A Furniture Manufacturing is considering producing 3 new products (tables, chairs, and dressers). The manufacturer should produce at least 2 chairs for each unit of table they produce. Time requirement and available capacity in the departments needed to produce these products is summarized in the following table:

Required time for each product at each department (hour)

Product

Sawing (hour)

Finishing (hour)

Assembling (hour)

Table

0.25

1

0.5

Chair

0.25

0.5

0.5

Dresser

0.5

1.5

1

Total available time per day (hour)

12

8

8

The sales department estimates that the profit per unit would be $100, $50, and $80, respectively for tables, chairs, and dressers. They also estimate that the sales potential for dressers is 5 units per day (production of dressers should not exceed this number). An LP model is formulated and solved using Excel Solver for the problem of maximizing the manufacturer’s profit.

If x1 is number of tables and x2 is number of chairs, which of the following inequalities ensures that at least two chairs are produced for each unit of table?

Question

A Furniture Manufacturing is considering producing 3 new products (tables, chairs, and dressers). The manufacturer should produce at least 2 chairs for each unit of table they produce. Time requirement and available capacity in the departments needed to produce these products is summarized in the following table:

Required time for each product at each department (hour)

Product

Sawing (hour)

Finishing (hour)

Assembling (hour)

Table

0.25

1

0.5

Chair

0.25

0.5

Dresser

0.5

1.5

1

Total available time per day (hour)

12

8

The sales department estimates that the profit per unit would be $100, $50, and $80, respectively for tables, chairs, and dressers.  They also estimate that the sales potential for dressers is 5 units per day (production of dressers should not exceed this number). An LP model is formulated and solved using Excel Solver for the problem of maximizing the manufacturer’s profit.

If x1 is number of tables and x2 is number of chairs, which of the following inequalities ensures that at least two chairs are produced for each unit of table?

BlackTom AI · Accepted Answer

To ensure the production plan respects the requirement that at least two chairs are produced for each unit of table, we translate the relationship into a linear inequality between x1 (tables) and x2 (chairs).
Option A: 2 x1 ≥ x2. This would allow scenarios where the number of chairs is less than twice the tables, which violates the stated requirement. For example, if x1 = 3, this would permit x2 up to 6, but it would also permit x2 = 5, which is not enough to satisfy the 'at least two chairs per table' rule. Therefore this option does not guarantee the constraint.
Option B: x1 ≤ 2 x2. This rearranges to x2 ≥ x1/2, which is a very loose condition and does not enforce the needed 2-to-1 ratio of chairs to tables. In fact, it allows many infeasible plans where chairs are far fewer than twice the tables, or even cases with zero chairs when x1 is positive, which is not acceptable.
Option C: x1 ≥ 2 x2. This flips the required relationship, implying there must be at least twice as many tables as chairs. That is directly opposite to the problem’s constraint and would never satisfy the requirement of at least two chairs per table.
Option D: 2 x1 ≤ x2. This inequality precisely encodes the intended requirement: for each table (x1), there must be at least two chairs (x2), i.e., x2 ≥ 2 x1. This is the correct way to enforce the stated production rule and ensures the ratio is not violated.
In summary, the key idea is to express the practical constraint “at least two chairs per table” as a direct lower bound on chairs in terms of tables, which is captured by 2 x1 ≤ x2. The other options either weaken the constraint, invert the relationship, or describe an incorrect ratio that does not align with the requirement.

类似问题

Which of the following gives the constraint for the demand met at a distribution center?

Linear optimization cannot be used on problems having multiple time periods.

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