Analyze the make-or-buy decision made by Acme
The company also has the option to obtain additional units from a subcontractor, who has offered to supply up to 20,000 units per month in any combination of electric and battery-operated models, at a charge of $21.50 per unit. For this price, the subcontractor will test and ship its models directly to the retailers without using Acme's production process.
a. What are the maximum profit and the corresponding make/buy levels? (Fractional decisions are acceptable.)
b. Suppose that Acme requires that the solution provided by the model be implementable without any rounding off. That is, the solution must contain integer decisions. What are the optimal make/buy levels?
c. Is the solution in part (b) a rounded-off version of the fractional solution in part (a)?
To solve this problem, we need to analyze the make-or-buy decision made by Acme. This involves calculating the costs associated with manufacturing the units in-house versus purchasing them from a subcontractor.
Part (a): Maximum Profit and Corresponding Make/Buy Levels
Given Information:
- Cost of manufacturing in-house: Assume it is represented by ( C_m ) (the actual cost needs to be provided for precise calculations).
- Cost of purchasing from subcontractor: $21.50 per unit.
- Maximum units available from subcontractor: 20,000 units/month.
- Total demand: Assume it is represented by ( D ) (the demand must be provided for precise calculations).
Steps:
1. Define variables:
- Let ( x ) be the number of units produced in-house.
- Let ( y ) be the number of units purchased from the subcontractor.
2. Constraints:
- ( x + y = D )
- ( y \leq 20,000 )
3. Profit Function:
- Profit = Revenue - Costs
- Revenue = Selling price per unit (assume ( P )) * Total number of units sold.
- Costs = In-house production cost + Cost of buying units.
4. Optimal Levels:
Based on these equations, we can calculate the maximum profit by balancing the production and purchase levels while taking into account the total demand and constraints.
Part (b): Optimal Make/Buy Levels with Integer Decisions
When requiring integer solutions, we need to round off the fractional values obtained in part (a) to the nearest whole numbers. The process involves:
1. Re-evaluating the profit function, but with integer constraints.
2. Using integer programming techniques to solve the optimization problem.
3. Checking for feasibility, ensuring that ( x + y ) still meets total demand and that ( y ) does not exceed 20,000.
Steps:
1. Use a linear programming solver or integer programming method to determine values for ( x ) (in-house production) and ( y ) (subcontracted units).
2. Ensure that both ( x ) and ( y ) are integers and satisfy all constraints.
Part (c): Comparison of Solutions
To determine if the solution in part (b) is a rounded-off version of the fractional solution from part (a):
1. Analyze both solutions:
- Compare the integer solution with the fractional solution derived in part (a).
- Check if rounding off the fractional solution leads directly to the integer solution or if adjustments were needed.
2. Conclusion:
If the integer solution can be derived directly from rounding off without violating any constraints and maximizing profit, then it can be considered a rounded-off version. If adjustments were made, it is not merely a rounding but rather an altered solution based on integer constraints.
Conclusion
The analysis above provides a structured approach to solving Acme's make-or-buy problem. By establishing profit functions, constraints, and utilizing optimization techniques, we can determine the best approach to maximize profits while adhering to production capabilities and demand requirements. Each part of this analysis builds upon the previous calculations, leading to informed decision-making for Acme.
To provide exact numeric answers, specific values for costs, selling price, and total demand need to be inputted into the equations.