線性規劃：限制下的最優選擇

1Summary 重點摘要

Constraints form a feasible region; the optimum hides at a corner point.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Define variables and write the constraintsThe first step of linear programming is always to define the variables, for example let x be the number of product A and y the number of product B, then translate the words into inequalities: 'each A takes 2 hours, each B takes 1 hour, and at most 100 machine hours are available' becomes 2x + y ≤ 100. Never forget x ≥ 0 and y ≥ 0, since production cannot be negative; without them the feasible region extends wrongly into negative territory, and every corner and answer found later is invalid.
2. Step 2: Draw the boundary lines and choose the shadingTreat each inequality as an equation to draw its boundary line; for 2x + y = 100 use the points (50, 0) and (0, 100). Then test a point to decide which side to shade, and the origin is the easiest test point: substituting gives 2 × 0 + 0 = 0 ≤ 100, which holds, so shade the side containing the origin; if the test fails, shade the other side. This method works for both ≤ and ≥. Watch the line style as well: a solid line when equality is included, a dashed line when it is not, because points on a dashed line are not solutions.
3. Step 3: Locate the feasible region and the corner pointsThe part where all the shaded regions overlap is the feasible region, collecting every plan that satisfies every constraint at once. Corner points are intersections of boundary lines found by solving simultaneous equations: subtracting x + y = 60 from 2x + y = 100 gives x = 40, and substituting back gives y = 20, so the corner is (40, 20). Find and label every corner on the graph, because the optimum must occur at a corner; missing one corner may mean the answer is not the true maximum or minimum.
4. Step 4: Substitute the corners into the objective functionSubstitute each corner into the objective function, for example P = 3x + 2y, and evaluate one by one: (0, 0) gives 0, (50, 0) gives 150, (40, 20) gives 160 and (0, 60) gives 120. The fundamental theorem of linear programming guarantees the maximum or minimum occurs at a corner, so comparing these values is enough: the largest is 160, giving the optimal solution (40, 20). Tabulate the values of the objective at the corners as you go, which prevents copying errors and missed corners, and makes the working easy for the examiner to follow.
5. Step 5: Check the integer requirementWhen the variables count discrete objects such as units of a product, the optimal solution must be integral. If the best corner has fractional coordinates, for example 8.5, test the nearest integer points still inside the feasible region and compare their objective values. This is a classic DSE trap: the corner itself is not allowed, while neighbouring integer points may both be feasible, and only calculation decides which is better; never estimate. Show this checking process clearly in the answer to earn the method marks.
6. Step 6: State the conclusion in wordsAfter the calculation, respond to the question with a complete sentence, for example: 'produce 40 units of product A and 20 units of product B for a maximum profit of 160 dollars.' Writing only the point (40, 20) and the number 160 loses presentation marks, because the final step of an applied question is to report the answer back in the context of the problem. The conclusion should name the variables, the quantities and the optimal value of the objective clearly, leaving the examiner nothing to guess, and it signals a complete solution.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Three steps: variables and constraints → feasible region → substitute corners.
• The optimum always sits at a corner point — a theorem.
• Never drop x≥0, y≥0.
• Check nearest integer points when items are whole.
• End with a sentence answering the question.