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L17 LP part3 Homework Review Multiple Solutions Degeneracy Unbounded problems Summary 1

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H16 8.35 2 Canonical… therefore Feasible!

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8.35 cont’d 3 Simplex Tableau rowbasicx1x2x3x4b b/a_pivo t ax321010-10n/a bx43201186min cc'-2000 First Tableau rowbasicx1x2x3x4b b/a_pivo t +Re to Radx302.6666710.3333316 /Rb by 3ex110.6666700.333336 2*Re+Rbfc'00.3333300.6666712 f+12=0 f= - 12

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8.39 4 Canonical… therefore Feasible!

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8.39 cont’d 5

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8.44 7 Canonical… therefore Feasible!

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8.44 cont’d 8

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8.44 9

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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 g i (x) and r enumber all x i 3.If b i < 0, then multiply constraint by (-1) 4.If constraint is ≤, then add slack s i 5.If constraint is ≥, then subtract surplus s i 10

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Std Form LP Problem 11 Matrix form All “ ≥0 ” i.e. non-neg. All “=“

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Canonical form Ex 8.4 & TABLEAU 12 basis all +1

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Simplex Method – Part 1 of 2 Single 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) 13

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Single-Phase Simplex Method 1. Set up LP prob in a SIMPLEX tableau add row for reduced cost, c j ’ 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

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Special cases? Multiple solutions Unbounded problems Degenerate solutions 15

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Multiple Solutions 16 Non-basic c i ’=0 Non-unique global solutions, ∆f = 0

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Unbounded problem 17 Pivot column coefficients a ij < 0

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Degenerate solution 18 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, slowly Sometimes it will “cycle” forever.

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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. 19

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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) 20

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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 21

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MIT and James Orlin © 2003 1 Chapter 3. The simplex algorithm Putting Linear Programs into standard form Introduction to Simplex Algorithm file Simplex2_AMII_05a_gr.

MIT and James Orlin © 2003 1 Chapter 3. The simplex algorithm Putting Linear Programs into standard form Introduction to Simplex Algorithm file Simplex2_AMII_05a_gr.

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