# BUS 220 Business Decision Making – Homework 3 Solution ## Problem Restatement **Tesla Motors produces 3 models: Model 3, Model S, and Model X.** - **Profits per unit:** - Model 3: \$35,000 - Model S: \$85,000 - Model X: \$90,000 - **Daily demands:** - At least 200 Model 3 - At least 125 Model S - At least 100 Model X - **Daily production capacity:** 3.8 hours per Model 3, 3 hours per Model S, 5 hours per Model X - **Total labor hours available per day:** 4,000 - **Daily output limit:** 1,400 units - **Objective:** Maximize profit --- ## (a) **Formulate the Linear Programming Model** Let: - \( x_1 \): Number of Model 3 produced - \( x_2 \): Number of Model S produced - \( x_3 \): Number of Model X produced **Objective Function (maximize profit):** \[ \text{Maximize } Z = 35{,}000x_1 + 85{,}000x_2 + 90{,}000x_3 \] **Subject to:** 1. ** demand constraints:** \( x_1 \geq 200 \) \( x_2 \geq 125 \) \( x_3 \geq 100 \) 2. **Labor hours constraint:** \( 3.8x_1 + 3x_2 + 5x_3 \leq 4,000 \) 3. **Total output constraint:** \( x_1 + x_2 + x_3 \leq 1,400 \) 4. **Non-negativity:** \( x_1, x_2, x_3 \geq \) --- ## (b) **Create Sensitivity Report: State Reduced Cost & Shadow Price** **To generate a sensitivity report, you would use Excel Solver or similar software.** But conceptually: - **Reduced Cost:** For a variable not in the optimal solution (at zero), reduced cost shows how much the profit coefficient must improve before it’s included in the solution. - **Shadow Price:** For a constraint, it represents how much the objective function (profit) would improve if the right-hand side of the constraint increased by one unit. **Assuming you solve the LP in Excel, you would get:** - **Reduced Cost for each variable** (if any are in the solution). - **Shadow Price for each constraint** (especially labor hours and output constraints). --- ## (c) **Interpret the Significance of Reduced Cost and Shadow Price** - **Reduced Cost:** - If a model’s reduced cost is zero, it is already being produced optimally. - If positive (for a maximization problem), it means that model’s profit would need to increase by the reduced cost value for it to be produced. - **Shadow Price:** - For labor hours: Shows how much additional profit is gained if you have one more labor hour. - For output constraint: Shows how much additional profit is gained if you can make one more unit in total. --- ## (d) **Impact of New Demand for Model 3: At least 475 units** - **If the optimal solution already produces more than 475 Model 3:** - No change in optimal solution. - **If the optimal solution produces less than 475 Model 3:** - The constraint becomes binding. - The solution will adjust to produce at least 475 Model 3, possibly decreasing the production of Model S or X to satisfy labor and output constraints. - The new solution should be recalculated. --- ## (e) **Impact of New Demand for Model 3: At least 476 units** - **Similar logic as (d):** - If the solution already meets this, no change. - If not, the new optimal solution must produce at least 476 Model 3, and you must resolve the LP. **Note:** If the constraints for Model 3 demand, contractual obligation, and marketing are to be combined, use only one constraint for the sum, as per the hint. --- ## **Final Summary** 1. **Formulated the LP model:** Defined variables, objective, and constraints. 2. **Sensitivity Report:** In practice, use Excel Solver to get exact values for Reduced Cost (for non-produced models) and Shadow Price (for constraints like labor and output). 3. **Interpretation:** Reduced Cost tells you how much profit must increase to start producing a model not currently produced; Shadow Price tells you how much profit increases with one more unit of a constrained resource. 4. **Demand changes:** If new demand for Model 3 (475 or 476 units) is less than or equal to what you’re already producing, the solution is unchanged; if more, re-solve the LP with this new constraint. 5. **Combined Constraints:** For contractual and marketing obligations, combine them into a single constraint. --- **To fully answer with numbers, you’d need to use a solver. This solution gives the structure, logic, and reasoning needed for your assignment.**