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    "# Sabrina Revisited\n",
    "## Analysis of the problem model\n",
    "\n",
    "Spellman´s Ltd is a company that produces two types of chilling soft drinks: A, and B. Both beverages use a semi-elaborate C, another expensive ingredient D and other ingredients that are not relevant for production planning.\n",
    "\n",
    "The selling price of drink A is 3€/liter and the selling price of drink B is 2€/liter.\n",
    "\n",
    "1 liter of drink A uses 3 grams of ingredient D. A liter of drink B uses 1 gram of ingredient D. There are only 3 grams of ingredient D available per day.\n",
    "\n",
    "The factory only has one mixer to elaborate both drink types and the semi-elaborate. It takes 1 hour to process a liter of drink A, 1 hour to process 1 liter of drink B, and 1 hour to process 1cl of semi-elaborate C. The mixer is available 6 hours per day.\n",
    "\n",
    "Drink A uses 2cl of semi-elaborate C and drink B uses 1cl of semi-elaborate C. The company has 3cl of semi-elaborate C plus the amount they decide to produce every day.\n",
    "\n",
    "The following problem model provides the optimal daily production:\n",
    "\n",
    "**Decision variables**:\n",
    "- $x_A$: Production of drink A in liters\n",
    "- $x_B$: Production of drink B in liters\n",
    "- $x_C$: Production of semi-elaborate C in centiliters\n",
    "\n",
    "$x_A, x_C, x_B \\in \\mathbb{R}$\n",
    "\n",
    "**Objective function**\n",
    "\n",
    "$\\max z = 3*x_A + 2*x_B$\n",
    "\n",
    "z is the profit in euros.\n",
    "\n",
    "**Constraints**\n",
    "\n",
    "- Availability of ingredient D in grams:\n",
    "\n",
    "$3*x_A + x_B \\leq 3$\n",
    "\n",
    "- Availability of mixer in hours:\n",
    "\n",
    "$x_A + x_B + x_C \\leq 6$\n",
    "\n",
    "- Availability of semi-elaborate C in centiliters:\n",
    "\n",
    "$2*x_A +  x_B -x_C  \\leq 3$\n",
    "\n",
    "- Logical constraint\n",
    "\n",
    "$ x_A, x_C, x_B \\geq 0$\n",
    "\n",
    "\n",
    "A solver provides the following solution:\n",
    "\n",
    "Optimal\n",
    "\n",
    "Total profit is 6.00 €\n",
    "\n",
    "The following table shows the decision variables:\n",
    "\n",
    "| j | Variables             | Solution | Reduced cost | Objective Coefficient | Objective Lower bound | Objective Upper bound |\n",
    "|:--|:----------------------|---------:|-------------:|----------------------:|----------------------:|----------------------:|\n",
    "| A | Production A (Liters) |        0 |           -3 |                     3 |                  -inf |                     6 |\n",
    "| B | Production B (Liters) |        3 |            0 |                     2 |                     1 |                   inf |\n",
    "| C | Production C (Liters) |        3 |            0 |                     0 |                    -0 |                   1.5 |\n",
    "\n",
    "The following table shows the constraints:\n",
    "\n",
    "| j | Constraint                              | Slack | Shadow Price | Right Hand Side | Min RHS | Max RHS |\n",
    "|--:|:----------------------------------------|------:|-------------:|----------------:|--------:|--------:|\n",
    "| 0 | Availability of ingredient D (Liters)   |     0 |            2 |               3 |       0 |     4.5 |\n",
    "| 1 | Availability of mixer (hours)           |     0 |            0 |               6 |       3 |     inf |\n",
    "| 2 | Availability of semi-elaborate (Liters) |     3 |            0 |               3 |       0 |     inf |\n",
    "\n",
    "Answer the following analysis questions:\n",
    "\n",
    "1. Operational problems in the mixer, like mechanical problems,  could reduce the available mixer. Based on the provided solution, how is the company able to deal with this type of problem? Motivate your response.\n",
    "\n",
    "2. The managers of the company want you to negotiate with a new supplier that could increase your daily supply of ingredient D. What would be your terms for negotiating with the new supplier,    ensuring that the basic variables remain the same?\n",
    "\n",
    "3. The managers of the company would like to review the prices of both products. Analyse how changes in the prices of both products could affect the production plan and provide useful feedback to guide the revision of the prices\n",
    "\n",
    "\n",
    "## Duality\n",
    "\n",
    "1. Write down the dual problem.\n",
    "\n",
    "## Problem Extension\n",
    "1. Spellman's Ltd wants to evaluate the possibility of subcontracting part of the production of semi-elaborate C to an external provider, at a cost of 0.35€ per cl. With the new set-up, the daily availability of semi-elaborate C will be the sum of the semi-elaborate C produced at Spellman's plus the amount purchased from the new provider. What changes are needed in the problem to evaluate the subcontracting of semi-elaborate C? (1.5 points)\n",
    "\n"
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