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    "# Production Mix example\n",
    "## Problem Definition\n",
    "Production Mix\n",
    "A company produces two kinds of products, P1 and P2.\n",
    "\n",
    "The selling price of P1 is 300€\n",
    "The selling price of P2 is 250€\n",
    "\n",
    "- P1 requires 2 operator man-hours and 1 hour of machine operating time\n",
    "- P2 requires 1 operator man-hours and 3 hours of machine time\n",
    "- Operator man-hours cannot be higher than 40 hours per day\n",
    "- Machine operating time cannot be higher than 45 hours per day\n",
    "- Marketing states that P1 units per day cannot be higher than 12\n",
    "\n",
    "\n",
    "***How many units of P1 and P2 should the company produce every day to maximise income?*** \n",
    "\n",
    "### Model\n",
    "We want to find the maximum solution to:\n",
    "\n",
    "$\\max Z = 300x_{1} + 250x_{2}$\n",
    "\n",
    "This is our objective function. The decision variables are:\n",
    "\n",
    "$x_{1}:$ units of Product 1\n",
    "\n",
    "$x_{2}:$ units of Product 2\n",
    "\n",
    "The objective function is subject to the following constraints:\n",
    "\n",
    "$2x_{1} + x_{2} \\leq 40$  \n",
    "$x_{1} + 3x_{2} \\leq 45$  \n",
    "$x_{1} \\leq 12$  \n",
    "\n",
    "We define the problem in its standard form as:\n",
    "\n",
    "$max Z = 300x_{1} + 250x_{2} + 0s_{1} + 0s_{2} + 0s_{3}$\n",
    "\n",
    "And the constraints as:\n",
    "\n",
    "$2x_{1} + x_{2} + s_{1} = 40$  \n",
    "$x_{1} + 3x_{2} + s_{2} = 45$  \n",
    "$x_{1} + s_{3} = 12$  \n",
    "\n"
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