Integer and Practical Constraints

Lesson 8 of 10

In this lesson:

• Recognise when a fractional optimum is not physically realisable
• Find the integer optimum by checking nearby integer points
• Add and apply practical constraints to an LP system

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Recognise when a fractional optimum is not physically realisable
2. Find the integer optimum by checking integer candidates near the fractional corner
3. Add practical constraints and re-identify corners

Hook: The Fractional Corner Problem

From LP-06, the cafeteria optimum was at .

"Serve 7.5 cups of rice and 4.5 cups of beans."

Is 7.5 cups of rice a real answer? It depends on the variable.

Continuous Variables vs. Integer-Required Variables

Ask: "Can this variable take a fractional value in the real world?"

The Diagnostic Question Before Every Answer

Before writing any LP answer, ask:

"Are these variables required to be whole numbers?"

• Yes → apply integer search
• No → accept the algebraic answer

LP-05 and LP-07 were lucky — their optima happened to be integer.

Check-In: Integer-Required or Continuous Variable?

Classify each variable:

1. Buses purchased
2. Hours worked
3. Chickens raised
4. Litres of milk produced

Write I (integer) or C (continuous) for each.

Workshop: Algebraic Optimum Is Fractional

Tables , stools . Maximize .

• (hours); (wood)
• must be whole numbers

Algebraic corner: ,

Can't make 8.1 tables — apply integer search.

Stage 5 Extended: The Fractional Corner

Four candidate integer points near :

• , , ,

Check each against ALL constraints.

Integer Search Procedure: Four Candidates Checked

Procedure — for each candidate:

1. Check
2. Check
3. If both pass → evaluate
4. If either fails → discard

The largest feasible P is the integer optimum.

Checking (8, 2): Feasible — P = 280

✓ (hours)

✓ (wood)

Both constraints pass → is feasible.

280

Checking (8, 3): Infeasible — Discard

✓ (hours)

✗ (wood)

Second constraint fails → is infeasible.

Do not evaluate P — this candidate is eliminated.

Checking (7, 3) and Comparing

• : ✓; ✓ →
• : ✓; ✓ →

Integer optimum: with

Integer Optimum Is Never Better

Algebraic optimum: at

Integer optimum: at

Restricting to integers can only reduce or equal the algebraic answer — never improve it.

Check-In: Find the Integer Optimum

From the candidate table for a fresh scenario:

• : , feasible
• : infeasible
• : , feasible
• : , feasible

Which candidate is the integer optimum? Write the interpreted answer.

Practical Constraints: Common Wording and Forms

Beyond resource limits — common patterns:

• "At least 3 tables" →
• "No more than 5 chairs" →
• "Twice as many tables as chairs" →
• "Total at least 10 items" →

Practical Constraint Added: x ≥ 4

Workshop system plus new constraint: at least 4 tables ()

• is now infeasible ()
• New corner: where meets x-axis

Adding a Constraint Can Only Shrink

Structural rule:

Adding any constraint to an LP system can only shrink or maintain the feasible region — never expand it.

• The new optimum ≤ the old optimum (for maximization)
• Some prior corners become infeasible

Constraints remove options — they never add new ones.

Check-In: Add a Practical Constraint

Add constraint to the workshop system.

Corner list before: , , ,

• Which corners survive ?
• Where does create new corners?

Identify surviving and new corners before advancing.

Variant: What If Variables Are Continuous?

What if x and y are continuous (no integer requirement)?

The algebraic optimum is the final answer: with .

No integer search needed. No rounding.

The diagnostic question determines which path you take.

Three Mistakes That Cost Marks

Watch out:

• Round without checking feasibility — (8,3) looked close but was infeasible
• Only round inward — the integer optimum may be at any of the four candidates
• Integer constraint applied to continuous variables — ask the diagnostic question per variable

Check-In: Complete the Integer Search

Fractional optimum at . Variables are integer-required.

Four candidates: , , , .

Check each against the constraints. Evaluate P at feasible candidates. State the integer optimum.

The Integer Check: Always Last

After every LP solution, ask one final question:

"Are these variables required to be whole numbers?"

If yes → apply the four-candidate integer search.
If no → accept the algebraic answer.

This question belongs in your solution routine — not as an afterthought.

Reading Checklist: Catch Every Constraint

Read the problem twice before forming the system:

1. Every resource limit (≤ and ≥)
2. Every practical constraint (at least, at most, ratio)
3. Each variable's type (integer or continuous)
4. Fractional optima: apply diagnostic question

Missed constraint → wrong system → wrong answer.

Key Takeaway: One Extra Step After Stage 5

✓ Ask the diagnostic question before finalizing any LP answer

✓ Integer search: check four candidates — verify all constraints, compare P

✓ Integer optimum ≤ algebraic optimum — always

✓ Adding constraints shrinks the region — never expands it

Coming Up: Design Your Own LP Problem

You can now solve LP problems end to end, including integer constraints.

Lesson 9 — Learner-Created LP Tasks:

• You design a context, define variables, and write constraints
• Integer classification is one of your design decisions
• Peer review ensures the problem is solvable