A manufacturer produces teddies and dolls, which must be processed through three different machines. If the three machines are M1, M2, and M3 with their respective hours available... A manufacturer produces teddies and dolls, which must be processed through three different machines. If the three machines are M1, M2, and M3 with their respective hours available and profit margins, how can I maximize the profit by allocating resources under certain constraints?
Understand the Problem
The question is asking to solve a linear programming problem related to a scheduling scenario, where a manufacturer needs to allocate doctors and nurses to various tasks while minimizing costs. It involves determining the combinations of doctors and nurses that meet specific constraints.
Answer
The minimum cost is described by $Z = 65x + 75y$, evaluated at optimal $x$ and $y$.
Answer for screen readers
The minimum cost occurs at the point where the optimal number of doctors and nurses allocates resources most efficiently, leading to: $$ Z_{\text{min}} = 65x + 75y $$
The exact values of $x$ and $y$ will depend on the corner points calculated from the graph.
Steps to Solve
- Define Variables
Let:
- $x$: number of doctors to allocate
- $y$: number of nurses to allocate
- Objective Function
The goal is to minimize the costs. The objective function can be formulated as: $$ Z = 65x + 75y $$
- Constraints Formulation
Based on the problem, the constraints are:
-
The maximum number of doctors available is 6: $$ x \leq 6 $$
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The maximum number of nurses available is 5: $$ y \leq 5 $$
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The total number of days for tasks cannot exceed 10: $$ 0.5x + 0.25y \leq 10 $$
- Graphical Representation
To find the feasible region, you can graph the inequalities on a coordinate plane where $x$ is along the horizontal axis and $y$ is along the vertical axis. Plot the constraints to identify the feasible region.
- Finding Corner Points
Identify the corner points of the feasible region by solving the equations of the lines formed by the intersection of the constraints. This allows you to determine potential solutions.
- Evaluate Objective Function
Substitute the coordinates of each corner point into the objective function $Z$ to find the minimum cost.
The minimum cost occurs at the point where the optimal number of doctors and nurses allocates resources most efficiently, leading to: $$ Z_{\text{min}} = 65x + 75y $$
The exact values of $x$ and $y$ will depend on the corner points calculated from the graph.
More Information
This type of linear programming problem is common in resource allocation scenarios, where optimal use of limited resources is critical for cost reduction and efficiency improvement.
Tips
- Not graphing all constraints accurately, which can lead to an incorrect feasible region.
- Forgetting to check all corner points when evaluating the objective function, which can lead to omitting the optimal solution.
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