Linear Programming. Preliminaries
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- Nathaniel Parsons
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1 Lier Progrmmig Prelimiries Optimiztio ethods: 3L
2 Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio ethods: 3L
3 Itroductio d Defiitio Lier Progrmmig (LP is the most useful optimiztio techique Objective fuctio d costrits re the lier fuctios of oegtive decisio vribles Thus, the coditios of LP problems re. Objective fuctio must be lier fuctio of decisio vribles. Costrits should be lier fuctio of decisio vribles 3 3. All the decisio vribles must be oegtive Optimiztio ethods: 3L
4 Emple imize Z = 6 5y p Objective Fuctio subject to 3y 5 p st Costrit 3y p d Costrit 4 y 5 p 3rd Costrit, y 0 p Noegtivity Coditio This is i geerl form 4 Optimiztio ethods: 3L
5 Stdrd form of LP problems Stdrd form of LP problems must hve followig three chrcteristics:. Objective fuctio should be of mimiztio type. All the costrits should be of equlity type 3. All the decisio vribles should be oegtive 5 Optimiztio ethods: 3L
6 Geerl form Vs Stdrd form 6 Geerl form iimize subject to Z = urestrict ed Violtig poits for stdrd form of LPP:. Objective fuctio is of miimiztio type. Costrits re of iequlity type 3. Decisio vrible,, is urestricted, thus, my tke egtive vlues lso. How to trsform geerl form of LPP to the stdrd form? Optimiztio ethods: 3L
7 Trsformtio Geerl form Stdrd form Geerl form. Objective fuctio Stdrd form. Objective fuctio iimize Z = 3 5 imize Z = Z = 3 5. First costrit. First costrit = Secod costrit 3. Secod costrit 3 4 = Vribles d re kow s slck vribles Optimiztio ethods: 3L
8 Trsformtio Geerl form Stdrd form Geerl form 4. Third costrit 4 5. Costrits for decisio vribles, d 5 Stdrd form 4. Third costrit 4 5 = Vrible is kow s surplus vrible 5. Costrits for decisio vribles, d 0 0 urestricted = d, 0 8 Optimiztio ethods: 3L
9 Coicl form of LP Problems The objective fuctio d ll the equlity costrits (stdrd form of LP problems c be epressed i coicl form. This is kow s coicl form of LPP Coicl form of LP problems is essetil for simple method (will be discussed lter Coicl form of set of lier equtios will be discussed 9 et. Optimiztio ethods: 3L
10 Coicl form of set of lier equtios Let us cosider the followig emple of set of lier equtios 3 y z = 0 y 3z = 6 y z = (A 0 (B 0 (C 0 The system of equtio will be trsformed through Elemetry Opertios. 0 Optimiztio ethods: 3L
11 Elemetry Opertios The followig opertios re kow s elemetry opertios:. Ay equtio E r c be replced by ke r, where k is ozero costt.. Ay equtio E r c be replced by E r ke s, where E s is other equtio of the system d k is s defied bove. Note: Trsformed set of equtios through elemetry opertios is equivlet to the origil set of equtios. Thus, solutio of trsformed set of equtios is the solutio of origil set of equtios too. Optimiztio ethods: 3L
12 Trsformtio to Coicl form: A Emple Set of equtio (A 0, B 0 d C 0 is trsformed through elemetry opertios (show iside brcket i the right side 0 y z = A = A y z = ( B = B 0 A y z = ( C = C 0 A Note tht vrible is elimited from B 0 d C 0 equtios to obti B d C. Equtio A 0 is kow s pivotl equtio. Optimiztio ethods: 3L
13 Trsformtio to Coicl form: Emple cotd. Followig similr procedure, y is elimited from equtio A d C cosiderig B s pivotl equtio: 0 z = 4 A = A B 3 0 y z = 3 B = B z = 6 C = C B 3 3 Optimiztio ethods: 3L
14 Trsformtio to Coicl form: Emple cotd. Filly, z is elimited form equtio A d B cosiderig C s pivotl equtio : ( A = A 0 0 = 3 C3 ( B = B 0 y 0 = 0 0 z = 3 C = 3 C3 3 C Note: Pivotl equtio is trsformed first d usig the trsformed pivotl equtio other equtios i the system re trsformed. 4 The set of equtios (A 3, B 3 d C 3 is sid to be i Coicl form which is equivlet to the origil set of equtios (A 0, B 0 d C 0 Optimiztio ethods: 3L
15 Pivotl Opertio Opertio t ech step to elimite oe vrible t time, from ll equtios ecept oe, is kow s pivotl opertio. Number of pivotl opertios re sme s the umber of vribles i the set of equtios. Three pivotl opertios were crried out to obti the coicl form of set of equtios i lst emple hvig three vribles. 5 Optimiztio ethods: 3L
16 Optimiztio ethods: 3L 6 Trsformtio to Coicl form: Geerlized procedure Cosider the followig system of equtios with vribles ( ( ( E b E b E b = = = LLL LLL LLL
17 Trsformtio to Coicl form: Geerlized procedure 7 Coicl form of bove system of equtios c be obtied by performig pivotl opertios ( Vrible i i = L is elimited from ll equtios ecept j th equtio for which is ozero. Geerl procedure for oe pivotl opertio cosists of followig two steps,. Divide j th equtio by. Let us desigte it s E, i.e., E =. Subtrct times of E equtio from ki ( j k th equtio k =,, L j, j, L,, i.e., ji ji ( ( j E E k Optimiztio ethods: 3L ki j j E j ji
18 Trsformtio to Coicl form: Geerlized procedure After repetig bove steps for ll the vribles i the system of equtios, the coicl form will be obtied s follows: 0 LLL 0 = b ( E c LLL 0 LLL = b = b It is obvious tht solutio of bove set of equtio such s is the solutio of origil set of equtios lso. ( E ( E c c Optimiztio ethods: 3L i = b i
19 Trsformtio to Coicl form: ore geerl cse Cosider more geerl cse for which the system of equtios hs m equtio with vribles ( m LLL = b ( E 9 m m LLL LLL m It is possible to trsform the set of equtios to equivlet coicl form from which t lest oe solutio c be esily deduced = b = b m ( E ( E Optimiztio ethods: 3L m
20 Trsformtio to Coicl form: ore geerl cse By performig pivotl opertios for y m vribles (sy,,, L m, clled pivotl vribles the system of equtios reduced to coicl form is s follows 0 LLL 0 m, m m LLL = b ( E c 0 LLL 0 m, m m LLL = b ( E c 0 0 LLL m m, m m LLL m = b m ( E c m 0 Vribles, m, L,, of bove set of equtios is kow s opivotl vribles or idepedet vribles. Optimiztio ethods: 3L
21 Bsic vrible, Nobsic vrible, Bsic solutio, Bsic fesible solutio Oe solutio tht c be obtied from the bove set of equtios is i i = b i for = 0 for This solutio is kow s bsic solutio. Pivotl vribles,,, L m, re lso kow s bsic vribles. Nopivotl vribles,,, re kow s obsic vribles. m, L i =, L, m i = ( m, L, Bsic solutio is lso kow s bsic fesible solutio becuse it stisfies ll the costrits s well s oegtivity criterio for ll the vribles Optimiztio ethods: 3L
22 Thk You Optimiztio ethods: 3L
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