agec test 3 (just 10 for now)

69) What is the optimal profit?

$1,240

27) 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 ________.

$28

67) What price for hit records will maximize the profit?

$44

28) 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.

$48

85) 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?

$5.50

93) Consider the curve 10x2 + 4x - 7. What is the second derivative at x = 8?

20

92) Consider the curve 10x2 + 4x - 7. What is the slope at x = 10?

204

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 profit at a price of 20 Rupees is ________.

4575 Rupees

90) 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?

5 square feet

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

6.25

75) Formulate the financial constraint for this scenario.

6X1 + 4X2 + 7X3 ≤ 15,000 where: X1 = 200 - 2.25p1 X2 = 300 - 3.00p2 X3 = 400 - 3.50p3 p1 = price of X1 p2 = price of X2 p3 = price of X3

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

90 (p = 5)

68) What is the optimal production quantity?

90 units

42) The slope of a curve at its highest point equals: A) 0. B) 1. C) 2. D) 3.

A

46) 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

A

50) The Lagrange multiplier is ________ to the dual variables in a linear programming problem. A) analogous B) contradictory C) inversely related D) opposite

A

53) It was a bumper crop for hominy this year, and The Hominy Man hoped to set a price for a case that maximized profit. The annual fixed cost for the hominy harvesting and other equipment is $10,000 and the variable cost per case is $0.50. The price is related to demand according to the following equation: v = 800 - 16p. What is optimal profit? A) -$199 B) $199 C) $808 D) $10,400

A

58) The model is entered in Excel and executes to reveal that p2 equals $51.50. Which of these conclusions is correct? A) The contribution to net profit from service X2 is $7,056.75. B) The per unit profit for service X2 is $51.50. C) There is excess demand for service X1. D) The demand for X2 is 194.

A

61) Schrute Farms has determined the following nonlinear model to determine the optimal pounds of potatoes (X1) and beets (X2) it should produce each day. Maximize Z = + 2 - 8X1 + 12X2 + 34 Subject To: X1 + 2X2 = 4 tons What quantities of potatoes and beets maximize profit? A) X1 = 0, X2 = 2 B) X1 = 1, X2 = 2 C) X1 = 2, X2 = 2 D) X1 = 2, X2 = 1

A

Zevon Enterprises Zevon Enterprises provides services for clients worldwide and to protect all parties to this course as well as Zevon, we shall refer to those services as X1, X2, and X3. Each of these services has its own special mix of needs for the resources the company has at its disposal. The X1 product requires three lawyers, seven guns, and $6,000; the X2 product requires two lawyers, five guns, and $4,000; and the X3 product requires four lawyers, six guns, and $7,000. Zevon has access to 5,000 lawyers, 10,000 guns, and $15,000,000. For ease of conversation, Zevon employees usually speak about dollars as "per thousand" so one of them asking for $7 means that they really need $7,000. Zevon's demand is variable depending on what they charge for it. For example, the X1 product's demand is 200 - 2.25p1. The demand for X2 is 300 - 3p2, and the demand for X3 is 400 - 3.5p3. The per unit profit for X1 through X3 can be calculated by subtracting the per unit cost from the sales price, so for X1, the profit is p1 - 2.25, for X2 the profit is p2 - 3, and for X3 the profit is p3 - 3.5. 54) What is an appropriate objective function for this scenario? A) Max Z = (p1 - 2.25)X1 + (p2 - 3)X2 + (p3 - 3.5)X3 B) Max Z = (p1 - 2.25p1) + (p2 - 3)(300 - 3.00p2) + (p3 - 3.5)(400 - 3.50p3) C) Max Z = 200X1 + 300X2 + 400X3 D) Max Z = 2.25(p1 - X1) + 3(p2 -X2) + 3.5(p3 - X3)

A

44) 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.

B

45) 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 - 5x1. B) 60x1 - 5x12 + 80x2 - 4x22. C) 80x1 - 5x12 + 60x2 - 4x22. D) 20 - (4x2)(5x1).

B

47) 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

B

52) It was a bumper crop for hominy this year, and The Hominy Man hoped to set a price for a case that maximized profit. The annual fixed cost for the hominy harvesting and other equipment is $10,000 and the variable cost per case is $0.50. The price is related to demand according to the following equation: v = 800 - 16p. What is the optimal price of a case of hominy that will maximize the profit? A) $16.16 B) $25.25 C) $37.37 D) $44.44

B

55) Which of these is the lawyer constraint for this scenario? A) 7X1 + 5X2 + 6X3 ≤ 10,000 B) 3X1 + 2X2 + 4X3 ≤ 5,000 C) 6X1 + 4X2 + 7X3 ≤ 15,000 D) X1 = 200 - 2.25p1

B

59) The model is entered in Excel and the sensitivity report reveals that all of the constraints' Lagrange multipliers are zero. The impact for Zevon is: A) The profit for this scenario cannot be maximized. B) Not all of the lawyers they have available will be used. C) The demand for service X1 exceeds Zevon's ability to supply it. D) The profit generated by service X1 is not a function of demand for X1.

B

43) 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

C

49) 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

C

51) 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.

C

56) Which of these is the money constraint for this scenario? A) 7X1 + 5X2 + 6X3 ≤ 10,000 B) 3X1 + 2X2 + 4X3 ≤ 5,000 C) 6X1 + 4X2 + 7X3 ≤ 15,000 D) X1 = 200 - 2.25p1

C

60) The analytics gurus at Zevon realize that they had misformulated their demand curves. They now believe that demand for X1 is given by 1000 - 2.25p1, demand for X2 is given by 2000 - 3p2, and demand for X3 is given by 3000 - 3.5p3. This model is entered in Excel and the sensitivity report contains the following: (better chart in the document) Constraints Cell Name Final Value Lagrange Multiplier $F$8 Lawyers 5000.00 101.052 $F$9 Guns 9275.74 0 $F$10 Money 9213.49 0 What is the best conclusion from the list below? A) The profit for this scenario cannot be maximized. B) Not all of the lawyers they have available will be used. C) If Zevon can retain the services of another lawyer for less than $101, they should do so. D) Lawyer jokes aside, Zevon cannot benefit from hiring additional lawyers at any cost.

C

62) What is profit when the optimal values of potatoes and beets are produced? A) Z = 22 B) Z = 44 C) Z = 66 D) Z = 88

C

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. 63) Write the appropriate expression for profit. A) Max Z = (p1 - 3)x1 + (p2 - 2)x2 B) Max Z = (p1 + 2)x1 + (p2 + 3)x2 C) Max Z = (p1 - 2)x1 + (p2 - 3)x2 D) Min Z = (p1 - 2)x1 + (p2 - 3)x2

C

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

Covariance

48) Both linear and nonlinear programming models are examples of: A) goal programming models. B) simplex tableaus. C) constrained likelihood models. D) constrained optimization models.

D

57) The model is entered in Excel and executes to reveal that p1 equals $45.57. Which of these conclusions is correct? A) The per unit profit for X1 is $45.57. B) The contribution to net profit from service X1 is $4,441.59. C) There is excess demand for service X1. D) The demand for X1 is 97.

D

64) 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

D

1) Nonlinear programming has the same format as linear programming, however either the objective function or the constraints (but not both) are nonlinear functions.

FALSE

10) 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.

FALSE

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

FALSE

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

FALSE

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

FALSE

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

FALSE

20) Constraints for nonlinear programs are usually nonlinear.

FALSE

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

FALSE

8) 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.

FALSE

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

FALSE

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

Lagrange multiplier

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

Lagrange multiplier

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. 86) Write the formulation for this problem

Max Z = (p1 - 2)x1 + (p2 - 3)x2 s.t. 0.10x1 + 0.25x2 ≤ 200 Alternatively, Z = 2100p1 - 150p12 - 2400p2 - 300p22 - 8100

Zevon Enterprises Zevon Enterprises provides services for clients worldwide and to protect all parties to this course as well as Zevon, we shall refer to those services as X1, X2, and X3. Each of these services has its own special mix of needs for the resources the company has at its disposal. The X1 product requires three lawyers, seven guns, and $6,000; the X2 product requires two lawyers, five guns, and $4,000; and the X3 product requires four lawyers, six guns, and $7,000. Zevon has access to 5,000 lawyers, 10,000 guns, and $15,000,000. For ease of conversation, Zevon employees usually speak about dollars as "per thousand" so one of them asking for $7 means that they really need $7,000. Zevon's demand is variable depending on what they charge for it. For example, the X1 product's demand is 200 - 2.25p1. The demand for X2 is 300 - 3p2, and the demand for X3 is 400 - 3.5p3. The per unit profit for X1 through X3 can be calculated by subtracting the per unit cost from the sales price, so for X1, the profit is p1 - 2.25, for X2 the profit is p2 - 3, and for X3 the profit is p3 - 3.5. 73) Formulate an appropriate objective function for this scenario.

Max Z = (p1 - 2.25)X1 + (p2 - 3)X2 + (p3 - 3.5)X3

76) Formulate the objective function and constraints for this scenario.

Max Z = (p1 - 2.25)X1 + (p2 - 3)X2 + (p3 - 3.5)X3 subject to: 3X1 + 2X2 + 4X3 ≤ 5,000 7X1 + 5X2 + 6X3 ≤ 10,000 6X1 + 4X2 + 7X3 ≤ 15,000 where: X1 = 200 - 2.25p1 X2 = 300 - 3.00p2 X3 = 400 - 3.50p3 p1 = price of X1 p2 = price of X2 p3 = price of X3

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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE

2) Nonlinear programming algorithms occasionally have difficulty distinguishing between local optima and the global optimum.

TRUE

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

TRUE

22) 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.

TRUE

3) The highest point on each peak of a surface can be considered a local optimum, but the highest point among all of the peaks is the only global optimum.

TRUE

4) The slope of a curve at any point is equal to the derivative of the curve's function.

TRUE

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

TRUE

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

TRUE

74) Formulate the lawyer constraint for this scenario.

The lawyer constraint is as follows: 3X1 + 2X2 + 4X3 ≤ 5,000 where: X1 = 200 - 2.25p1 X2 = 300 - 3.00p2 X3 = 400 - 3.50p3

77) The model was entered into an Excel spreadsheet and the table below shows part of the sensitivity report. Provide an interpretation. (better chart in the document) Constraints Cell Name Final Value Lagrange Multiplier $F$8 Lawyers 1358.906273 0 $F$9 Guns 2573.031361 0 $F$10 Money 2523.937562 0

The model provides a solution that calls for only 1358.9 lawyers, 2573.03 guns and 2523.9 thousands of dollars, far below the amount on hand for this endeavor. The Lagrange Multipliers are all zero, which reflects the lack of urgency in acquiring more lawyers, guns, and money. Since Zevon is not using all that they already have, there is no benefit to acquiring any more of these resources.

79) The model was entered into an Excel spreadsheet and the table below shows part of the answer report. Provide an interpretation.(better chart in the document) Constraints Cell Name Cell Value Formula Status Slack $F$8 Lawyers 1358.906273 $F$8<=$I$8 Not Binding 3641.093727 $F$9 Guns 2573.031361 $F$9<=$I$9 Not Binding 7426.968639 $F$10 Money 2523.937562 $F$10<=$I$10 Not Binding 12476.06244

The model provides a solution that calls for only 1359 lawyers out of the 5000 available, meaning Zevon has 3641 lawyers that are not assigned to this model. Similarly, only 2573 guns and $2,523,973.56 are needed out of the 10,000 guns and $15,000,000 available to them.

78) The model was entered into an Excel spreadsheet and the table below shows part of the sensitivity report. Calculate the expected per unit profit for the three services. (better chart in the document) Variable Cells Cell Name Final Value Reduced Gradient $C$3 P1_ 45.5694 0 $D$3 P2_ 51.5 0 $E$3 P3_ 58.8929 0

The model provides a solution that calls for only a sale price for P1 of $45.57 - coupled with its price of $2.25 means they make $43.32 per unit of X1. For item X2, the optimal price is $51.50, less the cost of $3, means Zevon makes $48.50 per unit. Finally, X3 will sell for $58.89, less the price of $3.50 means they realize a profit of $55.39 per unit.

Mad Over Donuts An entrepreneurial resident of the Oklahoma City metropolitan area is interested in securing a new franchise for Mad Over Donuts. Ideally this franchise would be centrally located so delivery could be economically handled and all citizens could enjoy fresh, delicious donuts delivered to the doorstep. The main cities and anticipated demand (in thousands per day) are shown in the table. (better chart in the document) City x-coord y-coord Demand Jones 6 28 45 Luther 13 35 56 Harrah 12 22 30 Edmond 0 32 25 Norman 2 0 33 Moore 3 8 22 82) What is the appropriate objective function for this scenario?

The objective function is: (he has weird equations for z and d and they won't copy and paste correctly so you'll have to look at the document for that) Min Z = where d = and ti = the number of trips (demand)

81) The analytics gurus at Zevon realize that they had misformulated their demand curves. They now believe that demand for X1 is given by 1000 - 2.25p1, demand for X2 is given by 2000 - 3p2, and demand for X3 is given by 3000 - 3.5p3. This model is entered in Excel and the sensitivity report contains the following: (better chart in the document) Constraints Cell Name Final Value Lagrange Multiplier $F$8 Lawyers 5000.00 101.052 $F$9 Guns 9275.74 0 $F$10 Money 9213.49 0 Provide an interpretation of all elements.

The only resource that Zevon could use more of is lawyers; the final value is 5000, which exhausts their entire supply. If additional lawyers could be retained for less than $101.05, then Zevon should pursue this possibility. There is no information available as to the range for the validity of this $101.05. The guns resource has only 725 units remaining and the money resource has $5,800 left. When one of those two resources is exhausted, the lawyer Lagrange multiplier will likely not be valid.

80) The model was entered into an Excel spreadsheet and the table below shows the answer report in its entirety. Show how the profit is calculated. (better chart in the document) Objective Cell (Max) Cell Name Original Value Final Value $F$4 Profit $5,638.13 $22,018.33 Objective Cell (Max) Cell Name Original Value Final Value Integer $C$3 P1_ $10.00 $45.57 Contin $D$3 P2_ $10.00 $51.50 Contin $E$3 P3_ $10.00 $58.89 Contin Constraints Cell Name Cell Value Formula Status Slack $F$8 Lawyers 1358.91 $F$8<=$I$8 Not Binding 3641.09 $F$9 Guns 2573.03 $F$9<=$I$9 Not Binding 7426.97 $F$10 Money 2523.94 $F$10<=$I$10 Not Binding 12476.06

The profit is a function of quantity sold and price, with the quantity sold a function of price. The X1 product has a demand of 200 - 2.25 × 45.57 = 97 The X2 product has a demand of 300 - 3 × 51.50 = 146 The X3 product has a demand of 400 - 3.5 × 58.89 = 194 The profit per X1 is $45.57 - 2.25 = $43.32 The profit per X2 is $51.50 - 3 = $48.50 The profit per X3 is $58.89 - 3.5 = $55.39 So 97 × $43.32 + 146 × $48.50 + 194 × 55.39 = $22,018.33

83) What are the appropriate constraints for this scenario?

There are no constraints for this model. It is a nonlinear unconstrained optimization problem.

89) 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?

There is no impact. Prices remain the same. Note: The optimal quantities are 750 balls and 300 mice for a profit of $4050.

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

X1 = 0, X2 = 2

Joe Jackson runs the ABC123 manufacturing company that produces hit records. The annual fixed cost is $2,000 and the variable cost per recording $8. The price is related to demand according to the following equation: 200 - 2.5p. 65) What is the nonlinear profit function for the ABC123 company? Simplify the terms as much as possible.

Z = -2.5p2 + 220p - 3,600

32) 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 ________.

Z = 125p - 2.5p2 - 6000

33) 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 ________.

Z = 15 p2 - 60p - 225

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

Z = 18

31) 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 ________.

Z = 208p - p2 - 11,600

84) 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.

Z = 2200p - 200p2 - 2500

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

constrained or nonlinear

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

higher, higher OR lower, lower

87) Determine the prices that Sara should charge to maximize profit.

p1 = 7 p2 = 4

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

straight line

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

straight line (direct, Euclidian)

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

the total distance traveled

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

unconstrained

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

variance

91) Consider the curve 10x2 + 4x - 7. What is the lowest point on this curve?

x = -0.2

72) 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 - 2x1x2 + 18x1 + 34x2 s.t. x1 + x2 = 2 Solve.

x1 = 1.66; x2 = 0.33, Lagrange multiplier = 24

88) 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.

x1 = 2.5x2 OR x1 -2.5x2 = 0

41) The first derivative of the fixed cost line is ________.

zero

66) What is the derivative of the profit function for the XYZ company? Simplify the terms as much as possible.

∂ Z/∂ p = -5p + 220