Quantitative Analysis for Management, 11e (Render)
Chapter 10 Integer Programming, Goal Programming, and Nonlinear Programming
1) If conditions require that all decision variables must have an integer solution, then the class of problem
described is an integer programming problem.
2) An integer programming solution can never produce a greater profit objective than the LP solution to the same
problem.
3) 0–1 integer programming might be applicable to selecting the best gymnastics team to represent a country from
among all identified teams.
4) Nonlinear programming is the case in which objectives and/or constraints are nonlinear.
5) The following objective function is nonlinear: Max 5X + (1/8)Y – Z.
6) In goal programming, if all the goals are achieved, then the value of the objective function will always be zero.
7) Unfortunately, multiple goals in goal programming are not able to be prioritized and solved.
8) The following objective function is nonlinear: Max 5X – 8YZ.
9) Goal programming permits multiple objectives to be satisfied.
10) The constraint X1 + X2 ≤ 1 with 0 –1 integer programming allows for either X1 or X2 to be a part of the optimal
solution, but not both.
11) Requiring an integer solution to a linear programming problem decreases the size of the feasible region.
12) The transportation problem is a good example of a pure integer programming problem.
13) The three types of integer programs are: pure integer programming, impure integer programming, and 0–1
integer programming.
14) When solving very large integer programming problems, we sometimes have to settle for a “good,” not
necessarily optimal, answer.
15) Quadratic programming contains squared terms in the constraints.
16) In goal programming, our goal is to drive the deviational variables in the objective function as close to zero as
possible.
17) There is no general method for solving all nonlinear problems.
18) A 0–1 programming representation could be used to assign sections of a course to specific classrooms.
19) In goal programming, the deviational variables have the same objective function coefficients as the surplus
and slack variables in a normal linear program.
20) Unfortunately, goal programming, while able to handle multiple objectives, is unable to prioritize these
objectives.
21) A model containing a linear objective function and linear constraints but requiring that one or more of the
decision variables take on an integer value in the final solution is called ________
A) a goal programming problem.
B) an integer programming problem.
C) a nonlinear programming problem.
D) a multiple objective LP problem.
E) a branch–and–bound programming problem.
22) Assignment problems solved previously by linear programming techniques are also examples of
A) pure–integer programming problems.
B) mixed–integer programming problems.
C) zero–one integer programming problems.
D) goal programming problems.
E) nonlinear programming problems.
23) A mathematical programming model that permits decision makers to set and prioritize multiple objective
functions is called a
A) pure–integer programming problem.
B) mixed–integer programming problem.
C) zero–one integer programming problem.
D) goal programming problem.
E) nonlinear programming problem.
24) Goal programming differs from linear programming in which of the following aspects?
A) It tries to maximize deviations between set goals and what can be achieved within the constraints.
B) It minimizes instead of maximizing as in LP.
C) It permits multiple goals to be combined into one objective function.
D) All of the above
E) None of the above
25) Which of the following is a category of mathematical programming techniques that doesn’t assume linearity
in the objective function and/or constraints?
A) integer programs
B) goal programming problems
C) nonlinear programs
D) multiple objective programming problems
E) None of the above
26) A type of integer programming is
A) pure.
B) mixed.
C) zero–one.
D) All of the above
E) None of the above
27) Which of the following functions is nonlinear?
A) 4X + 2Y + 7Z
B) –4X + 2Y
C) 4X + (1/2)Y + 7Z
D) Z
E) 4X/Y + 7Z
28) Goal programming is characterized by
A) all maximization problems.
B) setting of lower and upper bounds.
C) the deviation from a high–priority goal must be minimized before the next–highest–priority goal may be
considered.
D) All of the above
E) None of the above
29) An integer programming (maximization) problem was first solved as a linear programming problem, and the
objective function value (profit) was $253.67. The two decision variables (X, Y) in the problem had values of X =
12.45 and Y = 32.75. If there is a single optimal solution, which of the following must be true for the optimal
integer solution to this problem?
A) X = 12 Y = 32
B) X = 12 Y = 33
C) The objective function value must be less than $253.67.
D) The objective function value will be greater than $253.67.
E) None of the above
30) An integer programming (minimization) problem was first solved as a linear programming problem, and the
objective function value (cost) was $253.67. The two decision variables (X, Y) in the problem had values of X =
12.45 and Y = 32.75. If there is a single optimal solution, which of the following must be true for the optimal
integer solution to this problem?
A) X = 13 Y = 33
B) X = 12 Y = 32
C) The objective function value must be less than $253.67.
D) The objective function value will be greater than $253.67.
E) None of the above
31) In a goal programming problem with two goals at the same priority level, all the deviational variables are
equal to zero in the optimal solution. This means
A) there is no feasible solution to the problem.
B) all goals are fully achieved.
C) nonlinear programming must be used to solve this.
D) this problem was an integer programming problem.
E) None of the above
32) A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a
profit of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to
this problem resulted in a profit of $2,300 and no idle time. What was the value for the objective function for this
goal programming problem?
A) 2300
B) 100
C) –100
D) 0
E) None of the above
33) A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a
profit of $3,600 and goal number 2 was to have no wasted material. The optimal solution to this problem resulted
in a profit of $3,300 and no wasted material. What was the value for the objective function for this goal
programming problem?
A) 300
B) –300
C) 3300
D) 0
E) None of the above
34) In an integer programming problem, if it is desired to have variable X be exactly twice the value of variable Y,
the constraint would be written:
A) 2X + Y = 0.
B) X + 2Y = 0.
C) 2X – Y = 0.
D) X – 2Y = 0.
E) None of the above
Table 10–1
A company has decided to use 0–1 integer programming to help make some investment decisions. There are
three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to
be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one–half of a factory).
The integer programming model is as follows:
The optimal solution is X1 = 0, X2 = 1, X3 = 1
35) According to Table 10–1, which presents an integer programming problem, if the optimal solution is used,
what would the value of the objective function be ________.
A) 21,000
B) 12,000
C) 16,000
D) 2
E) None of the above
36) According to Table 10–1, which presents an integer programming problem, if the optimal solution is used,
how much of the budget would be spent?
A) $32,000
B) $29,000
C) $61,000
D) $62,000
E) None of the above
37) In Table 10–1, which presents an integer programming problem, using the optimal solution means only two of
the alternatives would be selected. How much slack is there in the third constraint?
A) 0
B) 3
C) 33
D) 36
E) None of the above
38) According to Table 10–1, which presents an integer programming problem, the optimal solution is to select
only two of the alternatives. Suppose you wished to add a constraint that stipulated that alternative 2 could only
be selected if alternative 1 is also selected (i.e., if alternative 1 is not selected, you may not select alternative 2;
however, you may select #1 and not select #2). How would this constraint be written?
A) X1 = X2
B) X1 ≤ X2
C) X1 ≥ X2
D) X1 + X2 = 2
E) None of the above
Table 10–2
39) According to Table 10–2, which presents a solution for an integer programming problem, at the optimal
solution, how much slack exists in the third constraint?
A) 0
B) 9
C) 5
D) 6
E) –1
40) We do not have a general method for solving all types of ________ problems.
A) mixed–integer programming
B) 0–1 integer programming
C) goal programming
D) nonlinear programming
E) pure integer programming
41) A capital budgeting problem involving the selection of possible projects under budget constraints is solved by
which of the following?
A) mixed–integer programming
B) 0–1 integer programming
C) goal programming
D) nonlinear programming
E) pure integer programming
42) A transportation problem is an example of
A) a pure–integer programming problem.
B) a mixed–integer programming problem.
C) a zero–one integer programming problem.
D) a goal programming problem.
E) a nonlinear programming problem.
43) If we wish to develop a stock portfolio wherein we maximize return and minimize risk, we would have to use
A) pure–integer programming.
B) goal programming.
C) zero–one integer programming.
D) mixed–integer programming.
E) nonlinear programming.
44) Another name for a 0–1 variable is a ________ variable.
A) either–or
B) binary
C) yes–no
D) quadratic
E) on–off
45) Terms that are minimized in goal programming are called ________.
A) deviational variables
B) global variables
C) decision variables
D) minimization variables
E) None of the above
46) The concept of a local optimum is affiliated with which of the following?
A) mixed integer programming
B) integer programming
C) linear programming
D) nonlinear programming
E) goal programming
47) The concept of “satisficing” is affiliated with which of the following?
A) mixed integer programming
B) integer programming
C) linear programming
D) nonlinear programming
E) goal programming
48) The following represents a:
A) goal programming problem.
B) mixed integer programming problem.
C) nonlinear programming problem.
D) 0–1 integer programming problem.
E) pure integer programming problem.
49) As part of a larger problem, you are trying to determine whether or not to open a plant with a capacity of
10,000 units (using binary variable Y). You also define X as the number of units (if any) produced at that plant.
How will you ensure that Y will equal 1 if the plant is open?
A) Y ≥ X
B) Y ≤ X
C) X + Y ≥ 2
D) X = 10000Y
E) X ≤ 10000Y
50) Which of the following is not considered nonlinear programming?
A) nonlinear objective and nonlinear constraints
B) linear objective with nonlinear constraints
C) nonlinear objective with linear constraints
D) binary decision variable with nonlinear constraints
E) integer decision variable with linear constraints
Topic: NONLINEAR PROGRAMMING
51) A quadratic programming problem involves which of the following conditions?
A) squared terms in the objective function and linear constraints
B) linear objective function and squared terms in the constraints
C) squared terms in both the objective function and constraints
D) a strictly goal programming problem with squared terms in the objective function
E) None of the above
52) Which of the following statements is false concerning goal programming?
A) The objective function is the main difference between linear programming and goal programming.
B) The objective in goal programming is to minimize deviational variables.
C) Deviational variables are zero if a goal is completely obtained.
D) It is not possible for two goals to have equal priority.
E) The priorities of each goal are reflected in the objective function.
53) Consider the following 0–1 integer programming problem:
If we wish to add the constraint that no more than two of these variables must be positive, how would this be
written?
A) 2X + 2Y + 2Z ≤ 3
B) X + Y + Z ≤ 2
C) X ≤ 2, and Y ≤ 2, and Z ≤ 2
D) X, Y, Z ≤ 2
E) None of the above
54) Consider the following 0 – 1 integer programming problem:
If we wish to add the constraint that X must be positive, and that only Y or Z but not both can be positive, how
would the additional constraint(s) be written?
A) X + Y + Z ≤ 3, Y + Z ≤ 1
B) X ≤ 1, Y + Z = 1
C) X ≤ 2, and Y ≤ 2, and Z ≤ 2
D) X = 1, Y + Z ≤ 1
E) None of the above
55) An integer programming (maximization) problem was first solved as a linear programming problem, and the
objective function value (profit) was $253.67. The two decision variables (X, Y) in the problem had values of X =
12.45 and Y = 32.75. Which of the following must be true for the optimal integer solution to this problem?
A) X = 12 Y = 32
B) X = 12 Y = 33
C) X = 12
D) Y = 32
E) None of the above
56) The overall best solution in a nonlinear program is a ________.
A) global optimum
B) local optimum
C) binary optimum
D) nonlinear optimum
E) goal optimum
57) A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a
cost of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to this
problem resulted in a cost of $2,400 and no idle time. What was the value for the objective function for this goal
programming problem?
A) 2300
B) 100
C) –100
D) 0
E) None of the above
58) A goal programming problem had two goals (with no priorities assigned). Goal number 1 was to achieve a
cost of $3,600 and goal number 2 was to have no wasted material. The optimal solution to this problem resulted
in a cost of $3,900 and no wasted material. What was the value for the objective function for this goal
programming problem?
A) 300
B) –300
C) 3300
D) 0
E) None of the above
Table 10–4
A company has decided to use 0−1 integer programming to help make some investment decisions. There are
three possible investment alternatives from which to choose, but if it is decided that a particular alternative is to
be selected, the entire cost of that alternative will be incurred (i.e., it is impossible to build one–half of a factory).
The integer programming model is as follows:
59) Table 10–4 presents an integer programming problem. What is the meaning of Constraint 1?
A) If X1 is selected, X2 must also be selected.
B) No more than two alternatives may be selected.
C) At least two alternatives must be selected.
D) If X2 is selected, X1 must also be selected.
E) None of the above
60) Table 10–4 presents an integer programming problem. What is the meaning of Constraint 2?
A) Both alternatives 1 and 2 must be selected.
B) If alternative 2 is selected, alternative 1 must also be selected.
C) Either alternative 1 or alternative 2 must be selected.
D) No more than one alternative may be selected.
E) None of the above
61) Table 10–4 presents an integer programming problem. If the optimal solution is used, then only two of the
alternatives would be selected. How much slack would there be in the third constraint?
A) 1000
B) 5000
C) 3300
D) 8000
E) None of the above
62) Table 10–4 presents an integer programming problem. Suppose you wish to add a constraint that stipulates
that both alternative 2 and alternative 3 must be selected, or neither can be selected. How would this constraint
be written?
A) X2 = X3
B) X2 ≤ X3
C) X2 ≥ X3
D) X2 + X3 = 1
E) None of the above