Further Mathematics · Pearson Edexcel 9FM0 · Option D1 · Topic 5

Linear programming

Writing real problems as linear programs and solving them graphically and with the Simplex algorithm.

  • 5 sub-topics
  • 2 in Year 12, 3 in Year 13
  • 69 practice questions
  • 5 animated lessons
Practice questions (69) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 5.1Formulating linear programming problemsY12
  2. 5.1Slack, surplus and artificial variablesY13
  3. 5.2Graphical solution of LP problemsY12
  4. 5.3The simplex algorithmY13
  5. 5.4Two-stage simplex and big-MY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 69 questions in this topic, free, no account needed.

Linear programming

Easy

Writing a constraint · Year 12 · 1 mark

  1. A workshop makes xx chairs and yy tables. Each chair takes 6 hours and each table takes 9 hours, and 180 hours are available. Which constraint models this?
    (1)

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  1. The hours used cannot be more than the hours available.DD: 6x+9y≤1806x + 9y \leq 180

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Linear programming

Medium

Formulating a linear programming problem · Year 12 · 5 marks

A workshop makes chairs and tables. A chair takes 2 hours of cutting and 1 hour of finishing; a table takes 3 hours of cutting and 2 hours of finishing. Each week there are 120 hours of cutting time and 70 hours of finishing time. At least 2 times as many chairs as tables must be made. The profit is £12 on a chair and £15 on a table.
  1. Formulate this as a linear programming problem, defining your variables.
    (5)
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  1. Let xx be the number of chairs and yy the number of tables made each week.
  2. maximise P=12x+15y\text{maximise } P = 12x + 15y
  3. One constraint for each resource.2x2x+3y≤120,  x{}+ 3y \leq 120, \ \ x+2y≤70{}+ 2y \leq 70
  4. x≥2y,  x,y≥0x \geq 2y, \ \ x, y \geq 0

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Linear programming

Hard

Graphical solution of a linear programming problem · Year 12 · 5 marks

Figure 1 shows the feasible region RR for the constraints x+y≤10x + y \leq 10, 3x+y≤213x + y \leq 21, x,y≥0x, y \geq 0.
xyR
Figure 1
  1. (a)
    Find the coordinates of the vertices of RR.
    (3)
  2. (b)
    Find the maximum value of P=2x+5yP = 2x + 5y on RR, and where it occurs.
    (2)
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Part (a)3 marks

  1. Solve the pairs of boundary equations simultaneously.(0,0), (0,10), (112,92), (7,0)(0, 0), \ (0, 10), \ (\frac{11}{2}, \frac{9}{2}), \ (7, 0)

Part (b)2 marks

  1. Test each vertex, or slide an objective line.P(0,0)P(0, 0)=0, P(0,10){}= 0, \ P(0, 10)=50, P(112,92){}= 50, \ P(\frac{11}{2}, \frac{9}{2})=672, P(7,0){}= \frac{67}{2}, \ P(7, 0)=14{}= 14
  2. Pmax⁡=50 at (0,10)P_{\max} = 50 \text{ at } (0, 10)

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