風結ぶ言葉たち

Decision-making is an important function of management, involving the process and results 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 help decision-makers make 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:

• $ 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 located, 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, which is the value function of the decision problem, generally called the benefit-loss value or utility value.

Deterministic Problem Analysis#

Problem Characteristics#

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

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

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

4. There exists one definite natural state;

Solution Methods#

1. When the number of plans is large, methods such as planning in operations research are commonly 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/item, a unit variable cost of 100 yuan/item, and fixed costs of 5000 yuan, with an annual production volume of 200 items. Solve:

1. What is the annual profit of the company?
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/item, if the unit selling price remains unchanged and cannot be changed, should the company stop production?

Annual profit of the company:

Break-even production volume:

Break-even price:

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

Target total cost:

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 under natural states can be calculated;

4. There are two or more natural states;

5. The probabilities of different natural states occurring 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-world 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 under natural states can be calculated;

4. There are two or more natural states;

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

Solution Methods#

1. Optimistic method (max-max principle);

2. Pessimistic method (min-max principle);

3. Regret value method (Savage criterion or maximum-minimum regret principle);

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 this new product for 10 years, and the profit and loss values 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 purchase | 400 | 250 | 90 | -50 |

Optimistic Method:

The maximum benefit of 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 gives:

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

Pessimistic Method:

The minimum benefit of 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 gives:

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

Regret Value Method:

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

Taking the minimum value among the maximum regret values of each plan gives:

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

Equal Probability Method:

Assuming equal probabilities for each state, the expected values of benefits for each plan are calculated and compared, taking the maximum expected value among the plans to select the corresponding plan. Due to the simplicity of the method, 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 under natural states can be calculated;

4. There are two or more natural states;

5. The probabilities of different natural states occurring 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.

• $ 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 on the production volume of a product for the next year to prepare for various pre-production tasks in advance. It is assumed that the size of the production volume mainly depends on 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 rising, remaining unchanged, and falling 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 of the product.

| | Price rises $ \theta_1 $ | Price remains unchanged $ \theta_2 $ | Price falls $ \theta_3 $ |
|-------------|------------------|------------------|------------------|
| | 0.3 | 0.6 | 0.1 |
| Large production $ A_1 $ | 40 | 36 | -6 |
| Medium production $ A_2 $ | 36 | 34 | 24 |
| Small 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 is needed, 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. The application of 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 complete certainty about 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 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 probabilities for 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 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), while 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 suggests that before producing the product, an additional "purification" process should be added to the chemical raw materials, 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.

It has been calculated 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 the chemical raw materials are inspected before production, it will be possible to determine the purity state of each batch of chemical raw materials, allowing for different strategies (i.e., to purify or not to purify) to be adopted based on the purity, thus 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 |
| Purify $ A_1 $ | 1000 | 1000 | 1000 | 1000 | 1000 |
| Do not purify $ 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% per year if successful, 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 obtain more information, the company seeks consulting services, with a consultation fee of 500 yuan, but the consultation 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 Successful | Investment Failed | 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:

This article was updated by Mix Space to xLog. The original link is https://nishikori.tech/posts/tech/Decision-Analysis