Linear Programming for Maximizing Profit in Jewelry Store
How can the optimal number of necklaces and bracelets be determined to maximize profit?
Given that the store has 18 units of gold and 20 units of platinum, each necklace requires 3 units of gold and 2 units of platinum, while each bracelet requires 2 units of gold and 4 units of platinum. The demand for bracelets is no more than four, and a necklace earns $300 in profit while a bracelet earns $400. What is the approach to finding the optimal production quantities?
If the store produces the optimal number of bracelets and necklaces, will the maximum demand for bracelets be met?
Considering the constraints of the available units of gold and platinum, as well as the maximum demand for bracelets, will the store be able to fulfill the maximum demand for bracelets with the optimal production quantities?
Optimizing Profit through Linear Programming
By setting up constraints and an objective function, we can use linear programming techniques to determine the optimal number of necklaces (N) and bracelets (B) that the jewelry store should produce to maximize profit.
Objective Function and Constraints
The objective function to maximize profit is Profit = 300N + 400B, considering the earnings from each necklace and bracelet.
The constraints are: - 3N + 2B ≤ 18 (gold constraint) - 2N + 4B ≤ 20 (platinum constraint) - B ≤ 4 (maximum demand for bracelets)
Solving the Linear Programming Model
By solving the linear programming model with the given constraints and objective function, we can find the optimal solution that maximizes profit. This solution will provide the store with the recommended quantities of necklaces and bracelets to produce.
Fulfilling Maximum Demand for Bracelets
If the store produces according to the optimal solution obtained from the linear programming model, the maximum demand for bracelets will be met. The constraint B ≤ 4 ensures that the production of bracelets does not exceed the maximum demand, eliminating the possibility of missing the demand.