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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 ?
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' 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.
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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?
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Ili 4 dj4 4 2.7 {1:15 For this new input, What can you say about the solution given above? Again, be
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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 nonnegativity
constraints on all variables). Also assume that all the right hand side coefﬁcients are
nonnegative. 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
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the optimal solution and explain why it is optimal. If the answer is not, ﬁnd a
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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
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#: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
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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
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