Chapter 6 – Optimization Models with Integer Variables
1. In a model with 8 changing cells, all of which are constrained to be binary, the number of potentially feasible solutions
is:
a.
4
b.
8
c.
64
d.
256
d
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2. In a linear model, the target cell and constraints are ultimately:
a.
b.
c.
d.
d
1
3. Many inherently nonlinear problems can be transformed into linear models with the use of:
a.
integer variables
b.
integer constraints
c.
binary variables
d.
binary constraints
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4. Rounding the solution of a linear programming to the nearest integer values provides an:
a.
integer solution that is optimal
b.
integer solution that is not optimal
c.
integer solution that might be optimal
d.
infeasible solution
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5. The branching part of the branch and bound algorithm that Solver uses to solve integer programming models means
that the algorithm:
a.
creates subsets of solutions through which to search
b.
searches through only a limited set of feasible integer solutions
c.
identifies an incumbent solution which is optimal
d.
uses a decision tree to find the optimal solution
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6. An algorithm which uses implicit enumeration:
a.
is impractical
b.
examines all possible solutions
Chapter 6 – Optimization Models with Integer Variables
c.
does not guarantee that some of the solutions not examined could be optimal
d.
excludes many possible solutions that guaranteed to be suboptimal
d
1
7. The branch and bound algorithm uses:
a.
lower bounds
b.
upper bounds
c.
both upper and lower bounds
d.
neither upper or lower bounds
c
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8. The default integer optimality setting in Solver is:
a.
0.1%
b.
0.5%
c.
1%
d.
5%
d
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9. Any integer program involving 0 − 1 variables with ____ constraint(s) is called a knapsack problem.
a.
three
b.
two
c.
one
d.
zero
c
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10. The binary variables in the fixed cost models correspond to:
a.
the number of units or products produced
b.
the total profit
c.
the amount of labor hours
d.
a process for which a fixed cost occurs
d
1
11. Optimization software, including Solver, typically has more difficulty solving an integer programming problem,
relative to a linear programming problem.
a.
True
b.
False
True
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12. Solver may be unable to solve some integer programming problems, even when they have an optimal solution.
Chapter 6 – Optimization Models with Integer Variables
a.
True
b.
False
True
1
13. Complete enumeration of all possible solutions in many integer programming problems is impractical.
a.
True
b.
False
True
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14. During the solution of a problem with the branch and bound algorithm, an incumbent solution represents an upper
bound on the solution.
a.
True
b.
False
False
1
15. The higher the upper bound produced in the branch and bound solution process, the faster the algorithm will be.
a.
True
b.
False
False
1
16. If Solver fails to find an optimal solution to an integer programming problem, we might be able to find a near optimal
solution by specifying a smaller number for the integer optimality setting.
a.
True
b.
False
False
1
17. The LP relaxation of an integer programming (IP) problem is typically easy to solve and provides a bound for the IP
model.
a.
True
b.
False
True
1
18. Problems with a fixed cost that is incurred only if an activity is undertaken at a positive level are inherently nonlinear.
a.
True
b.
False
True
1
19. Fixed costs imply that the divisibility assumption of linear models no longer holds.
a.
True
Chapter 6 – Optimization Models with Integer Variables
b.
False
False
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20. In a set-covering model, each member of a given set (set 1) must be “covered” by all members of another set (set 2).
a.
True
b.
False
False
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Exhibit 6-1
A recent college graduate is planning to move cross country from her college town to the city where her first job is
located. She has rented a truck that can haul up to 1100 cubic feet of furniture. The volume and value of each item she is
considering moving on the truck are given below.
21. Refer to Exhibit 6-1. This problem is an example of what type of integer programming model?
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22. Refer to Exhibit 6-1. Which items should she take with her to her new home? What is the value of these items?
1
23. Refer to Exhibit 6-1. Suppose she can save $400 by renting a smaller truck, which has only 800 cubic feed of capacity.
Would she be better off with the smaller truck?
Chapter 6 – Optimization Models with Integer Variables
Exhibit 6-2
Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which can produce up to
10,000 tons per month and the Lakeview plant which can produce up to 7,000 tons per month. Sinclair sells its
polyethylene to three different auto manufacturing plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge
(demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The variable unit cost of shipping a ton is
$1 per mile. The number of miles between the respective plants is shown in the table below:
Grand Rapids
Blue Ridge
Sunset
Ohio Valley
50
40
100
Lakeview
60
50
75
In addition to the variable unit costs of shipping, Sinclair incurs a fixed handling cost (of pickup and delivery) for any
shipping routes it uses. These costs are discounted for longer routes, as shown below:
24. Refer to Exhibit 6-2. This problem is an example of what type of integer programming model?
25. Refer to Exhibit 6-2. What problem will occur if we use an IF statement to model the fixed handling costs?
26. Refer to Exhibit 6-2. What is Sinclair’s optimal shipping plan? What is the total cost in that case?
27. Refer to Exhibit 6-2. Suppose the shipping capacity for all routes is 3000 tons per month. What is Sinclair’s optimal
shipping plan in that case? What is the total cost in that case?
Exhibit 6-3
PharmaCo wants to determine how to deploy sales representatives across its Western U.S. region to support a new drug
for obesity. Sales representatives will be located in a “home city”, which they serve, in addition to cities with feasible
commuting distance, with the objective that all cities must be served by at least one sales representative. The feasible
connections between each city in the region are listed below (1 indicates a feasible connection):
28. Refer to Exhibit 6-3. This problem is an example of what type of integer programming model?
29. Refer to Exhibit 6-3. Formulate an integer programming model to identify a feasible deployment plan to the sales
representatives. What is the minimum number of representatives that can cover this region? What are their home cities?
30. Refer to Exhibit 6-3. Suppose PharmaCo wants to have a backup for each city. What is the minimum number of
representatives that can cover this region in that case? What are their home cities?