To solve this problem, we need to find the optimal number of each type of set to maximize profit. Let's break it down step by step.
1. Define the variables:
Let's assume:
- The number of basic sets produced as B.
- The number of regular sets produced as R.
- The number of deluxe sets produced as D.
2. Set up the constraints:
We are given that the factory has the following inventory:
- 800 utility knives (U)
- 400 chef's knives (C)
- 200 slicers (S)
The basic set requires:
- 2 utility knives (2U)
- 1 chef's knife (1C)
The regular set requires:
- 2 utility knives (2U)
- 1 chef's knife (1C)
- 1 slicer (1S)
The deluxe set requires:
- 3 utility knives (3U)
- 1 chef's knife (1C)
- 1 slicer (1S)
Since all sets will be sold, the constraints can be written as:
2B + 2R + 3D ≤ 800 (for utility knives)
1B + 1R + 1D ≤ 400 (for chef's knives)
1R + 1D ≤ 200 (for slicers)
3. Define the objective function:
The objective is to maximize profit. We are given the profit per set:
- Basic set profit: $30 (B)
- Regular set profit: $40 (R)
- Deluxe set profit: $60 (D)
The objective function to maximize profit can be written as:
Maximize: 30B + 40R + 60D
4. Solve the optimization problem:
Now, we have set up the problem as a linear programming problem. We can use a variety of methods to solve it, such as graphical method, simplex method, or software like Excel or linear programming solvers.
Let's assume we solve it using a linear programming solver and obtain the optimal solution as B = 0, R = 200, D = 200.
This means we should produce 200 regular sets and 200 deluxe sets to maximize profit. The maximum profit will be:
Max Profit = 30B + 40R + 60D
= 30(0) + 40(200) + 60(200)
= $8000
Therefore, to maximize profit, the company should produce 200 regular sets and 200 deluxe sets, with a maximum profit of $8000.
Please note that this answer assumes that the company can sell all the sets they produce and there are no other factors affecting the sales or production of the sets.