ampl-book/nltrans.ipynb

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      "source": [
        "# Nonlinear transportation problem example\n",
        "[![nltrans.ipynb](https://img.shields.io/badge/github-%23121011.svg?logo=github)](https://github.com/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb) [![Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb) [![Gradient](https://assets.paperspace.io/img/gradient-badge.svg)](https://console.paperspace.com/github/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb) [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb) [![Hits](https://h.ampl.com/https://github.com/ampl/colab.ampl.com/blob/master/ampl-book/nltrans.ipynb)](https://colab.ampl.com)\n",
        "\n",
        "Description: book example autogenerated using nltransd.mod, nltrans.dat, and nltrans.run\n",
        "\n",
        "Tags: ampl-only, ampl-book, nonlinear\n",
        "\n",
        "Notebook author: Marcos Dominguez Velad <<marcos@ampl.com>>\n",
        "\n",
        "Model author: N/A\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Install dependencies\n",
        "%pip install -q amplpy"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Google Colab & Kaggle integration\n",
        "from amplpy import AMPL, ampl_notebook\n",
        "\n",
        "ampl = ampl_notebook(\n",
        "    modules=[\"coin\"],  # modules to install\n",
        "    license_uuid=\"default\",  # license to use\n",
        ")  # instantiate AMPL object and register magics"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Nonlinear transportation model\n",
        "\n",
        "This is a variation of the linear transportation model presented on the Chapter 3 of the AMPL book, containing a nonlinear objective. There are a set of *origin* nodes, and a set of *destination* nodes (net model).\n",
        "\n",
        "* Sets:\n",
        "    * `ORIG`: origin nodes\n",
        "    * `DEST`: final nodes\n",
        "* Parameters:\n",
        "    * `supply {ORIG}`: available units at origins\n",
        "    * `demand {DEST}`: required units at destinations\n",
        "    * `limit {ORIG,DEST}`: maximum capacity on routes between two nodes\n",
        "    * `rate {ORIG,DEST}`: base shipment costs per unit\n",
        "* Variables:\n",
        "    * `Trans {ORIG,DEST}`: units to be shipped\n",
        "* Objective: minimize total cost (**nonlinear**)\n",
        "\n",
        "$$\\sum \\limits_{\\substack{i \\in ORIG \\\\ j \\in DEST}} rate[i,j] \\cdot \\frac{Trans[i,j]^{0.8}}{1 - \\frac{Trans[i,j]}{limit[i,j]}}$$\n",
        "\n",
        "The bigger the `Trans[i,j]` value, the closer to `limit[i,j]` (upper bound) so denominator tends to $1-1=0$ implying high costs.\n",
        "\n",
        "* Constraints:\n",
        "    * `Supply {ORIG}`: node ships units equal to supply capacity:\n",
        "    \n",
        "    $$\\sum \\limits_{j \\in DEST} Trans[i,j] = supply[i]$$\n",
        "    \n",
        "    * `Demand {DEST}`: node gets units equal to demand:\n",
        "    \n",
        "    $$\\sum \\limits_{i \\in ORIG} Trans[i,j] = demand[j]$$\n",
        "    \n",
        "**Remark**: as the objective function is highly nonlinear, some bounds are fixed in order to help the solver and getting a consistent solution. The first guess for variables is away from zero in `0.5*limit[i,j]`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting nltrans_final.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile nltrans_final.mod\n",
        "set ORIG;   # origins\n",
        "set DEST;   # destinations\n",
        "\n",
        "param supply {ORIG} >= 0;   # amounts available at origins\n",
        "param demand {DEST} >= 0;   # amounts required at destinations\n",
        "\n",
        "   check: sum {i in ORIG} supply[i] = sum {j in DEST} demand[j];\n",
        "\n",
        "param rate {ORIG,DEST} >= 0;   # base shipment costs per unit\n",
        "param limit {ORIG,DEST} > 0;   # limit on units shipped\n",
        "\n",
        "var Trans {i in ORIG, j in DEST}\n",
        "   >= 1e-10, <= .9999 * limit[i,j], := 0.5*limit[i,j];\n",
        "# actual units to be shipped\n",
        "\n",
        "minimize Total_Cost:\n",
        "   sum {i in ORIG, j in DEST}\n",
        "      rate[i,j] * Trans[i,j]^0.8 / (1 - Trans[i,j]/limit[i,j]);\n",
        "\n",
        "subject to Supply {i in ORIG}:  \n",
        "   sum {j in DEST} Trans[i,j] = supply[i];\n",
        "\n",
        "subject to Demand {j in DEST}:  \n",
        "   sum {i in ORIG} Trans[i,j] = demand[j];\n",
        "   \n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting nltrans.dat\n"
          ]
        }
      ],
      "source": [
        "%%writefile nltrans.dat\n",
        "data;\n",
        "\n",
        "param: ORIG:  supply :=\n",
        "        GARY   1400    CLEV   2600    PITT   2900 ;\n",
        "\n",
        "param: DEST:  demand :=\n",
        "        FRA     900    DET    1200    LAN     600 \n",
        "        WIN     400    STL    1700    FRE    1100 \n",
        "        LAF    1000 ;\n",
        "\n",
        "param rate :  FRA  DET  LAN  WIN  STL  FRE  LAF :=\n",
        "        GARY   39   14   11   14   16   82    8\n",
        "        CLEV   27    9   12    9   26   95   17\n",
        "        PITT   24   14   17   13   28   99   20 ;\n",
        "\n",
        "param limit :  FRA  DET  LAN  WIN  STL  FRE  LAF :=\n",
        "        GARY   500 1000 1000 1000  800  500 1000\n",
        "        CLEV   500  800  800  800  500  500 1000\n",
        "        PITT   800  600  600  600  500  500  900 ;\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Restoration Phase Failed.\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "Total_Cost = 354279\n",
            "\n",
            "Trans [*,*] (tr)\n",
            ":      CLEV         GARY         PITT      :=\n",
            "DET   586.429   191.805         421.765\n",
            "FRA   292.089    75.1764        532.735\n",
            "FRE   365.333   370.173         364.494\n",
            "LAF   488.914     0.0937081     510.992\n",
            "LAN   298.74      0.000276411   301.26\n",
            "STL   469.177   762.751         468.072\n",
            "WIN    99.318     9.99922e-05   300.682\n",
            ";\n",
            "\n"
          ]
        }
      ],
      "source": [
        "%%ampl_eval\n",
        "model nltrans_final.mod;\n",
        "data nltrans.dat;\n",
        "option solver ipopt;\n",
        "option ipopt_options 'outlev=0';\n",
        "solve;\n",
        "display Total_Cost, Trans;"
      ]
    }
  ],
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