Practice: Optimal Solutions and Optimal Values

Recognition · Classification

A workshop maximizes profit and finds that making 6 benches and 2 stools earns 480, which is the best it can do.

Which of the following is the optimal value?

1 hint available, least help first.

Hint 1: Retrieval cue

One of these is a number the objective returns. The others are a choice you could act on, a function, and a restriction.

Classification · Explanation

For each problem, report the optimal solution and the optimal value. Where the optimal set has more than one point, say so and give two members of it. Where no optimal solution exists, say so and explain why.

(a) Minimise $2 x$ subject to $x \geq 5$ .

(b) Maximise $x + y$ subject to $x + y \leq 6$ , $x \geq 0$ , $y \geq 0$ .

(c) Minimise $x$ subject to $x > 1$ .

Write your answer

Write your answer, then compare it with the worked solution.

2 hints available, least help first.

Hint 1: Retrieval cue

For each part, ask separately: which choice or choices are best, and what number do they score?

Hint 2: Strategy cue

To test uniqueness, see whether a different feasible point scores the same. To test attainment, see whether every feasible point has a strictly better feasible point.

Compare with the worked solution

Comparing does not record a result. Judging your own written answer cannot show that you can do this without help.

(a) Unique optimal solution $x = 5$ ; optimal value 10. Decreasing $x$ below 5 is infeasible, and $2 x$ increases with $x$ , so 5 is the only minimiser.

(b) The optimal set has many points: every point with $x + y = 6$ and $x, y \geq 0$ is optimal, for instance $(6, 0)$ and $(3, 3)$ . The optimal value is 6 for all of them.

(c) No optimal solution. For any feasible $x > 1$ the point $(1 + x) / 2$ is also feasible and smaller, so no feasible point attains a minimum. The greatest lower bound is 1, but 1 is not feasible, so there is no optimal value either.

A complete answer does each of these:

reports correct object
uniqueness stated
attainment recognised

Error diagnosis · Explanation

A student solves maximize $5 x_{1} + 5 x_{2}$ subject to $x_{1} + x_{2} \leq 8$ , $x_{1}, x_{2} \geq 0$ and writes:

The optimal solution is 40, achieved at $x_{1} = 8$ .

Two things are wrong with this report. Identify both, and write a correct one.

Write your answer

Write your answer, then compare it with the worked solution.

2 hints available, least help first.

Hint 1: Retrieval cue

Check what kind of object each part of the report names: a point or a number?

Hint 2: Next step

Evaluate the objective at a second feasible point on the boundary and compare.

Compare with the worked solution

Comparing does not record a result. Judging your own written answer cannot show that you can do this without help.

First, 40 is the optimal value, not the optimal solution. The optimal solution is a point, and 40 is a number.

Second, the report implies a unique optimum. Because the objective $5 x_{1} + 5 x_{2}$ is constant along the line $x_{1} + x_{2} = 8$ , every feasible point on that segment is optimal, $(8, 0)$ , $(0, 8)$ and $(4, 4)$ all score 40.

A correct report: every point with $x_{1} + x_{2} = 8$ and $x_{1}, x_{2} \geq 0$ is an optimal solution, for instance $(8, 0)$ or $(4, 4)$ ; the optimal value is 40.

A complete answer does each of these:

reports correct object
uniqueness stated
attainment recognised

Practice: Optimal Solutions and Optimal Values

Recognition · Classification

Classification · Explanation

Error diagnosis · Explanation

Session complete

Settings

Focus timer

Today

Last seven days

Where the time went

Recent sessions

Settings

Notifications

Your history