Inequalities and linear programming - (8.4-8.6)

Represent the graphs of linear inequalities in two unknowns on a plane

  • Graph
  • Examples
  • Solutions

Solve systems of linear inequalities in two unknowns

  • Theory
  • Examples
    Find the maximum and minimum values of subject to the constraints \(\left\{ \begin{array}{l}3x + 11y - 66 \le 0\\x + y - 6 \ge 0\\5x - 3y - 46 \le 0\end{array} \right.\).
  • Solutions

    The feasible solutions of all constraints are represented graphically as below.


    Let \(f(x{\rm{ }},{\rm{ }}y) = x + 2y\).
    Draw the straight line \(f(x{\rm{ }},{\rm{ }}y) = 0\).
    From the graph, \(f(x{\rm{ }},{\rm{ }}y)\) attains its maximum / minimum values at the points \((8{\rm{ }},{\rm{ }} - {\rm{ }}2)\) and \((11{\rm{ }},{\rm{ }}3)\).
    At the point \((8{\rm{ }},{\rm{ }} - {\rm{ }}2)\), \(f(8{\rm{ }},{\rm{ }} - {\rm{ }}2) = 8 + 2( - {\rm{ }}2) = 4\).
    At the point \((11{\rm{ }},{\rm{ }}3)\), \(f(11{\rm{ }},{\rm{ }}3) = 11 + 2(3) = 17\).

    Maximum value\( = \underline{\underline {17}} \)
    Minimum value\( = \underline{\underline 4} \)

    Alternative method:
    Consider the vertices \((0{\rm{ }},{\rm{ }}6)\), \((8{\rm{ }},{\rm{ }} - {\rm{ }}2)\) and \((11{\rm{ }},{\rm{ }}3)\) of the feasible region.
    At the point \((0{\rm{ }},{\rm{ }}6)\), \(f(0{\rm{ }},{\rm{ }}6) = 0 + 2(6) = 12\).
    At the point \(({\rm{8 }},{\rm{ }} - {\rm{ }}2)\), \(f(8{\rm{ }},{\rm{ }} - {\rm{ }}2) = 8 + 2( - {\rm{ }}2) = 4\).
    At the point \((11{\rm{ }},{\rm{ }}3)\), \(f(11{\rm{ }},{\rm{ }}3) = 11 + 2(3) = 17\).

    Maximum value\( = \underline{\underline {{\rm{17}}}} \)
    Minimum value\( = \underline{\underline {\rm{4}}} \)

Solve linear programming problems

  • Graph

  • Theory
  • Examples
    A shop sells two different jewellery sets A and B. The relevant information of the two jewellery sets of is as follows.
    NecklacesWatchesRingsSelling Price ($/k)
    Set A31225
    Set B11420
    Mr. Chan wants to buy at least 150 necklaces, 100 watches and 240 rings.

    (a) Write down all the constraints about the number of sets A and sets B to be bought.
    (b) How many sets of each type should Mr. Chan buy to minimize the cost? Find the minimum cost.
  • Solutions

    a) Suppose x sets A and y sets B are to be bought,the constraints are:
    \(\left\{ \begin{array}{l}3x + y \ge 150\\x + y \ge {100^{}}\\2x + 4y \ge {240^{}}\\x{\rm{ \text{ and } }}y{\rm{ \text{ are non - negative integer} }}{{\rm{s}}^{}}\end{array} \right.\),

    which are equivalent to \(\left\{ \begin{array}{l}3x + y \ge 150\\x + y \ge {100^{}}\\x + 2y \ge {120^{}}\\x{\rm{ \text{ and } }}y{\rm{ \text{ are non - negative integer} }}{{\rm{s}}^{}}\end{array} \right.\).

    b) Cost \(\begin{array}{c}\$ C(x{\rm{ }},{\rm{ }}y) = \$ (25x + 20y)\\ = \$ 5{(5x + 4y)^{}}\end{array}\)
    Represent the feasible solutions graphically and draw the straight line \(5(5x + 4y) = 0\), i.e. the straight line \(5x + 4y = 0\).

    The ordered pairs representing all points with integral coordinates in the shaded region represent the feasible solutions.

    From the graph, the cost is the minimum when \(x = 25\) and \(y = 75\).
    Mr. Chan should buy 25 sets A and 75 sets B.
    Minimum cost \(\begin{array}{l} = \$ 5(5x + 4y)\\ = \$ 5{(5 \times 25 + 4 \times 75)^{}}\\ = {\underline{\underline {\$ 2{\rm{ }}125}} ^{}}\end{array}\)