment Decisions: Rules for Action and Criteria for Choice (Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 1962), p. 250.
2. "Long-Range Planning," Management Science, April 1959, p. 239.
3. We are expecting another article by Mr. Magee in a forthcoming issue.—The Editors
A version of this article appeared in the July 1964 issue of Harvard Business Review.
Source: https://hbr.org/1964/07/decision-trees-for-decision-making
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