03: LP_pyomo_gurobi_worker_assignment.ipynb

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      "source": [
        "<a href=\"https://colab.research.google.com/github/gulabpatel/LinearProgramming/blob/main/03%3A%20LP_pyomo_gurobi_worker_assignment.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
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      "cell_type": "markdown",
      "metadata": {
        "id": "VnKFWZ017j-r"
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      "source": [
        "## Problem Statement\n",
        "\n",
        "*Exercise 2.12 from Operations Research: Models and Methods by Jensen & Bard*\n",
        "\n",
        "Ten jobs are to be completed by three workers during the next week. Each worker has a 40-hour work week. The times for the workers to complete the jobs are shown in the table. The values in the cells assume that each job is completed by a single worker; however, jobs can be shared, with completion times being determined proportionally If no entry exists in a particular cell, it means that the corresponding job cannot be performed by the corresponding worker. Set up and solve an LP model that will determine the optimal assignment of workers to jobs. The goal is to minimize the total time required to complete all the jobs.\n",
        "\n",
        "| Workers \\ Tasks |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | 10 |\n",
        "|:---------------:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|\n",
        "| A               |  - |  7 |  3 |  - |  - | 18 | 13 |  6 |  - |  9 |\n",
        "| B               | 12 |  5 |  - | 12 |  4 | 22 |  - | 17 | 13 |  - |\n",
        "| C               | 18 |  - |  6 |  8 | 10 |  - | 19 |  - |  8 | 15 |"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W4GHV2Zl7j_B"
      },
      "source": [
        "## Import"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install pyomo\n",
        "# !apt install glpk-utils\n",
        "# !pip install glpk"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "A2lVu5rK7pKd",
        "outputId": "91d794c0-5df8-4a12-d87f-c1983e2e5849"
      },
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Requirement already satisfied: pyomo in /usr/local/lib/python3.7/dist-packages (6.4.0)\n",
            "Requirement already satisfied: ply in /usr/local/lib/python3.7/dist-packages (from pyomo) (3.11)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "t1VhwU7X7j_D"
      },
      "outputs": [],
      "source": [
        "import pandas as pd\n",
        "import pyomo.environ as pe\n",
        "import pyomo.opt as po"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "r98rMn097j_G"
      },
      "source": [
        "## Define Data"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "id": "nxlvY4297j_H"
      },
      "outputs": [],
      "source": [
        "workers = {'A', 'B', 'C'}\n",
        "\n",
        "tasks = set(range(1, 11))\n",
        "\n",
        "c = {\n",
        "    ('A',  2):  7,\n",
        "    ('A',  3):  3,\n",
        "    ('A',  6): 18,\n",
        "    ('A',  7): 13,\n",
        "    ('A',  8):  6,\n",
        "    ('A', 10):  9,\n",
        "    ('B',  1): 12,\n",
        "    ('B',  2):  5,\n",
        "    ('B',  4): 12,\n",
        "    ('B',  5):  4,\n",
        "    ('B',  6): 22,\n",
        "    ('B',  8): 17,\n",
        "    ('B',  9): 13,\n",
        "    ('C',  1): 18,\n",
        "    ('C',  3):  6,\n",
        "    ('C',  4):  8,\n",
        "    ('C',  5): 10,\n",
        "    ('C',  7): 19,\n",
        "    ('C',  9):  8,\n",
        "    ('C', 10): 15,\n",
        "}\n",
        "\n",
        "max_hours = 40"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fimyzKle7j_I"
      },
      "source": [
        "## Model\n",
        "Define $W$ as the set of workers and $T$ as the sets of tasks. Also, define $c_{wt}$ as the number of hours worker $w$ requires to complete task $t$. (Note that we do not explicitly prohibit a worker from completiting as task; rather, we make the cost arbitrarily large if worker $w$ is unable to perform task $t$.) Let $x_{wt}$ be the proportion of task $t$ that is completed by worker $j$. Let $H$ be the max number of hours that any single worker may log in a week. We formulate as follows.\n",
        "\n",
        "\n",
        "$$\n",
        "\\begin{alignat*}{3}\n",
        "\\text{minimize  }  & \\sum_{w \\in W} \\sum_{t \\in T} c_{wt} x_{wt} && \\\\\n",
        "\\text{subject to  }\n",
        "& \\sum_{t \\in T} c_{wt} x_{wt} \\le H,\n",
        "&& \\qquad \\forall w \\in W \\\\\n",
        "& \\sum_{w \\in W} x_{wt} = 1\n",
        "&& \\qquad \\forall t \\in T \\\\\n",
        "& 0 \\le x_{wt} \\le 1,\n",
        "&& \\qquad \\forall w \\in W, \\forall t \\in T\n",
        "\\end{alignat*}\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mHJH1-zu7j_K"
      },
      "source": [
        "## Implement"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "ucnTAEpl7j_M"
      },
      "outputs": [],
      "source": [
        "model = pe.ConcreteModel()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "id": "L3H5jdqD7j_N",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "c3eadacc-0c8d-4719-cbd1-d29549e158c3"
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          "text": [
            "WARNING: Initializing ordered Set workers with a fundamentally unordered data\n",
            "    source (type: set).  This WILL potentially lead to nondeterministic\n",
            "    behavior in Pyomo\n",
            "WARNING: Initializing ordered Set tasks with a fundamentally unordered data\n",
            "    source (type: set).  This WILL potentially lead to nondeterministic\n",
            "    behavior in Pyomo\n"
          ]
        }
      ],
      "source": [
        "model.workers = pe.Set(initialize=workers)\n",
        "model.tasks = pe.Set(initialize=tasks)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "Qpo78fhr7j_O"
      },
      "outputs": [],
      "source": [
        "model.c = pe.Param(model.workers, model.tasks, initialize=c, default=1000)\n",
        "model.max_hours = pe.Param(initialize=max_hours)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "id": "rtV1jQjp7j_P"
      },
      "outputs": [],
      "source": [
        "model.x = pe.Var(model.workers, model.tasks, domain=pe.Reals, bounds=(0, 1))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "id": "iiJYjGJl7j_Q"
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      "outputs": [],
      "source": [
        "expr = sum(model.c[w, t] * model.x[w, t]\n",
        "           for w in model.workers for t in model.tasks)\n",
        "model.objective = pe.Objective(sense=pe.minimize, expr=expr)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "5g1g4JQj7j_R"
      },
      "outputs": [],
      "source": [
        "model.tasks_done = pe.ConstraintList()\n",
        "for t in model.tasks:\n",
        "    lhs = sum(model.x[w, t] for w in model.workers)\n",
        "    rhs = 1\n",
        "    model.tasks_done.add(lhs == rhs)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "IzOSpgwW7j_S"
      },
      "outputs": [],
      "source": [
        "model.hour_limit = pe.ConstraintList()\n",
        "for w in model.workers:\n",
        "    lhs = sum(model.c[w, t] * model.x[w, t] for t in model.tasks)\n",
        "    rhs = model.max_hours\n",
        "    model.hour_limit.add(lhs <= rhs)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Zp6EARLa7j_S"
      },
      "source": [
        "## Solve and Postprocess"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "id": "A1mZSsr47j_T",
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        },
        "outputId": "db33c82c-ff9e-4515-d8ef-c04cff8e42ac"
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      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Restricted license - for non-production use only - expires 2023-10-25\n",
            "Read LP format model from file /tmp/tmpmek32hj0.pyomo.lp\n",
            "Reading time = 0.00 seconds\n",
            "x31: 14 rows, 31 columns, 61 nonzeros\n",
            "Gurobi Optimizer version 9.5.1 build v9.5.1rc2 (linux64)\n",
            "Thread count: 1 physical cores, 2 logical processors, using up to 2 threads\n",
            "Optimize a model with 14 rows, 31 columns and 61 nonzeros\n",
            "Model fingerprint: 0x2c3e42a4\n",
            "Coefficient statistics:\n",
            "  Matrix range     [1e+00, 1e+03]\n",
            "  Objective range  [3e+00, 1e+03]\n",
            "  Bounds range     [1e+00, 1e+00]\n",
            "  RHS range        [1e+00, 4e+01]\n",
            "Presolve removed 1 rows and 1 columns\n",
            "Presolve time: 0.00s\n",
            "Presolved: 13 rows, 30 columns, 60 nonzeros\n",
            "\n",
            "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
            "       0    1.8000000e+01   1.200000e+01   0.000000e+00      0s\n",
            "      10    8.8000000e+01   0.000000e+00   0.000000e+00      0s\n",
            "\n",
            "Solved in 10 iterations and 0.00 seconds (0.00 work units)\n",
            "Optimal objective  8.800000000e+01\n"
          ]
        }
      ],
      "source": [
        "solver = po.SolverFactory('gurobi')\n",
        "results = solver.solve(model, tee=True)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
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        "id": "dFQfDwjj7j_U"
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        "df = pd.DataFrame(index=pd.MultiIndex.from_tuples(model.x, names=['w', 't']))\n",
        "df['x'] = [pe.value(model.x[key]) for key in df.index]\n",
        "df['c'] = [model.c[key] for key in df.index]"
      ]
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              "t    1    2    3    4    5     6     7    8    9    10\n",
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              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-8f8c786b-48df-4327-b2cc-f27337cff6d6');\n",
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              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
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              "  "
            ]
          },
          "metadata": {},
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        }
      ],
      "source": [
        "(df['c'] * df['x']).groupby('w').sum().to_frame()"
      ]
    },
    {
      "cell_type": "code",
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          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "t   1    2    3    4    5    6    7    8    9    10\n",
              "x  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0"
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              "      <th>t</th>\n",
              "      <th>1</th>\n",
              "      <th>2</th>\n",
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              "      <th>4</th>\n",
              "      <th>5</th>\n",
              "      <th>6</th>\n",
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              "      <th>8</th>\n",
              "      <th>9</th>\n",
              "      <th>10</th>\n",
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              "      <th>x</th>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
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              "      <td>1.0</td>\n",
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              "      <td>1.0</td>\n",
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              "</div>\n",
              "      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-0084777c-e81d-460a-b45d-a713eb96acfb')\"\n",
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              "              style=\"display:none;\">\n",
              "        \n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M0 0h24v24H0V0z\" fill=\"none\"/>\n",
              "    <path d=\"M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z\"/><path d=\"M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z\"/>\n",
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              "\n",
              "    [theme=dark] .colab-df-convert {\n",
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              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-0084777c-e81d-460a-b45d-a713eb96acfb button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-0084777c-e81d-460a-b45d-a713eb96acfb');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
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              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 16
        }
      ],
      "source": [
        "df['x'].groupby('t').sum().to_frame().T"
      ]
    }
  ],
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