[管理學]運籌學課件4_線性規(guī)劃-預覽頁
【正文】 … + xn . a11x1 + a12x2 + … + a1nxn = b1 a21x1 + a22x2 + … + a2nxn = b2 … … am1x1 + am2x2 + … + amnxn = bm xj ≥ 0 （ j = 1,2,…,n） bi ≥ 0 （ i = 1,2,…,m） 特點： 目標函數(shù)為極 大化； 除決策變量的 非負約束外，所有 的約束條件都是等 式，且右端常數(shù)均 為非負； 所有決策變量 均非負 。 解 ： 令 x3= x4 x5 其中 x x5 ≥0 ； 對第一個約束條件加上松弛變量 x6 ； 對第二個約束條件減去松弛變量 x7 ； 對第三個約束條件兩邊乘以“ 1” ； 令 z’=z 把求 min z 改為求 max z’ max z’= x12x2+3x4 3x5 . x1+x2+x4x5+x6=7 x1x2+x4x5x7=2 3x1x22x4+2x5=5 x1,x2,x4,x5,x6,x7≥0 ),2,1(0),2,1(..m a x11njxmibxatsxczjinjjijnjjj???????????LP的幾種表示形式： ? ?12, , , nc c c c? ? ? 價 值 向 量m a x ( 1 ). . ( 2 )0 ( 3 )z c xs t A x bx???? ? 決策向量??? Tnxxxx , 21 ?? ? 右端向量???? 0, 21 iTm bbbbb ?系數(shù)矩陣?????????????????mnmmnnaaaaaaaaaA???????212222111211Terminology for Solutions of the Model A feasible solution (可行解 )is a solution for which all the constraints are satisfied. An infeasible solution（非可行解） is a solution for which at least one constraint is violated. The feasible region（可行域） is the collection of all feasible solutions. An optimal solution（最優(yōu)解） is a feasible solution that has the most favorable value of the objective function. The most favorable value is the largest value if the objective function is to be maximized, whereas it is the smallest value if the objective function is to be minimized. A cornerpoint feasible (CPF) solution(角點可行解 ) is a solution that lies at a corner of the feasible region. It is possible for a problem to have no feasible solutions. (Illustrated in figure .) Any problem having multiple optimal solutions will have an infinite number of them, each with the same optimal value of the objective function. (Illustrated in figure .) Another possibility is that a problem has no optimal solutions. This occurs only if (1)it has no feasible solutions or (2)the constraints do not prevent improving the value of the objective function. The latter case is referred to as having an unbounded Z or an unbounded objective. (Illustrated in figure .) 滿足所有約束條件的解被稱為 可行解 。 Min Z=3x1+4x22x3+5x4 .: 4x1 x2+2x3 x4=2 x1+ x2+3x3 x4=14 2x1+ 3x2 x3+2x4=2 x1, x2, x3=0, x4無約束
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