Integer constraints and interpreting an optimal solution | Year 12

01 / LEARN

The idea, step by step

A fractional decision may be impossible

A continuous model allows every real feasible point. Counts of workers or vehicles usually require integers. The continuous optimum gives a bound, but rounding both coordinates can break a constraint or miss a better plan. State integrality before solving and check the original conditions after choosing a candidate.

Turn a small integer problem into a finite search

For each permitted integer x, find the allowable integer y-range. If profit increases with y, test the largest permitted y for that x; otherwise choose the appropriate endpoint or compare all candidates. A table of values proves which tested decision is best. Report ties honestly and use another practical criterion only when it is explicitly stated.

A common trap

Round the continuous optimum to the nearest whole pair and stop.

Instead: The rounded pair may be infeasible or suboptimal; justify a search over the permitted integer decisions.

02 / WORKED EXAMPLES

See the thinking

Example 1

Rounding can fail

Maximise $P = 3 x + 4 y$ for non-negative integers with $2 x + y \leq 7$ and $x + 2 y \leq 7$ .

x	Largest feasible integer y	P
0	3	12
1	3	15
2	2	14
3	1	13

The continuous corner is $(7 / 3, 7 / 3)$ ; rounding up to $(3, 3)$ violates both constraints.
For x=0,1,2,3, the largest permitted y-values are 3,3,2,1.
The corresponding objectives are 12,15,14,13.

Integer maximum 15 at

(1, 3)

Example 2

Whole vehicles

At least 46 seats are needed. A small vehicle has 8 seats and costs AUD 70; a large vehicle has 14 seats and costs AUD 110. At most 5 vehicles are allowed. Minimise cost.

Let x and y count small and large vehicles: $8 x + 14 y \geq 46$ , $x + y \leq 5$ , both non-negative integers.
For y=0, no plan fits five vehicles. For y=1,2,3,4,5 the least required x-values are 4,3,1,0,0.
Costs are 390,430,400,440,550. Every cheaper x at each fixed y would fail seating.

Four small and one large vehicle; AUD 390, exactly 46 seats.

Check your understanding

A model gives 4.2 workers. Is 4 automatically acceptable?

Show answer

No.

Four may fail the workload requirement; original constraints determine the valid integer choices.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the greatest integer x with $3 x \leq 20$ and $x \geq 0$ .
2 marks
At least 25 items are packed in boxes of 6. Find the least number of boxes.
2 marks
Maximise $P = 2 x + 3 y$ for non-negative integers with $x + y \leq 5$ , $y \leq 3$ .
2 marks
Is $(2, 4)$ feasible for $2 x + y \leq 7$ , $x + 2 y \leq 9$ ?
2 marks
Two integer plans share the greatest profit. What should a recommendation explain?
2 marks
For x=0,1,2,3, the largest feasible integer y-values are 4,3,2,0. Maximise $P = 5 x + 4 y$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the greatest integer x with $3 x \leq 20$ and $x \geq 0$ .
2 marks
$x \leq 20 / 3$ ; choose the largest permitted integer below that bound.
6.
At least 25 items are packed in boxes of 6. Find the least number of boxes.
2 marks
$6 b \geq 25$ , so $b \geq 25 / 6$ ; round up to meet the minimum.
5.
Maximise $P = 2 x + 3 y$ for non-negative integers with $x + y \leq 5$ , $y \leq 3$ .
2 marks
Increasing y in place of x increases profit.
Use y=3 and the remaining capacity for x=2.
13 at $(2, 3)$ .
Is $(2, 4)$ feasible for $2 x + y \leq 7$ , $x + 2 y \leq 9$ ?
2 marks
The resource totals are 8 and 10, exceeding both limits.
No.
Two integer plans share the greatest profit. What should a recommendation explain?
2 marks
A secondary criterion such as fewer staff hours can break the tie only if it matters in the stated situation.
Report both optima and any explicitly justified practical preference.
For x=0,1,2,3, the largest feasible integer y-values are 4,3,2,0. Maximise $P = 5 x + 4 y$ .
2 marks
The four objective values are 16,17,18,15.
18 at $(2, 2)$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.

Official curriculum source ↗

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