Deck 8: Nonlinear Programming and Evolutionary Optimization

Question 1

The optimal solution to a LP problem is always at I.a cormer 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

Answer 1

I is true

Question 2

The optimal solution to a NLP problem can occur at a(n) I.cormer paint.
II.interior point

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

Answer 2

I and II are true

Question 3

The Risk Solver Platform (RSP) 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

Answer 3

B

Question 4

The GRG algorithm terminates when it

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

Question 5

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 6

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 7

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 8

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 9

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

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 10

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 11

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 12

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) $$\sqrt { \left( X _ { 1 } - X _ { 2 } \right) + \left( Y _ { 1 } - Y _ { 2 } \right) }$$
D) $$\sqrt { \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } + \left( Y _ { 1 } - Y _ { 2 } \right) ^ { 2 } }$$

Question 13

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 14

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 15

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 16

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 17

In the GRG algorithm the initial solution is called the

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

Question 18

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 19

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 20

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 21

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 22

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.
$\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 3 & 2 & 150 \\\text { Material (ounces/unit) } & 1 & 2 & 200 \\\text { Manufacturing cost }(\$ / \text { unit) } & 25 & 18 & \\\text { Demand (units) } & 20-0.10^{*} P_{1}& 30-0.07^{*} P_{2} &\end{array}$
Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
$\begin{array}{|c|l|c|c|c|c|} \hline& \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & & & & \\\hline 2 & & \mathrm{~A} & \mathrm{~B} & & \\\hline 3 & \text { Price } & 112.50 & 223.29 & & \\\hline 4 & \text { Marginal Cost } & 25.00 & 18.00 & & \\\hline 5 & \text { Profit Margin } & 87.50 & 205.29 & & \\\hline 6 & & & & & \\\hline 7 & \text { Demand } & 8.75 & 14.37 & & \\\hline 8 & & & & & \\\hline 9 & \text { Total Profit } & 3715.58 & & & \\\hline 10 & & & & & \\\hline 11 & \text { Constraints: } & &&\text { Used } & \text { Available }\\\hline 12 & \text { Labor } & 3.00 & 2.00 & 54.99 & 150.00 \\\hline 13 & \text { Material } & 1.00 & 2.00 & 37.49 & 200.00 \\\hline\end{array}$


-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 23

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? $$\begin{array} { | c | c | c | } \hline & \text { A } & \mathrm { B } \\\hline 1 & & \\\hline 2 & & \\\hline 3 & \text { Annual Demand: } & 20,000 \\\hline 4 & & \\\hline 5 & \text { Cost per Unit: } & \$ 25 \\\hline 6 & \text { Ordering Cost: } & \$ 30 \\\hline 7 & \text { Carrying Cust: } & 25 \% \\\hline 8 & & \\\hline 9 & \text { Order Quantity: } & 483.73 \\\hline 10 & & \\\hline 11 & \text { Total Cost: } & \$ 502,738.61 \\\hline\end{array}$$

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 24

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)

Question 25

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? $\begin{array}{|c|c|c|} \hline& \text { A } & \text { B } \\ \hline 1 & & \\\hline 2 & & \\\hline 3 & \text { Annual Demand: } & 20,000 \\\hline 4 & & \\\hline 5 & \text { Cost per Unit: } & \$ 25 \\\hline 6 & \text { Ordering Cost: } & \$ 30 \\\hline 7 & \text { Carrying Cost: } & 25 \% \\\hline 8 & & \\\hline 9 & \text { Order Quantity: } & 483.73 \\\hline 10 & & \\\hline 11 & \text { Total Cost: } & \$ 502,738.61 \\\hline\end{array}$

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

Question 26

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 27

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? $\begin{array}{|c|c|c|} \hline& \text { A } & \text { B } \\\hline 1 & & \\\hline 2 & & \\\hline 3 & \text { Annual Demand: } & 20,000 \\\hline 4 & & \\\hline 5 & \text { Cost per Unit: } & \$ 25 \\\hline 6 & \text { Ordering Cost: } & \$ 30 \\\hline 7 & \text { Carrying Cost: } & 25 \% \\\hline 8 & & \\\hline 9 & \text { Order Quantity: } & 483.73 \\\hline 10 & & \\\hline 11 & \text { Total Cost: } & \$ 502,738.61 \\\hline \end{array}$

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

Question 28

A company has collected the following inventory data for an item. What is the total annual cost for this item? $\begin{array}{llcc} \text { Annual demand for the item } & \text {D=500}\\ \text { Unit purchase cost for the item } &\text {C=10}\\ \text { Fixed cost of placing an order } &\text {S=20}\\ \text { Cost of holding one unit in inventory for a year } &\text {i=30\%}\\ \text { Order quantity } &\text {Q=50}\\\end{array}$

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

Question 29

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 30

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.
$\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 3 & 2 & 150 \\\text { Material (ounces/unit) } & 1 & 2 & 200 \\\text { Manufacturing cost }(\$ / \text { unit) } & 25 & 18 & \\\text { Demand (units) } & 20-0.10^{*} P_{1}& 30-0.07^{*} P_{2} &\end{array}$
Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
$\begin{array}{|c|l|c|c|c|c|} \hline& \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & & & & \\\hline 2 & & \mathrm{~A} & \mathrm{~B} & & \\\hline 3 & \text { Price } & 112.50 & 223.29 & & \\\hline 4 & \text { Marginal Cost } & 25.00 & 18.00 & & \\\hline 5 & \text { Profit Margin } & 87.50 & 205.29 & & \\\hline 6 & & & & & \\\hline 7 & \text { Demand } & 8.75 & 14.37 & & \\\hline 8 & & & & & \\\hline 9 & \text { Total Profit } & 3715.58 & & & \\\hline 10 & & & & & \\\hline 11 & \text { Constraints: } & &&\text { Used } & \text { Available }\\\hline 12 & \text { Labor } & 3.00 & 2.00 & 54.99 & 150.00 \\\hline 13 & \text { Material } & 1.00 & 2.00 & 37.49 & 200.00 \\\hline\end{array}$


-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 31

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 32

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.

Question 33

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 34

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 35

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 Risk Solver Platform (RSP) 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 36

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. $\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 3 & 2 & 150 \\\text { Material (ounces/unit) } & 1 & 2 & 200 \\\text { Manufacturing cost }(\$ / \text { unit) } & 25 & 18 & \\\text { Demand (units) } & 20-0.10^{*} P_{1}& 30-0.07^{*} P_{2} &\end{array}$
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 37

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 38

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? $$\begin{array} { | c | c | c | } \hline & \text { A } & \mathrm { B } \\\hline 1 & & \\\hline 2 & & \\\hline 3 & \text { Annual Demand: } & 20,000 \\\hline 4 & & \\\hline 5 & \text { Cost per Unit: } & \$ 25 \\\hline 6 & \text { Ordering Cost: } & \$ 30 \\\hline 7 & \text { Carrying Cost: } & 25 \% \\\hline 8 & & \\\hline 9 & \text { Order Quantity: } & 483.73 \\\hline 10 & & \\\hline 11 & \text { Total Cost: } & \$ 502,738.61 \\\hline\end{array}$$

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

Question 39

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 40

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.
$\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 3 & 2 & 150 \\\text { Material (ounces/unit) } & 1 & 2 & 200 \\\text { Manufacturing cost }(\$ / \text { unit) } & 25 & 18 & \\\text { Demand (units) } & 20-0.10^{*} P_{1}& 30-0.07^{*} P_{2} &\end{array}$
Let X₁ = demand for As and X₂ = demand for Bs.
Let P₁ = price for As and P₂ = price for Bs
$\begin{array}{|c|l|c|c|c|c|} \hline& \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & & & & \\\hline 2 & & \mathrm{~A} & \mathrm{~B} & & \\\hline 3 & \text { Price } & 112.50 & 223.29 & & \\\hline 4 & \text { Marginal Cost } & 25.00 & 18.00 & & \\\hline 5 & \text { Profit Margin } & 87.50 & 205.29 & & \\\hline 6 & & & & & \\\hline 7 & \text { Demand } & 8.75 & 14.37 & & \\\hline 8 & & & & & \\\hline 9 & \text { Total Profit } & 3715.58 & & & \\\hline 10 & & & & & \\\hline 11 & \text { Constraints: } & &&\text { Used } & \text { Available }\\\hline 12 & \text { Labor } & 3.00 & 2.00 & 54.99 & 150.00 \\\hline 13 & \text { Material } & 1.00 & 2.00 & 37.49 & 200.00 \\\hline\end{array}$


-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 41

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.
$\begin{array}{lllc}\text { Product } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor (hr/unit) } & 2 & 4 & 150 \\\text { Material (ounces/unit) } & 2 & 8 & 220 \\\text { Manufacturing cost(\$/unit) } & 30 & 20 & \\\text { Demand (units) } & 50-0.09 * P_{1} & 30-0.14 * P_{2} &\end{array}$

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:
$\text {MAX}$ : $52.70 P _ { 1 } - 0.09 P _ { 1 } ^ { 2 } + 32.80 P _ { 2 } - 0.14 P _ { 2 } ^ { 2 } - 2100$
$\text {Subject to:}$
$\begin{array}{l}\mathrm{X}_{1}-50+0.09 \mathrm{P}_{1}=0 \\\mathrm{X}_{2}-30+0.14 \mathrm{P}_{2}=0 \\2 \mathrm{X}_{1}+4 \mathrm{X}_{2} \leq 150 \\2 \mathrm{X}_{1}+8 \mathrm{X}_{2} \leq 220 \\90 \leq \mathrm{P}_{1}, \mathrm{P}_{2} \leq 140 \\\mathrm{X}_{1}, \mathrm{X}_{2} \geq 0 \\\text { and the solution }\left(\mathrm{P}_{1}, \mathrm{P}_{2}\right)=(140.0,117.14)\end{array}$ What values should go in cells B3:E18 of the spreadsheet for this problem?
$\begin{array}{|c|l|c|c|c|c|}\hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & & & & \\\hline 2 & & \text { A } & \text { B } & & \\\hline 3 & \text { Price } & & & & \\\hline 4 & \text { Min Price } & & & & \\\hline 5 & \text { Max Price } & & & & \\\hline 6 & \text { Marginal Cost } & & & & \\\hline 7 & \text { Profit Margin } & & & & \\\hline 8 & & & & & \\\hline 9 & \text { Demand } & & & & \\\hline 10 & & & & & \\\hline 11 & \text { Total Profit } & & & & \\\hline 12 & & & & & \\\hline 13 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 14 & \text { Labor } & & & & \\\hline 15 & \text { Material } & & & & \\\hline \end{array}$ .

Question 42

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?

Question 43

How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are produced based on the following sensitivity report? $\text { Changing Cells }$
$\begin{array}{llrr}&&\text { Final}& \text { Reduced }\\ \text { Cell } & \text { Name } & \text { Value } & \text { Gradient } \\\hline \$ \mathrm{~B} \$ 4 & \text { Corn } & 9.52 & 0 \\\$ \mathrm{C} \$ 4 & \text { Pumplsin } & 0 & -499.99 \\\$ \mathrm{D} \$ 4 & \text { Beans } & 10.79 & 0\end{array}$


$\text { Constraints }$
$\begin{array}{llrr}&&\text { Final}& \text { Reduced }\\ \text { Cell } & \text { Name } & \text { Value } & \text { Multiplier } \\\hline \$ E \$ 8 & \text { Corn } & 200000 & 0.016 \\\$ E \$ 9 & \text { Pumplkin } & 99 & 12 \\\$ E \$ 10 & \text { Beans } & 37777.78 & 0 \\\$ E \$ 11 & \text { Water } & 29.84 & 0 \\\$ E \$ 12 & \text { Fertilizer } & 8000 & 3.49\end{array}$

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

Question 44

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 45

Find the maximum solution on this graph of a function starting from X = 12. Mark its location on the graph.

Question 46

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 47

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.
Let X and Y represent the X, Y coordinates of the new warehouse. The NLP for this problem and solution is the following.

Question 48

Calculate the annual inventory costs for the following data.

Question 49

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 50

How much must the objective function coefficient of the variable X₂ increase before any X₂s are produced based on the following sensitivity report?

Question 51

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.
The average return for the stocks is:
The covariance matrix for the stocks is:
Let: P_i = proportion of total funds invested in i, i = A, B, C
Formulate the NLP for this problem.

Question 52

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 53

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 54

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 55

How much are additional units of Labor worth based on the following sensitivity report? $\text { Changing Cells }$

$\begin{array}{llrr}&& \text { Final}& \text { Rediuced}\\\text { Cell } & \text { Name } & \text { Value } & \text { Gradient } \\\hline \$ B \$ 4 & \text { Number to make: } \mathrm{X}_{1} & 9.42 & 0 \\\$ \mathrm{C} \$ 4 & \text { Number to make: } \mathrm{X}_{2} & 1.71 & 0\end{array}$


$\text { Constraints }$
$\begin{array}{llrr}& & \text {Final}&\text {Lagrange}\\\text { Cell } & \text { Name } & \text { Value } & \text { Multiplier } \\\hline \$ \mathrm{D} \$ 8 & \text { Wood } & 42 & 0 \\\$ \mathrm{D} \$ 9 & \text { Labor } & 132 & 1.21 \\\$ \mathrm{D} \$ 10 & \text { Plywood } & 24 & 2.57\end{array}$

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

Question 56

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 57

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 Risk Solver Platform (RSP) is used, which are the Objective, Variables and Constraint cells in the spreadsheet for this problem?

Question 58

How many local minimum solutions are there on this graph of a function Mark their locations on the graph.

Question 59

How many local maximum solutions are there on this graph of a function? Mark their locations on the graph.

Question 60

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 61

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 x 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.
$\begin{array}{|c|l|c|c|c|c|} \hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \text { Length } & \text { Width } & \text { Stories } & \text { Parking } \\ \hline 2 & & 108.39492 & 92.255243 & 6 & 80,000 \\\hline 3 & \text { Max Values } & 300 & 300 & 6 & \\\hline 4 & & & & & \\\hline 5 & \text { Total Floor Area } & 60,000 & \geq & 60,000 & \\\hline 6 & \text { Total Lot Area } & 90,000 & = & 90,000 & \\\hline 7 & \text { Min. Parking Area } & 65,000 & \geq & 0 & \\\hline 8 & \text { Max. Bld Height } & 60 & \leq & 65 & \\\hline 9 & & & & & \\\hline 10 & \text { Cost } & & & & \\\hline 11 & \text { Height Factor } & 0.6 & \text { Total Height } & 60 & \\\hline 12 & \text { Area Factor } & 0.01 & \text { Floor Area } & 10,000 & \\\hline 13 & & & \text { Cost } & \$ 136.0 & \text { millions } \$ \\\hline\end{array}$


-Refer to Exhibit 8.2. What formula would you place in cell B6 to calculate Total Lot Area?

Question 62

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 x 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.
$\begin{array}{|c|l|c|c|c|c|} \hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \text { Length } & \text { Width } & \text { Stories } & \text { Parking } \\ \hline 2 & & 108.39492 & 92.255243 & 6 & 80,000 \\\hline 3 & \text { Max Values } & 300 & 300 & 6 & \\\hline 4 & & & & & \\\hline 5 & \text { Total Floor Area } & 60,000 & \geq & 60,000 & \\\hline 6 & \text { Total Lot Area } & 90,000 & = & 90,000 & \\\hline 7 & \text { Min. Parking Area } & 65,000 & \geq & 0 & \\\hline 8 & \text { Max. Bld Height } & 60 & \leq & 65 & \\\hline 9 & & & & & \\\hline 10 & \text { Cost } & & & & \\\hline 11 & \text { Height Factor } & 0.6 & \text { Total Height } & 60 & \\\hline 12 & \text { Area Factor } & 0.01 & \text { Floor Area } & 10,000 & \\\hline 13 & & & \text { Cost } & \$ 136.0 & \text { millions } \$ \\\hline\end{array}$


-Refer to Exhibit 8.2. What values would you enter in the Risk Solver Platform (RSP) task pane for the above Excel spreadsheet?
Objective Cell:
Variables Cells:
Constraints Cells:

Question 63

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

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

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 x 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.
$\begin{array}{|c|l|c|c|c|c|} \hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \text { Length } & \text { Width } & \text { Stories } & \text { Parking } \\ \hline 2 & & 108.39492 & 92.255243 & 6 & 80,000 \\\hline 3 & \text { Max Values } & 300 & 300 & 6 & \\\hline 4 & & & & & \\\hline 5 & \text { Total Floor Area } & 60,000 & \geq & 60,000 & \\\hline 6 & \text { Total Lot Area } & 90,000 & = & 90,000 & \\\hline 7 & \text { Min. Parking Area } & 65,000 & \geq & 0 & \\\hline 8 & \text { Max. Bld Height } & 60 & \leq & 65 & \\\hline 9 & & & & & \\\hline 10 & \text { Cost } & & & & \\\hline 11 & \text { Height Factor } & 0.6 & \text { Total Height } & 60 & \\\hline 12 & \text { Area Factor } & 0.01 & \text { Floor Area } & 10,000 & \\\hline 13 & & & \text { Cost } & \$ 136.0 & \text { millions } \$ \\\hline\end{array}$


-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 66

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 x 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.
$\begin{array}{|c|l|c|c|c|c|} \hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \text { Length } & \text { Width } & \text { Stories } & \text { Parking } \\ \hline 2 & & 108.39492 & 92.255243 & 6 & 80,000 \\\hline 3 & \text { Max Values } & 300 & 300 & 6 & \\\hline 4 & & & & & \\\hline 5 & \text { Total Floor Area } & 60,000 & \geq & 60,000 & \\\hline 6 & \text { Total Lot Area } & 90,000 & = & 90,000 & \\\hline 7 & \text { Min. Parking Area } & 65,000 & \geq & 0 & \\\hline 8 & \text { Max. Bld Height } & 60 & \leq & 65 & \\\hline 9 & & & & & \\\hline 10 & \text { Cost } & & & & \\\hline 11 & \text { Height Factor } & 0.6 & \text { Total Height } & 60 & \\\hline 12 & \text { Area Factor } & 0.01 & \text { Floor Area } & 10,000 & \\\hline 13 & & & \text { Cost } & \$ 136.0 & \text { millions } \$ \\\hline\end{array}$


-Refer to Exhibit 8.2. What formula would you place into cell B5 to calculate Total Floor Area?

Question 67

Exhibit 8.2
The following questions pertain to the problem and spreadsheet below.
A construction company just purchased a 300 x 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.
$\begin{array}{|c|l|c|c|c|c|} \hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \text { Length } & \text { Width } & \text { Stories } & \text { Parking } \\ \hline 2 & & 108.39492 & 92.255243 & 6 & 80,000 \\\hline 3 & \text { Max Values } & 300 & 300 & 6 & \\\hline 4 & & & & & \\\hline 5 & \text { Total Floor Area } & 60,000 & \geq & 60,000 & \\\hline 6 & \text { Total Lot Area } & 90,000 & = & 90,000 & \\\hline 7 & \text { Min. Parking Area } & 65,000 & \geq & 0 & \\\hline 8 & \text { Max. Bld Height } & 60 & \leq & 65 & \\\hline 9 & & & & & \\\hline 10 & \text { Cost } & & & & \\\hline 11 & \text { Height Factor } & 0.6 & \text { Total Height } & 60 & \\\hline 12 & \text { Area Factor } & 0.01 & \text { Floor Area } & 10,000 & \\\hline 13 & & & \text { Cost } & \$ 136.0 & \text { millions } \$ \\\hline\end{array}$


-Refer to Exhibit 8.2. What formula would you place in cell D13 to calculate total cost?

Question 68

Project 8.1 - Truck Company Expansion
Kornfield Trucking handles private and commercial moves. They currently own 500 moving vans and employ 2000 full-time workers. Their trucks are used to pick-up and deliver office and household goods throughout the Eastern and Southeastern states. Kornfield mans each truck with three workers. This allows driver swaps providing increased miles-covered-per-hour ratio while staying within safety requirements for individual driving time. A 3-person crew also reduces company reliance on local help. Local distributors and warehouses provide a pool of laborers for loading and unloading the moving trucks. While in the past this arrangement has worked well, the arrangement has soured recently as the temporary workers have demanded higher wages while produced less work. Despite their pay and benefits package, Kornfield still finds that the nature of the work (lifting and time on the road) makes for a high rate of turnover. Thus, Kornfield maintains an excess of workers/drivers, but no more than 3.75 workers per truck at any time.
The following information is available on the trucks in the Kornfield inventory and their options for new purchases.
$$\begin{array}{l}\begin{array} { l c c c c c } & \begin{array} { c } \text { Purchase } \\p _ { \text {rice } }\end{array} & \begin{array} { c } \text { Salvage } \\\text { Value }\end{array} & \begin{array} { c } \text { Yearly } \\\text { Maintenance }\end{array} & \begin{array} { c } \text { Miles Per } \\\text { Year }\end{array} & \begin{array} { c } \text { Miles per } \\\text { Gallon" }\end{array} \\\hline \text { New Truck } & \$ 55,000 & \cdots & 750 & 100,000 & 12.5 \mathrm { mp } \\\text { Old Truck } & \cdots & \$ 25,000 & \$ 1500 & 100,000 & 10 \mathrm { mpg } \\\hline\end{array}\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text {*assume \$1.50/gal }\end{array}$$ The following information applies to Kornfield personnel actions.
$\begin{array}{lccc}&\text { Hiring Cost}&\text { Firing Cost }&\text { Salary }\\\hline \text { New Worker } & \$ 1000 & -- & \$ 28,000 \\\text { Old Worker } & -- & \$ 1500 & \$ 35,000\end{array}$ Kornfield has an operating budget of $ 75M next year and wants to expand their operations. As a part of this expansion, they are considering options for their truck fleet and may purchase new trucks, sell old trucks to salvage or some combination of the two. Any salvage money received is rolled into the operating budget. Kornfield is also considering possible changes to their work force. Their current force has a fairly high average salary and their operating budget is not greatly affected by releasing current employees. On the other hand, newer employees carry a much lower average salary and do not tax the operating budget heavily in hiring and training costs.
The Cobb-Douglas production function is used to model the number of vehicle miles driven per year. This function represents the quantity Kornfield management would like to maximize as they expand their operations. The general form of the Cobb-Douglas production function is the following:
$$y = a x _ { 1 } ^ { b } x _ { 2 } ^ { c } x _ { 3 } ^ { d }$$ where y is the output, each X_i represents an input and the letters represent constants. This function generalizes to fewer or a greater number of parameters than the three depicted above. The constants for the Kornfield Trucking production function are (a, b, c, d) = (9.1, 0.05, 0.40, 0.50).
Formulate the Kornfield Trucking problem as a non-linear programming problem. Implement the problem in Excel and use Risk Solver Platform (RSP) generalized reduced gradient (GRG) routing to obtain a solution to the problem. What is a recommended solution for Kornfield Trucking?