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    "# Manufacturing Solar Panels Revisited\n",
    "## Problem Definition\n",
    "Consider the following solution to a daily production mix optimization problem where:\n",
    "\n",
    "- The decision variables are the number of units to produce of three different types of products\n",
    "- The objective function coefficients (Objective coefficient in table) are the unitary profits (difference between production costs and sell price per unit) in euros\n",
    "- The three constraint represent the total availability of three sections in minutes\n",
    "\n",
    "**Decision variables**\n",
    "\n",
    "| Decision variable | Solution | Reduced cost | Objective Coefficient | Lower bound | Upper bound |\n",
    "|:------------------|---------:|-------------:|----------------------:|------------:|------------:|\n",
    "| product 1 units   |      300 |            0 |                    30 |       24.44 |         inf |\n",
    "| product 2 units   |    33.33 |            0 |                    20 |          -0 |          90 |\n",
    "| product 3 units   |        0 |        -8.33 |                    40 |        -inf |       48.33 |\n",
    "\n",
    "**Constraints**\n",
    "\n",
    "| Constraint              |   Right Hand Side |   Shadow Price |   Slack |   Min RHS |   Max RHS |\n",
    "|:------------------------|------------------:|---------------:|--------:|----------:|----------:|\n",
    "| Section 1 availability  |               400 |           6.67 |       0 |       300 |       525 |\n",
    "| Section 2  availability |               600 |          11.67 |       0 |         0 |       800 |\n",
    "| Section 3  availability |               600 |              0 |  166.67 |    433.33 |       inf |\n",
    "\n",
    "Answer the following questions. Motivate your response based on the provided results:\n",
    "\n",
    "- Is it profitable to produce units of the three types of products? If any of the products is not profitable, what changes are needed to make it profitable?\n",
    "- In the actual economic context, the production costs are rising. Discuss how the rise of the production costs can affect your profit and identify the maximum unitary production cost increase that your daily production mix can support without changes in the base solution\n",
    "\n",
    "- In order to increase your production capacity, you need to evaluate 2 different improvement proposals from an external consultant. The first one considers a 12% increase of the availability of section 1, and the second a 7.5% increase of the availability of section 2. Which one is more profitable for the company?\n",
    "\n",
    "## Solution\n",
    "- Is it profitable to produce units of the three types of products? If any of the products is not profitable, what changes are needed to make it profitable?\n",
    "\n",
    "Since the reduced cost of product 3 is negative, it is not profitable to produce product 3. To make it profitable, the objective coefficient of product 3 should be increased to at least 48.33.\n",
    "\n",
    "- In the actual economic context, the production costs are rising. Discuss how the rise of the production costs can affect your profit and identify the maximum unitary production cost increase that your daily production mix can support without changes in the base solution\n",
    "\n",
    "If the production costs increase, we can increase the selling costs ensuring that the resulting unitary profits is within the lower and upper bounds of the objective function coefficients. Note that in this case, product 2 provides the smaller margin to modify the selling price, since the difference between the upper and lower bounds is 90.\n",
    "\n",
    "We might also consider keeping the selling price constant, and analyze the difference between the objective coefficient and its lower bound to see how much the production costs can increase. In this case, product 1 provides the smaller margin to increase the production costs, since the difference between the objective coefficient and its lower bound is 30 - 24.44 = 5.56.\n",
    "\n",
    "\n",
    "- In order to increase your production capacity, you need to evaluate 2 different improvement proposals from an external consultant. The first one considers a 12% increase of the availability of section 1, and the second a 7.5% increase of the availability of section 2. Which one is more profitable for the company?  **(1 point)**\n",
    "\n",
    "Let us first consider the first proposal. The new availability of section 1 is 400 * 1.12 = 448. The new capacity is within the Minimum and Maximum values for the RHS, so this change is not going to change the base solution. The objective function is going to increase proportionally to the shadow price of section 1, which is 6.67. Therefore, the profit is going to increase by 6.67 * 400 * 0.12 = 6.67 * 48 = 320.16€.\n",
    "\n",
    "In the second proposal, on the other hand, the new availability of section 2 is 600 * 1.075 = 645. The new capacity is within the Minimum and Maximum values for the RHS, so this change is not going to change the base solution either. The objective function is going to increase proportionally to the shadow price of section 2, which is 11.67. Therefore, the profit is going to increase by 11.67 * 45 = 525,15€.\n",
    "\n",
    "Therefore, the second proposal is more profitable for the company.\n",
    "\n"
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