Minimum Cost Problems

Lesson 6 of 10

In this lesson:

• Recognise minimization contexts from wording
• Apply the five-stage method to minimize cost
• State the direction before every evaluation

What You Will Learn Today

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

1. Recognise minimization contexts from key wording cues
2. Apply all five stages to a minimum cost problem
3. State the optimization direction before evaluating

Hook: From Profit to Minimizing Cost

In LP-05, the carpenter maximized profit:

"Make 5 tables and 0 chairs for maximum profit of 100."

Today a delivery manager minimizes cost:

"Which vehicle mix delivers the batches most cheaply?"

Same method. Opposite direction.

Wording-Sort: Identify the Optimization Direction

Sort these five cues: profit · cost · revenue · fuel use · time

• Which cues mean "find more"?
• Which cues mean "find less"?

Sort before the next slide reveals the rule.

Wording-to-Direction Rule for LP Problems

Write the direction before any algebra.

The Direction-Stating Habit Prevents Errors

Before any algebra, write:

"I am minimizing cost C."

This 4-second commitment prevents the most common LP-06 error:

• Wrong: scanning for the largest value in a minimization table
• Right: scanning for the smallest value

Check-In: State the Direction and Quantity

Scenario: A hospital administrator wants to reduce daily drug waste.

Write the direction statement and name the quantity.

Pause and write before advancing.

Stage 1: Read the Transport Scenario

Context: A delivery service transports maize sacks.

• Minivans : 1 batch/trip, 2 h/trip, 3,000 UGX/trip
• Lorries : 1 batch/trip, 3 h/trip, 5,000 UGX/trip
• At least 8 batches needed; max 18 hours available

Direction: minimize total cost.

Stage 2: Form the System

Decision variables: = minivans, = lorries

Constraints:

• (batches delivered)
• (hours available)
• ,

Objective: minimize (thousands UGX)

Stage 3: Graph the Feasible Region

• : shade above (≥ means feasible side is above)
• : shade below (≤ means feasible side is below)

Mixed Constraint Directions: Test-Point Still Works

The system has one ≥ and one ≤ constraint:

• : feasible region is above the line
• : feasible region is below the line

Test-point method works identically — the sign determines which side.

Stage 4: Find Corner Candidates

Check all axis candidates against both constraints:

• (8, 0): C1 pass, C2 pass → feasible
• (9, 0): C1 pass, C2 pass → feasible
• (0, 8): C1 pass, C2 fail → infeasible
• (0, 6): C1 fail, C2 pass → infeasible

Two Infeasible Candidates: Why They Fail

• (0, 8): 24 hours exceeds the 18-hour limit — fail
• (0, 6): only 6 batches, minimum is 8 — fail

Corner list: (6, 2), (8, 0), (9, 0)

Verify every candidate against all constraints.

Interior Corner: Calculating (6, 2)

Solve and simultaneously:

Interior corner: (6, 2) ✓

Stage 5: Evaluation Table and Minimum

Minimum: C = 24 at (8, 0).

Check-In: Write the Interpreted Answer

Minimum at — in thousands UGX.

Write the full sentence:

"The delivery service should use ___ minivans and ___ lorries for a minimum cost of ___ UGX, delivering ___ batches within ___ hours."

The Surprise: No Lorries Is Cheapest

Minimum at — eight minivans, zero lorries.

Why does using lorries increase cost?

• Minivans: 3,000 UGX per batch delivered
• Lorries: 5,000 UGX per batch delivered

Lorries carry more hours per trip — they cost more per trip too.

Bounded vs. Unbounded Feasible Regions

Two types of feasible regions:

• Bounded: enclosed polygon — has a finite maximum AND minimum
• Unbounded: open in one direction — minimum may exist; maximum may not

The carpenter region (LP-03) was bounded; today's transport region is also bounded.

Bounded vs. Unbounded: Two Regions Compared

• Bounded (left): every objective has both a max and a min
• Unbounded (right): min typically exists at a corner; max may be infinite

On Unbounded Regions: What Still Works

For an unbounded feasible region:

• Minimization: the minimum usually exists at a corner
• Maximization: the maximum may not exist — check for boundedness first

Safe rule: if only ≥ constraints appear, the region is likely unbounded.

Cafeteria Worked Example: Fractional Corner Alert

Minimize . Corners: (7.5, 4.5), (30, 0), (0, 12)

• (7.5, 4.5): 3750 ← minimum
• (30, 0):
• (0, 12):

Algebraically correct — but can you serve 7.5 cups? See LP-08.

Variant: What If Minivan Cost Rises?

Now both vehicles cost 5,000 UGX (equal):

• (6, 2):
• (8, 0):
• (9, 0):

Two corners tie — the optimum shifts when cost rates equalise.

Three Mistakes That Cost Marks

Watch out:

• Default to largest in minimization — state direction first; scan for smallest
• Treat minimization as a different method — same 5 stages; only the scan rule changes
• Accept fractional answers for integer variables — flag and apply LP-08

Check-In: Apply All Five Stages

Scenario: A farmer minimizes fertiliser cost. Corner list: (4, 2), (6, 0), (0, 5).

Objective: minimize .

State direction. Build the table. Write the interpreted answer.

The Direction-Stating Habit Is Mandatory

Every LP problem from LP-06 onward must start:

"I am minimizing/maximizing [quantity]."

Write it. Every time. Before any algebra.

This sentence costs four seconds and prevents the most common LP error.

Key Takeaway: Same Method, Pick Smallest

✓ Recognize minimization from wording: cost, time, fuel, waste

✓ Same five stages as LP-05 — nothing new procedurally

✓ Evaluation table: scan for smallest value, not largest

✓ Write the direction before evaluating — every time

Coming Up: Lesson 7 — Maximum Profit

You can now solve LP problems in both directions.

Lesson 7 — Maximum Profit Problems:

• Fresh context: poultry farming
• Maximize revenue with an unbounded region
• Learn the diagnostic rule for unbounded max