{"id":122,"date":"2021-11-16T12:16:57","date_gmt":"2021-11-16T03:16:57","guid":{"rendered":"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=122"},"modified":"2025-04-10T14:00:10","modified_gmt":"2025-04-10T05:00:10","slug":"pulp%e3%81%a7%e9%85%8d%e9%80%81%e6%9c%80%e9%81%a9%e5%8c%96%e3%82%92%e8%a7%a3%e3%81%84%e3%81%a6%e3%81%bf%e3%82%8b%ef%bc%91","status":"publish","type":"post","link":"https:\/\/www.avelio.co.jp\/math\/wordpress\/?p=122","title":{"rendered":"PULP\u3067\u914d\u9001\u6700\u9069\u5316\u3092\u89e3\u3044\u3066\u307f\u308b\uff11"},"content":{"rendered":"\n

\u914d\u9001\u6700\u9069\u5316(Vehicle routing problem)\u3068\u306f\u3001\uff11\u3064\u307e\u305f\u306f\u8907\u6570\u306e\u62e0\u70b9\u306e\u8377\u7269\u3092\u3001\u9867\u5ba2\u304c\u3044\u308b\u5834\u6240\u306b\u904b\u3076\u969b\u3001\u79fb\u52d5\u30b3\u30b9\u30c8\u3092\u6700\u5c0f\u306b\u3059\u308b\u3088\u3046\u306a\u7d4c\u8def\u3092\u6c42\u3081\u308b\u554f\u984c\u3067\u3059\u3002<\/p>\n\n\n\n

VRP\u306b\u306f\u3044\u308d\u3044\u308d\u306a\u7a2e\u985e\u304c\u3042\u308a\u307e\u3059\u3002
CVRP\uff1a\u8eca\u4e21\u306e\u7a4d\u8f09\u91cf\uff08\u6570\uff09\u306b\u4e0a\u9650\u304c\u3042\u308b\u3082\u306e
VRPPD\uff1a\u96c6\u8377\u3068\u914d\u9001\u304c\u3042\u308b\u3082\u306e
VRPTW\uff1a\u914d\u9001\u306e\u5e0c\u671b\u6642\u9593(Time window\uff09\u304c\u6307\u5b9a\u3055\u308c\u308b\u3082\u306e
\u306a\u3069\u306a\u3069\u3002\uff08\u53c2\u8003\uff1awikipedia https:\/\/en.wikipedia.org\/wiki\/Vehicle_routing_problem\uff09
<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u554f\u984c<\/h2>\n\n\n\n

\u307e\u305a\u306f\u7c21\u5358\u306b\u3001\u914d\u9001\u30bb\u30f3\u30bf\u30fc\u306f\uff11\u3064\u3001\uff11\u53f0\u306e\u8eca\u4e21\u3067\uff15\u3064\u306e\u8377\u7269\u307e\u3067\u4e00\u7dd2\u306b\u904b\u3079\u308b\u3001\u3068\u3044\u3046\u6761\u4ef6\u3092\u8003\u3048\u3066\u307f\u307e\u3059\u3002
\u30fb\u4e0b\u56f3\u306e\u3088\u3046\u306b\u3001\u5f0a\u793e\u304c\u3042\u308b\u4e94\u53cd\u7530\u99c5\u3042\u305f\u308a\u306b\uff08\u67b6\u7a7a\u306e\uff09\u4e94\u53cd\u7530\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u304c\u3042\u308a\u307e\u3059\u3002
\u30fb\u305d\u306e\u5468\u8fba\u3067\u300120\u4eba\u306e\u9867\u5ba2\u5b85\u3092\u7def\u5ea6\u7d4c\u5ea6\u3092\u30e9\u30f3\u30c0\u30e0\u306b\u30d4\u30c3\u30af\u30a2\u30c3\u30d7\u3057\u307e\u3057\u305f\u3002\u3053\u306e20\u4eba\u306b\u8377\u7269\u3092\u5c4a\u3051\u307e\u3059\u3002
\u30fb\u5b9f\u969b\u306f\u3001\u79fb\u52d5\u6642\u9593\u3092\u30b3\u30b9\u30c8\u306b\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u79fb\u52d5\u6642\u9593\u306f\u5730\u56f3API\u306a\u3069\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u306e\u304c\u826f\u3044\u304b\u3068\u601d\u3044\u307e\u3059\u304c\u3001\u4eca\u56de\u306f\u7c21\u5358\u306e\u305f\u3081\u3001\u7def\u5ea6\u7d4c\u5ea6\u306e\u5dee\u304b\u3089\u6982\u7b97\u3057\u307e\u3057\u305f\u3002
\u30fb20\u4eba\u306e\u8377\u7269\u3092\u3001\u4e0a\u96505\u500b\u305a\u3064\u8eca\u4e21\u306b\u8f09\u305b\u3066\u904b\u3076\u3068\u304d\u3001\u3069\u308c\u3068\u3069\u308c\u3092\u307e\u3068\u3081\u3066\u3069\u306e\u3088\u3046\u306a\u7d4c\u8def\u3067\u884c\u3051\u3070\u4e00\u756a\u77ed\u3044\u6642\u9593\u3067\u6e08\u3080\u304b\u3001\u3068\u3044\u3046\u554f\u984c\u3067\u3059\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u3000\u3000\u3000<\/h2>\n\n\n\n

\u5909\u6570\uff11\uff1a\u8377\u7269\u304c\u901a\u308b\u30eb\u30fc\u30c8\u3092\u8868\u3059\u5909\u6570<\/h2>\n\n\n\n

\u4f55\u3092\u5909\u6570\u3068\u3059\u308b\u304b\u3092\u8003\u3048\u307e\u3059\u3002
\u307e\u305a\u306f\u300120\u4eba\u306e\u305d\u308c\u305e\u308c\u306e\u8377\u7269\u3092\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u304b\u3089\u9867\u5ba2\u5b85\u3078\u5c4a\u3051\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u306e\u3067\u3001\u305d\u308c\u305e\u308c\u306e\u8377\u7269\u304c\u901a\u308b\u30eb\u30fc\u30c8\u304c\u5909\u6570\u306b\u306a\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002
\\( p[i][j][k] \\) : \u9867\u5ba2i\u306e\u8377\u7269\u304c\u30dd\u30a4\u30f3\u30c8j\u2192\u30dd\u30a4\u30f3\u30c8k\u306e\u9593\u306e\u30d1\u30b9\u3092\u901a\u308b\u5834\u54081\u3001\u901a\u3089\u306a\u3044\u5834\u54080 \u3068\u3057\u307e\u3059\u3002
\\( i \\) \u306f\u9867\u5ba2\u306a\u306e\u30671\uff5e20\u3092\u52d5\u304d\u307e\u3059\u3002
\\( j,k \\) \u306f\u3001\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u30920\uff0b\u9867\u5ba21\uff5e20\u306e0\uff5e20\u3092\u52d5\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

<\/p>\n\n\n\n

\u5909\u6570\uff12\uff1a\u8eca\u4e21\u304c\u901a\u308b\u30eb\u30fc\u30c8\u3092\u8868\u3059\u5909\u6570<\/h2>\n\n\n\n

\u79fb\u52d5\u30b3\u30b9\u30c8\u306f\u3001\u8377\u7269\u3092\u904b\u3076\u8eca\u4e21\u306e\u30eb\u30fc\u30c8\u306b\u639b\u304b\u308b\u306e\u3067\u3001\u8eca\u4e21\u304c\u901a\u308b\u30eb\u30fc\u30c8\u304c\u5909\u6570\u306b\u306a\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002
\\( c[j][k] \\) : \u8eca\u4e21\u304c\u30dd\u30a4\u30f3\u30c8j\u2192\u30dd\u30a4\u30f3\u30c8k\u306e\u9593\u306e\u30d1\u30b9\u3092\u901a\u308b\u5834\u54081,\u901a\u3089\u306a\u3044\u5834\u54080<\/p>\n\n\n\n

<\/p>\n\n\n\n

\u5236\u7d04\u6761\u4ef6\uff08\u8377\u7269\u306e\u30eb\u30fc\u30c8\u306b\u95a2\u3059\u308b\u5236\u7d04\uff09<\/h2>\n\n\n\n

\u307e\u305a\u306f\u3001\u5404\u8377\u7269\u304c\u901a\u308b\u30eb\u30fc\u30c8\u306b\u3064\u3044\u3066\u7acb\u5f0f\u3057\u307e\u3059\u3002
\uff11\uff0e\u8377\u7269\u306e\u30b9\u30bf\u30fc\u30c8\u5730\u70b9\u306f\u305d\u306e\u30d1\u30c3\u30b1\u30fc\u30b8\u306e\u96c6\u914d\u30bb\u30f3\u30bf\u30fc
\u3000\u4eca\u56de\u306f\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u306f\uff11\u3064\u306e\u307f\u3067\u3059\u306e\u3067\u3001\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u306e\u30dd\u30a4\u30f3\u30c80\u304b\u3089\u51fa\u3066\u3044\u304f\u30d1\u30b9\u304c\uff11\u3064\u3060\u3051\u306b\u306a\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002
$$ \\sum_j {p[i][0][j]}=1 \\qquad(\\forall{i}) $$<\/p>\n\n\n\n

\uff12\uff0e\u30eb\u30fc\u30c8\u306e\u7d42\u4e86\u5730\u70b9\u306f\u3001\u9867\u5ba2\u5b85\u306b\u306a\u308a\u307e\u3059\u3002
\u305d\u306e\u305f\u3081\u3001\u3059\u3079\u3066\u306e\u9867\u5ba2\u5b85i\u306b\u5165\u308b\u30d1\u30b9\u306f\uff11\u3064\u3001\u51fa\u3066\u3044\u304f\u30d1\u30b9\u306f0\u306b\u306a\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002
$$ \\sum_j {p[i][j][i]}=1 \\qquad(\\forall{i}) \\\\
\\sum_j {p[i][i][j]}=0 \\qquad(\\forall{i}) $$
<\/p>\n\n\n\n

\uff13\uff0e\u30b9\u30bf\u30fc\u30c8\u5730\u70b9\u3001\u7d42\u4e86\u5730\u70b9\u4ee5\u5916\u306e\u5834\u6240\u3067\u306f\u3001\u3059\u3079\u3066\u306e\u30dd\u30a4\u30f3\u30c8\u3067\u3001\u5165\u3063\u3066\u304f\u308b\u30d1\u30b9\u3068\u51fa\u3066\u3044\u304f\u30d1\u30b9\u306e\u6570\u304c\u540c\u3058\u306b\u306a\u308b\u3088\u3046\u306b\u3057\u307e\u3059\u3002
$$ \\sum_{j\\backslash0,i}{p[i][j][i]}= \\sum_ {j\\backslash0,i} {p[i][i][j]} \\qquad(\\forall{i}) $$ <\/p>\n\n\n\n

\u3053\u3053\u307e\u3067\u306e\u7acb\u5f0f\u304c\u3042\u3063\u3066\u3044\u308b\u304b\u78ba\u8a8d\u3057\u307e\u3059\u3002
\u8377\u7269\u304c\u901a\u308b\u3059\u3079\u3066\u306e\u30d1\u30b9\u306b\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u30b3\u30b9\u30c8\u3092\u304b\u3051\u3066\u3001\u6700\u9069\u5316\u3092\u5b9f\u884c\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n

$$ minimize \\sum_i{\\sum_j{\\sum_k{p[i][j][k]\\cdot (j\u2192k\u3078\u306e\u79fb\u52d5\u306b\u304b\u304b\u308b\u6642\u9593)}}} $$ <\/p>\n\n\n\n

\u5f53\u305f\u308a\u524d\u3067\u3059\u304c\u3001\u4e0b\u56f3\u306e\u3088\u3046\u306b\u3069\u306e\u9867\u5ba2\u306b\u3082\u914d\u9001\u30bb\u30f3\u30bf\u30fc\u304b\u3089\u76f4\u63a5\u8377\u7269\u3092\u904b\u3076\u3088\u3046\u306a\u30d1\u30b9\u306b\u306a\u308a\u307e\u3059\u3002\u56de\u308a\u9053\u3082\u3057\u3066\u3044\u306a\u3044\u3057\u3001\u4e00\u5fdc\u3053\u3053\u307e\u3067\u306f\u5408\u3063\u3066\u3044\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


\u5236\u7d04\u6761\u4ef6\uff08\u8eca\u4e21\u306e\u30eb\u30fc\u30c8\u306b\u95a2\u3059\u308b\u5236\u7d04\uff09 <\/h2>\n\n\n\n

\u6b21\u306b\u3001\u8377\u7269\u3092\uff15\u3064\u307e\u3067\u8eca\u306b\u8f09\u305b\u3066\u904b\u3076\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n

\uff11\uff0e\u8eca\u4e21\u306b\u306f\u3001\uff15\u3064\u307e\u3067\u306e\u8377\u7269\u3092\u8f09\u305b\u3089\u308c\u307e\u3059\u3002\u305d\u308c\u3092\u5f0f\u306b\u3059\u308b\u3068
$$ \\sum_i{p[i][j][k] } \\leq 5c\\qquad(\\forall{j,k}) $$<\/p>\n\n\n\n

\uff12\uff0e\u8eca\u4e21\u306f\u3001\u96c6\u914d\u30bb\u30f3\u30bf\u30fc\u3092\u51fa\u767a\u3057\u3066\u8377\u7269\u3092\uff15\u500b\u305a\u3064\u8f09\u305b\u3066\u9867\u5ba2\u5b85\u3092\u56de\u308a\u3001\u6700\u5f8c\u306b\u914d\u9001\u30bb\u30f3\u30bf\u30fc\u306b\u5e30\u3063\u3066\u304f\u308b\u3082\u306e\u3068\u3057\u307e\u3059\u3002\u30eb\u30fc\u30c8\u306f\u3064\u306a\u304c\u3063\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308a\u3001\u914d\u9001\u30bb\u30f3\u30bf\u30fc\u306b\u306f\u4f55\u5ea6\u304b\u7acb\u3061\u5bc4\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7acb\u5f0f\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

$$ \\sum_{j \\backslash 0}{c[j][0]}== \\sum_{k \\backslash 0}{c[0][k]} \\\\
\\sum_{j}{c[j][k]}==1 \\qquad(k=1,\\cdots 20) \\\\
\\sum_{k}{c[j][k]}==1 \\qquad(j=1,\\cdots 20) $$
<\/p>\n\n\n\n

\u76ee\u7684\u95a2\u6570<\/h2>\n\n\n\n

\u6700\u5f8c\u306b\u3001\u76ee\u7684\u95a2\u6570\u3092\u8eca\u4e21\u306e\u79fb\u52d5\u306b\u304b\u304b\u308b\u30b3\u30b9\u30c8\u306b\u66f8\u304d\u63db\u3048\u307e\u3059\u3002<\/p>\n\n\n\n

$$ minimize \\sum_j{\\sum_k{c[j][k]\\cdot (j\u2192k\u3078\u306e\u79fb\u52d5\u306b\u304b\u304b\u308b\u6642\u9593)}} $$ <\/p>\n\n\n\n

\u3000\u3000<\/p>\n\n\n\n

\u7d50\u679c<\/h2>\n\n\n\n

\u3053\u306e\u7acb\u5f0f\u306b\u57fa\u3065\u304d\u3001PULP\u306e\u6700\u5c0f\u5316\u554f\u984c\u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u305f\u3002
\u7d50\u679c\u306b\u3064\u3044\u3066\u3001\u305d\u308c\u305e\u308c\u306e\u8377\u7269\u306e\u30eb\u30fc\u30c8\u3092\u4e0b\u56f3\u306b\u793a\u3057\u307e\u3059\u3002\u4e0b\u56f3\u306e\u3088\u3046\u306b\uff15\u500b\u307e\u3067\u306e\u8377\u7269\u3092\u307e\u3068\u3081\u3066\u904b\u3076\u3053\u3068\u304c\u3067\u304d\u3066\u3044\u308b\u3088\u3046\u3067\u3059\u3002
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u8eca\u4e21\u306e\u30eb\u30fc\u30c8\u306e\u65b9\u3092\u56f3\u793a\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u304a\u304a\u3088\u305d\u52b9\u7387\u3088\u304f\u56de\u308c\u3066\u3044\u308b\u304b\u306a\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n\n\n\n

\u4ee5\u4e0a\u306e\u3088\u3046\u306b\u3001\u7c21\u5358\u306a\u4f8b\u3067\u306f\u3042\u308a\u307e\u3059\u304c\u3001\u914d\u9001\u554f\u984c\u3092PULP\u3067\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002
\u6574\u6570\u8a08\u753b\u554f\u984c\u306f\u3001\u7acb\u5f0f\u304c\u697d\u3057\u3044\u3067\u3059\u3002
\u8377\u7269\u306e\u5e0c\u671b\u914d\u9001\u6642\u9593\u304c\u3042\u308b\u5834\u5408\u3084\u3001\u8907\u6570\u306e\u8eca\u4e21\u3067\u56de\u308b\u5834\u5408\u3001\u96c6\u8377\u304c\u3042\u308b\u554f\u984c\u306a\u3069\u3082\u3001\u8a18\u4e8b\u306b\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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