finalsol - ENGRI 1101 Engineering Applications of OR Fall...

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Unformatted text preview: ENGRI 1101 Engineering Applications of OR Fall ’08 Final Name: 1 Problem Points Possible fl Points Received fl You have 2 hour and 30 minutes to answer all the questions in the exam. The total number of points is 150. For questions with multiple parts, the point value of each part is given at the start of that part. Be sure to give full explanations for all of your answers. In most cases, the correct reasoning alone is worth more credit than the correct answer by itself. Your explanation need not be wordy7 but it should be sufficient to justify your answer. You receive much more credit for an incorrect answer if you recognize that something that you have computed is not correct than if you merely pretend that it is correct. Not all the parts are in the same level of difficulty! Please make sure that you do not Spend too much time on one part. Good Luck! 1. (20) it is Thursday noon7 and you and your partner are working in the lab to compute the max—flow on a huge graph as part of your homework in an introductory optimization course in a prestigious lvy University. However7 it is getting late as the due time of the homework is 2.30pm. Your friend is a computer freak and actually hates “theory”. He suggests that he will write a code to compute the max flow with no effort. It is already 1.45pm and your friend just finished the code and started to run it on the input graph. At 2.15 the code finishes computing and you print the solution in order to check if it has been computed correctly. Unfortunately, in the middle of printing the electricity goes oil" and you are able to print only a very partial list of the flow. The computer prints a list of the edges starting with edges going out of the origin (node 1) then edges going out of node 2 and so on. Question marks stand for parts in the print that are unreadable: Edge Capacity Flow (12) 10 5 (1.3) (1,4) 10 (2,5) 10 5 (3,4) V 5 5 (4,7) 10 10 (5,6) 5 g 5 (6,2) , 10 0 ? ? ? tr ? (7.3) E- L 7 ‘7 7 ' 7 This means that you know the flow on all edges going out from nodes 1, 2, 3, 4 and 5, but not necessarily for nodes 67 7. You are given that that there are no more arcs entering 1, 2, 3, 4, and 5. Also there might be many other nodes and edges that were not printed. In particular assume that the sink it does not Show up in the printed list. Your friend is now panicked as it is just 15 minutes before the due time. You on the other hand like “theory”. Looking at the partial print you tell your friend that he can relax7 since you have computed the optimal (maximum) flow on the graph. Your friend, though appreciating you very much, doubts your last statement. (a) (5) Assuming that the code has computed a valid flow, give the value of the current flow. 2*. i », w- i3 it huge iv , a t . W tkeg umer “‘1 M 2‘ as. ‘3‘ a rm % (a Kiss ? its! 1‘ W W ‘ (b) (15) If you think that the current valid flow is indeed optimal, provide definite evidence for its optimality and compute a, minimum cut. Otherwise indicate why the current flow may not be optimal (Hint: do not panic you know enough to figure out What is the right answer). 2. (25) Consider the following input to the transportation problem. .4 la? V? (b) (10) What can you say about the solution in which 270,2) 2 4, :E(1,4) = 2, :1:(2,1) = 4, $(2,4) = 5, 33(3, 3) = 2, and all remaining x values are equal to 0? Be sure to give as complete an explanation for your conclusion as possible. :4: a :i; 23 V e l 2% flea}?sz angel“: «23$; “wag? mg g :fiéfig all “A?” *v" i fix “5%” il at? a s r “3% K as}? “V A l é lag/23%; a}? Ewe 9;} a a l U 6/ i a gag“ M r“ '%3 “R fl“: g g 3&25’3“ "" % awe? aim? ‘ *3 ‘ ll“ l i %\ a W’ ; i lees“ EA: m m 13% " “I “if” i ' a} {:WV 3 Meg emaiseiw ' "‘ galg' ‘43??? (c) (10) Now consider the same solution for the following input to the transportation problem. I-li 4 dj4 4 2.7 {1:15 For this new input, What can you say about the solution given above? Again, be sure to give a complete explanation for your conclusion. (Hint: let the u values be 0.) N: {3% ‘ at"? \ WW vegan?” 6/. a Q“ W \31; ’2» e 1:“; a: {1:} “j 2 W56 lg: 4/ ’ if CM a see 23“"? ‘fimfi L/ {hiy‘f‘ 3’ ‘g” a l r): {2/ fix Q63 iii ill” 5 3. (20) Consider the following 3 dictionaries (recall that each dictionary includes also non— negativity constraints on all variables): Dictionary A $1 =2 2 —x4 +2365 .112 = 2 +394 "3.51:5 1173 -— l +2334 ~3$5 z z: 14 —x4 —$5 DictionaryB $3 -—- 9 “311:1 “$2 $4 = —3$1 “2152 £135 "‘ 4 -—.’I71 “"332 z — O +4121 +3132 DictionaryC {131 2 g “'g'ng +§IIJ4 $2 = 1 +373 “1134 i 1 x5 2 ¥ —-¥m'3 +gécc4 +3333 +§334 ml The above dictionaries partially describe an application of the simplex method to max- imization LP in canonical form (i.e.7 with 3 functional constraints and non-negativity constraints on all variables). Also assume that all the right hand side coefficients are non-negative. The simplex method is applied along the lines discussed in Class (i.e., introducing slack variables, working with dictionaries, etc). (a) (6) What is the original LP being solved? Write it in canonical form. max in (b) (7) In which order were these dictionaries encountered during the application of the simplex method? Explain. " " l » gaging: ‘ v G»ng {z 3% «la rigéifig as??? 9%: Q i a ‘ E ‘ 52 U n E 306% magma .1} x ‘ a a iifiéfiafi : s53 ng *V‘éfis V / \flfj I w (c) (7) Can you derive the optimal solution and its value? If the answer is yes, provide the optimal solution and explain why it is optimal. If the answer is not, find a better solution than the existed ones. R : Tiers SLKDGL [3% g: 32 i g 22...; if“ agar, is ii 3 Q? M \ g! 15;} Km“ ‘ {3?}? wall {BREW er i g x ‘ fig off‘ng‘: was was? ‘ x «L ‘1‘ rs m E 15mm boars agorast as V I we; a l ' 2&ka whale as mags Q {jg/mg 3‘? x i g a x: A Q A \ k l J? {:3 fiffifli. E ties he. Q i I a 4'3 5» >2 4% Xi} «3,- : E i X E 1 f as», j P“ } Kg ’7 g“ 3 3 $1 ‘0an 73' W’ 4. (15) In Class we gave an integer programming formulation of the traveling salesman problem. Suppose we are trying to solve the linear relaxation of this IP formulation. We have a proposed solution given by the figure below. Each edge is marked with the value of its decision variable. (The edges not drawn have value zero.) This proposed solution is not feasible for the LP relaxation. Name one degree constraint that is violated (that is7 say which is the offending node) and one subtour elimination constraint that is violated (that is, say which is the offending cut). } is. x o e: v i i y I (V (Lei? 3 :2 a E g 3 “(tag @53er ' i it? \9’35 / 33% {it ’ it has Jig; fifl’ggfil VCQ negfl £533; 5:2 j 1*? ’>/ W 3 K} 1 59' w firaaeifi f affi‘é’w“ 'i t; E N ‘ r J i y eve?" ‘ WVGE ‘ $6M- 2w 1 ‘ i “WW “I I) E. ’ ~ : gjnfizee’iw‘ lg in WW @ 9'? flagelegéifié 3 » A gm (2 QiEB'Cgééj \ z r \{g in mi 5 l T l V ’2 if 5} 3 i x #:9163833}; 3% \ ‘5; i J» 5. (40) Integer Programming Formulations (a) (25) Consider the following scheduling problem, which is called the generalized assignment problem. There are 71 jobs j 1,...,n that must be processed. There are m machines 2' = 1, . . . , m on which these jobs must be processed. Each job is to be processed by exactly one machine. We do not directly give a restriction on the number of jobs that a particular machine may process. Instead, for each 1' z 1, . . .,m, j = 1,... ,n, if job j is processed by machine 2', then it takes 10,, time units to be processed; for each machine 2' z: 1, . . . ,m, we do require that the total amount of processing that is assigned to be done by that machine is at most T. For each 2': 1,...,m, j=1,...,n,ifjob j is processed by machine 2‘, then it incurs a cost of 0,5. We wish to find an assignment of jobs to machines that minimizes the total cost incurred. Thus, the input for this problem consists of m, n, T, 19,5, for each 2': 1,...,m, j = 1,...,n, and cij, for each i: l,...,m, j = 1, . . .,n. Give an integer programming formulation of this problem. (Hint: the essential decision is, for each possible machine i 2: 1, . . . ,m on which each job j = 1, . . . , n might be scheduled, whether or not to have that job j processed by that machine g 2 3 is é meelamg to G’éfiiéaéfi, , = a 23 v , : \ b :3 X cs2, Qfl \igs‘eefiss2\83; 1 a ,s «l aes’l as i E ’Eiviiwr- ., ,, g é W1, 31 V f/j/M“)? mire , i_, 3 é i a??? gab la gm E“ «:99 g a"??? 332.,— s"; g all Eels“ fa" u:::<mg so a, V ‘ (r j: “2. 2 ,eeeasi‘zfié “*3 55 £53 X1; as ; x; ~ §;i ( 2 ( (33: L} 1 mien § ‘ {X’i gcfimfié ‘53 Mg N»- I i r l: 3"; :5 I“: ~ Xi? é‘ E ,9 gq CA :té sheeriéiyg (BEE: K} 7 $3,? ,3ny ::i V} L f \ ' “:3 " Q < 5:” 9% E? , r \3‘ {g} :i i a”? - a a Additional space... 10 (b) (15) Unfortunately, the following, more complicated, problem is actually more rel~ evant in many applications. Although there are m machines that may be used, in any particular schedule you might not use some machines at all. Furthermore, there is a fixed charge 12- that you must pay if you actually use a particular machine 71; this charge is not incurred if machine 2‘ is not used to process any jobs at all. The more complicated objective function is to minimize the sum of the total assignment costs and the total fixed costs incurred for the machines actually used. Adapt your integer programming formulation from (a) above to capture this problem. (Hint: the solution might remind you of the uncapacitated facility location problem; in this scheduling problem, you must decide whether or not to use machine 73 , for each 2‘ z 17 . . . ,m, and this directly determines the amount of processing that may be done on machine 2', which is consequently either T or 0 . i f 1.3 ‘i i i Eeggmfi “Omifi‘i‘ggl . : ‘ irreiigz‘fi’iirj‘ g3 éggé‘ ’4’: i meetimi 3’} gm x I k if r V a, 'm 6362!“; raving r at it m3“ . > “9" Vi '31" {f {333.5 {fie M fl §;,\l{ b it :ia‘fierr’?‘ (X. m it. 5“” g i :: xv “I P at r 9‘ ic‘i t We \’ 3 2"“ 35;" §Ti LI; 3 Y ,4: $331} 2}“ .un‘igréfi‘ l i,\ ‘3M} 5M”? x .. w? ' Ki” g: V; ,3 i i a: 3 a _ ‘E..§{% r , é 9‘ ‘imr‘i’ ‘ \ gf‘s 3; \# s} >< e E “i W 1-1“ 3 m “’3‘ in is“: :32? g g: iii lg } 11 6. (30) Consider the following two-person zero-sum game: 2—1 0 i 4, 5. 0 4-2 (a) (5) Does this game have a saddle point? Briefly explain why or why not. “‘3 . ‘ v » i “a; .; (ammo): } moist mmf} : L ‘3» maxed”? ““3 a {‘2 1‘3 M; e a5"? “*8” a m is: wax i x W «2 3:st “figs” W” t (b) (8) Are there any dominated strategies? Briefly explain why or why not. If there are dominated strategies, simplify the game to an equivalent one in which there are none. so n .3 i in ‘85“: Eva m mg} is ‘ may: {:70 s Q E “‘ :6 i ‘ Sex u 2: A @{g‘t‘ Ci" «5:: l R g y \ i l mu.) 3 when {1% mg” ream is ska mgyga PL f E XE“ om <3 3i @315" ‘lflfi Cg“ €§§§fifi§$ (8“) J {M kg A g r {:1 so 6. WWWMWW é EMMN‘FW‘AW A. i Q E z 9. w i : LE? :3 3 MW” W’V ““ Wflwmwwwwmww g 3 ‘ ‘2 :¥ 35‘s t {/6 lxéff‘xf‘x ‘ 9% Def g 73% a ‘9 £Q\\3 ma 1 g E {E (c) (9) Find the optimal strategy for one of the players {your choice) and the value of this game. 3%; J 5 g 3% r {3” k: 9K.) : fwi N r " ' “ \ % “we '“Wiafiéaggfijf 4 x r fl “€51 ' 5 ‘ may 3%: £33?“ é? "f [33" 5:: ( i V ‘ ‘ flag”, .L‘Va‘ii eggs; 4, v.3 5&6 5363‘ ‘ Thaw \ ‘ um‘fig i j. N; :9 W figs? {:23 e $5: 0"} “‘ iv fx’ 365%, , refiflk @ i} , g A: 3% £3?" a , "6% a = 3 age, 9* ‘a J?" 9 1’ j i W? ) g {jag/i {52 fix a ., LL '2} 1:“ 'i f I gwtfii [$3 "% E 3 \ ' h at 5:" ‘V V3 we? $53 is? X03 (E ' a (v f , a e“ _ :1 a g a»: 3 L5“ “I mg >3: m‘fl L m K, . m :3 ASWS " WM" L6% 3 E: (d) (8) For the player for which you did not compute the optimal strategy in (c) above, write an LP that computes it (you do not need to solve it). Is it possible that the optimal solution of this LP (i.e., the values of the decision variables) is integer? ‘ i Ami“ \ do 4 beef”? “i l 3V 9 Rage “fixes, gm: mieél? "i sag; we GEE V ' 2 .. ‘3 ~ , w, it; eef‘» §‘<}3\%§ k" ‘ 32 3A u ‘3 a i ~, ‘ ) ,‘w ii a SK} flow): {12‘ E the, We“, w» § A; EQXVFQ 73> i} was; Rita} 3a Ce * 0x2 x ‘ i mesa, mag“: {34? o r w e; Ci; {3% i, g to 2» x grit, «'2» it 6351,, w >/ Eli 3:31 - we ,3- i it we 01 a :5; ‘ C‘m } % ’3” it“? ‘i ‘x 413$" 1 \E x to we “*5” f :wegmokfiaafi “if Q ' m Wig b8 éz‘yzgfi Egaéfif’l a {by ‘ a . ‘ {EEK 3V1 \ a ‘7“ E 2": figlw‘fimfi ‘ ) "Tig‘iafiw x egg m 63:: . g . 8 x K ; C\izi— $32; Ah: wegggi L gym 1 \@ Qg‘tfi4g a} Q}, 1:" = 3 ¥\W€i¥ C325 . mag; —¥\x&i§ \ ‘ buggy ‘ U , l a: gk‘aoe 26‘! g? ...
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finalsol - ENGRI 1101 Engineering Applications of OR Fall...

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