Chapter 10 Integer Programming, Goal Programming, and 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 10-3

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:

Maximize 5000 X1 + 7000X2 + 9000X3

Subject to: X1 + X2 + X3 ? 2 Constraint 1

-X1 + X2 ? 0 Constraint 2

25,000 X1 + 32,000 X2 + 29,000 X3 ? 62,000 (budget limit)

16 X1 + 14 X2 + 19 X3 ? 36 (resource limitation)

all variables = 0 or 1

where X1 = 1 if alternative 1 is selected, 0 otherwise

X2 = 1 if alternative 2 is selected, 0 otherwise

X3 = 1 if alternative 3 is selected, 0 otherwise

Solution x1 = 1, x2 = 0, x3 = 1, objective value = 14,000.

59) Table 10-3 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 10-3 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 10-3 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 10-3 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

Table 10-4

63) Table 10-4 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Which goals are only partly achieved?

A) number 1 only

B) number 1 and number 2

C) number 2 and number 3

D) number 1 and number 3

E) None of the above

64) Table 10-4 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Goal number 3 represents a resource usage goal. How much of this resource would be used by this solution?

A) 50 units

B) 70 units

C) 2500 units

D) 240 units

E) None of the above

65) Table 10-4 represents a solution to a goal programming problem. There are three goals (each represented by a constraint). Which of the goals is assigned the highest priority?

A) goal 1

B) goal 2

C) goal 3

D) goals 2 and 3

E) All goals have the same priority.

Table 10-5

Maximize Z = 34 X1 + 43 X2 + 29 X3

Subject to: 5 X1 + 4 X2 + 7 X3 ? 50

1 X1 + 2 X2 + 2 X3 ? 16

3 X1 + 4 X2 + 1 X3 ? 9

all Xi are integer and non-negative

Final Integer Solution: Z = 208

Decision

Variable Solution

X1 1

X2 0

X3 6

66) Table 10-5 represents a solution for an integer programming problem. If this problem had been solved as a simple linear programming problem, what would you expect the value of the objective function to be?

A) less than 208

B) greater than 208

C) exactly 208

D) A or C

E) B or C

67) Table 10-5 represents a solution for an integer programming problem. If one uses the optimal solution presented, how much slack is there in the first equation?

A) 0 units

B) 5 units

C) 3 units

D) 2 units

E) None of the above

68) A model containing a linear objective function and requiring that one or more of the decision variables take on an integer value in the final solution is called

A) an integer programming problem.

B) a goal programming problem.

C) a nonlinear programming problem.

D) a multiple objective LP problem.

E) insufficient information.

Table 10-6

The profit function for a set of two electronics products can be expressed by the following expression: X12 -2X1 -3X2 + 2X22 where X1 = the number of product 1 units produced and X2 = the number of product 2 units produced. At least 10 units of X1 must be produced and at least 20 units of X2 must be produced. No more than 50 units total can be produced.

69) What type of mathematical programming model is required for the problem described in Table 10-6?

A) An integer programming model

B) A goal programming model

C) A nonlinear programming model

D) A zero-one integer programming model

E) A mixed-integer programming model

70) According to Table 10-6, how is the constraint expressed that no more than 50 units can be produced?

A) X1 + X2 ? 50

B) X1 + X2 ? 50

C) X12 + X22? 50

D) X12 + X22? 50

E) X1 + X2 = 50