## Monday 14 February 2022

Scientific decision-making is one of the main purposes of project evaluation. the scientific decision-making method is comparative judgment, that is, the selection of alternatives to the proposed project. however, there is a certain degree of risk in decision-making, and a large number of decisions in project evaluation work are basically risk-based decisions.

probability analysis is a method for deciding on the trade-offs under risk conditions, and decision tree analysis is also one of the commonly used risk decision-making methods.

the so-called decision tree analysis is a method of systematically analyzing and evaluating the solution by using the principle of probability analysis to describe the content, parameters, and status of alternative solutions and the interrelationship of different stages of the implementation process with a tree diagram. the application of decision tree analysis can not only make single-stage decisions, but also be effective for multi-stage decision-making.

### first, the structure of the decision tree

a decision tree is a tree-like figure made up of boxes and circles as nodes and connected by straight lines, which is composed of the following factors:

(1) decision points and program branches

the starting point of a decision, called the decision point, is represented by the box "mouth". symbols available within the box indicate that it is a decision point at several levels.
a decision should have a number of alternative schemes, represented by several straight lines "-" leading from the decision point, called the scheme branch. the meaning and parameters of the scheme can be noted on the upper and lower sides of the scheme branch.

(2) state nodes and state branches

due to the risks and uncertainties in the implementation process of the program, there may be a variety of opportunities or states, and the results (such as benefits or costs) that can be obtained by the program in various natural states are represented by a circle "○", which is called the state node or opportunity point.

the various states that may occur in each scheme are represented by several lines "-" led by the state nodes, called state branches. parameters such as codes and probabilities of various states can be marked on the upper and lower sides of the state, so it is also called the probability branch.

(iii) result point and present value

of profit and loss the result that the program may obtain in a certain state is represented by "△", which is called the result point after the result point, the present value of its profit and loss can be listed separately, and the so-called present value of profit and loss is the present value of the measurement result of the loss or gain of the program in a certain state, that is, the net present value of the state. the figure composed of the above symbols resembles a tree, so it is called a decision tree. as shown in the following figure:

### figure decision tree

as a decision-making tool, the decision tree analyzes simply and intuitively, and can visualize all the decision-making schemes of the decision-making problem and the various natural states that may appear, as well as the results of different natural states, in the entire decision-making process. it is a useful tool to help project evaluators analyze and compare programmes and has been widely used in project evaluation work in recent years.

### second, the process of decision tree analysis

(1) draw a decision tree diagram

the order in which the decision tree diagram is drawn is from left to right. the decision tree diagram is given according to the problems that need to be decided, the various schemes to be selected, and the natural state of the various schemes.

(2) calculate the present value and expected value
of earnings the calculation order of decision tree analysis is from right to left.

1. calculate the present value of the proceeds for each result point based on the relevant information and mark it after the result point.
2. according to the present value of the income of each state and the probability of occurrence, the expected value of the profit and loss of each program state is calculated, and it is marked on the status node.
3. calculate the expected value of the net present value of the scheme according to the status expectation value and the present value of the investment, and mark it on the upper side of the scheme branch.

expected value of the net present value of the scheme = state expectation value - present value of the investment.

(iii) decision-making choices

follow the principle of expectations when making decisions, that is, making decisions according to the expectations of each program.
1. if the p&l value is expressed as an expense, the scheme with the smallest expected value of the net present value should be chosen.
for the unsuccessful scheme, a "×" is drawn on the branch of the scheme, indicating that the branch has been "cut off", which is called pruning. in this way, only one branch of the scheme is left on the decision tree, which is the optimal scheme.
2. if the p&l value is expressed as a gain, the scheme with the largest expected value of the net present value should be chosen.

### third, a single-stage decision tree

when using the decision tree for decision-making, where only one decision-making activity is required to select the optimal solution, the decision to achieve the purpose of decision-making is called single-level decision-making. drawing the probabilities of the natural state of each scheme in the single-level decision-making process and the results produced is a single-stage decision tree.

### fourth, multi-stage decision trees

decisions that require more than two decision-making activities to select the optimal solution and achieve the purpose of decision-making are called multi-level decision-making. the natural state, probability and result of each program in the multi-level decision-making process are all drawn on a map to form a multi-stage decision tree.

multi-stage decision-making using decision trees should start from the last level of decision points, and replace the decision points at this level with the expectations of the post-decision plan at each level to reduce the decision tree, and then move the decision forward step by step.

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### decision tree approach

graphical representation of risk decision problems. because the graph is shaped like a tree, it is called a decision tree, and it has many advantages over the decision matrix representation. for example, the decision matrix representation can only represent unipolar decision-making problems and requires that all action plans face a completely consistent state of nature. when using the decision tree representation, the shortcomings of the decision matrix representation can be overcome, and the decision tree representation is also convenient and simple, the level is clear, and the decision-making process can be visualized.

### decision tree

the structure is shown in the figure. the squares in the diagram represent the decision nodes, and the branches derived from them are called scheme branches. each branch represents a scheme, and the number of branches is the equivalent number of scenarios that are possible. the circle represents the node of the scheme, from which the probability branches are derived, and each probability branch indicates the state of nature and the probability of its occurrence. the probability branching number reflects the number of possible states faced by the scheme. the triangle at the end is called the result point, which is marked with the result value of each scheme in the corresponding state.

the process of applying a decision tree to make a decision is to step back from right to left for analysis. according to the profit and loss value at the right end and the probability of the probability branch, the size of the expected value is calculated, the desired result of the scheme is determined, and then the choice is made according to the expected outcome of different schemes. the abandonment of the scheme is called pruning, and the discarded scheme is represented by the sign of "≠", and the final decision point leaves a branch, that is, the optimal solution.

when the decision problem can be solved with only one decision, it is called a single-stage decision problem. if the problem is more complex and a series of decisions need to be made to solve it, it is called a multi-stage decision-making problem, and the multi-stage decision-making problem is intuitive and easy to use the decision tree decision-making method.

#### example

in order to meet the needs of the market, two schemes to expand the production of television sets were proposed in a certain place. one option is to build large factories, and the second is to build small factories. the construction of a large factory requires an investment of 6 million yuan and can be used for 10 years. the construction of a small factory investment of 2.8 million yuan, such as good sales, 3 years later expansion, expansion needs to invest 4 million yuan, can be used for 7 years, annual profit of 1.9 million yuan. try the decision tree method to select a reasonable decision plan.

This is a decision tree in example 9.7 that calculates the expected value of each point:

points (2): 0.7×200×10+ .3× (-40) ×10-600 (investment) = 680 (million yuan)

points (5): 1.0×190×7-400 = 930 (million yuan)

point (6): 1.0×80×7 = 560 (million yuan)

comparing the situation of decision point 4, it can be seen that since the expected profit value of point (5) is larger than that of point (6) (5.6 million yuan) and point (6) (5.6 million yuan), the expansion scheme should be adopted instead of the non-expansion scheme. moving the 9.3 million yuan of point (5) to point 4 calculates the expected profit value of point (3).

points (3): 0.7×80×3+0.7×930+0.3×60× (3+7)-280 = 7.19 (million)

### legend of a decision tree

finally, compare the situation at decision point 1. since the expected profit value of pip (3) (7.19 million yuan) is larger compared to point (2) (6.8 million yuan), the point (3) is taken and the point (2) is rounded. in this way, in contrast, the plan for building a large factory is not the optimal solution, and a reasonable strategy should be to build a small factory in the first 3 years, such as good sales, and expand in the next 7 years.