Objective Function Evaluation

Lesson 5 of 10

In this lesson:

• Set up the objective function from context
• Evaluate at every corner point
• Identify and interpret the optimal solution

What You Will Learn Today

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

1. Set up the objective function with the correct direction (max or min)
2. Evaluate the objective at every corner point using a table
3. Identify the optimum and interpret the answer in context

The Corner List: Which Earns Most?

From LP-04, the carpenter region has four corners:

(0, 0), (5, 0), (3, 2), (0, 4)

Tables earn $20 each. Chairs earn $10 each.

Which corner gives the highest profit?

Two Corners Computed by Arithmetic

Before naming any formula — compute two rows:

• : profit
• : profit

The pattern: substitute the corner, multiply by profit rate, add.

The Objective Function: Form and Meaning

The objective function always has the form :

• = profit (or cost) per unit of
• = profit (or cost) per unit of

For the carpenter: — maximize

Wording Tells You the Direction

State the direction before you build the table.

Worked: Building P = 20x + 10y

Context: Tables earn $20 each; chairs earn $10 each.

1. Decision variables: = tables, = chairs
2. Contribution: from tables, from chairs
3. Objective: — maximize

Worked: Building C = 30x + 50y

A delivery service uses minivans ($30/trip) and lorries ($50/trip).

— minimize

Check-In: Set Up the Objective

Scenario: Each worker earns 15,000 UGX per shift; each machine saves 8,000 UGX. Maximize earnings.

Write the objective function. State the direction.

The Evaluation Table: Three-Column Structure

Fill every row — scan for the optimum.

Evaluating All Four Carpenter Corners

The Surprise: All Tables, No Chairs

Maximum at — five tables, zero chairs.

Why does the interior corner (3, 2) lose?

• Tables earn 20 each; chairs earn 10 each
• Coefficient ratio drives the optimum, not "balance"

Change the chair price and the optimum shifts.

The Interpretation Step: Write the Sentence

Never stop at a number. Always write:

"The carpenter should make 5 tables and 0 chairs for a maximum profit of 100."

Format: [decision] for a [max/min] [quantity] of [value]

Check-In: Read the Minimization Table

A minimization table shows:

• :
• :
• :

Which corner is optimal? Write the interpreted answer.

Worked: Minimization — Delivery Corners

Minimum: at .

The Complete Five-Stage LP Method

Five stages — one complete method:

1. Read — variables, constraints, direction
2. Form — inequalities and objective
3. Graph — boundaries and region
4. Corners — inspect and calculate
5. Evaluate — table, optimum, sentence

Method Roadmap: Stage 5 Active

1. Read — variables, constraints (LP-01)
2. Form — inequalities and objective (LP-02)
3. Graph — boundaries, region (LP-03)
4. Corners — find and verify (LP-04)
5. Evaluate — table, optimum, sentence (LP-05) ← today

Worked: Carpenter in 90 Seconds

1. Read: tables , chairs
2. Form: ; ; maximize
3. Graph: boundaries drawn, region shaded
4. Corners: , , ,
5. Evaluate: max at — 5 tables, 0 chairs

Check-In: Write the Interpreted Answer

Maximize :

• A (0, 0):
• B (4, 0):
• C (2, 3):
• D (0, 4):

Identify the optimum. Write the interpreted sentence.

Variant: Chair Price Rises to 30

Now . New P values:

• A (0, 0):
• B (5, 0):
• C (3, 2):
• D (0, 4):

Two corners tie — optimum has shifted.

Variant: The Coefficient Drives the Optimum

In original (): tables-only wins because

In variant (): balance wins because coefficients are closer

The optimum is not about "using both" — it is about the objective.

Always State the Direction First

Habit rule — always write:

"I am maximizing P = 20x + 10y"
— or —
"I am minimizing C = 30x + 50y"

before building the evaluation table.

Three Mistakes That Cost Marks

Watch out:

• Default to max when minimizing — state direction first; scan for smallest
• Stop at the number — $100 is not the answer; write the full sentence
• Evaluate interior points — the Fundamental Theorem guarantees corners

Check-In: Complete Evaluation Table Practice

Corner list given: , , ,

Objective: maximize

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

Key Takeaway: Table Converts LP to Arithmetic

✓ Objective function — direction from wording

✓ Evaluate at every corner — no skipping

✓ Scan for max or min — state direction first

✓ Write the interpreted sentence — not just a number

Coming Up: Lesson 6 — Minimum Cost

You have now solved a complete LP problem end to end.

Lesson 6 — Minimum Cost Problems:

• Fresh context: minimize transport cost
• Same five stages — direction is minimize
• Introduce the direction-stating step as the opening move