14.5

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diff --git a/14 综合评价方法/14.5 数据包络分析.ipynb b/14 综合评价方法/14.5 数据包络分析.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "> 数据包络分析 (DEA) 被认为是评价具有多指标输入和多指标输出的系统的较为有效的方法\n",
+    "---\n",
+    "### 14.5.1 数据包络分析的 C<sup>2</sup>R 模型\n",
+    "$\\boldsymbol \\alpha_i, \\boldsymbol\\beta_i$ 分别为评价对象 $i$ 的输入和输出向量，$\\boldsymbol u,\\boldsymbol v$ 分别表示输入和输出的权值向量，定义评价对象 $i$ 的效率评价指数为\n",
+    "$$h_i=\\left(\\boldsymbol\\beta_{i}^{\\rm T}\\boldsymbol v\\right)/\\left(\\boldsymbol\\alpha_{i}^{\\rm T}\\boldsymbol u\\right)$$\n",
+    "经过 Charnes-Cooper 变换，$\\boldsymbol\\omega=t\\boldsymbol u$，$\\boldsymbol\\mu=t\\boldsymbol v$，$t=\\dfrac{1}{\\boldsymbol\\alpha_{i_0}^{\\rm T}\\boldsymbol u}$，评价对象 $i_0$ 的数学规划模型为：\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "&\\max V_{i_0}=\\boldsymbol\\beta_{i_0}^{\\rm T}\\boldsymbol\\mu\\\\[1ex]\n",
+    "{\\rm s.t.}&\n",
+    "    \\begin{cases}\n",
+    "    \\boldsymbol\\alpha_{i}^{\\rm T}\\boldsymbol\\omega-\\boldsymbol\\beta_{i}^{\\rm T}\\boldsymbol\\mu\\geq 0,\\quad i=1,2,\\cdots,m\\\\[1ex]\n",
+    "    \\boldsymbol\\alpha_{i_0}^{\\rm T}\\boldsymbol\\omega = 1\\\\[1ex]\n",
+    "    \\boldsymbol\\omega \\geq 0,\\ \\boldsymbol\\mu\\geq 0\n",
+    "    \\end{cases}\n",
+    "\\end{align*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 14.5.2 数据包络分析案例\n",
+    "可持续发展评价\n",
+    "\n",
+    "输入变量：政府财政收入占 GDP 的比重、环保投资占 GDP 的比重、每千人科技人员数\n",
+    "\n",
+    "输出变量：经济发展（用人均 GDP 表示）、环境发展（用城市环境质量指数表示）\n",
+    "\n",
+    "目的：评价哪几年的发展是相对有效的；通过观察一段时间内的发展情况，判断可持续发展程度"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "第 1 个决策单元最优值： 0.2902\n",
+      "最优解: [0.     1.5385 0.    ] [0.0001 0.    ]\n",
+      "\n",
+      "第 2 个决策单元最优值： 0.2854\n",
+      "最优解: [0.     0.8997 0.0109] [0.0001 0.    ]\n",
+      "\n",
+      "第 3 个决策单元最优值： 0.2968\n",
+      "最优解: [0.     0.9029 0.011 ] [0.0001 0.    ]\n",
+      "\n",
+      "第 4 个决策单元最优值： 0.3425\n",
+      "最优解: [0.     0.8682 0.0106] [0.0001 0.    ]\n",
+      "\n",
+      "第 5 个决策单元最优值： 0.4595\n",
+      "最优解: [0.     1.3158 0.    ] [1.000e-04 1.914e-01]\n",
+      "\n",
+      "第 6 个决策单元最优值： 0.7183\n",
+      "最优解: [0.     1.4493 0.    ] [0.     1.2174]\n",
+      "\n",
+      "第 7 个决策单元最优值： 0.9069\n",
+      "最优解: [0.     1.6393 0.    ] [1.000e-04 2.385e-01]\n",
+      "\n",
+      "第 8 个决策单元最优值： 1.0\n",
+      "最优解: [0.     1.0279 0.0125] [0.0001 0.    ]\n",
+      "\n",
+      "第 9 个决策单元最优值： 1.0\n",
+      "最优解: [0.0281 0.8306 0.    ] [0.0001 0.    ]\n",
+      "\n",
+      "第 10 个决策单元最优值： 1.0\n",
+      "最优解: [0.     1.1905 0.    ] [0. 1.]\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import cvxpy as cp\n",
+    "\n",
+    "data = np.loadtxt('../../14第14章  综合评价方法/data14_5.txt')\n",
+    "a = data[:, :3]\n",
+    "b = data[:, 3:]\n",
+    "n = len(a[0])\n",
+    "m = len(b[0])\n",
+    "omg = cp.Variable(n, nonneg=True)    # omg = tu\n",
+    "mu = cp.Variable(m, nonneg=True)     # mu = tv\n",
+    "unit_num = 10\n",
+    "for i in range(unit_num):\n",
+    "    obj = cp.Maximize(b[i] @ mu)\n",
+    "    cons = [\n",
+    "        a @ omg - b @ mu >= 0,\n",
+    "        a[i] @ omg == 1,\n",
+    "    ]\n",
+    "    prob = cp.Problem(obj, cons)\n",
+    "    prob.solve(solver='GUROBI')\n",
+    "    print(f\"第 {i+1} 个决策单元最优值：\", np.round(prob.value, 4))\n",
+    "    print(\"最优解:\", np.round(omg.value, 4), np.round(mu.value, 4))\n",
+    "    print()"
+   ]
+  }
+ ],
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+}
diff --git a/tech stack.md b/tech stack.md
@@ -83,5 +83,6 @@
 |       13.3       |         偏最小二乘回归分析          | 多对多回归分析<br />▲ 体能训练数据回归建模<br />● 交通业和旅游业的回归分析 |          `skl.cross_decomposition.PLSRegression()`           |
 |       14.2       |             数据预处理              |      一致化<br />规范化（无量纲化）<br />定性指标定量化      |                   `sklearn.preprocessing`                    |
 |       14.3       |          综合评价数学模型           | *past: 主成分分析法，PageRank 法*<br />线性加权综合评价法<br />TOPSIS 法（理想解法）<br />灰色关联度分析<br />熵值法<br />秩和比法 |                `scipy.stats.rankdata()` 编秩                 |
-|       14.4       |            模糊综合评价             |                           非常好用                           |                              -                               |
+|       14.4       |            模糊综合评价             |      处理具有模糊性的评价因素<br />或评价对象，非常好用      |                              -                               |
+|       14.5       |            数据包络分析             | CCR模型 (C^2^R)<br />适用于具有多输入多输出的系统<br />● 可持续发展评价 |                              -                               |