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stackdump/knapsack.ipynb

Created January 3, 2025 21:57
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A continuous version of knapsack problem using Petri-net derived ODEs
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knapsack.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "e182136f-6d69-47a0-ace0-94c66acf7c9d",
"metadata": {},
"source": [
"\n",
"This foundational script uses Petri nets in Julia to model a knapsack problem. It defines states, transitions, and a simulation framework to explore how items with varying weights and values affect the system's behavior. By setting initial conditions and success rates, it lays the groundwork for analyzing transition dynamics and exclusions in subsequent simulations.\n",
"[![pflow](https://pflow.dev/img/zb2rheXF9ikouRESRNsR2d1ZcidxdvLuv6zuvxc7ERA5KfM5X.svg)](https://pflow.dev/p/zb2rheXF9ikouRESRNsR2d1ZcidxdvLuv6zuvxc7ERA5KfM5X/)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "203f0fda-e005-4d0f-82ec-755ea2e115e3",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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sIwjCxx9/TEQNGjRISkqyeOri4uIiIiJKDgNxdXVdtGiR6YMIpa8sU6xBccjunj17zAvFB4O8/PLL5oUffvih+XXXXr16RUVFUXnzCHfs2GEerbOz844dO8T/r7///ttU7c6dO126dDG1X8bKMtnZ2WL33cTd3X3Tpk3mdcR5hMUKBUEQL2ibFrWBWo4RMHQKbCc5OZnjOD8/v2KLbxFRVlaWOOQyKChIo9FkZma6u7uXHBmh0WiCgoLMewB//vnn1atXeZ4PDw9v27atXq+/d++eg4ND3bp1xQpz585dvHjxt99+++qrr0ZHR1+5coVl2Y4dO5YcZH/nzp0bN26kp6crlcrQ0NC2bduKt99MtFrtmTNn4uPjHR0du3TpEhAQkJub++DBgzp16pgG3Yg4jjt37tytW7ckEomfn1+nTp1M0/Xy8vKys7M9PT3FkTUmSUlJBoPB1dW1jMGfWq32ypUrycnJWVlZCoWifv36bdu2LXYyMzMzc3NzAwICTCNCExMTeZ4vNjw1Ozs7Ly/Px8fH0dHRVKjT6VJSUpycnMzn4RHR3bt3T548aTAYmjVr1r59e47jkpOTHR0dTcM4c3JysrKyfHx8zMetZGVlnTt3LjU1tW7duj169HB2dk5PT1er1YGBgcWGsBoMhvT0dI7jTG0mJCQYDIaS67DfvHnzwoULYiPdunUznwQphpGTk1Pyv+PevXt6vb5Y9x1qLSRCqHXME6GtYwEA28M9QgAAqNWQCAEAoFbD2CqodXr37u3s7NyqVStbBwIANQLuEQIAQK2GS6MAAFCrIRECAECthkQIAAC1GhIhAADUakiEAABQqyERAgBArVZDE+Fvv/1W7HE2ZcMkkAop9lRVKBu+XRWFL1iF4HRViLhMduW2WUMTYVRUlLievTUEQVCr1VUZjr0p93niYE6r1VrzqFgwwResQnC6KoTjuHIfV1lRNTQRAgAAVA8kQgAAqNWw1igAgC2lpaUdPnz4hRde8PDwMC/X6/U///xz8+bNGzZsaHFHjuM6der02WefdenSpYpiy8jIOHPmzO3bt9VqdVhY2ODBg01PdtTpdPv27TOvHBgY+Nxzz5VsJDk5WXzCs7l69eq1adNGfK3VamNiYpKSkho0aNCiRQuLkURHR9+5c6ddu3amB4tWIiRCAABbUigUr7322sKFC99//33z8oMHDw4fPvyPP/4obUee5y9dupSbm1t1sU2fPv3YsWMNGzaUSqUrV66cMWPG8ePHmzdvTkT5+fnDhw/39fU1PfA5IiLCYiK8evXqrFmzTG/FZ2XPnTtXTISrVq2aMWOGeBv+3XfftZgI09PTe/Xq9eDBg23btkVGRlb6x8SlUQAAW/Lw8Bg4cODGjRuLDYbcvHlz06ZNTd0mm/jss8/S09NPnTr122+/3b17193d3TylEdG3335755GVK1dabKRfv353zCxcuJCITI/Fbt269a5duxITE/39/UsLY9KkSQMGDKi8j1UcEiEAgI2NGTMmLi7uwoULppL79+8fOnRo7NixgiB8+eWXQ4YMadeuXd++fRcvXqzT6Uq2oNFohg8ffuXKFVPJ+vXrFy1aZHp78+bNcePGtW3btm/fvp9++qlp4KXRaIyOjr5//77FwOrWrcuyD9OEu7t7x44d792794QfduPGjV26dGnUqJH4tlOnTkOGDAkKCiqt/o4dO6Kjo+fPn/+Exy0DEiEAgI316dMnMDBwy5YtppJvv/2WYZhRo0YZDIbdu3d37dp15syZvXv3/vLLL994442SLYjV0tPTTSXR0dGm2dgxMTHt2rXLzs6eOnXqwIEDP//8c1MjeXl5zz77rPmhSxMTE/PLL78MGTLEvHDdunUjR4784IMPTpw4Yc0njYuLO3PmjMWPYFFWVta0adPWr1/v6Oho5S6PAfcI7ZOg0whGIxGRwPPah7OUBINe4HREZFSrOQcHQVNEVHxeKq8uLLtlXl1Q+eE+MUHgBW1VzSXV6/WcRGL6uxjKpdPpSKGwdRRPDfF0De/87Nfbti4c2FGlUBDR5jVf9G3XUhn7m5bowLxJREQkkEdA6LSxL33w0acv93ZUKbWcnog0MacLHDSFag0Raa6cLJDli83qU+4YH2QUHN9FRNPf+1+7hqFbJ79CRORbp+H7b/eeNu/dnm1C/evqtbrhL3QO1T8Qa5aUlp3Tfty7eoNRrdVNGzHov13DxZqFefkBdTwpO1WpVB764dzSpUvffWXw/DdeKfvDrt2w3UGp6OPJlDycoNNwSTeLlb+9eGXPlo2fY3MyTu0jIu31C+o6EmWzDuQXUpFzXA4kwhpK4HTGghxBU8hrinitWtAW/euFplDQFAmCIGgKiYjXFJJAvLaIBJ7XqonnGYWKkUiIiBiWVT76S0oqZeVKIjIajZxEwqgciZhix2UdnMoOjHVwruzPWgkYhmWUDlXUuGAw8CxLSIRWEziON1bylGc7Jp6ukd3afbZj7/7fTg/r2v7K7cSrdxLnjHxR/MM0My9/x4lz8en3H+QXanQcLwh34xPCQwJ4Tk9Egk7Lqwt5jZaIeJ3W9LesYNALRiOvLhQE4eKNuJe6tv/paNTDTYIgYZlrf8eFuDspidZPHUOl/xHsJmU2vzchp6Do7PW4L3cfCPJwHdO3GxG5S5mrGz5mGEZscM7GXZ/t2Pty17bP+Jc6qtNg5L//9fdhXdqpBEPJwwmCIOj15uWH/4j57XLshdX/49WFvKaIiHhOK/5WfKwzXSokQlsSdBpDTqYxN9OY/8CYc9+Yl23MzRJfCHqdxMWDUTmySgdW6cioHFmlI6t0ZJ3cpF5+rIMzo3RgWJZRORERq3IihlilIzEsq3Qo91d2QUGBaQw0lEuj0chkMqkUPyzWwhesQsTT1Zqo07eHdl9NfOOTdT9MmeLj4/PSvOVSqfTu3bvt27Zt0aJF7959PDw8cnJyjlyKFVq/4Nqpk0KrJZrg0K6n68CBbEEB0STH9r1d+/YVm5UfvCg1yFwHjs3Ozi7SjovJyEu9fMt00O7P93Dt2Nc1IsKaCF8cSkT0OpHLf/87d+3aiZ9/LZfLi9WZ06bfmr3+fzsFtRk4qrR29u3bl/4gd8LCj10tDS5llfMUYU1dB441lbw3Ofjl10bH1w2PJ3qgeUBE8XKvKy5hnQOfsSZs6+Fnu1rxRflc0k0u6Zb+XhyXFMdr1RJ3b4mrp8TNS+peRx5QX9L0OYlbHYmrJ+voYutgAaBajRkzZvz48Xfu3NmxY8cbb7wh/u21detWHx+fY8eOiRfnS07IEykUCpZltVqtqSQzM1N84eLiolQqR48ePX369CeMsGnTpkVFRVlZWX5+fhYrlL1c3MaNGxs1amRxioVFHMdt3759+/bt9GhF1lWrVv38888xMTEVDLwcSIRVS9BpuJS7+nu3uORb+nu3jLnZUt8QeWB9VYvOLhFjZD5BxBS/OAkAtdPw4cOnTp366quvZmVlvf7662JhUVERy7LMo18Ua9assbivXC739/e/cOHC4MGDiSg1NfXYsWPt2rUjIplMNmDAgDVr1owbN87UU9doNFKpVCaTicmmbdu24eHhJZtNT083TWA3GAzff/+9j4+Pj48PEWVmZnp5eZkujS5btoxhGNPU/qioqAcPHgwdOtTUVEZGxqFDhz766CPrT0haWpp5JL6+vmvXrq2KeYRIhFWC1xapL/+mPv+r/v49uX+YLKiBsnFbl94jpV5+yHwAYJGTk1NkZOSmTZs6duxoml0QGRn5+eefDxo0qH379idPnrQ4d0I0adKkuXPn3r9/38XFZd++fY0bNzZtWrlyZffu3Zs0aRIZGenq6nr79u0DBw789ddfdevWLSwsHDNmzLJlyywmwrCwsA4dOoSFhWm12pMnT6alpW3fvl0ikRDR8uXLf/rpp7Zt2zo5OV28ePHKlSuzZ89u0qSJuOOXX34ZExNjngi3bNkiCIJp+qDJH3/8MXv2bCLKysr64YcfYmJinJyc9uzZ85gn8bEgEVYy3d3rRecPa6+dUzRs7RLxuvKZlhhkAQBWeu+99xo2bNi5c2dTSbt27U6fPr1t27a4uLihQ4e+/PLL69atCwkJISKZTLZ06VJT7nn//fcDAgJOnDjB8/y+ffuSk5NNi874+/v/+eef27Ztu3DhQlZWVkhIyL59+7y9vYnIwcFh6dKl3bp1sxjPvn37fv/994SEBKVSOWHChGHDhoWFhYmbxo4d6+Xl9ffff6vV6oEDB3755ZedOnUy7fjaa6/16dPHvKmgoKCNGzeWXCDNy8urZ8+eRCT+S0QKS0OOnZ2dly5d2qpVK+tOZMUwNfNZa/Pnz5dIJPPmzbOmsvgYpiqdZVIuXl2ouXKy8PR+waB3bN/bsX1v1snNhvGUDWMZKgSDZSoKX7AKwemqEJ1OxzBMydE6TwI/209GEHS3rhT9cUx7/YKiQSu3wW8qnmmJi58AAE8RJMLHZ8i4l715MSOVOT7X1+2lif9M1wMAgKcHEuFj0lw7n7tzpUv//zh26GfrWAAA4PEhEVYcz+cd3KyOjvIcv0AeZPk5YQAA8LRAIqwYvig/e8tHJPA+735Rk4fDAACAlZAIK4C7F/dg02JVq26uA17HpAgAAPuARGitorOH8g5udh8xTdW8o61jAQCASoNEWD5Bz+X+sJpLvuU9faXUy9fW4QAAQGXC9b1yGHPuZ37xX8Gg9576ObIgAID9QY+wLLy68P6K6c69XnbqPNDWsQAAQJVAj7As+Ye2qJp1RBYEALBjSISl0qcnqq+ccun3mq0DAQCAKoREWKrcH1a79v8PHpALAGDfkAgtU0dH8doix+f62joQAACoWkiEFgicLu/ARrehb2PWPACA3cMvegvyf92uCGuuCG1q60AAAKDKIREWZ8hKKzr3i2vEGFsHAgAA1QGJsLjcn75yfmG4xNXT1oEAAEB1QCL8F+3flw2ZKU5dX7R1IAAAUE2QCP8hGA25P65xGzqBkcpsHQsAAFQTJMJ/FP72o6xusLJxW1sHAgAA1QeJ8CFjXnbBbz+6Dn7T1oEAAEC1QiJ8KG/veqfOA6WedW0dCAAAVCskQiIiLv66Lv4v5xcibR0IAABUNyRCIp7P+WG12+A3GbnS1qEAAEB1QyKkwrMHWQcXVYvOtg4EAABsoLYnQkHP5R/+1u2lt20dCAAA2EZtT4TcvVtSDx9Z3WBbBwIAALZR6xNhwl/ykMa2jgIAAGwGifAGEiEAQG1W6xNh4k1FSBNbRwEAADZTqxOh4UEGCYLEw9vWgQAAgM3U6kTIJfwlr4fuIABArVbrEyFuEAIA1G61OxHG/6VAIgQAqN1qbyIU9Jw+I0kWUN/WgQAAgC3V3kTI3YuT+YYwMrmtAwEAAFuSVlG7t2/fvnHjhtFobNasWf36//S6ioqKTpw4QUQ9evRwdHSsoqNbg4vHDUIAAKiaHuHt27d79eq1efPmnTt3tmvX7sMPPxTL79+/36JFi9WrV69Zs6ZFixb379+viqNbiUvADUIAAKiaRBgWFhYfH//TTz/t2LFj//79ixcv1ul0RPTll182b9788OHDv/zyS8uWLVetWlUVR7cSl/iXvB4SIQBAbVcliZBhGNNrnuednJykUikR7du3b/jw4WL58OHD9+7dWxVHt4YhO40YVuJWx1YBAABADVFV9wiJ6M0330xJSbl58+ZPP/0kkUiIKDk5OSAgQNwaEBCQkpJS2r45OTnx8fFr164V30okkpEjRyqVlh+cKwiC0Wg0Go3Wx6a9e10W3LhCu9iTip6uWs5oNLIsa/7nHZQNX7AKwemqEKPRyDCM9WfMmh/eKkyEo0aNSk1N3bx587Jly7p27cowjMFgEDMiEclkMo7jSts3Pz8/NTX10qVL4luWZQcMGGDatxhBEPR6vV6vtz423d0bkqAGFdrFnlT0dNVy4rkSBMHWgTw18AWrEJyuCtHr9QzDWP+HqUwmKy13mFRhIuzatSsRDR061NPT8+LFixP5NqAAACAASURBVO3bt/f19c3MzBS3ZmRk+Pn5lbZvcHBwaGjovHnzrDmQIAg8z5fWX7Qo716ce/te8orsYk/0en2FTlctJwiCTCYTL++DNfAFqxCcrgoRs6BcXpkz36p8HqFGo9Hr9eJMiS5duhw/flwsP378uJgpq5/A6Qz3k2X+YTY5OgAA1ChV8kfu999/f+zYsRYtWuh0um3btvXs2bNJkyZENHXq1I4dO3p5eTEMs3HjxjNnzlTF0cvFJcXJ/OphKj0AAFAV9Qh79OjRrl27+Pj4nJyc+fPn79u3j2VZIgoPDz979qxarS4qKjp79mzTpk2r4ujl4hJuYAYhAACIqqRH6OPj89Zbb1ncFB4e/tFHH1XFQa3HJfzl0PYF28YAAAA1RG1ca5RLvCkPRo8QAACIamEiNGSlkUQqcfOydSAAAFAj1LpEyMXfwMpqAABgUvsSIdbaBgAAM7UuEeoS8PQlAAD4R+1KhAKnNWSmYCo9AACY1K5EyCXelPmHMlKZrQMBAICaopYlQtwgBACAf6tdiRA3CAEAoJjalAgFgUv8G4kQAADM1aJEaMhKZeQKiaunrQMBAIAapBYlQtwgBACAkmpRItQl/CUPaWLrKAAAoGapRYmQw0gZAAAoobYkQkGnMWSlyfxDbR0IAADULLUlEXKJN+X+oYykSp6/CAAAT6/akggxgxAAACyqLYmQS7iBkTIAAFBS7UiEgsAl3pSHNLJ1HAAAUOPUikRoyExhlQ4SFw9bBwIAADVOrUiEuEEIAAClqRWJEDMIAQCgNLUjEcbfQCIEAACL7D8RCjqN4UGGzK+erQMBAICayP4ToSErTepZF1PpAQDAIvtPhAKnYRQOto4CAABqKPtPhLxOyyqUto4CAABqKPtPhAKnZeQKW0cBAAA1VC1IhDoto1DZOgoAAKih7D8R8pyGkePSKAAAWGb/iVDQaVn0CAEAoBS1IBFyWvQIAQCgNEiEAABQq9WCRKjTMpg+AQAApbD/RMhzWhY9QgAAKIX9J0JBh1GjAABQqlqQCDnMIwQAgFLZfyLkdRgsAwAApbL/RChwWGsUAABKVQsSIe4RAgBA6WpBIsQ8QgAAKF3tSIQYLAMAAKWw/0TIc1oWj2ECAIBS2HkiFPQcI5ESK7F1IAAAUEPZeyLEDUIAACiTvSdCLDQKAABlsvNEyHMaLDQKAABlsPNEKHBaRo4howAAUCp7T4Q6DS6NAgBAGew9EWKwDAAAlMnOEyGvw0KjAABQFjtPhOgRAgBA2ew9EeqwvhoAAJTFzhMhz+HREwAAUBY7T4SCTot5hAAAUAZ7T4S4RwgAAGWqBYkQ9wgBAKB09p4IdVoGz2ACAIDS2Xki5Dktix4hAACUzs4ToaDDqFEAACiLvSdC3CMEAIAy1YJEiB4hAACUzs4TIY95hAAAUCY7T4QChyfUAwBAWew9EWKwDAAAlMmuEyFvFHiekcltHQcAANRc9pwIeU7LYjY9AACUyZ4TIZ7BBAAA5bLrRIi5EwAAUB67ToQ6DBkFAIBy2HMi5DkNJhECAEDZLCfC2NjYmJgY8bVer1+2bNlLL720dOlSg8FQjbE9KUGHS6MAAFAOy4kwMjLy119/FV8vWbLk/fff/+uvvxYsWDBlypRqjO1JYaFRAAAol4VEqFar4+Linn/+eSISBGH9+vVvvvnmjRs3vv/++40bNxYWFlZ7kI8Js+kBAKBcFhJhbm4uEdWpU4eIYmNjU1NTX331VSLq27cvx3Hx8fHVHOJj4zkdi8EyAABQJmnJIk9PT4ZhEhMTg4ODf/rpJ0dHx/bt2xORVqslIkEQym3UaDSePn36woULRqOxS5cunTt3Nm2Kj4/fuXMnEY0YMaJevXqV9jksQY8QAADKZaFHqFAounfv/s477yxfvnzt2rUREREKhYKIrl+/zjBMYGBguY1u3bp1ypQpWVlZhYWFQ4cOXbJkiVh+69at1q1bZ2ZmZmVlPfvss7du3arcD1MM5hECAEC5LPQIiWj9+vWvvfba7NmzW7duvWzZMrFw27Zt4eHh7u7u5TY6bNiw119/nWEYImrXrt348eNnz55NRCtXrhw+fPjy5cuJqKioaMWKFatXr660j1KCwGlZ5/KjBQCA2sxyIqxfv/65c+eKFX7yyScsa9W8Q2dnZ9NrhmHEDiURnThxwtQ77Nev33vvvVfheCuC57RSjBoFAIAyWU6EFrm4uFS0dY1GM3fu3BkzZohv09LSvL29xdd169ZNS0srbcfk5OS///47Pz9ffMswzLvvvuvm5maxsiAIWq1WIpEUKzeoC1lGIt7aBHNarVYmk9k6iqeGVqs1Go1SaQV+WGo5fMEqBKerQnQ6HcMwPM9bWV8mk5XMDsWU+rN9+fLlvXv3JiYmajQa8/Jdu3ZZeXiO4yIjI8PDw6dOnSqWsCxrit5oNJYRnFKpVKlUHh4e4lu5XK5UKkurLwiCRCKxsFWvk6gcyz0FtZDl0wWlkDxi60CeGjhdFYLTVSESiYRhGOvPmHiTrmyWE+FHH300Z84cpVJZr149lepxri4aDIZXXnlFJpNt3brVdEHVz8/P1AtMTU318/MrbXcvLy8fHx/xzmK5BEGQyWQl/6Ri9JxU5YA/tUqyeLqgNAaDQSaToUdoPXzBKgSnq0J4nmcYpnLPmIWfbUEQPvrooxEjRnz99ddOTk6P0ajRaBw9enRRUdHevXvNw+3fv/+ePXsiIyOJaM+ePf3793/suK3Bc1oW9wgBAKBMFhLh/fv3CwoKZsyY8XhZkIg2bty4ffv2zp07R0REiCX79u1TqVTvvPNO+/btR4wYwTCMONHw8QO3AqZPAABAuSxPqHd3d3/w4MFjN9q3b9+jR4+al8jlciLy9/e/du3awYMHiWjt2rWlDX6pLIJOg8cwAQBA2SwkQqlU+uGHHy5cuLBdu3bWzBosKTAwsLR5925ubuKCbdVA4LSMHJdGAQCgLJbv/8fExNy+fTssLKxdu3bFZk1YP2rU5nidFmuNAgBA2SwnwsTExKCgICLKysrKysqq3pAqiSAIeh0jU9g6DgAAqNEsJ0LTwwifXoKeYyRSsm4pHAAAqLXsNk/gqbwAAGCNUucIx8XFrVmz5vr16zKZ7NChQ0S0c+dOJyenAQMGVGN4j4/XaVjMnQAAgPJY7hGePHmyVatWP/zwg1qtjo2NFQtTUlKmTZtWjbE9EfQIAQDAGpYT4aRJk55//vlbt24tXrzYVNi3b9/bt2+np6dXV2xPRNBhNj0AAJTPwqXR7Ozsa9eubdiwQaVSmS9XKo4jTU1NrVu3bvUF+LiwrAwAAFjDQo/QYDAQkekhgiaZmZn0aI2Ymo/nMIkQAADKZyERent7+/r67tmzh/79AIstW7a4uLg0bNiw+qJ7AoJOgx4hAACUy8KlUYZhZs2a9e6776rV6uDgYIPBcPbs2Z07d65evXrevHlPy+NCMFgGAACsYXn6xDvvvJOXl7dkyRLx8e6dOnWSSqVTpkyZM2dO9Yb3+NAjBAAAa5Q6j/D//u//Jk6cePr06YyMDBcXl86dOwcEBOj1+qflSco8p8M8QgAAKJflRLh8+fJ3333X09PzxRdfNBUajcb//Oc/33//fXXF9kQETssqHWwdBQAA1HSW5xEuX7589erV5iU8z48ZM2b//v3VElUlwPQJAACwhuVE+Pnnn0+bNu2nn34S3wqCMGnSpJ07d+7evbsaY3sieCovAABYw/Kl0REjRsTHx48aNerYsWMdO3Z87733vvnmmx9//LFfv37VHN9j4zkti6fyAgBAeUodLDNr1qykpKQXX3xxyJAhmzZt2rFjx8CBA6szsieEHiEAAFij1ERIRKtWrUpPT//mm2+2bNny0ksvVVtMlQL3CAEAwBr/JML9+/cvXLiw2GaNRqNSqVasWLFixQqx5NKlS9UX3RMQdDr0CAEAoFz/JEK5XO7u7l5ss7u7u5+fX/WGVDl4DhPqAQCgfP8kwj59+vTp08eGoVQuQadlscQaAACUx/L0CTsgoEcIAABWKDUR3rp16+233+7QoUNgYGDbtm3HjBkTExNTnZE9IUGHRbcBAKB8lhPh+fPnW7VqtXHjRoVC0alTJ1dX1127drVt2/bAgQPVHN/jEYwGQRAYSVljYgEAAKi06ROTJ08ODQ09ePBgYGCgWJKZmTl8+PAJEyYkJibW/HW3BQ43CAEAwCoWeoS5ubmXL1/+/PPPTVmQiOrUqbNmzZqUlJSbN29WY3iPSdBpGbnC1lEAAMBTwEIiFJ9BWHIqhaenp2lrDYen8gIAgJUsJEJvb29vb++1a9cWK1+3bp1CoWjQoEG1BPZEsKwMAABY6V/3CCdNmtSrV6/BgwfPnj172rRpN2/ejIyM9PPzy8zMPHDgwMGDB2fOnOnk5GSrWK3H6zR4Ki8AAFjjX4nwwIEDAQEBgwcPnjp1KsMwixcvnjJlirjJzc1twYIFc+bMsUWQFYZLowAAYKVSJxi88847kyZNiouLy8rKcnNza9SokUwmq87InoSgw6VRAHg63Lx584cffpg+fbqPj49YcubMmf3797/88sstW7YUS+7evbt+/fq33347ODi4tHamTJkSExNz8uTJsg+XkpISEBBw6tSpzp07l9yakZHBMIy3t7eVwScnJ1+6dCkmJkaj0cyaNcvNzc206fDhw1FRUeaVJ02aZD4G01xOTs6nn376559/enp6jh8/vmvXrlYGUCnKmmknkUgaN25cbaFUIp7TslhxGwCeBk5OTh9//HHTpk1HjRollmzYsGHLli08z5sS4d69e5cvXz579uwy2mnRooWrq+sTBjNhwgSVSrV9+3Yr67ds2TIvL69OnTppaWlvv/22eSI8efLkunXrevXqZSrR6XQWGykqKurcuTPHcW+++ebVq1eff/75H3/8cfDgwU/yQSqkeCI8ffr0Z599VsYOM2bMqMp4Koegw/pqAPB08Pf3r1+//u+//25KhFFRUREREb///rupTlRU1LPPPuvi4mIqMRqNxaZ0jxs3rmTjJauZ43meZZ9ooc1ff/21cePGBw8ejIyMLLk1JCRk165d5Tby1Vdf3b59+++//65Xrx4RabXa6dOnDxo06AljqwDBTFBQUIXqV5158+YtXLjQyso8zxcWFpqX5B/bmbtvQxXEZSfy8/NtHcLTRK1W6/V6W0fxNMEXrELy8/PHjRtXv3598W1SUhLDMOfPn5dKpXl5eYIgGI1GT0/P999/X6ywatWqkJAQIvL19Z04caJGoxHLFy1a9Oqrr5qaXbp0qYeHBxG1adPm4MGDoaGhsbGxgiAkJycT0c6dO/v168eyrIeHx9y5c3meFwRh8uTJjo6OTk5OoaGhoaGhO3bsEATh4sWLX331lVihNLt37yaihIQE88IPPvigZcuW1pyBLl269OnTx/T20KFDRPTnn39arKzVanU6nTXNWq94j3DatGkW/6x4umD6BAA8Rbp167Zhw4Z79+4FBgb+9ttvTZs2bd++va+v75kzZ/r16xcbG5udnd29e3ciWrZs2aJFi1asWNG9e/f4+Pg333xTp9Nt2LCBiDIyMpKSksQGd+3aNXfu3OXLlw8aNCg6OnrSpEkJCQnmVyb/+9//zp8/f/ny5SdOnJgyZUq7du0GDhw4ffr0a9euKRSKxYsXE5F4P3Lv3r2LFy8eP378Y6wpFh8f36JFCyJq3LjxO++807FjR4vVEhISRowYYXrbrFkzsdB0ZbiqFU+E3t7e4eHh1XPsypJamH4p+cqLjfuZSgSdlnXzsmFIT4gXKI8jrZE0RoGICvRk4EkgytUREXE8FRkEsaa4ySSnxBX4XE4QSrTPcVK53EhEnJGKDFXyEUz0PBXqq/YQVc1olLAsMYzR1oE8NQwGuVSK02Utg0Gul3cjosivooJeGHnp+yhJaJfhx43MM51mbI3aJO9968cTjES6Xv/chiPafQsXN3lt3pGQ148kEDEhfmM+27gw8sGQz6RKxz+ThdxcYfhxIxEdn/eJf/cRp8Mnnb5D5Bro1j+R1sz44A/ePc+oyTISkWvP8UdCXj+SStSogUejbTM2/rrNoT9R8C29q4RVfZzbkogol4iM2XX7t5zk9cpvApX+I5B8lSeiSWeMDrf/qXOnyK9un3GK4CZcYc6RX7fu6tK185L9ddv0Lr6zIKSkZRzOdkk8/nBfo86ViBZFpWx3tnREgZ3QUOhheczNY7KHZamz1NmHE06YJ0Ke00prWI8wj6N0jZCpoUytkK6hTC1laoRsHWkMVGgQ8jjSGEhtoBxO0BhIZyQ3BSlYcpAyROQkIxlLROSuICKSs+QoZcRmnWUkNbuK7l5iXTlXOVPyKrtOIIWcISK5hByr+CsgY8npqRlubBnH8RIJI5HY7TPLKp1GY1CpnvL/9Wqk0RhUDfwvBdd3ij8ZGfrqyRsnR763pEMo4/Z816gft0SGMp/cOVm/eZtXmrqk3r35ozrfJSOG++HhqBmH/FzBaOjI3AkObVHgQolKJjKUIaL99/5+6bU3eoU+/EVxt2/nK2voBX8KC2UeODAHiIb0eK7Zo60pYcG8Lk3cMcGR5AqKfLSJiCj0ORrwXNkf4fxN5hxR/yCmjr/ZjtMmml7qp0784MX2mT8smTK85FNvmf1yeWMXg+mgRfn6n4ja+6ueNw/D1JSer+dsofxJ2EMirOPgla19YF4i6LSMjUaNqg30d65wI1e4kSP8lUtJRUKGhrK0gpylug6Mt5K8lIyPinxU1MCV8VSSSkLOMtZFTioJOcrITc6oJKSq4v+WggKDszPmWVpLoxFkMkYqRSK0VkEB7+yM02Ut8XT92qv777//3kGWmpF0Z+bQLr6+bPjQbpsWvtPbo2Bi9Jnx48dH1mNPJWcRUQNPpZfro0zg6t76/fdfauxWrx570oUpUlJkPZbn+deM+ud85ZH1Hv4vXCtSzSLq6ce2qcemyFki6h2i6vxo688uUq3WIFbe7sCoVIxpRysx3gwRDQhkg4NL21F1Y9igL774wmLLc/x9fYwPTJvu3MkhokHN/PpaqqzTEVPJefDfiXDZsmVP43wJL5XHA22uQAJDD09PtT2Vt8hAN3KE6znCXw+TH6VphIauTCM3pqk789ozFOTE+qiojpJR1vQndgCALYm3Cb/77ruGDRv6+voSUePGjT08PNatW5eVldWtWzciEsfI9O7de/jw4WU0xbJsvXr1YmNjTSXWP02WZVme58uvV3FZWVmlTUZv1KjRmTNnTG/F140aNaqKMCz6V74dMWJE8+bNq+3YlUUmkamkyjxdvqmEr+IJ9Tk62nqLH/irwe87/dtnjMdTBXcFM64he6SfpGC07M8h0u+fl8xpyQ4NYdt4MYGOyIIAUI7nn3+eiJYvXy7mPCJiGKZz586ffvqpTCbr1KkTEQUGBvbu3XvBggWpqaliHUEQLly4ULK18ePHb9q06fz580SUlJS0ZMkSK8Pw9fW9c+eOeS48cODApEmTBAuDDcpx9OhRUzvHjx/fsmXLwIEDxbc5OTmTJk06deqU+Pa11167cuXKL7/8QkRFRUUrVqzo2rWrmPWrh51cvvBSeWSps01vq+h5hKb8F7JDvzuej6zHJr0iuzRYuq27ZFYL9sVgtr4LI6nsPjsA1AbibMLMzExTIiSirl27ZmZmtm3b1rTI8+bNmx0dHRs2bPjCCy/06dMnJCRk2LBhJVubNGlSnz59OnTo4OPj07Rp09GjRxORSlX+b8WxY8feuXPH29s7LCxs586dRHT+/Pk1a9aU1k3s378/wzDiJMKQkBCGYebPn2/a5OXl1bJly7CwsJ49e7Zu3Xr58uXipoKCgjVr1ly7dk18O2zYsNGjRw8ZMqRnz55NmzbNyMhYt26ddaetctjDPUIi8lR6ZKof1HcPFd9W7oT6bB0dTOJ3x/Mn04SuvkxkPfa751kXDAUAgEr1ww8/ZGZmtmvXzlQyatSo8PBwPz8/U4mvr++FCxd+//33q1evyuXysLAw02pk77zzTmFhofhaoVDs2rXr7t27GRkZ4eHhZ8+epUdXVr28vI4ePdq0aVNTmx988IEpz7Vu3TopKenWrVvZ2dninbI33njjhRdeKG3uxOLFi4utsiJOiieiu3fvnj9/PjU11dXVtUmTJm3btmUe3dzz8fE5ceJEw4YNxbcMw2zatGns2LGXL1/29vYeMGDAky+RUyHMY3R4q8H8+fMlEsm8efOsqSwIwtKzK5v6NB5Y/+F4pPRFY7wmLpF6+j5hGPEFwqSzxnMZQt9AdlgI0y+QdbCLvxwKCgqcnZ1tHcVTQ6PRyGQyqdQu/u+rBb5gFVIVp4vjuDt37oiZLCsra9CgQUQkpsOnnU6nYxhGLpdXYpt28rPtpfTI1vwzcJSvjB7httv8fy8YP2gh+akni5t8APAU0Wq1zZo1CwkJcXZ2jouL8/X13b9/v62DqrnsJBEOCO0lV/4zh07gtE/yPMJ8PU06Y4zOEo70lbb0xE0/AHjKuLi4ZGZmxsTE5OXl1atXr0mTJrikUQY7OTWuchdHlePDN4IgGLjH7hGevy+8FmXs4cf8MVhqHxdCAaAWcnd3F1dlg3LZ4W96Qa9jpPLHmHJp4Ol/V4zr/+bXd5ZGBKEjCABQK9hhIny8G4QJBcKoKKOjlC4Plvk6VEVcAABQE9nJPEIimn58rs7I0WNNItx6i2+31xBZjz3cT4osCABQq9hPjzCjKDNTnRXg7CdwOkZeYvHpUhgFevU341+5wm8DpOHuuBwKAFDr2E+PsI7KU1xcRtBprF9xe+stPkUtXHwRWRAAoJaynx6hl4NnpiabHj6V16pLo1ojLYjmd/SQKDBNEACgtrKjHqGDZ5b6ARHxOg1rXY9wzQ3+WS+mgzf6ggAAtZcd9QhVnqmF6WR1j7BQT59eNR7tZz9nAAAAHoNd9Qgz1Vlk9VN5P4k19glgcWsQAKCWs5/+UB0HzyxNNhHxXPkPI8zU0uob/MXB9vPxAQDg8dhPj9DH0VtvNBCRoNOWe4/wf38aX3uGDXVGdxAAoLazny6Rp8p9Q/8VJE6od3Aqo2ZiofDdbf76MDxREAAA7KhHaFLuU3nnXeYnh7M+lf8EewAAePrYT4/QROC0TOlLrF3LEY4k83HD0R0EAAAiO+sRqvUaPW/gdWU9jHDOJX5WC4kL8iAAABCRnfUIv4n9NsgloFPpo0YvZgox2cKuHlhIBgAAHrKrHqGbwvW+OkvgSp1HOOuicUFrFguqAQCAiV0lwjoOXpnq7NKeR/jLPSFVTaPq29VHBgCAJ2RXWaGOg2eWOtvi8wgFormXjUvbsVK7+sQAAPCk7CoteKk8MtXZFqdP7LjDSxh6MdiuPi8AADw5uxosU8fBK0uTLXDGYolQz9O8y/y6zhIsJAMAAMXYVQ/JQaYioiJj8cEyG27yYS7Uww95EAAAirOrHiEReak8cmRFjOSfz6Ux0OIr/N5eGCoKAAAW2FWPkIi6+7VTyP7VHYx9IPiq6FkvdAcBAMACe0uEr4dFeNO/EmFykRDkhCwIAACW2VsiFHTFFxpNUZO/o63CAQCAms7eEiHPaYotNJpSJPg7oEcIAACW2VsiLFTnxzgZzUuSiygAPUIAAChFVY0aPX369OXLl9PS0iZMmBASEmIqP3HixMaNG4lo7NixPXr0qPTjphSkfO+Q3cu8RC34O9pbvgcAgMpSVRliwoQJ0dHRX3zxRXJysqnw4sWLQ4YMeeGFF3r27Dl06NALFy5U+nE9BPkD1mBeklJE/g6VfhwAALATVdUjvHbtGhEdOnTIvHDlypUTJ04cM2YMEd26dWvlypXbt2+v3OO6GCWFZDDwRin7cOJgqlrwc8Q9QgAAsKxarxn+8ccfXbp0EV936dLl4sWLlX4IhtO4MIoH2hzxbY6O5Cw52tuyAQAAUGmqNUWkp6d7eHiIr728vNLS0kqrefPmzYsXL0ZFRYlvJRLJ2rVrfXx8LFYWBEGj0QiCQES6wnwPRnkvO9mBVxJRXC7jp5IVFhZW7gd52hUVFTEMesnW0mg0MplMKsXfU9bCF6xCcLoqRKfTMQwjl8utrK9UKsv94a3Wn20HBwedTie+1mg0Tk5OpdUMDAxUKpWjRo0S37IsGxISIpFYXiZNEASWZR0dHYmIJ95L6lQoqMXGc3KFQGdjGQeqnQRBwDmxnkQiQSKsEHzBKgSnq0JkMlmFEqE1qvVnOyAgIDExUXydmJgYEBBQWk0HB4fQ0NCePXtW9BC8Tusld87SZItvMYkQAADKVq33CCMjI7du3crzPM/zW7dujYyMrPRDCJzWS+GaqX6UCLGsDAAAlKmqEmHfvn3DwsJycnJGjBgRFhaWkJBARG+//XZhYWGrVq1atWqVn58/ceLESj+uoNN0dm3cNbCD+DalSPDHkFEAAChdVV0a3bVrl9H4zwovrq6uROTi4nL27NnY2Fgiat68OctWfhoWOK2/U12VV0PxbUqRMDAIs+kBAKBUVZUIXVxcLJazLNuyZcsqOigR8ZyWNVt0G5dGAQCgbPbWWxJ0WsZs0e2UIiEAl0YBAKB0dpcIOS2jUK66/LXWoNUZKV9PXsry9wIAgFrL7hKhTsPIlZfTY1MLM1LVgq8D5qkCAEBZ7C0R8pyWVSjrqDyz1NlYbhsAAMplb4tlCJyWkau8HDwyNdmFDOZOAABAOewrEQqCYNAzMnkdB88sdXY2oUcIAADlsKtLowKnZWQKYhgvlWemOjtFjR4hAACUw64SIa/TsnIlEdVx8MrSZCcXUQAmEQIAQJns6tKoOHeCiOo4eGSqs1OMgr+DXWV6AACodPaVCHUa5lGPMFebl6LHsjIAAFAOu+owCZyWUaiIyFXhsrrPJ+lqwRfPYAIAgDLZVSLkdRr20fpqEkkdJxkpLT/KFwAA4CG7SoQCpxPvERJRMlYZLxec9AAAHDNJREFUBQAAK9hZIvxnxW0sKwMAANawr0T4aLAMEf169wdH/lfbxgMAADWfXSVC84cRFuolckq1bTwAAFDz2VUiNO8RFhk9Wcq2bTwAAFDz2VciNLtHmGPwMBgf2DYeAACo+ewuET66NJqh9VRz6BECAEA57CoR8pyOlSvE1/fU7gW6HF4QbBsSAADUcHaVCAWdRpxHqDaQlpc6yhzydPm2DgoAAGo0+0qEnJaRq4gopUjwd2C8HDyzNLg6CgAAZbGvRPioR5iiJn9HesYj1MgbbR0UAADUaHb19Ame07JmPcJZz71j64gAAKCms7Me4cPnEeKRvAAAYCX7SoSP5hGmqAV/rLgNAABWsK9E+KhHKK64zQt8Un6KrYMCAIAaza4SIc9pGZmCHvUI7+WnzD+11NZBAQBAjWY/g2UEPcewLCORktgjdCQ3mXumGtMnAACgLPbTIzTdIOQFuq8R6qoYZ7mTntdrDVpbhwYAADWXfSVChYqIMjTkqSQZS0TkqfLI0mDpbQAAKJX9JEJe97BHmFwk+Ds8HDJaR+WZhaujAABQOvtJhAKnZRXF5054OXhmYpU1AAAonR0lwkdP5RVHyojqOHhmqXFpFAAASmVHidA0m97s0qiXyhMDRwEAoAz2M32C57SsQkVEKWrq4fewsKVPuIvCyZZhAQBAzWY/iVDQmfcIH/Z067uH1ncPtWlcAABQo9nXpVGzZzABAABYw44S4T+DZYQArLgNAADWsZ9EyHNaRqHM1xMROcv+Kf/l7nGtQWerqAAAoIazn0QocFpWrkopKv4Apj1xBxPykmwVFQAA1HB2lAh1WkahFB/AZK4OZlAAAEDp7CgRchpGriz5SF4vB88sLC4DAAClsJ9EyOt0rEKZXEQB/x4yiuVGAQCgDPaTCB/2CM2WlRF5OXhm4gEUAABQCjtKhDoto1CVnERYx8EzU51lo6AAAKCms6NEyGkf9gj/fY8wwNkvPjcRj+cFAACL7CcR8qVcGvVxrPNs3ZYH7xy1VWAAAFCT2dVao7xMlcORt6r4pnfbTTTwBlsEBQAANZ0dJUJOm25QeCsZSYnl1RxkJXIjAAAAEdnPpVGeF4yGFE6G5bYBAKBC7CQRCnodI1OkqMm/9OW2i/TqqKQz1RkVAADUfPaSCDktq1CVXF+tmBV/rE3KT6muoAAA4ClgJ4mQxLkTJdZXM+cocxjReMg3Md9WZ1wAAFDD2UkiFDgtY0WP8KWGETeybt7IulldcQEAQE1nN4lQV26PkIjkEvl/mo1Y9+eWagsMAABqOLtJhFqLK26XNCCsV44u73J6TLXEBQAANZ2dJELS6xi5Mk0t+DqU1SMkIpZhxzR7Zd2VLQIJ1RMaAADUZHaSCAWdhpMqFRJytGKFgO7BnUigU/fOV31cAABQ09nJyjKCXqdmVf6qcrqDIoaYmc9NxjLcAABA9pMIddoCRmn9sjLPuIdWZTgAAPDUsJNLo6TX5ZIioMwhowAAACXZSSIUOO0DXln2JEILe5GQpc6umogAAODpYC+JUKfNFBRlTyIsKVuTM+6X6UV6dRVFBQAANZ+9JEK9NtNQ4UTopfJ4zr/Nzr/2VFFUAABQ89lLItRpUw2qil4aJaIxzV7ZG3f4wO0jVRAUAAA8BewkEZJel1zxHiER+TjWWdX7o723Ds89uSRfV1AVoQEAQE1mJ4nQqNM+4JVeysfZN8glYG3fT8Pc6o05OOVC6uXKDg0AAGo0O0mEep3O0cG66fSWSBjJmOavzO44/dMLq88kX6zMyAAAoGazkwn1Rq3Wxemx+oNmnq3bYtvANSxjJ38cAACANewkEZJe6+akevJmlNInzaYAAPB0sZPeD8NpvVwqOYd9E/Pd3JNLcnV5ldssAADUKHaSCCVGnbdLJfQIzY1u9nKQS8AbB6euv7L15oPblds4AADUEPZwaVTQczwr9XOq5KQuZSVvtvxPz5BuxxN+//D0p0bB2C2wY7egTo29nmEIi5oCgB2Kjo52dnZ+5plnbB1ItbKHHqHAadWswr+8R/I+nlC34PEt//PdoLX/6zpbIZUvPb9yTfSmqjgQANRm+fn5CxYsCA8Pd3BwqFu3bvfu3bdt22YwGKo5jGnTpn3xxRfi62PHjh09erTSD8Fx3Pvvv+/r66tSqZ5//vlr166VVlMQhE8//TQkJESpVLZp0+b333+v9GBE1d0jXLNmzfr164nozTffnDhxYqW0yes0alYZYPUzmB5Pffd69d3rjW3+qnnhuZRLaoMm2MU/2CVQJpFVbQQAYKeys7O7deuWlZX17rvvtm/fXhCEs2fPzpo1S6FQDB8+vDojmTp1qpeXl/j666+/NhqNvXr1qtxDvPfee19//fWmTZuCg4PnzJnTs2fP69eve3p6lqy5atWq999/f9WqVe3bt//ss8/69esXHR1dr169yo2HqjkRHjx48H//+9/evXsZhhk0aFBQUFBERMSTN8tz2gJG1axqeoRlU+vVp5LPb8u7l1qQ5uXgGeoWHOwSEOwaFOYeEuYWUv3xAMDT6J133klNTb18+bLpt3y3bt0mTpyYkZEhvtVoNNevXy8oKAgODg4N/edxqnl5ebm5ucHBwXfv3r179+4zzzwTHBxs3nJCQkJiYqKjo2OjRo2cnJzMNwmCcOPGjbS0NA8PjxYtWkgkEiLq3r27TCYjovv37xcWFvI8f/fuXSLy8vIqLCyUSqXe3t6mFoqKijIyMoKDg8V9rZGVlbVmzZpFixaJCX737t1+fn7r1q2bPXt2sZpGo3Hx4sXjxo0Tu0ybN28+ffr0J598smbNGiuPVQFCNYqIiFiyZIn4eunSpQMGDCit5rx58xYuXGhls2l/3/jlgymVEN8TMPCGxLzkqMQzW6/uXHhq2cQjM4s4tbiJM3BfRW/aHPv9z3GHTiadi71/IykvuZArsmG0+fn5Njz6U0etVuv1eltH8TTBF6xC4uPjJRLJ3LlzS6uwf/9+BwcHLy+v4OBgqVTatWvXvLw8cdPnn3/u6+s7bdo0Z2dnPz8/lmXNf3N26NBBoVC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]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Petri\n",
"using LabelledArrays\n",
"using Plots\n",
"using OrdinaryDiffEq\n",
"\n",
"# Define the states (places in Petri net)\n",
"states = [\n",
" :item0, :item1, :item2, :item3, :weight, \n",
" :value, :capacity\n",
"]\n",
"\n",
"# Define the transitions (steps in Petri net)\n",
"transitions = LVector(\n",
" txn0=(LVector(item0=1, capacity=2), LVector(weight=2, value=10)),\n",
" txn1=(LVector(item1=1, capacity=4), LVector(weight=4, value=10)),\n",
" txn2=(LVector(item2=1, capacity=6), LVector(weight=6, value=12)),\n",
" txn3=(LVector(item3=1, capacity=9), LVector(weight=9, value=18))\n",
")\n",
"\n",
"# Create the Petri net model\n",
"knapsack_model = Petri.Model(states, transitions)\n",
"\n",
"# Define a function to create rates\n",
"function create_rates(success_rate)\n",
" return LVector(\n",
" txn0=success_rate, txn1=success_rate, \n",
" txn2=success_rate, txn3=success_rate\n",
" )\n",
"end\n",
"\n",
"# Define the initial state\n",
"initial_state = LVector(\n",
" item0=1, item1=1, item2=1, item3=1, \n",
" weight=0, value=0, capacity=15\n",
")\n",
"\n",
"# Define the simulation function\n",
"function simulate_petri_net(model, initial_state, rates, time_max)\n",
" tspan = (0.0, time_max)\n",
" prob = ODEProblem(model, initial_state, tspan, rates)\n",
" return solve(prob, Tsit5(), saveat=0.1)\n",
"end\n",
"\n",
"# Simulation parameters\n",
"time_max = 5.0\n",
"success_rates = [1] # Different success rates for testing\n",
"\n",
"# Run simulations\n",
"solutions = Dict()\n",
"for rate in success_rates\n",
" rates = create_rates(rate)\n",
" sol = simulate_petri_net(knapsack_model, initial_state, rates, time_max)\n",
" solutions[rate] = sol\n",
"end\n",
"\n",
"# Plot results\n",
"plot(title=\"Knapsack Simulation\", xlabel=\"Time\", ylabel=\"Tokens\", lw=2)\n",
"for (rate, sol) in solutions\n",
" plot!(sol.t, sol[5, :], label=\"Weight\")\n",
" plot!(sol.t, sol[6, :], label=\"Value\")\n",
" plot!(sol.t, sol[7, :], label=\"Capacity\", linestyle=:dash)\n",
"\n",
" # values at end of simulation rounded to 3 decimal places\n",
" weight = round(sol[5, end], digits=3)\n",
" value = round(sol[6, end], digits=3)\n",
" capacity = round(sol[7, end], digits=3)\n",
"\n",
" annotate!(4, sol[5, end], text(\"Weight: $weight\", 10, :left))\n",
" annotate!(4, sol[6, end], text(\"Value: $value\", 10, :left))\n",
" annotate!(4, sol[7, end], text(\"Capacity: $capacity\", 10, :left))\n",
"end\n",
"\n",
"# Display the plot no legend\n",
"plot!(legend=false)\n"
]
},
{
"cell_type": "markdown",
"id": "e9592b20-83c1-462d-aff5-00ebb818baeb",
"metadata": {},
"source": [
"This is a python implementation that calculates an optimal solution\n",
"```python3\n",
"class Item:\n",
"\tdef __init__(self, value, weight, idx):\n",
"\t\tself.value = value\n",
"\t\tself.weight = weight\n",
"\t\tself.idx = idx\n",
"\n",
"class Node:\n",
"\tdef __init__(self):\n",
"\t\tself.ub = 0 # Upper bound\n",
"\t\tself.lb = 0 # Lower bound\n",
"\t\tself.level = 0 # Level in the decision tree\n",
"\t\tself.flag = False # Flag to indicate if the item is selected or not\n",
"\t\tself.tv = 0 # Total value of selected items\n",
"\t\tself.tw = 0 # Total weight of selected items\n",
"\n",
"def assign(a, ub, lb, level, flag, tv, tw):\n",
"\t# Helper function to assign values to a Node object\n",
"\ta.ub = ub\n",
"\ta.lb = lb\n",
"\ta.level = level\n",
"\ta.flag = flag\n",
"\ta.tv = tv\n",
"\ta.tw = tw\n",
"\n",
"def upper_bound(tv, tw, idx, arr, capacity):\n",
"\t# Calculate the upper bound of the current node\n",
"\tvalue, weight = tv, tw\n",
"\tfor i in range(idx, len(arr)):\n",
"\t\tif weight + arr[i].weight <= capacity:\n",
"\t\t\tweight += arr[i].weight\n",
"\t\t\tvalue -= arr[i].value\n",
"\t\telse:\n",
"\t\t\tvalue -= ((capacity - weight) / arr[i].weight) * arr[i].value\n",
"\t\t\tbreak\n",
"\treturn value\n",
"\n",
"def lower_bound(tv, tw, idx, arr, capacity):\n",
"\t# Calculate the lower bound of the current node\n",
"\tvalue, weight = tv, tw\n",
"\tfor i in range(idx, len(arr)):\n",
"\t\tif weight + arr[i].weight <= capacity:\n",
"\t\t\tweight += arr[i].weight\n",
"\t\t\tvalue -= arr[i].value\n",
"\t\telse:\n",
"\t\t\tbreak\n",
"\treturn value\n",
"\n",
"def solve(arr, capacity):\n",
"\t# Main function to solve the 0/1 Knapsack problem using Branch and Bound\n",
"\n",
"\t# Sort items by value-to-weight ratio in descending order\n",
"\tarr.sort(key=lambda x: x.value / x.weight, reverse=True)\n",
"\n",
"\t# Initialize the root node\n",
"\tcurrent = Node()\n",
"\tcurrent.tv = current.tw = current.ub = current.lb = 0\n",
"\tcurrent.level = 0\n",
"\tcurrent.flag = False\n",
"\n",
"\t# Initialize variables to track the best solution\n",
"\tmin_lb = 0\n",
"\tfinal_lb = float('inf')\n",
"\tcurrent.tv = current.tw = current.ub = current.lb = 0\n",
"\tcurrent.level = 0\n",
"\tcurrent.flag = False\n",
"\n",
"\t# Lists to track the current and final selected items\n",
"\tcurr_path = [False] * len(arr)\n",
"\tfinal_path = [False] * len(arr)\n",
"\n",
"\t# Priority queue for node exploration\n",
"\tpq = []\n",
"\tpq.append(current)\n",
"\n",
"\twhile pq:\n",
"\t\tcurrent = pq.pop(0)\n",
"\n",
"\t\t# Prune the node if its upper bound is greater than the current minimum lower bound\n",
"\t\tif current.ub > min_lb or current.ub >= final_lb:\n",
"\t\t\tcontinue\n",
"\n",
"\t\tif current.level != 0:\n",
"\t\t\tcurr_path[current.level - 1] = current.flag\n",
"\n",
"\t\t# Check if the current node is a leaf node\n",
"\t\tif current.level == len(arr):\n",
"\t\t\t# Update the final solution if the current lower bound is better\n",
"\t\t\tif current.lb < final_lb:\n",
"\t\t\t\tfor i in range(len(arr)):\n",
"\t\t\t\t\tfinal_path[arr[i].idx] = curr_path[i]\n",
"\t\t\t\tfinal_lb = current.lb\n",
"\t\t\tcontinue\n",
"\n",
"\t\tlevel = current.level\n",
"\n",
"\t\t# Explore right child (item is not included in the knapsack)\n",
"\t\tright = Node()\n",
"\t\tright.ub = upper_bound(current.tv, current.tw, level + 1, arr, capacity)\n",
"\t\tright.lb = lower_bound(current.tv, current.tw, level + 1, arr, capacity)\n",
"\t\tassign(right, right.ub, right.lb, level + 1, False, current.tv, current.tw)\n",
"\n",
"\t\t# Explore left child (item is included in the knapsack)\n",
"\t\tleft = Node()\n",
"\t\tif current.tw + arr[current.level].weight <= capacity:\n",
"\t\t\tleft.ub = upper_bound(\n",
"\t\t\t\tcurrent.tv - arr[level].value,\n",
"\t\t\t\tcurrent.tw + arr[level].weight,\n",
"\t\t\t\tlevel + 1,\n",
"\t\t\t\tarr,\n",
"\t\t\t\tcapacity\n",
"\t\t\t)\n",
"\t\t\tleft.lb = lower_bound(\n",
"\t\t\t\tcurrent.tv - arr[level].value,\n",
"\t\t\t\tcurrent.tw + arr[level].weight,\n",
"\t\t\t\tlevel + 1,\n",
"\t\t\t\tarr,\n",
"\t\t\t\tcapacity\n",
"\t\t\t)\n",
"\t\t\tassign(\n",
"\t\t\t\tleft,\n",
"\t\t\t\tleft.ub,\n",
"\t\t\t\tleft.lb,\n",
"\t\t\t\tlevel + 1,\n",
"\t\t\t\tTrue,\n",
"\t\t\t\tcurrent.tv - arr[level].value,\n",
"\t\t\t\tcurrent.tw + arr[level].weight\n",
"\t\t\t)\n",
"\t\telse:\n",
"\t\t\t# If adding the current item exceeds the capacity, set bounds to 1\n",
"\t\t\tleft.ub = 1\n",
"\t\t\tleft.lb = 1\n",
"\n",
"\t\tmin_lb = min(min_lb, left.lb, right.lb)\n",
"\n",
"\t\t# Add valid children to the priority queue for further exploration\n",
"\t\tif min_lb >= left.ub:\n",
"\t\t\tpq.append(left)\n",
"\t\tif min_lb >= right.ub:\n",
"\t\t\tpq.append(right)\n",
"\n",
"\t# Print the selected items and maximum profit\n",
"\tprint(\"Items taken into the knapsack are\")\n",
"\tmax_profit = -final_lb\n",
"\tfor i in range(len(arr)):\n",
"\t\tif final_path[i]:\n",
"\t\t\tprint(\"1\")\n",
"\t\telse:\n",
"\t\t\tprint(\"0\")\n",
"\tprint(\"Maximum profit is\", max_profit)\n",
"\n",
"# Main function\n",
"def main():\n",
"\t# Example usage\n",
"\tcapacity = 15\n",
"\tarr = [\n",
"\t\tItem(10, 2, 0),\n",
"\t\tItem(10, 4, 1),\n",
"\t\tItem(12, 6, 2),\n",
"\t\tItem(18, 9, 3),\n",
"\t]\n",
"\n",
"\tsolve(arr, capacity)\n",
"\n",
"if __name__ == \"__main__\":\n",
"\tmain()\n",
"```\n",
"Output:\n",
"```\n",
"Items taken into the knapsack are\n",
"0\n",
"1\n",
"0\n",
"1\n",
"Maximum profit is 38\n",
"```"
]
}
],
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@stackdump
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Read part 2 to see an analysis comparing the result when suppressing 1 item at a time

https://gist.github.com/stackdump/af151355d414491ac2eff9160e134892

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Python code referenced from https://www.geeksforgeeks.org/0-1-knapsack-using-least-count-branch-and-bound/

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Screenshot 2025-01-04 at 9 54 42 AM

Above is a model of the nodes in the branch-and-bound approach

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