CE 561 Lecture Notes. Optimal Timing of Investment. Set 3. Case A- C is const. cost in 1 st yr, benefits start at the end of 1 st yr
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1 CE 56 Letue otes Set 3 Optmal Tmg of Ivestmet Case A- C s ost. ost st y, beefts stat at the ed of st y C b b b3 0 3 Case B- Cost. s postpoed by oe yea C b b3 0 3 (B-A C s saved st yea C C, b b 0 3 Savg by postpog by oe yea, S( C( -C b at the ed of st y S( C b If the st y s beeft s less tha C, the postpoemet s wothwhle ad addtoal postpoemet should be tested.
2 Ga by postpoemet by oe y, S(t 0 - S(t C - b t PVMax PV 0 - feas PV0 opt Optmal Tmg Ctea FYB FYRR S ( t C b 0 t defe < 0 stat bt / C MARR defe > MARR stat The hoe stat meas to ope the poet fo use yea t. Thus ost. must be sheduled fo ompleto po to yea t so that the beeft yea t, b t, wll be fully ealzed. Costuto Moe Tha Oe Y The ash flows of ostuto osts must be ompouded to the equvalet sgle value the fal y (the y peeedg the fst beeft y YR 3 4 Cost x Y 4 Value ,450( MARR 0%
3 Y et Beefts x Beeft as % of C Usg the FYB teo, C 0. x The fst yea b t exeeds 45 s 997 (o C b t beomes ve. The poet should stat 993 Usg the FYRR teo, 997 s the fst yea whh b t /C exeeds 0% (MARR. Stage Costuto Should the falty be bult ow at the full seve level o should t be bult stages as the demad develops? Sgle stage (4-lae $,866,000 -stage (-laes ow $,00,000, addl. -laes yeas fom ow $,500,000 7% Aual beefts fo both ases ae the same.
4 PW of d stage ost $,69,500 (5,96,450 (0,866,000,587,600 ( (0,397,00 ( Y of d stage If the d stage s eeded befoe yea, sgle stage ostuto s bette. Optmzg Stage Costuto Pogessve Impovemets ove a exteded peod of tme. I stage ostuto, the poblem of detemg optmal tmg s todued at eah stage of mpovemet. C C Total Cost a b t a b t a 3 b 3 t Stage 0 (exstg Stage Stage 0 t t C ost of upgadg to stage a, b oeffets of eug ost futos (lea futo of yea t t optmal tme of upgadg to stage Assumptos:. Eah upgadg takes y.. Wth eah stage, lealy easg osts ou (use ad mateae. 3. Taff eases lealy wth tme. 4. Cotuous ompoudg.
5 Obetves: To deteme the values of t ad t whh wll mmze the total PW of all osts. Ths s equvalet to alulatg max PV of a poet whh ost edutos ae teated as beefts. Computato: PW t ( a b t 0 t t t e dt Ce ( a b t t α t t e dt Ce ( a b 3t t t 3 e dt By dffeetatg the above expesso fo PW w..t. t ad settg the esults equal to 0, C ( a a t b b ( dpw 0 dt Fd t, t C t C t t log ( a a3 ( b b geealzed fom: 3 ( a a ( b b Assumg expoetal taff gowth athe tha lea, usg k as the gowth ate % pe y, otuous osts a b e kt [ C ( a a ] e b b k
6 Example: 3 stages, 4.3 mle gavel oad. Fom gavel to stablzed $30,000/mle. Fom stablzed to asphalt $400,000/mle Reug ost futos (lea Ital t ( $00,000 Stablzed t Asphalt.7.88t MARR 5.5% a o b o a b a b Soluto: 0 ( a a C 0 t b b 30, ( ( yeas fom ow t yeas fom ow Tmg fo Pavg: Beak Eve vs. Optmal Yea C R R α Pt. Of ef. C tal R Rehab peod pepetuty Route Mat aual Veh Op Cost aual
7 s the yea of pavg PV of ost osts PV of Rehab osts PV of savgs oute mateae PV of Veh Op Cost savgs ( C ( ( R ( ( m m ( ( 365 < Whee, PV of pavg ( 0 ( ( ( R C m m 365 Vehle Opeatg Costs ( ( ( ( ( ( ( ( ( 365 At the Yea of Pavg, PV of st Y savgs PV of d Y savgs ( Σ 365
8 Puttg x Summato of a geomet sees x x x To bg the Veh Op Cost savgs to the peset PV of Veh Op Costs 365 ( Optmal Y of Pavg et Beeft the st Y of Pavg 365 ( ( m m Substtutg opt fo R C ad opt opt loge log C ( m m ( 365( ( opt / 0 ( e R ( Fo 0 < opt Othewse 0
9 Colusos B/E s ot a desable appoah PV s maxmzed whe the st Yea s et beeft s equal to zeo Fst Yea Beeft Rule tme to pave s whe the beefts dug the fst yea expessed as a p.. of the ostuto ost s geate tha the oppotuty ost of aptal. Rehab Feq. ( B-E Vol. ( be Opt Vol. ( opt Base Y. Vol. ( /7* 8/4 6/3 00 5/ 3/9 / /5** 0/ 0/ 400 0/0 0/7 0/6 *B-E/Opt Y of pavg **Whe 0 > be, be 0
10 PV of Cost. Def. to Opt Y ($/km ,95 8,74 8,447 00,587 3,36 3, ,98,499 8, ,434 6,073 3,444
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