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Unformatted text preview: Lecture 16 Linear Programming LP examples • A post office requires different numbers of full time employees on different days of the week. The number of fulltime employees required on each day is given in the table. Union rules state that each fulltime employee must work five consecutive days and then receive two days off . The post office wants to meet its daily requirements using only fulltime employees. Formulate an LP that the post office can use to minimize the number of fulltime employees that must be hired. 60 3 2 ≤ + y x y x z 5 4 + = Feasible domain Optimal occurs at a vertex!!! . , 9 , 60 3 2 s.t. 5 4 max ≥ ≥ ≥ = + + + = w y x w y x y x z Slack Form . ) ( rank . s.t. max m A x b Ax cx z = ≥ = = What’s a vertex? . , ), ( 2 1 if vertex a called is polyhadren a in point A z y x z y z y x x = = Ω ∈ + = Ω ⇒ . of s in vertice found be can it then solution, optimal an has over max If . }  { Let Ω Ω ∈ ≥ Ω x cx Ax = b, x x = Fundamental Theorem . constraint one least at violates ' is, that , in not ' point a have must line the Thus, line. any contain not does However, solutions. optimal are *) ( line on points feasible all that follows It solutions. optimal also are and that means This . have must we 2, ) ( and , , * Since distinct. are , *, and 2 / ) ( * such that , exist there is, that not, is * suppose on, contraditi By . of vertex a is * that show will We solutions. optimal all among components zero of number maximum with * solution optimal an Consider x x yx x*+ z y cz cx* = cy = / cy+cz cx* = cz cx* cy cx z y x z y x z y x x x Ω Ω ≥ ≥ + = Ω ∈ Ω α Proof. ion.ion....
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This note was uploaded on 11/03/2010 for the course COMPUTER S CS 6363 taught by Professor Dingzhudu during the Fall '10 term at University of Texas at Dallas, Richardson.
 Fall '10
 DingzhuDu
 Algorithms

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