airbus.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 5,
  "metadata": {
    "kernelspec": {
      "display_name": "Qiskit v0.31.0 (ipykernel)",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.8.10"
    },
    "colab": {
      "name": "airbus.ipynb",
      "provenance": [],
      "collapsed_sections": []
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "Author: Xiang Zhao magelead@outlook.com"
      ],
      "metadata": {
        "id": "s_24D8T4H3YW"
      },
      "id": "s_24D8T4H3YW"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wq0bs3Vd8l5W"
      },
      "source": [
        "TODO:\n",
        "1. Qbsolv https://github.com/dwavesystems/qbsolv/ ✅\n",
        "2. Include linear coefficients in diagonal of QUBO quadratic matrix ✅ \n",
        "3. implement figure 1 of paper && determine penalty factors\n",
        "4. consider other constraints"
      ],
      "id": "wq0bs3Vd8l5W"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "NV2aUnnxTG5b",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "6b9d691d-07c2-4e6c-d7c6-c1b3ac95ae67"
      },
      "source": [
        "!pip install qiskit qiskit-optimization dwave-qbsolv"
      ],
      "id": "NV2aUnnxTG5b",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
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          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V5cEVCOnJLJ9"
      },
      "source": [
        "# Airbus challenge\n",
        "References:\n",
        "\n",
        "[1] https://arxiv.org/pdf/2102.09621.pdf\n",
        "\n",
        "[2] https://github.com/magelead/QGSS-2021/blob/main/Lab%202/lab-2.ipynb"
      ],
      "id": "V5cEVCOnJLJ9"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-2Ricwgt5JlA"
      },
      "source": [
        "# Part I Setup the QUBO model"
      ],
      "id": "-2Ricwgt5JlA"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "30ea0e02-ddee-443c-a5c4-d78003d6f44c"
      },
      "source": [
        "import sympy as sym\n",
        "import numpy as np"
      ],
      "id": "30ea0e02-ddee-443c-a5c4-d78003d6f44c",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Problem 1: This problem's size is optimal for qiskit to simulate, so I solve it via a classical algorithm and a quantum algorithm."
      ],
      "metadata": {
        "id": "4fyA80tno6hI"
      },
      "id": "4fyA80tno6hI"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kDtTyTs3eNQf"
      },
      "source": [
        "# Settings for a toy experiment correct solution 3\n",
        "\n",
        "n=3\n",
        "N=2\n",
        "W=3 # max weight unit is 100kg\n",
        "m=[2.0, 1.0, 3.0] # unit is 100kg, type 1 container, type 2 container, type 3 container\n",
        "t=[1.0,1.0,0.5] # t[i] = 1 for type 1 or 2 container; 1/2 for type 3 container \n",
        "d=[1.0,0.5,1.0] # d[i] = 1 for type 1 or 3 container; 1/2 for type 2 container\n",
        "\n",
        "\n",
        "K=n+N+2 # number of slack variables: K = n + N + number of bits of binary expansion of W\n",
        "\n",
        "    \n",
        "z=[sym.Symbol(f'z_{i}') for i in range(n*N+K)]\n",
        "z"
      ],
      "id": "kDtTyTs3eNQf",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Problem 2: This problem's size is too big to handle via qiskit, so I only solve it via a classical algorithm."
      ],
      "metadata": {
        "id": "qvpbzkuWnGvR"
      },
      "id": "qvpbzkuWnGvR"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3bb5bea3-54a8-471b-948f-d831639c46da"
      },
      "source": [
        "# Settings for experiments on D-Wave 2000Q, section V.C. of [1]\n",
        "# This can only be solved by Qbsolv due to large problem size\n",
        "n=6\n",
        "N=4\n",
        "W=8000 # max weight unit is kg\n",
        "m=[2134.0, 3455.0, 1866.0, 1699.0, 3500.0, 3332.0] # TABLE VII of [1], type 1 container , unit is kg\n",
        "t=[1.0,1.0,1.0,1.0,1.0,1.0] # t[i] = 1 for type 1 container \n",
        "d=[1.0,1.0,1.0,1.0,1.0,1.0] # d[i] = 1 for type 1 container\n",
        "\n",
        "\n",
        "K = 24 # number of slack variables: K = n + N + number of bits of binary expansion of W\n",
        " \n",
        "# len(z) equals the number of qubits in QAOA circuit\n",
        "z=[sym.Symbol(f'z_{i}') for i in range(n*N+K)]\n",
        "z"
      ],
      "id": "3bb5bea3-54a8-471b-948f-d831639c46da",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Problem 3:\n",
        "This problem's size is too big to handle via sympy. QUBO model will be constructed via the Mathematica."
      ],
      "metadata": {
        "id": "UMacrRHGnJOh"
      },
      "id": "UMacrRHGnJOh"
    },
    {
      "cell_type": "code",
      "source": [
        "# Settings for experiments of Table II and I, trying to repeat results of Fig 1\n",
        "# This can only be solved by Qbsolv due to large problem size\n",
        "n=35\n",
        "N=20\n",
        "W=40000\n",
        "m=[2134.0, 3455.0, 1866.0, 1699.0, 3500.0, 3332.0, 2578.0, 2315.0, 1888.0, 1786.0, \n",
        "   3277.0, 2987.0, 2534.0, 2111.0, 2607.0, 1566.0, 1765.0, 1946.0, 1732.0, 1641.0, \n",
        "   1800.0, 986.0, 873.0, 1764.0, 1239.0, 1487.0, 769.0, 836.0, 659.0, 765.0, 3132.0, \n",
        "   3530.0, 3892.0, 3464.0, 3282.0] \n",
        "t=[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,\n",
        "   1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,\n",
        "   1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,\n",
        "   0.5, 0.5, 0.5, 0.5, 0.5]\n",
        "d=[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,\n",
        "   1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,\n",
        "   0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,\n",
        "   1.0, 1.0, 1.0, 1.0, 1.0]\n",
        " \n",
        "\n",
        "K = n + N + (len(bin(W))-2) # number of slack variables: K = n + N + number of bits of binary expansion of W\n",
        "\n",
        "# len(z) equals the number of qubits in QAOA circuit\n",
        "z=[sym.Symbol(f'z_{i}') for i in range(n*N+K)]"
      ],
      "metadata": {
        "id": "gI4PMlfPu9o_"
      },
      "execution_count": null,
      "outputs": [],
      "id": "gI4PMlfPu9o_"
    },
    {
      "cell_type": "code",
      "source": [
        "# This problem is too large for sympy, I will just load results obtained from Mathematica notebook airbus.nb\n",
        "# Run this cell to get `quadratic` and `linear`, then you can skip to Part III\n",
        "def read_quadratic(path):\n",
        "    lines=[]\n",
        "    with open(path) as f:\n",
        "        for line in f:\n",
        "            cols = line.strip().split(',')\n",
        "            lines.append(np.array([float(x) for x in cols]))\n",
        "    quadratic = np.stack(lines)\n",
        "    return quadratic\n",
        "\n",
        "def read_linear(path):\n",
        "    lst = []\n",
        "    with open(path) as f:\n",
        "        for line in f:\n",
        "            lst.append(float(line.strip()))\n",
        "    linear = np.array(lst)\n",
        "    return linear\n",
        "\n",
        "quadratic = read_quadratic('quadratic.csv')\n",
        "linear = read_linear('linear.csv')\n",
        "# constant does not matter\n",
        "\n",
        "def f_obj():\n",
        "    \"\"\" objective fucntion \"\"\"\n",
        "    exp = 0 # init a expression\n",
        "    for i in range(n):\n",
        "        for j in range(N):\n",
        "            exp = exp - t[i] * m[i] * z[i*N+j]\n",
        "\n",
        "    return exp\n",
        "\n",
        "payload = f_obj()\n",
        "qubo = None"
      ],
      "metadata": {
        "id": "7ruFE737fOLN"
      },
      "id": "7ruFE737fOLN",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Run following cells again everytime you choose a different problem."
      ],
      "metadata": {
        "id": "pXQ-mWvxpM0n"
      },
      "id": "pXQ-mWvxpM0n"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d9271e9c-3b0a-46c0-8a3a-68db8758d7f7"
      },
      "source": [
        "def f_obj():\n",
        "    \"\"\" objective fucntion \"\"\"\n",
        "    exp = 0 # init a expression\n",
        "    for i in range(n):\n",
        "        for j in range(N):\n",
        "            exp = exp - t[i] * m[i] * z[i*N+j]\n",
        "\n",
        "    return exp\n",
        "            "
      ],
      "id": "d9271e9c-3b0a-46c0-8a3a-68db8758d7f7",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c067a60c-87df-48a5-93f5-a1b51153f9d8"
      },
      "source": [
        "f_obj_ = f_obj()"
      ],
      "id": "c067a60c-87df-48a5-93f5-a1b51153f9d8",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_obj_"
      ],
      "metadata": {
        "id": "ROGdk20F6dAH"
      },
      "id": "ROGdk20F6dAH",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "00699f68-d9fd-478e-94de-d9e64b2d0fb1"
      },
      "source": [
        "def f_O_j(j):\n",
        "    \"\"\" no-overlaps constriant \"\"\"\n",
        "    exp = 0\n",
        "    for i in range(n):\n",
        "        exp = exp + d[i] * z[i*N+j]\n",
        "    exp = (exp + z[n*N+j] - 1)**2\n",
        "    \n",
        "    return sym.simplify(sym.expand(exp))"
      ],
      "id": "00699f68-d9fd-478e-94de-d9e64b2d0fb1",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "f482588b-45a8-4908-a6f8-2fdec12bb58d"
      },
      "source": [
        "f_O_j(0)"
      ],
      "id": "f482588b-45a8-4908-a6f8-2fdec12bb58d",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "977124e2-de7d-4046-b4e3-a788d33c649e"
      },
      "source": [
        "def f_O():\n",
        "    exp = 0\n",
        "    for j in range(N):\n",
        "        exp = exp + f_O_j(j)\n",
        "    return sym.simplify(sym.expand(exp))"
      ],
      "id": "977124e2-de7d-4046-b4e3-a788d33c649e",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7c881c8d-213f-41f5-90a0-9e95ef837abc"
      },
      "source": [
        "f_O_ = f_O()"
      ],
      "id": "7c881c8d-213f-41f5-90a0-9e95ef837abc",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_O_"
      ],
      "metadata": {
        "id": "TA2Nj7k9IwOn"
      },
      "id": "TA2Nj7k9IwOn",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "sqrt = math.sqrt\n",
        "subs = [ (x, sqrt(3)) for x in z]"
      ],
      "metadata": {
        "id": "_4y1rOGaJd5l"
      },
      "id": "_4y1rOGaJd5l",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_O_.subs(subs)"
      ],
      "metadata": {
        "id": "NR5fdnHaJCO9"
      },
      "id": "NR5fdnHaJCO9",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d09232af-9abf-4402-ac2b-8a29cf0658d6"
      },
      "source": [
        "def f_D_i(i):\n",
        "    \"\"\" no-duplicates constriant \"\"\"\n",
        "    exp = 0\n",
        "    for j in range(N):\n",
        "        exp = exp + z[i*N+j]\n",
        "    exp = (t[i] * exp + z[n*N+N+i] - 1)**2\n",
        "    \n",
        "    return sym.simplify(sym.expand(exp))"
      ],
      "id": "d09232af-9abf-4402-ac2b-8a29cf0658d6",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d1f3d4c8-5abf-4d0d-a25f-99ac45654aa4"
      },
      "source": [
        "f_D_i(0)"
      ],
      "id": "d1f3d4c8-5abf-4d0d-a25f-99ac45654aa4",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "b394d163-d60e-4e20-8bca-0125d406142a"
      },
      "source": [
        "def f_D():\n",
        "    exp = 0\n",
        "    for i in range(n):\n",
        "        exp = exp + f_D_i(i)\n",
        "    return sym.simplify(sym.expand(exp))"
      ],
      "id": "b394d163-d60e-4e20-8bca-0125d406142a",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "808b15d4-5257-4686-925f-e87f503c4079"
      },
      "source": [
        "f_D_ = f_D()"
      ],
      "id": "808b15d4-5257-4686-925f-e87f503c4079",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_D_"
      ],
      "metadata": {
        "id": "makk5MykLxFu"
      },
      "id": "makk5MykLxFu",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_D_.subs(subs)"
      ],
      "metadata": {
        "id": "kUsm7k8FLzms"
      },
      "id": "kUsm7k8FLzms",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def f_C_i(i):\n",
        "    \"\"\"Contiguity for big containers\"\"\"\n",
        "    exp = 0\n",
        "    for j in range(N):\n",
        "        exp = exp + z[i*N+j]\n",
        "    exp = 1/2 * exp\n",
        "    for j in range(N-1):\n",
        "        exp = exp - z[i*N+j] * z[i*N+j+1]\n",
        "\n",
        "    return exp\n"
      ],
      "metadata": {
        "id": "lKbSU0aFJ-A-"
      },
      "id": "lKbSU0aFJ-A-",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_C_i(2)"
      ],
      "metadata": {
        "id": "AFL_UE0uLodj"
      },
      "id": "AFL_UE0uLodj",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def f_C():\n",
        "    \"\"\"sum over all type 3 containers\"\"\"\n",
        "    exp = 0\n",
        "    for i in range(n):\n",
        "        if t[i] == 1/2:\n",
        "            exp = exp + f_C_i(i)\n",
        "    return sym.simplify(sym.expand(exp))\n",
        "    "
      ],
      "metadata": {
        "id": "fWS-Ou0IL-2_"
      },
      "id": "fWS-Ou0IL-2_",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_C_ = f_C()"
      ],
      "metadata": {
        "id": "sLyywlsvWi6S"
      },
      "id": "sLyywlsvWi6S",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_C_"
      ],
      "metadata": {
        "id": "RkWmWANYOLEc"
      },
      "id": "RkWmWANYOLEc",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4208f365-6e34-440e-8822-68fa1a24f149"
      },
      "source": [
        "def f_W():\n",
        "    exp = 0\n",
        "    for i in range(n):\n",
        "        for j in range(N):\n",
        "            exp = exp + t[i] * m[i] * z[i*N+j]\n",
        "    for k in range(K-n-N):\n",
        "        exp = exp + 2**k * z[n*N+N+n+k]\n",
        "        \n",
        "    exp = (exp - W)**2\n",
        "    \n",
        "    return sym.simplify(sym.expand(exp))\n",
        "        "
      ],
      "id": "4208f365-6e34-440e-8822-68fa1a24f149",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "37fffd32-45c9-4241-b79c-9e48b7175b5e"
      },
      "source": [
        "f_W_ = f_W()"
      ],
      "id": "37fffd32-45c9-4241-b79c-9e48b7175b5e",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f_W_.subs(subs)"
      ],
      "metadata": {
        "id": "UAVXU5eCPWYV"
      },
      "id": "UAVXU5eCPWYV",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c33ba3d7-5a6e-4a8c-b870-b6002a9731a2"
      },
      "source": [
        "def f_tot(P_O, P_D, P_C, P_W=None):\n",
        "    if P_W is not None:\n",
        "        exp = f_obj_ + P_O*f_O_ + P_D*f_D_ + P_C*f_C_ + P_W*f_W_\n",
        "    else:\n",
        "        exp = f_obj_ + P_O*f_O_ + P_D*f_D_ + P_C*f_C_\n",
        "\n",
        "    return sym.simplify(exp)\n",
        "    \n",
        "    "
      ],
      "id": "c33ba3d7-5a6e-4a8c-b870-b6002a9731a2",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8f9b60dd-3c4e-4c1b-80c1-096a2d466089"
      },
      "source": [
        "# global variable qubo is necessary for the rest of notebook\n",
        "qubo = f_tot(1.0, 1.0, 1.0, 1.0)"
      ],
      "id": "8f9b60dd-3c4e-4c1b-80c1-096a2d466089",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "qubo.subs(subs)"
      ],
      "metadata": {
        "id": "kQsCJmuSpTUJ"
      },
      "id": "kQsCJmuSpTUJ",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "i-XsOnk1-TIm"
      },
      "source": [
        "def get_constant(exp):\n",
        "    subs = [(z[i], 0) for i in range(len(z))]\n",
        "    constant = exp.subs(subs)\n",
        "    return float(constant)\n"
      ],
      "id": "i-XsOnk1-TIm",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2QM-Dx5_Fyg5"
      },
      "source": [
        "get_constant(qubo)"
      ],
      "id": "2QM-Dx5_Fyg5",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2f11f480-668b-4588-8bc6-9c0ffcdbb209"
      },
      "source": [
        "def get_qubo_coefficients(qubo, tolist=True):\n",
        "    size = len(z)\n",
        "    quadratic = np.zeros((size, size), dtype=np.float)\n",
        "    linear = np.zeros(size, dtype=np.float)\n",
        "    constant = get_constant(qubo)\n",
        "    \n",
        "    for i in range(size):\n",
        "        for j in range(size):\n",
        "            if j>=i:\n",
        "                quadratic[i, j] = float(qubo.coeff(z[i]*z[j]))\n",
        "\n",
        "    for i in range(size):\n",
        "        linear[i] = float(get_constant(qubo.coeff(z[i])))\n",
        "\n",
        "    if tolist:\n",
        "        return quadratic.tolist(), linear.tolist(), constant\n",
        "    else:\n",
        "        return quadratic, linear, constant"
      ],
      "id": "2f11f480-668b-4588-8bc6-9c0ffcdbb209",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-CLY2O7rJ4Eo"
      },
      "source": [
        "**Result:**"
      ],
      "id": "-CLY2O7rJ4Eo"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d2406c11-6216-448b-82e3-8ce05b768ee8"
      },
      "source": [
        "quadratic, linear, constant = get_qubo_coefficients(qubo, tolist=False)"
      ],
      "id": "d2406c11-6216-448b-82e3-8ce05b768ee8",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "quadratic"
      ],
      "metadata": {
        "id": "yaLqNhcne1ig"
      },
      "id": "yaLqNhcne1ig",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "linear"
      ],
      "metadata": {
        "id": "iJomq3AWhCgI"
      },
      "id": "iJomq3AWhCgI",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "constant"
      ],
      "metadata": {
        "id": "OX9HNEflhJJK"
      },
      "id": "OX9HNEflhJJK",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aG3daOhxJIY1"
      },
      "source": [
        "**Check**"
      ],
      "id": "aG3daOhxJIY1"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dSr8gVGbA9bG"
      },
      "source": [
        "def reconstruct_qubo(quadratic, linear, constant):\n",
        "    size = len(z)\n",
        "    exp = constant\n",
        "    for i in range(size):\n",
        "        for j in range(size):\n",
        "            exp += quadratic[i, j]*z[i]*z[j]\n",
        "    \n",
        "    for i in range(size):\n",
        "        exp += linear[i]*z[i]\n",
        "\n",
        "    return exp "
      ],
      "id": "dSr8gVGbA9bG",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KZ9aI2sOBoVp"
      },
      "source": [
        "qubo2 = reconstruct_qubo(quadratic, linear, constant)"
      ],
      "id": "KZ9aI2sOBoVp",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "sym.simplify(qubo-qubo2)"
      ],
      "metadata": {
        "id": "ulzbS5b6uhKu"
      },
      "id": "ulzbS5b6uhKu",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2_FgxlA55Vdu"
      },
      "source": [
        "# Part II： Quantum Algorithm\n",
        "\n"
      ],
      "id": "2_FgxlA55Vdu"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Va1Xj2Nd5gvx"
      },
      "source": [
        "The QAOA and MinimumEigenOptimizer class\n",
        "\n",
        "Qiskit provides its own implementation of QAOA which allows us to solve optimization problems with only a few lines of code. <br>\n",
        "The [`QAOA`](https://qiskit.org/documentation/stubs/qiskit.algorithms.QAOA.html) class in Qiskit is located in `qiskit.algorithms` and directly inherits from the Variational Quantum Eigensolver class [`VQE`](https://qiskit.org/documentation/stubs/qiskit.aqua.algorithms.VQE.html). The initializer for `QAOA` takes the following input parameters, amongst others\n",
        "- `optimizer`: This argument refers to the classical optimizer used for updating the circuit parameters. Qiskit offers a number of different [optimizers](https://qiskit.org/documentation/apidoc/qiskit.aqua.components.optimizers.html) and you can also define your own by subclassing Qiskit's native [`Optimizer`](https://qiskit.org/documentation/stubs/qiskit.aqua.components.optimizers.Optimizer.html#qiskit.aqua.components.optimizers.Optimizer) class. Some of the basic optimizers offered by Qiskit are the following: <br>\n",
        "    - [COBYLA](https://qiskit.org/documentation/stubs/qiskit.aqua.components.optimizers.COBYLA.html#qiskit.aqua.components.optimizers.COBYLA): Constrained Optimization By Linear Approximation optimizer.\n",
        "    - [SLSQP](https://qiskit.org/documentation/stubs/qiskit.aqua.components.optimizers.SLSQP.html#qiskit.aqua.components.optimizers.SLSQP): Sequential Least SQuares Programming optimizer.\n",
        "    - [ADAM](https://qiskit.org/documentation/stubs/qiskit.aqua.components.optimizers.ADAM.html#qiskit.aqua.components.optimizers.ADAM):  A gradient-based optimization algorithm that is relies on adaptive estimates of lower-order moments.\n",
        "- `reps`: The number of layers $p$ in the QAOA variational form, i.e. the depth of the algorithm. For higher values of $p$, QAOA can theoretically obtain better results but the quantum circuit becomes deeper and there are more parameters to optimize.\n",
        "- `initial_point`: Optional initial parameter values (values for $\\beta$ and $\\gamma$) to start the optimization with.\n",
        "- `quantum_instance`: The quantum instance to run the algorithm on. This can be a simulator or a real hardware device.\n",
        "- `callback`: A callback function that can be used to monitor the optimization process.\n",
        "\n",
        "\n",
        "The [`MinimumEigenOptimizer`](https://qiskit.org/documentation/tutorials/optimization/3_minimum_eigen_optimizer.html) object provides a wrapper for the optimization process that handles the conversion from a quadratic program to a qubit operator as well as calling a given [`MinimumEigenSolver`](https://qiskit.org/documentation/stubs/qiskit.aqua.algorithms.MinimumEigensolver.html), such as QAOA to obtain the corresponding optimization result. The initializer takes as input the `MinimumEigenSolver` to use for the optimization, and the optimization is run by calling the `optimize` method with a quadratic program as input. "
      ],
      "id": "Va1Xj2Nd5gvx"
    },
    {
      "cell_type": "code",
      "source": [
        "quadratic, linear, constant = get_qubo_coefficients(qubo, tolist=True)"
      ],
      "metadata": {
        "id": "jDIGElFQ2j9L"
      },
      "id": "jDIGElFQ2j9L",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(f'We need {len(z)} qubits')"
      ],
      "metadata": {
        "id": "O7uodwsGchne"
      },
      "id": "O7uodwsGchne",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KdInl-zh9F3_"
      },
      "source": [
        "from qiskit_optimization import QuadraticProgram\n",
        "\n",
        "quadratic_program = QuadraticProgram('airbus')\n",
        "\n",
        "for i in range(len(z)):\n",
        "    quadratic_program.binary_var(name = f'z_{i}')\n",
        "\n",
        "\n",
        "quadratic_program.minimize(quadratic=quadratic, linear=linear, constant=constant)\n",
        "\n",
        "print(quadratic_program.export_as_lp_string())"
      ],
      "id": "KdInl-zh9F3_",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BrGJQHyy6uwL"
      },
      "source": [
        "# qiskit only supports 15 qubits\n",
        "\n",
        "from qiskit import Aer\n",
        "from qiskit.algorithms.optimizers import COBYLA, SLSQP, ADAM\n",
        "from qiskit.algorithms import QAOA\n",
        "from qiskit_optimization.algorithms import MinimumEigenOptimizer\n",
        "\n",
        "backend = Aer.get_backend('statevector_simulator')\n",
        "qaoa = QAOA(optimizer=ADAM(), quantum_instance=backend, reps=1, initial_point=[0.1,0.1])\n",
        "eigen_optimizer = MinimumEigenOptimizer(min_eigen_solver = qaoa)\n",
        "\n",
        "result = eigen_optimizer.solve(quadratic_program)\n",
        "print(result)"
      ],
      "id": "BrGJQHyy6uwL",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HCFu0uMbUA0a"
      },
      "source": [
        "# Part III： Classical Algorithm"
      ],
      "id": "HCFu0uMbUA0a"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QM8uN8u0_mG9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 168
        },
        "outputId": "eb39dddb-f4e7-4cc3-d352-fe7a3f97cb78"
      },
      "source": [
        "quadratic, linear, constant = get_qubo_coefficients(qubo, tolist=False)"
      ],
      "id": "QM8uN8u0_mG9",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "error",
          "ename": "NameError",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-19-0340d06a94bd>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mquadratic\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlinear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mconstant\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mget_qubo_coefficients\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mqubo\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtolist\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
            "\u001b[0;31mNameError\u001b[0m: name 'get_qubo_coefficients' is not defined"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "75-sJNDCAOM6"
      },
      "source": [
        "import copy\n",
        "def mix_linear_quadratic(linear, quadratic, dict_=True):\n",
        "    \"\"\"\n",
        "    for bianry variable z_i, z_i = z_i^2\n",
        "    \"\"\"\n",
        "    quadratic = copy.deepcopy(quadratic)\n",
        "    for i in range(len(z)):\n",
        "        quadratic[i,i] += linear[i]\n",
        "    \n",
        "    quadratic_dict = {}\n",
        "    for i in range(len(z)):\n",
        "        for j in range(len(z)):\n",
        "            if j >= i:\n",
        "                quadratic_dict[(i, j)] = quadratic[i, j]\n",
        "\n",
        "    if dict_:\n",
        "        return quadratic_dict\n",
        "    else:\n",
        "        return quadratic\n",
        "   \n",
        "    "
      ],
      "id": "75-sJNDCAOM6",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AyG5LGoVUF90"
      },
      "source": [
        "**Qbsolv**\n",
        "\n",
        "A decomposing solver that finds a minimum value of a large quadratic unconstrained binary optimization (QUBO) problem by splitting it into pieces. The pieces are solved using a classical solver running the tabu algorithm. Qbsolv also enables configuring a D-Wave system as the solver.\n",
        "\n",
        "Note: Access to a D-Wave system must be arranged separately.\n",
        "\n",
        "https://github.com/dwavesystems/qbsolv\n",
        "\n",
        "\n",
        "https://docs.dwavesys.com/docs/latest/c_gs_3.html#qubo\n",
        "\n",
        "\n",
        "$z_i=z_i^2 $ for binary variables $z_i$'s\n",
        "\n",
        "\n",
        "\n",
        "In this case, the proper format of QUBO matrix is upper-triangular with linear terms included in the diagonal. For example:\n",
        "\n",
        "\n",
        "```python\n",
        "# [1 2 4\n",
        "#    2 3\n",
        "#      1]\n",
        "\n",
        "from dwave_qbsolv import QBSolv\n",
        "Q = {(0,0):1, (0,1):2, (0,2):4, (1,1):2, (1,2):3, (2,2):1}\n",
        "response = QBSolv().sample_qubo(Q)\n",
        "```\n",
        "\n"
      ],
      "id": "AyG5LGoVUF90"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xIIVFv0CFpMl"
      },
      "source": [
        "Airbus example"
      ],
      "id": "xIIVFv0CFpMl"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "SwQjxGu4UUS2",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "aa2534c1-0694-4f73-feda-b8bdc414ae3d"
      },
      "source": [
        "from dwave_qbsolv import QBSolv\n",
        "Q = mix_linear_quadratic(linear, quadratic)\n",
        "response = QBSolv().sample_qubo(Q)\n",
        "\n",
        "print(\"energies=\" + str(list(response.data_vectors['energy'])))  # list of optimal function value\n",
        "print(\"samples=\" + str(list(response.samples()))) # list of optimal value, [{z_0: 0, z_1:0, z_2:1, ...}, ...]"
      ],
      "id": "SwQjxGu4UUS2",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:3: DeprecationWarning: QBSolv is deprecated as of the end of 2021 and support will be discontinued after March 2022. Please update your code to use dwave-hybrid or Leap's quantum-classical hybrid solvers instead.\n",
            "  This is separate from the ipykernel package so we can avoid doing imports until\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "energies=[-1600039989.0, -1600039837.75, -1600039806.25, -1600039801.25, -1600039801.0, -1600039788.0, -1600039761.25, -1600039753.5, -1600039751.5, -1600039750.25, -1600039744.25, -1600039733.5, -1600039733.0, -1600039721.75, -1600039702.5, -1600039700.0, -1600039691.75, -1600039690.75, -1600039688.75, -1600039680.5]\n",
            "samples=[{0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0, 10: 0, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 1, 17: 0, 18: 0, 19: 0, 20: 0, 21: 0, 22: 0, 23: 0, 24: 0, 25: 0, 26: 0, 27: 0, 28: 0, 29: 0, 30: 0, 31: 0, 32: 0, 33: 0, 34: 1, 35: 1, 36: 0, 37: 0, 38: 0, 39: 0, 40: 0, 41: 0, 42: 0, 43: 0, 44: 0, 45: 1, 46: 0, 47: 0, 48: 0, 49: 0, 50: 0, 51: 0, 52: 0, 53: 0, 54: 0, 55: 0, 56: 0, 57: 0, 58: 0, 59: 0, 60: 1, 61: 0, 62: 0, 63: 0, 64: 0, 65: 0, 66: 0, 67: 0, 68: 0, 69: 1, 70: 0, 71: 0, 72: 0, 73: 0, 74: 0, 75: 0, 76: 0, 77: 0, 78: 0, 79: 0, 80: 0, 81: 0, 82: 0, 83: 0, 84: 1, 85: 0, 86: 0, 87: 0, 88: 0, 89: 0, 90: 0, 91: 0, 92: 0, 93: 0, 94: 0, 95: 0, 96: 1, 97: 0, 98: 0, 99: 0, 100: 0, 101: 1, 102: 0, 103: 0, 104: 0, 105: 0, 106: 0, 107: 0, 108: 0, 109: 0, 110: 0, 111: 0, 112: 0, 113: 0, 114: 0, 115: 0, 116: 0, 117: 0, 118: 0, 119: 0, 120: 0, 121: 0, 122: 0, 123: 0, 124: 0, 125: 0, 126: 0, 127: 0, 128: 0, 129: 0, 130: 0, 131: 0, 132: 0, 133: 0, 134: 0, 135: 0, 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367: 0, 368: 0, 369: 0, 370: 0, 371: 0, 372: 0, 373: 0, 374: 0, 375: 0, 376: 0, 377: 0, 378: 0, 379: 0, 380: 0, 381: 0, 382: 0, 383: 0, 384: 0, 385: 0, 386: 0, 387: 0, 388: 0, 389: 0, 390: 0, 391: 0, 392: 0, 393: 0, 394: 0, 395: 0, 396: 0, 397: 0, 398: 0, 399: 0, 400: 0, 401: 0, 402: 0, 403: 0, 404: 0, 405: 0, 406: 0, 407: 0, 408: 0, 409: 0, 410: 0, 411: 0, 412: 0, 413: 0, 414: 0, 415: 0, 416: 0, 417: 0, 418: 0, 419: 0, 420: 1, 421: 1, 422: 0, 423: 0, 424: 0, 425: 0, 426: 0, 427: 0, 428: 0, 429: 1, 430: 0, 431: 1, 432: 1, 433: 0, 434: 0, 435: 1, 436: 0, 437: 0, 438: 0, 439: 1, 440: 0, 441: 0, 442: 0, 443: 1, 444: 1, 445: 0, 446: 1, 447: 0, 448: 0, 449: 0, 450: 0, 451: 0, 452: 0, 453: 0, 454: 1, 455: 0, 456: 0, 457: 0, 458: 0, 459: 1, 460: 0, 461: 0, 462: 0, 463: 0, 464: 0, 465: 0, 466: 0, 467: 0, 468: 0, 469: 0, 470: 0, 471: 0, 472: 0, 473: 0, 474: 0, 475: 0, 476: 0, 477: 0, 478: 0, 479: 0, 480: 0, 481: 0, 482: 0, 483: 0, 484: 0, 485: 0, 486: 0, 487: 0, 488: 0, 489: 0, 490: 0, 491: 0, 492: 0, 493: 0, 494: 1, 495: 0, 496: 0, 497: 0, 498: 0, 499: 0, 500: 0, 501: 0, 502: 0, 503: 0, 504: 0, 505: 0, 506: 0, 507: 0, 508: 0, 509: 0, 510: 0, 511: 0, 512: 0, 513: 0, 514: 0, 515: 0, 516: 0, 517: 0, 518: 0, 519: 0, 520: 0, 521: 0, 522: 1, 523: 0, 524: 0, 525: 0, 526: 0, 527: 0, 528: 0, 529: 0, 530: 1, 531: 0, 532: 0, 533: 0, 534: 0, 535: 1, 536: 0, 537: 0, 538: 0, 539: 0, 540: 0, 541: 0, 542: 0, 543: 1, 544: 1, 545: 1, 546: 0, 547: 0, 548: 0, 549: 1, 550: 0, 551: 0, 552: 1, 553: 0, 554: 0, 555: 1, 556: 0, 557: 0, 558: 0, 559: 1, 560: 0, 561: 0, 562: 1, 563: 1, 564: 0, 565: 0, 566: 0, 567: 0, 568: 0, 569: 0, 570: 0, 571: 1, 572: 0, 573: 0, 574: 0, 575: 0, 576: 0, 577: 0, 578: 0, 579: 0, 580: 0, 581: 0, 582: 0, 583: 1, 584: 0, 585: 0, 586: 0, 587: 0, 588: 0, 589: 0, 590: 0, 591: 0, 592: 1, 593: 0, 594: 0, 595: 1, 596: 0, 597: 0, 598: 0, 599: 0, 600: 0, 601: 0, 602: 0, 603: 0, 604: 0, 605: 0, 606: 0, 607: 0, 608: 0, 609: 0, 610: 0, 611: 0, 612: 0, 613: 0, 614: 0, 615: 0, 616: 0, 617: 0, 618: 0, 619: 0, 620: 0, 621: 0, 622: 0, 623: 0, 624: 0, 625: 0, 626: 0, 627: 0, 628: 0, 629: 0, 630: 0, 631: 0, 632: 0, 633: 0, 634: 0, 635: 0, 636: 0, 637: 0, 638: 0, 639: 0, 640: 0, 641: 0, 642: 0, 643: 0, 644: 0, 645: 0, 646: 0, 647: 0, 648: 0, 649: 1, 650: 0, 651: 0, 652: 0, 653: 0, 654: 0, 655: 0, 656: 0, 657: 0, 658: 0, 659: 0, 660: 0, 661: 0, 662: 0, 663: 0, 664: 0, 665: 0, 666: 0, 667: 0, 668: 0, 669: 0, 670: 0, 671: 0, 672: 0, 673: 0, 674: 0, 675: 0, 676: 0, 677: 0, 678: 0, 679: 0, 680: 0, 681: 0, 682: 0, 683: 0, 684: 0, 685: 0, 686: 0, 687: 0, 688: 0, 689: 0, 690: 0, 691: 0, 692: 0, 693: 0, 694: 0, 695: 0, 696: 0, 697: 0, 698: 0, 699: 0, 700: 1, 701: 1, 702: 0, 703: 0, 704: 0, 705: 0, 706: 0, 707: 1, 708: 1, 709: 0, 710: 0, 711: 0, 712: 0, 713: 1, 714: 0, 715: 0, 716: 1, 717: 0, 718: 1, 719: 0, 720: 0, 721: 0, 722: 1, 723: 0, 724: 1, 725: 1, 726: 1, 727: 1, 728: 1, 729: 1, 730: 1, 731: 1, 732: 1, 733: 0, 734: 1, 735: 1, 736: 1, 737: 1, 738: 1, 739: 1, 740: 1, 741: 0, 742: 0, 743: 1, 744: 0, 745: 1, 746: 0, 747: 0, 748: 0, 749: 0, 750: 1, 751: 1, 752: 1, 753: 1, 754: 1, 755: 0, 756: 0, 757: 0, 758: 1, 759: 0, 760: 0, 761: 0, 762: 0, 763: 1, 764: 0, 765: 0, 766: 0, 767: 0, 768: 0, 769: 0, 770: 0}]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "results = list(response.samples())"
      ],
      "metadata": {
        "id": "USpwMvv72YeL"
      },
      "id": "USpwMvv72YeL",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "9VHHqjHI5T1d"
      },
      "id": "9VHHqjHI5T1d",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "for i, result in enumerate(results):\n",
        "      replacements = []\n",
        "      for key, val in result.items():\n",
        "          replacements.append((z[key], val))\n",
        "      print(f'{i}: {payload.subs(replacements)}')\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "vdeCqWPF17j6",
        "outputId": "0fa213aa-c785-4e08-b1fc-971364d9098f"
      },
      "id": "vdeCqWPF17j6",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0: 39941.0000000000\n"
          ]
        },
        {
          "output_type": "error",
          "ename": "KeyboardInterrupt",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-103-2ab9684fb243>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      3\u001b[0m       \u001b[0;32mfor\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mval\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m           \u001b[0mreplacements\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mz\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mval\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m       \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34mf'{i}: {payload.subs(replacements)}'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36msubs\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    946\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    947\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msequence\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 948\u001b[0;31m                 \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    949\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    950\u001b[0m                     \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     70\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     71\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 72\u001b[0;31m                 \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     73\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     74\u001b[0m                 \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1060\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1061\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1062\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1063\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1064\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mfallback\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m   1032\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'_eval_subs'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1033\u001b[0m                     \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1034\u001b[0;31m                 \u001b[0marg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1035\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0m_aresame\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1036\u001b[0m                     \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     70\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     71\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 72\u001b[0;31m                 \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     73\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     74\u001b[0m                 \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1058\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1059\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1060\u001b[0;31m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1061\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1062\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/local/lib/python3.7/dist-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_eval_subs\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m   1529\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1530\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunctions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melementary\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcomplexes\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1531\u001b[0;31m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mntheory\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfactor_\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mmultiplicity\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1532\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpowsimp\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mpowdenest\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1533\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mradsimp\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfraction\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m/usr/lib/python3.7/importlib/_bootstrap.py\u001b[0m in \u001b[0;36m_handle_fromlist\u001b[0;34m(module, fromlist, import_, recursive)\u001b[0m\n",
            "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KlVPxHiLJmC8"
      },
      "source": [
        "result = results[15]  # here we choose result 0, better try different element in the list, sometimes Qbsolv may give several feasible soultion\n",
        "replacements = []\n",
        "for key, val in result.items():\n",
        "    replacements.append((z[key], val))"
      ],
      "id": "KlVPxHiLJmC8",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eT6-q9M2I622",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "480a75ce-2de9-4271-bb02-1f73b1f563ef"
      },
      "source": [
        "if qubo is not None:\n",
        "    print('optimal function value:', qubo.subs(replacements))\n",
        "else:\n",
        "    print('optimal function value:', payload.subs(replacements))\n",
        "    \n",
        "optimal_values = [j for i, j in replacements]\n",
        "array = np.zeros((35,20))\n",
        "for i in range(n):\n",
        "    for j in range(N):\n",
        "        array[i,j] = optimal_values[i*N+j] \n",
        "\n",
        "print('optimal value:', optimal_values)\n",
        "print('status: SUCCESS')"
      ],
      "id": "eT6-q9M2I622",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "optimal function value: 39670.0000000000\n",
            "optimal value: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]\n",
            "status: SUCCESS\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "%matplotlib inline\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np"
      ],
      "metadata": {
        "id": "f9359tVivmg8"
      },
      "id": "f9359tVivmg8",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "array[7, :]"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cB3PrbpRAXnV",
        "outputId": "7e32de16-4efb-4b1b-9005-da9fe86f73c2"
      },
      "id": "cB3PrbpRAXnV",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\n",
              "       0., 0., 0.])"
            ]
          },
          "metadata": {},
          "execution_count": 101
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(12, 12))\n",
        "plt.imshow(array, cmap='gray', vmin=0, vmax=1.)\n",
        "plt.xticks(np.arange(0, 20, 1))\n",
        "plt.yticks(np.arange(0, 35, 1))\n",
        "plot.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 700
        },
        "id": "_chbVW0K0Pe9",
        "outputId": "be786a62-d80d-48df-fb37-4bfba44da4a2"
      },
      "id": "_chbVW0K0Pe9",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 864x864 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JhwD4JTmSsjp"
      },
      "source": [
        "For problem 2, refer to TABLE IX of [1], the eaxt solution weight is indeed 7500 kg. (n=6, N =4)\n",
        "\n",
        "Recall that the frst $n\\times N$ elemnets of $z$ is $p_{ij}$, i.e. $z[i\\times N+j]=p_{ij}$.\n",
        "\n",
        " if container $i$ occupies position $j$, $p_{ij}=1$, otherwise $p_{ij}=0$."
      ],
      "id": "JhwD4JTmSsjp"
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FdnwlztFVcgD"
      },
      "source": [
        ""
      ],
      "id": "FdnwlztFVcgD",
      "execution_count": null,
      "outputs": []
    }
  ]
}