風結ぶ言葉たち

Decision-making is an important function of management, involving the process and outcomes of decisions made by decision-makers regarding system solutions, as well as the behaviors and responsibilities of decision-makers during the decision-making process.

Management decision analysis is a set of reasoning methods, logical steps, and specific techniques provided to assist decision-makers in making correct decisions under changing environmental conditions, as well as the process of selecting satisfactory action plans using these techniques and methods.

Types of Decision Analysis#

Basic pattern of decision problems:

$ W_{ij} = f(A_i,\theta_j) \qquad i=1,2,\cdots,m \qquad j=1,2,\cdots,n $

• $ A_i $ is the $i$-th strategy or plan of the decision-maker, belonging to decision variables, which are controllable factors for the decision-maker;

• $ \theta_j $ is the $j$-th environmental condition or natural state in which the decision-maker and the decision object (decision problem) are situated, belonging to state variables, which are uncontrollable factors for the decision-maker;

• $ W_{ij} $ is the result of the decision-maker choosing the $i$-th strategy under the $j$-th environmental condition, representing the value function of the decision problem, generally referred to as the benefit-loss value or utility value.

Deterministic Problem Analysis#

Problem Characteristics#

1. There exists a goal that the decision-maker wishes to achieve (maximize profit and minimize loss);

2. There are two or more action plans available for selection;

3. The benefit-loss values of different action plans can be calculated under natural states;

4. There exists one definite natural state;

Solution Methods#

1. When the number of plans is large, methods from operations research such as planning are often used for analysis and resolution (linear programming, dynamic programming, goal programming, etc.);

2. For solving multi-stage deterministic decision problems—dynamic programming methods;

3. Strictly speaking, deterministic problems are knowledge optimization calculation problems, rather than true management decision analysis problems.

Typical Example#

A certain company plans to produce Product A, with a unit selling price of 150 yuan/piece, a unit variable cost of 100 yuan/piece, and fixed costs of 5000 yuan, with an annual production volume of 200 pieces. Solve:

1. What is the company's annual profit?
2. What is the break-even production volume?
3. What is the minimum price at which the company will not incur losses?
4. If the target profit is 10,000 yuan, what is the target cost?
5. If the raw material prices rise and labor wages increase, causing the unit variable cost to rise to 140 yuan/piece, and if the unit selling price remains unchanged and cannot be changed, should the company stop production?

Company's annual profit:

$ E=(P-V)N-F = (150-100)\times200-5000=5000 yuan $

Break-even production volume:

$ N^{*}=\frac{F}{P-V}=\frac{5000}{150-100} = 100 pieces $

Break-even price:

$ P^{*}=\frac{VN+F}{N}=100+\frac{5000}{200} = 125 yuan/piece $

Sales volume for a target profit of 10,000 yuan:

$ N_{target}=\frac{F+E}{P-V}=\frac{5000+10000}{150-100} = 300 pieces $

Target total cost:

$ VN_{target}+F=100\times300+5000 = 35000 yuan $

Risk-type Problem Analysis#

Problem Characteristics#

1. There exists a goal that the decision-maker wishes to achieve (maximize profit or minimize loss);

2. There are two or more action plans available for selection;

3. The benefit-loss values of different action plans can be calculated under natural states;

4. There are two or more natural states;

5. The probabilities of different natural states can be predicted.

Solution Methods#

1. Expected value;

2. Matrix method;

3. Decision tree method.

Risk-type decision analysis problems are the main content of decision analysis in general real situations. Based on basic methods, attention should be paid to grasping the value of information and its analysis, as well as important issues such as the decision-maker's utility perspective.

Uncertainty-type Problem Analysis#

Problem Characteristics#

1. There exists a goal that the decision-maker wishes to achieve (maximize profit or minimize loss);

2. There are two or more action plans available for selection;

3. The benefit-loss values of different action plans can be calculated under natural states;

4. There are two or more natural states;

5. The probabilities of different natural states cannot be predicted.

Solution Methods#

1. Optimistic method (maximax principle);

2. Pessimistic method (maximin principle);

3. Regret value method (Savage criterion or maximum minimization of regret values);

4. Equal probability method (Laplace criterion, a special type of risk decision).

Typical Example#

A certain company plans to produce a new product. It estimates that the sales volume of the product can be categorized into four situations: high, medium, low, and very low, but the probabilities of each state occurring cannot be predicted. To produce this product, the company has three implementation plans: build a new workshop for production; renovate an existing workshop for production; produce some parts in the existing workshop and purchase some parts externally. The company plans to produce the new product for 10 years, and the profit and loss values (after deducting investment costs) under different states within 10 years are shown in Table 5-1. Please use the optimistic method, pessimistic method, and regret value method to decide on the implementation plan.

| | Market sales volume high | Market sales volume medium | Market sales volume low | Market sales volume very low |
|-----------|---------|---------|---------|---------|
| Build new workshop | 850 | 420 | -150 | -400 |
| Renovate existing workshop | 600 | 400 | -100 | -350 |
| Partial production, partial external purchase | 400 | 250 | 90 | -50 |

Optimistic Method:

The maximum benefit for each different plan under different states is:

$max_{A_1}{850,420,-150,-400} = 850$

$max_{A_2}{600,400,-100,-350} = 600$

$max_{A_3}{400,250,90,-50} = 400$

Taking the maximum value among the maximum benefit values of each plan, we get:

$ max\{850,600,400\} = 850 $

Thus, the corresponding plan is $ A_1 $, to build a new workshop.

Pessimistic Method:

The minimum benefit for each different plan under different states is:

$min_{A_1}{850,420,-150,-400} = -400$

$min_{A_2}{600,400,-100,-350} = -350$

$min_{A_3}{400,250,90,-50} = -50$

Taking the maximum value among the minimum benefit values of each plan, we get:

$ max\{-400,-350,-50\} = -50 $

Thus, the corresponding plan is $ A_3 $, partial production, partial external purchase.

Regret Value Method:

Taking the maximum benefit value for each state, subtracting it from the benefit values of the other plans, and then comparing the maximum regret values of each plan.

Taking the minimum value among the maximum regret values of each plan, we get:

$ max\{350,300,450\} = 300 $

Thus, the corresponding plan is $ A_2 $, to renovate the existing workshop.

Equal Probability Method:

Assuming that the probabilities of each state occurring are the same, by calculating the expected value of the benefits for each plan and comparing them, we take the maximum expected value of the benefits to choose the corresponding plan. As this method is too simple, it will not be elaborated further.

Risk-type Problem Analysis#

Problem Characteristics#

1. There exists a goal that the decision-maker wishes to achieve (maximize profit or minimize loss);

2. There are two or more action plans available for selection;

3. The benefit-loss values of different action plans can be calculated under natural states;

4. There are two or more natural states;

5. The probabilities of different natural states can be predicted.

Solution Methods#

1. Expected value;

2. Matrix method;

3. Decision tree method.

Risk-type decision problems are the main content of general decision analysis. Based on basic methods, attention should be paid to grasping the value of information and its analysis, as well as important issues such as the decision-maker's utility perspective.

Expected Value Method#

The expected value method calculates the expected benefit-loss value of each action plan using the mathematical expectation formula of random variables in probability theory and compares them. If the decision goal (criterion) is to maximize expected profit, then the action plan with the highest expected profit value is chosen as the optimal plan; conversely, if the decision goal is to minimize expected costs, then the plan with the lowest expected cost value is chosen as the optimal plan.

$ E(X) = \sum{P_iX_i}$

• $ X_i $ is the $i$-th value of the discrete random variable $ X $, where $i=1,2,\cdots,m $;

• $ P_i $ is the probability when $ X = X_i $.

Decision Tree Method#

The decision tree method uses a tree diagram model to describe decision analysis problems and directly conducts decision analysis on the decision tree diagram based on its decision goals (criteria), which can also be the expected benefit-loss value or other transformed indicator values.

Typical Example:

A certain light industry company needs to decide the production volume of a product for the next year to prepare for various production preparations in advance. It is assumed that the size of the production volume is mainly determined by the sales price of the product. Based on past statistical data on market sales prices and market forecast information, it is known that the probabilities of future product sales prices increasing, remaining unchanged, and decreasing are 0.3, 0.6, and 0.1, respectively. If the product is produced in three different batch sizes (i.e., three different plans), the benefit-loss values under different price states for the next year can be estimated, as shown in the table below. Now, it is required to determine the production volume for the next year through decision analysis to maximize the expected profit for the product.

| | Price Increase $ \theta_1 $ | Price Unchanged $ \theta_2 $ | Price Decrease $ \theta_3 $ |
|-------------|------------------|------------------|------------------|
| | 0.3 | 0.6 | 0.1 |
| Large Batch Production $ A_1 $ | 40 | 36 | -6 |
| Medium Batch Production $ A_2 $ | 36 | 34 | 24 |
| Small Batch Production $ A_3 $ | 20 | 16 | 14 |

Based on the problem description, a decision tree can be drawn as follows:

Multi-level Decision Tree Method#

If only one decision needs to be made, the analysis and solution are completed, then this type of decision analysis problem is called a single-level decision. Conversely, some decision problems require multiple decisions to be completed, then this type of decision problem is a multi-level decision problem. Using the decision tree method for multi-level decision analysis is called the multi-level decision tree method.

Value of Information#

The relationship between information and decision-making is very close. To obtain correct decisions, sufficient and reliable information must be relied upon. The classification of information required for decision-making: one type is complete information, which allows for completely certain natural states and aids in correct decision-making; another type is sampling information, which is a type of incomplete and unreliable information.

Value of Complete Information#

Typical Example#

A certain chemical factory produces a type of chemical product. Analysis of statistical data indicates that the defect rate of the product can be divided into five levels (i.e., five states), with the probability of each level (state) as follows:

| Defect Rate | $ S_1 $（0.02） | $ S_2 $（0.05） | $ S_3 $（0.10） | $ S_4 $（0.15） | $ S_5 $（0.20） |
|-----|---------------|---------------|---------------|---------------|---------------|
| Probability | 0.20 | 0.20 | 0.10 | 0.20 | 0.30 |

Further analysis reveals that the defect rate of the product is related to the purity of the main raw materials used for the product. It is known that high purity of chemical raw materials leads to low defect rates (e.g., $ S_1 $ is 0.02), and conversely, low purity leads to high defect rates. The purity of chemical raw materials is also related to factors such as transportation and storage dates. Therefore, the production department manager of the factory suggests adding a "purification" process to the production of the product, which can ensure that all raw materials are in the $ S_1 $ state, thereby reducing the defect rate. However, adding the purification process will incur additional processing costs.

After calculation, it is found that the purification cost for each batch of raw materials is 3400 yuan. It is estimated that the benefit-loss values under different purity states are shown in the table below. If, before production, the chemical raw materials are inspected, and through inspection, it is possible to determine the purity state of each batch of chemical raw materials, different strategies can be adopted for raw materials of different purities, i.e., to purify or not to purify, thereby maximizing the expected benefit-loss value.

| Defect Rate | $ S_1 $（0.02） | $ S_2 $（0.05） | $ S_3 $（0.10） | $ S_4 $（0.15） | $ S_5 $（0.20） |
|------------|---------------|---------------|---------------|---------------|---------------|
| Probability | 0.20 | 0.20 | 0.10 | 0.20 | 0.30 |
| Purification $ A_1 $ | 1000 | 1000 | 1000 | 1000 | 1000 |
| No Purification $ A_2 $ | 4400 | 3200 | 2000 | 800 | -400 |

Based on the problem description, a decision tree can be drawn as follows:

Value of Sampling Information#

Typical Example#

A certain company has 50,000 yuan of surplus funds. If used for a project development, the estimated success rate is 96%, with a profit of 12% in case of success, but there is a risk of losing all funds in case of failure. If the funds are deposited in a bank, a stable annual interest of 6% can be obtained. To gather more information, the company seeks consulting services, with a consultation fee of 500 yuan, but the consulting advice is only for reference. Based on the results of similar 200 consulting cases in the past, the specific situation is shown in the table below.

| | Investment Success | Investment Failure | Total |
|------|------|------|-----|
| Can Invest | 154 | 2 | 156 |
| Not Suitable for Investment | 38 | 6 | 44 |
| Total | 192 | 8 | 200 |

Based on the problem description, a decision tree can be drawn as follows:

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