Deck 10: Nonlinear Programming

Question 1

Maximum profit is achieved everywhere the first derivative of the profit function equals zero.

Answer 1

False

Question 2

If a nonlinear programming model consists of a single nonlinear objective function and a single linear constraint, it is called a(n) ________ optimization problem.

Answer 2

constrained or nonlinear

Question 3

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

Answer 3

False

Question 4

The first derivative of a cost function equals zero at the point V = 100. This is definitely the worst output volume for the producer to choose.

Question 5

Both linear and nonlinear programming models have the general form of an objective function subject to more than 1 constraint.

Question 6

Decision variables cannot be multiplied by each other in the objective function of a nonlinear program.

Question 7

Both linear and nonlinear programming models are examples of constrained optimization models.

Question 8

If a nonlinear program has been correctly formulated, procedures guarantee a solution.

Question 9

Constraints for nonlinear programs are usually nonlinear.

Question 10

In solving the facility location problem, the objective is to locate a centralized facility that serves customers or other facilities such that the distance traveled between the facility and customers or other facilities is minimized.

Question 11

Classical optimization is the use of calculus to determine the optimal value of a variable.

Question 12

A firm has a cost function of 3x²- 25x + 374. Without having two examples of their output volume and total cost, we cannot determine their fixed cost.

Question 13

Classical optimization uses calculus to determine the optimal values of a variable.

Question 14

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

Question 15

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

Question 16

A profit function of Z = 3x² - 12x + 5 reaches maximum profit at x = 2 units of output.

Question 17

In portfolio selection problems, risk is measured by the variance of the return on the portfolio.

Question 18

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

Question 19

An optimal solution to a nonlinear programming problem will always occur at the boundary of the feasible solution space formed by the constraint.

Question 20

In an unconstrained nonlinear programming problem, we have a single nonlinear objective function and no constraints..

Question 21

The objective of a facility location problem is to minimize ________.

Question 22

If a nonlinear programming model consists of a single nonlinear objective function and no constraints, it is called a(n) ________ optimization problem.

Question 23

The first derivative of the fixed cost line is ________.

Question 24

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the derivative of the profit function for the XYZ company? Simplify the terms as much as possible.

Question 25

Assume price and demand are related by the following function: v = 100 - 2.5p. If fixed cost = $5000 and variable cost = $10, then the expression for profit is ________.

Question 26

________, a measure of correlation between returns on investment i and returns on investment j is used to reflect risk.

Question 27

Assume price and demand are related by the following function: v = 200 - p. If fixed cost = $10,000 and variable cost = $8, then the expression for profit is ________.

Question 28

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.

-What is the nonlinear profit function for the XYZ company? Simplify the terms as much as possible.

Question 29

Assume a nonlinear programming problem with a single constraint has been solved. The value of the Lagrange multiplier is $0.75 and the value of the optimal profit (Z) is $25. If the right-hand side of the constraint is increased from 38 to 42, the new value of Z will be ________.

Question 30

The distance formula of d =
will find the ________ distance between two locations.

Question 31

The ________ measure of distance between two points on a set of X and Y coordinates is the hypotenuse of a right triangle.

Question 32

If a firm's profit is Z = 20p -2p² + 40, then the optimal value of I yields a maximum profit of ________.

Question 33

The ________ the variability in an investment portfolio, the ________ the risk of the investment portfolio.

Question 34

If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the expression for profit is ________.

Question 35

The ________ reflects the approximate change in the objective function resulting from a unit change in the quantity (right-hand-side) value of the constraint.

Question 36

If price and demand are related by the function v = 15 + 15p and the fixed cost is $150 while the variable cost is $5, then the profit at a price of 20 Rupees is ________.

Question 37

If a nonlinear programming problem results in profit (Z) of $50, and the Lagrange multiplier for a constraint is -2, the new profit will be ________ if the right-hand side of the constraint is increased by 1 unit.

Question 38

The ________ of the value of investment is a measure of risk.

Question 39

The dual value of a resource in a nonlinear programming model is given by the ________.

Question 40

If a firm's profit is Z = 100p -8p² +16, then the maximum profit occurs where p = ________.

Question 41

A custom 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. An expression for the total contribution is:

A) 20 - 4x₂ - 5 x₁
B) 60 x₁ - 5x1² + 80x₂ - 4x2²
C) 80x₁ - 5x₁²2 - 4x2²
D) 20 - (4x₂)(5x₁)

Question 42

Lush Lawns, Inc. provides a lawn fertilizer and weed control service. They are adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A budget of $2000 is to be used on this promotional campaign over the next quarter. Based on past experience in promoting its other services, Lush Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media:
s = 2x1² - 10x2² - 2_x1x2 + 18_x1 + 34x2
s.t. x₁ + x₂ = 2
Solve.

Question 43

Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. Write the new constraint.

Question 44

Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. What price should Zoey charge to maximize profit?

Question 45

Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. How will this impact the prices that she should charge to maximize profit?

Question 46

The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X1² + 2X2² - 8X₁ - 12X₂ + 34
Subject to: X₁ + 2X₂ = 4 lbs

-Determine the quantity of soap and shampoo that should be produced to maximize profit.

Question 47

If a firm's profit is Z = 12x - 6x² + 30, and their minimum production level of x is equal to 0.5, then the level of x that maximizes profit is:

A) .5
B) 1
C) 1.5
D) 2

Question 48

Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the formulation for this problem

Question 49

The slope of a curve at its highest point equals:

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

Question 50

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 - 200. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?

A) 3
B) 4
C) 5
D) 1 or 9

Question 51

Zoey's Catnip Toys faces the following relationship between price and demand: v = 2000 - 200p. The fixed cost is $500 and variable cost is $1. Write an expression for the total profit.

Question 52

Consider the curve 7x² - 14x + 28. What is the highest point on this curve?

Question 53

Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Determine the prices that Sara should charge to maximize profit.

Question 54

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, how many square feet of shelf space is required to maximize sales?

Question 55

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What price for the ball bearings will maximize the profit?

Question 56

The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X1² + 2X2² - 8X₁ - 12X₂ + 34
Subject to: X₁ + 2X₂ = 4 lbs

-Determine the profit for the optimal production quantities of soap and shampoo.

Question 57

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal production quantity?

Question 58

Consider the curve 7x² - 14x + 28. What is the slope at x = 5?

Question 59

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1000 - 8p.
What is the optimal profit?

Question 60

Consider the curve 7x² - 14x + 28. What is the second derivative at x = 10?

Question 61

Sara's Sensible Critters makes two kinds of catnip toys: balls (x₁) and mice (x₂). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for the demand constraint.

A) 2.5x₁ = x₂
B) x₁ - 2.5x₂ ? 0
C) x₁ + 2.5x₂ ? 0
D) x₁ = 2.5x₂

Question 62

The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X1² + 2X2² - 8X1 - 12X₂ + 34
Subject to: X₁ + 2X₂ = 4 lbs

-What is profit when the optimal values of soap and shampoo are produced?

A) Z = 12
B) Z = 14
C) Z = 16
D) Z = 18

Question 63

The Lagrange multiplier is

A) the shadow price for the constraint coefficients.
B) valid over a range of changes in the RHS.
C) the rate of change in the objective value as the RHS of the constraint increases.
D) the minimum threshold for decision variables to enter the solution.

Question 64

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is the optimal price of the ball bearings that will maximize the profit?

A) $47.99
B) $53.99
C) $58.99
D) $63.99

Question 65

Both linear and nonlinear programming models are examples of

A) goal programming models.
B) simplex tableaus.
C) constrained likelihood models.
D) constrained optimization models.

Question 66

Sara's Sensible Critters makes two kinds of catnip toys: balls (x₁) and mice (x₂). The relationship between demand and price for balls and mice is:
x₁ = 1800 - 150p₁
x₂ = 1500 - 300p₂
The cost for a catnip ball is $2 and for the mouse, $3.
Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce.

-Write the appropriate expression for profit.

A) Max Z = (p₁ - 3)x₁ + (p₁ - 2)_x2
B) Max Z = (p1 + ₁)x₁ + (p₁ + 3)_x2
C) Max Z = (p1 - ₁)x₁ + (p₁ - 3)_x2
D) Min Z = (p1 -₁)x₁ + (p₁ - 3)_x2

Question 67

The Salt Creek Soap Company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.
Maximize Z = X1² + 2X2² - 8X1 - 12X₂ + 34
Subject to: X₁ + 2X₂ = 4 lbs

-What quantities of soap and shampoo maximize profit?

A) X₁ = 0, X₂ = 2
B) X₁ = 1, X₂ = 2
C) X₁ = 2, X₂ = 2
D) X₁ = 2, X₂ = 1

Question 68

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 69

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

A) analogous
B) contradictory
C) inversely related
D) opposite

Question 70

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 71

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation:
V = 1000 - 8p. What is optimal profit?

A) $6668
B) $7668
C) $8668
D) $9768

Question 72

Classical optimization is the use of ________ to determine the optimal value of a variable.

A) calculus
B) linear programming
C) nonlinear programming
D) goal programming