RESOURCE MANAGEMENT TECHNIQUES NOTES MC9242 RESOURCE MANAGEMENT TECHNIQUESUNIT ILINEAR PROGRAMMING MODELS 9Mathematical Fom!latio" # Ga$hical Sol!tio" o% li"ea $o&ammi"& mo'el( ) Sim$le* metho' ) Ati%icial +aia,le Tech"i-!e(# .aia"t( o% Sim$le* metho'UNIT IITRANSPORTATION AND ASSIGNMENT MODELS9Mathematical %om!latio" o% ta"($otatio" $o,lem# Metho'( %o %i"'i"& i"itial ,a(ic %ea(i,le (ol!tio" ) o$tim!m (ol!tio" # 'e&e"eac/ ) Mathematical %om!latio" o% a((i&"me"t mo'el( ) H!"&aia" Al&oithm ) .aia"t( o% the A((i&"me"t $o,lemUNIT III INTEGER PROGRAMMING MODELS9Fom!latio" ) Gomo/0( IPP metho' ) Gomo/0( mi*e' i"te&e metho' ) 1a"ch a"' ,o!"' tech"i-!e2UNIT I. SCHEDULING 13 PERT AND CPM 9Net4o5 Co"(t!ctio" ) Citical Path Metho' ) Po6ect E+al!atio" a"' Re+ie4 Tech"i-!e ) Re(o!ce A"al/(i( i" Net4o5 Sche'!li"&UNIT . QUEUEING MODELS9Chaactei(tic( o% Q!e!i"& Mo'el( ) Poi((o" Q!e!e( # 7M 8 M 8 9: ; 7FIFO 8 < 82 De%i"e M!lti$le O$timal (ol!tio"@A LPP ha+i"& moe tha" o"e o$timal (ol!tio" i( (ai' to ha+e alte"ati+e o m!lti$le o$timal (ol!tio"(22C2 ?hat i( (lac5 +aia,le@I% the co"(tai"t a( &e"eal LPP ,e NH t/$e the" a "o" "e&ati+e +aia,le i( i"to'!ce' to co"+et the i"e-!alitie( i"to e-!alitie( ae calle' (lac5 +aia,le(2 The +al!e( o% the(e +aia,le( ae i"te$ete'a( the amo!"t o% !"!(e' e(o!ce(22D2 ?hat ae (!$l!( +aia,le(@I% the co"(tai"t a( &e"eal LPP ,e OH t/$e the" a "o" "e&ati+ei( i"to'!ce' to co"+et the i"e-!alitie( i"to e-!alitie( ae calle' the (!$l!( +aia,le(22E2 De%i"e 1a(ic (ol!tio"@Gi+e" a (/(tem o% m li"ea e-!atio"( 4ith " +aia,le(7mN":2The (ol!tio" o,tai"e' ,/ (etti"& 7"#m: +aia,le( e-!al to Aeo a"' (ol+i"& %o the emai"i"& m +aia,le( i( calle' a ,a(ic (ol!tio"2292 De%i"e "o" De&e"eate 1a(ic %ea(i,le (ol!tio"@The ,a(ic (ol!tio" i( (ai' to ,e a "o" 'e&e"eate ,a(ic (ol!tio"i% No"e o% the ,a(ic +aia,le( i( Aeo2BF2 De%i"e 'e&e"eate ,a(ic (ol!tio"@A ,a(ic (ol!tio" i( (ai' to ,e a 'e&e"eate ,a(ic (ol!tio" i% o"e omoe o% the ,a(ic +aia,le( ae Aeo2B92 ?hat i( the %!"ctio" o% mi"im!m atio@ To 'etemi"e the ,a(ic +aia,le to lea+e To 'etemi"e the ma*im!m i"cea(e i" ,a(ic +aia,le To mai"tai" the %ea(i,ilit/ o% %ollo4i"& (ol!tio"B22 Fom the o$tim!m (im$le* ta,le ho4 'o /o! i'e"ti%/ that LPP ha( !",o!"'e' (ol!tio"@To %i"' the lea+i"& +aia,le( the atio i( com$!te'2 The atio i( NHFthe" thee i( a" !",o!"'e' (ol!tio" to the &i+e" LPP2BB2 Fom the o$tim!m (im$le* ta,le ho4 'o /o! i'e"ti%/ that the LPP ha( "o(ol!tio"@I% atlea(t o"e ati%icial +aia,le a$$ea( i" the ,a(i( at Aeo le+el 4ith a I+e +al!e i" the P, col!m" a"' the o$timalit/ co"'itio" i( (ati(%ie'the" the oi&i"al $o,lem ha( "o %ea(i,le (ol!tio"2B42 Ho4 'o /o! i'e"ti%/ that LPP ha( "o (ol!tio" i" a t4o $ha(e metho'@I% all G6 ) C6 K F M the" atlea(t o"e ati%icial +aia,le a$$ea( i" the o$tim!m ,a(i( at "o" Aeo le+el the LPP 'oe( "ot $o((e(( a"/ (ol!tio"2B>2 ?hat 'o /o! !"'e(ta"' ,/ 'e&e"eac/@The co"ce$t o% o,tai"i"& a 'e&e"eate ,a(ic %ea(i,le (ol!tio" i" LPP i( 5"o4" a( 'e&e"eac/2 Thi( ma/ occ! i" the i"itial (ta&e 4he" atlea(t o"e ,a(ic +aia,le i( Aeo i" the i"itial ,a(ic %ea(i,le (ol!tio"2BC2 ?ite the (ta"'a' %om o% LPP i" the mati* "otatio"@I" mati* "otatio" the ca"o"ical %om o% LPP ca" ,e e*$e((e' a( Ma*imiAe G H CP7o,6 %"2: S!, to AP NH ,7co"(tai"t(: a"' P OH F 7"o" "e&ati+e e(tictio"(: ?hee C H 7C9=C2=J22C":= A H a99 a92 J22 a9"P H*9, H,9 a29 a22J22a2"=*2 = ,2222222am9am2J2 am"*" ,"BD2 De%i"e ,a(ic +aia,le a"' "o"#,a(ic +aia,le i" li"ea $o&ammi"&2A ,a(ic (ol!tio" to the (et o% co"(tai"t( i( a (ol!tio" o,tai"e' ,/ (etti"& a"/ " +aia,le( e-!al to Aeo a"' (ol+i"& %o emai"i"& m +aia,le( "ot e-!al to Aeo2 S!ch m +aia,le( ae calle' ,a(ic +aia,le( a"' emai"i"& " Aeo +aia,le( ae calle' "o"#,a(ic +aia,le(2BE2Sol+e the %ollo4i"& LP $o,lem ,/ &a$hical metho'2 7MA3 0FE:Ma*imiAe A HC*9 I4*2 S!,6ect tot the co"(tai"t(;*9 I *2 K >*2L E*9 =*2L FB92 De%i"e !"e(ticte' +aia,le a"' ati%icial +aia,le2 7NO. 0FD: U"e(ticte'.aia,le;A+aia,lei( !"e(ticte'i% it i( allo4e'tota5eo" $o(iti+e= "e&ati+e o Aeo +al!e( Ati%icial +aia,le ;O"e t/$e o% +aia,le i"to'!ce' i" a li"ea $o&am mo'el i" o'e to %i"' a" i"itial ,a(ic %ea(i,le (ol!tio"Q a" ati%icial +aia,le i( !(e' %o e-!alit/ co"(tai"t( a"' %o &eate#tha" o e-!al i"e-!alit/ co"(tai"t(UNIT#II92 De%i"e ta"($otatio" $o,lem2It i( a ($ecial t/$e o% li"ea $o&ammi"& mo'el i" 4hich the &oo'( ae (hi$$e' %om +aio!( oi&i"( to 'i%%ee"t 'e(ti"atio"(2 The o,6ecti+e i( to %i"' the ,e(t $o((i,le allocatio" o% &oo'( %om +aio!( oi&i"( to 'i%%ee"t 'e(ti"atio"( (!ch that the total ta"($otatio" co(t i( mi"im!m2B2 De%i"e the %ollo4i"&; Fea(i,le (ol!tio"A (et o% "o"#"e&ati+e 'eci(io" +al!e( *i6 7iH9=2=J2mQ 6H9=2J": (ati(%ie( the co"(tai"t e-!atio"( i( calle' a %ea(i,le (ol!tio"242 De%i"e the %ollo4i"&;,a(ic %ea(i,le (ol!tio"A ,a(ic %ea(i,le (ol!tio" i( (ai' to ,e ,a(ic i% the "!m,e o% $o(iti+e allocatio"( ae mI"#927 m#oi&i" a"' "#'e(ti"atio":2I% the "!m,e o% allocatio"( ae le(( tha" 7mI"#9:it i( calle' 'e&e"eate ,a(ic %ea(i,le (ol!tio"2>2 De%i"e o$timal (ol!tio" i" ta"($otatio" $o,lemA %ea(i,le (ol!tio" i( (ai' to ,e o$timal= i% it mi"imiAe( the total ta"($otatio" co(t2C2 ?hat ae the metho'( !(e' i" ta"($otatio" $o,lem to o,tai" the i"itial ,a(ic %ea(i,le (ol!tio"2 Noth#4e(t co"e !le Lo4e(t co(t e"t/ metho' o mati* mi"ima metho' .o&el0( a$$o*imatio" metho'D2 ?ite 'o4" the ,a(ic (te$( i"+ol+e' i" (ol+i"& a ta"($otatio" $o,lem2 To %i"' the i"itial ,a(ic %ea(i,le (ol!tio" To %i"' a" o$timal (ol!tio" ,/ ma5i"& (!cce((i+e im$o+eme"t( %om the i"itial ,a(ic %ea(i,le (ol!tio"2E2= ?hat 'o /o! !"'e(ta"' ,/ 'e&e"eac/ i" a ta"($otatio" $o,lem@ 7NO. 0FD:I% the "!m,e o% occ!$ie' cell( i" a m * " ta"($otatio" $o,lem i( le(( tha"7 mI"#9: the" the $o,lem i( (ai' to ,e 'e&e"eate292?hat i( ,ala"ce' ta"($otatio" $o,lemM !",ala"ce' ta"($otatio" $o,lem@?he" the (!m o% (!$$l/ i( e-!al to 'ema"'(= the" the $o,lem i( (ai' to ,e ,ala"ce' ta"($otatio" $o,lem2A ta"($otatio" $o,lem i( (ai' to ,e !",ala"ce' i% the total (!$$l/ i( "ot e-!al to the total 'ema"'29F2 Ho4 'o /o! co"+et a" !",ala"ce' ta"($otatio" $o,lem i"to a ,ala"ce' o"e@The !",ala"ce' ta"($otatio" $o,lem i( co"+ete' i"to a ,ala"ce' o"e ,/ a''i"& a '!mm/ o4 7(o!ce: o '!mm/ col!m" 7'e(ti"atio": 4hiche+e i( "ece((a/2 The !"it ta"($otatio" co(t o% the '!mm/ o48 col!m" eleme"t( ae a((i&"e' to Aeo2 The" the $o,lem i( (ol+e' ,/ the !(!al $oce'!e2992 E*$lai" ho4 the $o%it ma*imiAatio" ta"($otatio" $o,lem ca" ,e co"+ete' to a" e-!i+ale"t co(t mi"imiAatio" ta"($otatio" $o,lem2 7MA3 0FE:I% the o,6ecti+e i( to ma*imiAe the $o%it o ma*imiAe the e*$ecte' (ale( 4e ha+e to co"+et the(e $o,lem( ,/ m!lti$l/i"& all cell e"tie( ,/ #92No4 the ma*imiAatio" $o,lem ,ecome( a mi"imiAatio" a"' it ca" ,e (ol+e' ,/ the !(!al al&oithm922 Detemi"e ,a(ic %ea(i,le (ol!tio" to the %ollo4i"& ta"($otatio" $o,lem !(i"& lea(t co(t metho'2 7MA3 0F9:A 1 C D SUPPL3P929 4BFQ BB29 >FR 42>92FDema"' 2F4FBF9F9B2 De%i"e ta"((hi$me"t $o,lem(@A $o,lem i" 4hich a+aila,le commo'it/ %e-!e"tl/ mo+e( %om o"e (o!ce to a"othe (o!ce o 'e(ti"atio" ,e%oe eachi"& it( act!al 'e(ti"atio" i( calle' ta"((hi$me"t $o,lem(2942?hat i( the 'i%%ee"ce ,et4ee" Ta"($otatio" $o,lem M Ta"((hi$me"t Po,lem@I" a ta"($otatio" $o,lem thee ae "o i"teme'iate (hi$$i"& $oi"t( 4hile i" ta"((hi$me"t $o,lem thee ae i"teme'iate (hi$$i"& $oi"t(9>2 ?hat i( a((i&"me"t $o,lem@A" a((i&"me"t $o,lem i( a $atic!la ca(e o% a ta"($otatio" $o,lem i" 4hich a "!m,e o% o$eatio"( ae a((i&"e' to a" e-!al "!m,e o% o$eato( 4hee each o$eato $e%om( o"l/ o"e o$eatio"= the o+eall o,6ecti+e i( to ma*imiAe the total $o%it o mi"imiAe the o+eall co(t o%the &i+e" a((i&"me"t2 9C2 E*$lai" the 'i%%ee"ce ,et4ee" ta"($otatio" a"' a((i&"me"t $o,lem(@

Ta"($otatio" $o,lem(A((i&"me"t $o,lem(9:(!$$l/ at a"/ (o!ce ma/ ,e aS!$$l/ at a"/ (o!ce 4ill a"/ $o(iti+e -!a"tit/2,e 922:Dema"' at a"/ 'e(ti"atio" ma/ Dema"' at a"/ 'e(ti"atio",e a $o(iti+e -!a"tit/24ill ,e 92B:O"e o moe (o!ce to a"/ "!m,e O"e (o!ce o"e 'e(ti"atio"2 o% 'e(ti"atio"29D2 De%i"e !",o!"'e' a((i&"me"t $o,lem a"' 'e(ci,e the (te$( i"+ol+e' i" (ol+i"& it@I% the "o2 o% o4( i( "ot e-!al to the "o2 o% col!m" i" the &i+e" co(t mati* the $o,lem i( (ai' to ,e !",ala"ce'2 It i( co"+ete' to a ,ala"ce' o"e ,/ a''i"& '!mm/ o4 o '!mm/ col!m" 4ith Aeo co(t2

9E2 E*$lai" ho4 a ma*imiAatio" $o,lem i( (ol+e' !(i"& a((i&"me"t mo'el@The ma*imiAatio" $o,lem( ae co"+ete' to a mi"imiAatio" o"e o% the %ollo4i"& metho'27i: Si"ce ma* A H mi"7#A:7ii: S!,tact all the co(t eleme"t( all o% the co(t mati* %om theHi&he(t co(t eleme"t i" that co(t mati*2992 ?hat 'o /o! !"'e(ta"' ,/ e(ticte' a((i&"me"t@ E*$lai" ho4 /o! (ho!l' o+ecome it@The a((i&"me"t tech"i-!e= it ma/ "ot ,e $o((i,le to a((i&" a $atic!la ta(5 to a $atic!la %acilit/ '!e to tech"ical 'i%%ic!ltie( o othe e(tictio"(2 Thi( ca" ,e o+ecome ,/ a((i&"i"& a +e/ hi&h $oce((i"& time o co(t 7it ca" ,e 2 ?hat i( $!e IPP@I" a li"ea $o&ammi"& $o,lem= i% all the +aia,le( i" the o$timal (ol!tio" ae e(ticte' to a((!me "o"#"e&ati+e i"te&e +al!e(= the" it i( calle' the $!e 7all: IPP2C2 ?hat i( Mi*e' IPP@I" a li"ea $o&ammi"& $o,lem= i% o"l/ (ome o% the +aia,le( i" the o$timal (ol!tio" ae e(ticte' to a((!me "o"#"e&ati+e i"te&e +al!e(= 4hile the emai"i"& +aia,le( ae %ee to ta5e a"/ "o"#"e&ati+e +al!e(= the" it i( calle' A Mi*e' IPP2D2 ?hat i( Geo#o"e $o,lem@I% all the +aia,le( i" the o$tim!m (ol!tio" ae allo4e' to ta5e +al!e( eithe F o 9 a( i" R'o0 o R"ot to 'o0 t/$e 'eci(io"(= the" the $o,lem i( calle'Geo#o"e $o,lem o (ta"'a' 'i(cete $o&ammi"& $o,lem2E2?hat i( the 'i%%ee"ce ,et4ee" P!e i"te&e $o&ammi"& M mi*e' i"te&e i"te&e $o&ammi"&2?he" a" o$timiAatio" $o,lem= i% all the 'eci(io" +aia,le( ae e(ticte' to ta5e i"te&e +al!e(= the" it i( e%ee' a( $!e i"te&e $o&ammi"&2 I% (ome o% the +aia,le( ae allo4e' to ta5e i"te&e +al!e(= the" it i( e%ee' a( mi*e' i"te&e i"te&e $o&ammi"&292 E*$lai" the im$ota"ce o% I"te&e Po&ammi"&@I" li"ea $o&ammi"& $o,lem= all the 'eci(io" +aia,le( allo4e' to ta5e a"/ "o"#"e&ati+e eal +al!e(= a( it i( -!ite $o((i,le a"' a$$o$iate to ha+e %actio"al +al!e( i" ma"/ (it!atio"(2 Ho4e+e i" ma"/ (it!atio"(= e($eciall/ i" ,!(i"e(( a"' i"'!(t/= the(e 'eci(io" +aia,le( ma5e (e"(e o"l/ i% the/ ha+e i"te&e +al!e( i" the o$timal (ol!tio"2 He"ce a "e4 $oce'!e ha( ,ee" 'e+elo$e' i" thi( 'iectio" %o the ca(e o% LPP (!,6ecte' to a''itio"al e(tictio" that the 'eci(io" +aia,le( m!(t ha+e i"te&e +al!e(29F2 ?h/ "ot o!"' o%% the o$tim!m +al!e( i" (tea' o% e(oti"& to IP@ 7MA3 0FE:Thee i( "o &!aa"tee that the i"te&e +al!e' (ol!tio" 7o,tai"e' ,/ (im$le* metho': 4ill (ati(%/ the co"(tai"t(2 i2e2 2= it ma/ "ot (ati(%/ o"e o moe co"(tai"t( a"' a( (!ch the "e4 (ol!tio" ma/ "ot %ea(i,le2 So thee i( a "ee' %o 'e+elo$i"& a (/(tematic a"' e%%icie"t al&oithm %o o,tai"i"& the e*act o$tim!m i"te&e (ol!tio" to a" IPP2992 ?hat ae metho'( %o IPP@ 7MA3 0FE:I"te&e $o&ammi"& ca" ,e cate&oiAe' a(7i: C!tti"& metho'(7ii: Seach Metho'(2922 ?hat i( c!tti"& metho'@A (/(tematic $oce'!e %o (ol+i"& $!e IPP 4a( %i(t 'e+elo$e' ,/ R2E2Gomo/ i" 99>E2 Late o"= he e*te"'e' the $oce'!e to (ol+e mi*e' IPP= "ame' a( c!tti"& $la"e al&oithm= the metho' co"(i(t( i" %i(t (ol+i"& the IPP a( o'i"a/ LPP21/ i&"oi"& the i"te&it/ e(tictio" a"' the" i"to'!ci"& a''itio"al co"(tai"t( o"e a%te the othe to c!t cetai" $at o% the (ol!tio" ($ace !"til a" i"te&al (ol!tio" i( o,tai"e'29B2 ?hat i( (each metho'@It i( a" e"!meatio" metho' i" 4hich all %ea(i,le i"te&e $oi"t( ae e"!meate'2 The 4i'el/ !(e' (each metho' i( the 1a"ch a"' 1o!"' Tech"i-!e2 It al(o (tat( 4ith the co"ti"!o!( o$tim!m= ,!t (/(tematicall/ $atitio"( the (ol!tio" ($ace i"to (!, $o,lem( that elimi"ate $at( that co"tai" "o %ea(i,le i"te&e (ol!tio"2 It 4a( oi&i"all/ 'e+elo$e' ,/ A2H2La"' a"' A2G2Doi&2942 E*$lai" the co"ce$t o% 1a"ch a"' 1o!"' Tech"i-!e@The 4i'el/ !(e' (each metho' i( the 1a"ch a"' 1o!"' Tech"i-!e2 It(tat( 4ith the co"ti"!o!( o$tim!m= ,!t (/(tematicall/ $atitio"( the (ol!tio" ($ace i"to (!, $o,lem( that elimi"ate $at( that co"tai" "o %ea(i,le i"te&e (ol!tio"2 It 4a( oi&i"all/ 'e+elo$e' ,/ A2H2La"' a"' A2G2Doi&29>2 Gi+e the &e"eal %omat o% IPP@The &e"eal IPP i( &i+e" ,/Ma*imiAe G H CPS!,6ect to the co"(tai"t(AP K ,=P L F a"' (ome o all +aia,le( ae i"te&e29C2 ?ite a" al&oithm %o Gomo/0( Factio"al C!t al&oithm@92 Co"+et the mi"imiAatio" IPP i"to a" e-!i+ale"t ma*imiAatio" IPP a"' all the coe%%icie"t( a"' co"(tai"t( (ho!l' ,e i"te&e(222 Fi"' the o$tim!m (ol!tio" o% the e(!lti"& ma*imiAatio" LPP ,/ !(i"& (im$le*metho'2B2 Te(t the i"te&it/ o% the o$tim!m (ol!tio"242 Re4ite each P1i>2 E*$e(( each o% the "e&ati+e %actio"( i% a"/= i" the 5th o4 o% the o$tim!m (im$le* ta,le a( the (!m o% a "e&ati+e i"te&e a"' a "o"#"e&ati+e %actio"2C2 Fi"' the %actio"al c!t co"(tai"tD2 A'' the %actio"al c!t co"(tai"t at the ,ottom o% o$tim!m (im$le* ta,le o,tai"e' i" (te$ 22E2 Go to (te$ B a"' e$eat the $oce'!e !"til a" o$tim!m i"te&e (ol!tio" i( o,tai"e'29D2 ?hat i( the $!$o(e o% Factio"al c!t co"(tai"t(@I" the c!tti"& $la"e metho'= the %actio"al c!t co"(tai"t( c!t the !"!(e%!l aea o% the %ea(i,le e&io" i" the &a$hical (ol!tio" o% the $o,lem2 i2e2 c!t that aea 4hich ha( "o i"te&e#+al!e' %ea(i,le (ol!tio"2 Th!( the(e co"(tai"t( elimi"ate all the "o"#i"te&al (ol!tio"( 4itho!t loo(i"& a"/ i"te&e#+al!e' (ol!tio"29E2A ma"!%act!e o% ,a,/ 'oll( ma5e( t4o t/$e( o% 'oll(= 'oll P a"' 'oll 32 Poce((i"& o% the(e 'oll( i( 'o"e o" t4o machi"e( A a"' 12 Doll P e-!ie( 2 ho!( o" machi"e A a"' C ho!( o" Machi"e 12 Doll 3 e-!ie( > ho!( o" machi"e A a"' > ho!( o" Machi"e 12 Thee ae 9C ho!( o% time $e 'a/ a+aila,le o" machi"e A a"' BF ho!( o" machi"e 12 The $o%it i( &ai"e' o" ,oth the 'oll( i( (ame2 Fomat thi( a( IPP@Let the ma"!%act!e 'eci'e to ma"!%act!e *9 the "!m,e o% 'oll P a"' *2"!m,e o% 'oll 3 (o a( to ma*imiAe the $o%it2 The com$lete %om!latio" o% the IPP i( &i+e" ,/Ma*imiAe G H *9I*2S!,6ect to2 *9 I > *2 K9CC *9I > *2 KBFa"' LF a"' ae i"te&e(2992 E*$lai" Gomo/0( Mi*e' I"te&e Metho'@The $o,lem i( %i(t (ol+e' ,/ co"ti"!o!( LPP ,/ i&"oi"& the i"te&it/ co"'itio"2 I% the +al!e( o% the i"te&e co"(tai"e' +aia,le( ae i"te&e(= the" the c!e"t (ol!tio" i( a" o$timal (ol!tio" to the &i+e" mi*e' IPP2 El(e (elect the (o!ce o4 4hich coe($o"'( to the la&e(t %actio"al $at amo"& the(e ,a(ic +aia,le( 4hich ae co"(tai"e' to ,e i"te&e(2 The" co"(t!ct the Gomaia" co"(tai"t %om the (o!ce o42 A'' thi( (eco"'a/ co"(tai"t at the ,ottom o% the o$tim!m (im$le* ta,le a"' !(e '!al (im$le* metho' to o,tai" the "e4 %ea(i,le o$timal (ol!tio"2 Re$eat thi( $oce'!e !"til the +al!e( o% the i"te&e e(ticte' +aia,le( ae i"te&e( i" the o$tim!m (ol!tio" o,tai"e'22F2 ?hat i( the &eometical mea"i"& o% $otio"e' o ,a"che' the oi&i"al $o,lem@Geometicall/ it mea"( that the ,a"chi"& $oce(( elimi"ate( $otio" o% the %ea(i,le e&io" that co"tai"( "o %ea(i,le#i"te&e (ol!tio"2 Each o% the (!,#$o,lem( (ol+e' (e$aatel/ a( a LPP2292 ?hat i( (ta"'a' 'i(cete $o&ammi"& $o,lem@I% all the +aia,le( i" the o$tim!m (ol!tio" ae allo4e' to ta5e +al!e( eithe F o 9 a( i" R'o0 o R"ot to 'o0 t/$e 'eci(io"(= the" the $o,lem i( calle' (ta"'a' 'i(cete $o&ammi"& $o,lem2222 ?hat i( the 'i(a'+a"ta&e o% ,a"che' o $otio"e' metho'@It e-!ie( the o$tim!m (ol!tio" o% each (!, $o,lem2 I" la&e $o,lem( thi( co!l' ,e +e/ te'io!( 6o,22B2 Ho4 ca" /o! im$o+e the e%%icie"c/ o% $otio"e' metho'@The com$!tatio"al e%%icie"c/ o% $otio"e' metho' i( i"cea(e' ,/ !(i"& the co"ce$t o% ,o!"'i"&2 1/ thi( co"ce$t 4he"e+e the co"ti"!o!( o$tim!m (ol!tio" o% a (!, $o,lem /iel'( a +al!e o% the o,6ecti+e %!"ctio" lo4e tha" that o% the ,e(t a+aila,le i"te&e (ol!tio" it i( !(ele(( to e*$loe the $o,lem a"/ %!the co"(i'eatio"2 Th!( o"ce a %ea(i,le i"te&e (ol!tio" i( o,tai"e'= it( a((ociati+e o,6ecti+e %!"ctio" ca" ,e ta5e" a( a lo4e ,o!"' to 'elete i"%eio (!,#$o,lem(2 He"ce e%%icie"c/ o% a ,a"ch a"' ,o!"' metho' 'e$e"'( !$o" ho4 (oo" the (!cce((i+e (!,#$o,lem( ae %athome'2UNIT#I.92 ?hat 'o /o! mea" ,/ $o6ect@A $o6ect i( 'e%i"e' a( a com,i"atio" o" i"te elate' acti+itie( 4ith limite' e(o!ce( "amel/ me"= machi"e( mateial(= mo"e/ a"' time all o% 4hich m!(t ,e e*ec!te' i" a 'e%i"e' o'e %o it( com$letio"222 ?hat ae the thee mai" $ha(e( o% $o6ect@ Pla""i"&= Sche'!li"& a"' Co"tolB2 ?hat ae the t4o ,a(ic $la""i"& a"' co"tolli"& tech"i-!e( i" a "et4o5 a"al/(i(@ Citical Path Metho' 7CPM: Po&amme E+al!atio" a"' Re+ie4 Tech"i-!e 7PERT:42 ?hat ae the a'+a"ta&e( o% CPM a"' PERT tech"i-!e(@ It e"co!a&e( a lo&ical 'i(ci$li"e i" $la""i"&= (che'!li"& a"' co"tol o% $o6ect( It hel$( to e%%ect co"(i'ea,le e'!ctio" o% $o6ect time( a"' the co(t It hel$( ,ette !tiliAatio" o% e(o!ce( li5e me"=machi"e(=mateial( a"' mo"e/ 4ith e%ee"ce to time It mea(!e( the e%%ect o% 'ela/( o" the $o6ect a"' $oce'!al cha"&e( o" the o+eall (che'!le2>2 ?hat i( the 'i%%ee"ce CPM a"' PERTCPM Net4o5 i( ,!ilt o" the ,a(i( o% acti+it/ Detemi"i(tic "at!e O"e time e(timatio"PERT A" e+e"t oie"te' "et4o5 Po,a,ili(tic "at!e Thee time e(timatio"C2 ?hat i( "et4o5@A "et4o5 i( a &a$hical e$e(e"tatio" o% a $o6ect0( o$eatio" a"' i( com$o(e' o% all the e+e"t( a"' acti+itie( i" (e-!e"ce alo"& 4ith thei i"te elatio"(hi$ a"' i"te 'e$e"'e"cie(2D2 ?hat i( E+e"t i" a "et4o5 'ia&am@A" e+e"t i( ($eci%ic i"(ta"t o% time 4hich ma5( the (tat( a"' e"' o% a" acti+it/2 It "eithe co"(!me( time "o e(o!ce(2 It i( e$e(e"te' ,/ a cicle2E2 De%i"e acti+it/@A $o6ect co"(i(t( o% a "!m,e o% 6o, o$eatio"( 4hich ae calle' acti+itie(2 It i( the eleme"t o% the $o6ect a"' it ma/ ,e a $oce((= mateial ha"'li"&= $oc!eme"t c/cle etc292 De%i"e Citical Acti+itie(@I" a Net4o5 'ia&am citical acti+itie( ae tho(e 4ho(e i% co"(!me moe tha" e(timate' time the $o6ect 4ill ,e 'ela/e'2

9F2 De%i"e "o" citical acti+itie(@Acti+itie( 4hich ha+e a $o+i(io" (!ch that the e+e"t i% the/ co"(!me a ($eci%ie' time o+e a"' a,o+e the e(timate' time the $o6ect 4ill "ot ,e 'ela/e' ae teme' a( "o" citical acti+itie(2992 De%i"e D!mm/ Acti+itie(@?he" t4o acti+itie( (tat at a (ame time= the hea' e+e"t ae 6oi"e' ,/ a 'otte' ao4 5"o4" a( '!mm/ acti+it/ 4hich ma/ ,e citical o "o" citical2922 De%i"e '!atio"@It i( the e(timate' o the act!al time e-!ie' to com$lete a ta'e o a" acti+it/2

9B2 De%i"e total $o6ect time@It i( time ta5e" to com$lete to com$lete a $o6ect a"' 6!(t %o!"' %om the (e-!e"ce o% citical acti+itie(2 I" othe 4o'( it i( the '!atio" o% the citical $ath2942 De%i"e Citical Path@ It i( the (e-!e"ce o% acti+itie( 4hich 'eci'e( the total $o6ect '!atio"2 It i( %ome' ,/ citical acti+itie( a"' co"(!me( ma*im!m e(o!ce( a"' time29>2 De%i"e %loat o (lac5@ 7MA3 0FE:Slac5 i( 4ith e($ect to a" e+e"t a"' %loat i( 4ith e($ect to a" acti+it/2 I" othe 4o'(= (lac5 i( !(e' 4ith PERT a"' %loat 4ith CPM2 Float o (lac5 mea"( e*ta time o+e a"' a,o+e it( '!atio" 4hich a "o"#citical acti+it/ ca" co"(!me 4itho!t 'ela/i"& the $o6ect29C2 De%i"e total %loat@ 7MA3 0FE:The total %loat %o a" acti+it/ i( &i+e" ,/ the total time 4hich i( a+aila,le %o $e%oma"ce o% the acti+it/= mi"!( the '!atio" o% the acti+it/2 The total time i( a+aila,le %o e*ec!tio" o% the acti+it/ i( &i+e" ,/ the late(t %i"i(h time o% a" acti+it/ mi"!( the ealie(t (tat time %o the acti+it/2 Th!(Total %loat H Late(t (tat time ) ealie(t (tat time29D2 De%i"e %ee %loat@ 7MA3 0FE:Thi( i( that $at o% the total %loat 4hich 'oe( "ot a%%ect the (!,(e-!e"t acti+itie(2 Thi( i( the %loat 4hich i( o,tai"e' 4he" all the acti+itie( ae (tate' at the ealie(t29E2 De%i"e I"'e$e"'e"t %loat@ 7MA3 0FD: 7MA3 0FE:I% all the $ece'i"& acti+itie( ae com$lete' at thei late(t= i" (ome ca(e(= "o %loat a+aila,le %o the (!,(e-!e"t acti+itie( 4hich ma/ thee%oe ,ecome citical2 I"'e$e"'e"t %loat H %ee ) tail (lac52992 De%i"e I"te%ei"& %loat@Sometime( %loat o% a" acti+it/ i% !tiliAe' 4holl/ o i" $at= ma/ i"%l!e"ce the (tati"& time o% the (!ccee'i"& acti+itie( i( 5"o4" a( i"te%ei"& %loat2I"te%ei"& %loat H late(t e+e"t time o% the hea' # ealie(t e+e"t time o% the e+e"t22F2 De%i"e O$timi(tic@O$timi(tic time e(timate i( the '!atio" o% a"/ acti+it/ 4he" e+e/thi"& &oe( o" +e/ 4ell '!i"& the $o6ect2 292 De%i"e Pe((imi(tic@Pe((imi(tic time e(timate i( the '!atio" o% a"/ acti+it/ 4he" almo(te+e/thi"& &oe( a&ai"(t o! 4ill a"' a lot o% 'i%%ic!ltie( i( %ace' 4hile 'oi"& a $o6ect2222 De%i"e mo(t li5el/ time e(timatio"@Mo(t li5el/ time e(timate i( the '!atio" o% a"/ acti+it/ 4he" (ometime( thi"& &o o" +e/ 4ell= (ometime( thi"&( &o o" +e/ ,a' 4hile 'oi"& the $o6ect2242 ?hat i( a $aallel citical $ath@?he" citical acti+itie( ae ca(he' a"' the '!atio" i( e'!ce' othe $ath( ma/ al(o ,ecome citical (!ch citical $ath( ae calle' $aallel citical $ath22>2 ?hat i( (ta"'a' 'e+iatio" a"' +aia"ce i" PERT "et4o5@ 7NO. 0FD:The e*$ecte' time o% a" acti+it/ i" act!al e*ec!tio" i( "ot com$letel/ elia,le a"' i( li5el/ to +a/2 I% the +aia,ilit/ i( 5"o4" 4e ca" mea(!e the elia,ilit/ o% the e*$ecte' time a( 'etemi"e' %om thee e(timate(2 The mea(!e o% the +aia,ilit/ o% $o((i,le acti+it/ time i( &i+e" ,/ (ta"'a' 'e+iatio"= thei $o,a,ilit/ 'i(ti,!tio".aia"ce o% the acti+it/ i( the (-!ae o% the (ta"'a' 'e+iatio"2C2 Gi+e the 'i%%ee"ce ,et4ee" 'iect co(t a"' i"'iect co(t@ 7NO. 0FD:Diect co(t i( 'iectl/ 'e$e"'i"& !$o" the amo!"t o% e(o!ce( i"+ol+e' i" the e*ec!tio" o% all acti+itie( o% the $o6ect2 I"cea(e i" 'iect co(t 4ill 'ecea(e i" $o6ect '!atio"2 I"'iect co(t i( a((ociate' 4ith &e"eal a"' a'mi"i(tati+e e*$e"(e(= i"(!a"ce co(t= ta*e( etc2 I"cea(e i" i"'iect co(t 4ill i"cea(e i" $o6ect '!atio"2UNIT#.92 De%i"e Se"'al0( "otatio" %o e$e(e"ti"& -!e!i"& mo'el(2A -!e!i"& mo'el i( ($eci%ie' a"' e$e(e"te' (/m,olicall/ i" the %om 7a8,8c: ; 7'8e:?hee a# i"te ai+al time,#(e+ice mecha"i(mc#"!m,e o% (e+ice'#the ca$acit/ o% the (/(teme#the -!e!e 'i(ci$li"e22 I" a (!$e ma5et= the a+ea&e ai+al ate o% c!(tome i( > i" e+e/ BF mi"!te( %ollo4i"& Poi((o" $oce((2 The a+ea&e time i( ta5e" ,/ the ca(hie to li(t a"' calc!late the c!(tome0( $!cha(e i( 42> mi"!te(Q %ollo4i"& e*$o"e"tial 'i(ti,!tio"2 ?hat i( the $o,a,ilit/ that the -!e!e le"&th e*cee'( >@Ai+al ateH >8BF mi"Se+ice ateH289mi"Po,a,ilit/ that the -!e!e le"&th e*cee'( > H 7T: "I2 H 72D>: DHF29BBB2 E*$lai" Q!e!e 'i(ci$li"e a"' it( +aio!( %om(27i: FIFO o FCFS #Fi(t I" Fi(t O!t o Fi(t Come Fi(t Se+e'27ii:LIFO o LCFS #La(t I" Fi(t O!t o La(t Come Fi(t Se+e'27iii: SIRO#Selectio" %o (e+ice i" a"'om o'e27i+:PIR# Pioit/ i" (electio"42 Di(ti"&!i(h ,et4ee" ta"(ie"t a"' (tea'/ (tate -!e!i"& (/(tem2A (/(tem i( (ai' to ,e i" ta"(ie"t (tate 4he" it( o$eati"& chaactei(tic( ae 'e$e"'e"t o" time2 A (tea'/ (tate (/(tem i( o"e i" 4hich the ,eha+io o% the (/(tem i( i"'e$e"'e"t o% time2>2 De%i"e (tea'/ (tate@A (/(tem i( (ai' to ,e i" (tea'/ (tate 4he" the ,eha+io o% the (/(tem i"'e$e"'e"t o% time2 Let $"7t: 'e"ote the $o, that thee ae R"0 !"it( i" the (/(tem at time t2 the" i" (tea'/ (tateHO lim $"U7 t :HF

tV