Deck 20: Nonlinear Programming: Solution Techniques

Question 1

The Lagrangian function is differentiated with respect to each variable, and the resulting equations are solved simultaneously.

Answer 1

True

Question 2

The slope of a curve at its highest point equals 0.

Answer 2

True

Question 3

The derivative of a function equals the slope of the curve defined by that function.

Answer 3

True

Question 4

Lagrange multiplier is another name for shadow price.

Question 5

In the method of Lagrange multipliers, constraints as multiples of a multiplier are subtracted from the objective function, which is then differentiated with respect to each variable and solved.

Question 6

If a Lagrange multiplier equals 3, a one-unit increase in the right-hand side of a constraint, will result in an increase of 3 in the objective function.

Question 7

The Lagrange multiplier is analogous to the dual variables in a linear programming problem.

Question 8

In the method of ________, constraints as multiples of multiplier λ are subtracted from the objective function, which is then differentiated with respect to each variable and solved.

Question 9

In the ________ method, the constraint equation is solved for one variable in terms of another and then substituted into the objective function.

Question 10

The slope of a curve at any point is equal to the derivative of the curves function.

Question 11

The Lagrangian function is the original objective function plus the Lagrange multiplier times the difference of the left and right sides of the constraint.

Question 12

The method of Lagrange multipliers can only be used in solving nonlinear programming problems with one equality constraint.

Question 13

The Lagrange multiplier reflects the appropriate change in the objective function resulting from a unit change in the quantity value of the constraint equation.

Question 14

In the substitution method, the constraint equation is solved for one variable in terms of another and then substituted into the objective function.

Question 15

The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value.

Question 16

In the method of Lagrange multipliers, the model constraints are multiplied by Lagrange multipliers and subtracted from the objective function.

Question 17

The slope of a curve at its highest point equals 1.

Question 18

The Lagrange multiplier reflects a change in the objective function from a unit change in the right-hand-side value of a constraint.

Question 19

Linear programming is a special case of nonlinear programming.

Question 20

The Lagrangian function is ________ with respect to each variable and the resulting equations are solved simultaneously.

Question 21

If a nonlinear programming problem results in profit (Z) of $250, and the Lagrange multiplier for a constraint is 5, the new profit will be ________ if the right-hand side of the constraint is decreased by 1 unit.

Question 22

The slope of a curve at its highest point equals:

A) 0
B) 1
C) 2
D) 3

Question 23

In solving a maximization problem, the optimal profit associated with the relaxed solution is always less than or equal to the value of the optimal profit associated with the ________ solution.

Question 24

A customer molder produces 6-ounce juice glasses and 10-ounce cocktail glasses. The per unit contribution for the juice glasses (x1) is equal to 60 - 5x1, and the per unit contribution for the cocktail glasses (x2) is 80 - 4x2 for a total contribution of 60x1 - 5x12 + 80x2 - 4x22. Demand requires that the production of juice glasses is twice the production of cocktail glasses.(x1 = 2x2). What is the Lagrangian function for this problem?

Question 25

Given the nonlinear programming model Max Z = 4x1 - 0.1x1x2 + 5x2 - 0.2 x22
Subject to: x1 + 2x2 = 4
What is the Lagrangian function?

Question 26

Given the nonlinear programming model Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What is the Lagrangian function?

Question 27

A customer molder produces 6-ounce juice glasses and 10-ounce cocktail glasses. The per unit contribution for the juice glasses (x₁) is equal to 60 - 5x₁, and the per unit contribution for the cocktail glasses (x₂) is 80 - 4 x₂ for a total contribution of 60x₁ - 5x₁² + 80x₂ - 4x₂2. Demand requires that the production of juice glasses is twice the production of cocktail glasses (x₁ = 2x₂). Write the objective function for this problem which illustrates the technique of substitution.

Question 28

Write the Lagrangian function for the following nonlinear program:
Min x12 + 2x22 - 8x1 - 12x2 + 34
subject to: x12 + 2x22 = 5

Question 29

Given the nonlinear programming model Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What is the optimal value of the Lagrange multiplier?

Question 30

If the substitution method cannot be used to solve a nonlinear problem, then the equations which result from taking partial derivatives must be solved using the method of ________.

Question 31

The Lagrange multiplier, λ, reflects a change in the objective function from a unit change in the ________ value of a constraint.

Question 32

The slope of a curve at its highest point equals ________.

Question 33

Given the nonlinear programming model Max Z = 5x1 - 2x22
Subject to: x1 + 4x2 = 6
What is the Lagrangian function?

Question 34

Given the nonlinear programming model Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What is the optimal profit?

Question 35

Solve the following nonlinear program:
Min x12 + 2x22 - 8x1 - 12x2 + 34
subject to: x12 + 2x22 = 5

Question 36

Given the nonlinear programming model Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What are the optimal values of x1 and x2?

Question 37

The slope of a curve at any point is equal to the ________ of the curves function.

Question 38

The Lagrangian function is differentiated with respect to each variable and the resulting equations are solved ________ to obtain the value of each variable.

Question 39

Given the nonlinear programming model Max Z = 5x1 - 2x2- 3x22
Subject to: x1 + 4x2 = 6
What is the Lagrangian function?

Question 40

If total contribution of is expressed as C = 60x₁ - 5x₁² + 80x₂ - 4x₂² and x₁ = 2x₂, then the Lagrangian function is 60x₁ - 5x₁² + 80x₂ - 4x₂² ________.

Question 41

In the substitution method, the constraint equation is solved for one variable ________ and then substituted into the objective function.

A) only
B) after another
C) at a time
D) in terms of another

Question 42

Solving nonlinear optimization problems using Lagrange multipliers requires

A) calculus.
B) algebra.
C) computers.
D) A and B
E) A and C

Question 43

The ________ method for solving nonlinear programming problems is referred to as substitution.

A) least complex
B) most complex
C) most difficult
D) least difficult

Question 44

The Lagrange multiplier reflects the appropriate change in the objective function resulting from a unit change in the ________ of the constraint equation.

A) coefficient
B) objective function
C) right-hand side
D) shadow price

Question 45

Consider the following objective function and constraint: C(x) = 2x12 - 15x + 3x23 - 9x2 + 30, such that x1 = 3x2. Which of the following is the correct Lagrangian function?

A) L = 2x₁² - 15x + 3x₂3 - 9x₂ + 30 - ? (3x₂ - x1)
B) L = 2x₁² - 15x + 3x₂3 - 9x₂ + 30 - ? (x₂ - 3x1)
C) L = 2x₁² - 15x + 3x₂3 - 9x₂ + 30 - ? (3x₁ - x2)
D) L = 2x₁² - 15x + 3x₂3 - 9x₂ + 30 - ? (3x₂ - 3x₁)

Question 46

The least complex method for solving nonlinear programming problems is referred to as

A) the substitution method.
B) Lagrange multipliers.
C) nonlinear programming.
D) none of the above

Question 47

Given the nonlinear programming model:
Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What is the optimal value of the objective function?

A) Z = 19.625
B) Z = 20.625
C) Z = 21.625
D) Z = 22.625
E) Z =23.625

Question 48

The slope of a curve at any point is ________ the derivative of the curves function.

A) smaller than
B) larger than
C) equal to
D) none of the above

Question 49

The Lagrangian function is differentiated with respect to each variable, and the resulting equations are solved

A) in the order of the variables.
B) one after the other.
C) one at a time.
D) simultaneously.

Question 50

Given the nonlinear programming model:
Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What are the optimal values of x1 and x2?

A) x1 = 4.75, x2 = 1.25
B) x1 = 5.00, x2 = 3.25
C) x1 = 4.50, x2 = 2.25
D) x1 = 4.50, x2 = 1.25
E) x1 = 5.75, x2 = 0.75

Question 51

Given the nonlinear programming model:
Max Z = 5x1 - 2x22
Subject to: x1 + x2 = 6
What is the optimal value of the Lagrange multiplier?

A) λ = 1
B) λ = 2
C) λ = 3
D) λ = 4
E) λ = 5

Question 52

In the method of Lagrange multipliers, the model constraints are multiplied by Lagrange multipliers and ________ the objective function.

A) divided by
B) multiplied by
C) added to
D) subtracted from

Question 53

The derivative of a function ________ the slope of the curve defined by that function.

A) is larger than
B) equals
C) is smaller than
D) is similar to

Question 54

The Lagrange multiplier is ________ the dual variables in a linear programming problem.

A) analogous to
B) contradictory to
C) inversely related to
D) the opposite of

Question 55

A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20x² - 10C² + 40Cx + 120x - 200C + 600. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, which of these methods represents the simplest way to determine the optimal number of square feet of shelf space for the product?

A) Lagrangian functions
B) Excel Solver
C) substitution method
D) solution of simultaneous equations
E) none of the above