Linear Equation by nkc8ft

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									Maximum and Minimum
       Chapter 6




                      1
   Find inequalities from region
In the graph, find inequalities for regions A, B, C, D, E




                                                            2
                          4 x  3 y  24
                          
Region A has 3 boundaries  x  2 y  10
                               x0
                          
So it is
 4 x  3 y 24
 
  x  2 y 10
 
      x 0




                                            3
                           x  2 y  10
                          4 x  3 y  24
                          
                          
Region B has 5 boundaries  x  y  4
                               x0
                          
                          
                               y0
So it is
  x  2 y 10
 4 x  3 y 24
 
 
  x y 4
      x 0
 
 
      y 0




                                            4
                           x  2 y  10
                          4 x  3 y  24
                          
Region C has 4 boundaries 
                           x y 4
                          
                               x0
So it is
  x  2 y 10
 4 x  3 y 24
 
 
  x y 4
  x 0
 




                                            5
                           x  2 y  10
                          
Region D has 3 boundaries 4 x  3 y  24
                           x y 4
                          
So it is
  x  2 y 10
 
 4 x  3 y 24
  x y 4
 




                                            6
                           x y 4
                          
Region E has 3 boundaries 4 x  3 y  24
                           y0
                          
So it is
  x y 4
 
 4 x  3 y 24
  y 0
 




                                            7
                        Exercises
1. In the graph, find inequalities for regions F and G




                                                         8
            Positive linear functions
If a  0, b  0 then z  ax  by is called a positive linear function.
If z  c then the line ax  by  c is called a z-value line.
1) All z-value lines are parallel.
2) Value of z  ax  by is increasing from southwest to northeast
For example z  2 x  3 y, draw 0-value, 6-value and  6  -value lines.




                                                                            9
z-value lins of z  2 x  3 y




                                10
            Maximum and Minimum
In a finite polygon region, any positive linear function z  ax  by will
reach its maximum and minimum values at the corner points.

                                               Maximum value point




    Minimum value point

                                                                            11
 No maximum in unbounded region
In a unbounded region (of Quadrant I), a positive linear function z  ax  by
only has the minimum value, but NO MAXIMUM



                                                  No Maximum value




  Minimum value point

                                                                           12
                          Example
Find maximum value of the subject function z  3x  4 y in the region
 2x  y  4
 x  2 y  4


 x0
 y0

First we draw 4 lines
 2x  y  4
 x  2 y  4


 x0
 y0

Then we shadow the region


                                                                    13
Next we have to decide the corner points. Among 4 corner points,
3 of them are already known. They are (0,0), (0,2) and (2,0).
We need to solve the last one, which is the common point of lines
 2 x  y  4 and  x  2 y  4
Therefore , we setup
 2 x  y  4    (1)

 x  2 y  4    (2)
(1)  2(2) to get 5 y  12
               12
         y            (3)
                5
Plug into (2)
         12               4
x  2       4 x 
          5               5

Corner points are (0,0), (0,2), (2, 0) and    
                                             4 12
                                              ,
                                             5 5
                                                                    14
Now we calculate value of z  3x  4 y at those 4 corner points and
draw the following table

   Corner
             Value z = 3x + 4y
   points
   (0, 0)                      0
   (0, 2)              30+42 = 8
   (2, 0)              32+40 = 6

   
   4 12
    ,
   5 5
             3
                4
                5
                     4
                         12
                          5
                               12 Maximum



 The maximum value of z is 12 at point              4 12
                                                      ,
                                                     5 5    
                                                                      15
                            Example
Find maximum and minimum values of z  x  10 y in the region
 x  4 y  12
 x  2y  0


 2y  x  6
 x6

First we draw 4 lines
 x  4 y  12
 x  2y  0


 2y  x  6
 x6

Then we shadow the region


                                                                16
Next we have to decide the corner points. Among 4 corner points,
3 of them are easy to be found. They are (0,3), (6,3) and (6,6).
The last one is the common point of lines
x  4 y  12 and x  2 y  0. Therefore we setup
 x  4 y  12    (1)

 x  2 y  0    (2)
(1)  (2), we can get
   6 y  12  y  2
Plug into (2),
   x40 x  4
So the last coner point
is (4, 2)

                                                               17
Now we calculate value of z  x  10 y at those 4 corner points and
draw the following table

 Corner
           Value z = x + 10y
 points
  (0, 3)       0+3(10)=30
  (6, 3)      6+10(3) = 36
              6+10(6) = 66
  (6, 6)
                maximum
  (4, 2)      4+10(2) = 24
                minimum

 The maximum value of z is 66 at point (6,6)
 The minimum value of z is 24 at point (4,2)
                                                                      18
                             Exercises
Find maximum and minimum values in the region for the functions
1. z  3 x  2 y                 2. z  0.4 x  0.75 y
           (2, 8)
                                   (0, 12)

                         (4, 6)                    (4, 8)
  (1, 4)



                                                             (7, 3)



                    (4, 1)
                                       (0, 0)               (8, 0)


                                                                      19
3. Find maximum and minimum of z  4 x  y if they exist




 (0, 10)




     (2, 4)

              (5, 2)
                              (15, 0)




                                                           20
4. Find maximum and minimum of z  3x  6 y in region
    x  y  10
   
   5 x  2 y  20
    x  2y  4
   




                                                        21

								
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