Deck 8: Nonlinear Programming Evolutionary Optimization

Question 1

An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the variable cell in this problem?

A) B3
B) B7
C) B9
D) B11

Answer 1

B9

Question 2

An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell G4 of the Excel spreadsheet? ​

A) =COVAR(B4:B13)
B) =VARP(B4:B13)
C) =COVAR(B4:D13,$D$4:$B$13)
D) =COVAR(B4:B13,$B$4:$B$13)

Answer 2

=COVAR(B4:B13,$B$4:$B$13)

Question 3

Which of the following is not an assumption of an EOQ problem.

A) Demand for a product is fairly constant.
B) Inventory depletion rates vary non-linearly.
C) Each new order is delivered in full when the inventory level reaches 0.
D) All are valid assumptions.

Answer 3

Inventory depletion rates vary non-linearly.

Question 4

The process of formulating an NLP problem is virtually the same as formulating an LP problem

Question 5

Nonzero reduced gradient values indicate the exact impact on the objective function value of large changes in the value of a given variable

Question 6

What is the straight line (Euclidean) distance between the points (5,7) and (1, 11)?

A) 2.83
B) 5.65
C) 8
D) 16

Question 7

This selection of the initial feasible solution, often referred to as starting point, has no impact on the determination of the optimal solution to the NLP problem

Question 8

An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell H11 of the Excel spreadsheet? ​

A) =SUMPRODUCT(B15:D15,G9:I9)
B) =SUMPRODUCT(B15:D15,B4:D13)
C) =SUMPRODUCT(B15:D15,I9)
D) =SUMPRODUCT(B15:D15,G4:I6)

Question 9

In the GRG algorithm the initial solution is called the

A) originating point.
B) insertion point.
C) zero point.
D) starting point.

Question 10

A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D9 of the spreadsheet?

A) =MIN(D4:D7)^2
B) =SUM(D4:D7)
C) =SQRT(D4^2+D5^2+D6^2+D7^2)
D) =SQRT(MMULT(B3:C3,D4:D7))

Question 11

​If Solver produces the completion message: "Solver found a solution. All constraints and optimality conditions are satisfied," a global optimal solution has been found

Question 12

Which point or points are local optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function. ​ ​

A) D
B) E
C) F
D) D, F

Question 13

​The solution procedure Solver uses to solve NLP problems is called the generalized reduced gradient (GRG) algorithm

Question 14

An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 5% return but with minimum risk. What is the objective function for the NLP formulation of this problem?

A)
B)
C)
D)

Question 15

The Analytic Solver Platform solution strategy for NLP problems is the GRG method. What does GRG stand for?

A) Graphical Residual Gradient
B) Generalized Reduced Gradient
C) Goal Restricted Gradient
D) Gradually Reduced Gradient

Question 16

When using the GRG algorithm to solve NLPs one should try multiple starting points because

A) If two different starting points return the same solution, that solution is optimal.
B) the solution returned depends upon the starting point.
C) the solution returned is always near the starting point.
D) a random element of GRG requires multiple starting points.

Question 17

​A given local optimal solution is also a global optimal solution to a NLP problem

Question 18

Exhibit 8.1
The following questions pertain to the problem and spreadsheet below.
A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200. Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
Refer to Exhibit 8.1. What formula is used in cell B9 of the spreadsheet for this problem?

A) =B3*B7+C3*C7
B) =B5*B7+C5*C7
C) =(B5-B4)*B7+(C5-C4)*B7
D) =B3*B7+C3*B7+B4*B7+C4*B7

Question 19

A NLP problem can have nonlinear objective and a nonlinear constraints

Question 20

​The main difference between shadow prices and Lagrange multipliers involves the range of RHS values over which the shadow price or Lagrange multiplier remains valid

Question 21

Exhibit 8.1
The following questions pertain to the problem and spreadsheet below.
A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200. Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
Refer to Exhibit 8.1. What formula is used in cell B7 of the spreadsheet for this problem?

A) = 30 − .07 * C3
B) = 20 − .1 * B3
C) = B3 − B4
D) = B5 * B7 + C5 * C7

Question 22

The Lagrange Multiplier is similar to which of these terms from linear programming?

A) Shadow Price
B) Allowable Increase
C) Allowable Decrease
D) Reduced Cost

Question 23

A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. Which cell(s) in the spreadsheet represent the decision variables in the problem?

A) B3
B) B3:C3
C) B4:C7
D) D9

Question 24

The optimal trade-off between risk and return for a given portfolio problem can be summarized by the

A) efficient frontier.
B) investment frontier.
C) portfolio boundary.
D) variance boundary.

Question 25

A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of product A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of product B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200. Let X₁ = demand for As and X₂ = demand for Bs. Let P₁ = price for As and P₂ = price for Bs.
The objective function for this problem is?

A) MAX 25 X₁ + 18 X₂
B) MAX (20 − 0.10*P₁) X₁ + (30 − 0.07*P₂) X₂
C) MAX 20 P₁ − 0.1 P₁² + 30 P₂ − 0.07 P₂² − 1040
D) MAX 22.5 P₁ − 0.1 P₁² + 31.26 P₂ − 0.07 P₂² − 1040

Question 26

In a maximization problem, the GRG algorithm's search for a feasible direction of improvement equates to ____ in the simplex algorithm for LP problems.

A) an adjacent corner point
B) a zero reduced cost
C) a positive shadow price
D) a positive reduced cost

Question 27

Why does the GRG algorithm not provide allowable increase or allowable decrease information with the Reduced Gradient and Lagrange multiplier information?

A) Because all constraints have slack.
B) Because the constraints may not be linear.
C) Because the solution is not necessarily a global optimum.
D) Because the solutions are not necessarily corner points.

Question 28

A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D4 of the spreadsheet?

A) =(B4-$B$3)^2+(C4-$C$3)^2
B) =SQRT((B4-$B$3)+(C4-$C$3))
C) =SQRT((B4-$B$3)^2+(C4-$C$3)^2)
D) =(B4-$B$3)+(C4-$C$3)

Question 29

What is the search path for the following feasible solution space? The dashed line represents the objective function and the objective is to maximize the value of the objective function. ​ ​

A) A, D
B) A, F, E, D
C) A, B, C, D
D) A, F, E

Question 30

In NLP a local optimum is best described as

A) an interior point.
B) a corner point.
C) a point yielding no improving direction.
D) a good starting point for subsequent searches.

Question 31

In solving the NLP problem, Solver produced a message "Solver has converged to the current solution. All constraints are satisfied." This means that:

A) Solver found a local optimal solution
B) Solver found a global optimal solution
C) the objective function changed very slowly for the last few iterations
D) Your model is degenerate

Question 32

The Lagrange multipliers can be used to

A) estimate the impact of a small change in the RHS of the constraints on the objective function
B) estimate the impact of an arbitrary change in the RHS of the constraints on the objective function
C) estimate the impact of a small change in the RHS of the constraints on the shadow prices
D) estimate the impact of a large change in the RHS of the constraints on the shadow prices

Question 33

How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are produced based on the following sensitivity report?

A) increase by 12
B) decrease by 12
C) increase by 499.99
D) decrease by 499.99

Question 34

The main difference between LP and NLP problems is that NLPs will have a

A) linear objective function and nonlinear or linear constraints.
B) minimum of one nonlinear constraint or a nonlinear objective function.
C) multilevel objective functions.
D) multilevel objective function and nonlinear constraints.

Question 35

The Reduced Gradient is similar to which of these terms from linear programming?

A) Shadow Price
B) Allowable Decrease
C) Allowable Increase
D) Reduced Cost

Question 36

An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the objective cell in this problem?

A) B3
B) B7
C) B9
D) B11

Question 37

The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be minimized, is:

A) always larger than the local optimum solution
B) always smaller than the local optimum solution
C) is smaller or equal to the local optimum solution
D) is larger or equal to the local optimum solution

Question 38

The main difference between linear (LP) and nonlinear programming problems (NLP) is that

A) NLP must have a nonlinear objective function
B) Some constraints in NLP may be nonlinear
C) No interaction terms are allowed in NLP
D) Only one constraint in NLP can be nonlinear

Question 39

In solving the NLP problem, Solver found a degenerate model. This means that:

A) Solver is cycling and redundant constraints must be removed
B) a global optimal solution was found
C) the objective function changed very slowly for the last few iterations
D) a local optimum was found

Question 40

An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formula goes in cell C7 of the spreadsheet?

A) =B7*(1+$C$3/4)^4-B7
B) =B7*(1+$C$3)-B7
C) =B7*($C$3/4)^4-B7
D) =B7*(1+$C$3/4)^4

Question 41

The GRG algorithm terminates when it

A) has completed 100 iterations.
B) has reached the global optimal solution.
C) detects no feasible direction for improvement.
D) reaches the steepest gradient.

Question 42

If the "Standard LP/Quadratic Engine" option is chosen in the Analytic Solver Platform (ASP) task pane, ASP

A) will test for non-linearity in the model and apply GRG.
B) will test for non-linearity in the model and apply Simplex.
C) uses the B & B algorithm to solve the problem.
D) will apply Simplex if the linearity test passes.

Question 43

In solving the NLP problem, Solver produced a message "Solver found a solution. All constraints and optimality conditions are satisfied." This means that Solver found:

A) a local optimal solution
B) a global optimal solution
C) the objective function changed very slowly for the last few iterations
D) a degenerate model

Question 44

The main difference between linear (LP) and nonlinear programming problems (NLP) is that

A) NLP may have a nonlinear objective function
B) All constraints in NLP must be nonlinear
C) No interaction terms are allowed in NLP
D) Only one constraint in NLP can be nonlinear

Question 45

The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be maximized, is:

A) always larger than the local optimum solution
B) always smaller than the local optimum solution
C) is smaller or equal to the local optimum solution
D) is larger or equal to the local optimum solution

Question 46

Exhibit 8.1
The following questions pertain to the problem and spreadsheet below.
A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200. Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
Refer to Exhibit 8.1. What formula is used in cell D12 of the spreadsheet for this problem?

A) =SUMPRODUCT(B3:C3, B12:C12)
B) =SUMPRODUCT(B4:C4, B12:C12)
C) =SUMPRODUCT(B5:C5, B12:C12)
D) =SUMPRODUCT(B7:C7, B12:C12)

Question 47

An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What formula goes in cell B11 in this problem?

A) =B3*B7+B9/B3*B6+B9*2+B5*B7
B) =B3*B5+B3/B9*B6+B9/2*B5*B7
C) =SQRT(B3*B5+B3/B9*B6+B9/2*B5*B7)
D) =B3*B3+B3/B9+B6*B9/2*B5*B7

Question 48

In solving the NLP problem, Solver produced a message "Solver cannot improve the current solution. All constraints are satisfied." This means that Solver found:

A) a local optimal solution
B) a global optimal solution
C) the objective function changed very slowly for the last few iterations
D) a degenerate model

Question 49

A nonzero reduced gradient value indicates the approximate impact

A) on the objective function of very small changes in the value of a given variable
B) on the RHS of a constraint of a unitary change in the objective function value
C) on the objective function of very large changes in the value of a given variable
D) on the objective function of unitary change in the RHS of associated constraint

Question 50

The total annual cost for the economic order quantity model is

A) purchase cost + ordering cost + holding cost.
B) fixed ordering cost + holding cost.
C) purchase cost + fixed ordering cost.
D) unit cost + variable ordering cost + holding cost.

Question 51

A company has collected the following inventory data for an item. What is the total annual cost for this item?

A) 275
B) 450
C) 5,275
D) 5,450

Question 52

The optimal solution to a LP problem is always at I.
A corner point.
II)
An interior point
III)
The origin.

A) I is true
B) II is true
C) III is true
D) I and III are true

Question 53

An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What constraint must be entered in the Analytic Solver Platform (ASP) task pane?

A) $D$7:$D$16 ≤ $E$7:$E$16
B) $C$7:$D$16 ≥ $E$7:$E$16
C) $C$7:$C$16 ≥ $E$7:$E$16
D) $D$7:$D$16 ≥ $E$7:$E$16

Question 54

The main difference between shadow prices and Lagrange multipliers is

A) the range of RHS values over which the Lagrange multipliers or the shadow prices remain valid
B) that shadow prices overestimate the impact of a small change in the RHS of the constraints on the objective function
C) that shadow prices underestimate the impact of a small change in the RHS of the constraints on the objective function
D) they are equivalent

Question 55

Which point or points are global optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function. ​ ​

A) B
B) D
C) E
D) F

Question 56

The GRG algorithm operates by

A) moving in the direction of most rapid improvement in the objective function.
B) choosing a search direction at random.
C) searching directly for the optimum solution.
D) moving in a clockwise direction.

Question 57

How much are additional units of Labor worth based on the following sensitivity report?

A) 0
B) 1.21
C) 2.57
D) 9.42

Question 58

An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What constraint would you impose on this problem to ensure that at least one order is placed per year?

A) MAX B9
B) B9 = 1
C) B9 ≥ 1
D) MIN B11

Question 59

NLP problems which have slack in all the constraints

A) are infeasible.
B) are only at local optimal solutions.
C) have zeros for all shadow prices.
D) are at an interior point of the feasible region.

Question 60

The GRG and Simplex algorithms are similar in that

A) each algorithm process continues until there is no further improvement in the objective function.
B) both algorithms take their starting solution from the spreadsheet.
C) both return the globally optimal solution.
D) both algorithms return a solution that satisfies at least one constraint at equality.

Question 61

An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income.
What formulas should go in cells B7:F7 of the spreadsheet for this problem?

Question 62

A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − .09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − .14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140. Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs.
Formulate the NLP for this company

Question 63

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft² of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).
The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions.
Refer to Exhibit 8.2. What formula would you place in cell D13 to calculate total cost?

Question 64

A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft² of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).
​
Formulate the NLP for the problem.

Question 65

An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income.
If the Analytic Solver Platform is used, which are the Objective, Variables and Constraint cells in the spreadsheet for this problem?

Question 66

An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk.
Let: P_i = proportion of total funds invested in i, i = A, B, C
The NLP for this problem is: What formulas should go in cells G4:J14 of the spreadsheet for this problem? NOTE: Formulas are not required in all of these cells. ​

Question 67

A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the below. Formulate the objective function for this problem. Let X and Y represent the X, Y coordinates of the new warehouse.

Question 68

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft² of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).
The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions.
Refer to Exhibit 8.2. The company wishes to have a relatively square building. Thus, they wish neither the building length nor the building width exceed the other by more than 25%. Add constraint(s) to enforce this design constraint.

Question 69

The optimal solution to a NLP problem can occur at a(n) I.
Corner point.
II)
Interior point.

A) I is true
B) II is true
C) I and II are true
D) neither I or II is true

Question 70

A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below. Formulate the NLP for this problem. Let X and Y represent the X, Y coordinates of the new warehouse.

Question 71

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft² of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).
The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions.
Refer to Exhibit 8.2. What formula would you place into cell B5 to calculate Total Floor Area?

Question 72

Calculate the annual inventory costs for the following data.

Question 73

An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item.
​
Formulate the objective function for this problem. Let Q indicate the order quantity.

Question 74

A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − 0.09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − 0.14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140. Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs.
The NLP for the problem is: What values should go in cells B3:E18 of the spreadsheet for this problem?

Question 75

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft² of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).
The following is the NLP formulation for the problem. The spreadsheet implementation of this formulation applies to the following questions.
Refer to Exhibit 8.2. What formula would you place in cell B6 to calculate Total Lot Area?

Question 76

An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item.
What values should go in cells B3:B11 of the spreadsheet for this problem if Q = 223.61?

Question 77

The straight line (Euclidean) distance between two points (X₁, Y₁) and (X₂, Y₂) is calculated as

A) X₁ − Y₁ + X₂ − Y₂
B) (X₁ − X₂)² + (Y₁ − Y₂)²
C)
D)

Question 78

The Sweet Water beverage company is designing a new soft drink can. The designers wish to minimize the manufacturing cost of the can, a cost that is directly related to the amount of aluminum used in the can. The can must hold at least 350 ml (or cm³) of beverage, have a diameter between 3 and 7 cm, and have a height between 7 and 19 cm.
​
Formulate the NLP for Sweet Water.

Question 79

How much are additional units of labor worth based on the following sensitivity report?

Question 80

A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formulas should go in cells D4:D9 of the spreadsheet for this problem?