Decision Model Construction and Applications | 决策模型的构建与应用

📚 Decision Model Construction and Applications | 决策模型的构建与应用

Decision models are mathematical frameworks used to support rational choice under uncertainty or conflicting objectives. In A-level Mathematics, decision modelling lies at the heart of Decision Mathematics, where real-world problems are translated into structured forms – tables, trees, networks, or inequalities – that can be solved systematically.

决策模型是数学框架，用于在不确定性或目标冲突下支持理性选择。在A-level数学中，决策建模是决策数学的核心，将现实问题转化为结构化形式——表格、决策树、网络或不等式——从而可以系统求解。

1. What Is a Decision Model? | 什么是决策模型？

A decision model is a simplified representation of a decision problem. It captures the essential components: the decision maker, the available actions, the possible outcomes, their probabilities, and the resulting payoffs. By quantifying these elements, mathematical tools can identify the optimal strategy or at least support better judgement.

决策模型是对决策问题的简化表示。它包含基本要素：决策者、可行行动、可能结果、概率以及相应的收益。通过量化这些要素，数学工具可以确定最优策略，或至少支持更明智的判断。

For example, a shop owner deciding how many newspapers to stock each day can model uncertain demand as a probability distribution. The model would compare profit for each stock level and recommend the quantity that maximises expected profit.

例如，店主决定每天进货多少份报纸，可将不确定需求建模为概率分布。模型比较每种进货量对应的利润，并推荐使期望利润最大的进货量。

2. Key Components of a Decision Model | 决策模型的关键要素

Every decision model must contain three building blocks:

每个决策模型都必须包含三个基本构件：

Actions (alternatives): the choices available to the decision maker, such as invest or not invest, advertise or not advertise.

行动（备选方案）：决策者可选的方案，如投资或不投资、广告或不广告。
States of nature: the possible future scenarios that may occur and are outside the decision maker’s control, e.g. high market growth or recession.

自然状态：未来可能发生的、不受决策者控制的情形，例如市场高增长或衰退。
Payoffs: the numerical consequences of choosing each action under each state, measured in profit, cost, utility, or time.

收益：在每种自然状态下选择各行动所产生的数值结果，以利润、成本、效用或时间来衡量。

A model may also include probabilities assigned to each state, allowing expected-value calculations.

模型还可为每种自然状态赋予概率，从而进行期望值计算。

3. Building a Decision Model – Step by Step | 构建决策模型的步骤

Constructing a reliable model is a disciplined process. A commonly used sequence involves:

构建可靠模型是一个严谨的过程。常用步骤如下：

Define the problem clearly and identify the objective (maximise profit, minimise cost, etc.).

清晰定义问题并明确目标（最大化利润、最小化成本等）。
List all feasible actions and all relevant states of nature.

列出所有可行行动和相关的自然状态。
Estimate probabilities for each state, based on historical data or expert opinion.

基于历史数据或专家意见估计每种状态的概率。
Assign payoffs for every action–state combination.

为每个行动–状态组合赋予收益。
Choose a decision criterion (expected value, maximin, minimax regret) and evaluate the alternatives.

选择决策准则（期望值、最大最小原则、最小最大后悔值）并评估各方案。
Perform sensitivity analysis to test how changes in probabilities or payoffs affect the decision.

进行敏感性分析，检验概率或收益变化如何影响决策。

The final step is essential because real-world estimates are rarely exact.

最后一步至关重要，因为现实中的估计很少精确。

4. Expected Value and Expected Utility | 期望值与期望效用

When probabilities are known, the expected value of an action is calculated as:

当概率已知时，行动的期望值按下式计算：

E(X) = Σ pᵢxᵢ

where xᵢ is the payoff for outcome i, and pᵢ is its probability. The action with the highest expected value is selected under the expected-value criterion.

其中xᵢ是结果i的收益，pᵢ为其概率。在期望值准则下，选择期望值最高的行动。

However, expected value ignores risk. In deciding between a guaranteed £100 and a 50% chance of £250, the expected values are £100 and £125, so the risky option appears better. Yet risk-averse individuals may prefer the certainty. Expected utility theory replaces monetary payoffs with utilities to handle this.

然而，期望值忽略了风险。若在确定的100英镑与50%概率获得250英镑之间选择，期望值分别为100英镑和125英镑，因此风险选项看似更优。但厌恶风险者可能偏好确定性。期望效用理论用效用替代货币收益来处理此问题。

Expected utility = Σ pᵢu(xᵢ)

where u(xᵢ) is the utility attached to payoff xᵢ.

其中u(xᵢ)是收益xᵢ对应的效用。

5. Decision Trees | 决策树

A decision tree is a graphical model that maps every action, chance event, and outcome chronologically. Square nodes represent decision points, and circular nodes represent chance events. Each branch is labelled with a probability or a payoff.

决策树是一种按时间顺序展示所有行动、随机事件和结果的图形模型。方形节点代表决策点，圆形节点代表随机事件。每条分支标有概率或收益。

To solve a decision tree, we work from right to left. For each chance node, calculate the expected payoff. For each decision node, select the branch with the best expected payoff and ‘fold back’ the tree.

求解决策树时，自右向左计算。对每个随机节点，计算期望收益；对每个决策节点，选择期望收益最优的分支并“折叠”树。

Example: A firm can build a large plant (cost £2m) or a small plant (cost £1m). Demand is high with probability 0.6, giving net revenues £5m for large and £3m for small; low demand with probability 0.4 gives £1m for large and £2m for small.

例如：一家公司可建大工厂（成本200万）或小工厂（成本100万）。需求高概率0.6，大厂净收入500万，小厂300万；需求低概率0.4，大厂净收入100万，小厂200万。

Expected payoff for large = 0.6×5 + 0.4×1 − 2 = 2m. For small = 0.6×3 + 0.4×2 − 1 = 1.6m. Thus the large plant is preferred.

大厂期望收益=0.6×5+0.4×1−2=200万。小厂期望收益=0.6×3+0.4×2−1=160万。因此大厂更优。

6. Linear Programming as a Decision Model | 线性规划决策模型

When a decision must respect limited resources, linear programming (LP) is a powerful tool. LP models have decision variables, a linear objective function, and linear constraints.

当决策必须受限于有限资源时，线性规划（LP）是强大工具。LP模型包含决策变量、线性目标函数和线性约束。

The general form for a maximisation problem is:

最大化问题的一般形式如下：

Maximise Z = c₁x₁ + c₂x₂ + … + cₙxₙ

subject to: a₁₁x₁ + a₁₂x₂ + … ≤ b₁, xᵢ ≥ 0

For example, a workshop produces two products. Product A gives £4 profit per unit, product B gives £3. Machine time is limited to 100 hours, with A using 2 hours and B using 1 hour. Labour is limited to 80 hours, with A using 1 hour and B using 2 hours.

例如，某车间生产两种产品。产品A每单位利润4英镑，产品B每单位3英镑。机器时间限制100小时，A需2小时，B需1小时。劳动力限制80小时，A需1小时，B需2小时。

The model is:

模型如下：

Maximise Z = 4x + 3y

subject to: 2x + y ≤ 100, x + 2y ≤ 80, x, y ≥ 0

Solving graphically or with the simplex method gives the optimal production plan.

通过图解法或单纯形法求解，得到最优生产计划。

7. Game Theory Models | 博弈论模型

Game theory models decisions where two or more players have conflicting interests. In a zero-sum game, one player’s gain equals the other’s loss. The payoff matrix lists all outcomes.

博弈论建模的是两个或多个参与者利益冲突的决策。在零和博弈中，一方的收益等于另一方的损失。收益矩阵列出所有结果。

Consider two firms choosing to advertise (A) or not advertise (N). The matrix shows Firm 1’s market share gain:

考虑两家公司选择做广告（A）或不做广告（N）。矩阵显示公司1的市场份额增益：

	Firm 2: A	Firm 2: N
Firm 1: A	10	8
Firm 1: N	6	12

If Firm 2 advertises, Firm 1’s best response is to advertise (10 > 6). If Firm 2 does not advertise, Firm 1 still prefers to advertise (8 > 12? No – 8 < 12, so best response is not advertising). This game has no pure Nash equilibrium, so mixed strategies may be needed.

若公司2做广告，公司1的最佳应对是做广告（10>6）。若公司2不做广告，公司1更偏好不做广告（12>8）。该博弈没有纯策略纳什均衡，因此可能需要混合策略。

In a mixed strategy, each player randomises to make the opponent indifferent. The equilibrium probabilities are found by solving simultaneous equations.

在混合策略中，每个参与者随机化以使对手无差异。通过联立方程求均衡概率。

8. Critical Path Analysis (CPA) | 关键路径分析

Critical Path Analysis is a decision model for project scheduling. Activities are represented by nodes and arrows, with durations and dependencies. The model identifies the longest path through the network, which determines the minimum project completion time.

关键路径分析是项目排程的决策模型。活动用节点和箭头表示，包含持续时间和依赖关系。模型识别网络中的最长路径，该路径决定项目最短完成时间。

Constructing a CPA model involves three steps:

构建CPA模型包括三个步骤：

List every activity and its immediate predecessors along with the duration.

列出每项活动、其紧前活动及持续时间。
Draw the activity-on-node network.

绘制节点型网络图。
Perform forward and backward passes to find earliest and latest start times.

进行正向和反向遍历，求最早和最晚开始时间。

Activities with zero total float are critical; any delay in them delays the whole project.

总时差为零的活动是关键活动；其任何延误都会延误整个项目。

9. Sensitivity Analysis | 敏感性分析

A robust decision model must be tested for sensitivity. Sensitivity analysis asks: ‘How much can probabilities or payoffs change before the optimal decision changes?’

稳健的决策模型必须经过敏感性检验。敏感性分析提出问题：“概率或收益变化多少才会改变最优决策？”

For a two-action problem with a probability p of a high outcome, the crossover point is found by setting expected values equal:

对于具有高结果概率p的两行动问题，令期望值相等可求得临界点：

p·α + (1−p)·β = p·γ + (1−p)·δ

Solving for p tells the decision maker how robust the choice is. If the optimal decision changes only when p is far from its estimated value, the model is robust.

解出p后，决策者可判断该选择有多稳健。若只有p远离估计值时最优决策才改变，说明模型是稳健的。

10. Applications of Decision Models | 决策模型的应用

Decision models are used in a wide range of fields. In finance, they optimise investment portfolios and assess risk. In operations research, they schedule deliveries, manage inventories, and design supply chains. In healthcare, they help decide treatment strategies and allocate scarce resources.

决策模型广泛应用于众多领域。金融中，用于优化投资组合和评估风险；运筹学中，用于调度货物、管理库存和设计供应链；医疗中，帮助决定治疗方案和配置稀缺资源。

One classic application is the newsvendor model: a vendor must decide how many copies of a magazine to order before knowing demand. The optimal quantity balances the cost of overstocking against the lost profit of understocking.

一个经典应用是报童模型：报童必须在知道需求前决定订购多少份杂志。最优数量需平衡库存过多的成本与缺货的损失利润。

Optimal order satisfies: P(demand < Q) ≤ (selling price − cost) / (selling price − salvage value)

最优订购量满足：需求小于订购量的概率 ≤ (售价−成本)/(售价−残值)。

11. Limitations and Practical Considerations | 局限性与现实考量

Mathematical models are only as good as their inputs. Probabilities are often subjective, payoffs can be hard to quantify, and real-world constraints may be nonlinear or dynamic. Furthermore, models treat consequences as numerical values, yet ethical or qualitative factors may dominate.

数学模型的好坏取决于输入。概率常常是主观的，收益难以量化，而现实约束可能是非线性或动态的。此外，模型将后果视为数值，但伦理或定性因素可能更为主导。

Therefore, decision models are decision aids, not decision replacements. A skilled analyst combines model outputs with judgement, context, and sensitivity checks before committing to a course of action.

因此，决策模型是决策辅助工具，而非决策替代品。熟练的分析师在采取行动之前，会将模型输出与判断、情境和敏感性检验相结合。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Decision Model Construction and Applications | 决策模型的构建与应用