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Download Basics for Operation Research (LPP and Non Linear Programming) and more Summaries Operational Research in PDF only on Docsity!
al) oi a Research (OR) Homdy A‘Taha hy (neneral oR Model Macienize | ot [inlenee Objeck ve function Sub jet fo Conshai fs Y Impwve EFFideny a Produdhivity in Real-world probes, ; J Defining the prblern covrectly is the most impatant prrae of pracy OR. We should alisarys ‘ acound unquanttfiable factors (such at human pehavior) io final dectaion along with mathematias modeling ; => Mao Component of an OR modal - a (1) Alternative, (iD objective Catena Wis Restathion ap Reaaible Sol”: TL ik satisfies all the Lonstrad ats: Optimal dol”: feasible sol? wahich giv best valiac of objecH ve function. 2p “Though oR models are designed objoukve criterion subject to 0 seh of resulting 441° iacephionoter off depends on the depree “of complehencns of the model in representing the real dysem, The Optional Gol” : if model roprasesnts real aysten reasonably well, estes until then a tk. A ub optimal 40{?, to Option ize a ape dfic of cohsfredots , dhe quately y Guving the OR Medel + [2 Thy type and. complex ity of the mathematical model clictate, the naire of the Solution method, delerenined by Agoritims, Some OR TtechMqus: y- Linear “Pragrarrornies & D) Integer Prograrnmiag * 3 Dynamic ‘Prog rarnniog gma UJ Nehoork Pregrarooniag ¢ Prob 3) Non bine Resrarsmings functions of model ae nom Ling, dain mputadtonal ules that are with Yi each iberation Models with Linear objective 2 woshriats yaniables, assume intyger valu er, *, Original nnadel can be elecomposed info Her nore mangseable subproblens. tern can be Madeleol oA a nehoork . HE An Algorithm provides fixed co applied repofitively +o the problem, mois te towords the option. Ae Ada Lovelace : athe iret —ever Algorithen Meduve (Hie) st Ouauing & Siren Mes oft vaalting L084. (Not optirnt zabion fe) ahh parpornree of woaiting dine Lie for service “Deals with the study 7a cleherming. ‘medures such as O%G waiting Ame i qe « Ong oats 2 dilicatin of service facibte arms offers. Simulitions _ Quewi, Dude prebabi ity a stochastic modeh| 8) Ry * peattation’ af belnoviaus of veal WW) Plirely enedthernatical joe ) flexible & can be usec! fo Riot spe 6F applicahion analy ze practly any query situation Phases oF an oR Study: fe Md ’ ue F Be Rar oR dy OR is both Science 2 Ark’, to I] oQ Studies are rooted in ' teamwork', where the OR aralips | A the cont voovk side hy bide , ) Z| Paincipal Phores for implementing Of? in practice : Bo Definition of the Problem 7) Csnspructton of the rode] : x3) Solutio of the model: of the pnoclel ; ne Validation er : th odel wa 2) Study 5 be me Ags. .0fh* are more act a) Problem defi oveat geen. ey Ain: Ldentify fore. principle (yy Descr pon of dedsion alter naHyes (ti) deter ni nation of objectve ar Ahidy aD) Cpecifi cain af Limitation under iahich modelect yshern oper, ( omatanh ), nition’ debineatiy He scope of the probe under atie OR tearm. elements of? decien problem 3 Cnet ue an oF made] ’ b “Wanalate the prbler definihan nto Matherratical eelabinayy, . Ly { 49 frat caulby tedel Can be Avlved by cw meld 1) pee. wy (€) pode! Solutions simppleah of all OR-Phaes be cue it cots the we of welll defined — optiontzation alppr ton "Seneitiviby analysiA’ yan jenper fant aspect Ge model af” © ten model un dexyoea gome parameber change. (Rehaniour of Mele! ) ) rhode) validity, checks wheter or net the pro poral adel does what it purports +p do- 3) eplementeation . a the dol” of 0 validated model favoher dhe franslation of the rusults inty understandable, operating issued to perple cho will admwister the. inatruckon, to be ve cormmended ayshe™ . Rated on type of decision Variables Dynamic Prog raromigg (nenmenc ‘Prog rar 4 Dlgee Pers (Quadrah'c ; Prgrammigy. <\ Seporable Prograror Stochath ¢ Pear les Pei Objective furckry . > (tt - oljech ve Programmiaf —>(noal Programing 2 yl lead Progen? Facey Peg Ci Goltionley ir ee 7 a ( Ne Po ramm on é ; Gochoshre Pgormney Sema ai = Bvolutronay Srategig Serer ere Co eS 6 ee Al\ 0 modely consis} of three hadic Laenpon ents, : 1: Decision Vartabl that we seek to deherminc. a “Object.” that we neeol to optimize. : Se Omaha” that the Alution rust dakisfy. : r Feauble 40}) > uf it aabigfen “all “Hoe con shatrk, + Optimal Lal? option: Peasibe gol" wahich gives bes splice: oF objective function . The Optiencd C19 JF model vopresents veal Aysfer reuorably well, until thn if uo suboptimal $017, tH LP Model an Equation form Cteundation of 'sienp ex | ae nl Wehbe) by AN the comatrainds are equations with nen-nepative wignd band cide. Lis AN fhe variables, are nonnegative Convo i ‘24 slock vanablefoy> = ~ ' -- So pucpls nariablegoy > = Foe unrestricted wattle’ Le Pak 2 hE FO Wes) 5 variably HE Atgebrac Shuler) { Bosc 2 Non Baste Var) Repruent fre sdubion bpae hy equations in and rain als variables ty non-negative values OS) al ays Wo, inficsty of feasible pluto, f dhe faite, bonic soluti od of the eqpation by a fintbe PUBUCUEUGECCECOTS GUCUGOOCEE GF tems fre bye function Fo. determine fine. option leit fron or ld the eandidahes, They are debexmined by Adting “Y= aula as ie oth Pes ok solvi4s the Mm egmanre for tre remaining yn’ variabley Provided the resulting Kolution 2 unique. Trae nen’ variable, are Called Non Basic Veriab ke, Remaining m1 variables are Called Raric “SeeheeeRe eee Bute = Basic Solujran” sei e.eiy . = drerexa] LPP: Decision vowiabley %,, Aa,» Pp iJ Minion zef Maxionce fixe Com TO%t': HenMy Sulgect to | ) ae Oy Ht ptqgt 7 £4N% = by , $ foo , ae ; : ‘ ' | . : ‘ ii) se Oe y% Ue Avg Ka. ei? i t Qin An S by Me Pe ess \ ’ : 3 } oe ret 5 : « Amy t Ory ee Op «© tm %Xn 3 The Sxmprex Metnop (Birth of Optimization a > rr ‘esge @ Dantzig ary-2e0s! duang ww Tl q TF erative Process ; iewestigabes only a ‘slect fevs’ ca Bie —> Starts wt the ongin uhere al) the decision variables 1G fe are zero $0 J the value of dbjcdive z, (Non- Gsic) > Gr we imprve objective , by changing, values of Non asic variables. > Targets the variable one ata Bime, Not Ainnultaneens wid respect of groplucl 40)” > Eady corner pont along the path £ assovubed votth= te : i) aoe tmocds Comer poivts. always Moved alongside edger of Ao{? Space , without tole Ossi the Soh? bpace » Cdon'b Susp irecHy) Some terminolor - ii €ateny, vorably ( moat negekive coeff ty Sige Me) sho Rosi Leaving yvanobley Emin woo ~ eis 7°) Phase 4 Updake fhe foble, (identity nabs x for Rosie wasiables } Ringex feesibilfy (ncdibon, Optimality lie AP objective ro hor zem entia io @lumn ~(gbelesl ba kuic vadobles 4 9° negative entoa in Com laloeled by ran~ basic voniahiay Remarks: “The simplex tableay offers a wealth of S atp att tp . additional information © ( Lates) j- Sensi bivity Analy sis > deals with determing the cond” = pat will keep Curent soletion ctchonsed. = Posk- Optimal Anatysis’, deals with finding a neo optional solution wher data of the modes are chonged . Bubcomes of Sinpler Methd ae (oaran ont J | ged se a doesnt exist ‘zen’ ening fab Caste => 11 . : - _ jail Besic feasible => No postive min-ahi ae Oo naqative for Neon- — . are ar. = = a Conapaindy we subtoc surplus variable LS get equabigtren 2. fhe ale of alacks oF ada axtifi dal vasiables”’ > ply ghe fist thexation. fatigicia yonable objective function coffiden i. aa > 0 maxinrtation powlerr M. | ho minieni tat prions 2 my~ 20 mse Qeynasks: he vomoval of the orifidal, varialles & BS ee ee Hee eolumns ot the end of Phate-T can take place enly ake they are all non-basic. | GF me or mare addifical variables are basic af the end of Phase-T, ther do a follovs to vemove : Slep- 5 Select a zero artiGiddl yorable to leave the basic — i wen en prot rw, “The enheint words Qn «be any non basic non at fi cial fey “7h 2s Gonplex shee ‘ib, Aunties “eo 1 ee ee oe , id bie, wilh it Kees afhed alan of pO, rar aX ), Step-D. ‘Remove cal umn of tyat leaving) atifidal vanable fron the fee i ol Preaeteey Ee have beed romovel , 3° Jo prose 3 ofa Bhp L. colution & designabe a Special Cases 19 Siopplex Methel’ Deg enera cy Alternative Opting Unbourced Selutay Aion eis ting (or infeasible) solutions “when a Ge fot minimum ratio occas , if con be byte ostitrily. when His happens, of leuk one banic vore Gill be wo i the ext theration L new I @ dell fo be “Degenerte’ Tt on aute jhe simplex iteatioy to aie dndefinately coer teeminatyy He algeritiry, © pesibitity of at lecst 21 redundant censhaic) toto an te canued Pwd » < ranging kre glutton spate] «a j i @ HE Attenative Optima: = fn LPP a have On infinite number cf albermaivve | optima when te objective fundion ss parallel fo aa a non-wedundant binding comtaink lie a cpubert aa | | aa a | | fot B wtifid a8 9 gpation Ls the optimal sl7,j forex QNDAar er sf Ago optrattrn, — wey WEE oper tb Unbanded Sslodien: ——— eer e aime, the seldom Space Is unbounded in ot lett One vars — meaning tral variable, may be sncreasd ioihefins Feb. wilh vidatiy ay of he constearty making objet < udue alto unisundal - toteoy gods Vari thle caval leby< least 4 variable . But * hall ' Def”: Deal Problem is defined ays temeb cally from fhe Pim ia por original) LP model. Loptimnal golution of one poblers audornahicoly provides tre optimal ad? to the other. (Prime! [pat J ‘ania! eC onihaicts 7 Neng = ; —» * Contant > \rarjabl a ee cot of dual objerve fo. Matelarl LP | At=b> oe , > RUS of omstolal : Val Vaey cea penton > Minit > ese) Wy aun = Csnspoiats Bre, 5 ee Unrestiched. imal yarable tolumn info row a Pla ajedie dt —> RHE oF dual consheabl T wwwVvwwvwywwv~s ~~ ? . GD (gtait Co 5 dual (eft aide cH pee 0 , <0 SoS anrespac f coupam 3 = R reece 2 ee 2 : a & xe ) -_ 7 Zo ao i) minioni cata) ( >) ss da eS Zo | Bro Sago . J Var [. aan \% Be Psimol G Conapee Duct wy. 44 Sons Maxine Z= SU + 12% +4%g Minioze J = 104 +84, Suijeck ho wbyect fo UMARgt% <10 Y) +2 3 9% —-x = 4 ~~ 24 - U2 7 0 %9/%e 20 Yi tS$q = y | S729 4 Yo unrespichef | Pimad Pru] ~ Minimize Zea ISGF2%5 Max inir® LO = Sgt Sy Subject to Subject fo Ut%. BS Yyr2¥2< \S 2U UD <> DY -4$2 S12 4% Zo 4) 70 , a0 Ty Primal Dus) Was Haim te T= SYA, Miami ees SY +90 t Bue, subjyect to et I =F subject to Yt Sha 23 Oy Wet Wa = S : UNMET. Sb GLY + 5Yy 411 2 6 Ay unweiched We 20 > wore ched 
