Linear Programming by Michel Sakarovitch (auth.), John B. Thomas (eds.)

By Michel Sakarovitch (auth.), John B. Thomas (eds.)

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M 1 1 x;;' 0 cx = z (Max) And we can app ly t o ( PS ') the pro ce s s of Def i ni tion 1 t o fi n d i t s dual . constraints o f (PS') (o t he r than x ~ 0 ) a r e in one of t wo groups . The Let us associate dual va r i ab l e y , i to t he i th cons t r ai nt of th e fi rs t gro up and y" i to t he i th con stra int o f t he s e con d gr oup . Le t y ' (res p . y ") be th e m-row th componen t of wh ich is y , i (resp. y " i ) . 4 ) . Remark 6 : Hore ge ne ra l ly , it i s conve nien t t o be able t o write t he dua l o f a linear pro gram without pass i n g t h r ough th e canon ical form (ho we ver, fo r exe r cise an d fo r checking , we recommend th at the be ginners a lways fo llow t his p r ocess) .

1 . 3} . respectively. l) (E and (E are obviously the "same" system. Since we want t o say that 3) 4) J (E is a solution of (E as well as (E we introduced the f ac t that is l) 4) 3), "up to a permutation of r ows or col umns" the unit matrix. A Remark 5: Suppose that ( 3) is a s olution of (1) with respect to ba s ic se t J. (3) can be written. by separating basic columns from nonbasic on e s, (3') Since AJ b is by assumption. up to a permutation of rows. the ill X m-unit matrix, one is tempted to write (3') in the following way: (3") b which makes it appear clearly the basic set J that the system has been so lved with respect to of variables ( give arbitrary values to the nonbasi c vari ables and deduce.

P be n po i n ts of a ne t wo r k . A commodi t y i s p roduced Z' n i n PI and consumed in P. To e ach couple (p . , P . ) we associate a nonn e gat i ve n 1 J number c .. : t he maximum q uanti t y o f commodity th at ca n be shLpped from i t o j 1J in one day . 34 Chapter II. Dual Linear Programs (a) Wri t e , as a l i ne ar p r og ram , the prob lem t hat consis ts of s endi ng a maximum amount of commo di t y f ro m PI t o P ( th e f low i s cons e rva t ive , i. e . , t he r e n is no accumulat ion of commodity at any node ).

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