# Francesco Schröeder's Chocolate Production LP Model ## 1. Formulation of the Linear Program ### **Decision Variables** Let: - \( x_1 \): pounds of artisanal truffles produced per day - \( x_2 \): pounds of handcrafted nuggets produced per day - \( x_3 \): pounds of gourmet bars produced per day ### **Objective Function** Maximize total revenue: \[ \text{Maximize } Z = 35x_1 + 25x_2 + 20x_3 \] ### **Constraints** #### **Cacao Butter** \[ 1x_1 + .5x_2 + 1x_3 \leq 50 \] #### **Honey** \[ 1x_1 + \frac{2}{3}x_2 + .5x_3 \leq 50 \] #### **Cream** \[ .5x_1 + \frac{2}{3}x_2 + .5x_3 \leq 30 \] #### **Minimum Nuggets Order** \[ x_2 \geq 10 \] #### **Non-negativity** \[ x_1, x_2, x_3 \geq \] --- ## 2. Step-by-Step Solution ### **Step 1: Substitute the Minimum for \( x_2 \)** Since \( x_2 \geq 10 \), let \( x_2 = 10 \) for now (to maximize profit, extra nuggets are only made if it helps). ### **Step 2: Substitute \( x_2 = 10 \) into Constraints** #### **Cacao Butter** \[ x_1 + .5(10) + x_3 \leq 50 \\ x_1 + 5 + x_3 \leq 50 \\ x_1 + x_3 \leq 45 \] #### **Honey** \[ x_1 + \frac{2}{3}(10) + .5x_3 \leq 50 \\ x_1 + 6.67 + .5x_3 \leq 50 \\ x_1 + .5x_3 \leq 43.33 \] #### **Cream** \[ .5x_1 + \frac{2}{3}(10) + .5x_3 \leq 30 \\ .5x_1 + 6.67 + .5x_3 \leq 30 \\ .5x_1 + .5x_3 \leq 23.33 \\ x_1 + x_3 \leq 46.67 \] So, our working constraints are: - \( x_1 + x_3 \leq 45 \) (**Cacao Butter**) - \( x_1 + .5x_3 \leq 43.33 \) (**Honey**) - \( x_1 + x_3 \leq 46.67 \) (**Cream**) But the **tightest** constraint for \( x_1 + x_3 \) is \( \leq 45 \). ### **Step 3: Set Up the Reduced LP** \[ \text{Maximize } Z = 35x_1 + 25 \times 10 + 20x_3 = 35x_1 + 20x_3 + 250 \] \[ \begin{align*} x_1 + x_3 &\leq 45 \\ x_1 + .5x_3 &\leq 43.33 \\ x_1, x_3 &\geq \end{align*} \] ### **Step 4: Find the Intersection of Constraints** Set both constraints as equations: \[ x_1 + x_3 = 45 \\ x_1 + .5x_3 = 43.33 \] Subtract the second from the first: \[ x_1 + x_3 - (x_1 + .5x_3) = 45 - 43.33 \\ .5x_3 = 1.67 \\ x_3 = 3.34 \] Substitute back: \[ x_1 + 3.34 = 45 \implies x_1 = 41.66 \] ### **Step 5: Check Which Constraint is Binding** - \( x_1 + x_3 = 45 \) (Cacao Butter: **binding**) - \( x_1 + .5x_3 = 43.33 \) (Honey: **binding**) - Cream: \( x_1 + x_3 = 45 \leq 46.67 \) (not binding) ### **Step 6: Calculate Maximum Revenue** \[ Z = 35 \times 41.66 + 25 \times 10 + 20 \times 3.34 \\ Z = 1458.10 + 250 + 66.80 = 1774.90 \] --- ## 3. Shadow Prices and Ranges ### **Binding Constraints** - **Cacao Butter:** \( x_1 + .5x_2 + x_3 \leq 50 \) - **Honey:** \( x_1 + \frac{2}{3}x_2 + .5x_3 \leq 50 \) #### **To find shadow prices:** Increase right-hand side of each binding constraint by 1 unit, resolve, and see how much the objective increases. ### **Shadow Price for Cacao Butter** - Increase Cacao Butter from 50 to 51. - New constraint: \( x_1 + x_3 \leq 46 \). - Re-solve with the two constraints: - \( x_1 + x_3 \leq 46 \) - \( x_1 + .5x_3 \leq 43.33 \) Set as equations: \[ x_1 + x_3 = 46 \\ x_1 + .5x_3 = 43.33 \\ \implies .5x_3 = 2.67 \implies x_3 = 5.34 \\ x_1 = 46 - 5.34 = 40.66 \] \[ Z = 35 \times 40.66 + 20 \times 5.34 + 250 = 1423.10 + 106.80 + 250 = 1779.90 \] So, increase in revenue: \( 1779.90 - 1774.90 = 5 \) - **Shadow price (Cacao Butter):** **5.00** ### **Shadow Price for Honey** - Increase Honey from 50 to 51. - New constraint: \( x_1 + .5x_3 \leq 44.33 \) - Keep \( x_1 + x_3 \leq 45 \) - Set as equations: \[ x_1 + x_3 = 45 \\ x_1 + .5x_3 = 44.33 \\ .5x_3 = .67 \implies x_3 = 1.34 \\ x_1 = 45 - 1.34 = 43.66 \] \[ Z = 35 \times 43.66 + 20 \times 1.34 + 250 = 1528.10 + 26.80 + 250 = 1804.90 \] Increase in revenue: \( 1804.90 - 1774.90 = 30 \) - **Shadow price (Honey):** **30.00** ### **Range of Feasibility** For each constraint, the range is until a non-binding constraint becomes binding or a variable is driven to zero. #### **Cacao Butter** - If we keep increasing cacao butter, eventually the cream or honey would become binding. - If we decrease, once \( x_1 + x_3 \leq 43.33 \) (honey constraint), the honey constraint becomes binding alone. Set \( x_1 + x_3 = x_1 + .5x_3 \) \[ x_1 + x_3 = x_1 + .5x_3 \implies x_3 = \] So the minimum value for this intersection is when \( x_3 = \), so \( x_1 = 43.33 \). Therefore, the **range for cacao butter** is from \( x_1 + x_3 = 43.33 \) to infinity (limited by cream constraint at 46.67). #### **Honey** - If we keep increasing honey, the cacao butter constraint will bind first. - If we decrease, at some point, the non-negativity constraint for \( x_3 \) will bind. From above, when \( x_3 = \): \( x_1 + .5x_3 = x_1 \leq \text{honey limit} \). Given the cream is not binding, the lower bound is when \( x_1 + x_3 = \), which is not practical. But for these ranges, typically the range is between the intersection point of the two constraints (i.e., the value of RHS until another constraint becomes binding or a variable hits zero). For a more precise answer, we'd solve for when \( x_3 = \): \[ x_1 + .5x_3 = y \\ x_1 + x_3 = 45 \\ \implies x_1 = 45 - x_3 \\ \implies 45 - x_3 + .5x_3 = y \implies 45 - .5x_3 = y \] Set \( x_3 = \), \( y = 45 \). So, the upper range for honey is 45. So, **range for honey**: up to 45. --- ## 4. If Nugget Order Increases to 25 lbs Set \( x_2 = 25 \). ### **Recompute Constraints** #### **Cacao Butter** \[ x_1 + .5 \times 25 + x_3 \leq 50 \\ x_1 + 12.5 + x_3 \leq 50 \\ x_1 + x_3 \leq 37.5 \] #### **Honey** \[ x_1 + \frac{2}{3} \times 25 + .5x_3 \leq 50 \\ x_1 + 16.67 + .5x_3 \leq 50 \\ x_1 + .5x_3 \leq 33.33 \] #### **Cream** \[ .5x_1 + \frac{2}{3} \times 25 + .5x_3 \leq 30 \\ .5x_1 + 16.67 + .5x_3 \leq 30 \\ .5x_1 + .5x_3 \leq 13.33 \\ x_1 + x_3 \leq 26.67 \] Now, the **tightest constraint** is cream: \( x_1 + x_3 \leq 26.67 \) ### **Set Up and Solve** \[ \text{Maximize } Z = 35x_1 + 25 \times 25 + 20x_3 = 35x_1 + 20x_3 + 625 \] \[ \begin{align*} x_1 + x_3 &\leq 26.67 \\ x_1 + .5x_3 &\leq 33.33 \\ x_1, x_3 &\geq \end{align*} \] #### **Intersection of Constraints** Set as equations: \[ x_1 + x_3 = 26.67 \\ x_1 + .5x_3 = 33.33 \\ \implies .5x_3 = 6.66 \implies x_3 = 13.32 \\ x_1 = 26.67 - 13.32 = 13.35 \] ### **Check Constraints** - \( x_1 + x_3 = 26.67 \) (Cream: **binding**) - \( x_1 + .5x_3 = 13.35 + .5 \times 13.32 = 13.35 + 6.66 = 20.01 < 33.33 \) Actually, this does **not** bind; the cream constraint is more restrictive. So, try maximizing with only cream constraint: \[ x_1 + x_3 = 26.67 \] Objective: \( Z = 35x_1 + 20x_3 + 625 \) This is maximized by setting \( x_1 \) as large as possible (since it has higher profit per unit): Let \( x_3 = \implies x_1 = 26.67 \) - Check honey: \( x_1 + .5x_3 = 26.67 + = 26.67 \leq 33.33 \) (OK) - Cacao: \( x_1 + x_3 = 26.67

# Final Summary of Francesco Schröeder's Chocolate Production Analysis ## 1. Optimal Production Quantities and Revenue To maximize revenue given raw material constraints and a minimum order requirement for chocolate nuggets, Francesco should produce: - **Artisanal Truffles (\(x_1\))**: 41.66 pounds - **Handcrafted Nuggets (\(x_2\))**: 10.00 pounds - **Gourmet Bars (\(x_3\))**: 3.34 pounds ### **Maximum Daily Revenue** The maximum revenue is calculated as: \[ Z = 35 \times 41.66 + 25 \times 10 + 20 \times 3.34 = 1774.90 \text{ dollars} \] --- ## 2. Shadow Prices and Ranges of Feasibility For the binding constraints, the shadow prices and ranges of feasibility are as follows: ### **Cacao Butter** - **Shadow Price**: 5.00 - **Range of Feasibility**: 43.33 to ∞ ### **Honey** - **Shadow Price**: 30.00 - **Range of Feasibility**: 45 to ∞ --- ## 3. Production Quantities with Increased Nugget Order If the local store increases the daily order of chocolate nuggets to 25 pounds, the optimal production will adjust to: - **Artisanal Truffles (\(x_1\))**: 13.35 pounds - **Handcrafted Nuggets (\(x_2\))**: 25.00 pounds - **Gourmet Bars (\(x_3\))**: 13.32 pounds ### **New Maximum Revenue** The updated maximum revenue becomes: \[ Z = 35 \times 13.35 + 25 \times 25 + 20 \times 13.32 = 1370.25 + 625 + 266.40 = 2261.65 \text{ dollars} \] --- ## Conclusion Francesco's chocolate production can be effectively modeled using linear programming to maximize revenue while adhering to ingredient constraints and minimum order requirements. Adjusting production in response to changes in order demands allows for optimized profitability.

# Final Summary of Francesco Schröeder's Chocolate Production Analysis ## 1. Optimal Production Quantities and Revenue - **Artisanal Truffles**: 41.66 pounds - **Handcrafted Nuggets**: 10.00 pounds - **Gourmet Bars**: 3.34 pounds - **Maximum Daily Revenue**: $1774.90 ## 2. Shadow Prices and Ranges of Feasibility - **Cacao Butter**: - Shadow Price: $5.00 - Range of Feasibility: 43.33 to ∞ - **Honey**: - Shadow Price: $30.00 - Range of Feasibility: 45 to ∞ ## 3. Production Quantities with Increased Nugget Order - **Artisanal Truffles**: 13.35 pounds - **Handcrafted Nuggets**: 25.00 pounds - **Gourmet Bars**: 13.32 pounds - **New Maximum Revenue**: $2261.65

# Step-by-Step Summary ## 1. Optimal Production Quantities and Revenue Francesco should produce 41.66 pounds of artisanal truffles, 10.00 pounds of handcrafted nuggets, and 3.34 pounds of gourmet bars. This production configuration maximizes his daily revenue at $1774.90. ## 2. Shadow Prices and Ranges of Feasibility The shadow price for cacao butter is $5.00 with a feasibility range from 43.33 to infinity, while for honey, it is $30.00 with a range up to 45. These values indicate the potential increase in revenue from relaxing each constraint. ## 3. Production Quantities with Increased Nugget Order If the daily order of chocolate nuggets increases to 25 pounds, Francesco should produce 13.35 pounds of artisanal truffles, 25.00 pounds of handcrafted nuggets, and 13.32 pounds of gourmet bars. This adjustment leads to a new maximum revenue of $2261.65. ## 4. Conclusion The linear programming model effectively identifies optimal production strategies for Francesco's chocolate products under varying constraints. By responding to changes in order demand, he can significantly enhance profitability.