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Unformatted text preview: Qiizriii'yiiig Exam (\Nihter 2006): Operations Research \"uu have 4 huurs t0 do this i mm Rumimlvr: Th2: r‘;:::n i cimml notes and Ch)st hooks.
De ‘2 out of prohhtiiis 1,13
Du 2 out of [thii't'Hi‘ 15:10. Do 301113 of pluiiwlim '1’1519Y10111112)13114.
Ali pruliltiiis Lli't' Weighted eqmily. On this cover page write. which seven problenm \‘till mini graded. problems to be graded: Ai.LLh‘I11iC illtl 'H‘ i:. CXIH‘HH ii of all students at all times, whether iii the ]il('ht‘Jlr'i5 or absence of members “J .
of the faculty 1 iiiiliateiini'u: this7 I declare that I shall not give, use, or receive lll'iiiutiMJl'iZCd aid in this
exuiiiiiiaLioll. Name (PRINT CLEARLY), ID number Signature 1). lll'wvci lil lilll'lt 3.x: :3 0, Ag]; 3’ b7 OR 7hr! 1? U. and if Uy and 11*?) <; (l. but not both. xi is an arbitrary
tin X 11) matrix, and b is an never tor. Note: lt is curv to show that .r and w as specified cannot both exist.
You must al: show that it :1: does not 1‘.\.i‘il.i then m ilillrl t‘kirl, (or vice versa). (‘onsfdr‘r the following optimal tableau of a in.i,\.n’ni/..tl;on problem where the constraints are of the type, and s), 53. .93 are the lillfl'; ruiinhlw of the constraints. (1171,1‘2.:F3.1:".,r: me the original variables.) [ z Tl‘l I 272 :11; 1'47 1'5 s1 32 53 l RH"
t ' r t r i
z i 0 I 0 I 0 2 ) 0 2 1—? 2 0 1
11 0 1 t) l 0 1 _ 0 i 1 2
t: 0 0 1 t) 2 l 1 1 0 § 3 I
la. 0 l 0 11 1 ‘ 1) 2 5 “:5 2 1 I (it). Find the optimal objective value 6. (b). What is the allowable increase and decrease of ()1 (the right hand side of the first constraint) that keep
the. opliutil Huldlnltl hursibile‘? (c). What is the optimal tableau if a new activity 1'0 is added with czovilirieiits (2,0. 3)" in the constraints
and price of 5 were added to the problem? (id). (lgnoring part (c).) Suppose that we add the : unstraint 1.1 r 17; l 31'; S 10 to the problem What is
the new optimal tableau? 3). (a). Prove the following constructively: li'o: vrcry basic feasible solution it, there exists a vm lul' e. such
that :1: is optimal to the LP min {tar l ALI," : b. a: 2% U}. (“Prove constructively” means give the values of the
component . in such a Vector c.) (b). In chi: » we have shown several rult .~ that prevent cycling when applying Simplex to an LP that may be
degenerate. (Recall the lexicographic)"perturbation method and Bland’s method.) Prove that cycling can never occur even in the pl‘t‘m‘lll c of degeneracy when applviue' the Simplex method
without any cycling prevention rules, provided that the minimum Litio test always? has a unique. solution.
(in other words. show that il‘ the minimum ratio test has a unique winner) Simplex will not cycle.) 4). Let (Xn),.;31 be independent and identically distributed random variables with [)(Xﬂ : 1) : PtX,” :
r 1) t 0.5. Let 5'0 :2 0, 5.” :7 X1 +  ~  + X”. and Ml.“ : Ih‘dX{Sk : 0 3' A: _<. Dow each of the following :i\11]1'lll‘t: l
(T) ~ bl”),lj:)1 form a h‘larkov chain? lixplain your answers. 5). Consider a single—server h’larhovian queue with unlimited waiting spam The service intensity is )1 and
the arrival rate is A. If an arrival sees 71 iIislonscrs in the system: this arrival joins the queue with probalulity Find the average amount of time Il1.11 an arrival who joins the queue spends in Llllr avstem. Consider a single—server bank for which CllSEOllltil'o in rive in accordance with a l’orwon pun in»: \n'uh nu l'
A. A customer will enter the bank only if there is no other custcu’ners there. The. customer service time:
.ev independent and identically dist:?lmlml random variables with the distribution function (3’. Find the
pi umbility that an arriving customer l'Illl‘lk the bank. Explain your answer. 7). Stony Brook Univeruv 1‘.lr$l1€S to _I:¢.I:[11 professors to courws for llll' rimLt academic year. Suppose that
there cnw a total of m won‘ tors and 72. dih‘erent courses to be offered. For course j, at least Sj met ions 0 should be oliered to 111:1:11111111. .3411: the 110111111111, tor _j 7 l,...,/1. For 13:11:11 prolix 1111‘ 1,. let (.7) denote the set
ol~ vourses she (“.111 teach. '2' : 1. . . . .111. 1311111 prol} 111‘ can teach zit most 1:, (courses in 1111 Htiflth1lllt your.
15111'tlier111o1'e. 21 prob 1 111; 1 .1111.111 111.11:h1'nore than 11 sections of the szune course. 11":1 professor is not assigned
any courses to teach, then she will be on leave (without pay) for the entire ttLﬁzulelllic your. Let PL denote the
2111mm] salary 11pr‘of1‘r'11'i, t : 17...,771. 1301111111111 .111 integer 1')J'(')g;1‘i1111111111gPl‘ulJlt’lll to find 1111 assignment
of the courses to p1‘ol11:.~11's so as to minimize the 111 .11 rainy paid during the 111 :.1 .11.1de1nie your. Consider the generiv nonlinear opti1niz.1tion problem 1111111 j‘ptf) st y(;r) 5' U. 11(1') : l where f : 111:” a l: y i“  :1 "", and h : R” 7+ .'. Suppose that the 1\langasariAirprm'iitxwitv (~1111st1'11h1t
i.e., Witt/(1") has full column 1111111' and there exists (1 such that Wig/(1')”! < 0
11nd VI {'1')Td 77 t). Stlp])(1.~t‘ that i‘ is at 111111:
multipliers. Show that the «11, of vectors (u, satisfying the KK‘T 1:1.111ditions is bounded, 1.8., there exists
:1 linite 1111111ber 1M > 0 such that $1 11/], (pinlilication holds at a. feasible 111i11in1izer of (P). Let 711) be the c<.1rrespo111’lii1g Lagrange 111)) g .M for all Lagmnge multipliers (11.111). 9). Consider the CDF O r :L' <: 0
F(:L‘) : (L 1:);1:  (l 71.1).1'2 O 5: LL' 5: l 1 :L‘ > 1 11111111 is a parruneter, with 0 < a < 1. Give two different 111ethod’s 711. which we 121111 1.11111111‘.111~ independent
\‘.1:'i.1111:. lroni this distribution. For each method. be as detailed and pieeise as you can: your answers should
come as close to providing a, detailed, specific .1.;=l111'itl1n1 for this particular distribution as you can, rather
than 21 germ .1l (l1:1‘ription ot the method. 10). Suppw 1 1w want to si11111l11te the lullmving random walk problem. A gambler has :1: dollztrs. 111 earth
1.11111 '7'111 :1111‘1._‘1i'.'!\.' oi winning 1 dolhtr is p. the probability of 1051111, 1 dollar is (1 VA 1 — [1. The gambler
will stop either when he reaches 1) > 11' dollars or when he reaches a < 0 dollars. {21). Suppom the 1111;11:1bility for the 13.1.:nb'111r to lose, (stops at (r) is What is the I‘Cl‘dt ion between 1/1111?)
and 1.11:1: i 1).cf11\.r  l)? (b). VVhat is the 1111:1111l11 for 1319(1)? (1t). Desei 11111 the .1.“lul‘l[lllll for simulating the gmnbling in details. 11). “1'0 11;11.t to solve the li‘1)ll0\\7lllg query problem: Given a set 'P of 71. disjoint line segunnts in the plane,
determine .1 a. query line I? 111:1 1'seets all ll of the segments (in which case we C‘rtll it a. “51 11l)be1"" ). We want
to prepro1'1 :«r the input data so that we c1111 answer queries very ﬁrst. The output to :1 query will be “yes” if
the line I} intersects all n seg111ents and will be “no” otherwise. 111). Describe brie11y Ll 111ethod for solvinu, this problem. Try to be 11:7 111111 ie11t as possible in both spam: and
query time. (i). Preproeessi11g time is ()t )
tii). Stornui s;1.1ee (111emory11..1;_g1) is O( (721.1171)’ 1111111} 15 (b). A: 1111111 now that we know that 11' :5 .1>;isparallel (horiyuntal or vertical). How 113L1'ie1'1tly now .111 11. 1111511111. 111 ~‘1il‘.11l‘? [Give Lll)1‘1£'lJil."‘.1l.1:111111.) (i). Preprm .Z:1:_1i111e'.s01\ )
(ii). Stormr 1111.111} (memory usage) is O( (iii). Queiy time is U( ) w). Absllllh’f, new that we lumuc that the 1:, r  ments are all vertieah and we allow {I to be arbitrary {any urientation). How niin'mntly nwv.’ can the pin.rh‘m be solved? (Give a l)1’l1:3f_lll‘3~lllit .d‘und (i). Prcprm t‘: ing time O( tn). SLOIJjth‘ .~.paee (memory Matty?) 0( (iii). Query tll’lle is ()( 12). Let H : “1,119,. . . «pull; be a set of n in tints in the plane. How efficiently can one determine a pair; tpmpj ), of the points that minimizes the interpeint (,listanee,
m Ihul “brgjmytq :: mian (12(plipjl7 \vhere (lg/(3 denotes usual Euclidean (lihtdlite between two points?
State the time lmulld in bigoh notation, and justify billHr. (bl How eth<_iently can une determine all pairs of points that minin’iize tln intwrpoint distance.) (There can
be many pairs “' :ul' luj lung nlosest pairs.) (e) How efﬁciently can one determine a pair, (pl.731} 1)., of the points that rnrrr’iinict .9 the interpoint distance1
so that d;(pp‘pp) : maij (tritiphpﬂ? State the time bound in big—Oh numtion7 and jnstify brielly. Let X1, X3, and X3 be independent and i<:lenti:;1‘:'.§.' distributml Jlol‘nml random variables with the
variance 1. Show that X1 t X2 Xx v/1 t Xi has the same distribution, 14). Let (,\',L),,:1 and (Kilo, all be deﬁiml on the same probability Spar'l'. .Cllllllltlrl‘ X” 0211\1‘1113“; in distribution to X and )7”, COllVGX't‘J‘H in probability to 0. Show that X” + Y“ converges in distribution to X. ...
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 Fall '08
 Mitchell,J
 Probability theory, Quantification, Universal quantification, Existential quantification, optimal tableau

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