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L17 LP part3
• Homework• Review• Multiple Solutions• Degeneracy• Unbounded problems• Summary
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H16 8.35
2
1 2
1 2 3 4
1 2 3 4
( ) 2. .1 2 1 0 103 2 0 1 18
0i
Min f x xs tx x x xx x x x
x
x
1 2
1 2
1 2
( ) 2. .
2 103 2 18
0i
Max f x xs tx xx x
x
x
Canonical…thereforeFeasible!
8.35 cont’d
3
Simplex Tableau
row basic x1 x2 x3 x4 bb/
a_pivota x3 -1 2 1 0 10 -10 n/ab x4 3 2 0 1 18 6 minc c' -2 -1 0 0 0
First Tableau
row basic x1 x2 x3 x4 bb/
a_pivot+Re to Ra d x3 0 2.66667 1 0.33333 16/Rb by 3 e x1 1 0.66667 0 0.33333 6 2*Re+Rb f c' 0 0.33333 0 0.66667 12
f+12=0f= - 12
8.39
4
1 2
1 2 3 4
1 2 3 4
( ) 2. .2 1 0 50 0 1 2
0i
Min f x xs tx x x xx x x x
x
x
Canonical…thereforeFeasible!
8.39 cont’d
5
1 2
1 2 3 4
1 2 3 4
( ) 2. .2 1 0 50 0 1 2
0i
Min f x xs tx x x xx x x x
x
x
8.39 cont’d
6
8.44
7
1 2
1 2
1 2
1 2
( ). .4 3 9
2 62 6
0i
Max z x xs tx xx xx xx
x
1 2
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
( ). .4 3 1 0 0 91 2 0 1 0 62 1 0 0 1 6
0i
Min f x xs tx x x x xx x x x xx x x x x
x
x
Canonical…thereforeFeasible!
8.44 cont’d
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Simplex Tableaurow basic x1 x2 x3 x4 x5 b b/a_pivota x3 4 3 1 0 0 9 2.25 minb x4 1 2 0 1 0 6 6c x5 2 1 0 0 1 6 3d c' -1 -1 0 0 0
First Tableaurow basic x1 x2 x3 x4 x5 b b/a_pivot
e x1 1 0.75 0.25 0 0 2.25 3f x4 0 1.25 -0.25 1 0 3.75 3 ming x5 0 -0.5 -0.5 0 1 1.5 -3 n/ah c' 0 -0.25 0.25 0 0 2.25
f=-2.25
8.44
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First Tableaurow basic x1 x2 x3 x4 x5 b b/a_pivot
e x1 1 0.75 0.25 0 0 2.25 3f x4 0 1.25 -0.25 1 0 3.75 3 ming x5 0 -0.5 -0.5 0 1 1.5 -3 n/ah c' 0 -0.25 0.25 0 0 2.25
f=-2.25
Second Tableaurow basic x1 x2 x3 x4 x5 b b/a_pivot
h x1 1 0 0.4 -0.6 0 0i x2 0 1 -0.2 0.8 0 3j x5 0 0 -0.6 0.4 1 3k c' 0 0 0.2 0.2 0 3
f = - 3
1
2
5
3 4
033
, 0( ) 3
3
xxxx xfz f
x
Transforming LP to Std Form LP
1. If Max, then f(x) = - F(x)2. If x is unrestricted, split into x+ and x-, and
substitute into f(x) and all gi(x) and renumber all xi
3. If bi < 0, then multiply constraint by (-1)
4. If constraint is ≤, then add slack si5. If constraint is ≥, then subtract surplus si10
Std Form LP Problem
11
ntojxmtoib
bxaxa
bxaxabxaxa
tsxcxcxcfMin
j
i
mnmnm
nn
nn
nn
1,01,0
..)(
11
22121
11111
2211
x Matrix form
All “≥0” i.e. non-neg.
0x0bbAx
xcx T
..
)(tsfMin
All “=“
Canonical form Ex 8.4 & TABLEAU
12
124
1
14
1
114
1
28
116
521
421
321
xxx
xxx
xxx
basisall +1
Simplex Method – Part 1 of 2Single Phase Simplex Method
When the Standard form LP Problem has only≤ inequalties…. i.e. only slack variables, we can solve using the Single-Phase Simplex Method!(i.e. canonical form!)
If surplus variables exist… we need the Two-Phase Simplex Method –with artificial variables… Sec 8.6-7 (after Spring Break)
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Single-Phase Simplex Method1. Set up LP prob in a SIMPLEX tableau
add row for reduced cost, cj’ and column for min-ratio, b/a label the rows (using letters) of each tableau
2. Check if optimum, all non-basic c’≥0? 3. Select variable to enter basis(from non-basic)
Largest negative reduced cost coefficient/ pivot column
4. Select variable to leave basis Use min ratio column / pivot row
5. Use Gauss-Jordan elimination on rows to form new basis, i.e. identity columns
6.Repeat steps 2-5 until opt solution is found!14
Special cases?
• Multiple solutions• Unbounded problems• Degenerate solutions
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Multiple Solutions
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Non-basic ci’=0
Non-unique global solutions, ∆f = 0
Unbounded problem
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Pivot column coefficients aij < 0
1 2 3
2 3 1
2 1 1 00 2
x x xx x x
Degenerate solution
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First Tableaurow basic x1 x2 x3 x4 x5 b b/a_pivot
e x1 1 2 1 0 0 3 3/1f x4 0 3 2 1 0 0 0/2 ming x5 0 4 -1 0 1 0 neg n/ah c' 0 -2 -4 0 0 2.25
f=-2.25
Want to bring in x3 for x4… but the min ratio rule says no amount of x3!...
Therefore no change in f either.
Simplex method will move to a solution, slowlySometimes it will “cycle” forever.
More Terms• Degererate basic solution - one or more basic
variables has a zero value in a basic solution (i.e. b=0)• Degererate basic feasible solution - one or more basic
variables has a zero value in a basic feasible solution (i.e. b=0)
• Optimum basic solution – basic feasible solution with minimum f.
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Test 3• T/F 15 pts• M/choice (terms????) 10 pts• Excel Curve Fitting -set up Excel equations, for
one of five analytical equations, using cell labels only e.g. B4, C6 (i.e. no naming of variables) (25 pts)
• Transform prob to Standard LP Form (25 pts)• Solve LP problem using Simplex (25 pts)
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Summary• Simplex Method moves efficiently from one
feasible combination of basic variables to another.• Use Single-Phase Simplex Method when only
“slack” type constraints.• Multiple solutions• Unbounded solutions/problems• Degenerate Basic Solution• Degenerate Basic Feasible Solution
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