# Homework 3 Solution ## BUS 220 Business Decision Making **Professor: Christian de Rojas** **Due Date: October 8, 2025** --- ## 1. Tesla Motors Production Decision Problem ### **Given Data** - **Vehicles:** Model 3, Model S, Model X - **Profits per unit:** - Model 3: $35,000 - Model S: $85,000 - Model X: $90,000 - **Daily production min:** - Model 3: 200 units - Model S: 125 units - Model X: 125 units - **Daily production max:** - Model 3: 438 - Model S: 318 - Model X: 201 - **Company must produce combined total of at least 800 units per day** - **Contractual obligation:** At least 300 Model 3, 150 Model S, and 150 Model X per day - **Marketing department:** Max combined output: **1,400 units per day** Define: - \( x_1 \) = units of Model 3 - \( x_2 \) = units of Model S - \( x_3 \) = units of Model X --- ### **a) Formulate the Linear Programming Model** #### **Objective Function:** Maximize Profit: \[ \text{Maximize } Z = 35000x_1 + 85000x_2 + 90000x_3 \] #### **Constraints:** 1. **Production Limits:** - \( 200 \leq x_1 \leq 438 \) - \( 125 \leq x_2 \leq 318 \) - \( 125 \leq x_3 \leq 201 \) 2. **Combined Output:** - \( x_1 + x_2 + x_3 \geq 800 \) - \( x_1 + x_2 + x_3 \leq 140 \) 3. **Contractual Obligations:** - \( x_1 \geq 300 \) - \( x_2 \geq 150 \) - \( x_3 \geq 150 \) 4. **Non-negativity:** - \( x_1, x_2, x_3 \geq \) **Note:** The Combined Output and Marketing Constraint are combined: \( x_1 + x_2 + x_3 \leq 140 \) Contractual obligations are individual minimums. --- ### **b) Solve for Optimal Production Quantities** #### **Set Up the Problem:** Given the high profits for S and X, prioritize their production within constraints. - **Maximize S and X:** - \( x_2 \) max: 318 - \( x_3 \) max: 201 - Remaining for \( x_1 \): \( 140 - (318 + 201) = 881 \), but upper bound for \( x_1 \) is 438. Therefore, the feasible maximums are: - \( x_1 = 438 \) - \( x_2 = 318 \) - \( x_3 = 201 \) - Combined: \( 438 + 318 + 201 = 957 \leq 140 \) All minimums are satisfied. #### **Optimal Quantities:** \[ x_1 = 438, \quad x_2 = 318, \quad x_3 = 201 \] #### **Maximum Profit:** \[ Z = 35000(438) + 85000(318) + 90000(201) \\ Z = 15,330,000 + 27,030,000 + 18,090,000 \\ Z = \$60,450,000 \] --- ### **c) Sensitivity Report: Reduced Cost and Shadow Price** - **Reduced Cost:** For a variable not at its bound, this is zero. If at its lower/upper bound, it tells how much profit coefficient must improve for the variable to enter solution. - **Shadow Price (Dual Value):** For a constraint, this tells how much the objective would improve if the right-hand side increased by one unit. **Interpretation:** - For constraints that are not binding, shadow price is zero. - For constraints that are binding (e.g., upper limit on x1, x2, x3), the shadow price shows the value of relaxing that constraint. - Reduced cost = for all variables that are positive and not at their limit; otherwise it’s the amount objective would change if variable entered solution. --- ### **d) New Demand for Model 3: At least 475 units** - **Current solution:** \( x_1 = 438 \) - **Required:** \( x_1 \geq 475 \) - **Feasible to increase x1 to 475?** - \( x_1 + x_2 + x_3 = 475 + 318 + 201 = 994 \leq 140 \) - \( x_1 \leq 438 \) (given previously), but now must be at least 475. However, this violates the upper bound for \( x_1 \) (438). So, it’s **not feasible** to produce at least 475 Model 3s without relaxing the upper limit. --- ### **e) New Demand for Model 3: At least 476 units** Same as above, but worse: \( x_1 = 476 \), which is even further above the upper limit (438). **Also not feasible**. --- ## **Final Summary** - **Optimal daily production** for maximum profit is **438 Model 3s, 318 Model S, and 201 Model X**, with a total profit of **$60,450,000**. - **All constraints are satisfied** at this solution. - **Reducing or increasing the minimum requirement for Model 3 to 475 or 476 units is not feasible without changing the upper bound of Model 3**, as it exceeds the allowed maximum of 438. - **Reduced cost and shadow price** provide insight into how changes in constraints or profit coefficients would affect the optimal solution. - This solution ensures both contractual obligations and marketing constraints are met optimally. --- ### **If you have access to Excel Solver or another LP tool, you can verify the exact shadow prices and reduced costs. This answer outlines the mathematical structure and logic for each step as required.**