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LP Examples: another Max and a min
Dr. Ron Lembke
Slide2Example 2
mp3  4
min electronics
 2
min assembly
DVD
 3
min electronics
 1
min assembly
Min available
: 240 (elect) 100 (
assy
)
Profit / unit: mp3 $7, DVD $5
X
1
= number of mp3 players to make
X
2
= number of DVD players to make
Slide3Standard Form
Max 7x
1
+ 5x
2
s.t. 4x1 + 3x2 <= 240 2x1 + 1x2 <= 100 x1 >= 0 x2 >= 0
electronics
assembly
Slide4Graphical Solution
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
2
X
1
Slide5Graphical Solution
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
1
= 0, X
2
= 80
X
1
= 60, X
2
= 0
Electronics Constraint
X
2
X
1
4x
1
+
3x
2
<=
240
x
1
=0, x
2
=80
x
2
=0, x
1
=60
Slide6Graphical Solution
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
1
= 0, X
2
= 100
X
1
= 50, X
2
= 0
Assembly Constraint
X
2
X
1
2x
1
+
1x
2
<=
100
x
1
=0, x
2
=100
x
2
=0, x
1
=50
Slide7Graphical Solution
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
Assembly Constraint
Electronics Constraint
Feasible Region – Satisfies all constraints
X
2
X
1
Slide80 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
Isoprofit Line:
$7X
1
+ $5X
2
= $210
(0, 42)
(30,0)
Isoprofit Lnes
X
2
X
1
Slide9Isoprofit Lines
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
$210
$280
X
2
X
1
Slide10Isoprofit Lines
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
$210
$280
$350
X
2
X
1
Slide11Isoprofit Lines
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
(0, 82)
(58.6, 0)
$7X
1
+ $5X
2
= $410
X
2
X
1
Slide12Mathematical Solution
Obviously, graphical solution is slow
We can prove that an optimal solution always exists at the intersection of constraints.
Why not just go directly to the places where the constraints intersect?
Slide13Constraint Intersections
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
1
= 0 and 4X
1
+ 3X
2
<= 240
So X
2
= 80
X
2
X
1
4X
1
+ 3X
2
<= 240
(0, 0)
(0, 80)
Slide14Constraint Intersections
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
2
= 0 and 2X
1
+ 1X
2
<= 100
So X
1
= 50
X
2
X
1
(0, 0)
(0, 80)
(50, 0)
Slide15Constraint Intersections
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
4X
1
+ 3X
2
=
240
2X
1
+ 1X
2
=
100 – multiply by 2
X
2
X
1
(0, 0)
(0, 80)
(50, 0)
4X
1
+ 3X
2
=
240
4X
1
2X
2
=
200 add rows together
0X
1
+ 1X
2
=
40 X
2
= 40 substitute into #2
2
X
1
+ 40
=
100 So X
1
= 30
Slide16Constraint Intersections
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
2
X
1
(0, 0)
$0
(0, 80)
$400
(50, 0)
$350
(30,40)
$410
Find profits of each point.
Substitute into
$
7X
1
+ $5X
2
Slide17
Do we have to do this?
Obviously, this is not much fun: slow and tedious
Yes, you have to
know
how to do this to solve a twovariable problem.
We won’t solve every problem this way.
Slide18Constraint Intersections
0 20 40 60 80
80
20
40
60
0
100
DVD players
mp3
X
2
X
1
Start at (0,0), or some other easy feasible point.
Find a profitable direction to go along an edge
Go until you hit a corner, find profits of point.
If new is better, repeat, otherwise, stop.
Good news:
Excel can do
this for us
.
Using the
Simplex Algorithm
Slide19Minimization Example
Min
8x
1
+ 12x2 s.t. 5x1 + 2x2 ≥ 20 4x1 + 3x2 ≥ 24 x2 ≥ 2
x1 , x2 ≥ 0
Slide20Minimization Example
Min
8x
1
+ 12x2 s.t. 5x1 + 2x2 ≥ 20 4x1 + 3x2 ≥ 24 x2 ≥ 2
x1 , x2 ≥ 05x1 + 2x
2
=20
If x
1
=0, 2x
2=20, x2=10 (0,10)If x2=0, 5x1=20, x
1=4 (4,0)
4x
1 + 3x2 =24
If x1=0, 3x2=24, x2=8 (0,8)If x2=0, 4x1
=24, x1=6 (6,0)x2= 2
If x
1=0, x2=2No matter what x1 is, x2=2
Slide21Graphical Solution
0
2
4
6
8
8
2
4
6
0
10
5x
1
+ 2x
2
=20
X
2
X
1
4x
1
+
3x
2
=24
x
2
=2
Slide220
2
4
6
8
8
2
4
6
0
10
5x
1
+ 2x
2
=20
X
2
X
1
4x
1
+3x
2
=24
x
2
=2
(0,10)
[5x
1
+
2x
2
=
20]*3
[4x
1
+3x
2
=
24]*2
1
5x
1
+
6x
2
= 60
8x
1
+6x
2
= 48

7x
1
= 12
x
1
= 12/7= 1.71
5x
1
+2x
2
=
20
5*1.71 + 2x
2
=
20
2x
2
= 11.45
x
2
= 5.725
(1.71,5.73)
(1.71,5.73)
Slide230
2
4
6
8
8
2
4
6
0
10
5x
1
+ 2x
2
=20
X
2
X
1
4x
1
+3x
2
=24
x
2
=2
(0,10)
(1.71,5.73)
4x
1
+3x
2
=24
x
2
=2
4x
1
+3*2 =24
4x
1
=18
x
1
=18/4 = 4.5
(4.5,2)
(4.5,2)
Slide240
2
4
6
8
8
2
4
6
0
10
5x
1
+ 2x
2
=20
X
2
X
1
4x
1
+3x
2
=24
x
2
=2
(0,10)
(1.71,5.73)
Z=8x
1
+12x
2
8*0 + 12*10 =
120
(4.5,2)
Z=8x
1
+12x
2
8*1.71 + 12*5.73 =
82.44
Z=8x
1
+12x
2
8*4.5+ 12*2 =
60
Lowest Cost
Slide25IsoCost
Lines
0
2
4
6
8
10 12
8
2
4
6
0
10
5x
1
+ 2x
2
=20
X
2
X
1
4x
1
+
3x
2
=24
x
2
=2
Z=8x
1
+
12x2Try 8*12 = 96x1=0
12x
2
=96, x
2
=8
x
2
=0
8x
1
=96
,
x
1
=12
Slide26Summary
Method for solving a twovariable problem graphically
Find end points of each constraint
Draw constraints
Figure out which intersections are interesting
Use algebra to solve for intersection ptsFind profits (or costs) of intersectionsChoose the best one Isoprofit (or IsoCost) lines can help find the most interesting points
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