線形計画をPythonで解いてみる(2)

PuLP2.ipynb

{
 "cells": [
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 線形計画をPythonで解いてみる(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "「入門オペレーションズ・リサーチ」の演習問題を解いてみるシリーズ。\n",
    "\n",
    "## 演習問題 p.165\n",
    "\n",
    "冷蔵庫を開けたら、豚肉300g、鶏肉500g、キャベツ300g、人参300g、椎茸200gがありました。あなたはこの材料で今日の夕食を作らなければなりません。作れるメニューは以下の通りで、100g作るのに必要な材料が書いてあります。\n",
    "\n",
    "| メニュー100gあたりの分量 | 豚肉 (300g) | 鶏肉 (500g) | キャベツ (300g) | 人参 (300g) | 椎茸 (200g) |\n",
    "|:---|:---|:---|:---|:---|:---|\n",
    "| 鳥のスープ |  | 40g |  | 20g | 10g |\n",
    "| キャベツのスープ |  |  | 30g | 20g | 20g |\n",
    "| 鶏肉の炒め物 |  | 60g |  | 20g | 20g |\n",
    "| 豚肉の炒め物 | 40g |  | 40g | 10g | 10g |\n",
    "\n",
    "このメニューで、スープを500g以上、炒め物を500g以上作らなければなりません。どのような線形計画法の問題を作れば、この条件を満たすように、それぞれのメニューを作る量が求められるでしょうか。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pulp\n",
    "\n",
    "problem = pulp.LpProblem('夕食計画', pulp.LpMaximize)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 鳥のスープの量を x1、キャベツのスープの量を x2 とする。\n",
    "* 鶏肉の炒め物の量を y1、豚肉の炒め物の量を y2 とする。\n",
    "\n",
    "これらはそれぞれ最小値0、最大値500となる。（単位：グラム）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 鳥のスープの量\n",
    "x1 = pulp.LpVariable('x1', 0, 500, 'Integer')\n",
    "# キャベツのスープの量\n",
    "x2 = pulp.LpVariable('x2', 0, 500, 'Integer')\n",
    "# 鶏肉の炒め物の量\n",
    "y1 = pulp.LpVariable('y1', 0, 500, 'Integer')\n",
    "# 豚肉の炒め物の量\n",
    "y2 = pulp.LpVariable('y2', 0, 500, 'Integer')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 豚肉、鶏肉、キャベツ、人参、椎茸の使用量を、それぞれ a, b, c, d, e とする。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "| メニュー100gあたりの分量 | a豚肉 (300g) | b鶏肉 (500g) | cキャベツ (300g) | d人参 (300g) | e椎茸 (200g) |\n",
    "|:---|:---|:---|:---|:---|:---|\n",
    "| x1鳥のスープ |  | 40g |  | 20g | 10g |\n",
    "| x2キャベツのスープ |  |  | 30g | 20g | 20g |\n",
    "| y1鶏肉の炒め物 |  | 60g |  | 20g | 20g |\n",
    "| y2豚肉の炒め物 | 40g |  | 40g | 10g | 10g |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 豚肉の使用量\n",
    "a = pulp.LpVariable('a', 0, 300, 'Integer')\n",
    "# 鶏肉の使用量\n",
    "b = pulp.LpVariable('b', 0, 500, 'Integer')\n",
    "# キャベツの使用量\n",
    "c = pulp.LpVariable('c', 0, 300, 'Integer')\n",
    "# 人参の使用量\n",
    "d = pulp.LpVariable('d', 0, 300, 'Integer')\n",
    "# 椎茸の使用量\n",
    "e = pulp.LpVariable('e', 0, 200, 'Integer')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 何を最大化（or最小化）するか（目的関数）\n",
    "# problem += x1 + x2 + y1 + y2\n",
    "problem += c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* スープを500g以上作る\n",
    "* 炒め物を500g以上作る"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 調理する量\n",
    "problem += x1 + x2 >= 500\n",
    "problem += y1 + y2 >= 500"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "| メニュー100gあたりの分量 | a豚肉 (300g) | b鶏肉 (500g) | cキャベツ (300g) | d人参 (300g) | e椎茸 (200g) |\n",
    "|:---|:---|:---|:---|:---|:---|\n",
    "| x1鳥のスープ |  | 40g |  | 20g | 10g |\n",
    "| x2キャベツのスープ |  |  | 30g | 20g | 20g |\n",
    "| y1鶏肉の炒め物 |  | 60g |  | 20g | 20g |\n",
    "| y2豚肉の炒め物 | 40g |  | 40g | 10g | 10g |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# それぞれの料理に使う食材の量の定義\n",
    "# 豚肉\n",
    "problem += 40/100*y2 == a\n",
    "# 鶏肉\n",
    "problem += 40/100*x1 + 60/100*y1 == b\n",
    "# キャベツ\n",
    "problem += 30/100*x2 + 40/100*y2 == c\n",
    "# 人参\n",
    "problem += 20/100*x1 + 20/100*x2 + 20/100*y1 + 10/100*y2 == d\n",
    "# 椎茸\n",
    "problem += 10/100*x1 + 20/100*x2 + 20/100*y1 + 10/100*y2 == e\n",
    "\n",
    "# それぞれの料理に使える食材の量の上限の制約\n",
    "# 豚肉\n",
    "problem += 40/100*y2 <= 300\n",
    "# 鶏肉\n",
    "problem += 40/100*x1 + 60/100*y1 <= 500\n",
    "# キャベツ\n",
    "problem += 30/100*x2 + 40/100*y2 <= 300\n",
    "# 人参\n",
    "problem += 20/100*x1 + 20/100*x2 + 20/100*y1 + 10/100*y2 <= 300\n",
    "# 椎茸\n",
    "problem += 10/100*x1 + 20/100*x2 + 20/100*y1 + 10/100*y2 <= 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "status = problem.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimal\n"
     ]
    }
   ],
   "source": [
    "print(pulp.LpStatus[status])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "夕食計画:\n",
      "MAXIMIZE\n",
      "1*c + 0\n",
      "SUBJECT TO\n",
      "_C1: x1 + x2 >= 500\n",
      "\n",
      "_C2: y1 + y2 >= 500\n",
      "\n",
      "_C3: - a + 0.4 y2 = 0\n",
      "\n",
      "_C4: - b + 0.4 x1 + 0.6 y1 = 0\n",
      "\n",
      "_C5: - c + 0.3 x2 + 0.4 y2 = 0\n",
      "\n",
      "_C6: - d + 0.2 x1 + 0.2 x2 + 0.2 y1 + 0.1 y2 = 0\n",
      "\n",
      "_C7: - e + 0.1 x1 + 0.2 x2 + 0.2 y1 + 0.1 y2 = 0\n",
      "\n",
      "_C8: 0.4 y2 <= 300\n",
      "\n",
      "_C9: 0.4 x1 + 0.6 y1 <= 500\n",
      "\n",
      "_C10: 0.3 x2 + 0.4 y2 <= 300\n",
      "\n",
      "_C11: 0.2 x1 + 0.2 x2 + 0.2 y1 + 0.1 y2 <= 300\n",
      "\n",
      "_C12: 0.1 x1 + 0.2 x2 + 0.2 y1 + 0.1 y2 <= 200\n",
      "\n",
      "VARIABLES\n",
      "0 <= a <= 300 Integer\n",
      "0 <= b <= 500 Integer\n",
      "0 <= c <= 300 Integer\n",
      "0 <= d <= 300 Integer\n",
      "0 <= e <= 200 Integer\n",
      "0 <= x1 <= 500 Integer\n",
      "0 <= x2 <= 500 Integer\n",
      "0 <= y1 <= 500 Integer\n",
      "0 <= y2 <= 500 Integer\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(problem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x1鳥のスープ: 20\n",
      "x2キャベツのスープ: 480\n",
      "y1鶏肉の炒め物: 140\n",
      "y2豚肉の炒め物: 390\n",
      "\n",
      "a豚肉 (300g): 156\n",
      "b鶏肉 (500g): 92\n",
      "cキャベツ (300g): 300\n",
      "d人参 (300g): 167\n",
      "e椎茸 (200g): 165\n"
     ]
    }
   ],
   "source": [
    "print(\"x1鳥のスープ: %d\" % x1.value())\n",
    "print(\"x2キャベツのスープ: %d\" % x2.value())\n",
    "print(\"y1鶏肉の炒め物: %d\" % y1.value())\n",
    "print(\"y2豚肉の炒め物: %d\" % y2.value())\n",
    "print(\"\")\n",
    "\n",
    "print(\"a豚肉 (300g): %d\" % a.value())\n",
    "print(\"b鶏肉 (500g): %d\" % b.value())\n",
    "print(\"cキャベツ (300g): %d\" % c.value())\n",
    "print(\"d人参 (300g): %d\" % d.value())\n",
    "print(\"e椎茸 (200g): %d\" % e.value())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ※答え合わせ\n",
    "\n",
    "演習の解答に「ここに適当な目的関数を設定すると、線形計画問題になります」とあるけれど、その「目的関数」を知りたいのです。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "x1 + x2 + y1 + y2 を maximize （最大限の量を作るには）\n",
    "```\n",
    "x1鳥のスープ: 500\n",
    "x2キャベツのスープ: 0\n",
    "y1鶏肉の炒め物: 500\n",
    "y2豚肉の炒め物: 500\n",
    "\n",
    "a豚肉 (300g): 200\n",
    "b鶏肉 (500g): 500\n",
    "cキャベツ (300g): 200\n",
    "d人参 (300g): 250\n",
    "e椎茸 (200g): 200\n",
    "```\n",
    "\n",
    "x1 + x2 + y1 + y2 を minimize （最小限の量を作るには）\n",
    "```\n",
    "x1鳥のスープ: 500\n",
    "x2キャベツのスープ: 0\n",
    "y1鶏肉の炒め物: 500\n",
    "y2豚肉の炒め物: 0\n",
    "\n",
    "a豚肉 (300g): 0\n",
    "b鶏肉 (500g): 500\n",
    "cキャベツ (300g): 0\n",
    "d人参 (300g): 200\n",
    "e椎茸 (200g): 150\n",
    "```\n",
    "\n",
    "a + b を maximize （肉を食べたい！）\n",
    "```\n",
    "x1鳥のスープ: 500\n",
    "x2キャベツのスープ: 0\n",
    "y1鶏肉の炒め物: 500\n",
    "y2豚肉の炒め物: 500\n",
    "\n",
    "a豚肉 (300g): 200\n",
    "b鶏肉 (500g): 500\n",
    "cキャベツ (300g): 200\n",
    "d人参 (300g): 250\n",
    "e椎茸 (200g): 200\n",
    "```\n",
    "\n",
    "a + b を minimize （肉は高い！）\n",
    "```\n",
    "x1鳥のスープ: 0\n",
    "x2キャベツのスープ: 500\n",
    "y1鶏肉の炒め物: 130\n",
    "y2豚肉の炒め物: 370\n",
    "\n",
    "a豚肉 (300g): 148\n",
    "b鶏肉 (500g): 78\n",
    "cキャベツ (300g): 298\n",
    "d人参 (300g): 163\n",
    "e椎茸 (200g): 163\n",
    "```\n",
    "\n",
    "c を maximize （安いキャベツを最大限使う！）\n",
    "```\n",
    "x1鳥のスープ: 20\n",
    "x2キャベツのスープ: 480\n",
    "y1鶏肉の炒め物: 140\n",
    "y2豚肉の炒め物: 390\n",
    "\n",
    "a豚肉 (300g): 156\n",
    "b鶏肉 (500g): 92\n",
    "cキャベツ (300g): 300\n",
    "d人参 (300g): 167\n",
    "e椎茸 (200g): 165\n",
    "```\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": []
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