{
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  {
   "cell_type": "markdown",
   "id": "a07e739f",
   "metadata": {},
   "source": [
    "# Knapsack Problem\n",
    "\n",
    "In the knapsack problem, we are given the set of items $N = \\{ 0, ..., n-1 \\}$ with weights $w_i$ and profits $p_i$ for $i \\in N$ and a knapsack with capacity $c$.\n",
    "We want to maximize the total profit of the items in the knapsack.\n",
    "\n",
    "## DP Formulation\n",
    "\n",
    "Let $V(r, i)$ be the maximum profit of selecting items to pack from $\\{ i, ..., n - 1 \\}$ into a knapsack with capacity $r$.\n",
    "The DP formulation is as follows:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\text{compute } & V(c, 0) \\\\\n",
    "    & V(r, i) = \\begin{cases}\n",
    "        \\max\\{ p_i + V(r - w_i, i + 1), V(r, i + 1) \\} & \\text{if } i < n \\land r \\geq w_i \\\\\n",
    "        V(r, i + 1) & \\text{if } i < n \\land r < w_i \\\\\n",
    "        0 & \\text{otherwise.}\n",
    "    \\end{cases}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "319326c4",
   "metadata": {},
   "source": [
    "## Install DIDPPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "86a40ebc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: didppy in /home/kuro/tidel/code/didp-rs/didppy/.venv/lib/python3.10/site-packages (0.7.1)\r\n"
     ]
    }
   ],
   "source": [
    "!pip install didppy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1348cd06",
   "metadata": {},
   "source": [
    "## Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bb78b2a6",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 4\n",
    "weights = [10, 20, 30, 40]\n",
    "profits = [5, 25, 35, 50]\n",
    "capacity = 50"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3dd70db8",
   "metadata": {},
   "source": [
    "## Modeling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4f1905cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "import didppy as dp\n",
    "\n",
    "model = dp.Model(maximize=True, float_cost=False)\n",
    "\n",
    "item = model.add_object_type(number=n)\n",
    "r = model.add_int_var(target=capacity)\n",
    "i = model.add_element_var(object_type=item, target=0)\n",
    "\n",
    "w = model.add_int_table(weights)\n",
    "p = model.add_int_table(profits)\n",
    "\n",
    "pack = dp.Transition(\n",
    "    name=\"pack\",\n",
    "    cost=p[i] + dp.IntExpr.state_cost(),\n",
    "    effects=[(r, r - w[i]), (i, i + 1)],\n",
    "    preconditions=[i < n, r >= w[i]],\n",
    ")\n",
    "model.add_transition(pack)\n",
    "\n",
    "ignore = dp.Transition(\n",
    "    name=\"ignore\",\n",
    "    cost=dp.IntExpr.state_cost(),\n",
    "    effects=[(i, i + 1)],\n",
    "    preconditions=[i < n],\n",
    ")\n",
    "model.add_transition(ignore)\n",
    "\n",
    "model.add_base_case([i == n])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2608cba4",
   "metadata": {},
   "source": [
    "## Solving with ForwardRecursion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5f5f3122",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solver: ForwardRecursion from DIDPPy v0.7.1\n",
      "pack 1\n",
      "pack 2\n",
      "profit: 60\n"
     ]
    }
   ],
   "source": [
    "solver = dp.ForwardRecursion(model)\n",
    "solution = solver.search()\n",
    "\n",
    "for i, t in enumerate(solution.transitions):\n",
    "    if t.name == \"pack\":\n",
    "        print(\"pack {}\".format(i))\n",
    "\n",
    "print(\"profit: {}\".format(solution.cost))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "563920fc",
   "metadata": {},
   "source": [
    "## Solving with CABS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5118404b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "pack 1\n",
      "pack 2\n",
      "\n",
      "Profit: 60\n"
     ]
    }
   ],
   "source": [
    "solver = dp.CABS(model, quiet=True)\n",
    "solution = solver.search()\n",
    "\n",
    "for i, t in enumerate(solution.transitions):\n",
    "    if t.name == \"pack\":\n",
    "        print(\"pack {}\".format(i))\n",
    "\n",
    "print(\"\")\n",
    "print(\"Profit: {}\".format(solution.cost))"
   ]
  }
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