Deck 3: Modeling and Solving Lp Problems in a Spreadsheet

Question 1

Which type of spreadsheet cell represents the left hand sides LHS)formulas in an LP model?

A)Target or set cell
B)Changing variable cell
C)Constraint cell
D)Constant cell

Answer 1

C

Question 2

In the Analytic Solver Platform dialog box simple upper and lower bounds for decision variables are specified by

A)referring directly to the decision variable cells in the Constraints-Bound area.
B)requiring the addition of the bounds above and below the variable cells.
C)resolving the problem with the bounds added.
D)incorporating the bounds in the objective function.

Answer 2

A

Question 3

An LP problem with a feasible region will have

A)an optimal solution at some interior point.
B)an optimal solution at some extreme point.
C)an optimal solution only at the origin.
D)an optimal solution at two interior points.

Answer 3

B

Question 4

What does the Excel "=SUMPRODUCTA1:A5,C6;C10)" function do?

A)Sums each range and multiplies the sums.
B)Sum each pair of cells and multiples each sum.
C)Multiplies the contents of cells containing the =SUM)command.
D)Multiplies each pair of cells in two arrays matched by position and sums the products.

Question 5

Models which are setup in an intuitively appealing,logical layout tend to be the most

A)Reliable
B)Modifiable
C)Auditable
D)Organized

Question 6

The "Analyze Without Solving" tool in Analytic Solver Platform is useful for

A)verifying the equations in a spreadsheet model.
B)toggling between absolute and relative cell referencing.
C)executing the Excel spreadsheet layout Wizard.
D)naming cells and cell ranges for easier modifiability.

Question 7

The "Objective Value of" option in the Analytic Solver Platform task pane may be used to

A)find a solution at a maximum value.
B)find a solution at a minimum value.
C)find a solution for a specific objective function value.
D)returns the best feasible solution.

Question 8

What action is required to make Analytic Solver Platform ASP)solve a specified problem?

A)Type go in cell A1.
B)Click the "Optimize" button on the ASP Ribbon,or the green arrow "Solve" in the Task Pane.
C)Click the Close button in the ASP Parameters dialog box.
D)Click the Guess button in the ASP Parameters dialog box.

Question 9

Numeric constants should be

A)embedded in formulas.
B)placed in individual cells
C)placed in separate workbooks.
D)entered manually every time a model is solved.

Question 10

The constraints X₁ ≥ 0 and X₂ ≥ 0 are referred to as

A)positivity constraints.
B)optimality conditions.
C)left hand sides.
D)nonnegativity conditions.

Question 11

Which tab in the Analytic Solver Platform task pane is used to define an optimization problem?

A)Guess
B)Model
C)Change
D)Delete

Question 12

Which type of spreadsheet cell represents the decision variables in an LP model?

A)Target or set cell
B)Variable cell
C)Constraint cell
D)Constant cell

Question 13

Problems which have only integer solutions are called

A)discrete programming problems
B)integer programming problems
C)discrete programming problems
D)infeasible programming problems

Question 14

Microsoft Excel contains a built-in optimizer called

A)what-if engines.
B)calculators.
C)solvers.
D)risk analyzers.

Question 15

What function is used to add the contents of cells A1,A2 and A3?

A)=A1+A2+A3
B)=ADDA1:A3)
C)=TOTALA1:A3)
D)=PRODUCTA1:A3)

Question 16

Which type of spreadsheet cell represents the objective function in an LP model?

A)Objective cell
B)Changing variable cell
C)Constraint cell
D)Constant cell

Question 17

The built-in Solver in Excel is found under which tab on the ribbon?

A)Tools
B)Insert
C)Data
D)Window

Question 18

Spreadsheet modeling is an acquired skill because

A)there is generally only one correct way to build a model.
B)the spreadsheet is free-form providing many modeling options.
C)using Analytic Solver Platform requires lots of experience.
D)spreadsheets are not very easy to use.

Question 19

The "Objective Sense" option in the Analytic Solver Platform task pane may be used to

A)return a heuristic solution to the problem.
B)tell the Solver what value it should seek for your optimization objective.
C)determine the value of the objective based on specified decision variable cells.
D)always works correctly.

Question 20

Which function is equivalent to =SUMPRODUCTA1:A3,B1:B3)?

A)=SUMPRODUCTA1:A3,B1:B3))
B)=PRODUCTSUMA1:A3,B1:B3))
C)=PRODUCTA1+B1,A2+B2,A3+B3))
D)=A1*B1+A2*B2+A3*B3

Question 21

Data Envelopment Analysis DEA)is an LP-based methodology in which weighted sums of inputs and outputs are calculated and

A)the constraints capture the maximum effectiveness of each unit.
B)the objective is to maximize every units output.
C)the constraints ensure the sum of the weighted outputs is one.
D)the objective for each unit is to maximize the weighted sum of its outputs.

Question 22

Exhibit 3.2
The following questions are based on this problem and accompanying Excel windows.
The Byte computer company produces two models of computers,Plain and Fancy.It wants to plan how many computers to produce next month to maximize profits.Producing these computers requires wiring,assembly and inspection time.Each computer produces a certain level of profits but faces only a limited demand.There are also a limited number of wiring,assembly and inspection hours available in each month.The data for this problem is summarized in the following table.
$\begin{array}{cccccc} & & \text { Maximum } & & \text { Assembly } & \text { Inspection } \\\text { Computer } & \text { Profit per } & \text { demand for } & \text { Wiring Hours } & \text { Hours } & \text { Hours } \\\text { Model } & \text { Model } \$ \text { ) } & \text { product } & \text { Required } & \text { Required } & \text { Required } \\\hline \text { Plain } & 30 & 80 & .4 & .5 & .2 \\\text { Fancy } & 40 & 90 & .5 & .4 & .3 \\\hline&&\text {hours avaible}&50&50&22\\\hline\end{array}$ Let ^X₁ ^{= Number of Plain computers to produce} X₂ = Number of Fancy computers to produce
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{.4 X}₁ ^{+ .5 X}₂ ^{? 50 wiring hours)}
.5 X₁ + .4 X₂ ? 50 assembly hours)
.2 X₁ + .2 X₂ ? 22 inspection hours)X₁ ? 80 Plain computers demand)X₂ ? 90 Fancy computers demand)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \begin{array}{c}\text { Byte } \\\text { Computer } \\\text { Company }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \text { Plain } & \text { Fancy } & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & \text { Wiring } & 0.4 & 0.5 & & 50 \\\hline 9 & \text { Assembly } & 0.5 & 0.4 & & 50 \\\hline 10 & \text { Inspection } & 0.2 & 0.3 & & 22 \\\hline 11 & \text { Plain Demand } & 1 & & & 80 \\\hline 12 & \text { Fancy Demand } & & 1 & & 90 \\\hline\end{array}$

-Refer to Exhibit 3.2.Which of the following statements will represent the constraint for just assembly hours?

A)B4:C4 ? B5:C5
B)D9 ? E9
C)D8:D10 ? E8:E10
D)E8:E10 ? D8:D10

Question 23

Using Data Envelopment Analysis DEA)for an inefficient unit,a more efficient composite unit can be found by

A)Solving its DEA problem and retrieving the weights from the answer report.
B)Solving its DEA problem and examining those units whose final value is non-zero.
C)Solving its DEA problem and using the resulting shadow prices as composite weights.
D)Solving its DEA problem and using the positive resulting shadow prices as composite weights.

Question 24

Exhibit 3.3
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons.One of his customers has requested a special blend of scotch targeted as a bar scotch.The customer wants the blend to involve two scotch products,call them A and B.Product A is a higher quality scotch while product B is a cheaper brand.The customer wants to make the claim the blend is closer to high quality than the alternative.The customer wants 50 1500 ml bottles of the blend.Each bottle must contain at least 48% of Product A and at least 500 ml of B.The customer also specified that the blend have an alcohol content of at least 85%.Product A contains 95% alcohol while product B contains 78%.The blend is sold for $12.50 per bottle.Product A costs $7 per liter and product B costs $3 per liter.The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let ^X₁ ^{= Number of liters of product A in total blend delivered} X₂ = Number of liters of product B in total blend delivered
MIN: ^{7 X}₁ ^{+ 3 X}₂
Subject to: ^X₁ ^{+ X}₂ ^{= 1.5 * 50 Total liters of mix)}X₁ ? 0.48 * 1.5 * 50 X₁ minimum)
X₂ ? 0.5 * 50 X₂ minimum)
.0.95 X₁ + 0.78 X₂ ? 0.85 * 1.5 * 50 85% alcohol minimum)X₁,X₂ ? 0
$$\begin{array} { | l | l | c | c | c | c |} \hline & \text { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \\\hline 1 & & \begin{array} { c } \text { Jacks' } \\\text { Distill } \\\text { ery }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm {~A} & \mathrm {~B} & & \\\hline 4 & \text { Liters to use } & & & & \text { Total Cost: } \\\hline 5 & \text { Unit cost: } & 10.5 & 4 & & \\\hline 6 & & 5 & & \\\hline 7 & \text { Constraints: } & & & \text { Supplied } & \text { Requirement } \\\hline \mathbf { 8 } & \text { Total Liters } & 1 & 1 & & 75 \\\hline 9 & \text { A required } & 1 & & & 36 \\\hline 1 & \text { B required } & & 1 & & 25 \\\hline 0 & & & & & \\\hline 1 & \text { 85\% alcohal } & 0.95 & 0.78 & & 63.7 \\1 & & & & & \\\hline\end{array}$$

-Refer to Exhibit 3.3.What formula should be entered in cell D11 in the accompanying Excel spreadsheet to compute the total liters of alcohol supplied?

A)=B4*B5+C4*C5
B)=SUMPRODUCTB11:C11,$B$4:$C$4)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 25

Which of the following describes Data Envelopment Analysis DEA).

A)DEA finds the most effective company among some set of companies.
B)DEA determines if a company is converting inputs to outputs as effectively as possible.
C)DEA determines how effective a company converts inputs to outputs compared to other companies.
D)DEA compares how effective a company converts inputs to outputs compared to a benchmark composite of all companies.

Question 26

A heuristic solution is

A)used by Analytic Solver Platform ASP)when the Guess button is used.
B)guaranteed to produce an optimal solution.
C)used by Analytic Solver Platform ASP)if Standard GRG Nonlinear method is selected.
D)a rule-of-thumb for making decisions.

Question 27

Exhibit 3.2
The following questions are based on this problem and accompanying Excel windows.
The Byte computer company produces two models of computers,Plain and Fancy.It wants to plan how many computers to produce next month to maximize profits.Producing these computers requires wiring,assembly and inspection time.Each computer produces a certain level of profits but faces only a limited demand.There are also a limited number of wiring,assembly and inspection hours available in each month.The data for this problem is summarized in the following table.
$\begin{array}{cccccc} & & \text { Maximum } & & \text { Assembly } & \text { Inspection } \\\text { Computer } & \text { Profit per } & \text { demand for } & \text { Wiring Hours } & \text { Hours } & \text { Hours } \\\text { Model } & \text { Model } \$ \text { ) } & \text { product } & \text { Required } & \text { Required } & \text { Required } \\\hline \text { Plain } & 30 & 80 & .4 & .5 & .2 \\\text { Fancy } & 40 & 90 & .5 & .4 & .3 \\\hline&&\text {hours avaible}&50&50&22\\\hline\end{array}$ Let ^X₁ ^{= Number of Plain computers to produce} X₂ = Number of Fancy computers to produce
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{.4 X}₁ ^{+ .5 X}₂ ^{? 50 wiring hours)}
.5 X₁ + .4 X₂ ? 50 assembly hours)
.2 X₁ + .2 X₂ ? 22 inspection hours)X₁ ? 80 Plain computers demand)X₂ ? 90 Fancy computers demand)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \begin{array}{c}\text { Byte } \\\text { Computer } \\\text { Company }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \text { Plain } & \text { Fancy } & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & \text { Wiring } & 0.4 & 0.5 & & 50 \\\hline 9 & \text { Assembly } & 0.5 & 0.4 & & 50 \\\hline 10 & \text { Inspection } & 0.2 & 0.3 & & 22 \\\hline 11 & \text { Plain Demand } & 1 & & & 80 \\\hline 12 & \text { Fancy Demand } & & 1 & & 90 \\\hline\end{array}$

-Refer to Exhibit 3.2.Which cells should be changing cells in this problem?

A)B4:C4
B)E5
C)D8:D10
D)E8:E10

Question 28

How many decision variables are there in a transportation problem which has 5 supply points and 4 demand points?

A)4
B)5
C)9
D)20

Question 29

Exhibit 3.2
The following questions are based on this problem and accompanying Excel windows.
The Byte computer company produces two models of computers,Plain and Fancy.It wants to plan how many computers to produce next month to maximize profits.Producing these computers requires wiring,assembly and inspection time.Each computer produces a certain level of profits but faces only a limited demand.There are also a limited number of wiring,assembly and inspection hours available in each month.The data for this problem is summarized in the following table.
$\begin{array}{cccccc} & & \text { Maximum } & & \text { Assembly } & \text { Inspection } \\\text { Computer } & \text { Profit per } & \text { demand for } & \text { Wiring Hours } & \text { Hours } & \text { Hours } \\\text { Model } & \text { Model } \$ \text { ) } & \text { product } & \text { Required } & \text { Required } & \text { Required } \\\hline \text { Plain } & 30 & 80 & .4 & .5 & .2 \\\text { Fancy } & 40 & 90 & .5 & .4 & .3 \\\hline&&\text {hours avaible}&50&50&22\\\hline\end{array}$ Let ^X₁ ^{= Number of Plain computers to produce} X₂ = Number of Fancy computers to produce
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{.4 X}₁ ^{+ .5 X}₂ ^{? 50 wiring hours)}
.5 X₁ + .4 X₂ ? 50 assembly hours)
.2 X₁ + .2 X₂ ? 22 inspection hours)X₁ ? 80 Plain computers demand)X₂ ? 90 Fancy computers demand)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \begin{array}{c}\text { Byte } \\\text { Computer } \\\text { Company }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \text { Plain } & \text { Fancy } & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & \text { Wiring } & 0.4 & 0.5 & & 50 \\\hline 9 & \text { Assembly } & 0.5 & 0.4 & & 50 \\\hline 10 & \text { Inspection } & 0.2 & 0.3 & & 22 \\\hline 11 & \text { Plain Demand } & 1 & & & 80 \\\hline 12 & \text { Fancy Demand } & & 1 & & 90 \\\hline\end{array}$

-Refer to Exhibit 3.2.What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of wiring used?

A)=B4*B5+C4*C5
B)=SUMPRODUCTB8:C8,$B$4:$C$4)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 30

Exhibit 3.3
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons.One of his customers has requested a special blend of scotch targeted as a bar scotch.The customer wants the blend to involve two scotch products,call them A and B.Product A is a higher quality scotch while product B is a cheaper brand.The customer wants to make the claim the blend is closer to high quality than the alternative.The customer wants 50 1500 ml bottles of the blend.Each bottle must contain at least 48% of Product A and at least 500 ml of B.The customer also specified that the blend have an alcohol content of at least 85%.Product A contains 95% alcohol while product B contains 78%.The blend is sold for $12.50 per bottle.Product A costs $7 per liter and product B costs $3 per liter.The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let ^X₁ ^{= Number of liters of product A in total blend delivered} X₂ = Number of liters of product B in total blend delivered
MIN: ^{7 X}₁ ^{+ 3 X}₂
Subject to: ^X₁ ^{+ X}₂ ^{= 1.5 * 50 Total liters of mix)}X₁ ? 0.48 * 1.5 * 50 X₁ minimum)
X₂ ? 0.5 * 50 X₂ minimum)
.0.95 X₁ + 0.78 X₂ ? 0.85 * 1.5 * 50 85% alcohol minimum)X₁,X₂ ? 0
$$\begin{array} { | l | l | c | c | c | c |} \hline & \text { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \\\hline 1 & & \begin{array} { c } \text { Jacks' } \\\text { Distill } \\\text { ery }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm {~A} & \mathrm {~B} & & \\\hline 4 & \text { Liters to use } & & & & \text { Total Cost: } \\\hline 5 & \text { Unit cost: } & 10.5 & 4 & & \\\hline 6 & & 5 & & \\\hline 7 & \text { Constraints: } & & & \text { Supplied } & \text { Requirement } \\\hline \mathbf { 8 } & \text { Total Liters } & 1 & 1 & & 75 \\\hline 9 & \text { A required } & 1 & & & 36 \\\hline 1 & \text { B required } & & 1 & & 25 \\\hline 0 & & & & & \\\hline 1 & \text { 85\% alcohal } & 0.95 & 0.78 & & 63.7 \\1 & & & & & \\\hline\end{array}$$

-Refer to Exhibit 3.3.What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total cost?

A)=B4*C4+B5*C5
B)=SUMPRODUCTB4:C4,B5:C5)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 31

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{? 36 carpentry)}
4 X₁ + 8 X₂ ? 40 varnishing)X₂ ? 8 demand for desks)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & \text { Jones } & & & \\\text { Furnit } & & & & \\\hline 2 & & \text { ure } & & & \\\hline 3 & & & & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \begin{array}{c}\text { Use } \\\mathrm{d}\end{array} & \text { Available } \\\hline 8 & \text { Carpentry } & 6 & 4 & & 36 \\\hline 9 & \text { Varnishing } & 4 & 8 & & 40 \\\hline 10 & \text { Desk demand } & & 1 & & 8\\\hline\end{array}$

-Refer to Exhibit 3.1.What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total profit?

A)=B4*B5+C4*C5
B)=SUMPRODUCTB8:C8,$B$4:$C$4)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 32

Exhibit 3.2
The following questions are based on this problem and accompanying Excel windows.
The Byte computer company produces two models of computers,Plain and Fancy.It wants to plan how many computers to produce next month to maximize profits.Producing these computers requires wiring,assembly and inspection time.Each computer produces a certain level of profits but faces only a limited demand.There are also a limited number of wiring,assembly and inspection hours available in each month.The data for this problem is summarized in the following table.
$\begin{array}{cccccc} & & \text { Maximum } & & \text { Assembly } & \text { Inspection } \\\text { Computer } & \text { Profit per } & \text { demand for } & \text { Wiring Hours } & \text { Hours } & \text { Hours } \\\text { Model } & \text { Model } \$ \text { ) } & \text { product } & \text { Required } & \text { Required } & \text { Required } \\\hline \text { Plain } & 30 & 80 & .4 & .5 & .2 \\\text { Fancy } & 40 & 90 & .5 & .4 & .3 \\\hline&&\text {hours avaible}&50&50&22\\\hline\end{array}$ Let ^X₁ ^{= Number of Plain computers to produce} X₂ = Number of Fancy computers to produce
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{.4 X}₁ ^{+ .5 X}₂ ^{? 50 wiring hours)}
.5 X₁ + .4 X₂ ? 50 assembly hours)
.2 X₁ + .2 X₂ ? 22 inspection hours)X₁ ? 80 Plain computers demand)X₂ ? 90 Fancy computers demand)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \begin{array}{c}\text { Byte } \\\text { Computer } \\\text { Company }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \text { Plain } & \text { Fancy } & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & \text { Wiring } & 0.4 & 0.5 & & 50 \\\hline 9 & \text { Assembly } & 0.5 & 0.4 & & 50 \\\hline 10 & \text { Inspection } & 0.2 & 0.3 & & 22 \\\hline 11 & \text { Plain Demand } & 1 & & & 80 \\\hline 12 & \text { Fancy Demand } & & 1 & & 90 \\\hline\end{array}$

-Refer to Exhibit 3.2.Which cells should be the constraint cells in this problem?

A)B4:C4
B)E5
C)D8:D12
D)E8:E12

Question 33

Exhibit 3.2
The following questions are based on this problem and accompanying Excel windows.
The Byte computer company produces two models of computers,Plain and Fancy.It wants to plan how many computers to produce next month to maximize profits.Producing these computers requires wiring,assembly and inspection time.Each computer produces a certain level of profits but faces only a limited demand.There are also a limited number of wiring,assembly and inspection hours available in each month.The data for this problem is summarized in the following table.
$\begin{array}{cccccc} & & \text { Maximum } & & \text { Assembly } & \text { Inspection } \\\text { Computer } & \text { Profit per } & \text { demand for } & \text { Wiring Hours } & \text { Hours } & \text { Hours } \\\text { Model } & \text { Model } \$ \text { ) } & \text { product } & \text { Required } & \text { Required } & \text { Required } \\\hline \text { Plain } & 30 & 80 & .4 & .5 & .2 \\\text { Fancy } & 40 & 90 & .5 & .4 & .3 \\\hline&&\text {hours avaible}&50&50&22\\\hline\end{array}$ Let ^X₁ ^{= Number of Plain computers to produce} X₂ = Number of Fancy computers to produce
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{.4 X}₁ ^{+ .5 X}₂ ^{? 50 wiring hours)}
.5 X₁ + .4 X₂ ? 50 assembly hours)
.2 X₁ + .2 X₂ ? 22 inspection hours)X₁ ? 80 Plain computers demand)X₂ ? 90 Fancy computers demand)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & & \begin{array}{c}\text { Byte } \\\text { Computer } \\\text { Company }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \text { Plain } & \text { Fancy } & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & \text { Wiring } & 0.4 & 0.5 & & 50 \\\hline 9 & \text { Assembly } & 0.5 & 0.4 & & 50 \\\hline 10 & \text { Inspection } & 0.2 & 0.3 & & 22 \\\hline 11 & \text { Plain Demand } & 1 & & & 80 \\\hline 12 & \text { Fancy Demand } & & 1 & & 90 \\\hline\end{array}$

-Refer to Exhibit 3.2.What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total profit?

A)=B4*C4+B5*C5
B)=SUMPRODUCTB4:C4,B5:C5)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 34

Scaling problems

A)can cause Analytic Solver Platform to consider a linear problem as nonlinear.
B)can cause problems in accuracy of solutions returned.
C)are caused by small numbers and large numbers used in the same problem.
D)all of these.

Question 35

How many constraints are there in a transportation problem which has 5 supply points and 4 demand points? ignore the non-negativity constraints)

A)4
B)5
C)9
D)20

Question 36

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{? 36 carpentry)}
4 X₁ + 8 X₂ ? 40 varnishing)X₂ ? 8 demand for desks)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & \text { Jones } & & & \\\text { Furnit } & & & & \\\hline 2 & & \text { ure } & & & \\\hline 3 & & & & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \begin{array}{c}\text { Use } \\\mathrm{d}\end{array} & \text { Available } \\\hline 8 & \text { Carpentry } & 6 & 4 & & 36 \\\hline 9 & \text { Varnishing } & 4 & 8 & & 40 \\\hline 10 & \text { Desk demand } & & 1 & & 8\\\hline\end{array}$

-Refer to Exhibit 3.1.Which cells should be changing cells in this problem?

A)B4:C4
B)E5
C)D8:D10
D)E8:E10

Question 37

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{? 36 carpentry)}
4 X₁ + 8 X₂ ? 40 varnishing)X₂ ? 8 demand for desks)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & \text { Jones } & & & \\\text { Furnit } & & & & \\\hline 2 & & \text { ure } & & & \\\hline 3 & & & & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \begin{array}{c}\text { Use } \\\mathrm{d}\end{array} & \text { Available } \\\hline 8 & \text { Carpentry } & 6 & 4 & & 36 \\\hline 9 & \text { Varnishing } & 4 & 8 & & 40 \\\hline 10 & \text { Desk demand } & & 1 & & 8\\\hline\end{array}$

-Refer to Exhibit 3.1.Which of the following statements represent the carpentry,varnishing and limited demand for desks constraints?

A)B4:C4 ? B5:C5
B)E5 ? 0
C)D8:D10 ? E8:E10
D)E8:E10 ? D8:D10

Question 38

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{? 36 carpentry)}
4 X₁ + 8 X₂ ? 40 varnishing)X₂ ? 8 demand for desks)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & \text { Jones } & & & \\\text { Furnit } & & & & \\\hline 2 & & \text { ure } & & & \\\hline 3 & & & & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \begin{array}{c}\text { Use } \\\mathrm{d}\end{array} & \text { Available } \\\hline 8 & \text { Carpentry } & 6 & 4 & & 36 \\\hline 9 & \text { Varnishing } & 4 & 8 & & 40 \\\hline 10 & \text { Desk demand } & & 1 & & 8\\\hline\end{array}$

-Refer to Exhibit 3.1.What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of carpentry used?

A)=B4*B5+C4*C5
B)=SUMPRODUCTB8:C8,$B$4:$C$4)
C)=SUMB5:C5)
D)=SUME8:E10)

Question 39

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{? 36 carpentry)}
4 X₁ + 8 X₂ ? 40 varnishing)X₂ ? 8 demand for desks)X₁,X₂ ? 0
$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & \text { Jones } & & & \\\text { Furnit } & & & & \\\hline 2 & & \text { ure } & & & \\\hline 3 & & & & & \\\hline 4 & \text { Number to make: } & & & & \text { Total Profit: } \\\hline 5 & \text { Unit profit: } & 30 & 40 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \begin{array}{c}\text { Use } \\\mathrm{d}\end{array} & \text { Available } \\\hline 8 & \text { Carpentry } & 6 & 4 & & 36 \\\hline 9 & \text { Varnishing } & 4 & 8 & & 40 \\\hline 10 & \text { Desk demand } & & 1 & & 8\\\hline\end{array}$

-Refer to Exhibit 3.1.Which cells should be the constraint cells in this problem?

A)B4:C4
B)E5
C)D8:D10
D)E8:E10

Question 40

What is the significance of an absolute cell reference in Excel?

A)The cell reference will not change if the formula containing the reference is copied to another location
B)The cell will always contain the absolute value of any number entered into it
C)The cell reference changes if the formula containing the reference is copied to another location
D)It is the only formula used to refer to a cell on another spreadsheet

Question 41

A hospital needs to determine how many nurses to hire to cover a 24 hour period.The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts.They are paid different wages depending on when they start their shifts.The number of nurses required per 4-hour time period and their wages are shown in the following table.
$\begin{array}{cc}\text { Time period } &\text { Required \# of Nurses }& \text {Wage \$ / h r ) }\\\hline12 \mathrm{am}-4 \mathrm{am} & 20 & 15 \\4 \mathrm{am}-8 \mathrm{am} & 30 & 16 \\8 \mathrm{am}-12 \mathrm{pm} & 40 & 13 \\12 \mathrm{pm}-4 \mathrm{pm} & 50 & 13 \\4 \mathrm{pm}-8 \mathrm{pm} & 40 & 14 \\8 \mathrm{pm}-12 \mathrm{am} & 30 & 15\end{array}$

What values would you enter in the Risk Solver Platform RSP)task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem?
Objective Cell: Variables Cells: Constraints Cells:

Question 42

State Farm Supply has just received an order for 10,000 pounds of chicken feed.The farmer has specified certain that the feed meet minimum requirements for Protein,Carbohydrate,Fat and Vitamins.State Farm can blend four different feeds to produce the required mix.The farmer would like to pay the lowest possible price for the feed.The data for the problem is summarized in the following table.
State Farm Supply
Percent of Nutrient in: Minimum
$$\begin{array} { l c c c c c } \text { Nutrient } & \text { Feed } 1 & \text { Feed 2 } & \text { Feed 3 } & \text { Feed 4 } & \text { Req'd Ant } \\\hline \text { Pratein } & 15 & 20 & 30 & 15 & 18 \\\text { Carbahydirate } & 20 & 10 & 10 & 15 & 12 \\\text { Fat } & 20 & 30 & 15 & 20 & 20 \\\text { Vitamin } & 1 & 1.50 & 0.75 & 0.50 & 1 \\\hline \text { Cast'1,000 lbs } & \$ 500 & \$ 600 & \$ 550 & \$ 450 &\end{array}$$
Formulate the LP for this problem.

Question 43

Exhibit 3.3
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons.One of his customers has requested a special blend of scotch targeted as a bar scotch.The customer wants the blend to involve two scotch products,call them A and B.Product A is a higher quality scotch while product B is a cheaper brand.The customer wants to make the claim the blend is closer to high quality than the alternative.The customer wants 50 1500 ml bottles of the blend.Each bottle must contain at least 48% of Product A and at least 500 ml of B.The customer also specified that the blend have an alcohol content of at least 85%.Product A contains 95% alcohol while product B contains 78%.The blend is sold for $12.50 per bottle.Product A costs $7 per liter and product B costs $3 per liter.The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let ^X₁ ^{= Number of liters of product A in total blend delivered} X₂ = Number of liters of product B in total blend delivered
MIN: ^{7 X}₁ ^{+ 3 X}₂
Subject to: ^X₁ ^{+ X}₂ ^{= 1.5 * 50 Total liters of mix)}X₁ ? 0.48 * 1.5 * 50 X₁ minimum)
X₂ ? 0.5 * 50 X₂ minimum)
.0.95 X₁ + 0.78 X₂ ? 0.85 * 1.5 * 50 85% alcohol minimum)X₁,X₂ ? 0
$$\begin{array} { | l | l | c | c | c | c |} \hline & \text { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \\\hline 1 & & \begin{array} { c } \text { Jacks' } \\\text { Distill } \\\text { ery }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm {~A} & \mathrm {~B} & & \\\hline 4 & \text { Liters to use } & & & & \text { Total Cost: } \\\hline 5 & \text { Unit cost: } & 10.5 & 4 & & \\\hline 6 & & 5 & & \\\hline 7 & \text { Constraints: } & & & \text { Supplied } & \text { Requirement } \\\hline \mathbf { 8 } & \text { Total Liters } & 1 & 1 & & 75 \\\hline 9 & \text { A required } & 1 & & & 36 \\\hline 1 & \text { B required } & & 1 & & 25 \\\hline 0 & & & & & \\\hline 1 & \text { 85\% alcohal } & 0.95 & 0.78 & & 63.7 \\1 & & & & & \\\hline\end{array}$$

-Refer to Exhibit 3.3.Which cells should be changing cells in this problem?

A)B4:C4
B)E5
C)D8:D10
D)E8:E10

Question 44

You have been given the following linear programming model and Excel spreadsheet to solve this problem.What cell references would you enter in the Risk Solver Platform RSP)task pane for the following?
Objective Cell:
Variables Cells:
Constraints Cells:

$\begin{array}{|l|l|l|l|l|l|} \hline& \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm{X}_{1} & \mathrm{X}_{2} & & \\\hline 4 & \text { Number to make: } & & & & \begin{array}{l}\text { OBJ. FN. } \\\text { VALUE }\end{array} \\\hline 5 & \text { Unit profit: } & 1 & 9 & & \\&&2\\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & 1 & 9 & 10.5 & & \begin{array}{c}12 \\6\\\end{array} \\\hline 9 & 2 & 1 & 0 & & 5 \\\hline 10 & 3 & 0 & 1 & & 6 \\\hline\end{array}$

Question 45

Exhibit 3.5
The following questions are based on this problem and accompanying Excel windows.
A company is planning production for the next 4 quarters.They want to minimize the cost of production.The production cost is stable but demand and production capacity vary from quarter to quarter.The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand.Quarterly
inventory holding cost is 3% of the cost of production.The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month.There are currently 5,000 units in inventory.The company wants to produce at no less than one half of its maximum capacity in any quarter.
$$\begin{array}{l}\begin{array} { l r r r r r } &&\text { Quarter }\\& { 1 } & 2 & { 3 } & { 4 } \\\hline \text { Unit Production Cost } & \$ 300 & \$ 300 & \$ 300 & \mathbf { \$ 3 0 0 } \\\text { Units Demanded } & 2,000 & 9,000 & 12,000 & 11,000 \\\text { Marimum Production } & 8,000 & 7,000 & 8,000 & 9,000\end{array}\end{array}$$ Let ^P_i ^{= number of units produced in quarter i,i = 1,... ,4} B_i = beginning inventory for quarter i
MIN: ^{300 P}₁ ^{+ 300 P}₂ ^{+ 300 P}₃ ^{+ 300 P}₄ ⁺
9B₁ + B₂)/2 + 9B₂ + B₃)/2 + 9B₃ + B₄)/2 + 9B₄ + B₅)/2
Subject to: ^{4000 ? P}₁ ^{? 8000}
3500 ? P₂ ? 7000
4000 ? P₃ ? 8000
4500 ? P₄ ? 9000
3000 ? B₁ + P₁ ? 2000 ? 12000
3000 ? B₂ + P₂ ? 9000 ? 12000
3000 ? B₃ + P₃ ? 12000 ? 12000
3000 ? B₄ + P₄ ? 11000 ? 12000 B₂ = B₁ + P₁ ? 2000
B₃ = B₂ + P₂ ? 9000
B₄ = B₃ + P₃ ? 12000
B₅ = B₄ + P₄ ? 11000
P_i,B_i ? 0
$\begin{array}{|c|l|c|c|c|c|c|}\hline&A&B&C&D&E&F\\\hline 1&&&\text { Quarter }\\\hline 2 & & & 1 & 2 & 3 & 4 \\\hline 3 & \text { Beginning Inventory } & & 5,000 & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} \\\hline 4 & \text { Units Produced } & & 8,000 & \begin{array}{c}7,000 \\\end{array} & \begin{array}{c}8,000 \\\end{array} & \begin{array}{c}9,000 \\\end{array} \\\hline 5 & \text { Units Demanded } & & 2,000 & \begin{array}{c}9,000 \\\end{array} & 12,000 & 11,000 \\\hline 6 & \text { Ending Inventory } & & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} & \begin{array}{c}3,000 \\\end{array} \\\hline 7\\\hline 8 & \text { Minimum Production } & & 4,000 & 3,500 & 4,000 & 4,500 \\\hline 9 & \text { Maximum Production } & & 8,000 & 7,000 & 8,000 & 9,000 \\\hline10\\\hline11& \text { Minimum Inventory } && 3,000 & 3,000 & 3,000 & 3,000 \\\hline\end{array}$ $\begin{array}{|c|l|c|c|c|c|c|}\hline12 & \text { Maximum Inventory } & & 12,000 & 12,000 & 12,000 & 12,000 \\\hline 13 & & & & & & \\\hline 14 & \text { Unit Production Cost } & & \$ 300 & \$ 300 & \$ 300 & \$ 300\\\hline 15 & \text { Unit Carrying Cost } & 3.00 \% & \$ 9.00 & \$ 9.00 & \$ 9.00 & \$ 9.00 \\\hline16\\\hline 17 & \text { Quarterly Production Cost } & & \$ 2,400,000 & \$ 2,100,000 & \$ 2,400,000 & \$ 2,700,000 \\\hline 18 & \text { Quarterly Carrying Cost } & & \$ 72,000 & \$ 90,000 & \$ 63,000 & \$ 36,000 \\\hline 19 & & & & & & \\\hline 20 & & & & & \text { Total Cost } & \$ 9,861,000 \\\hline\end{array}$

-Refer to Exhibit 3.5.Which cells are changing cells in the accompanying Excel spreadsheet?

A)C4:F4
B)C9:F9
C)F20
D)C12:F12

Question 46

A company needs to purchase several new machines to meet its future production needs.It can purchase three different types of machines A,B,and C.Each machine A costs $80,000 and requires 2,000 square feet of floor space.Each machine B costs $50,000 and requires 3,000 square feet of floor space.Each machine C costs $40,000 and requires 5,000 square feet of floor space.The machines can produce 200,250 and 350 units per day respectively.The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines.The company wants to buy as many machines as possible to maximize daily production.
What values would you enter in the Risk Solver Platform RSP)task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem?
Objective Cell: Variables Cells: Constraints Cells:

Question 47

You have been given the following linear programming model and Excel spreadsheet to solve this problem.What cell references would you enter in the Risk Solver Platform RSP)task pane for the following?
Objective Cell:
Variables Cells:
Constraints Cells:


$\begin{array}{|l|l|c|c|c|c|} \hline& \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm{X}_{1} & \mathrm{X}_{2} & & \\\hline 4 & \text { Number to make: } & & & & \begin{array}{l}\text { OBJ. FN. } \\\text { VALUE }\end{array} \\\hline 5 & \text { Unit profit: } & 8 & 5 & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & 1 & 3 & 5 & & 54 \\\hline 9&2 & 1 & 1 & & 14 \\&& 1 & 0 && 4 \\\hline 10 & 3 & 1 & 0 & & 12 \\\hline\end{array}$

Question 48

A farmer is planning his spring planting.He has 20 acres on which he can plant a combination of Corn,Pumpkins and Beans.He wants to maximize his profit but there is a limited demand for each crop.Each crop also requires fertilizer and irrigation water which are in short supply.There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available.The following table summarizes the data for the problem.

Formulate the LP for this problem.

Question 49

You have been given the following linear programming model and Excel spreadsheet to solve this problem.What cell references would you enter in the Risk Solver Platform RSP)task pane for the following?

$\begin{array}{|l|l|l|l|l|l|} \hline& \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} \\\hline 1 & & & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm{x}_{1} & \mathrm{X}_{2} & & \\\hline 4 & \text { Number to make: } & & & & \begin{array}{l}\text { OBJ. FN. } \\\text { VALUE }\end{array} \\\hline 5 & \text { Unit profit: } & & & & \\\hline 6 & & & & & \\\hline 7 & \text { Constraints: } & & & \text { Used } & \text { Available } \\\hline 8 & 1 & & 1 & & 8 \\\hline 9 & 2 & 8 & 5 & & 80 \\\hline 10 & 3 & 3 & 5 & & 60 \\\hline\end{array}$

Question 50

Exhibit 3.3
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons.One of his customers has requested a special blend of scotch targeted as a bar scotch.The customer wants the blend to involve two scotch products,call them A and B.Product A is a higher quality scotch while product B is a cheaper brand.The customer wants to make the claim the blend is closer to high quality than the alternative.The customer wants 50 1500 ml bottles of the blend.Each bottle must contain at least 48% of Product A and at least 500 ml of B.The customer also specified that the blend have an alcohol content of at least 85%.Product A contains 95% alcohol while product B contains 78%.The blend is sold for $12.50 per bottle.Product A costs $7 per liter and product B costs $3 per liter.The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let ^X₁ ^{= Number of liters of product A in total blend delivered} X₂ = Number of liters of product B in total blend delivered
MIN: ^{7 X}₁ ^{+ 3 X}₂
Subject to: ^X₁ ^{+ X}₂ ^{= 1.5 * 50 Total liters of mix)}X₁ ? 0.48 * 1.5 * 50 X₁ minimum)
X₂ ? 0.5 * 50 X₂ minimum)
.0.95 X₁ + 0.78 X₂ ? 0.85 * 1.5 * 50 85% alcohol minimum)X₁,X₂ ? 0
$$\begin{array} { | l | l | c | c | c | c |} \hline & \text { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \\\hline 1 & & \begin{array} { c } \text { Jacks' } \\\text { Distill } \\\text { ery }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm {~A} & \mathrm {~B} & & \\\hline 4 & \text { Liters to use } & & & & \text { Total Cost: } \\\hline 5 & \text { Unit cost: } & 10.5 & 4 & & \\\hline 6 & & 5 & & \\\hline 7 & \text { Constraints: } & & & \text { Supplied } & \text { Requirement } \\\hline \mathbf { 8 } & \text { Total Liters } & 1 & 1 & & 75 \\\hline 9 & \text { A required } & 1 & & & 36 \\\hline 1 & \text { B required } & & 1 & & 25 \\\hline 0 & & & & & \\\hline 1 & \text { 85\% alcohal } & 0.95 & 0.78 & & 63.7 \\1 & & & & & \\\hline\end{array}$$

-Refer to Exhibit 3.3.Which cells should be the constraint cells in this problem?

A)B4:C4
B)E5
C)D8:D11
D)E8:E10

Question 51

Exhibit 3.5
The following questions are based on this problem and accompanying Excel windows.
A company is planning production for the next 4 quarters.They want to minimize the cost of production.The production cost is stable but demand and production capacity vary from quarter to quarter.The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand.Quarterly
inventory holding cost is 3% of the cost of production.The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month.There are currently 5,000 units in inventory.The company wants to produce at no less than one half of its maximum capacity in any quarter.
$$\begin{array}{l}\begin{array} { l r r r r r } &&\text { Quarter }\\& { 1 } & 2 & { 3 } & { 4 } \\\hline \text { Unit Production Cost } & \$ 300 & \$ 300 & \$ 300 & \mathbf { \$ 3 0 0 } \\\text { Units Demanded } & 2,000 & 9,000 & 12,000 & 11,000 \\\text { Marimum Production } & 8,000 & 7,000 & 8,000 & 9,000\end{array}\end{array}$$ Let ^P_i ^{= number of units produced in quarter i,i = 1,... ,4} B_i = beginning inventory for quarter i
MIN: ^{300 P}₁ ^{+ 300 P}₂ ^{+ 300 P}₃ ^{+ 300 P}₄ ⁺
9B₁ + B₂)/2 + 9B₂ + B₃)/2 + 9B₃ + B₄)/2 + 9B₄ + B₅)/2
Subject to: ^{4000 ? P}₁ ^{? 8000}
3500 ? P₂ ? 7000
4000 ? P₃ ? 8000
4500 ? P₄ ? 9000
3000 ? B₁ + P₁ ? 2000 ? 12000
3000 ? B₂ + P₂ ? 9000 ? 12000
3000 ? B₃ + P₃ ? 12000 ? 12000
3000 ? B₄ + P₄ ? 11000 ? 12000 B₂ = B₁ + P₁ ? 2000
B₃ = B₂ + P₂ ? 9000
B₄ = B₃ + P₃ ? 12000
B₅ = B₄ + P₄ ? 11000
P_i,B_i ? 0
$\begin{array}{|c|l|c|c|c|c|c|}\hline&A&B&C&D&E&F\\\hline 1&&&\text { Quarter }\\\hline 2 & & & 1 & 2 & 3 & 4 \\\hline 3 & \text { Beginning Inventory } & & 5,000 & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} \\\hline 4 & \text { Units Produced } & & 8,000 & \begin{array}{c}7,000 \\\end{array} & \begin{array}{c}8,000 \\\end{array} & \begin{array}{c}9,000 \\\end{array} \\\hline 5 & \text { Units Demanded } & & 2,000 & \begin{array}{c}9,000 \\\end{array} & 12,000 & 11,000 \\\hline 6 & \text { Ending Inventory } & & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} & \begin{array}{c}3,000 \\\end{array} \\\hline 7\\\hline 8 & \text { Minimum Production } & & 4,000 & 3,500 & 4,000 & 4,500 \\\hline 9 & \text { Maximum Production } & & 8,000 & 7,000 & 8,000 & 9,000 \\\hline10\\\hline11& \text { Minimum Inventory } && 3,000 & 3,000 & 3,000 & 3,000 \\\hline\end{array}$ $\begin{array}{|c|l|c|c|c|c|c|}\hline12 & \text { Maximum Inventory } & & 12,000 & 12,000 & 12,000 & 12,000 \\\hline 13 & & & & & & \\\hline 14 & \text { Unit Production Cost } & & \$ 300 & \$ 300 & \$ 300 & \$ 300\\\hline 15 & \text { Unit Carrying Cost } & 3.00 \% & \$ 9.00 & \$ 9.00 & \$ 9.00 & \$ 9.00 \\\hline16\\\hline 17 & \text { Quarterly Production Cost } & & \$ 2,400,000 & \$ 2,100,000 & \$ 2,400,000 & \$ 2,700,000 \\\hline 18 & \text { Quarterly Carrying Cost } & & \$ 72,000 & \$ 90,000 & \$ 63,000 & \$ 36,000 \\\hline 19 & & & & & & \\\hline 20 & & & & & \text { Total Cost } & \$ 9,861,000 \\\hline\end{array}$

-Refer to Exhibit 3.5.What formula should be entered in cell C18 in the accompanying Excel spreadsheet to compute the quarterly carrying costs?

A)=C15*C3+C6
B)=C15*C3+C6)
C)=C15*C3/2
D)=C15*C3+C6)/2

Question 52

You have been given the following linear programming model and Excel spreadsheet to solve this problem.What numbers should be entered into cells B5:C5 and B8:C10 to implement this model?
MAX: ^{4 X}₁ ^{+ 3 X}₂
Subject to: ^{6 X}₁ ^{+ 7 X}₂ ^{≤ 84}
X₁ ≤ 10
X₂ ≤ 8 X₁,X₂ ≥ 0

Question 53

Exhibit 3.3
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons.One of his customers has requested a special blend of scotch targeted as a bar scotch.The customer wants the blend to involve two scotch products,call them A and B.Product A is a higher quality scotch while product B is a cheaper brand.The customer wants to make the claim the blend is closer to high quality than the alternative.The customer wants 50 1500 ml bottles of the blend.Each bottle must contain at least 48% of Product A and at least 500 ml of B.The customer also specified that the blend have an alcohol content of at least 85%.Product A contains 95% alcohol while product B contains 78%.The blend is sold for $12.50 per bottle.Product A costs $7 per liter and product B costs $3 per liter.The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let ^X₁ ^{= Number of liters of product A in total blend delivered} X₂ = Number of liters of product B in total blend delivered
MIN: ^{7 X}₁ ^{+ 3 X}₂
Subject to: ^X₁ ^{+ X}₂ ^{= 1.5 * 50 Total liters of mix)}X₁ ? 0.48 * 1.5 * 50 X₁ minimum)
X₂ ? 0.5 * 50 X₂ minimum)
.0.95 X₁ + 0.78 X₂ ? 0.85 * 1.5 * 50 85% alcohol minimum)X₁,X₂ ? 0
$$\begin{array} { | l | l | c | c | c | c |} \hline & \text { A } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { E } \\\hline 1 & & \begin{array} { c } \text { Jacks' } \\\text { Distill } \\\text { ery }\end{array} & & & \\\hline 2 & & & & & \\\hline 3 & & \mathrm {~A} & \mathrm {~B} & & \\\hline 4 & \text { Liters to use } & & & & \text { Total Cost: } \\\hline 5 & \text { Unit cost: } & 10.5 & 4 & & \\\hline 6 & & 5 & & \\\hline 7 & \text { Constraints: } & & & \text { Supplied } & \text { Requirement } \\\hline \mathbf { 8 } & \text { Total Liters } & 1 & 1 & & 75 \\\hline 9 & \text { A required } & 1 & & & 36 \\\hline 1 & \text { B required } & & 1 & & 25 \\\hline 0 & & & & & \\\hline 1 & \text { 85\% alcohal } & 0.95 & 0.78 & & 63.7 \\1 & & & & & \\\hline\end{array}$$

-Refer to Exhibit 3.3.Which of the following statements could represent a constraint in this problem?

A)B4:C4 ? B5:C5
B)E5 ? 0
C)D8 = E8
D)E8:E11 ? D8:D11

Question 54

A company needs to purchase several new machines to meet its future production needs.It can purchase three different types of machines A,B,and C.Each machine A costs $80,000 and requires 2,000 square feet of floor space.Each machine B costs $50,000 and requires 3,000 square feet of floor space.Each machine C costs $40,000 and requires 5,000 square feet of floor space.The machines can produce 200,250 and 350 units per day respectively.The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines.The company wants to buy as many machines as possible to maximize daily production.
Formulate the LP for this problem.

Question 55

A hospital needs to determine how many nurses to hire to cover a 24 hour period.The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts.They are paid different wages depending on when they start their shifts.The number of nurses required per 4-hour time period and their wages are shown in the following table.
$\begin{array}{lll}\text { Time period }&\text { Required \# of Nurses }&\text { Wage } \$ / h r)\\\hline12 \mathrm{am}-4 \mathrm{am} & 20 & 15 \\4 \mathrm{am}-8 \mathrm{am} & 30 & 16 \\8 \mathrm{am}-12 \mathrm{pm} & 40 & 13 \\12 \mathrm{pm}-4 \mathrm{pm} & 50 & 13 \\4 \mathrm{pm}-8 \mathrm{pm} & 40 & 14 \\8 \mathrm{pm}-12 \mathrm{am} & 30 & 15\end{array}$
Formulate the LP for this problem.

Question 56

Exhibit 3.5
The following questions are based on this problem and accompanying Excel windows.
A company is planning production for the next 4 quarters.They want to minimize the cost of production.The production cost is stable but demand and production capacity vary from quarter to quarter.The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand.Quarterly
inventory holding cost is 3% of the cost of production.The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month.There are currently 5,000 units in inventory.The company wants to produce at no less than one half of its maximum capacity in any quarter.
$$\begin{array}{l}\begin{array} { l r r r r r } &&\text { Quarter }\\& { 1 } & 2 & { 3 } & { 4 } \\\hline \text { Unit Production Cost } & \$ 300 & \$ 300 & \$ 300 & \mathbf { \$ 3 0 0 } \\\text { Units Demanded } & 2,000 & 9,000 & 12,000 & 11,000 \\\text { Marimum Production } & 8,000 & 7,000 & 8,000 & 9,000\end{array}\end{array}$$ Let ^P_i ^{= number of units produced in quarter i,i = 1,... ,4} B_i = beginning inventory for quarter i
MIN: ^{300 P}₁ ^{+ 300 P}₂ ^{+ 300 P}₃ ^{+ 300 P}₄ ⁺
9B₁ + B₂)/2 + 9B₂ + B₃)/2 + 9B₃ + B₄)/2 + 9B₄ + B₅)/2
Subject to: ^{4000 ? P}₁ ^{? 8000}
3500 ? P₂ ? 7000
4000 ? P₃ ? 8000
4500 ? P₄ ? 9000
3000 ? B₁ + P₁ ? 2000 ? 12000
3000 ? B₂ + P₂ ? 9000 ? 12000
3000 ? B₃ + P₃ ? 12000 ? 12000
3000 ? B₄ + P₄ ? 11000 ? 12000 B₂ = B₁ + P₁ ? 2000
B₃ = B₂ + P₂ ? 9000
B₄ = B₃ + P₃ ? 12000
B₅ = B₄ + P₄ ? 11000
P_i,B_i ? 0
$\begin{array}{|c|l|c|c|c|c|c|}\hline&A&B&C&D&E&F\\\hline 1&&&\text { Quarter }\\\hline 2 & & & 1 & 2 & 3 & 4 \\\hline 3 & \text { Beginning Inventory } & & 5,000 & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} \\\hline 4 & \text { Units Produced } & & 8,000 & \begin{array}{c}7,000 \\\end{array} & \begin{array}{c}8,000 \\\end{array} & \begin{array}{c}9,000 \\\end{array} \\\hline 5 & \text { Units Demanded } & & 2,000 & \begin{array}{c}9,000 \\\end{array} & 12,000 & 11,000 \\\hline 6 & \text { Ending Inventory } & & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} & \begin{array}{c}3,000 \\\end{array} \\\hline 7\\\hline 8 & \text { Minimum Production } & & 4,000 & 3,500 & 4,000 & 4,500 \\\hline 9 & \text { Maximum Production } & & 8,000 & 7,000 & 8,000 & 9,000 \\\hline10\\\hline11& \text { Minimum Inventory } && 3,000 & 3,000 & 3,000 & 3,000 \\\hline\end{array}$ $\begin{array}{|c|l|c|c|c|c|c|}\hline12 & \text { Maximum Inventory } & & 12,000 & 12,000 & 12,000 & 12,000 \\\hline 13 & & & & & & \\\hline 14 & \text { Unit Production Cost } & & \$ 300 & \$ 300 & \$ 300 & \$ 300\\\hline 15 & \text { Unit Carrying Cost } & 3.00 \% & \$ 9.00 & \$ 9.00 & \$ 9.00 & \$ 9.00 \\\hline16\\\hline 17 & \text { Quarterly Production Cost } & & \$ 2,400,000 & \$ 2,100,000 & \$ 2,400,000 & \$ 2,700,000 \\\hline 18 & \text { Quarterly Carrying Cost } & & \$ 72,000 & \$ 90,000 & \$ 63,000 & \$ 36,000 \\\hline 19 & & & & & & \\\hline 20 & & & & & \text { Total Cost } & \$ 9,861,000 \\\hline\end{array}$

-Refer to Exhibit 3.5.What formula should be entered in cell C6 in the accompanying Excel spreadsheet to compute ending inventory?

A)=C3-C4+C5
B)=C3+C4-C5
C)=C3-C4-C5)
D)=C5-C4-C3

Question 57

Exhibit 3.5
The following questions are based on this problem and accompanying Excel windows.
A company is planning production for the next 4 quarters.They want to minimize the cost of production.The production cost is stable but demand and production capacity vary from quarter to quarter.The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand.Quarterly
inventory holding cost is 3% of the cost of production.The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month.There are currently 5,000 units in inventory.The company wants to produce at no less than one half of its maximum capacity in any quarter.
$$\begin{array}{l}\begin{array} { l r r r r r } &&\text { Quarter }\\& { 1 } & 2 & { 3 } & { 4 } \\\hline \text { Unit Production Cost } & \$ 300 & \$ 300 & \$ 300 & \mathbf { \$ 3 0 0 } \\\text { Units Demanded } & 2,000 & 9,000 & 12,000 & 11,000 \\\text { Marimum Production } & 8,000 & 7,000 & 8,000 & 9,000\end{array}\end{array}$$ Let ^P_i ^{= number of units produced in quarter i,i = 1,... ,4} B_i = beginning inventory for quarter i
MIN: ^{300 P}₁ ^{+ 300 P}₂ ^{+ 300 P}₃ ^{+ 300 P}₄ ⁺
9B₁ + B₂)/2 + 9B₂ + B₃)/2 + 9B₃ + B₄)/2 + 9B₄ + B₅)/2
Subject to: ^{4000 ? P}₁ ^{? 8000}
3500 ? P₂ ? 7000
4000 ? P₃ ? 8000
4500 ? P₄ ? 9000
3000 ? B₁ + P₁ ? 2000 ? 12000
3000 ? B₂ + P₂ ? 9000 ? 12000
3000 ? B₃ + P₃ ? 12000 ? 12000
3000 ? B₄ + P₄ ? 11000 ? 12000 B₂ = B₁ + P₁ ? 2000
B₃ = B₂ + P₂ ? 9000
B₄ = B₃ + P₃ ? 12000
B₅ = B₄ + P₄ ? 11000
P_i,B_i ? 0
$\begin{array}{|c|l|c|c|c|c|c|}\hline&A&B&C&D&E&F\\\hline 1&&&\text { Quarter }\\\hline 2 & & & 1 & 2 & 3 & 4 \\\hline 3 & \text { Beginning Inventory } & & 5,000 & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} \\\hline 4 & \text { Units Produced } & & 8,000 & \begin{array}{c}7,000 \\\end{array} & \begin{array}{c}8,000 \\\end{array} & \begin{array}{c}9,000 \\\end{array} \\\hline 5 & \text { Units Demanded } & & 2,000 & \begin{array}{c}9,000 \\\end{array} & 12,000 & 11,000 \\\hline 6 & \text { Ending Inventory } & & 11,000 & \begin{array}{c}9,000 \\\end{array} & \begin{array}{c}5,000 \\\end{array} & \begin{array}{c}3,000 \\\end{array} \\\hline 7\\\hline 8 & \text { Minimum Production } & & 4,000 & 3,500 & 4,000 & 4,500 \\\hline 9 & \text { Maximum Production } & & 8,000 & 7,000 & 8,000 & 9,000 \\\hline10\\\hline11& \text { Minimum Inventory } && 3,000 & 3,000 & 3,000 & 3,000 \\\hline\end{array}$ $\begin{array}{|c|l|c|c|c|c|c|}\hline12 & \text { Maximum Inventory } & & 12,000 & 12,000 & 12,000 & 12,000 \\\hline 13 & & & & & & \\\hline 14 & \text { Unit Production Cost } & & \$ 300 & \$ 300 & \$ 300 & \$ 300\\\hline 15 & \text { Unit Carrying Cost } & 3.00 \% & \$ 9.00 & \$ 9.00 & \$ 9.00 & \$ 9.00 \\\hline16\\\hline 17 & \text { Quarterly Production Cost } & & \$ 2,400,000 & \$ 2,100,000 & \$ 2,400,000 & \$ 2,700,000 \\\hline 18 & \text { Quarterly Carrying Cost } & & \$ 72,000 & \$ 90,000 & \$ 63,000 & \$ 36,000 \\\hline 19 & & & & & & \\\hline 20 & & & & & \text { Total Cost } & \$ 9,861,000 \\\hline\end{array}$

-Refer to Exhibit 3.5.What formula could be entered in cell F20 in the accompanying Excel spreadsheet to compute the Total Cost for all four quarters?

A)SUMPRODUCT$C$4:$F$4,C17:F17)
B)SUMC17:F17)+ SUMC18:F18)
C)PRODUCTSUMC14:F15,C17:F18)
D)SUMPRODUCTC4:F4,C14:F14)+ SUMPRODUCTC6:F6,C15:F15)

Question 58

You have been given the following linear programming model and Excel spreadsheet to solve this problem.What numbers should be entered into cells B5:C5 and B8:C10 to implement this model?
MIN: ^{8 X}₁ ^{+ 3 X}₂
Subject to: ^X₂ ^{≥ 8}
8 X₁ + 5 X₂ ≥ 80
3 X₁ + 5 X₂ ≥ 60 X₁,X₂ ≥ 0

Question 59

A farmer is planning his spring planting.He has 20 acres on which he can plant a combination of Corn,Pumpkins and Beans.He wants to maximize his profit but there is a limited demand for each crop.Each crop also requires fertilizer and irrigation water which are in short supply.There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available.The following table summarizes the data for the problem.

Enter the numbers in the appropriate cells of ranges B12:D12 and E8:F12 in the Excel spreadsheet to solve this problem based on the following formulation.
Let ^X₁ ^{= aces of corn}
X₂ = acres of pumpkin X₃ = acres of beans
MAX: ^2100X₁ ^{+ 900X}₂ ^{+ 1050X}₃
Subject to: ^21X₁ ^{≤ 200}
10X₂ ≤ 180
3.5X₃ ≤ 80
X₁ + X₂ + X₃ ≤ 20 2X₁ + 3X₂ + 1X₃ ≤ 50
5X₁ + 4X₂ + 3X₃ ≤ 80 X₁,X₂,X₃ ≥ 0

Question 60

A hospital needs to determine how many nurses to hire to cover a 24 hour period.The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts.They are paid different wages depending on
when they start their shifts.The number of nurses required per 4-hour time period and their wages are shown in the following table.
$\begin{array}{lll}\text { Time period }&\text { Required \# of Nurses }&\text { Wage } \$ / h r)\\\hline12 \mathrm{am}-4 \mathrm{am} & 20 & 15 \\4 \mathrm{am}-8 \mathrm{am} & 30 & 16 \\8 \mathrm{am}-12 \mathrm{pm} & 40 & 13 \\12 \mathrm{pm}-4 \mathrm{pm} & 50 & 13 \\4 \mathrm{pm}-8 \mathrm{pm} & 40 & 14 \\8 \mathrm{pm}-12 \mathrm{am} & 30 & 15\end{array}$
Enter the numbers in the appropriate cells of ranges B6:G11 and B13:G13 in the Excel spreadsheet to solve this problem based on the following formulation.
Let ^X_i ^{= number of nurses working in time period i;i = 1,6} MIN: ^1X₁ ^{+ 1X}₂ ^{+ 1X}₃ ^{+ 1X}₄ ^{+ 1X}₅ ^{+ 1X}₆
Subject to: ^1X₁ ^{+ 1X}₂ ^{? 30}
1X₂ + 1X₃ ? 40
1X₃ + 1X₄ ? 50
1X₄ + 1X₅ ? 40
1X₅ + 1X₆ ? 30
1X₁ + 1X₆ ? 20
X_i ? 0

Question 61

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{≤ 36 carpentry)}
4 X₁ + 8 X₂ ≤ 40 varnishing)
X₂ ≤ 8 demand for desks)
X₁,X₂ ≥ 0

You have been given the following linear programming model.
MAX: ^{8 X}₁ ^{+ 5 X}₂
Subject to: ^{3 X}₁ ^{+ 5 X}₂ ^{= 54}
11 X₁ + 10 X₂ ≤ 144 X₁ ≥12 X₂
X₁,X₂ ≥ 0
This formulation is:

A)not an LP formulation.Pre-processing is required to make it linear.
B)a transportation problem
C)a transshipment problem
D)the assignment problem

Question 62

Robert Hope received a welcome surprise in this management science class;the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class.These instruments were: homework,an individual project,a mid-term exam,and a final exam.Robert's grades on these instruments were 75,94,85,and 92,respectively.However,the instructor complicated Robert's task somewhat by adding the following stipulations:
• homework can account for up to 25% of the grade,but must be at least 5% of the grade;
• the project can account for up to 25% of the grade,but must be at least 5% of the grade;
• the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined;and
• the project and final exam grades may not collectively constitute more than 50% of the grade.Formulate an LP model for Robert to maximize his numerical grade.

Question 63

Exhibit 3.1
The following questions are based on this problem and accompanying Excel windows.
Jones Furniture Company produces beds and desks for college students.The production process requires carpentry and varnishing.Each bed requires 6 hours of carpentry and 4 hour of varnishing.Each desk requires 4 hours of carpentry and 8 hours of varnishing.There are 36 hours of carpentry time and 40 hours of varnishing time available.Beds generate $30 of profit and desks generate $40 of profit.Demand for desks is limited,so at most 8 will be produced.
Let ^X₁ ^{= Number of Beds to produce} X₂ = Number of Desks to produce
The LP model for the problem is
MAX: ^{30 X}₁ ^{+ 40 X}₂
Subject to: ^{6 X}₁ ^{+ 4 X}₂ ^{≤ 36 carpentry)}
4 X₁ + 8 X₂ ≤ 40 varnishing)
X₂ ≤ 8 demand for desks)
X₁,X₂ ≥ 0

You have been given the following linear programming model.
MAX: ^{8 X}₁ ^{+ 5 X}₂
Subject to: ^{3 X}₁ ^{+ 5 X}₂ ^{= 54}
11 X₁ + 10 X₂ ≤ 144 X₁ / X₂ ≥12
X₁,X₂ ≥ 0
This formulation:

A)can be solved without preprocessing using LP
B)cannot be solved using LP
C)requires that the third constraint be first converted to a linear form: X₁ - 12 X₂ ≤ 0
D)must be first converted to a minimization problem

Question 64

A grain store has six types of grain,each varying in cost,quality,and nutritional content.Periodically,excess inventory of these grains are consolidated into two local products,Feed-M-All and Supreme-Feed.Feed-M-All sells for $6.50 for a 10-pound bag while Supreme-Feed sells for $8.50 for a 10-pound bag.These feeds are advertised as having the following nutritional content:
$\begin{array}{lcrc}\text { Grain } & \text { Minimum Protein } & \text { Minimum Fat } & \text { Maximum Carbohydrates } \\\hline \text { Feed-M-All } & 16 \% & 18 \% & 10 \% \\\text { Supreme-Feed } & 18 \% & 18 \% & 9 \%\end{array}$

The component grains have the following content characteristics:
$$\begin{array} { c c c c c c c } \text { Gran } & \text { Cost/10 lbs } & \text { Quality } & \text { Pratein } & \text { Fat } & \text { Carbohydrates } & \text { Paunds Avail. } \\\hline \text { A } & \$ 4.75 & 4 & 15 \% & 10 \% & 10 \% & 90 \\\mathrm {~B} & \$ 4.00 & 2 & 20 \% & 20 \% & 8 \% & 120 \\\mathrm { C } & \$ 3.75 & 1 & 10 \% & 25 \% & 5 \% & 150 \\\mathrm { D } & \$ 4.25 & 3 & 15 \% & 20 \% & 10 \% & 125 \\\mathrm { E } & \$ 4.50 & 3 & 20 \% & 20 \% & 10 \% & 85 \\\mathrm {~F} & \$ 5.00 & 4 & 25 \% & 15 \% & 12 \% & 165\end{array}$$
Targets for Feed-M-All are a cost of $ 4.35 per 10-pound bag,a quality rating of 2.25,along with the minimum percentages of protein and fat,and the maximum percentage of carbohydrates.Similar targets are set for Supreme- Feed with cost set at $ 4.60 and quality at 2.45.There must be at least a 70%-30% mix among these two local feeds.
Formulate an LP model for this product mix problem.

Question 65

A company is planning production for the next 4 quarters.They want to minimize the cost of production.The production cost,demand and production capacity vary from quarter to quarter.The maximum amount of inventory which can be held is 100 units and management wants to keep at least 50 units on hand.Quarterly inventory holding cost is 4% of the cost of production.There are currently 50 units in inventory.The company wants to produce at no less than one half of its maximum capacity in any quarter.
Quarter
$$\begin{array} { r r r r r } &1&2&3&4\\\hline \text { Unit Praduction Cost } & \$ 55 & \$ 50 & \$ 50 & \$ 45 \\\text { Units Demanded } & 100 & 150 & 180 & 120 \\\text { Maramum Production } & 150 & 150 & 160 & 130 \\\text { Holding cost } & \$ 2.2 & \$ 2 & \$ 2 & \$ 1.8\end{array}$$
Let ^P_i ^{= number of units produced in quarter i,i = 1,... ,4} B_i = beginning inventory for quarter i
MIN: ^{55 P}₁ ^{+ 50 P} ₂ ^{+ 50 P}₃ ^{+ 45 P}₄ ⁺
2.2 B₁ + B₂)/2 + 2 B₂ + B₃)/2 + 2 B₃ + B₄)/2 + 1.8 B₄ + B₅)/2
Subject to: ^{75 ? P}₁ ^{? 150}
75 ? P₂ ? 150
80 ? P₃ ? 160
65 ? P₄ ? 130
50 ? B₁ + P₁ ? 100 ? 100
50 ? B₂ + P₂ ? 150 ? 100
50 ? B₃ + P₃ ? 180 ? 100
50 ? B₄ + P₄ ? 120 ? 100
B₂ = B₁ + P₁ ? 100
B₃ = B₂ + P₂ ? 150
B₄ = B₃ + P₃ ? 180
B₅ = B₄ + P₄ ? 120
P_i,B_i ? 0

What formulas are required for cells D3,D6,D8,D15,D17 and D18 in the Excel spreadsheet implementation of the formulation?

Question 66

Carlton construction is supplying building materials for a new mall construction project in Kansas.Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period.Carlton's supply depot has access to three modes of transportation: a trucking fleet,railway delivery,and air cargo transport.Their contract calls for 120,000 tons delivered by the end of week one,80% of the total delivered by the end of week two,and the entire amount delivered by the end of week three.Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking,at least 40% of the total delivered be delivered by railway,and up to 15% of the total delivered be delivered by air cargo.Unfortunately,competing demands limit the availability of each mode of transportation each of the three weeks to the following levels all in thousands of tons):
$\begin{array}{cccc}\text { Week } & \text { Trucking Limits } & \text { Railway Limits } & \text { Air Cargo Limits } \\\hline 1 & 45 & 60 & 15 \\2 & 50 & 55 & 10 \\3 & 55 & 45 & 5 \\\hline \text { Costs } \$ \text { per } 1000 \text { tons) } & \$ 20 & \$ 14 & \$ 40\\&0&0&0\end{array}$

The following is the LP model for this logistics problem.
$\text { Let }$



What values would you enter in the Risk Solver Platform RSP)task pane for the cells in this Excel spreadsheet implementation of this problem?
Objective Cell: Variables Cells: Constraints Cells:

Question 67

Carlton construction is supplying building materials for a new mall construction project in Kansas.Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period.Carlton's supply depot has access to three modes of transportation: a trucking fleet,railway delivery,and air cargo transport.Their contract calls for 120,000 tons delivered by the end of week one,80% of the total delivered by the end of week two,and the entire amount delivered by the end of week three.Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking,at least 40% of the total delivered be delivered by railway,and up to 15% of the total delivered be delivered by air cargo.Unfortunately,competing demands limit the availability of each mode of transportation each of the three weeks to the following levels all in thousands of tons):
$\begin{array}{cccc}\text { Week } & \text { Trucking Limits } & \text { Railway Limits } & \text { Air Cargo Limits } \\\hline 1 & 45 & 60 & 15 \\2 & 50 & 55 & 10 \\3 & 55 & 45 & 5 \\\hline \text { Costs } \$ \text { per } 1000 \text { tons) } & \$ 20 & \$ 14 & \$ 40\\&0&0&0\end{array}$

Formulate an LP model for this logistics problem.

Question 68

Pete's Plastics manufactures plastic at plants in Miami,St.Louis and Cleveland.Pete needs to ship plastic to customers in Pittsburgh,Atlanta and Chicago.He wants to minimize the cost of shipping the plastic from his plants to his customers.The data for the problem is summarized in the following table.
Distance From Plants to Customers
$$\begin{array} { l c c c c } \text { Plant } & \text { Pittsburgh } & \text { Atlanta } & \text { Chicago } & \text { Supply } \\\hline \text { Miani } & 1,200 & 700 & 1,300 & 30 \\\text { St. Lauis } & 700 & 550 & 300 & 40 \\\text { Cleveland } & 125 & 675 & 350 & 50 \\\hline \text { Damand } & 40 & 60 & 20 &\end{array}$$
Formulate the LP for this problem.

Question 69

A financial planner wants to design a portfolio of investments for a client.The client has $400,000 to invest and the planner has identified four investment options for the money.The following requirements have been placed on the planner.No more than 30% of the money in any one investment,at least one half should be invested in long-term bonds which mature in six or more years,and no more than 40% of the total money should be invested in B or C since they are riskier investments.The planner has developed the following LP model based on the data in this table and the requirements of the client.The objective is to maximize the total return of the portfolio.
$$\begin{array} { l c c c } \text { Investment } & \text { Return } & \text { Years to Maturity } & \text { Rating } \\\hline \text { A } & 0.45 \% & 6 & 1 \text {-Facellent } \\\text {B }& 8.5 \% & 5 & \text { 3-Good } \\\text { C } & 9.00 \% & 8 & 4 - \text { Fair } \\\text { D } & 7.75 \% & 4 & 2 - V e r y \text { Good }\end{array}$$



What formulas are required for the following cells in the Excel spreadsheet implementation of the formulation? B7
D7 F7 H7

Question 70

The hospital administrators at New Hope,County General,and City East recently received notice of an impending state inspection of their facilities.Under new guidelines established to improve the overall health care system,state inspectors will be assessing the efficiency of each hospital.The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility.The data collected thus far is summarized in the following table.All data reflects averages compiled over the past six months.
Hospital
$\begin{array}{cccc}&&\text { New Hope}&\text { County General }&\text { City East }\\\hline\text { Input Measures }\\&\text { Bed days unused 1000s) } & 83.0 & 105.0 & 104.1 \\&\text { Supply expense } \$ 1000 \mathrm{~s}) & 123.8 & 162.3 & 154.0 \\&\text { Full-time staff } & 225.0 & 200.0 & 231.0 \\& & & \\\text { Output Measures }\\&\text { Patient-days 1000s) } & 105.0 & 71.0 & 82.7 \\&\text { Nurses qualified } & 253.0 & 92.0 & 175.0 \\&\text { Assistants on staff } & 125.0 & 45.0 & 65.0\end{array}$

a.Formulate a DEA LP model to evaluate the efficiency of City East.
b.Implement a spreadsheet model for this problem and compute the DEA efficiency for each facility.Which facilities are efficient?

Question 71

A financial planner wants to design a portfolio of investments for a client.The client has $400,000 to invest and the planner has identified four investment options for the money.The following requirements have been placed on the planner.No more than 30% of the money in any one investment,at least one half should be invested in long-term bonds which mature in six or more years,and no more than 40% of the total money should be invested in B or C since they are riskier investments.The planner has developed the following LP model based on the data in this table and the requirements of the client.The objective is to maximize the total return of the portfolio.
$$\begin{array} { l c c c } \text { Inve5trment } & \text { Retuin } & \text { Years to Maturity } & \text { Rating } \\\hline \text { A } & 6.45 \% & 6 & \text { 1-Eacellent } \\\text { B } & \mathbf { 8 . 5 0 \% } & 5 & \text { 3-Good } \\\text { C } & 9.00 \% & 8 & 4 - F a i r \\D & 7.75 \% & 4 & 2 - V e r y \text { Good }\end{array}$$
Formulate the LP for this problem.

Question 72

Robert Hope received a welcome surprise in this management science class;the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class.These instruments were: homework,an individual project,a mid-term exam,and a final exam.Robert's grades on these instruments were 75,94,85,and 92,respectively.However,the instructor complicated Robert's task somewhat by adding the following stipulations:
• homework can account for up to 25% of the grade,but must be at least 5% of the grade;
• the project can account for up to 25% of the grade,but must be at least 5% of the grade;
• the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined;and
• the project and final exam grades may not collectively constitute more than 50% of the grade.The following LP model allows Robert to maximize his numerical grade.
Let W₁= weight assigned to homework W₂= weight assigned to the project W₃= weight assigned to the mid-term
W₄= weight assigned to the final


12 Both Exams 0.65 0.70
13 Final & Project 0.5 0.50
What values would you enter in the Risk Solver Platform RSP)task pane for the cells in this Excel spreadsheet implementation of this problem?
Objective Cell: Variables Cells: Constraints Cells:

Question 73

Carlton construction is supplying building materials for a new mall construction project in Kansas.Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period.Carlton's supply depot has access to three modes of transportation: a trucking fleet,railway delivery,and air cargo transport.Their contract calls for 120,000 tons delivered by the end of week one,80% of the total delivered by the end of week two,and the entire amount delivered by the end of week three.Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking,at least 40% of the total delivered be delivered by railway,and up to 15% of the total delivered be delivered by air cargo.Unfortunately,competing demands limit the availability of each mode of transportation each of the three weeks to the following levels all in thousands of tons):
Week Trucking Limits Railway Limits Air Cargo Limit

$$\begin{array} { c c c c } 1 & 45 & 60 & 15 \\2 & 50 & 55 & 10 \\3 & 55 & 45 & 5 \\\hline \text { Costs } \$ \text { per } 1000 \text { tons) } & \$ 200 & \$ 140 & \$400\end{array}$$
The following is the LP model for this logistics problem.


What formula goes in cells F10,E3,E4,E5,and B6 of this Excel spreadsheet?

Question 74

A paper mill has received an order for rolls of paper.The customer wants 400 12" wide rolls,300 18" rolls and 200 24" rolls.The company has 40" wide rolls of paper which it can slit to the appropriate width.The company wants to minimize the number of rolls it must use to fill the order.
Formulate the LP for this problem.

Question 75

Project 3.1 − The Diet Problem: Ordering Meals from McDonald's
Based on: Robert A.Bosch,"Big Mac Attack: The Diet Problem revisited,Eating at McDonald's," OR/MS Today,
August 1993,pp 30-31.
Tina Simpson is a new fourth-grade teacher at Forest Ridge Elementary.The first teacher workshop for the upcoming school year is next Monday and by majority vote,McDonald's was selected as the food of choice.As the new person,Tina is tasked with developing the meal for the workshop.McDonald's has graciously offered to deliver whatever food Tina decides to order,along with a variety of condiments applicable to whatever is ordered.Rather than offer a menu choice,Tina has decided to simply order the same meal for each person in the workshop.
To get started,Tina took a trip to McDonald's and obtained their published information on the nutritional content of their food.That data is summarized in the table below.


Tina wants the meal to be nutritionally complete.The National Research Council publishes their Recommended Daily Allowances.In this publication,they contend that a diet in this case the meal)should provide at least 100 percent of the U.S.RDA of numerous nutrients.The specific amount of the RDA depends on such factors as age,weight and gender.In addition,the council recommends daily sodium and cholesterol intakes be kept to at most 2.4 grams of sodium and 300 milligrams of cholesterol.Further,at most 30 percent of the calories consumed should come from fat,and at most 10 percent from saturated fat.Each gram of fat contains 9 calories.
Based on the above information,Tina wants to design a least-cost meal that provides at least 100% of the U.S.RDA of vitamins A,C,B₁,B₂,niacin,calcium,and iron;supplies at least 55 grams of protein;contains at most 3 grams of sodium;and contains at most 30 percent of its calories from fat.Only those foods list in the table above are available for the meal.
Formulate the LP model for Tina's problem.Develop a spreadsheet model of the problem and use Excel Solver to determine the least-cost meal that meets all the stated requirements.
What is the recommended meal? Is this meal reasonable? If not,modify the model to obtain what you believe to be a reasonable meal that meets the stated requirements.