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International ASMaths: Decision 1EdexcelRevision NotesLinear ProgrammingGraphical Solution of LP ProblemsSketching the Feasible Region

Sketching the Feasible Region (Edexcel International AS Maths: Decision 1)

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Paul

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Paul

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Introduction to Solving an LP Problem Graphically

How do I solve a linear programming problem graphically?

  • For problems with two decision variables
    • Plot the constraints (inequalities) accurately on a graph
    • This leads to the feasible region
    • The optimal solution will be one of the vertices of the feasible region
  • In harder problems
    • There may be a third decision variable
      • There will be a connection to one (or both) of the other decision variables
      • All constraints can be rewritten in terms of just two of the decision variables
      • E.g. x comma space y comma space z are the numbers of chairs, tables and desks made by a furniture manufacturer
        The number of desks produced is twice the number of tables, i.e. z equals 2 y
    • The optimal solution may not give integer values for the decision variables but the context demands they are
      • E.g. Is it possible to make 3.65 chairs and 4.2 tables per day?

Feasible Region

What is the feasible region?

  • The feasible region is the set of all values that satisfy all the constraints in a linear programming problem
    • In practice, this is the area on a graph that satisfies all of the inequalities
      • This includes the non-negativity inequality

How do I find the feasible region?

  • To find the feasible region
    • Accurately plot each inequality (constraint) on a graph
      • Plot each inequality as a straight line
        • Rearranging to the form y less or equal than m x plus c or y greater or equal than m x plus c may help
        • It can be easier to determine two points that lie on each line, plot and join them up
      • Draw the line solid for inequalities involving ≤ or ≥, or dotted for inequalities involving > or <
        • < and > are rare in linear programming problems
    • Shade the part of the graph not satisfied by each inequality
      • It is easier to see a 'blank' area rather than an area shaded several times
    • The feasible region is the area on the graph left unshaded
      • It is the area that satisfies all of the inequalities
      • It is usually labelled with the capital letter R
  • Label the line of each inequality around the edge of the graph

Examiner Tip

  • Exam questions will provide a graph for you to accurately plot the inequalities
    • Alternatively they may provide an accurate graph with some or all of the inequalities already plotted
  • It is helpful to complete the shading for each inequality as you add it to the graph
    • If you leave the shading to the end it can become confusing

Worked example

A linear programming problem is formulated as

Maximise

P equals 30 x plus 40 y

subject to

table row cell x plus y end cell less or equal than 10 row cell 3 x plus 2 y end cell less or equal than 24 row cell x plus 2 y end cell less or equal than 18 row cell x comma space y end cell greater or equal than 0 end table

Show graphically the feasible region, R, of the linear programming problem.

The objective function is not needed to plot the feasible region

Plot each inequality as a straight line graph with the 'unwanted' side shaded
For this problem, all lines will be solid lines
The first constraint will be the line y equals negative x plus 10 (gradient -1 and y-axis intercept 10)
(You may find it easier to 'see' that points like (0, 10) and (10, 0) lie on the line, which you can plot and join up)

feasible-1

Plot and label the rest of the inequalities in the same way

feasible-2

Label the feasible region with R

feasible-3-green

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.