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lecture10 - Simplex Algorithm Degeneracy and Cycling x3...

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e62: lecture 10 18/10/10 x 1 x 2 x 3 Simplex Algorithm 1 pick initial BFS encoding: B let m be index of maximizing (nonbasic) variable compute reduced profits increase until a basic variable becomes 0 add m to B and remove new nonbasic variable maximum positive? yes no N =3, M =1 vertex: ! 1 basic variable ! 2 nonbasic variables m = 3 B = { 1 } B = { 3 } e62: lecture 10 18/10/10 Degeneracy and Cycling Def. degeneracy = when there is a zero-valued basic variable Example where N =5, M =3 (3 basic variables) Intersection of N+1 constraint boundaries Def. cycling = keep swapping basic variables without changing solution 2 0 1.5 3 4.5 6 1 2 3 4 5 e62: lecture 10 18/10/10 Initialization Simple case Symmetric form LP If b nonnegative then the origin is a vertex General case Multiple methods One approach: big-M method 3 max cx s . t . Ax b x 0 e62: lecture 10 18/10/10 Minimum Cost Network Flows Network nodes Directed edges Decision variables = flows Objective = flow cost 4 1 2 5 8 7 4 3 6 V = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
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