Faaliyetler İş Araştırma ve İktisat Linear Programming Applications Chapter 9 Linear Programming Applications • Blending Problem--- • Portfolio Planning Problem • Product Mix Problem • Transportation Problem • Data Envelopi:nent Analysis • Revenue Management C 2006 Tho~on ScuU.-Wesll'ffi. All Rights Reserved. Slide 1 Blending P;oblem Vitamin C Protein Iron ': Craii1 . · .L!nits/ lb Units/lb .Units/lb Cost j Ib . .1. . 9 12 0 .75 .' .. 2 16 10 14 .90 ,. . c· . .. 8 . 0 . '""": ,.. , . " .. .J.; 8 )0 . 15. ;'-"'-."- .. "'4 '. .. 10 8 7 .70 . .. ~. . " .' :_ .:.;.',\ ,c: '.: .,.; .". Ferdinand is interested in producing the 8-ounce ~,~ ~/ : '\i·~;:[·~;~·~:~ .M~.~t '~i.un io~t ~hiie in~etfug ih~ ~~~ " "?;;~, !f;;-7;~~;fi~~1::e;:: ~/iI:t;?fVit~mID·C'5 "ni ,lSof' ..... . Slide 3 · ': .. , ;.: .. ; ... "' :: .. :; ' .. ," ';"':~ . . ", -. -. '-. .; :< . Blending Problem Ferdm.al Feed Company rece!ves four raw grains froOl-tuch il blends its dry pel food. The pel food advertiiles thai each 8-ounce packet .III'U"I'>.ia meets the Dliinimum daily requirements for vitamin c.. protein and iron. The cost of eachaw grain as well as the vitamin C. potein, and iron units per pound of eadlgrs'in are su.mmarized °rt .. £o~~-4i~ the next 'slj~ 0 2006 Thomsons...t>--We!ilem. AU Ri ghu Reserved. B~dingP~oblem . . Xi 0: _ pounOs ' of grain-j (j'; }.;L,3.4) -.sed in the .&-ounc-e mixture -._ .... - ... _, . ... -~- ,-,- ---- " , ;. " .' .' .. ' • Define the oIpdive fUrtdioti.· . . - .- -- .- . :~:~~.,.. ..... ~ Slide 2 MirUmiZeih! "iol~t~Qst'f~J:i~-8:Z~n~e mixture: ~-:?~ ;~·;~~}j~J-.~.·:~d~~~ij?x~ . ''-: ~:~ ~~~-~~~·~~~~~·~;~·~t~~·~:·~?\~~/ ·: • • -< . ' - • ~ ~.-- - ::-~- -. - . ~ . " . -- _ ._ '.' '/~;-<~e~eiliecoru:::mgprOblem ' . . a . . ~Mana8e~::/g::::~ , il ' . ..-: :> ;,.", "." Total weight of the DUX 15 8-ounces (.S pounds). . '==- r+> - R1-E'-CllVE- . ' - . -RJN -- C:nCC" " -Oc,N-'V~ .· '-A~ 'L-U-E-~-·-0-.4-0'-6-==' . - . . '-~,'_. . (1) Xl + x 2 + Xl + i~ .... 5 ~- - '':'- i;;;;=f~":' -:"- :" -.-.- .=.'!DlOunt6fVitamin ·Cinthemixisatlea......--- . _ '~_ .;...._ '·- F · ~LE VALUE 0.099 0.213 0.088 REDllf>SD COSTS 0.000 Xl (2) 9x 1 + 16x2 + 8x) + lOr. > 6 X2 0.000 Total amount of protein in the mix is at least 5 units: X3 0.000 "'-,,=='§o~ (3) 12x]+10x 2 +10x,+8x.>5 -_.::--- --: X4 O.099 ---2 --.~ 0.000 Tool 3moWlt of:iwJHR the mix is at least 5 units: == _ _ + __ J~ __ :':': _ _ __ =:':': ___ ,::::=_=:,:":,,,, _ _ .J .,;;:::. (4) 14x2 + lSx) + 7X4 > 5 Nonnegativity of variables: xi?: 0 for all j C 2QP6 Tlwunson South-Western. AU Righb R~ed. "SlideS - Thus, the oJllii-al blend is about _10 tb:'of gTain 1, .21 lb. of grain 2, .«8" of grain 3, ~n? .10. 1~ . of grain 4. The mixture cosls&ederic.k's dlLb.~IYi~'~'~' !l:==-- 0 2006 Thomson ~estem. All Rights ~~. Slide 6 1 . . ," '.: '. Portfolio PlannirIg Problem fo r investment. 11 wishes to invest over the next fbur months in such a way that it will maximize the total interest earned.over the four month period as well as have at least $10 million available al the start of the fifth month for a mgnrise hUildu:g venture in which it will be participating. (1 2006 Thomson South-WeS tern . . AU Righ~ Reserved. SUde 7 porti9h~Plaruting Problem ~ Formulat~ ·a ;~ear .prograO"l that will help ~ W imIow sa: v . iflgs '~e~e how to invest over !he n~xt f~~I ~~n~ if ~-t ~o 'fun~ d'~s it wi~h to h'a~e more tlta ,n '$8:'~'Op ~ :,ei~}\I~r-gqveriunent bonds or cons~c.ti~ri .r¢~~ "' _ ~' . ~ - -- ~ ~ .---- -~ -- -.. •.. ' • ...... Slid~ 9 ., ~'.!':.'~ ':-";. " - :" \:::=. ·i~l :..~~~.;~)~ ;i~.):' ". ' .... ... " . ' . · port.te\i9~fl.~gProblem 41 ·· ··· ···· . . .,'. , . • . Define the o:bj~~\I;e ~ ~!=ti 2006 Tho~on 5oooIIo-We;tem. All Righll;,Reserved .. Slide 14 Portfolio 'Plarutirtg Problem No more than-sB million'iic()nsttUttion·loans at·a ny time: '1 + C:J . . ~. ·8;000;000 (10) c, (11) (12) (13) (I ~~~ c~--·:+~{~,{s,{@~~:.~ C:z + (3 1- ( • . ~ ·8,000,000 . ~ . ., .. : . . ....... SJid~ 1·~ : ],2,3;4'" . . . . ' . - ";;-:~;';:; '; .;~~~.; ::.- . .... ;. ~.: .~ ,." -. -:~ -' .. Slide 19 - Sl,ide 21 P :r?~.il;~t~!0,Problem· - . . . . c-,ea'~il:ial 'c9inpOsite unit. ., .. ,U £. ~: 1.. unit k.is.less ~cient than the composite unit " '.. '.:': ·:);~:'i~~f~:q~~~Q:#J~,~velyiI)eifi~.~t; . '.: , ' .. :' ,: . ': . ',: . . ' .' .' , ;', '~.:.,·~:~',~::it' .(¥J~·~~(~.~~Q;eYide,.c .e Ut~'t P ' IVI): ~ ~~ci~nt, .~~1 -:: ::· 1<~.o)i~.qnnot-"q:)Odude that k is absolutely. efficient. ·. , ' 1 '1 '1 :~ " ., " .\ \ ., " " . . , . ~ 'i ,.; ~ The Langley County School District.is.bying to .,' . :..\.:d~te:rn.1iJ:tiUhei:eIative efficiency 'of ' ':.' ~. i~ ~three high schools. In particular, . ., "-: - ii"w1iltS to ev~.!late Rciosevelt High. . Theaistrlct'is evaluating performances on SAT scores, the rwmber of seniors finishing high - number of students who enter college as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and.,lhp n l@j sf students in the senior class. C:mP6 ThOmsDn 5ovth-West~ AU Righu Reserved. Slide 41 . Tr.msportation Problem • Solution Su..-ary ----- • San Diego..ruJ receive 1000 Ibs. by truck and 3000 Jb.. by airplane. • Norfolk ~receive2000 Ibs. by-truck and 500 1b5.. by railroad, • Pensacola will receive 2500 lbs, by railroad, - - • The total shipping cost will be- $142.000. t~-~ d 1- [ Data Envelop~~~t::;,;;alysis · • The DEA Modd MIN E -' .' .: < . ' ,; 5.1. w~ Ou.tp _uJ~.~JJ1V! KS_ f'H.tp .4.t. , (for each h,ea~u~ irutput) W"'&I>Ied inpuJS : i ti\lha,,' s:iriJ>~t) Slide 38 ~i(::~::~ ·:;~~·f!~f~~:~~,t).:.~' .. : ' -:_- · ' E; _~ts~o: ::~!:~~:\::' ~-;:'::- .: 'C: ': ., ' ,.',,. , .~ ~-~ ., ,~,!,; :.:: ., , ., ._ " "" ->~:-=:'+'-1~-!:'~";': '::,~ ~;"-i. " 1 '.j' ",,' '-f '.,- '., .• . , ., ., - _ Slide 40 ' ... .... ~. ' .- '.: :., '.',... . _ Da~ ~ve:lop~.e:·~f~.~~r.s~s,,' . • . Input - Roosevelt _ i.IDcdn 'Washington 'i;e.;ror Facul.,!: Budget ($100'-5) Senior Eruou-,.ts "J.' 6,' 850 > 25 · 23· 5.0 - 4.7 700 600 ~-~ - --- - - 02006 Thomson~lem. AURighb~erved. SUde42 T ':. 7 " ., ". Data Envelopment Analysis ~ "" --- • Output p Roosevelt Lincoln ~ Washlng!on - - Average SAT Score 800 830 900 High School Graduates 450 500 , 400 College Admissions 140 250 370 ._ - - C 2006 Thomson50uth-Wes!~= AU Rigt.ts Reserved_ .. Sljd~ 43 Data Envelop!11ent AnalysIs • ObJecbve FunchOn ~ J ~ ' Minimize the fracbon of. Roosevelt H.lgh- School's mpu resources required by the composite high school: '" ~ . . ~ . ...... - - MINE .' !' i SUde 45 · i .. - .' . '~. ': .. : - , . .. ...... .. -..... . -~ii~-.~~~~~~~~~'--~~]S • Constraints. Input ConstraintS: '.. ~ - - "'~ . t resow:ces ~~,~,bl,e}~ the composite school is .' r-===- .- 'a ~onal multiple, £;- .ohheTesources availalMe to ' Roosevelt. Since the composite mgh sdlool cannot use more input than that available to it, the input constraints are: ,:: 1-- (5) 3~Wl +_ 25w 2 + ~!.~ .3-..J?E (Faculty) - (.6) 6.4w 1 + 5.Ow 2 + 4.7w3 ~ -6,41: (13udget) - (7) 8SoW 1 + 70Ow 2 + 600w) S 850E (Seniors) Nonnegativity of variables: E, lO l , w 2 • w) ~ 0 - C 2006 Thomson South-Western. All Rights Eese~. Slide 47 . ~ Data Envelopment Analysis • Decision Variables E = Fraction of Roosevelt's input resources required by the composik:.hig.b.schooJ w 1 = Weight applied to Roosevelt's input/output resources ~ the composite high school 10 2 = Weight applied to Lincoln's input/output -resources ~ the composite h,i ghschool wl- = Weight a~o Washington's input/output resources t _ . 2 COWPQfjte high school (l2()()6 Thomson Sou!l.-'llWesum. AU Rights Reservl"d. - Slide 44 Data Envelopment Analysis .~ ... . • Constraints Sum of the Weipits;iS1; - (I ) Wl+W2+~= 1 _ Output Constr.rirds: ~ '. '. '- ". '. Since WI -=1 is passante, each output of the compOsite school must be ill least as grea t as that of RoOs-evelt .~:·.;:;;~;~:~ (2) 800w l + ~ + 900w):: 800 (SAT Scor~s) . (~) 45~1 + ~"!" :40OwJ:: 450 {G .r~d~a.~.~} _ : ; .. :;:~ "".:j.:~!, .. "(4) 14Ow 1 + 2so..z +37Ow):: ]40 , (College Ad".~~~~}; \.:~:; .::; -:::" ':;~ ':':-< ':i::-r:;_~. , l > , , • '--__ " _2_0_ 06_Th _'_~_ ,"_5_~_ .. _"_=_m_ . _ _ A_"_R~i g~h_ "~R_ ~ ~'_~~ ."'_'.;..-';"";''''''''''' -~ '~ - ~Sti~ ·d~ ~~ ~'_'~ .;..J ~ .'. Data ~velopment Analysis. Th<. M".nal(em~ Scientist Outpu I E WJ W2 ~ ::. 0]65 0.000 MOO 0500- REDUCED C05IS'_ 0_000 0.235 0.000 0.000 Reserved. Slide 48 " _ .. _ ._- 8 Data Envelopment Analysis I~~ • The MaMgrnunt Scientist Output CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 -0.23'5 2 65.000 0.000 3 0.000 ,0.001 4 170.000 0.000 5 4.294 0.000 6 0.044 0.000 .. 7 0.000 0.ll8i i 02006 ThOD\SOI\ SoUth·Wblem. AU Rights Reserved. Slide 49 ... :, .. ', ~evenue Managemenl - . :'Ailp~~.r LP application is revenue management. :. ,:.Re';';enu· e·management involves·ma.Ilagmg the short- - .'. t"erm'(fe'man'd'fcii- 8 fixed .·periShable inventory in ' . '. _ .... :,C?,.>});.\ ': ,:,-: ... Slidl' SO . _,slide 52 - - --- 9