LAB3_Sudoku_Local_POT.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Sudoku\n",
    "\n",
    "This tutorial includes everything you need to set up decision optimization engines, build constraint programming models.\n",
    "\n",
    "\n",
    "When you finish this tutorial, you'll have a foundational knowledge of _Prescriptive Analytics_.\n",
    "\n",
    ">This notebook is part of the **[Prescriptive Analytics for Python](https://rawgit.com/IBMDecisionOptimization/docplex-doc/master/docs/index.html)**\n",
    "\n",
    ">It requires a valid subscription to **Decision Optimization on the Cloud** or a **local installation of CPLEX Optimizers**. \n",
    "Discover us [here](https://developer.ibm.com/docloud)\n",
    "\n",
    "\n",
    "Table of contents:\n",
    "\n",
    "-  [Describe the business problem](#Describe-the-business-problem)\n",
    "*  [How decision optimization (prescriptive analytics) can help](#How--decision-optimization-can-help)\n",
    "*  [Use decision optimization](#Use-decision-optimization)\n",
    "    *  [Step 1: Download the library](#Step-1:-Download-the-library)\n",
    "    *  [Step 2: Set up the engines](#Step-2:-Set-up-the-prescriptive-engine)\n",
    "    -  [Step 3: Model the Data](#Step-3:-Model-the-data)\n",
    "    -  [Step 4: Set up the prescriptive model](#Step-4:-Set-up-the-prescriptive-model)\n",
    "        * [Define the decision variables](#Define-the-decision-variables)\n",
    "        * [Express the business constraints](#Express-the-business-constraints)\n",
    "        * [Express the objective](#Express-the-objective)\n",
    "        * [Solve with Decision Optimization solve service](#Solve-with-Decision-Optimization-solve-service)\n",
    "    *  [Step 5: Investigate the solution and run an example analysis](#Step-5:-Investigate-the-solution-and-then-run-an-example-analysis)\n",
    "*  [Summary](#Summary)\n",
    "****"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Describe the business problem\n",
    "\n",
    "* Sudoku is a logic-based, combinatorial number-placement puzzle.\n",
    "* The objective is to fill a 9x9 grid with digits so that each column, each row,\n",
    "and each of the nine 3x3 sub-grids that compose the grid contains all of the digits from 1 to 9.\n",
    "* The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a unique solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### References\n",
    "* See https://en.wikipedia.org/wiki/Sudoku for details"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "*****\n",
    "## How  decision optimization can help\n",
    "* Prescriptive analytics technology recommends actions based on desired outcomes, taking into account specific scenarios, resources, and knowledge of past and current events. This insight can help your organization make better decisions and have greater control of business outcomes.  \n",
    "\n",
    "* Prescriptive analytics is the next step on the path to insight-based actions. It creates value through synergy with predictive analytics, which analyzes data to predict future outcomes.  \n",
    "\n",
    "* Prescriptive analytics takes that insight to the next level by suggesting the optimal way to handle that future situation. Organizations that can act fast in dynamic conditions and make superior decisions in uncertain environments gain a strong competitive advantage.  \n",
    "<br/>\n",
    "\n",
    "+ For example:\n",
    "    + Automate complex decisions and trade-offs to better manage limited resources.\n",
    "    + Take advantage of a future opportunity or mitigate a future risk.\n",
    "    + Proactively update recommendations based on changing events.\n",
    "    + Meet operational goals, increase customer loyalty, prevent threats and fraud, and optimize business processes.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Use decision optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Step 1: Download the library\n",
    "\n",
    "Run the following code to install Decision Optimization CPLEX Modeling library.  The *DOcplex* library contains the two modeling packages, Mathematical Programming and Constraint Programming, referred to earlier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "* system is: Linux 64bit\n",
      "* Python is present, version is 2.7.13\n",
      "* docplex is present, version is (2, 4, 61)\n",
      "* CPLEX wrapper is present, version is 12.7.0.0, located at: /opt/conda/lib/python2.7/site-packages\n"
     ]
    }
   ],
   "source": [
    "from docplex.mp.environment import Environment\n",
    "env = Environment()\n",
    "env.print_information()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import sys\n",
    "try:\n",
    "    import docplex.cp\n",
    "except:\n",
    "    if hasattr(sys, 'real_prefix'):\n",
    "        #we are in a virtual env.\n",
    "        !pip install docplex\n",
    "    else:\n",
    "        !pip install --user docplex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Note that the more global package <i>docplex</i> contains another subpackage <i>docplex.mp</i> that is dedicated to Mathematical Programming, another branch of optimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from docplex.cp.model import *\n",
    "from sys import stdout"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Step 2: Set up the prescriptive engine\n",
    "\n",
    "* Subscribe to the [Decision Optimization on Cloud solve service](https://developer.ibm.com/docloud).\n",
    "* Get the service URL and your personal API key.\n",
    "\n",
    "__Set your DOcplexcloud credentials:__\n",
    "0. A first option is to set the DOcplexcloud url and key directly in the model source file *(see below)*\n",
    "1. For a persistent setting, create a Python file __docloud_config.py__ somewhere that is visible from the __PYTHONPATH__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Initialize IBM Decision Optimization credentials\n",
    "SVC_URL = None # \"ENTER YOUR URL HERE\"\n",
    "SVC_KEY = None # \"ENTER YOUR KEY HERE\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Step 3: Model the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Grid range"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "GRNG = range(9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Different problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "_zero means cell to be filled with appropriate value_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "SUDOKU_PROBLEM_1 = ( (0, 0, 0,  0, 9, 0,  1, 0, 0),\n",
    "                     (2, 8, 0,  0, 0, 5,  0, 0, 0),\n",
    "                     (7, 0, 0,  0, 0, 6,  4, 0, 0),\n",
    "                     (8, 0, 5,  0, 0, 3,  0, 0, 6),\n",
    "                     (0, 0, 1,  0, 0, 4,  0, 0, 0),\n",
    "                     (0, 7, 0,  2, 0, 0,  0, 0, 0),\n",
    "                     (3, 0, 0,  0, 0, 1,  0, 8, 0),\n",
    "                     (0, 0, 0,  0, 0, 0,  0, 5, 0),\n",
    "                     (0, 9, 0,  0, 0, 0,  0, 7, 0),\n",
    "                   )\n",
    "\n",
    "SUDOKU_PROBLEM_2 = ( (0, 7, 0,  0, 0, 0,  0, 4, 9),\n",
    "                     (0, 0, 0,  4, 0, 0,  0, 0, 0),\n",
    "                     (4, 0, 3,  5, 0, 7,  0, 0, 8),\n",
    "                     (0, 0, 7,  2, 5, 0,  4, 0, 0),\n",
    "                     (0, 0, 0,  0, 0, 0,  8, 0, 0),\n",
    "                     (0, 0, 4,  0, 3, 0,  5, 9, 2),\n",
    "                     (6, 1, 8,  0, 0, 0,  0, 0, 5),\n",
    "                     (0, 9, 0,  1, 0, 0,  0, 3, 0),\n",
    "                     (0, 0, 5,  0, 0, 0,  0, 0, 7),\n",
    "                   )\n",
    "\n",
    "SUDOKU_PROBLEM_3 = ( (0, 0, 0,  0, 0, 6,  0, 0, 0),\n",
    "                     (0, 5, 9,  0, 0, 0,  0, 0, 8),\n",
    "                     (2, 0, 0,  0, 0, 8,  0, 0, 0),\n",
    "                     (0, 4, 5,  0, 0, 0,  0, 0, 0),\n",
    "                     (0, 0, 3,  0, 0, 0,  0, 0, 0),\n",
    "                     (0, 0, 6,  0, 0, 3,  0, 5, 4),\n",
    "                     (0, 0, 0,  3, 2, 5,  0, 0, 6),\n",
    "                     (0, 0, 0,  0, 0, 0,  0, 0, 0),\n",
    "                     (0, 0, 0,  0, 0, 0,  0, 0, 0)\n",
    "                    )\n",
    "\n",
    "\n",
    "SUDOKU_PROBLEM_WORLDS_HARDEST = \\\n",
    "        ( (8, 0, 0,  0, 0, 0,  0, 0, 0),\n",
    "          (0, 0, 3,  6, 0, 0,  0, 0, 0),\n",
    "          (0, 7, 0,  0, 9, 0,  2, 0, 0),\n",
    "          (0, 5, 0,  0, 0, 7,  0, 0, 0),\n",
    "          (0, 0, 0,  0, 4, 5,  7, 0, 0),\n",
    "          (0, 0, 0,  1, 0, 0,  0, 3, 0),\n",
    "          (0, 0, 1,  0, 0, 0,  0, 6, 8),\n",
    "          (0, 0, 8,  5, 0, 0,  0, 1, 0),\n",
    "          (0, 9, 0,  0, 0, 0,  4, 0, 0))\n",
    "\n",
    "    \n",
    "SUDOKU_PROBLEM_UNSOLVABLE = \\\n",
    "            ((1, 0, 0, 0, 0, 0, 0, 0, 0),\n",
    "            (0, 0, 0, 1, 0, 0, 0, 0, 0),\n",
    "            (0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
    "            (0, 1, 0, 0, 0, 0, 0, 0, 0),\n",
    "            (0, 0, 0, 0, 1, 0, 0, 0, 0),\n",
    "            (0, 0, 0, 0, 0, 0, 0, 1, 0),\n",
    "            (0, 0, 1, 0, 0, 0, 0, 0, 0),\n",
    "            (0, 0, 0, 0, 0, 1, 0, 0, 0),\n",
    "            (0, 0, 0, 0, 0, 0, 0, 0, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "try:\n",
    "    import numpy as np\n",
    "    import matplotlib.pyplot as plt\n",
    "    VISU_ENABLED = True\n",
    "except ImportError:\n",
    "    VISU_ENABLED = False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def print_grid(grid):\n",
    "    \"\"\" Print Sudoku grid \"\"\"\n",
    "    for l in GRNG:\n",
    "        if (l > 0) and (l % 3 == 0):\n",
    "           stdout.write('\\n')\n",
    "        for c in GRNG:\n",
    "            v = grid[l][c]\n",
    "            stdout.write('   ' if (c % 3 == 0) else ' ')\n",
    "            stdout.write(str(v) if v > 0 else '.')\n",
    "        stdout.write('\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def draw_grid(values):\n",
    "    %matplotlib inline\n",
    "    fig, ax = plt.subplots(figsize =(4,4))\n",
    "    min_val, max_val = 0, 9\n",
    "    R =  range(0,9)\n",
    "    for l in R:\n",
    "        for c in R:\n",
    "            v = values[c][l]\n",
    "            s = \" \"\n",
    "            if v > 0:\n",
    "                s = str(v)\n",
    "            ax.text(l+0.5,8.5-c, s, va='center', ha='center')\n",
    "        ax.set_xlim(min_val, max_val)\n",
    "    ax.set_ylim(min_val, max_val)\n",
    "    ax.set_xticks(np.arange(max_val))\n",
    "    ax.set_yticks(np.arange(max_val))\n",
    "    ax.grid()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def display_grid(grid, name):\n",
    "    stdout.write(name)\n",
    "    stdout.write(\":\\n\")\n",
    "    if VISU_ENABLED:\n",
    "        draw_grid(grid)\n",
    "    else:\n",
    "        print_grid(grid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PROBLEM 1:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nz6kAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f52840decd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PROBLEM 2:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ntPGV8iFtfKV8SBtfKR/SxlfKh/4fkVOreIn/ew8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f527fc77b50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PROBLEM 3:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nFHyRACn4IgFS8EUC9L+ykwcvqvn8QgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f527fc77ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "WORLDS HARDEST:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ngi8SQQq+SAT9HxnjR3KEyHx5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f527744e0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "UNSOLVABLE:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAADetJREFUeJzt3X/sVfV9x/HnW74aFbDUiR0TU1ycBiQZoqk6HSkyGm1N\nm8iWSWwXq0uXbHNat3TrksX1r/1TG/mjW2LEzviDhqImi3NOG2lMzWonoFNBlmqpQLVqGgu4pGr7\n3h/34Cgo3/P97nzu997zeT6Sm9z75Xw/788FXvece+65709kJpLqcsxMT0DS8Bl8qUIGX6qQwZcq\nZPClChl8qUIGX6qQwZcqZPClCk2UGHTevHl55plnlhgagLfeeovZs2cXG98ao1WjD89hWDW2bNny\nRmbOn3TDzOz8dtZZZ2VJmzdvLjq+NUarRh+ew7BqAE9li4x6qC9VyOBLFTL4UoUMvlQhgy9VyOBL\nFZrR4F977bWceuqpLF26dCanIVVnRoN/zTXX8PDDD8/kFKQqzWjwV6xYwcknnzyTU5Cq5Ht8qUIG\nX6qQwZcqZPClCs1o8NeuXctFF13Ezp07WbhwIevXr5/J6UjVKPJ9/LY2bNgwk+WlanmoL1XI4EsV\nMvhShQy+VKFWwY+IL0bE8xHxXERsiIjjS09MUjmTBj8iTgP+Ajg/M5cCs4CrSk9MUjltD/UngBMi\nYgI4EfhxuSlJKm3S4GfmXuCrwMvAK8DPMvOR0hOTVE4MWnEfZYOIDwP3AX8IvAl8C9iUmXcftt0X\ngC8AzJ8//7yNGzcWmTDAgQMHmDNnTrHxrTFaNfrwHIZVY+XKlVsy8/xJN5ys8T7wB8D6Qx7/EfCP\nR/sdF9SwxjiN36cadLigxsvAhRFxYkQEsArYMd1XJEkzr817/CeBTcBW4Nnmd24rPC9JBbX6kk5m\n3gzcXHgukobEK/ekChl8qUIGfwy4/oC6ZvDHgOsPqGsGfwy4/oC6ZvClChl8qUIGX6qQwZcqZPDH\ngOsPqGsz2ldf7bj+gLrmHl+qkMGXKmTwpQoZfKlCbdprnx0RTx9y2xcRNw5jcpLKmPSsfmbuBJYB\nRMQsYC/wQOF5SSpoqof6q4AXM/NHJSYjaTimGvyrAD9UlsbcpH3139sw4jgGK+ick5k/eZ8/t6++\nNcZy/D7V6Kyvfv5fP/3PAI+02da++tYYp/H7VIMO++oftBYP86VeaLtM9mxgNXB/2elIGoa2ffXf\nAn6t8FwkDYlX7kkVMvhShQy+esU1CNox+OoV1yBox+CrV1yDoB2DL1XI4EsVMvhShQy+VCGDr15x\nDYJ27KuvXnENgnbc40sVMvhShQy+VCGDL1WobSOOeRGxKSJeiIgdEXFR6YlJKqftWf11wMOZ+ftN\n080TC85JUmGTBj8iPgSsAK4ByMy3gbfLTktSSW0O9c8AXge+ERHbIuL2pgefpDE1aV/9iDgf+B5w\ncWY+GRHrgH2Z+XeHbWdffWuM5fh9qtFZX33g14Fdhzz+XeBfj/Y79tW3xjiN36cadNVXPzNfBXZH\nxNnNj1YB26f5giRpBLQ9q389cE9zRv8l4PPlpiSptLZ99Z8GJn/fIGkseOWeVCGDL1Wo98G3z7p0\npN4H3z7r0pF6H3z7rEtH6n3wJR3J4EsVMvhShQy+VKHeB98+69KRet9X3z7r0pF6v8eXdCSDL1XI\n4EsVMvhShVqd3IuIXcB+4BfAu9mmp5ekkTWVs/orM/ONYjORNDQe6ksVahv8BL4dEVuaNtqSxtik\nffUBIuK0zNwbEacCjwLXZ+bjh21jX31rjOX4farRWV/9w2/A3wN/dbRt7KtvjXEav0816KqvfkTM\njoi5B+8DnwCe+/+8KkmaWW3O6n8EeCAiDm5/b2bay0oaY5MGPzNfAn57CHORNCR+nCdVyOBLFTL4\nAlx/oDYGX4DrD9TG4Atw/YHaGHypQgZfqpDBlypk8KUKGXwBrj9Qm9731Vc7rj9QF/f4UoUMvlQh\ngy9VyOBLFWod/IiYFRHbIuLBkhOSVN5U9vg3ADtKTUTS8LQKfkQsBD4F3F52OpKGoe0e/1bgS8Av\nC85F0pBM2lc/Iq4APpmZfxoRH2fQWvuK99nOvvrWGMvx+1Sjs776wD8Ae4BdwKvA/wB3H+137Ktv\njXEav0816KqvfmZ+OTMXZuYi4Crgscz87LRfkiTNOD/Hlyo0pS/pZOZ3gO8UmYmkoXGPL1XI4EsV\nMvhShQy+NEV9WHzE4EtT1IfFRwy+NEV9WHzE4EsVMvhShQy+VCGDL1XI4EtT1IfFR1xQQ5qiPiw+\n4h5fqpDBlypk8KUKGXypQpMGPyKOj4jvR8QzEfF8RHxlGBOTVE6bs/o/By7NzAMRcSzw3Yj4t8z8\nXuG5SSpk0uA3nTsPNA+PbW5H78ktaaRN2lcfBuvmAVuAM4GvZ+Zfv8829tW3xliO36canfXVz1/t\nsT8P2AwsPdp29tW3xjiN36cadNVX/7AXiTeb4F829dciSaOizVn9+RExr7l/ArAaeKH0xCSV0+as\n/gLgzuZ9/jHAxsx8sOy0JJXU5qz+fwHnDmEukobEK/ekChl8qUIGvwN96LOuuhj8DvShz7rqYvA7\n0Ic+66qLwZcqZPClChl8qUIGX6qQwe9AH/qsqy721e9AH/qsqy7u8aUKGXypQgZfqpDBlyrUpgPP\n6RGxOSK2N331bxjGxCSV0+as/rvAX2bm1oiYC2yJiEczc3vhuUkqZNI9fma+kplbm/v7gR3AaaUn\nJqmcVn3139s4YhHwOIP22vsO+zP76ltjLMfvU43O++oDcxgsqnHlZNvaV98a4zR+n2rQZV/9Zs28\n+4B7MvP+6b8eSRoFbc7qB7Ae2JGZXys/JUmltdnjXwx8Drg0Ip5ubp8sPC9JBbXpq/9dIIYwF0lD\n4pV7UoUMvlQhg6+hcf2B0WHwNTSuPzA6DL6GxvUHRofBlypk8KUKGXypQgZfqpDB19C4/sDosK++\nhsb1B0aHe3ypQgZfqpDBlypk8KUKtenAc0dEvBYRzw1jQpLKa7PH/2fgssLzkDREbfrqPw78dAhz\nkTQkvseXKtRqQY1mIY0HM/MDOyi4oIY1xnX8PtXodEENYBHwXJtt0wU1rDFm4/epBl0uqCGpX9p8\nnLcB+A/g7IjYExHXlZ+WpJLa9NVfO4yJSBoeD/WlChl8qUIGXxoxu3fvZuXKlSxZsoRzzjmHdevW\ndV7DRhzSiJmYmOCWW25h+fLl7N+/n/POO4/Vq1ezZMmSzmq4x5dGzIIFC1i+fDkAc+fOZfHixezd\nu7fTGgZfGmG7du1i27ZtXHDBBZ2Oa/ClEXXgwAHWrFnDrbfeykknndTp2AZfGkHvvPMOa9as4eqr\nr+bKK6/sfHyDL42YzOS6665j8eLF3HTTTUVqGHxpxDzxxBPcddddPPbYYyxbtoxly5bx0EMPdVrD\nj/OkEXPJJZcc/FZsMe7xpQoZfKlCBl+qkMGXKtQq+BFxWUTsjIgfRMTflJ6UpLLadOCZBXwduBxY\nAqyNiO6+LSBp6Nrs8T8G/CAzX8rMt4FvAp8pOy1JJbUJ/mnA7kMe72l+JmlMdXYBz6F99YGfF15r\n7xTgjYLjW2O0avThOQyrxkfbbNQm+HuB0w95vLD52a/IzNuA2wAi4qls09R/mkqPb43RqtGH5zCs\nGm21OdT/T+C3IuKMiDgOuAr4l7LTklRSm/ba70bEnwP/DswC7sjM54vPTFIxrd7jZ+ZDwFS+HnTb\n9KYzMuNbY7Rq9OE5DKtGK60WzZTUL16yK1Wo0+CXvrQ3Iu6IiNdKflQYEadHxOaI2B4Rz0fEDQVq\nHB8R34+IZ5oaX+m6RlNnVkRsi4gHC42/KyKejYinI+KpQjXmRcSmiHghInZExEUdj392M/+Dt30R\ncWPHNb7Y/Ds/FxEbIuL4LsefljZL6ra5MTjx9yLwm8BxwDPAkq7Gb2qsAJYzhSW7p1FjAbC8uT8X\n+O8CzyOAOc39Y4EngQsLPJebgHuBBwv9Xe0CTin1b9HUuBP44+b+ccC8grVmAa8CH+1wzNOAHwIn\nNI83AteU/Dtrc+tyj1/80t7MfBz4aZdjvk+NVzJza3N/P7CDjq9UzIEDzcNjm1unJ1siYiHwKeD2\nLscdpoj4EIMX+/UAmfl2Zr5ZsOQq4MXM/FHH404AJ0TEBHAi8OOOx5+yLoPfu0t7I2IRcC6DPXLX\nY8+KiKeB14BHM7PrGrcCXwJ+2fG4h0rg2xGxpblys2tnAK8D32jestweEbML1DnoKmBDlwNm5l7g\nq8DLwCvAzzLzkS5rTIcn9z5ARMwB7gNuzMx9XY+fmb/IzGUMroT8WEQs7WrsiLgCeC0zt3Q15ge4\npHkOlwN/FhErOh5/gsFbu3/KzHOBt4AiXwtvLk77NPCtjsf9MIMj3zOA3wBmR8Rnu6wxHV0Gv9Wl\nveMgIo5lEPp7MvP+krWaQ9fNwGUdDnsx8OmI2MXgLdelEXF3h+MD7+3NyMzXgAcYvN3r0h5gzyFH\nQ5sYvBCUcDmwNTN/0vG4vwf8MDNfz8x3gPuB3+m4xpR1GfxeXNobEcHgPeWOzPxaoRrzI2Jec/8E\nYDXwQlfjZ+aXM3NhZi5i8O/wWGZ2upeJiNkRMffgfeATQKeftmTmq8DuiDi7+dEqYHuXNQ6xlo4P\n8xsvAxdGxInN/61VDM4bzajOvp2XQ7i0NyI2AB8HTomIPcDNmbm+yxoM9pafA55t3oMD/G0Orl7s\nygLgzqbJyTHAxsws8pFbQR8BHhj8X2YCuDczHy5Q53rgnmZn8hLw+a4LNC9cq4E/6XrszHwyIjYB\nW4F3gW2MwBV8XrknVciTe1KFDL5UIYMvVcjgSxUy+FKFDL5UIYMvVcjgSxX6X6eC1Wkt/ODgAAAA\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f527fd93b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display_grid(SUDOKU_PROBLEM_1, \"PROBLEM 1\")\n",
    "display_grid(SUDOKU_PROBLEM_2, \"PROBLEM 2\")\n",
    "display_grid(SUDOKU_PROBLEM_3, \"PROBLEM 3\")\n",
    "display_grid(SUDOKU_PROBLEM_WORLDS_HARDEST, \"WORLDS HARDEST\")\n",
    "display_grid(SUDOKU_PROBLEM_UNSOLVABLE, \"UNSOLVABLE\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Choose your preferred problem (SUDOKU_PROBLEM_1 or SUDOKU_PROBLEM_2 or SUDOKU_PROBLEM_3)\n",
    "If you change the problem, ensure to re-run all cells below this one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#problem = SUDOKU_PROBLEM_1\n",
    "#problem = SUDOKU_PROBLEM_2\n",
    "#problem = SUDOKU_PROBLEM_3\n",
    "problem = SUDOKU_PROBLEM_WORLDS_HARDEST\n",
    "#problem = SUDOKU_PROBLEM_UNSOLVABLE\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Step 4: Set up the prescriptive model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "mdl = CpoModel(name=\"Sudoku\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Define the decision variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "grid = [[integer_var(min=1, max=9, name=\"C\" + str(l) + str(c)) for l in GRNG] for c in GRNG]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Express the business constraints"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Add alldiff constraints for lines"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "for l in GRNG:\n",
    "    mdl.add(all_diff([grid[l][c] for c in GRNG]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Add alldiff constraints for columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "for c in GRNG:\n",
    "    mdl.add(all_diff([grid[l][c] for l in GRNG]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Add alldiff constraints for sub-squares"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "ssrng = range(0, 9, 3)\n",
    "for sl in ssrng:\n",
    "    for sc in ssrng:\n",
    "        mdl.add(all_diff([grid[l][c] for l in range(sl, sl + 3) for c in range(sc, sc + 3)]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initialize known cells"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "for l in GRNG:\n",
    "    for c in GRNG:\n",
    "        v = problem[l][c]\n",
    "        if v > 0:\n",
    "            grid[l][c].set_domain((v, v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Solve with Decision Optimization solve service"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Solving model....\n"
     ]
    }
   ],
   "source": [
    "print(\"\\nSolving model....\")\n",
    "msol = mdl.solve(url=SVC_URL, \n",
    "                 key=SVC_KEY, TimeLimit=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NoneType"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(SVC_URL)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Step 5: Investigate the solution and then run an example analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial problem:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ngi8SQQq+SAT9HxnjR3KEyHx5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f52840ded50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solve time: 0.400348901749\n",
      "Solution:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP4AAAD8CAYAAABXXhlaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n71K7AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5299e43610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display_grid(problem, \"Initial problem\")\n",
    "if msol:\n",
    "    sol = [[msol[grid[l][c]] for c in GRNG] for l in GRNG]\n",
    "    stdout.write(\"Solve time: \" + str(msol.get_solve_time()) + \"\\n\")\n",
    "    display_grid(sol, \"Solution\")\n",
    "else:\n",
    "    stdout.write(\"No solution found\\n\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Summary\n",
    "\n",
    "You learned how to set up and use the IBM Decision Optimization CPLEX Modeling for Python to formulate a Constraint Programming model and solve it with IBM Decision Optimization on the cloud."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### References\n",
    "* [CPLEX Modeling for Python documentation](https://rawgit.com/IBMDecisionOptimization/docplex-doc/master/docs/index.html)\n",
    "* [Decision Optimization on Cloud](https://developer.ibm.com/docloud/)\n",
    "* Need help with DOcplex or to report a bug? Please go [here](https://developer.ibm.com/answers/smartspace/docloud)\n",
    "* Contact us at dofeedback@wwpdl.vnet.ibm.com"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<div class=\"alert alert-block alert-info\"> Note: To save resources and get the best performance please use the code below to stop the kernel before exiting your notebook.</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "%%javascript\n",
    "Jupyter.notebook.session.delete();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<hr>\n",
    "Copyright &copy; IBM Corp. 2017. Released as licensed Sample Materials."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python2.7 with DSX Spark 2.0.2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}