Regionalization using exact optimization model called "Tree" in Duque et al. (2011)

prm_tree.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "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": {
    "collapsed": true
   },
   "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": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# lat_w.full()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# networkx_graph = lat_w.to_networkx()\n",
    "# networkx_graph.adj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Adding data to the lattice (_hardcoded_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": 6,
     "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": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from pyomo.environ import RangeSet, Param, Var, \\\n",
    "                          ConcreteModel, Binary, \\\n",
    "                          PositiveIntegers, \\\n",
    "                          NonNegativeIntegers, \\\n",
    "                          Objective, minimize, \\\n",
    "                          Constraint, \\\n",
    "                          Set, value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model = ConcreteModel()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model.n = Param(initialize=len(value_dict))  # number of areas\n",
    "model.p = Param(initialize=2)  # number of regions\n",
    "# use value(model.n) to get the value of parameter n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model.I = RangeSet(0, model.n-1)\n",
    "\n",
    "model.II = model.I * model.I\n",
    "\n",
    "II_upper_triangle = [(i, j) for i, j in model.II if i < j]\n",
    "model.II_upper_triangle = Set(initialize=II_upper_triangle,\n",
    "                              ordered=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def c_init(model, i, j):\n",
    "    \"\"\"\\\n",
    "    Returns 1, if areas i and j share a border and\n",
    "    0 otherwise.\n",
    "    \"\"\"\n",
    "    return int(i in neighbor_dict[j])\n",
    "\n",
    "model.c = Param(model.I, model.I, initialize=c_init)\n",
    "\n",
    "c_init(None, 1,2), c_init(None, 2,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 5]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def N_init(model, i):\n",
    "    \"\"\"\\\n",
    "    Returns list of areas that are adjacent to \n",
    "    area i.\n",
    "    \"\"\"\n",
    "    return neighbor_dict[i]\n",
    "\n",
    "model.N = Param(model.I, initialize=N_init)\n",
    "\n",
    "N_init(None, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def d_init(model, i, j):\n",
    "    \"\"\"\\\n",
    "    Returns the dissimilarity between areas i and j.\n",
    "    \"\"\"\n",
    "    return dissim_measure(value_dict[i], \n",
    "                        value_dict[j])\n",
    "\n",
    "model.d = Param(model.I, model.I, initialize=d_init)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Decision variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model.t = Var(model.II, domain=Binary,\n",
    "              doc=\"1 if areas i and j belong to the \"\n",
    "                  \"same region.\\n\"\n",
    "                  \"0 otherwise.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model.x = Var(model.II, domain=Binary,\n",
    "              doc=\"1 if link (i, j) is selected as a \"\n",
    "                  \"part of a tree.\\n\"\n",
    "                  \"0 otherwise.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model.u = Var(model.I, domain=PositiveIntegers,\n",
    "              doc=\"Order assigned to each area in a \"\n",
    "                  \"tree. This variable is used to build\"\n",
    "                  \" constraints preventing cycles.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Objective function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (3) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def obj_rule(model):\n",
    "    return sum(model.d[i,j] * model.t[i,j]\n",
    "               for i, j in model.II_upper_triangle)\n",
    "\n",
    "model.Obj = Objective(rule=obj_rule, sense=minimize)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constraints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (4) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _four(model):\n",
    "    return sum(model.x[i, j] \n",
    "               for i in model.I\n",
    "               for j in model.N[i]) == model.n - model.p\n",
    "\n",
    "model._four_constr = Constraint(rule=_four)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (5) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _five(model, i):\n",
    "    return sum(model.x[i, j] for j in model.N[i]) <= 1\n",
    "\n",
    "model._five_constr = Constraint(model.I, rule=_five)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (6) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "III_not_equal = [(i, j, m) \n",
    "                 for i, j, m in model.I * model.I * model.I\n",
    "                 if i != j and i != m and j != m]\n",
    "model.III_not_equal = Set(initialize=III_not_equal, ordered=True)\n",
    "\n",
    "def _six(model, i, j, m):\n",
    "    return model.t[i, j] + model.t[i, m] - model.t[j, m] <= 1\n",
    "\n",
    "model._six_constr = Constraint(model.III_not_equal, rule=_six)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (7) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _seven(model, i, j):\n",
    "    return model.t[i, j] - model.t[j, i] == 0\n",
    "\n",
    "model._seven_constr = Constraint(model.II, rule=_seven)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def I_with_neighbors_init(model):\n",
    "    return ((i, ni) for i in model.I \n",
    "                    for ni in model.N[i]) \n",
    "\n",
    "model.I_with_neighbors = Set(dimen=2,\n",
    "                             initialize=I_with_neighbors_init)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "# (8) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _eight(model, i, j):\n",
    "    return model.x[i, j] <= model.t[i, j]\n",
    "\n",
    "model._eight_constr = Constraint(model.I_with_neighbors,\n",
    "                                 rule=_eight)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (9) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _nine(model, i, j):\n",
    "    n, p, u, x = model.n, model.p, model.u, model.x\n",
    "    return u[i] - u[j] + (n-p) * x[i, j] \\\n",
    "                + (n-p-2) * x[j, i] <= model.n - model.p - 1 # typo in Duque et al.\n",
    "\n",
    "model._nine_constr = Constraint(model.I_with_neighbors,\n",
    "                                rule=_nine)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (10) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "def _ten(model, i):\n",
    "    return 1 <= model.u[i] <= model.n - model.p\n",
    "\n",
    "model._ten_constr = Constraint(model.I, rule=_ten)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (11) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "# already in Var()-definition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# (12) in Duque et al. (2011): \"The p-Regions Problem\"\n",
    "# already in Var()-definition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve the optimization problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "from pyomo.opt import SolverFactory\n",
    "opt = SolverFactory(\"cbc\")\n",
    "results = opt.solve(model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 0, 0, 1, 1, 0, 1, 1, 1], dtype=object)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def find_index_of_sublist(el, lst):\n",
    "    for idx, sublst in enumerate(lst):\n",
    "        if el in sublst:\n",
    "            return idx\n",
    "    raise LookupError(\"{} not found any of the sublists of {}\".format(el, lst))\n",
    "\n",
    "# build a list of regions like [[0, 1, 2, 5], [3, 4, 6, 7, 8]]\n",
    "idx_copy = model.I.data().copy()\n",
    "num_regions = value(model.p)\n",
    "regions = [[] for i in range(num_regions)]\n",
    "for i in range(num_regions):\n",
    "    area = idx_copy.pop()\n",
    "    regions[i].append(area)\n",
    "    \n",
    "    for other_area in idx_copy:\n",
    "        if model.t[area, other_area].value == 1:\n",
    "            regions[i].append(other_area)\n",
    "    \n",
    "    idx_copy.difference_update(regions[i])\n",
    "\n",
    "# build a column with region information\n",
    "region_col = np.arange(model.n)\n",
    "for i in range(len(model.I)):\n",
    "    region_col[i] = find_index_of_sublist(i, regions)\n",
    "    \n",
    "region_col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "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, region in enumerate(region_col):\n",
    "    print(\"area\", i, \"is in region\", region)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1222.8"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value(model.Obj)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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