Maximum Profit Problems

Lesson 7 of 10

In this lesson:

• Recognise maximization contexts from wording
• Apply the five-stage method to maximize profit
• Detect when an unbounded region has no maximum

What You Will Learn Today

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

1. Recognise maximization contexts and set up the objective function
2. Apply all five stages to a maximum profit problem
3. Identify when a maximization problem on an unbounded region has no solution

Hook: From Minimizing to Maximizing

LP-06 minimized transport cost:

"8 minivans, 0 lorries — minimum cost of 24,000 UGX."

Today the poultry farmer maximizes profit:

"Which mix of chickens and ducks earns the most?"

Same method. Opposite direction.

Wording Cues: Maximize or Minimize

The goal determines the direction — not a fixed rule.

Direction-Stating Habit Applies to Max Too

Even when maximization feels obvious, write:

"I am maximizing profit P."

Habits work because they apply uniformly — not just when uncertain.

Check-In: Name the Direction and Quantity

Scenario: A school wants to maximise total attendance hours given limited classroom space.

Write the direction statement. Name the quantity.

Pause and write before advancing.

Stage 1 and 2: Read and Form

Poultry farmer: chickens crates, ducks crates

Constraints:

• (market capacity)
• (days available)
• ,

Objective: maximize (thousands UGX)

Stage 3: Graph the Feasible Region

• Both constraints are ≤ — feasible region is below both lines
• Region is bounded in the first quadrant

Stage 4: Verify Axis Candidates

Check axis intersections against both constraints:

• (10, 0): C1 pass, C2 pass → feasible
• (12, 0): C1 fail → infeasible
• (0, 9): C1 pass, C2 pass → feasible
• (0, 10): C2 fail → infeasible

Interior Corner (4, 6) by Calculation

Solve and :

Interior corner: (4, 6) ✓

Stage 5: Build the Evaluation Table

Maximize :

• (0, 0):
• (10, 0):
• (4, 6):
• (0, 9): 630 ← maximum

Surprise: All Ducks Wins the Maximum

Maximum at — zero chickens, nine ducks.

Why does the interior corner (4, 6) lose?

• Ducks earn 70,000 UGX per crate; chickens earn 50,000 UGX
• Higher duck coefficient drives the optimum toward ducks

Change chicken profit and the answer shifts.

Interpretation: Write the Full Answer with Units

Full interpreted answer:

"The farmer should raise 0 crates of chickens and 9 crates of ducks for a maximum profit of 630,000 UGX, using all 36 days and meeting the 10-crate limit."

Include units. Include both constraints checked.

Check-In: Read the Table and Interpret

Maximize . Completed table:

• (0, 0):
• (10, 0):
• (4, 6):
• (0, 9):

State direction. Identify the maximum. Write the full sentence with units.

When the Feasible Region Is Unbounded

A feasible region is unbounded when it extends to infinity.

Example:

The region extends upper-right without bound.

Maximization on this region: does a maximum exist?

Worked: No Maximum on Unbounded Region

Maximize on region :

• Region extends to infinity upper-right
• P grows without bound as x and y increase
• No maximum exists

Diagnostic Rule for Unbounded Maximum

Check before evaluating:

1. Is the feasible region unbounded in some direction?
2. Does the objective function grow in that direction?

If both yes → no maximum exists — state this.

If the objective decreases in the unbounded direction → minimum exists at a corner.

Bounded vs. Unbounded: Max Comparison

• Bounded (left): maximum exists at a corner
• Unbounded (right): P grows without limit — no maximum

Counter-Example: Min Exists on Unbounded

For — minimize :

Corners: (5, 0), (0, 5)

• (5, 0):
• (0, 5):

Minimum at both corners — finite minimum exists.

Check-In: Apply the Diagnostic Rule

Given: The feasible region is unbounded upper-right.

Objective: Maximize .

Does a maximum exist? Apply the two-step diagnostic rule.

Write your reasoning before advancing.

Variant: Chicken Profit Rises to 80,000 UGX

New objective :

• (0, 0):
• (10, 0): ← new maximum
• (4, 6):
• (0, 9):

Coefficient change shifts the optimum from (0,9) to (10,0).

Three Mistakes That Cost Marks

Watch out:

• Largest-coordinate error — the corner with the biggest x or y is not necessarily optimal
• Balance heuristic — the optimum follows coefficients, not a "use-both" intuition
• Unbounded means unsolvable — apply the diagnostic rule; unbounded maximization may have no max, but minimization usually does

Check-In: Apply All Five Stages

School store: pencils , erasers . Maximize .

Corner list provided: , , , .

State direction. Build the evaluation table. Write the sentence.

Key Takeaway: Check for Bounded Region

✓ Maximize: profit, revenue, output, yield — scan for largest value

✓ Same five-stage method as LP-05 and LP-06

✓ Bounded region: maximum always exists at a corner

✓ Unbounded region + max growing: no maximum — state this explicitly

Coming Up: Lesson 8 — Integer Constraints

You can now solve LP problems in both directions.

Lesson 8 — Integer and Practical Constraints:

• When the algebraic optimum is fractional
• Integer search near the fractional corner
• Practical feasibility beyond the algebra