LP—Graphical Solution Method
MSci331—Week 2-3
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Convex Set and Extreme Points
2
LP: Example (Papa Louis )
• Papa Louis manufacturers wooden tables and chairs for small kids. Each "table" built: Sells for $27 and uses $10 worth of raw materials, increases Papa Louis’s variable labor/overhead costs by $14. Requires 2 hours of finishing labor and 1 hour of carpentry labor. Each "chair" built: Sells for $21 and uses $9 worth of raw materials, increase Papa Louis’s variable labor/overhead costs by $10. Requires 1 hours of finishing labor AND 1 hour of carpentry labor. Each week Papa Louis can obtain only 100 finishing hours and only 80 carpentry hours. Also demand for the chairs is unlimited. However, at most 40 tables are bought each week. Papa Louis wants to maximize weekly profit (revenues - expenses).
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LP: Example (Papa Louis )
4
LP: Example (Papa Louis )
5
LP: Example (Papa Louis )
6
LP: Example (Papa Louis )
7
8
100
90
80
70
60
50
40
30
20
10
10 20 30 40 50 60 70 80 90 100 BP100
BP200N
umbe
r of
Bod
yPlu
s 20
0
Number of BodyPlus 100
Example: 2
Machining and Welding
Painting and Finishing
Assembly, Test and Packaging
BodyPlus 200 Requirement
Feasible region
(0,0)
(0,45), Z= 20,745
(30,30), Z=24,960
(50,50/ 3), Z=26,233
Max 371BP100 + 461BP200 s.t.
8BP100 + 12BP200 600 Machining and Welding 5BP100 + 10BP200 450 Painting and Finishing 2BP100 + 2BP200 140 Assembly, Test, and Packaging -0.25BP100 + 0.75BP200 0 BodyPlus 200 Requirement
BP100, BP200 > 0
Example 2: Multiple Optimal Solutions
• Consider the LP model
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Example 2: Multiple Optimal Solutions
10
Infeasible Solution
11
Unbounded Solution
12
LP Model to Standard Form
• Convert LP format to a standard form1 21 2
11
2 2
1 2 1 2
1 2 1 2
1 2 51 2 3 4 6
3
4
5
6
, ,
max 12 9max 12 9s.t.s.t.
10001000
1500 1500
1750 1750
4 2 4800 4 2 4800
, 0 , , , 0
x xx x
xx
x x
x x x x
x x x x
x x x
x
x
x x
x
x
x x x
1 21 2
1 21 2
1 41
1 21 2
51 21 2
1 21 2
51 2 3 4 6
3
6
, ,
min 9 6min 9 6 s.t.s.t. 2 10
2 1050
5040
402 100
100 2 152 15
, 0, , , 0
w ww w
w ww w
w ww
w ww w
w w ww w
w ww w
w w w w w w
w
w
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Insight from the Geometric Procedure
1 2
1 2
1 2
1
1 2
Max 2 3
. .
5
3 35
20
, 0
Z x x
s t
x x
x x
x
x x
For a constraint to be reasonable, all terms in the constraints must have the same units.
14
40
35
30
25
20
15
10
5
0 5 10 15 20 25 30 35 40
Insights from the Geometric Procedure
x111
22
33
99
55
x2
66
Constraint 1
Constraint 2
Constraint 3
88
1010
Z
77
11 (x1=0,x2=0,Z=0)
22 (x1=0,x2=5,Z=15)
88 (x1=5,x2=10,Z=40)
1010 (x1=20,x2=5,Z=55)
77 (x1=20,x2=0,Z=40)
1 2Max 2 3Z x x
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Basic and Nonbasic Variables
4040
3535
3030
2525
2020
1515
1010
55
0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111
222
333
999
555
xx22
666
Constraint 1
Constraint 2
Constraint 3
888
101010
Z
777
1 2Max 2 3Z x x
4040
3535
3030
2525
2020
1515
1010
55
0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111
222
333
999
555
xx22
666
Constraint 1
Constraint 2
Constraint 3
888
101010
Z
777
1 2Max 2 3Z x x
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0
2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15
3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No
4 (x2,s1,s2) (x1,s3) No solution n/a n/a
5 (x1,s2,s3) (x2,s1) (-5,0,0,40,25) No
6 (x1,s1,s3) (x2,s2) (35,0,40,0,-15) No
7 (x1,s1,s2) (x2,s3) (20,0,25,15,0) Yes 40
8 (x1,x2,s3) (s1,s2) (5,10,0,0,10) Yes 40
9 (x1,x2,s2) (s1,s3) (20,25,0,-60,0) No
10 (x1,x2,s1) (s2,s3) (20,5,20,0,0) Yes 55
16
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
123456789
10
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0
23456789
10
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0
2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15
3456789
10
Basic and Nonbasic Variables
4040
3535
3030
2525
2020
1515
1010
55
0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111
222
333
999
555
xx22
666
Constraint 1
Constraint 2
Constraint 3
888
101010
Z
777
1 2Max 2 3Z x x
4040
3535
3030
2525
2020
1515
1010
55
0 5 10 15 20 0 5 10 15 20 25 30 35 40 25 30 35 40 xx11111
222
333
999
555
xx22
666
Constraint 1
Constraint 2
Constraint 3
888
101010
Z
777
1 2Max 2 3Z x x
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0
2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15
3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No
456789
10
Point Basic VariableNonbasic Variable
Solution (x1,x2,s1,s2,s3)
FeasibilityObjective
Function Value
1 (s1,s2,s3) (x1,x2) (0,0,5,35,20) Yes 0
2 (x2,s2,s3) (x1,s1) (0,5,0,20,20) Yes 15
3 (x2,s1,s3) (x1,s2) (0,11.7,-6.7,0,20) No
4 (x2,s1,s2) (x1,s3) No solution n/a n/a
5 (x1,s2,s3) (x2,s1) (-5,0,0,40,25) No
6 (x1,s1,s3) (x2,s2) (35,0,40,0,-15) No
7 (x1,s1,s2) (x2,s3) (20,0,25,15,0) Yes 40
8 (x1,x2,s3) (s1,s2) (5,10,0,0,10) Yes 40
9 (x1,x2,s2) (s1,s3) (20,25,0,-60,0) No
10 (x1,x2,s1) (s2,s3) (20,5,20,0,0) Yes 55
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