Learner-Created LP Tasks

Lesson 9 of 10

In this lesson:

• Design an LP problem from a real-world context
• Solve a classmate's LP using all five stages
• Critique a design using the structural validity rubric

What You Will Learn Today

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

1. Design a complete LP problem with all structural features
2. Solve a peer-designed LP using the full five-stage method
3. Critique an LP design using the structural validity rubric

Hook: Now You Are the Problem-Setter

For 8 lessons you solved LP problems.

Today the question flips:

"Can you write a problem that requires LP to solve — and has a non-trivial answer?"

Designing is harder than solving.

Design Process: Five Steps for LP Creation

The solving method reversed becomes the design process:

1. Choose a real context
2. Define decision variables
3. Write constraints (at least 4)
4. Write objective with direction
5. Verify — solve your own problem

Design Step 1: Choose a Real Context

The context needs two activities competing for shared resources:

• A market trader selling two types of goods
• A household choosing between two purchases
• A school event with two food items
• A farmer choosing between two crops

Two decision variables, at least two shared resource limits.

Design Step 2: Define the Variables

State clearly:

• = what unit and what activity (e.g., "x = number of samosas made")
• = what unit and what activity (e.g., "y = number of chapati made")
• Variable type: integer or continuous?

Vague variables produce ambiguous problems.

Design Step 3: Write the Constraints

At least four constraints:

1. First resource limit:
2. Second resource limit:
3. Non-negativity: ,
4. Optional: practical constraint (at least, ratio)

Design Step 4: Write the Objective Function

State:

• The function: or
• The direction: maximize or minimize
• Per-unit values: choose non-trivial values (neither dominates completely)

Non-trivial means the optimum is not determined by inspection.

Design Step 5: Verify the Problem

Solve your LP. The optimum must:

• Not be at origin — "do nothing" is trivial
• Have at least 3 corners — Fundamental Theorem needs them
• Show a real trade-off — interior corner must be competitive

Fail any check → revise before submitting.

Live Demo: Designing the Samosa-Chapati Fundraiser

• Samosas (): 2 min baking, 300 UGX each; chapati (): 3 min, 200 UGX each
• Limits: 120 min baking time, 80 items total
• Objective: maximize revenue

Check-In: Diagnose a Design Failure

Your partner's LP has a feasible region with only one corner.

What design failure does this signal? What must be redesigned?

Structural Validity Test: Three Features

A valid LP problem must have:

1. Non-trivial optimum — not at the origin; not obvious by inspection
2. Multiple corners — at least three (Fundamental Theorem needs them)
3. Both constraints binding — each constraint eliminates some feasible points

Fail any feature → redesign.

Common LP Design Failures to Avoid

Four design traps:

• Trivial origin optimum — objective says "do nothing is best"
• Single-dominant variable — one variable always wins regardless of constraints
• Redundant constraint — one constraint is always weaker than another
• Ambiguous wording — the solver must infer what the problem means

Check-In: Trivial vs. Non-Trivial Objective

On the carpenter region (corners , , , ):

Which objective produces a trivial optimum?

• Option A: (maximize)
• Option B: (maximize)

Evaluate both at all four corners. Which is trivial?

Variable-Type Decision Is Part of the Design

Before writing the problem, decide:

• Integer-required variables → add "whole numbers only" to the problem statement
• Continuous variables → the fractional algebraic optimum is the final answer

The variable-type choice must appear in the problem statement.

The Design Template: Five Required Fields

1. Context: real-world scenario
2. Variables: ___, ___ (integer or continuous)
3. Constraints: each inequality with interpretation
4. Objective: maximize or minimize
5. Verified solution: optimum at ___, ___

Check-In: Start the Design Template

Fill in fields 1, 2, and 3 of the design template for your chosen context.

• Context: ___
• Variables: ___, ___
• First constraint: ___

Pause and write before the peer-review phase.

Peer Solve: Read a Classmate's LP

When solving a classmate's LP:

1. Identify the three structural features
2. Run all five stages without asking the designer questions
3. Write your complete solution including the interpreted sentence

If you cannot solve without asking — the design has an ambiguity failure.

Peer Review Rubric: Six Criteria

1. Non-trivial optimum (not origin)
2. At least 3 feasible corners
3. Both resource constraints active
4. Variable type stated
5. Wording unambiguous
6. Verified solution provided

Check-In: Apply the Rubric Systematically

Sample LP: A farmer plants maize () and beans () to maximize . Constraints: , , , .

Which rubric criterion fails? State why.

When Solver and Designer Disagree

If the solver and designer get different optimal answers, two causes:

• Designer error — the constraints or objective are wrong
• Ambiguous wording — the solver interpreted the problem differently

Both are design failures. Neither is the solver's fault.

Revise the wording until one unambiguous interpretation is possible.

Variant: Adding a Constraint to a Design

After completing the design, you add one more constraint.

Does the optimum change? Reason qualitatively:

• What happens to the feasible region?
• Can the optimum only stay the same or get worse (max) or better (min)?

Three Design Pitfalls That Invalidate LP Problems

Watch out:

• Design for the answer — steps run forward; the solution is what comes out, not what you put in
• Trivial feasible region — too many constraints collapses the region to a point
• Skip verification — design without solved verification is an incomplete deliverable

Key Takeaway: Valid LP Design Requirements

✓ Five design steps — context, variables, constraints, objective, verify

✓ Three structural features — non-trivial optimum, 3+ corners, active constraints

✓ Six rubric criteria — check every peer design against all six

✓ Verification is mandatory — design without it is incomplete

Coming Up: Lesson 10 — Full Consolidation

You have now designed, solved, and critiqued LP problems.

Lesson 10 — Linear Programming Consolidation:

• Two full assessment problems end to end
• All five stages for each
• Error classification rubric