Exam Operations Research (WI4051TU)
Date:
February 1st, 2007, 9.00 – 12.00.
Teacher: Dr. J.B.M. Melissen
During this exam books and notes may be consulted. Calculators are allowed, no
computers or cell phones.
Lots of success!
The next multiple choice questions are 2 points each. The final mark is the total number
of points divided by 10 plus 1, rounded to an integer.
M1. The number of optimal solutions in a BIP (binary) problem can be
a. any finite non-negative integer.
b. any finite positive integer.
c. finite or infinite.
M2 An LP problem with n variables and k > n constraints has
a. at least one CPF (Corner Point Feasible) solution.
b. at most nk CPF solutions.
c. possibly no CPF solutions.
M3. The pivot element of a simplex tableau is 0.
a. This means that the problem is unbounded.
b. This means that the problem is non-feasible
c. This cannot happen.
M4. The number of iterations that Prim’s algoritme (the algorithm in the book) must
perform to find a minimal spanning tree in a connected graph with n  2 nodes is
a. at most n-1.
b. exactly n.
c. different answer.
M5. Optimal crashing of a project can be done with
a. the Hungarian algorithm.
b. the simplex method.
c. Dijkstra’s algorithm.
M6. All optimal solutions of a transportation problem are integer
a. if all costs are non-negative and integer.
b. if all supply and demand quantities are non-negative and integer.
c. otherwise.
M7. A shortest path problem is a special case of a
a. minimal cost flow problem.
b. assignment problem.
c. maximal flow problem.
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M8. For an LP problem the simplex method finds an optimal solution. In this solution a
number of constraints is active (equality is assumed), and others are not. The non-active
constraints are removed and the problem is solved again.
a. The simplex method finds the same optimal solution in the same way as before.
b. The simplex method may find a different solution, possibly in a different way.
c. The simplex method may find a better solution.
M9. The shadow prices of an LP problem are
a. optimal solutions of the dual problem.
b. the right-hand sides in the primal problem.
c. something different.
M10. The possible number of cuts in a maximal flow problem on a graph with n nodes is:
a. 2n.
b. 2n-2.
c. something else.
M11. The Big-M method is used to solve an LP problem. The optimal solution is 2M+27.
a. This means that the problem is unbounded.
b. This means that the problem is infeasible.
c. This may be correct.
M12. During the Branch-and-Bound process one of the relaxed sub-problems turns out to
be infeasible.
a. This cannot happen.
b. This means that the original problem is infeasible.
c. This will speed up the process.
M13. An IP problem can be rewritten as a BIP problem.
a. This is always true.
b. This is only true if the feasible region is bounded.
c. This is never true.
M14. The solution process for a linear integer problem can be speeded up by:
a. sharpening constraints.
b. adding suitable constraints.
c. decreasing the number of constraints.
M15. A project can be speeded up by two weeks by:
a. decreasing the duration of two activities on a critical path each by one week.
b. decreasing the duration of all activities on all critical paths each by one week.
c. different measures.
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O1. An LP problem is solved with the simplex method. The “Big-M” approach is used.
During the algorithm the following simplex tableau occurs:
basis
Z
x5
x6
x7
Z
-1
0
0
0
x1
-M+1
1
0
1
x2
-M+5
1
1
-1
x3
M-5
-1
-1
1
x4
M
-1
0
0
x5
0
1
0
0
x6
0
0
1
0
x7
0
0
0
1
RHS
-M
1
3
2
a. (2 pts.) Is this a minimization or a maximization problem?
b. (2 pts.) Give the coordinates x1, … , x7 of the current corner point.
c. (2 pts.) Does this corner point correspond to a feasible point in the original problem?
d. (2 pts.) What is the minimum number of zeros that must occur in a simplex tableau for
this problem?
e. (2 pts.) How many constraints and unknowns could the original problem have had?
f. (10 pts.) Find the next simplex tableaus.
O2. a. (5 pts.) The numbers in the
graph on the right are distances.
Use the algorithm from the book
(Dijkstra) to find the shortest
distance between A and F.
b. (6 pts.) The numbers in the
graph on the right are upper
bounds for a flow. Use the
algorithm from the book (FordFulkerson) to determine the
maximum flow from A to F.
c. (2 pts.) Show how the result of
the last iteration in b.) can be used
to find a minimal cut.
d. (2 pts.) Nodes C and D are removed from the original graph together with all arcs that
are attached to them. They are replaced by an arc BE with value 1 and an arc EB with
value 1. Does this new network behave exactly like the old one with respect to flows
between nodes A, B, E and F? Explain.
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O3 Mary bought a CD at an internet auction for € 3.50. She decides to send the money to
the seller in Germany by mail. She wonders which combination of coins has the smallest
weight, in order to save postage cost. The table shows the weight of each coin.
a. (2 pts.) Formulate this problem as an integer linear
€
weight
optimization problem (IP).
(gramms)
b. (2 pts.) Argue why in the optimal solution each 0.01
2.3
denomination can occur at most once.
0.02
3.0
c. (2 pts.) Does this mean that the integer constraint in a.) 0.05
3.9
can be replaced by binarity? Why?
0.10
4.1
d. (7 pts.) Find the optimal solution using the Branch and 0.20
5.7
Bound algorithm. Hint: First find the (probably) best 0.50
7.8
solution by trial and error. Start Branch and Bound at the 1.00
7.5
coin with the highest denomination. Find an upper bound 2.00
8.5
of the relaxed problem without solving it explicitly, or show
that there is no feasible solution (unless you enjoy solving LP problems with 7
variables).
e. (2 pts.) Mary realizes that the problem is not formulated weight (gr) postage
properly, because the main issue is minimal total cost and
(€)
not minimal weight of the coins. The postage is given in
0-20
0,54
the table on the right. An envelope weighs 3 gramms. Is
20-50
1,08
the solution that you found in d.) still optimal under these
50-500
1,62
new conditions? Why?
O4a. (4 pts.) Three brothers inherit from their father n objects of art with values ci (I = 1,
…, n). The total value of their inheritance is C (= c1 + c2 + … + cn). They want to divide
the objects in a way that is as fair as possible. In general an equal division may prove
impossible (cutting up an object of art reduces its value to 0), but they strive for a division
in which the value of the least share is as large as possible. Formulate a linear integer
programming problem that models their intentions.
b. (3 pts.) The brothers, having no knowledge about linear programming, come up with
the following algorithm. Each of the objects is in turn assigned to the brother that has
obtained, up to that moment, the least total value. Does this algorithm always find the
optimal solution? Explain!
c. (3 pts.) In general the brothers will not receive an equal value of C/3. Is there some
constant  with 0 <  < 1/3 such that each brother will receive al least C, no matter how
the objects are valued? Explain!
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