L6_10 - ESI 6314 Deterministic Methods in Operations...

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1/1 ESI 6314 Deterministic Methods in Operations Research Lecture Notes 7 University of Florida Department of Industrial and Systems Engineering / REEF
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2/1 Directions of Unboundedness Consider an LP in a standard form with the feasible region described by Ax = b and x 0. Let S denote this feasible region. A nonzero vector d is a direction of unboundedness if for all x S and any c 0: x + cd S . That is, from any point in the feasible region, it is possible to move infinitely far along the direction d and remain in the feasible region.
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3/1 Direction of Unboundedness Example Standard form of the Dorian Auto LP problem (recall the problem that we formulated earlier): min z = 50 x 1 + 100 x 2 s.t. 7 x 1 + 2 x 2 - e 1 = 28 2 x 1 + 12 x 2 - e 2 = 24 x 1 , x 2 , e 1 , e 2 0
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4/1 Direction of Unboundedness Example The direction of unboundedness will be, for ex- ample, defined by a 45-degree angle from the origin. That is, d = [1 , 1 , 9 , 14] T . In general, it can be shown that d is a direction of unboundedness if and only if Ad = 0 and d 0.
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5/1 Important fact Theorem: Consider an LP in a standard form: Ax = b and x 0, having basic feasible solutions f 1 , f 2 , ..., f k . Any point x in the LP’s feasible region may be written in the form x = d + k X i =1 σ i f i , where d is 0 or a direction of unboundedness, and k i =1 σ i = 1, σ i 0. The expression k i =1 σ i f i (where k i =1 σ i = 1, σ 0) is referred to as a convex combination of vectors f 1 , f 2 , ..., f k .
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6/1 Illustration Consider the following LP:
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L6_10 - ESI 6314 Deterministic Methods in Operations...

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