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    "# Blending Craft Beer"
   ]
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    "## Problem Definition\n",
    "\n",
    "Duff Beer is an american brewery that sells alcoholic beverages worldwide. Duff beer has a very large market share in the US. One of the new business lines of Duff Beer is in the craft beer sector. Duff beer buys 3 types of wort from local suppliers. The availability of wort (in liters) and the price for each of them is specified in Table 1. \n",
    "\n",
    "**Table 1:** daily availability and cost of wort\n",
    "\n",
    "| Wort   | Availability (liters) | Cost (€cents/liter) | \n",
    "|--------|-----------------------|---------------------|\n",
    "| Type 1 | 2500                  | 120                 |\n",
    "| Type 2 | 1200                  | 095                 |\n",
    "| Type 3 | 2000                  | 150                 |\n",
    "\n",
    "The blend of the three types of wort must comply with the following quality requirements: \n",
    "\n",
    "- For Beer Angels, no less than 60% of Type 3 and no more of 20% of Type 2. The price of one bottle of Beer Angel is 4.10€s/liter. \n",
    "\n",
    "- For Beer Beast, no less than 15% of Type 3 and no more than 60% of Type 2. The price of one bottle of Beer Beast is 2.80€s/liter.\n",
    "\n",
    "- For Beer Cactus, no more than 50% of type 2. The selling price is 2.45€/liter.\n",
    "\n",
    "**Write a linear program to find the most profitable blend of wort to elaborate the three kinds of beers**"
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    "## Problem Model\n",
    "### Indices\n",
    "We can define the following indices to express the problem in a compact form:\n",
    "- i: wort type $i \\in [1, 2, 3]$\n",
    "- j: Beer brand $j \\in [A, B, C]$ where A notes Angels, B notes Beast, and C notes Cactus\n",
    "\n",
    "### Decision variables\n",
    "The decision variables are: \n",
    "\n",
    "$x_{ij}$: Amount of wort of type i (1, 2, 3) in Beer j: A (Angels), B (Beast), C (Cactus), (ie i=[1,2,3], j=[A,B,C])\n",
    "\n",
    "\n",
    "### Objective Function\n",
    "The objective function is:\n",
    "\n",
    "$\\max z = \\sum_{i=1}^{3}\\sum_{j=A}^{C}(d_j - c_i)*x_{ij}$\n",
    "\n",
    "where $d_j$ is the selling price of a bottle of beer j and $c_i$ is the cost per liter of wort type i.\n",
    "\n",
    "$\\max z =4 10(x_{1A}+x_{2A}+x_{3A})+280(x_{1B}+x_{2B}+x_{3B})+245(x_{1C}+x_{2C}+x_{3C})-120(x_{1A}+x_{1B}+x_{1C})-95(x_{2A}+x_{2B}+x_{2C})-150(x_{3A}+x_{3B}+x_{3C})$\n",
    "\n",
    "$\\max z = 290x_{1A}+315x_{2A}+260x_{3A}+160x_{1B}+185x_{2B}+130x_{3B}+125x_{1C}+150x_{2C}+95x_{3C}$\n",
    "\n",
    "\n",
    "### Constraints\n",
    "Subject to the following constraints:\n",
    "\n",
    "**Availability constraints**\n",
    "The availability constraints can be expressed in a compact form as: \n",
    "\n",
    "$\\sum_jx_{ij} \\leq  a_i \\forall i$  \n",
    "\n",
    "Where $a_i$ is the availability of wort type i:\n",
    "\n",
    "$x_{1A}+x_{1B}+x_{1C} \\leq  2500$     \n",
    "\n",
    "$x_{2A}+x_{2B}+x_{2C} \\leq  1200$\n",
    "\n",
    "$x_{3A}+x_{3B}+x_{3C} \\leq  2000$\n",
    "\n",
    "**Quality constraints**\n",
    "The proportion of a given type of wort i' in a type of beer j' can be expressed as:\n",
    "\n",
    "$\\frac{x_{i'j'}}{\\sum_i(x_{ij'})} \\forall i', j'$\n",
    "\n",
    "If we compare this to the minimum proportion $R_{min_{i'j'}}$ we obtain:\n",
    " \n",
    "$\\frac{x_{i'j'}}{\\sum_i(x_{ij'})} \\geq R_{min_{i'j'}} \\forall i', j'$\n",
    "\n",
    "Or \n",
    "\n",
    "$x_{i'j'} \\geq R_{min_{i'j'}}*\\sum_i(x_{ij'}) \\forall i', j'$\n",
    "\n",
    "If we take the denominator to the RHS.\n",
    "We can obtain a similar expression for the maximum proportion $R_{max_{i'j'}}$:\n",
    "\n",
    "$x_{i'j'} \\geq R_{max_{i'j'}}*\\sum_i(x_{ij'}) \\forall i', j'$\n",
    "\n",
    "From these expressions, we can derive the constraints for every type of beer: \n",
    "\n",
    "**Angels quality requirement constraints**\n",
    "\n",
    "$x_{3A} \\geq 0.6(x_{1A}+x_{2A}+x_{3A})$\n",
    "\n",
    "$x_{2A} \\leq 0.2(x_{1A}+x_{2A}+x_{3A})$\n",
    "\n",
    "Beast quality requirement constraints\n",
    "\n",
    "$x_{3B} \\geq 0.15(x_{1B}+x_{2B}+x_{3B})$\n",
    "\n",
    "$x_{2B} \\leq 0.6(x_{1B}+x_{2B}+x_{3B})$\n",
    "\n",
    "Cactus quality requirement constraints\n",
    "\n",
    "$x_{2C} \\leq 0.5(x_{1C}+x_{2C}+x_{3C})$"
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