Linear Optimization in Context

Explore objective function, feasible vertices, model limitations.

Lesson 5 of 53 parts Practice Flashcards

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Objective Function

Objective: Explain objective function and use it to solve or interpret a problem.

A linear objective P=ax+by gives a score, profit, or cost for a feasible choice. Units and coefficient meanings must be stated. They tell what an extreme value actually measures.

Worked examples

Try itLinear Optimization in Context — practice 1.1: For P=2x+4y, find P at vertex (5,2).Show answer

Step 1: Substitute x=5, y=2. Step 2: Compute 2(5)+4(2). Step 3: The objective value is 18.

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Objective Function

Objective: Explain objective function and use it to solve or interpret a problem.

A linear objective P=ax+by gives a score, profit, or cost for a feasible choice. Units and coefficient meanings must be stated. They tell what an extreme value actually measures.

Worked examples

Try itLinear Optimization in Context — practice 1.1: For P=2x+4y, find P at vertex (5,2).Show answer

Step 1: Substitute x=5, y=2. Step 2: Compute 2(5)+4(2). Step 3: The objective value is 18.

Feasible Vertices

Objective: Explain feasible vertices and use it to solve or interpret a problem.

For a bounded polygonal feasible region, a linear objective reaches an extreme at a vertex, possibly along an entire edge. Evaluate every feasible vertex. Yes, when the objective is parallel to that edge.

Worked examples

Try itLinear Optimization in Context — practice 2.1: For P=3x+5y, find P at vertex (8,2).Show answer

Step 1: Substitute x=8, y=2. Step 2: Compute 3(8)+5(2). Step 3: The objective value is 34.

Model Limitations

Objective: Explain model limitations and use it to solve or interpret a problem.

Real decisions may require integer counts or extra conditions absent from the simple feasible region; interpret and check the final candidate. Rounding can violate one of the constraints.

Worked examples

Try itLinear Optimization in Context — practice 3.1: For P=4x+6y, find P at vertex (11,2).Show answer

Step 1: Substitute x=11, y=2. Step 2: Compute 4(11)+6(2). Step 3: The objective value is 56.

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