Optimize within a Simple Feasible Region
An objective assigns a modeled value to each feasible choice.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: For x≥0, y≥0, x+y≤5, maximize P=2x+3y. Candidate answers: (A) 14 (B) 15 (C) 16 (D) 30.
- Independent check inputs: For x≥0, y≥0, x+y≤6, maximize P=2x+3y. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36.
Learn the idea
A mathematical plan can require simultaneous equalities or inequalities. The feasible set contains inputs satisfying all constraints, and a linear objective asks which allowed choice gives the greatest or least modeled value. An objective assigns a modeled value to each feasible choice. When one resource can be allocated to variables with unequal positive contributions, the larger contribution can determine the optimal corner.
Here is the method on the supplied teaching case. Each unit assigned to y contributes 3 rather than 2 to P, so the maximum uses x=0,y=5. P=15. Equivalently P=2(x+y)+y≤2(5)+5.
An optimum must satisfy the original constraints, not merely make the expression large. The continuous model has no additional hidden caps here; adding such restrictions could change the best allocation.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: For x≥0, y≥0, x+y≤5, maximize P=2x+3y. Candidate answers: (A) 14 (B) 15 (C) 16 (D) 30. Solution: Each unit assigned to y contributes 3 rather than 2 to P, so the maximum uses x=0,y=5. P=15. Equivalently P=2(x+y)+y≤2(5)+5. The final answer is 15. An optimum must satisfy the original constraints, not merely make the expression large. The continuous model has no additional hidden caps here; adding such restrictions could change the best allocation.
Check your reasoning
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Helpful tips
- An objective assigns a modeled value to each feasible choice.
- An optimum must satisfy the original constraints, not merely make the expression large.
See the answer and explanation
18
Each unit assigned to y contributes 3 rather than 2 to P, so the maximum uses x=0,y=6. P=18. Equivalently P=2(x+y)+y≤2(6)+6.
Take it further
Changed-input task: For x≥0, y≥0, x+y≤12, maximize P=2x+3y. Candidate answers: (A) 35 (B) 36 (C) 37 (D) 72. Show the mathematical reason, not only the selected answer. Expected result: 36. Each unit assigned to y contributes 3 rather than 2 to P, so the maximum uses x=0,y=12. P=36. Equivalently P=2(x+y)+y≤2(12)+12.
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More practice: try new cases
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