Simplex Method

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Solution to Simplex Method Problem
  SIMPLEX METHOD – SOLVED PROBLEMHighTech Industries imports electronic components tht re used to ssem!le t o di##erentmodels o# personl computers$ One model is clled the Des%Pro& nd the other model is clledthe Port!le$ HighTech's mngement is currentl( interested in de)eloping  ee%l( productionschedule #or !oth products$The Des%Pro genertes  pro#it contri!ution o# *+, per unit& nd the Port!le genertes  pro#itcontri!ution o# *-, per unit$ .or ne/t ee%'s production&  m/imum o# 0+, hours o# ssem!l(time cn !e mde )il!le$ Ech unit o# the Des%Pro re1uires 2 hours o# ssem!l( time& ndech unit o# the Port!le re1uires + hours o# ssem!l( time$ In ddition& HighTech currentl( hsonl( 3, Port!le displ( components in in)entor(4 thus no more thn 3, units o# the Port!lem( !e ssem!led$ .inll(& onl( 2,, s1ure #eet o# rehouse spce cn !e mde )il!le #or ne production$ 5ssem!l( o# ech Des%Pro re1uires 6 s1ure #eet o# rehouse spce4 similrl(&ech Port!le re1uires + s1ure #eet$ Ho mn( units o# the Des%Pro nd Port!le shouldHighTech industries ssem!le to m/imi7e pro#it8The pro!lem is #ormulted s #ollo s9M/ 03 +,-,  x x + &  s t  03 2+0+,  x x +  � 5ssem!l( time  3 03,  x  � Displ( components 03 6+2,,  x x +  � :rehouse cpcit(  03 &,  x x  � SolutionIn stndrd #orm&M/ 03032 +,-,,,,  x x s s s + + + +  &  s t  030 2+0 0+,  x x s + + =   33 0 0 3,  x s + = 032 6+ ;0s2,,  x x + = 03032 &&&&,  x x s s s  � Let num!er o# )ri!les !e n $i$e$ + n  = Let num!er o# e1utions !e clled n i$e$ 2 m  = Bsic SolutionTo determine  !sic solution& set n m -  o# the )ri!les e1ul to 7ero& nd sol)e the m  liner constrint e1utions #or the remining n  )ri!les$Set +23 n m - = - =  )ri!les to 7ero$In terms o# the pro!lem !o)e&  !sic solution cn !e o!tined !( setting n( t o )ri!lese1ul to 7ero nd then sol)ing the s(stem o# three liner e1utions #or the remining three)ri!les$ :e re#er to the n m -  )ri!les tht e set to 7ero s the non<!sic )ri!les nd theremining n )ri!les s the !sic )ri!les$Bsic .esi!le Solution5 !sic solution cn either !e #esi!le or in#esi!le$ 5 !sic #esi!le solution is  !sic solutiontht lso stis#ies the non<negti)it( conditions$Suppose 0 ,  x  =  nd 3 ,  x  = & then the three constrint e1utions reduce to 0 0 0+,  s  =   3 0 3,  s  =   2 0s2,, = The complete solution ith 0 ,  x  =  nd 3 ,  x  =  is  03032 ,,0+,3,2,,  x x s s s ===== This is  !sic #esi!le solution !ecuse ll the )ri!les stis#( the non<negti)it( conditions$2 Steps necessr( to prepre n LP pro!lem #or solution using the Simple/ MethodStep 09 .ormulte the pro!lemStep 39 Set up the stndrd #orm !( dding slc% nd=or su!trcting surplus )ri!lesStep 29 Set up the initil t!leu$Setting up the Initil T!leu Part of the inial simplex tableau is a table containing all the coecients shown in the tableau form of alinear program. The general notaon is as follows c ro  A mtri/  b column  j c  = O!>ecti)e #unction coe##icient #or )ri!le  j  j b  = Right< hnd side )lue #or constrint i ij a  = ?oe##icient ssocited ith )ri!le  j in constrint i  B c  = ?oe##icient o# !sic )ri!lesThe initil simple/ t!leu #or the pro!lem is s sho n !elo 0  x   3  x   0  s   3  s   2  s    Basis    B c ?pcit(+, -, , , , 0  s   , 3  s   , 2  s   , 0+,3, 2  + 0 , , ,  0 ,  0 ,  2,,6 + , , 0  j  z  ,, , , , ,  j j c z  - +,  -, , , , @ote tht #or ech !sic )ri!le&  corresponding column hs  0 in the onl( non7ero position$Such columns re %no n s unit columns or unit )ectors$ 5lso ech ro o# the t!leu isssocited ith ech !sic )ri!le$ This ro hs  0 in the unit column corresponding to the !sic )ri!le$ @e/t& clculte the  j  z  )lues$ This represents the decrese in the )lue o# the o!>ecti)e #unctiontht ill result i# one unit o# the )ri!le corresponding to the  jth  column o# the  A  mtri/ is !rought into the !sis$ @e/t is to #ind the )lues& hich represents the net chnge in the )lue o# the o!>ecti)e #unction i# one unit o# the )ri!le corresponding to the  jth column o# the  A mtri/is !rought into the solution$  j j c z  -  is re#erred to s the net e)lution ro $Belo is ho the  j  z  s re clculted 032-+ ,A2B,A,B,A6B,,A+B,A0B,A+B,,A0B,A,B,A,B,,A,B,A0B,A,B,,A,B,A,B,A0B,  z  z  z  z  z  = + + == + + == + + == + + == + + = The completed initil !sic #esi!le solution A 0303 ,&,&0+,&3,  x x s s = = = =  nd 2 2,,  s  =  hs n o!>ecti)e #unction )lue& or pro#it o# *,$IMPROVI@C THE SOLTIO@ – 0 st  ITER5TIO@.irst loo% t the  j j c z  - )lues nd select the lrgest )lue$ In this cse the lrgest  j j c z  - )lue is+,& s sho n in the t!le !o)e$
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