Lecture5

Problem maxt unbounded 42x xigs2 if at any iteration

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Unformatted text preview: are non-positive, and the objective coefkient is (-) for maximization [or (+) for minimization] for that variable then, the problem is unbounded in the direction of that variable l l ernate Optimal Problem Special Cases Unbounded uftxr-20~~+$04 Iteration 0 Sk -.4yt.I*50 0 2~+4~s 100 @ Basic Xl ?I 5 x4 Soln I -2 -1 0 0 0 x3 1 -1 1 0 10 4 1 0 0 1 40 nr . . Special Cases Alternate Optimal Aiternate Optimal Iteration 0 If in an optimal tableau The coeffkient of any non-basic variable is zero in the z-row then, the probkm has alternate optimal solutions with equioptimal objective values Basic l z l x1 I I x2 x4 0 -4 -2 saln x3 010 I xJ 1 2 10 5 4 1 1 0 4 . . EM-602 / QM-710 (NJ) Lecture 5 Page 5-7 1 Alternate Optimal (iterationa 1 L 2) IL-h 0 2 0 1 1 -1 0 -1 2 see problem 2-l 7 1 1 Infeasible Problem X 24 3 3 I[q ~~~ I 01101 Special Cases Infeasible Iteration8 0 & 1 Infeasible Basic if, in an optimal tableau, An artifictai variable is basic and has value > zero, and The objective function value contains M if minimizing [or (A) if maximizing] then the probtem is infeasible Note: if the artiftciai variable is basic but with zero value, the problem is feasible, but had one or more redundant constraints L l x1 ~2 q ~q R, SoIn u 0 0 0 01 0 112 2 0 M 2+4M 0 4-SY 1 0 0 -1 1 -4 0 1 2 4 -3-3M_2-4M ;I:: 4-l 1 -1215 l L l+Skl g2 2 Ri -5 .. Homework Assignment Due in 1 we&k l Note: Textbook problem # 341 l l The remainder of the slides are presented for your reference only They were not discussed in class and are thus not included in the scope of the course EM-602 / QM-710 (NJ) Lecture 5 Page 5-8 Dual Problem Degenerate For every LP problem (called the Primal) we can derive a related symmetrkal LP (called the Dual) l The solution of the Primal LP automatically gives the optlmal solution to the Dual l The solution of the Dual LP automatkally gives the optimal solution to the Primal l The decision variables and objective for the primal are XJ and r; and for the dual they are y and w If In successive iterations of the tableau l The objective function value stays the same l the non-basic variable’s z-row entry is nonzero l the bask variable’s z-row entries are zero then the solution at this point is degenerate Note: Degeneracy may be temporary (degenerate point is suboptimal) or permanent (degenerate at optlmallty) Dual Problem Dual Problem l l l For every prlmal constraint there is a dual variable For every primal variable there is a dual constraint The co...
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