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Problem Set 3 Solutions

Problem Set 3 Solutions - cm 33°01‘30” gm Pfai’ M...

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Unformatted text preview: cm 33°01‘30” gm Pfai’) M. : gumm 2(a) Algebralc tormulatlon max . 0.043.134 + 0.027%; + 0.025220 + 0.022223 + 0.045.173 subject to : “#kwaafi-‘x {EB +$0 + (ED 2 4 /jk$ pf $A+$B+$C+$D+$ESIO @MI 2xA+2$B+$c+$D+5$ESl.4($A+£EB+£Ec+ID+$E) 933A+l5$3 +4xc+3$D +2$E S 5($A+CEB+£BC+$D +183) 37A 2 01373 20,560201333 20,2727 2 0 This can be simplified to max 0.043334 + 0.0272213 + 0.025235: + 0.022333 + 0.04533}; subject to : $B+$o+x324 $A+$B+$c+zp+rgsm 0.6%; + 0.6.933 - 0.4.30 —- 0.4273 + 3.61123 S 0 423A + 10%; — 1:130 —— 23:13 — 33:3 _<_ O $A202x320330203$D20)$E20 ““x‘ sum {j in BONDS} (years_to_maturity[j] - avg,maturity_,max) * INVEST[j] <= 0 # maximum time to maturity requirement (linearization of proportion constra 2(0) Solution given by AMPL: objective 0 . 2983636364 K?A INVEST [*J := C I; f/ ' 2.18182 0 7.36364 0 0.454545 “WOODS J 2(d) Note that this is different from the solution in Bradley, flax, and Magnanti. Their solution is: Objective = 0 . 294 A 3.36 B 0 C O D 6.48 E 0 16 \EEEDWhiCh is different from What We obtained using AMPL. In fact, the textbook result is W“ effig- *”2(e) ,nppose we increase the total available cash from $10 million to $20 or $100 million, lather than $10 million. if g0: y; :5 f If the total available cash 18 increased, then we should be able to buy more bonds and Kk ,1“ increase the objective function value. When we check with AMPL, this IS indeed true 1 gr When we increase available cash to $20 million, we have: {M l ,5 \obj ect ive 0 . 5967272727 INVEST [=1] := m i) #2» HM! M “4,; A 4.36364 0 - W3 "M Q. -m 1%?) $15.31" (w (33‘) 14. 7273 B C E 0 smelt My 13:1, fifieia 0.909091 and when we increase available cash to $100 million, we have: \objective 2.983636364 INVEST [*J := A 21.8182 B 0 ' C 73.6364 D 0 E 4.54545 5\ 31k ma- .5 ‘(fmw‘fi EWM w ::W +15, [X W iwmkmm )5 a“ CEM’H‘aM Katha ”V M mww> M a“ WWW WEWM\%’€“E \Qfigkfikffingfim Atyfi'iq- Ma§ 03mg?" M A” M Nw-mfi‘f’ in”; MW}: kmefi fiafififi . 9) Changes: — — Add parameters param borrow_rate and parain borroinmit. — — Add variable var BORROW. — — Modify cash constraint to consider also borrowed money. — — Modify objective. The model file is: set BONDS ; # possible bonds for investment set GDVTANDAGENCY_BDNDS; # subset of government and agency bonds param quality_rating {BONDS} >=O; # 3rd column in table param years_to_maturity {BONDS} >=0; # 4rth column in table param yield {BONDS} >=0; # 6th column in table param cash_avail >=0; # amount of money to invest param avg_quality_max >=0; # threshold (3) in problem param avg_maturity_max >=0; # threshold (2) in problem param govtandagency_bonds_min >=O ; # threshold (1) in problem [pfifiamfi‘ifiemmmfifi’é‘i’teflzw :QHEQEJQQ;%QI_IGQF no Airgun-3:“ W Wfimsde‘I'OWillmlt >=0,v.#l-im it. ‘ 5.130, amount iborrofie d var INVEST{BDNDS} >= 0; # decision variables of how much to invest in each bond fidfiéfiBDRRGW>;OJ-. ;',-: < =bofrewi-1ii-miit“? ‘5 # decision variable of how much to borrowr maximize total_earnings: sum {j in BONDS} (yieldEjj * INVEST[j])es min (”esseeemws figflgfi?mu%stfifiite%§EEEePax returE;93ngD§DW§dmmflflfifif i‘" .3. "‘0‘ .77 subject to cash: sum {j in BONDS} INVESTEj] gfiBORROWBk= cash_avail ; seesannospendmorethan Rash. f hos-'0‘? a, subject to govtandagency_requirement: sum {j in GUVTANDAGENCY_BONDS} INVESTEj] >= govtandagency_bonds_min; # minimum investment in government and agency bonds subject to avg_quality: sum {j in BONDS} (quality_rating[j] - avg_quality_max) * INVESTEj] <= 0; # maximum quality requirement (linearization of proportion constraint) subject to avg_maturity: sum {j in BONDS} (years_to_maturity[j] - avg_maturity_max) * INVESTEj] <= 0; # maximum time to maturity requirement (linearization of proportion constraint) The data file is: set BONDS := A B C D E; set GDVTANDAGENCY_BDNDS := B C D; param : quality_rating years_to_maturity yield := A 2 9 .043 B 2 15 .027 C 1 4 .025 D 1 3 .022 E 5 2 .045; param cash_avail := 10; # in millions dollars param avg_qnality_max := 1.4; param avg_maturity_max := 5; #in years Hfififififififirrow3rate“?é”fO275T”W“ tseseerrtaxzinterest*ratenon:borrowedrfpnds_, Eggwamwhernow;limit-z= 1 ; #;in millions of dollars 4 param govtandagency_bonds_min := 4; # in millions dollars # yields are after-tax The output is: 6 variables, all linear 4 constraints, all linear; 19 nonzeros 1 linear objective; 6 nonzeros. MINDS 5.5: MINUS 5.5: optimal solution found. 5 iterations, objective 0.3007 display total_earnings; total_earnings = 0.3007 == 92 ================ d isplay INVEST; INVEST [*J := A 2.4 B 0 m U m n o H == 93 ================ display BORROW ; BORROW = 1 - Weaowcanmake profit 'of $0.3007M. ‘ (2 EN; fl) < Optimal policy: Borrow $1M and invest $2.4M in bond A, $8.1M in bond C and $0.5M “or“ in bond E Note that the percentage allocation is the same as in part (5:). This makes “N sense, in light of part (4:) since once we choose to borrow $1M (note that the after—tax 5‘" ‘33}? cost of borrowing is lower than the after—tax yield from our portfolio), we will invest ‘W “(I the extra money the same way is if it was an unobligated addition to our endowment. Gd Changes: - - Get rid of govtandagenchequirement constraint (make the threshold=0 in data 7 ye”? file). @‘f — — Get rid of upper bound on BORROW by making threshold to infinity. — — Add new constraint. The model file is: set BUNDS ; # possible bonds for investment set GDVTANDAGENCY_BONDS; # subset of government and agency bonds set MUNICIPALHBDNDS; # subset of municipal bonds param quality_rating {BONDS} >=0; # 3rd column in table param years_to_maturity {BONDS} >=0; # arth column in table param yield {BONDS} >=0; # 6th column in table param cash_avail >=O; # amount of money to invest param avg_quality_max >=O; # threshold (3) in problem param avg_maturity_max >=0; # threshold (2) in problem param govtandagency_bonds_min >=0 ; # threshold (1) in problem param borrow_rate >=O; # intrest on borrowed money param borrow_limit >=O; # limit to amount borrowed param municipal_bonds_limit >= 0; # limit on investment in municipal bonds var INVEST{BONDS} >= 0; # decision variables of how much to invest in each bond var BORROW >=0 , <=borrow_limit; # decision variable of how much to borrow maximize tota1_earnings: sum {j in BONDS} (yield[j] * INVESTEj])- borrow_rate * BORROW; # maximum profit after tax - return of borrowed money subject to cash: sum {j in BONDS} INVESTEjJ-BORROW <= cash_avail ; # can not invest more than cash and borrowed subject to govtandagency_requirement: sum {j in GOVTANDAGENCY_BONDS} INVEST[j] >= govtandagency_bonds_min; # minimum investment in government and agency bonds subject to avg_qua1ity: sum {j in BONDS} (quality_rating[j] - avg_quality_max) * INVESTEj] <= 0; # maximum quality requirement (linearization of proportion constraint) subject to avgnmaturity: sum {j in BONDS} (years_to_maturity[j] - avg_maturity_max) * INVESTEj] <= 0; # maximum time to maturity requirement (linearization of proportion constraint) figfigmctstovmunicipal;limit: asumsfifiwifiVMUNICIPAL;BDNDS}*INVESTEj]'<: municipa1_bonds;limit; # limit qgmimyestment.in municipal-bonds_ -. 'fififlmmwwsw The data file is: set BONDS := A B C D E; set MUNICIPAL_BUNDS := A E; set GDVTANDAGENCY_BDNDS := B C D; param : quality_rating years_to_maturity yield := A 2 9 .043 B 2 15 .027 C 1 4 .025 D 1 3 .022 E 5 2 .045; param cash_avail := 10; # in millions dollars param avg_quality_max := 1.4; param avg_maturity_max := 5; # in years param borrow_rate := .02?5; # after—tax interest rate on borrowed funds paramborr‘owjlimit="If1f1f11ty;”“# 'i'fihfiii'll'i'o'ns' 'of dollars Bgmamagovtanddgency_bonds_min := O; #"in millions dollars # yields are after-tax figifimmmunicipal;bonds_limit :=-6; i-’ # limit in millions on investment in municipal bonds The output is: Presolve eliminates 1 constraint. Adjusted problem: 6 variables, all linear 4 constraints, all linear; 18 nonzeros 1 linear objective; 6 nonzeros. MINDS 5.5: MINDS 5.5: optimal solution found. 3 iterations, objective 0.3281724138 display total_earnings; tota1_earnings = 0 . 328172 display INVEST; INVEST [*] := 4.96552 0 16.7586 0 1.03448 ; WUC‘JW‘F- Mo») MMWf’r-t \rxvifl‘“ { Nit-$51M? <3 “.3"? <3; 0. "2, 2'7 x F) 2' WERE-a INK-.1. waltz/Q (a) An algebraic pr m is: max 3280221 + 2000M + 112033 f,“ L.) . Qflhfi ' subject to : 200m1 + 50032 + 80033 5 180000 *" 0.8271 + 0.5$2 + 0.233 S 120 $1 2 0:272 2 0:33 2 0 ,b) 200201 + 500m2 + 800503 +1324 = 180000 0.8111 + 0.5312 + 0.21133 + 1315 = 120 mm = 2m = 5 ....... 1-, 0.8 0.5 0.2 0 1 b 1.. 180000 _ 120 i, (C‘Wty *..5'\ A=(200 500 800 1 0) c=(3280 2000 1120 0 0) (3?“ P = ( —i//73%%0 4??) ,d) 2;,- -b= Pb: ( —11//2?3%0 11/?) ( 181020000 ) ”’13” 35% “J =00) (6) 3:1 = 100, :33 = 200, 1.3., produce 100 tons of AIloyl and 200 tons of Alloy3. we- .merofit: $328, 000 + $224, 000 = $552, 000 (f) z = 32802:; + 2000022 + 112033 + 03:4 + 0:05 (-Tflfiz reports total profit in dollars. 2:3 2 = 3280501 + 200% + 1120.1:3 —0.4(180000 = 200m1 + 500022 + 80033 + 134) (g: _ 0(120: 0.8$1+ 0.5592+ 0.21:3-1- 19:5) 5 2H 2 4000 , fiflz — 552, 000 = Owl — 200332 + 0333 -— 0.4m — 4000505 2 - 552,000 3 0 =>- 3 S 552,000 fl every feasible solution satisfies C”), and since 501,...,:c5 Z 0, this implies: \“E d,- \ For all feasible solutions => :31 = 100, $2 = 0,503 = 200 is optimal. {3.7- “var” ...
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