對(duì)偶理論和靈敏度2009

1、線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 對(duì)偶問(wèn)題的提出對(duì)偶問(wèn)題的提出對(duì)偶關(guān)系對(duì)偶關(guān)系對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶單純形法對(duì)偶單純形法對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的提提出出ExampleKgKg對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的提提出出primal對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的提提出出對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的提提出出（*）對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的提提出出dual線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 對(duì)偶問(wèn)題的提出對(duì)偶問(wèn)題的提出對(duì)偶關(guān)系對(duì)偶關(guān)系 對(duì)稱形式的對(duì)偶關(guān)系對(duì)稱形式的對(duì)偶關(guān)系 非對(duì)稱形式的對(duì)偶關(guān)系非對(duì)稱形式的對(duì)偶關(guān)系對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶單純形法對(duì)偶單純形法對(duì)對(duì)稱稱形

2、形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系定理定理證明：證明：非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系定理定理非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系證明：證明：非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)

3、對(duì)偶偶關(guān)關(guān)系系非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系定理定理非非對(duì)對(duì)稱稱形形式式的的對(duì)對(duì)偶偶關(guān)關(guān)系系對(duì)偶關(guān)系相互對(duì)照表 原問(wèn)題（對(duì)偶問(wèn)題） 對(duì)偶問(wèn)題（原問(wèn)題） 目標(biāo)函數(shù)形式 zmax 目標(biāo)函數(shù)形式 min 變 量 n個(gè) 0 0 無(wú)正負(fù)限制 n個(gè) = 約 束 條 件 約 束 條 件 m個(gè) = m個(gè) 0 0 無(wú)正負(fù)限制 變 量 約束方程右端項(xiàng) 目標(biāo)函數(shù)中的價(jià)值系數(shù) 目標(biāo)函數(shù)中的價(jià)值系數(shù) 約束方程右端項(xiàng) 相同相同相反相反線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 對(duì)偶問(wèn)題的提出對(duì)偶問(wèn)題的提出對(duì)偶關(guān)系對(duì)偶關(guān)系對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶單純形法對(duì)偶單純形法對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)

4、對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)（弱對(duì)偶性）（弱對(duì)偶性）對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì) 0,024 2max21212121xxxxxxxxZ 0,012 24min21212121yyyyyyyyW對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性

5、質(zhì)質(zhì)例例 對(duì)偶問(wèn)題無(wú)可行解，原問(wèn)題無(wú)可行解。對(duì)偶問(wèn)題無(wú)可行解，原問(wèn)題無(wú)可行解。0,1 -1 min 21212121yyyyyyyy0,11 max 21212121xxxxxxxxZ當(dāng)原問(wèn)題（對(duì)偶問(wèn)題）無(wú)可行解時(shí)，其對(duì)偶當(dāng)原問(wèn)題（對(duì)偶問(wèn)題）無(wú)可行解時(shí)，其對(duì)偶問(wèn)題（原問(wèn)題）或具有無(wú)界解或無(wú)可行解。問(wèn)題（原問(wèn)題）或具有無(wú)界解或無(wú)可行解。對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)關(guān)于無(wú)界解、無(wú)可行解有如下結(jié)論：關(guān)于無(wú)界解、無(wú)可行解有如下結(jié)論：無(wú)界解無(wú)界解無(wú)可無(wú)可行解行解無(wú)可無(wú)可行解行解無(wú)界解無(wú)界解對(duì)偶問(wèn)題對(duì)偶問(wèn)題原問(wèn)題原問(wèn)題對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)（強(qiáng)對(duì)偶性）（強(qiáng)對(duì)偶性）對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的

6、基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)互補(bǔ)松弛性對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)例例對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性

7、性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)例例對(duì)對(duì)偶偶問(wèn)問(wèn)題題的的基基本本性性質(zhì)質(zhì)對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋影子價(jià)格影子價(jià)格對(duì)對(duì)偶偶解解的的經(jīng)經(jīng)濟(jì)濟(jì)解解釋釋|影影子子價(jià)價(jià)格格對(duì)對(duì)偶偶解解的的經(jīng)經(jīng)濟(jì)濟(jì)解解釋釋|影影子子價(jià)價(jià)格格對(duì)對(duì)偶偶解解的的經(jīng)經(jīng)濟(jì)濟(jì)解解釋釋|影影子子價(jià)價(jià)格格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格KgKg對(duì)偶

8、解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格對(duì)偶解的經(jīng)濟(jì)解釋對(duì)偶解的經(jīng)濟(jì)解釋-影子價(jià)格影子價(jià)格14)Z(X*T*)4 , 0 , 0 , 2 , 4(X 線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 線性規(guī)劃的對(duì)偶理論線性規(guī)劃的對(duì)偶理論 對(duì)偶問(wèn)題的提出對(duì)偶問(wèn)題的提出對(duì)偶關(guān)系對(duì)偶關(guān)系對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶問(wèn)題的基本性質(zhì)對(duì)偶單純形法對(duì)偶單純形法對(duì)對(duì)偶偶單單純純形形法法原始單純形法當(dāng)當(dāng)P與與DP均有可行解時(shí)，均有可行解時(shí)， P與與DP均得到最優(yōu)解。均得到最優(yōu)解。單純形方法：?jiǎn)渭冃畏?/p>

9、法：保持保持P可行解的同時(shí)，可行解的同時(shí)，逐步將逐步將DP非可行解變?yōu)榉强尚薪庾優(yōu)榭尚薪??？尚薪狻?duì)偶單純形方法：對(duì)偶單純形方法：保持保持DP可行解的同時(shí)，可行解的同時(shí)，逐步將逐步將P非可行解變非可行解變?yōu)榭尚薪?。為可行解。?duì)對(duì)偶偶單單純純形形法法計(jì)算步驟： (1) 根據(jù)線性規(guī)劃問(wèn)題，列出初始單純形表。 檢查b列的數(shù)字， 若都為非負(fù)，檢驗(yàn)數(shù)都為非正，則已得到最優(yōu)解。停止計(jì)算。 若至少還有一個(gè)負(fù)分量，檢驗(yàn)數(shù)保持非正，那么進(jìn)行以下計(jì)算。 對(duì)對(duì)偶偶單單純純形形法法 (2) 確定離基變量lx liibBbBbB1110|min 對(duì)對(duì)偶偶單單純純形形法法 (3) 確定進(jìn)基變量kx 在單純形表中檢查lx所

10、在行的各系數(shù)),.,2 , 1(njalj。 若所有0lja，則無(wú)可行解，停止計(jì)算。 若存在0lja (j=1,2,，n), 計(jì)算 lkkljljjjaaa0min 對(duì)對(duì)偶偶單單純純形形法法 (4) 以lka為主元素，按原單純形法在表中進(jìn)行迭代運(yùn)算， 得到新的計(jì)算表。 重復(fù)步驟(1)(4)。 對(duì)對(duì)偶偶單單純純形形法法例例 用對(duì)偶單純形法求解用對(duì)偶單純形法求解0, 432 32 s.t.432z min221321321321xxxxxxxxxxxx對(duì)對(duì)偶偶單單純純形形法法5,.,2 , 1, 0 4 32- 3 2- s.t.432z max53214321321jxxxxxxxxxxxxjj

11、c -2 -3 -4 0 0 BC BX b 1x 2x 3x 4x 5x 0 0 4x 5x -3 -4 -1 -2 -2 1 -1 -3 1 0 0 1 Z -2 -3 -4 0 0 對(duì)對(duì)偶偶單單純純形形法法jc -2 -3 -4 0 0 BC BX b 1x 2x 3x 4x 5x 0 -2 4x 1x -1 2 0 1 -5/2 -1/2 1/2 3/2 1 0 -1/2 -1/2 Z 0 -4 -1 0 -1 對(duì)對(duì)偶偶單單純純形形法法jc -2 -3 -4 0 0 BC BX b 1x 2x 3x 4x 5x -3 -2 2x 1x 2/5 11/5 0 1 1 0 -1/5 7/5

12、 -2/5 -1/5 1/5 -2/5 Z 0 0 -3/5 -8/5 -1/5 TX)0005/25/11(*528511*252*3*z5/15/8*2*1*yyY對(duì)對(duì)偶偶單單純純形形法法優(yōu)點(diǎn)優(yōu)點(diǎn)當(dāng)初始基解不是基可行解且檢驗(yàn)數(shù)都為當(dāng)初始基解不是基可行解且檢驗(yàn)數(shù)都為負(fù)數(shù)時(shí)，可以采用對(duì)偶單純形法，不需要負(fù)數(shù)時(shí)，可以采用對(duì)偶單純形法，不需要加入人工變量，簡(jiǎn)化了計(jì)算。加入人工變量，簡(jiǎn)化了計(jì)算。對(duì)對(duì)偶偶單單純純形形法法優(yōu)點(diǎn)優(yōu)點(diǎn)在靈敏度分析和求解整數(shù)規(guī)劃的分支定在靈敏度分析和求解整數(shù)規(guī)劃的分支定界法中，有時(shí)需要對(duì)偶單純形法，使問(wèn)界法中，有時(shí)需要對(duì)偶單純形法，使問(wèn)題簡(jiǎn)化。題簡(jiǎn)化。對(duì)對(duì)偶偶單單純純形形法法

13、缺點(diǎn)缺點(diǎn)對(duì)大多數(shù)線型規(guī)劃問(wèn)題，很難滿足對(duì)對(duì)大多數(shù)線型規(guī)劃問(wèn)題，很難滿足對(duì)偶單純形法的適用條件。所以不是一偶單純形法的適用條件。所以不是一種主要方法。種主要方法。對(duì)對(duì)偶偶單單純純形形法法靈敏度分析靈敏度分析靈靈敏敏度度分分析析靈敏度分析所要解決的問(wèn)題：靈敏度分析所要解決的問(wèn)題：v系數(shù)在什么范圍內(nèi)變化，不會(huì)影響已獲系數(shù)在什么范圍內(nèi)變化，不會(huì)影響已獲得的最優(yōu)解得的最優(yōu)解。v如果系數(shù)的變化超過(guò)以上范圍，如何在如果系數(shù)的變化超過(guò)以上范圍，如何在原來(lái)最優(yōu)解的基礎(chǔ)上求得新的最優(yōu)解。原來(lái)最優(yōu)解的基礎(chǔ)上求得新的最優(yōu)解。v當(dāng)線性規(guī)劃問(wèn)題增加一個(gè)新的變量或新當(dāng)線性規(guī)劃問(wèn)題增加一個(gè)新的變量或新的約束，如何在原來(lái)最優(yōu)解

14、的基礎(chǔ)上獲得的約束，如何在原來(lái)最優(yōu)解的基礎(chǔ)上獲得新的最優(yōu)解。新的最優(yōu)解。. s.t.max0XbAXCXz jc BC NC BC BX b BX NX bB1 mI NB1 檢驗(yàn)數(shù) NBCCBN1 靈靈敏敏度度分分析析 jc BC NC BC BX b BX NX bB1 mI NB1 檢驗(yàn)數(shù) NBCCBN1 的的變變化化分分析析b例 1 某工廠在計(jì)劃期內(nèi)要安排生產(chǎn) I、II 兩種產(chǎn)品。已知生產(chǎn)單位 產(chǎn)品所需要的設(shè)備臺(tái)時(shí)及 A、B 兩種原材料的消耗如表所示。該工 廠每生產(chǎn)一件產(chǎn)品 I 可獲利 2 元，每生產(chǎn)一件產(chǎn)品 II 可獲利 3 元。 問(wèn)應(yīng)如何制定生產(chǎn)計(jì)劃使該廠獲利最多？ I II 設(shè)備

15、 1 2 8 臺(tái)時(shí) 原材料 A 4 0 16Kg 原材料 B 0 4 12Kg 0, 124 614 8 2 s.t.00032max54321524132154321xxxxxxxxxxxxxxxxxZ 的的變變化化分分析析b問(wèn)題：?jiǎn)栴}：若該廠從其它處抽調(diào)若該廠從其它處抽調(diào)4臺(tái)時(shí)用于生產(chǎn)產(chǎn)品臺(tái)時(shí)用于生產(chǎn)產(chǎn)品I、II。求該廠的最優(yōu)生產(chǎn)計(jì)劃。求該廠的最優(yōu)生產(chǎn)計(jì)劃。142*34*2*z最優(yōu)單純形表最優(yōu)單純形表 的的變變化化分分析析b28000408/12/112/1204/101bB解： 08/12/112/1204/101B 的的變變化化分分析析b 的的變變化化分分析析b 的的變變化化分分析析b

16、173*34*2*z jc BC NC BC BX b BX NX bB1 mI NB1 檢驗(yàn)數(shù) NBCCBN1 的的變變化化分分析析C 的的變變化化分分析析C最優(yōu)解不變，最優(yōu)值變化！132c jc BC NC BC BX b BX NX bB1 mI NB1 檢驗(yàn)數(shù) NBCCBN1 的的變變化化分分析析kP 的的變變化化分分析析kPkkkPPPn)1,2,.,(j 0 m)1,2,.,(i b. s.t.z maxi,22,11 ,2211jnniiinnxxaxaxaxcxcxc原問(wèn)題 最優(yōu)基B 最優(yōu)解TnxxxX*2*1*. 增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析1)nn,1

17、,2,.,(j 0 m)1,2,.,(i b. s.t.z maxi11,22 ,11 ,112211jnninniiinnnnxxaxaxaxaxcxcxcxc新問(wèn)題增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析 jc BC NC BC BX b BX NX bB1 mI NB1 檢驗(yàn)數(shù) NBCCBN1 若 01n，則新問(wèn)題的最優(yōu)性準(zhǔn)則仍滿足。 最優(yōu)基不變 B 基變量不變bBXXBB1_ 最優(yōu)值不變bBCzzB1_ 最優(yōu)解TnxxxX0 ,.*2*1* 1111nnBnPBCC增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析增增加加一一

18、個(gè)個(gè) 新新變變量量的的變變化化分分析析例 某工廠在計(jì)劃期內(nèi)要安排生產(chǎn) I、II 兩種產(chǎn)品。已知生產(chǎn)單位 產(chǎn)品所需要的設(shè)備臺(tái)時(shí)及 A、B 兩種原材料的消耗如表所示。該工 廠每生產(chǎn)一件產(chǎn)品 I 可獲利 2 元，每生產(chǎn)一件產(chǎn)品 II 可獲利 3 元。 該廠計(jì)劃安排生產(chǎn)一種新產(chǎn)品 III。已知生產(chǎn)產(chǎn)品 III，每件需消耗原材 料 A、B 各為 6kg，3kg，使用設(shè)備 2 臺(tái)時(shí)；每件可獲利 5 元。問(wèn)該廠 是否應(yīng)生產(chǎn)該產(chǎn)品和生產(chǎn)多少? I II III 資源 設(shè)備 1 2 2 8 臺(tái)時(shí) 原材料 A 4 0 6 16Kg 原材料 B 0 4 3 12Kg 增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析解 ： (1) 設(shè)生產(chǎn)產(chǎn)品 III 為3x臺(tái)， TP)3 , 6 , 2(3， 025. 1362)0125. 05 . 1 (53133PBCCB 增增加加一一個(gè)個(gè) 新新變變量量的的變變化化分分析析解 ： (1) 設(shè)生產(chǎn)產(chǎn)品 III 為3x臺(tái)， TP)3 , 6 , 2(3， 025. 1362)0125. 05 . 1 (53133PBCCB (2) 計(jì)算產(chǎn)品在新問(wèn)題單純形表中對(duì)應(yīng) x3的列向量 25. 025 . 13620125. 05 . 015 . 02025. 0031PB 增增加加一一個(gè)個(gè) 新新變變量量