Regionalization with PuLP: exact optimization model called "Flow" in Duque et al. (2011)

prm_flow_PuLP.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import libpysal as ps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Making the lattice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are generating a 3x3 lattice: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "lat_w = ps.weights.util.lat2W(nrows=3, ncols=3, rook=True, id_type=\"int\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [3, 1],\n",
       " 1: [0, 4, 2],\n",
       " 2: [1, 5],\n",
       " 3: [0, 6, 4],\n",
       " 4: [1, 3, 7, 5],\n",
       " 5: [2, 4, 8],\n",
       " 6: [3, 7],\n",
       " 7: [4, 6, 8],\n",
       " 8: [5, 7]}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neighbor_dict = lat_w.neighbors\n",
    "neighbor_dict"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Defining the data (_hardcoded_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 726.7,\n",
       " 1: 623.6,\n",
       " 2: 487.3,\n",
       " 3: 200.4,\n",
       " 4: 245.0,\n",
       " 5: 481.0,\n",
       " 6: 170.9,\n",
       " 7: 225.9,\n",
       " 8: 226.9}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value_list = [726.7, 623.6, 487.3, 200.4, 245.0, 481.0, 170.9, 225.9, 226.9]\n",
    "value_dict = {area: value for area, value in zip(neighbor_dict.keys(), value_list)}\n",
    "value_dict"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Defining the optimization model from Duque et al. (2011): \"The p-Regions Problem\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from pulp import LpProblem, LpMinimize, LpVariable, \\\n",
    "                 LpInteger, lpSum, GLPK, LpStatus, \\\n",
    "                 value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "prob = LpProblem(\"Tree\", LpMinimize)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = len(value_dict)\n",
    "p = 2  # TODO: turn this into a part of the API\n",
    "I = list(value_dict.keys())\n",
    "II = [(i, j) for i in I \n",
    "             for j in I]\n",
    "II_upper_triangle = [(i, j) for i, j in II if i < j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K = range(p)  # index for regions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def dissim_measure(v1, v2):\n",
    "    \"\"\"\\\n",
    "    Returns the dissimilarity between the values v1 and\n",
    "    v2.\n",
    "    \"\"\"\n",
    "    return abs(v1 - v2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "d = {(i, j): dissim_measure(value_dict[i], value_dict[j])\n",
    "     for i, j in II_upper_triangle}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Decision variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t = LpVariable.dicts(\n",
    "    \"t\",\n",
    "    ((i, j) for i, j in II_upper_triangle),\n",
    "    lowBound=0, upBound=1, cat=LpInteger)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# The amount of flow (non-negative integer) from area i to j \n",
    "# in region k.\n",
    "f = LpVariable.dicts(\n",
    "    \"f\",\n",
    "    ((i, j, k) for i, j in II for k in K),\n",
    "    lowBound=0, cat=LpInteger)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 1 if area i is assigned to region k. 0 otherwise.\n",
    "y = LpVariable.dicts(\n",
    "    \"y\",\n",
    "    ((i, k) for i in I for k in K),\n",
    "    lowBound=0, upBound=1, cat=LpInteger)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 1 if area i is chosen as a sink. 0 otherwise.\n",
    "w = LpVariable.dicts(\n",
    "    \"w\",\n",
    "    ((i, k) for i in I for k in K),\n",
    "    lowBound=0, upBound=1, cat=LpInteger)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Objective function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "prob += lpSum(d[i, j] * t[i, j] for i, j in II_upper_triangle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constraints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (21) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i in I:\n",
    "    prob += sum(y[i, k] for k in K) == 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (22) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i in I:\n",
    "    for k in K:\n",
    "        prob += w[i, k] <= y[i, k]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (23) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for k in K:\n",
    "    prob += sum(w[i, k] for i in I) == 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (24) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i in I:\n",
    "    for j in neighbor_dict[i]:\n",
    "        for k in K:\n",
    "            prob += f[i, j, k] <= y[i, k] * (n-p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (25) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i in I:\n",
    "    for j in neighbor_dict[i]:\n",
    "        for k in K:\n",
    "            prob += f[i, j, k] <= y[j, k] * (n-p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (26) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i in I:\n",
    "    for k in K:\n",
    "        lhs = sum(f[i, j, k] - f[j, i, k] for j in neighbor_dict[i])\n",
    "        prob += lhs >= y[i, k] - (n-p) * w[i, k]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (27) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "for i, j in II_upper_triangle:\n",
    "    for k in K:\n",
    "        prob += t[i, j] >= y[i, k] + y[j, k] -1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve the optimization problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob.solve(GLPK(msg=0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Optimal'"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LpStatus[prob.status]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "#for v in prob.variables():\n",
    "#    print(v.name, \"=\", v.varValue)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "area 0 is in region 0\n",
      "area 1 is in region 0\n",
      "area 2 is in region 0\n",
      "area 3 is in region 1\n",
      "area 4 is in region 1\n",
      "area 5 is in region 0\n",
      "area 6 is in region 1\n",
      "area 7 is in region 1\n",
      "area 8 is in region 1\n"
     ]
    }
   ],
   "source": [
    "for i in I:\n",
    "    for k in K:\n",
    "        if y[i, k].varValue == 1:\n",
    "            print(\"area\", i, \"is in region\", k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1222.8"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value(prob.objective)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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