對(duì)偶理論一、原問(wèn)題與對(duì)偶問(wèn)題例1、某廠在計(jì)劃期內(nèi)要安排生產(chǎn)A、B兩種產(chǎn)品，這兩種產(chǎn)品分別需要在甲、乙丙三種不同的設(shè)備上加工，有關(guān)數(shù)據(jù)如表3-14所示。問(wèn)應(yīng)如何安排生產(chǎn)計(jì)劃才能使利潤(rùn)最大？45利潤(rùn)（元/件）4511丙8012乙9031甲有效臺(tái)時(shí)BA設(shè)備產(chǎn)品該問(wèn)題的數(shù)學(xué)模型如果設(shè)A、B兩種產(chǎn)品生產(chǎn)的件數(shù)分別為,則這個(gè)問(wèn)題可以歸結(jié)為求解下列線(xiàn)性規(guī)劃問(wèn)題：其對(duì)偶問(wèn)題的數(shù)學(xué)模型設(shè)分別表示設(shè)備甲、乙、丙每臺(tái)時(shí)的價(jià)格（或租金）,則

例2、每頭牲畜每日對(duì)各種維生素的需求量及飼料商提供的營(yíng)養(yǎng)飼料M和N中各種維生素含量及定價(jià)如下表所示，牧場(chǎng)主在保證牲畜維生素需求條件下，每日為每頭牲畜購(gòu)M、N各多少可使總費(fèi)用最少？MN日需要量ABCD0.100.10.200.10.20.10.40.62.01.7定價(jià)104維生素營(yíng)養(yǎng)飼料設(shè)每日每頭牲畜需營(yíng)養(yǎng)飼料M、N分別為，則該問(wèn)題的線(xiàn)性規(guī)劃模型為：已知條件同上例，某藥品商想提供畜用維生素A、B、C、D四種營(yíng)養(yǎng)藥品，在滿(mǎn)足牲畜營(yíng)養(yǎng)要求且可與飼料商競(jìng)爭(zhēng)條件下，四種藥品如何定價(jià)可使總收入最大？對(duì)稱(chēng)的對(duì)偶問(wèn)題原問(wèn)題對(duì)偶問(wèn)題對(duì)稱(chēng)的對(duì)偶問(wèn)題原問(wèn)題對(duì)偶問(wèn)題非對(duì)稱(chēng)的對(duì)偶問(wèn)題原問(wèn)題對(duì)偶問(wèn)題混合形式的對(duì)偶問(wèn)題原問(wèn)題和對(duì)偶問(wèn)題的關(guān)系

原問(wèn)題（對(duì)偶問(wèn)題）約束條件對(duì)偶問(wèn)題（原問(wèn)題）決策變量max約束≤為正向約束≥為反向約束＝為雙向變量≥0為正向變量≤0為反向變量無(wú)約束為雙向min約束≥為正向約束≤為反向約束＝為雙向原問(wèn)題和對(duì)偶問(wèn)題的關(guān)系二、對(duì)偶問(wèn)題的基本性質(zhì)

1.弱對(duì)偶性：若X和Y分別是原問(wèn)題和對(duì)偶問(wèn)題的任一可行解，則必有該性質(zhì)告訴我們，最大化問(wèn)題的任一可行解的目標(biāo)函數(shù)值都是其對(duì)偶最小化問(wèn)題目標(biāo)函數(shù)的下界；而最小化問(wèn)題的任一可行解的目標(biāo)函數(shù)值都是其對(duì)偶最大化問(wèn)題目標(biāo)函數(shù)的上界。2.強(qiáng)對(duì)偶性：若分別為原問(wèn)題和對(duì)偶問(wèn)題的可行解，且可行解對(duì)應(yīng)的原問(wèn)題和對(duì)偶問(wèn)題的目標(biāo)函數(shù)值相等，即，則分別為原問(wèn)題和對(duì)偶問(wèn)題的最優(yōu)解。（最優(yōu)性準(zhǔn)則）（對(duì)偶可行性）二、潑對(duì)偶蚊問(wèn)題拍的基鋼本性偵質(zhì)（續(xù)勺）6.若原調(diào)問(wèn)題前的最牽?jī)?yōu)解瞧為3.無(wú)界流性若原往問(wèn)題話(huà)（對(duì)咬偶問(wèn)犯題）辦的目辜標(biāo)函祥數(shù)無(wú)廢界，勸則其傅對(duì)偶問(wèn)題態(tài)（原絮問(wèn)題果）必故無(wú)可句行解斃。該性鐮質(zhì)說(shuō)愛(ài)明，鋒原問(wèn)煙題和理對(duì)偶膜問(wèn)題儀之一乘無(wú)最盡優(yōu)解腿，則增另一褲個(gè)也無(wú)最飄優(yōu)解腎。4.對(duì)偶光定理若原壩問(wèn)題皇和對(duì)任偶問(wèn)傍題之垂一有情最優(yōu)販解，深則另辛一個(gè)冬也也有競(jìng)最優(yōu)齒解，巷且兩蜓者的車(chē)最優(yōu)香目標(biāo)聽(tīng)函數(shù)告值相坡等。5.若原陜問(wèn)題轟和對(duì)僅偶問(wèn)蛾題同栗時(shí)有她可行春解，搶則他剛們必鳥(niǎo)都有挎最優(yōu)往解。7.根據(jù)凝原問(wèn)讓題最啦優(yōu)單拒純形但表中釣的檢鴨驗(yàn)數(shù)仙可以竭讀出男對(duì)偶旦問(wèn)題扯的最優(yōu)桐解。例1、絨原問(wèn)送題對(duì)偶蜂問(wèn)題原問(wèn)首題標(biāo)準(zhǔn)彩型xjx1x2x3x4x5B-1bx3x1x20012-51001-1010-12253510-f000-1-3-215xjx1x2x3x4x5bx3x4x5121002101011001908045-f540000初始崗單純瘋形表最優(yōu)周單純軌形表原問(wèn)妹題5嫩4裁0環(huán)0狗0054常用狀單純垂形表商的矩恨陣形絲式式

XB

XN

XSbXSBNIb-f

CBCN00CBCN00XBIB-1NB-1B-1b-f0CN-CBB-1N

-CBB-1

-CBB-1bCB……儉……冒……厲……綢……盲……擺……對(duì)偶伙問(wèn)題y1y2y3y4y5B-1by2y3-210-115011-213-g’-2500-35-10215對(duì)偶設(shè)問(wèn)題閱最優(yōu)略單純番形表工：綜上介所述視，一取對(duì)對(duì)擇偶問(wèn)松題的未解必基然是浩下列狼三種鴉情況觀之一距：1.原問(wèn)侄題和隆對(duì)偶紐奉問(wèn)題濱都有洽最優(yōu)煌解，峰且最胞優(yōu)目過(guò)標(biāo)函吧數(shù)值談相等朽。3.原問(wèn)映題和局對(duì)偶?jí)艈?wèn)題轟都無(wú)情可行鈴解。2.一個(gè)遵問(wèn)題吼具有華無(wú)界屠解，歷則另貪一個(gè)堵問(wèn)題擾無(wú)可駱行解榴。Cj58600CBXBX1X2X3X4X5B-1b58X1X21001202-1-1124λj00-4-2-3-42ma覆xf=5酬X1+8宮X2+6對(duì)X3X1+遮X2+2畏X3≤6X1+2促X2+2宜X3≤1竊0X1，X2,X2≥0例3已知揮線(xiàn)性拳規(guī)劃綱問(wèn)題試用舍對(duì)偶比理論寫(xiě)證明抄上述情問(wèn)題菜無(wú)最柴優(yōu)解夢(mèng)。三、捕對(duì)偶心解的寶經(jīng)濟(jì)咐涵義——影子蝦價(jià)格通過(guò)殿求解坐：原養(yǎng)問(wèn)題謝和對(duì)鼠偶問(wèn)決題的高最優(yōu)閉解分板別為1.影子柴價(jià)格種的定倦義對(duì)偶蚊問(wèn)題塌是資茂源定月價(jià)問(wèn)榨題，傷對(duì)偶婦問(wèn)題男的最農(nóng)優(yōu)解y1、y2、..抄.、ym稱(chēng)為m種資矩源的影子竹價(jià)格（Sh事ad控ow船P濫ri上ce）。影子譜價(jià)格是指霧在最堪優(yōu)解鉤的基她礎(chǔ)上咳，當(dāng)浪第i個(gè)約房誠(chéng)束條某件的販右端詠?lái)?xiàng)bi增加尋一個(gè)匙單位些時(shí)，緣瑞目標(biāo)歲函數(shù)秤的變給化量畢。由對(duì)爪偶定偽理可自知，當(dāng)達(dá)給到最滿(mǎn)優(yōu)解雙時(shí)，柏原問(wèn)刺題與謝對(duì)偶各問(wèn)題糟的目衡標(biāo)函昂數(shù)值緊相等劑，即揮有f*=C芝X*=Y*b=敢y1*b1+y2*b2+…頁(yè)+ym*bm現(xiàn)考曉慮在仔最優(yōu)綱解處改，右禁端項(xiàng)bi的微言小變蘆動(dòng)對(duì)耽目標(biāo)獅函數(shù)劇值的凱影響駐，由謀上式束，將f*對(duì)bi求偏寬導(dǎo)數(shù)眠：該式胡表明放了，協(xié)若原獄問(wèn)題扁的某賭一個(gè)寧約束拍條件鍛的右嘉端項(xiàng)bi每增燦加一罰個(gè)單湖位，橡則由化此引詞起的蠻最優(yōu)痕目標(biāo)控函數(shù)站值的彎增加賽量，雀就等焦于與煙該約慚束條望件相引對(duì)應(yīng)嗓的對(duì)紡偶變夸量的涼最優(yōu)零解的這值。2.影子悼?jī)r(jià)格檔的求翻法例4某工種廠生業(yè)產(chǎn)三晃種產(chǎn)福品，繼三種及產(chǎn)品倘對(duì)于銹原材汗料、渣勞動(dòng)鐘力、艙電力亂的單禿位消低耗系寸數(shù)，年資源娛限量退和單羽位產(chǎn)仁品價(jià)優(yōu)格如有下表厲所示凱：ABC資源限量原材料（kg）勞動(dòng)力（人）電力（度）2652.5154810320640750單位價(jià)格（元）4610資源產(chǎn)品1.求最夫佳生槍產(chǎn)方迫案使割總產(chǎn)勸值最躲大。2.求各陜資源芝的影驗(yàn)子價(jià)炸格，憲并解黑釋其上經(jīng)濟(jì)顫意義團(tuán)。xjx1x2x3x4x5x6B-1bx2x5x301020-0.820061-3.21/201-100.54016055-f-100-20-0.2-790xjx1x2x3x4x5x6B-1bx2x5x301020-0.820061-3.21/201-100.54016055-f-100-20-0.2-7903.影子繪價(jià)格秩的作耍用①影子葡價(jià)格烈可以蘇告訴泊管理失人員屑，增醋加哪襲一種纏資源勁對(duì)增惑加經(jīng)華濟(jì)效喜益最朽有益星。②影子仁價(jià)格歡可以神告訴兇管理苗人員土，花亭多大哀的代謙價(jià)增牲加資渡源才泊是合丹算的憤。③影子噸價(jià)格舞在新催產(chǎn)品伙開(kāi)發(fā)榴決策祖中的阿應(yīng)用勤。④影子添價(jià)格娃在資煉源購(gòu)秘銷(xiāo)決繭策中堡的應(yīng)伏用。⑤利用價(jià)影子捆價(jià)格辦分析虧工藝懇改變?nèi)竞髮?duì)餃資源美節(jié)約界的收親益。如在連上例接中，紀(jì)當(dāng)工深藝改腦進(jìn)后求，使也原材言料節(jié)本約10食%，則份帶來(lái)棋的經(jīng)朋濟(jì)效居益為殲：232天010茶%=授64（元壩）在利站潤(rùn)最壁大化折的生性產(chǎn)計(jì)糟劃中（1）邊嚇際利昨潤(rùn)大高于0的資伸源沒(méi)建有剩窩余（2）有類(lèi)剩余晨的資便源邊綢際利栽潤(rùn)等月于0關(guān)于超影子責(zé)價(jià)格爆的幾柴點(diǎn)說(shuō)塔明：影子杠價(jià)格應(yīng)越大胃，說(shuō)凍明這網(wǎng)種資錢(qián)源越聽(tīng)是相堆對(duì)緊爛缺影子佩價(jià)格票越小吉，說(shuō)馬明這督種資械源相悔對(duì)不饅緊缺如果珠最優(yōu)得生產(chǎn)妥計(jì)劃垮下某朵種資愿源有繪剩余驚，這累種資愚源的鞠影子泳價(jià)格貪一定轎等于0影子序價(jià)格閱為0，資系源并嘉不一悟定有押剩余影子筆價(jià)格荷是資己源最旺優(yōu)配喘置下顏資源次的理糖想價(jià)釣格，位資源掩的影晴子價(jià)歇格與忘資源打的緊潔缺度僑有關(guān)思路擴(kuò)：(ma侵x型)單純坊形法：找舊基B，滿(mǎn)足B-1b0，但檢綠驗(yàn)數(shù)C-CBB-1A不全0。迭代保持B-1b0，使C

-CBB-1

A0。對(duì)偶屑單純邊形法：找芹基B，滿(mǎn)足C-CBB-1A庫(kù)0，但B-1b不全0。迭代保持C

-CBB-1

A0，使B-1b0。四、泉對(duì)偶亂單純熱形法舉例原問(wèn)爸題對(duì)偶肆問(wèn)題xjx1x2x3x4x5B-1bx3x4x505100620101100115245-f210000x3x1x50510011/301/6002/30-1/611541-f01/30-1/30-8x3x1x20015/4-15/21001/4-1/2010-1/43/215/27/23/2-f000-1/4-1/2-17/2yiy1y2y3y4y5B-1by4y50-6-110-5-2-101-2-1-f-15-24-5000y2y5011/6-1/60-50-2/3-1/311/3-1/3-f-150-1-40-8y2y3-5/410-1/41/415/2011/2-3/21/41/2-f-15/200-7/2-3/2-17/2例1maxf=2x1

+x2x1+x2+x3=

52x2+x354x2+6x3

9x1,x2,x30maxf=2x1+x2x1+x2+x3=

52x2+x3+x4=5-4x2–6x3+x5=-9x1…x5

0xjx1x2x3x4x5B-1bx1x4x511100021100-4-60155-9-f0-1-200-102堂1奸0講0缸0斯0200x1x4x210-1/201/400-21-1/2013/20-1/411/41/29/4-f00-1/20-1/4-31/4201例2旦.標(biāo)準(zhǔn)化找初始基變量xjx1x2x3x4x5bix3x4x522100-3-2010120013-41-f-1-30000xjx1x2x3x4x5B-1bx3x1x5

-f02/認(rèn)312/康301/斜312/牧30-1森/304/雖304/旨301/罵31-1創(chuàng)/30-7步/30-1灰/304/臉3Xjx1x2x3x4x5B-1bx3x4x5-2-1100-3-2010-1-2001-3-4-1-f-1-30000x3x1x501/31-2/3012/30-1/300-4/30-1/31-1/34/31/3-f0-7/30-1/304/3x4x1x50-1/2-3/21011/2-1/2000-3/2-1/2011/23/21/2-f0-5/2-1/2003/2例3(課本宵）練習(xí)minf=2x1+3x2+4x3x1+2x2+x3

32x1-x2+3x34x1,x2,x30minf=2x1+3x2+4x3

-x1–2x2-x3+x4=-

3-2x1+x2–3x3+x5=-

4x1…x5

0xix1x2x3x4x5B-1bx4x5-1-2-110-21-301-3-4-f-2-3-4000x4x10-5/21/21-1/21-1/2