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Unformatted text preview: Qualifying Exam (Spring 2003): Operations Research You have 4 hours to do this exam. Do ‘2 out of problems 1,2,3. Do 2 out of problems 4,5,6. Do 3 out of problems 7,8,§,10,11,12. All problems are weighted equally. On the front page write clearly which seven
problems you want graded. Reminder: This {exam is closed notes and closed books. 1). Let a: be a nondegenerate BFS for an LP, and suppose that an improving non basic variable enters
the basis, and that the minimum ratio test for exiting from the basis has a unique solution. Show that the
resulting BFS is nondegenerate. 2). Recall the transshipment (minimum cost network ﬂow) problem: z = 2;. 2;; ca.. xij F In = bi
Iii Where b, is the given supply/demand of node i, 2‘. b; = f). cqJ 2 0 is the cost. of shipping one unit on arc
[id], and 35: the variables, are the amounts shipped on the arcs. (a). Write down the dual of the transshipment problem.
(b). Show that the dual of the transshipment problem is always feasible. 3). Consider the following LP: max 2 = 2031 + 10932 11 +22 : 150
$1 5 40
:2 2 20
31,32 2 0 Let the slack (excms) mriabla he 5, (eg) for the constraints, and the artiﬁcial variables be oi. The optimal
tableau is given below (some entries are missing): 1 U 0 0 1900
D 0 D ~1 l 1 —1 90
0 1 ' 0 l U 0 0 40
0 U l —1 U 1 U 110 (a) Which variables are basic in the current BPS? What is the current B‘l? (b) Complete the optimal tableau. (c) Find the dual to the original LP and its optimal solution. Make sure to give the optimal value of the
dual variables and the objective function. 4]. There are 6 nodes on the plane. Five are arranged on a circle and are put at the vertices ofa pentagon and
the sixth is the center of this circle. There are 10 edges. Five connect the outside nodes and form a pentagon
and the other ﬁve connect the center with each of the verticus of the pentagon. A particle moves from one
node to another according to the following rule: If it is on the outside node, then it moves with probability ,..;rHw19e, with probability q counterclockwise and with probability 1' to the center. (p + q + r z 1). If
the particle is in the center it moves with probability 1/5 to each of the remaining nodes_ Let X“ be the
position of the particle after 11 steps. (a) Find the twostep transitiod ‘rnatrix of this Markov chain.
(1)) Find the steady state distribution. 5). The participants to the Internet duplicate bridge tournament arrive to the web site at Poisson rate A.
To start a tournament it is necessary to have N pairs. Each new arrival is either paired .with the person who
arrived just before him or if there are already evennumber of participants, he waits for the next person to
arrive to form a pair. When N pairs are formed, they start a tournament and after that any new arrivals
wait to form another N pairs to start another tournament, and so on. (:1) Find the average number of pairs waiting on the web site to start a tournament. (We count only full pairs, ie., if there are 4 people, then we have two pairs, if there is one person, then we
have 0 pairs, and if there are 3 people we have 1 pair, etc.) ' (b) What is the average waiting time until the beginning of the tournament for a participant? 6). (A queue with reneging). Customers are arriving to the service'station with Poisson rate A. The service
is exponential with parameter ii. If there are 11 customers already in the system (either waiting or being
served) then the newly arrived customer joins the queue with probability 1/(11 + 1). Find the mean and the variance of the number of customers in the system after it was under operation for
a long period of time. 7). Two armies of queens (black and white) peaceably coexist on an N x N chessboard if no two queens
from opposing armies can attack one another (two queens from opposing arnries can attack one another if
they share the same row, column, or diagonal on the chessboard). Formulate an integer program to ﬁnd the
maximum size of two equal—sized peaceably coexisting armies. 8). Let f be a convex function deﬁned on D C; R". Suppose that ﬂi) is ﬁnite. Deﬁne f(2:) = +00 if; g D_
Recall that g E JR" is called a subgradient of f at i“ if . fly) 2 N53) + 9TH;  i“) for all y E R". Let Elﬁn?) denote the set of subgradients of f at 5:.
(a) Prove that 8ft?) is bounded if there exists an open neighborhood U 9 D of i such that f is ﬁnite on U. (b) Show that ifdﬂfs) is not bounded. then there exists a nonzero direction d E IR“ such that f(5:+ca‘) : +00
for all z > D. 9). Consider the max ﬂow problem in a directed graph D : (N,A) with source node 3 and sink (target)
node t, and are capacities 11,}. As usual, let the number of arcs be m, and assume m 2 n, the number of
nodes. (a) Suppose a max flow 3,3 is given. Show how to find a min cut in time 0(m). (1)) Suppose that after solving the max flow problem, we realize that the capacity of arc (p,q) was under—
estimated hy k units. (1.e., the true capacity of arc (p,q) is 1: units more than we used in our solution
11;” : um + 1:.) Show that we can ﬁnd an optimal solution in time 0(mk). Hint: use the labeling algorithm. 10). The cossrnmnsn ssorrrcsr FATll problem is as follows: Given a directed graph D : (N ,A) with
an. lengths iii, and are weights mg, and integers L and Vi". (You may assume all lengths and weights are
non negative.) Let s,t E N. is there a path from s to t of length at most L and weight at most W. :.. .u that the CONSTHA'INED SHORTEST PATH problem is INTcomplete. You may use a reduction from n'i‘ITioN problem: Given integers a1,u2,...,am is there a subset 5' C {1,...,n}, such that 25690:; = XiES a" (1;) Now consider a. special case'of the CONSTRAINED snrmrssr PATH problem in which all weights of arts
are 1 (wij : 1 for all arcs)” Show that this problem can be solved in polynomial time. Hint: Think Belman
Ford. 11). Consider a small bank with two tellers. Teller 1 deals only with business accounts, while Teller 2 deals
exclusively with general accounts. Each teller has his/her own separate queue. Clients arrive at the bank
according to a PoissOn procms at a rate of one customer every 5 minutes. Of the clients, 33% are business
accounts and the rest are general accounts. A business—account client take 7 :i: 5 minutes to complete service
at his/her teller (is, uniformly distributed in the range [ 2 , 12 1 minutes), and generalaccount clients'3 :t
2 minutes. (a) identify the state variables and events you would need to implement an eventoriented simulation model
of this facility. For each event, write pseuddcode that shows the actions that the event undertakes. Note
that you are not required to write a fullblown program, but simply a pseudo~code description of what each
event does, together with an explanation of the variables and data structures your events use. Assume that
you have at your disposal a function Random( ) which returns to you uniformly distributed random variates
in the range L 0.0 , 1.0 ), but no other random number generation capabilities. Also assume that you have
available a function Schedule(etype , t} which schedules an event of type etype for time instant t. (b) Carefully describe how you would go about gathering statistics so that the simulation program to be
implemented calculates, at the end of its run, the mean number of (business) customers in the queue for
Teller 1. Assume that the program will be encoded in some high—level language such as C or C++, and
not in general purpose simulation language such as SIMSCRJPT. Thus, you may not assume that any
statistics—gathering capabilities are available to you in the programming language. 1 12). Consider a random variable X which has a triangular distribution with pdf 1', if US$51
ffz): 2—9:, if 1(352
0, otherwise Generate the ﬁrst random variate value from the distribution for X . You may use any method you choose,
but you must state what that method is, and clearly shOw and explain each step involved in the development
of your ﬁnal answer. You will need to generate uniformly distributed random variates in the range ( 0.0 , 1.0 ). Use the (Mixed)
LCG (Linear Congruential Generator) X“. = (Us. + 29) mod 100 with intial seed value XE. : 37 . ...
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 Fall '08
 Mitchell,J
 Optimization, Randomness, web site, Transshipment problem, bridge tournament arrive

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