CHAPTER II: LINEAR PROGRAMMING

Transcription

1 CHAPTER II: LINEAR PROGRAMMING CHAPTER II: LINEAR PROGRAMMING. The Bsic LP Prolem... Bsic LP Exmple... Other Forms of the LP Prolem...4 Assumptios of LP Ojective Fuctio Appropriteess Decisio Vrile Appropriteess 4.4. Costrit Appropriteess Proportiolity Additivity Divisiility Certity.. 6 The most fudmetl optimiztio prolem treted i this ook is the lier progrmmig (LP) prolem. I the LP prolem, decisio vriles re chose so tht lier fuctio of the decisio vriles is optimized d simulteous set of lier costrits ivolvig the decisio vriles is stisfied.. The Bsic LP Prolem A LP prolem cotis severl essetil elemets. First, there re decisio vriles( j ) the level of which deotes the mout udertke of the respective ukows of which there re ( j=,, ). Next is the lier ojective fuctio where the totl ojective vlue (Z) equls c + c c. Here c j is the cotriutio of ech uit of j to the ojective fuctio. The prolem is lso suject to costrits of which there re m. A lgeric expressio for the i th costrit is i + i i i (I=,,, m) where i deotes the upper limit or right hd side imposed y the costrit d ij is the use of the items i the i th costrit y oe uit of j. The c j, i, d ij re the dt (exogeous prmeters) of the LP model. Give these defiitios, the LP prolem is to choose,,, so s to B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING

2 Mx s. t. c m c m c m c m This formultio my lso e expressed i mtrix ottio. Mx C Suject to A 0 My vrits hve ee posed of the ove prolem d pplictios sp wide vriety of settigs. For exmple, the sic prolem could ivolve settig up: ) livestock diet determiig how much of ech feed stuff to uy so tht totl diet cost is miimized suject to costrits o miimum d mximum levels of utriets; ) productio pl where the firm chooses the profit mximizig level of productio suject to resource (lor d rw mterils) costrits; or c) miimum cost trsporttio pl determiig the mout of goods to trsport cross ech ville route suject to costrits o supply vilility d demd.. Bsic LP Exmple For further expositio of the LP prolem it is coveiet to use exmple. Cosider the decisio prolem of Joe's v coversio shop. Suppose Joe tkes pli vs d coverts them ito custom vs d c produce either fie or fcy vs. The decisio modeled is how my of ech v type to covert this week. The umer coverted this week y v type costitutes the decisio vriles. We deote these vriles s fie d fcy. Both types require $5,000 pli v. Fcy vs sell for $7,000 d Joe uses $0,000 i prts to customize them yieldig profit mrgi of $,000. Fie vs use $6,000 i prts d sell for $,700 yieldig profits of $,700. Joe figures the shop c work o o more th vs i week. Joe hires 7 people icludig himself d opertes 8 hours per dy, 5 dys week d thus hs t most 80 hours of lor ville i week. Joe lso estimtes tht fcy v will tke 5 hours of lor, while fie v will tke 0 hours. I order to set up Joe's prolem s LP, we must mthemticlly express the ojective d costrit fuctios. Sice the estimted profit per fcy vs is $,000 per v, the,000 fcy is the totl profit from ll the fcy vs produced. Similrly,,700 fie is the totl profit from fie v productio. The totl profit from ll v coversios is,000 fcy +,700 fie. This equtio mthemticlly descries the totl profit cosequeces of Joe's choice of the decisio vriles. Give tht Joe wishes to mximize totl profit, his ojective is to determie the levels of B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING m 0

3 the decisio vriles tht Mximize Z = 000 fcy fie This is the ojective fuctio of the LP model. Joe's fctory hs limited mouts of cpcity d lor. I this cse, cpcity d lor re resources which limit the llowle (lso clled fesile) vlues of the decisio vriles. Sice the decisio vriles re defied i terms of vs coverted i week, the totl vs coverted is fcy + fie. This sum must e less th or equl to the cpcity ville (). Similrly, totl lor use is give y 5 fcy + 0 fie which must e less th or equl to the lor ville (80). These two limits re clled costrits. Filly, it mkes o sese to covert egtive umer of vs of either type; thus, fcy d fie re restricted to e greter th or equl to zero. Puttig it ll together, the LP model of Joe's prolem is to choose the vlues of fcy d fie so s to: Mximize Z = 000 fcy fie s.t. fcy + fie 5 fcy + 0 fie 80 fcy, fie 0 This is formultio of Joe's LP prolem depictig the decisio to e mde (i.e. the choice of fcy d fie ). The formultio lso idetifies the rules, commoly clled costrits, y which the decisio is mde d the ojective which is pursued i settig the decisio vriles.. Other Forms of the LP Prolem Not ll LP prolems will turlly correspod to the ove form. Other legitimte represettios of LP models re: ) Ojectives which ivolves miimize isted of mximize i.e., Miimize Z = c + c + + c. ) Costrits which re "greter th or equl to" isted of "less th or equl to"; i.e., i + i + + i. ) Costrits which re strict equlities; i.e., i + i + + i =. 4) Vriles without o-egtivity restrictio i.e., j c e urestricted i sig i.e., <=> 0 B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING

4 5) Vriles required to e o-positive i.e., j 0..4 Assumptios of LP LP prolems emody seve importt ssumptios reltive to the prolem eig modeled. The first three ivolve the ppropriteess of the formultio; the lst four the mthemticl reltioships withi the model..4. Ojective Fuctio Appropriteess This ssumptio mes tht withi the formultio the ojective fuctio is the sole criteri for choosig mog the fesile vlues of the decisio vriles. Stisfctio of this ssumptio c ofte e difficult s, for exmple, Joe might se his v coversio pl ot oly o profit ut lso o risk exposure, vilility of vctio time, etc. The risk modelig d multi-ojective chpters cover the relxtio of this ssumptio..4. Decisio Vrile Appropriteess A key ssumptio is tht the specifictio of the decisio vriles is pproprite. This ssumptio requires tht ) The decisio vriles re ll fully mipultle withi the fesile regio d re uder the cotrol of the decisio mker. ) All pproprite decisio vriles hve ee icluded i the model. The ture d relxtio of su-ssumptio () is discussed i the Advced modelig cosidertios chpter i the "Commo Mistkes" sectio, s is su-ssumptio (). Su-ssumptio ) is lso highlighted i Chpters I d VI..4. Costrit Appropriteess The third ppropriteess ssumptio ivolves the costrits. Agi, this is est expressed y idetifyig su-ssumptios: ) The costrits fully idetify the ouds plced o the decisio vriles y resource vilility, techology, the exterl eviromet, etc. Thus, y choice of the decisio vriles, which simulteously stisfies ll the costrits, is dmissile. ) The resources used d/or supplied withi y sigle costrit re homogeeous items tht c e used or supplied y y decisio vrile pperig i tht costrit. c) Costrits hve ot ee imposed which improperly elimite dmissile vlues of the decisio vriles. d) The costrits re iviolte. No cosidertios ivolvig model vriles other th those icluded i the model c led to the relxtio of the costrits. Relxtios d/or the implictios of violtig these ssumptios re discussed 4 B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING 4

5 throughout the text..4.4 Proportiolity Vriles i LP models re ssumed to exhiit proportiolity. Proportiolity dels with the cotriutio per uit of ech decisio vrile to the ojective fuctio. This cotriutio is ssumed costt d idepedet of the vrile level. Similrly, the use of ech resource per uit of ech decisio vrile is ssumed costt d idepedet of vrile level. There re o ecoomies of scle. For exmple, i the geerl LP prolem, the et retur per uit of j produced is c j. If the solutio uses oe uit of j, the c j uits of retur re ered, d if 00 uits re produced, the returs re 00c j. Uder this ssumptio, the totl cotriutio of j to the ojective fuctio is lwys proportiol to its level. This ssumptio lso pplies to resource usge withi the costrits. Joe's lor requiremet for fie vs ws 5 hours/v. If Joe coverts oe fie v he uses 5 hours of lor. If he coverts 0 fie vs he uses 50 hours (5*0). Totl lor use from v coversio is lwys strictly proportiol to the level of vs produced. Ecoomists ecouter severl types of prolems i which the proportiolity ssumptio is grossly violted. I some cotexts, product price depeds upo the level of productio. Thus, the cotriutio per uit of ctivity vries with the level of the ctivity. Methods to relx the proportiolity ssumptio re discussed i the olier pproximtios, price edogeous, d risk chpters. Aother cse occurs whe fixed costs re to e modeled. Suppose there is fixed cost ssocited with vrile hvig y o-zero vlue (i.e., costructio cost). I this cse, totl cost per uit of productio is ot costt. The iteger progrmmig chpter discusses relxtio of this ssumptio..4.5 Additivity Additivity dels with the reltioships mog the decisio vriles. Simply put their cotriutios to equtio must e dditive. The totl vlue of the ojective fuctio equls the sum of the cotriutios of ech vrile to the ojective fuctio. Similrly, totl resource use is the sum of the resource use of ech vrile. This requiremet rules out the possiility tht iterctio or multiplictive terms pper i the ojective fuctio or the costrits. For exmple, i Joe's v prolem, the vlue of the ojective fuctio is,000 times the fcy vs coverted plus,700 times the fie vs coverted. Covertig fcy vs does ot lter the per v et mrgi of fie vs d vice vers. Similrly, totl lor use is the sum of the hours of lor required to covert fcy vs d the hours of lor used to covert fie vs. Mkig lot of oe v does ot lter the lor requiremet for mkig the other. I the geerl LP formultio, whe cosiderig vriles j d k, the vlue of the ojective fuctio must lwys equl c j times j plus c k times k. Usig j does ot ffect the per uit et retur of k d vice vers. Similrly, totl resource use of resource I is the sum of ij j d ik k. 5 B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING 5

6 Usig j does ot lter the resource requiremet of k. The olier pproximtio, price edogeous d risk chpters preset methods of relxig this ssumptio..4.6 Divisiility The prolem formultio ssumes tht ll decisio vriles c tke o y o-egtive vlue icludig frctiol oes; (i.e., the decisio vriles re cotiuous). I the Joe's v shop exmple, this mes tht frctiol vs c e coverted; e.g., Joe could covert. fcy vs d 0.8 fie vs. This ssumptio is violted whe o-iteger vlues of certi decisio vriles mke little sese. A decisio vrile my correspod to the purchse of trctor or the costructio of uildig where it is cler tht the vrile must tke o iteger vlues. I this cse, it is pproprite to use iteger progrmmig..4.7 Certity The certity ssumptio requires tht the prmeters c j, i, d ij e kow costts. The optimum solutio derived is predicted o perfect kowledge of ll the prmeter vlues. Sice ll exogeous fctors re ssumed to e kow d fixed, LP models re sometimes clled o-stochstic s cotrsted with models explicitly delig with stochstic fctors. This ssumptio gives rise to the term "determiistic" lysis. The exogeous prmeters of LP model re ot usully kow with certity. I fct, they re usully estimted y sttisticl techiques. Thus, fter developig LP model, it is ofte useful to coduct sesitivity lysis y vryig oe of the exogeous prmeters d oservig the sesitivity of the optiml solutio to tht vritio. For exmple, i the v shop prolem the et retur per fcy v is $,000, ut this vlue depeds upo the v cost, the cost of mterils d the sle price ll of which could e rdom vriles. Cosiderle reserch hs ee directed towrd icorportig ucertity ito progrmmig models. We devote chpter to tht topic. 6 B.A. McCrl d T.H. Spree, 0 LINEAR PROGRAMMING 6