APPM 4120/5120
Exam #2 Practice
Spring 2015
You are not allowed to use textbooks, class notes.
Problem #1 (20 points): Consider the following activity-on-arc project network, where the 12 arcs
(arrows) represent the 12 activities (tasks) that must be performed to complete the project and the
network displays the order in which the activities need to be performed. The number next to each arc
(arrow) is the time required for the corresponding activity. Consider the problem of finding the
longest path (the largest total time) through this network from start (node 1) to finish (node 9), since
the longest path is the critical path. Formulate a BIP model for this problem.
Figure 1: Network for Problem 1
Problem #2 (10 points): Decora Accessories manufactures a variety of bathroom accessories,
including decorative towel rods and shower curtain rods. Each of the accessories includes a rod made
out of stainless steel. However, many different lengths are needed: 12", 18", 24", 40", and 60". Decora
purchases 60" rods from an outside supplier and then cuts the rods as needed for their products. Each
60" rod can be used to make a number of smaller rods. For example, a 60" rod could be used to make a
40" and an 18" rod (with 2" of waste), or 5 12" rods (with no waste). For the next production period,
Decora needs 25 12" rods, 52 18" rods, 45 24" rods, 30 40" rods, and 12 60" rods. What is the fewest
number of 60? rods that can be purchased to meet their production needs? Formulate an integer
programming model in algebraic form for this problem.
Problem #3 (10 points): Suppose you are given a graph G = (V, E ) with edge weights w(e) and a
minimum spanning tree T of G. Now, suppose a new edge {u, v} is added to G. Describe (in words) a
method for determining if T is still a minimum spanning tree for G.
Problem #4 (20 points): The residual graph shown below is for some flow f on a flow graph G.
(a) What is the capacity of the edge connecting e and c in G? Justify your answer.
(b) Is the edge in G directed from c to e or from e to c. Justify your answer. (Hint: consider the total
incoming flow at e.) You may assume that there is no more than one edge joining any two
vertices, but you should not assume anything about the direction of the edges at s and t (that is,
G may have edges entering s or leaving t).
(c) Find a shortest augmenting path relative to f. Draw the residual graph that results from adding
as much flow as possible to this path.
Figure 2: Network for Problem 4
Problem #5 (20 points): The Build-Em-Fast Company has agreed to supply its best customer with
three widgits during each of the next 3 weeks, even though producing them will require some
overtime work. The relevant production data are as follows:
Week
1
2
3
Max Production/Regular Time
2
3
1
Max Production/Overtime
2
2
2
Cost per Unit/Regular
$300
$500
$400
Table 1: Data
The cost per unit produced with overtime for each week is $100 more than for regular time. The cost
of storage is $50 per unit for each week it is stored. There is already an inventory of two widgets on
hand currently, but the company does not want to retain any widgets in inventory after the 3 weeks.
Management wants to know how many units should be produced in each week to minimize the total
cost of meeting the delivery schedule. Formulate this problem as a transportation problem by
constructing the appropriate parameter table.
Problem #6 (20 points): Consider an assignment problem with four workers, A, B, C, and D, and three
jobs, 1, 2, and 3:
A
B
C
D
1
4
3
9
7
2
6
-1
2
6
3
5
7
8
9
Table 2: Data for problem 6
(a) Put into standard form for the Hungarian Method, increasing or decreasing costs and creating
dummy workers or dummy jobs as necessary. (For ease of grading, please do your costs
increases or decreases before creating dummy workers or jobs.) Explain why your cost increases
or decreases don’t distort the optimal assignment of workers to jobs. Explain why your
assignment of costs to dummy workers or jobs doesn’t distort the optimal assignment of real
workers to real jobs.
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(b) Can you solve this using the Hungarian algorithm? Why to why not? Explain why row and
column reduction don’t distort the optimal assignment of workers to jobs. If it is solvable, solve
it.
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