You have formulated a problem with three constraints: (1) 2X1 + 3X2 + 4X3 $\le$ 300; (2) X1 + X2 $\ge$40; (3) X1 + X2 + X3 = 100.Which of the following states that at least 2 of these 3 constraints must hold? (M = a large value)

A) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 - MY2 $\ge$40 X1 + X2 + X3 - MY3 = 100 Y1 + Y2 + Y3 $\ge$2 B) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 - MY2 $\ge$ 40 X1 + X2 + X3 + MY3 $\ge$ 100 Y1 + Y2 + Y3 $\ge$ 2 C) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 + MY2 $\ge$ 40 X1 + X2 + X3 - MY3 $\le$ 100 X1 + X2 + X3 + MY3 $\ge$100 Y1 + Y2 + Y3 $\le$ 1 D) 2X1 + 3X2 + 4X3 - MY1 $\le$300 X1 + X2 – MY2$\ge$ 40 X1 + X2 + X3 – MY3 $\le$ 100 Y1 + Y2 + Y3$\le$ 1 A) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 - MY2 $\ge$40 X1 + X2 + X3 - MY3 = 100 Y1 + Y2 + Y3 $\ge$2 B) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 - MY2 $\ge$ 40 X1 + X2 + X3 + MY3 $\ge$ 100 Y1 + Y2 + Y3 $\ge$ 2 C) 2X1 + 3X2 + 4X3 - MY1 $\le$ 300 X1 + X2 + MY2 $\ge$ 40 X1 + X2 + X3 - MY3 $\le$ 100 X1 + X2 + X3 + MY3 $\ge$100 Y1 + Y2 + Y3 $\le$ 1 D) 2X1 + 3X2 + 4X3 - MY1 $\le$300 X1 + X2 – MY2$\ge$ 40 X1 + X2 + X3 – MY3 $\le$ 100 Y1 + Y2 + Y3$\le$ 1

Heavenly Casket Company is trying to choose sites for the production of its "mail order" caskets.It is considering plants in Chicago, Dallas, and Atlanta.Finished caskets will then be sent to their two distribution sites in Trenton and Tacoma, which take orders over the internet.Heavenly expects demand of 4000 caskets per year in Trenton and 2500 in Tacoma.The table below gives annual plant capacity, fixed yearly operating expenses, unit production costs, and unit transportation costs between possible plant locations and the distribution sites:$\begin{array}{cccccc} \text { Site } & \text { Capacity } & \begin{array}{c} \text { Fixed } \\ \text { Annual } \\ \text { Operating } \\ \text { Costs } \end{array} & \begin{array}{c} \text { Production } \\ \text { Cost Per } \\ \text { Unit } \end{array} & \begin{array}{c} \text { Shipping } \\ \text { to Trenton } \end{array} & \begin{array}{c} \text { Shipping } \\ \text { to Tacoma } \end{array} \\ \text { Chicago } & 3500 & \$ 40,000 & \$ 200 & 75 & 50 \\ \text { Dallas } & 3200 & \$ 42,000 & \$ 160 & 95 & 70 \\ \text { At lanta } & 3700 & \$ 45,000 & \$ 170 & 82 & 98 \end{array}$ Which plants should be operational, and what should the production quantities and shipping pattern be to minimize Heavenly's annual expenses?

(Let X ij = amount shipped from plant i (1

Two constraints in a model with binary variables Y1, Y2, Y3 representing whether or not project 1, 2, or 3 will be performed are: Y1 - Y2 $\le$ 0 and Y1 - Y3 $\le$ 0.Taken together, what can be inferred from these constraints?

A) Projects 2 and 3 cannot both be performed. B) Projects 2 and 3 must be performed if project 1 is performed. C) Projects 2 and 3 may be performed is project is performed. D) Project 1 must be performed. A) Projects 2 and 3 cannot both be performed. B) Projects 2 and 3 must be performed if project 1 is performed. C) Projects 2 and 3 may be performed is project is performed. D) Project 1 must be performed.

Clancy's Casino, in Muledeer, Nevada, is open 24 hours a day, seven days a week.Along with all the other attractions and diversion, Clancy's operates a variety of gaming tables.Dealers at these tables are interchangeable.The casino has the following daily requirements for dealers:$\begin{array} { c c } \text { Time } & \text { Minimum \# of dealers } \\ 0100 - 0500 & 7 \\ 0500 - 0900 & 4 \\ 0900 - 1300 & 9 \\ 1300 - 1700 & 12 \\ 1700 - 2100 & 15 \\ 2100 - 0100 & 17 \end{array}$ A dealer may start work at the beginning of any one of the six shifts and, having begun, works eight consecutive hours.Find the employee schedule that minimizes the total number of dealers required, meeting the minimum level of each shift's requirements.

(X i = the number of dealers beginning wo

Exam 3: Applications of Linear and Integer Programming Models

Nike must build a factory at either Millville or Greenfield, but not both.The appropriate linear constraint to express this restriction using binary variables Y₁ and Y₂ is:

(Multiple Choice)

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Question 21

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.

(True/False)

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Question 22

Suppose in problem 4, Clancy's Casino pays a wage differential depending on the hours worked.In particular, between midnight and 0700, it pays dealers $16 per hour, between 0700 and 1900 $10 per hour, and between 1900 and midnight $12 per hour.Modify your formulation to problem 4, and determine the minimum cost shift schedule for Clancy's.

(Essay)

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Question 23

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

(True/False)

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Question 24

Eastern Engineering Company is trying to decide which of 6 projects to perform during the next quarter.The net present value, the estimated cost, and the number of engineers and staff personnel required for each project are given in the following table. Project Net Present Cost Engineers Staff Value \ 100s Required Required (\ 100) 1 100 35 5 2 2 145 65 8 3 3 200 95 11 2 4 250 180 4 2 5 500 250 16 7 6 695 475 19 11 Eastern has a $550,000 budget and 30 engineers and 15 staff available.Which projects should Eastern perform during the quarter?

(Essay)

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Question 25

One approach for solving an integer linear programming problem is simply to enumerate all feasible points and select the one yielding the "best" value for the objective function.However, the number of feasible integer points is usually so large, even for small problems, that this approach is inefficient for solving most models even with a computer.

(True/False)

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Question 26

What is the initial step in the process of building linear models?

(Multiple Choice)

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Question 27

The objective function coefficient for X₁ is currently $18 and for X₂ is $29, and the ranges of optimality for these coefficients are between $15 and $20 and between $25 and $35, respectively.If the objective function coefficients for X₁ and X₂ decline by $2 each, since both coefficients are still within their ranges of optimality, the optimal solution is guaranteed to remain the same.

(True/False)

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Question 28

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

(True/False)

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Question 29

What is the difference in the interpretation of reduced cost for an unbounded variable versus a bounded variable?

(Essay)

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Question 30

Why use hidden cells in an Excel spreadsheet representation of a linear programming model?

(Essay)

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Question 31

You have formulated a problem with three constraints: (1) 2X₁ + 3X₂ + 4X₃ $\le$ 300; (2) X₁ + X₂ $\ge$ 40; (3) X₁ + X₂ + X₃ = 100.Which of the following states that at least 2 of these 3 constraints must hold? (M = a large value)

(Multiple Choice)

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Question 32

Heavenly Casket Company is trying to choose sites for the production of its "mail order" caskets.It is considering plants in Chicago, Dallas, and Atlanta.Finished caskets will then be sent to their two distribution sites in Trenton and Tacoma, which take orders over the internet.Heavenly expects demand of 4000 caskets per year in Trenton and 2500 in Tacoma.The table below gives annual plant capacity, fixed yearly operating expenses, unit production costs, and unit transportation costs between possible plant locations and the distribution sites: Site Capacity Fixed Annual Operating Costs Production Cost Per Unit Shipping to Trenton Shipping to Tacoma Chicago 3500 \ 40,000 \ 200 75 50 Dallas 3200 \ 42,000 \ 160 95 70 At lanta 3700 \ 45,000 \ 170 82 98 Which plants should be operational, and what should the production quantities and shipping pattern be to minimize Heavenly's annual expenses?

(Essay)

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Question 33

X₁ is limited to 40% of the total, as modeled by the constraint .6X₁ - .4X₂ - .4X₃ $\le$ 0.Rewrite this as two constraints using a summation variable.

(Essay)

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Question 34

Why use summation variables, which make the linear programming model larger? The model can be completed without summation variables.

(Essay)

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Question 35

Two constraints in a model with binary variables Y₁, Y₂, Y₃ representing whether or not project 1, 2, or 3 will be performed are: Y₁ - Y₂ $\le$ 0 and Y₁ - Y₃ $\le$ 0.Taken together, what can be inferred from these constraints?

(Multiple Choice)

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Question 36

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

(True/False)

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Question 37

Clancy's Casino, in Muledeer, Nevada, is open 24 hours a day, seven days a week.Along with all the other attractions and diversion, Clancy's operates a variety of gaming tables.Dealers at these tables are interchangeable.The casino has the following daily requirements for dealers: Time Minimum \# of dealers 0100-0500 7 0500-0900 4 0900-1300 9 1300-1700 12 1700-2100 15 2100-0100 17 A dealer may start work at the beginning of any one of the six shifts and, having begun, works eight consecutive hours.Find the employee schedule that minimizes the total number of dealers required, meeting the minimum level of each shift's requirements.

(Essay)

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Question 38

The Data Envelopment Analysis model gives the following equation: $\frac {\text { Relative Output Value }} { \text {Relative Input Value } }$ = $\frac{5 Y_{1}}{2 X_{1}}\frac{+7 Y_{2}+9 X_{2}}{+4X_{4}+3 X_{3}}$ Create a linear programming model for this DEA problem by converting this equation to a linear objective function and two linear constraints.

(Essay)

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Question 39

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

(True/False)

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Question 40

Nike must build a factory at either Millville or Greenfield, but not both.The appropriate linear constraint to express this restriction using binary variables Y₁ and Y₂ is:

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

What is the initial step in the process of building linear models?

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

What is the difference in the interpretation of reduced cost for an unbounded variable versus a bounded variable?

Why use hidden cells in an Excel spreadsheet representation of a linear programming model?

You have formulated a problem with three constraints: (1) 2X₁ + 3X₂ + 4X₃ $\le$ 300; (2) X₁ + X₂ $\ge$ 40; (3) X₁ + X₂ + X₃ = 100.Which of the following states that at least 2 of these 3 constraints must hold? (M = a large value)

X₁ is limited to 40% of the total, as modeled by the constraint .6X₁ - .4X₂ - .4X₃ $\le$ 0.Rewrite this as two constraints using a summation variable.

Why use summation variables, which make the linear programming model larger? The model can be completed without summation variables.

Two constraints in a model with binary variables Y₁, Y₂, Y₃ representing whether or not project 1, 2, or 3 will be performed are: Y₁ - Y₂ $\le$ 0 and Y₁ - Y₃ $\le$ 0.Taken together, what can be inferred from these constraints?

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

Introduction to Management Science Models

Introduction to Management Science Models

Network Models

Project Scheduling Models

Decision Models

Forecasting

Inventory Models

Queuing Models

Simulation Models

Quality Management Models

Markov Process Models

Nonlinear Models:dynamic, Goal, and Nonlinear Programming

Filters

Exam 3: Applications of Linear and Integer Programming Models

Nike must build a factory at either Millville or Greenfield, but not both.The appropriate linear constraint to express this restriction using binary variables Y1 and Y2 is:

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X1 + 80X2 - 1000Y2, where Y2 is a binary variable.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X1 = 23 and X2 = 15.This must also be the optimal solution to the integer linear model.

What is the initial step in the process of building linear models?

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

What is the difference in the interpretation of reduced cost for an unbounded variable versus a bounded variable?

Why use hidden cells in an Excel spreadsheet representation of a linear programming model?

You have formulated a problem with three constraints: (1) 2X1 + 3X2 + 4X3 ≤\le≤ 300; (2) X1 + X2 ≥\ge≥ 40; (3) X1 + X2 + X3 = 100.Which of the following states that at least 2 of these 3 constraints must hold? (M = a large value)

X1 is limited to 40% of the total, as modeled by the constraint .6X1 - .4X2 - .4X3 ≤\le≤ 0.Rewrite this as two constraints using a summation variable.

Why use summation variables, which make the linear programming model larger? The model can be completed without summation variables.

Two constraints in a model with binary variables Y1, Y2, Y3 representing whether or not project 1, 2, or 3 will be performed are: Y1 - Y2 ≤\le≤ 0 and Y1 - Y3 ≤\le≤ 0.Taken together, what can be inferred from these constraints?

Joe Chan is modeling the installation of smoke alarms.The constraint Y1 - Y2 ≥\ge≥ 0 uses the binary variables Y1 for upstairs installation and Y2 for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

Introduction to Management Science Models

Introduction to Management Science Models

Network Models

Project Scheduling Models

Decision Models

Forecasting

Inventory Models

Queuing Models

Simulation Models

Quality Management Models

Markov Process Models

Nonlinear Models:dynamic, Goal, and Nonlinear Programming

Filters

Nike must build a factory at either Millville or Greenfield, but not both.The appropriate linear constraint to express this restriction using binary variables Y₁ and Y₂ is:

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

You have formulated a problem with three constraints: (1) 2X₁ + 3X₂ + 4X₃ $\le$ 300; (2) X₁ + X₂ $\ge$ 40; (3) X₁ + X₂ + X₃ = 100.Which of the following states that at least 2 of these 3 constraints must hold? (M = a large value)

X₁ is limited to 40% of the total, as modeled by the constraint .6X₁ - .4X₂ - .4X₃ $\le$ 0.Rewrite this as two constraints using a summation variable.

Two constraints in a model with binary variables Y₁, Y₂, Y₃ representing whether or not project 1, 2, or 3 will be performed are: Y₁ - Y₂ $\le$ 0 and Y₁ - Y₃ $\le$ 0.Taken together, what can be inferred from these constraints?

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.