簡介
對線性規劃的標準型:
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min
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s.t.
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記一個基礎可行解的基為 B , A 其餘的列為 N , 相應地,記 。有
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這是一個很重要的式子,說明了對一個非基變數,僅當量 為負時,才有可能變成新的基變數。這些係數在下面的單純形表中稱為判別數或檢驗數。
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當 全部大於等於零時,已經是最優基。對基 B ,以下的表稱為單純形表:
![]() | 0 (基變數判別數) | ![]() |
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舉例
設有線性規劃問題如下:
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min
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s.t.
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其對應的單純形表為:
max | 0 | 0 | 0 | 0 | 0 | -1 | -1 | RHS |
x₁ | x₂ | x₃ | x₄ | x₅ | x₆ | x₇ | ||
x₆ | 1 | 2 | 1 | -1 | 0 | 1 | 0 | 3 |
x₇ | 2 | -1 | 3 | 0 | -1 | 0 | 1 | 4 |
檢驗數σ | 3 | 1 | 4 | -1 | -1 | 0 | 0 | |
x₆ | 1/3 | 7/3 | 0 | -1 | 1/3 | 1 | -1/3 | 5/3 |
x₃ | 2/3 | -1/3 | 1 | 0 | -1/3 | 0 | 1/3 | 4/3 |
檢驗數σ | 1/3 | 7/3 | 0 | -1 | 1/3 | 0 | -4/3 | |
x₂ | 1/7 | 1 | 0 | -3/7 | 1/7 | 3/7 | -1/7 | 5/7 |
x₃ | 5/7 | 0 | 1 | -1/7 | -2/7 | 1/7 | 2/7 | 11/7 |
檢驗數σ | 0 | 0 | 0 | 0 | 0 | -1 | -1 | |
x₂ | 1/7 | 1 | 0 | -3/7 | 1/7 | 3/7 | -1/7 | 5/7 |
x₃ | 5/7 | 0 | 1 | -1/7 | -2/7 | 1/7 | 2/7 | 11/7 |
檢驗數σ | 0 | 0 | 0 | 0 | 0 | -1 | -1 |