How Many Of Each Desk Can Be Built Maximum Construction
How Many Of Each Desk Can Be Built Maximum Cut100 Construction 100
How many of each desk can be built given the constraints:
- Maximum cut hours = 100
- Maximum construction hours = 100
- Maximum finishing hours = 65
Desk A requires:
- 2 hours cut
- 2 hours construction
- 1 hour finishing
Desk B requires:
- 3 hours cut
- 1 hour construction
- 1 hour finishing
Desk C requires:
- 2 hours cut
- 3 hours construction
- 2 hours finishing
Determine the maximum number of each desk (A, B, C) that can be built without exceeding these constraints.
Paper For Above instruction
To determine the maximum number of each desk (A, B, C) that can be built within the given resource constraints, we can formulate this as a linear programming problem. The purpose is to maximize the production quantities of each desk while respecting the limits on the cutting, construction, and finishing hours.
Problem Formulation
Let:
- \( x_A \) = number of Desk A to produce
- \( x_B \) = number of Desk B to produce
- \( x_C \) = number of Desk C to produce
Our goal is to maximize the total production, but since the problem doesn't specify an objective function (such as profit), it appears the primary aim is to find the maximum possible combination that does not violate resource constraints. Therefore, this can be viewed as a feasibility problem with bounds.
The constraints based on resource availability:
1. Cutting hours:
\[
2x_A + 3x_B + 2x_C \leq 100
\]
2. Construction hours:
\[
2x_A + 1x_B + 3x_C \leq 100
\]
3. Finishing hours:
\[
1x_A + 1x_B + 2x_C \leq 65
\]
Non-negativity constraints:
\[
x_A, x_B, x_C \geq 0
\]
Solution Approach
To analyze this system, linear programming methods such as graphical solution (for two variables), simplex method, or computational solvers can be applied. Since this is a three-variable problem, using LP solvers or software like Excel Solver, MATLAB, or Python's scipy.optimize.linprog would be practical.
Step-by-Step Solution
1. Set up the LP model: The problem is largely about feasibility, so an objective function can be defined as maximizing the sum \( x_A + x_B + x_C \), subject to the constraints, to find the greatest total number of desks. Alternatively, we can analyze the constraints to find feasible corner points.
2. Identify corners of the feasible region: The solution space is bounded by the constraints, and the maximum is likely at intersection points.
3. Calculate intersection points:
- Set two constraints equal to each other and solve for variables.
- Check the feasibility of those solutions within all constraints.
For simplicity, I will illustrate the calculations for key points:
- Case 1: Maximize \( x_A \), set \( x_B = x_C = 0 \):
\[
2x_A \leq 100 \Rightarrow x_A \leq 50
\]
\[
2x_A \leq 100 \Rightarrow x_A \leq 50
\]
\[
x_A \leq 65
\]
The most restrictive is \( x_A \leq 50 \).
Similarly, for other cases, we can find combinations to maximize \( x_A + x_B + x_C \).
Optimal Solution
By analyzing the constraints, the solution suggests that the maximum number of desks is achieved at the boundary points where resource constraints are fully utilized.
- When producing only Desk A:
- \( x_A = 50 \)
- Total resource usage:
- Cut: \( 2 \times 50 = 100 \) hours
- Construction: \( 2 \times 50 = 100 \) hours
- Finishing: \( 1 \times 50 = 50 \) hours
Finishing hours are less than 65, feasible.
- When producing only Desk B:
- \( x_B = \frac{100}{3} \approx 33.33 \), but since the number of desks must be integer, \( x_B = 33 \)
- Check resource usage:
- Cut: \( 3 \times 33 = 99 \)
- Construction: \( 1 \times 33 = 33 \)
- Finishing: \( 1 \times 33 = 33 \)
Satisfies all constraints.
- When producing only Desk C:
- \( x_C = \frac{65}{2} = 32.5 \), so maximum integer \( x_C = 32 \)
- Resource usage:
- Cut: \( 2 \times 32 = 64 \)
- Construction: \( 3 \times 32 = 96 \)
- Finishing: \( 2 \times 32 = 64 \)
Construction hours exceed 100? No, it is 96, so feasible.
By examining mix combinations or using LP software for optimization, the combination that maximizes total count might involve a mixture of these production plans.
Conclusion
For a straightforward approximation, the manufacturing can be optimized by production of:
- 50 desks of type A, or
- 33 desks of type B, or
- 32 desks of type C
or a combination thereof that sums close to resource limits and maximizes desks produced.
Using computational methods, the precise optimal mix can be found. For practical manufacturing, these constraints suggest producing the maximum possible units based on each individual production line and combining them within resource constraints for optimal output.
References
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