Question 1

The global optimum point may be superior to nearby points, but the local optimum point is the true optimal solution-n to a nonlinear programming problem.

Accepted Answer

The statement is false. The global optimum point is the true optimal solution to a nonlinear programming problem, not just the local optimum point. The global optimum point is the point that yields the highest (or lowest) objective function value across the entire feasible region, while a local optimum point is the point that yields the highest (or lowest) objective function value within a local neighborhood of feasible points.

Question 2

The constraint X₁ - X₂ ≤ 0 with 0 -1 integer programming allows for X₁ to be selected as part of the optimal solution only if X₂ is selected to be a part of the optimal solution, but not both.

Accepted Answer

The constraint ensures that X₁ can only be selected if X₂ is also selected, but it does not prevent both from being part of the optimal solution; it only restricts the selection of X₁ without X₂.

Question 3

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.

Accepted Answer

Since the original solution to the linear programming problem was not in integers, rounding to the nearest integers (either up or down) will change the solution. Given that this is a maximization problem and the original solution likely utilized the fractional parts of X and Y to reach the maximum, any adjustment to integers will likely result in a lower maximum, hence the objective function value must be less than $253.67.

Question 4

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

Accepted Answer

The objective function value in goal programming is the sum of the deviations from the target goals. Since the first goal was missed by $100 ($2400 - $2300 = $100) and the second goal was achieved exactly (no deviation), the value of the objective function is the deviation from the first goal, which is $100.

Question 5

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

Accepted Answer

Nonlinear programming problems can have a wide range of possible solution methods, but there is no general method that can solve all types of nonlinear programming problems. The other options (mixed-integer programming, 0-1 integer programming, and goal programming) are all specific types of optimization problems, and while there may not be a single method that can solve all instances of these problems, there are well-established methods for solving many instances of each type.

Question 6

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

Accepted Answer

The problem of selecting projects under budget constraints can be modeled as a 0-1 integer program, where the decision variable represents whether a project is selected or not. The objective function is to maximize the total value of the selected projects, subject to the budget constraint. The decision variables are binary, which means they can only take the values of 0 (not selected) or 1 (selected). Mixed-integer programming can also be used, but it involves more complex algorithms and is not necessary for this problem. Goal programming and nonlinear programming are not appropriate for this type of problem.

Question 7

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.

Accepted Answer

In goal programming, decision makers can set and prioritize multiple objective functions or goals, often taking into account trade-offs between them. The other options listed are all types of integer programming problems, which do not necessarily allow for multiple objectives to be prioritized.

Question 8

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)It maximizes instead of minimizing as in LP.

Accepted Answer

Goal programming allows for multiple goals to be incorporated into a single objective function, unlike linear programming which typically focuses on optimizing a single objective.

Question 9

The constraint X₁ - X₂ = 0 with 0 -1 integer programming allows for either both X₁ and X₂ to be selected to be a part of the optimal solution, or for neither to be selected.

Accepted Answer

When the constraint X₁ - X₂ = 0 is included in the 0-1 integer programming model, it means that the difference between the binary variables X₁ and X₂ must be zero. This can only happen when both variables are either both 0 or both 1. Therefore, both variables can be included in the optimal solution or both can be excluded, but they cannot have different values.

Question 10

The following objective function is nonlinear: Max 5X + X/Y - Z.

Accepted Answer

The objective function contains the term X/Y which is a ratio of two variables and thus nonlinear.

Question 11

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

Accepted Answer

The objective function in goal programming measures the deviation from the set goals. Since the first goal was to achieve a profit of $3,600 and the actual profit was $3,300, the deviation is $300 ($3,600 - $3,300). The second goal was fully met (no wasted material), so there's no deviation. The value of the objective function is the total deviation from all goals, which is $300.

Question 12

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.

Accepted Answer

Assignment problems involve assigning tasks to individuals or machines. The decision variable is often binary (0 or 1) representing whether a certain task is assigned to a certain person/machine or not. This is known as zero-one integer programming. Linear programming can be used to solve such problems, but it requires the addition of specific constraints to enforce the binary nature of the decision variables. Pure-integer programming involves decision variables that are integer but not necessarily binary, while mixed-integer programming involves decision variables that may be a mix of continuous and integer values. Goal programming involves multiple objectives that need to be optimized simultaneously.

Question 13

The constraint X₁ + X₂ + X₃+ X₄ ≤ 2 with 0 -1 integer programming allows at most two of the items X_1, X_2, X_3, and X₄ to be selected to be a part of the optimal solution.

Accepted Answer

The constraint states that the sum of the decision variables X₁, X₂, X₃, and X₄ cannot exceed 2. Since these decision variables are binary (can only take on values of 0 or 1), this constraint ensures that at most two of them can be selected (i.e. have a value of 1). Therefore, the answer is A.

Question 14

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

Accepted Answer

Nonlinear programs are the category of mathematical programming techniques that do not assume linearity in the objective function and/or constraints. Integer programs focus on variables that are restricted to integer values, goal programming problems aim to find a solution that minimizes deviations from multiple objectives, and multiple objective programming problems deal with finding solutions that optimize multiple objectives simultaneously.

Question 15

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.

Accepted Answer

The correct constraint to express that variable X should be exactly twice the value of variable Y is D) X - 2Y = 0. This equation rearranges to X = 2Y, directly stating that X is twice Y.

Question 16

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.

Accepted Answer

An integer programming problem is a type of linear programming problem where one or more decision variables are required to take on integer values in the final solution. This distinguishes it from a standard linear programming problem where all decision variables can take on any real value. The objective function and constraints, however, are still linear. Goal programming problems involve multiple objectives, nonlinear programming problems have nonlinear objective functions or constraints, and multiple objective LP problems have linear objectives but multiple conflicting objectives to be optimized simultaneously.

Question 17

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.

Accepted Answer

When all the deviational variables are equal to zero in a goal programming problem, it means that all the goals have been fully achieved. Therefore, option B is the correct choice. Option A is incorrect because if there was no feasible solution, then there would have been deviational variables. Option C is irrelevant as the method used to solve the problem is not mentioned. Option D is also incorrect as there is no mention that this problem was an integer programming problem.

Question 18

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.

Accepted Answer

Since the original solution to the linear programming problem was not in integers, rounding to the nearest integers (either up or down) to meet the integer constraint will necessarily change the objective function value. Given that this is a minimization problem, and the original solution likely represents the minimum possible value without the integer constraints, any adjustment to meet the integer constraints will result in a solution that is less optimal (i.e., has a higher cost) than the original. Therefore, the objective function value in the optimal integer solution will be greater than $253.67.

Question 19

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)deviation from certainty.

Accepted Answer

Goal programming involves setting multiple goals with different levels of priority. The deviation between the actual and desired values for each goal is measured, and the objective is to minimize these deviations subject to constraints. In goal programming, the deviation from a high-priority goal must be minimized before the next-highest-priority goal can be considered. Therefore, option C is the correct choice. Option A is incorrect because goal programming can also involve minimization problems. Option B is incorrect because setting bounds is a feature of linear programming, not necessarily goal programming. Option D is incorrect because goal programming does not necessarily involve deviations from certainty.

Question 20

Which of the following functions is nonlinear?
A)4X + 2Y + 7Z
B)-4X + 2Y
C)4X + (1/2)Y + 7Z
D)4X/Y + 7Z

Accepted Answer

The function 4X/Y + 7Z is nonlinear because it contains a term with a variable in the denominator, namely X/Y. Nonlinear functions do not exhibit a constant rate of change and cannot be graphed as straight lines. The other options are all linear functions because they do not contain any terms with variables in the denominator, and therefore exhibit a constant rate of change and can be graphed as straight lines.

Quiz 10: Integer Programming, Goal Programming, and Nonlinear Programming

The global optimum point may be superior to nearby points, but the local optimum point is the true optimal solution-n to a nonlinear programming problem.

The constraint X1 - X2 ≤ 0 with 0 -1 integer programming allows for X1 to be selected as part of the optimal solution only if X2 is selected to be a part of the optimal solution, but not both.

We do not have a general method for solving all types of ________ problems.

A capital budgeting problem involving the selection of possible projects under budget constraints is solved by which of the following?

A mathematical programming model that permits decision makers to set and prioritize multiple objective functions is called a

Goal programming differs from linear programming in which of the following aspects?

The constraint X1 - X2 = 0 with 0 -1 integer programming allows for either both X1 and X2 to be selected to be a part of the optimal solution, or for neither to be selected.

The following objective function is nonlinear: Max 5X + X/Y - Z.

Assignment problems solved previously by linear programming techniques are also examples of

The constraint X1 + X2 + X3+ X4 ≤ 2 with 0 -1 integer programming allows at most two of the items X1, X2, X3, and X4 to be selected to be a part of the optimal solution.

Which of the following is a category of mathematical programming techniques that doesn't assume linearity in the objective function and/or constraints?

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 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

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

Goal programming is characterized by

Which of the following functions is nonlinear?

The constraint X₁ - X₂ ≤ 0 with 0 -1 integer programming allows for X₁ to be selected as part of the optimal solution only if X₂ is selected to be a part of the optimal solution, but not both.

The constraint X₁ - X₂ = 0 with 0 -1 integer programming allows for either both X₁ and X₂ to be selected to be a part of the optimal solution, or for neither to be selected.

The constraint X₁ + X₂ + X₃+ X₄ ≤ 2 with 0 -1 integer programming allows at most two of the items X_1, X_2, X_3, and X₄ to be selected to be a part of the optimal solution.