Deck 13: Introduction to Optimization Modeling

Question 1

In using Excel^® to solve linear programming problems, the objective cell represents the:

A) value of the objective function
B) constraints
C) decision variables
D) total cost of the model

Answer 1

value of the objective function

Question 2

Reduced costs indicate how much the objective coefficient of a decision variable that is currently 0 or at its upper bound must change before that the value of that variable changes.

Answer 2

True

Question 3

The set of all values of the decision variable cells that satisfy all constraints, not including the nonnegativity constraints, is called the feasible region.

Answer 3

False

Question 4

(A) Determine how to minimize the total cost of meeting the next 3 quarters' demand. Assume that 1000 usable units are available at the beginning of quarter 1.
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(B) Referring to (A), the company wants to know how much money it would be worth to decrease the percentage of unsuitable items and/or the percentage of items that spoil. Write a short report that provides relevant information. Base your report on two uses of the SolverTable add-in: one where the percentage of unsuitable items decreases and the percentage of items that spoil stays at 10%; and one where the percentage of unsuitable items stays at 20% and the percentage of items that spoil decreases.

Question 5

The divisibility property of LP models simply means that we allow only integer levels of the activities.

Question 6

(A) Determine how to minimize the net cost incurred in meeting the demands for the next four months.
(B) Starting with the optimal solution to (A), use SolverTable add-in to see what happens to the decision variables and the total cost when the initial inventory varies from 0 (the implied value in (A)) to 100 in 10-units increments. How much lower would the total cost be if the company started with 10 units in inventory, rather than none? Would the same cost decrease occur for every 10-init increase in initial inventory?

Question 7

As related to sensitivity analysis in linear programming, when the profit increases with a unit increase in labor, this change in profit is referred to as the:

A) add-in price
B) sensitivity price
C) shadow price
D) additional profit

Question 8

(A) Find an optimal solution to the problem. What is the production plan, and what is the total revenue?
(B) Obtain a sensitivity report for the solution reported in (A). Which constraints are binding?
(C) What is the incremental contribution associated with adding an hour of assembly time? Over what range of increase is the marginal value valid?
(D) What is the value of additional capacity on the polisher? How much increase and decrease in this capacity is possible before a change occurs in the optimal production schedule?
(E) An advertising agency has devised a marketing plan for the Western Chassis Company that will increase the market for Deluxe chassis. The plan will increase demand by 75 Deluxe chassis per month at a cost of $100 per month. Should Western adopt the plan? Briefly explain why.
(F) Suppose that four more hours of chassis assembly time could be made available. How much would profit change?
(G) Suppose next that Western's marketing department proposes lowering the price for a standard chassis from $12 to $11.50 so that more can be sold (since there is slack under the demand constraint). Would the optimal solution change? Explain why, or why not.
(H) If Western could obtain 1,000 pounds more of raw material (steel or aluminum), which should it procure? How much should they be willing to pay per pound for the steel or aluminum? Explain your answer.
(I) In doing some contingency planning, Western thinks that the aging stamping machine will soon need to be taken down for repairs that could last 2 months and will cost $10,000. During that time, they can continue to operate by outsourcing the stamping at $2.50 per chassis (deluxe or standard), although the capacity will be reduced from 2,500 to 1,500. What will be the total cost to repair the stamping machine?

Question 9

A farmer in Egypt owns 50 acres of land. He is going to plant each acre with cotton or corn. Each acre planted with cotton yields $400 profit; each with corn yields $200 profit. The labor and fertilizer used for each acre are given in the table below. Resources available include 150 workers and 200 tons of fertilizer.
(A) Formulate a linear programming model that will enable the farmer to determine the number of acres that should be planted cotton and/or corn in order to maximize his profit.
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(B) Find an optimal solution to the model in (A) and determine the maximum profit.
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(C) Implement the model in (A) in Excel Solver and obtain an answer report. Which constraints are binding on the optimal solution?
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(D) Obtain a sensitivity report for the model in (A). How much should the farmer be willing to pay for an additional worker?
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(E) Suppose the farmer hires 10 additional workers. Can you use the sensitivity analysis obtained for (D) to determine his expected profit? Would his planting plan change? Explain your answer.
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(F) Suppose the farmer now wants to hire 20 additional workers, instead of just 10. Can you use the sensitivity analysis obtained for (D) to determine his expected profit? Explain your answer.

Question 10

A shadow price indicates how much a company would pay for more of a scarce resource.​

Question 11

Proportionality, additivity, and divisibility are three important properties that LP models possess that distinguish them from general mathematical programming models.

Question 12

An efficient algorithm for finding the optimal solution in a linear programming model is the:

A) spreadsheet method
B) solution mix method
C) complex method
D) simplex method

Question 13

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $60, the optimal value of the objective function would not change.

Question 14

Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase) and $10 (decrease). If the right hand side of that constraint were to increase by $10, the optimal value of the objective function would change.

Question 15

A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below: Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: The company's goal is to maximize the revenue obtained from selling units during the next two months.
What are the decision variables in this problem?

Question 16

Linear programming is a subset of a larger class of models called:

A) mathematical programming models
B) mathematical optimality models
C) linear regression models
D) linear simplex models

Question 17

(A) Determine how to minimize the cost of meeting the demand for the next three weeks.
(B) Revise the model in (A) so that the demands are of the form , where is the original demand in month t, k is a factor, and is an amount of change in month t. Formulate the model in such away that you can use the SolverTable add-in to analyze changes in the amounts produced and the total cost when k varies from 0 to 10 in 1-unit increments, for any fixed values of the 's. For example, try this when = 2, = 5, and = 3. Describe the behavior you observe in the table. Can you find any "reasonable" 's that induce positive production levels in week 3?

Question 18

It is often useful to perform sensitivity analysis to see how, or if, the optimal solution to a linear programming problem changes as we change one or more model inputs.

Question 19

Shadow prices are associated with nonbinding constraints, and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Question 20

(A) Verify that Mary should purchase 12 units of food 2 each day and thus oversatisfy the vitamin C requirement by 6 units.
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(B) Mary's husband has put his foot down and demanded that Mary fulfill the family's daily nutritional requirement exactly by obtaining precisely 12 units of vitamin A and 6 units of vitamin C. The optimal solution to the new problem will involve ingesting less vitamin C, but it will be more expensive, why?
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(C) Starting with the optimal solution to (B), use the SolverTable add-in to see what happens to the total cost when the vitamin A and vitamin C requirements both vary (independently) from 3 to 18 in 3-unit increments. That is, from a two-way table. Describe the behavior you observe. In particular, are the changes in total cost the same as you look across each row of the table? Are they the same as you look across each column of the table?

Question 21

In using Excel^® to solve linear programming problems, the decision variable cells represent the:

A) value of the objective function
B) constraints
C) decision variables
D) total cost of the model

Question 22

All optimization problems have:

A) an objective function and decision variables
B) an objective function and constraints
C) decision variables and constraints
D) an objective function, decision variables and constraints

Question 23

There is often more than one objective in linear programming problems.

Question 24

Every linear programming problem involves optimizing a:

A) linear regression model subject to several linear constraints
B) linear function subject to several linear constraints
C) linear function subject to several non-linear constraints
D) non-linear function subject to several linear constraints

Question 25

When formulating a linear programming spreadsheet model, there is one target (objective) cell that contains the value of the objective function.

Question 26

A chemical manufacturer produces two products, chemical X and chemical Y. Each product is manufactured by a two-step process that involves blending and mixing in machine A and packaging on machine B. Chemical X provides a $60/unit contribution to profit, while Chemical Y provides a $50 contribution to profit. The processing times for the two products on the mixing machine (A) and the packaging machine (B) are as follows: For the upcoming two-week period, machine A has available 80 hours and machine B has available 60 hours of processing time. Forecasts of the markets indicate that the manufacturer can expect to sell a maximum of 16 units of chemical X and 18 units of chemical Y.
What are the decision variables in this problem?

Question 27

Adam Enterprises manufactures two products. Each product can be produced on either of two machines. The time (in hours) required to make each product on each machine is shown below: Each month, 500 hours of time are available on each machine, and also customers are willing to buy up to the quantities of each product at the prices shown below: The company's goal is to maximize the revenue obtained from selling units during the next two months.
(A) Determine how the company can meet its goal. Assume that Adam will not produce any units in either month that it cannot sell in that month.
(B) Referring to (A), suppose Adam wants to see what will happen if customer demands for each product in each month simultaneously change by a factor 1 + k. Revise the model so that you can use the SolverTable add-in to investigate the effect of this change on total revenue as k varies from -0.3 to 0.3 in increments of 0.1. Does revenue change in a linear manner over this range? Can you explain intuitively why it changes in the way it does?

Question 28

It is instructive to look at a graphical solution procedure for LP models with three or more decision variables.

Question 29

Write out an algebraic expression for the objective function in this problem.

Question 30

There are generally two steps in solving an optimization problem: model development and optimization.

Question 31

When using the graphical solution method to solve linear programming problems, the set of points that satisfy all constraints is called the:

A) optimal region
B) feasible region
C) constrained region
D) logical region

Question 32

When formulating a linear programming spreadsheet model, there is a set of designated cells that play the role of the decision variables. These are called the objective cells.

Question 33

All linear programming problems should have a unique solution, if they can be solved.

Question 34

The feasible region in all linear programming problems is bounded by:

A) corner points
B) hyperplanes
C) an objective line
D) all of these options

Question 35

In most cases, when solving linear programming problems, we want the decision variables to be:

A) equal to zero
B) nonnegative
C) nonpositive
D) continuous

Question 36

In an optimization model, there can only be one:

A) decision variable
B) constraint
C) objective function
D) shadow price

Question 37

The term nonnegativity refers to the condition in which the:

A) objective function cannot be less that zero
B) decision variables cannot be less than zero
C) right hand side of the constraints cannot be greater then zero
D) reduced cost cannot be less than zero

Question 38

The most important solution method for linear programming problems is known as the:

A) spreadsheet method
B) solution mix method
C) complex method
D) simplex method

Question 39

(A) Write out algebraic expressions for all of the constraints in this problem.
(B) Construct a graph of the feasible region for this problem, given the constraints you identified in (A).
(C) Describe how you would find the location of the optimal solution in the feasible region you graphed in (B).
(D) Use the procedure you described in (C) to identify the optimal production plan. Confirm your solution using Solver. What is the maximized profit?
(E) What constraints are binding on the optimal solution? Use your graphical solution to explain your answer.

Question 40

All optimization problems include decision variables, an objective function, and constraints.

Question 41

Binding constraints are constraints that hold as an equality.​

Question 42

Suppose a firm must at least meet minimum expected demands of 60 for product x and 80 of product y. An algebraic formulation of these constraints is:

A)
B)
C)
D)

Question 43

If a manufacturing process takes 3 hours per unit of x and 5 hours per unit of y and a maximum of 100 hours of manufacturing process time are available, then an algebraic formulation of this constraint is:

A)
B)
C)
D)

Question 44

In general, the complete solution of a linear programming problem involves three stages: formulating the model, invoking Solver to find the optimal solution, and performing sensitivity analysis.

Question 45

The solution of a linear programming problem using Excel^® typically involves the following three stages:

A) formulating the problem, invoking Solver, and sensitivity analysis
B) formulating the problem, graphing the problem, and sensitivity analysis
C) designing the decision variable cells, the target cells, and the constraints
D) designing the inputs, the decision variable cells, and the outputs

Question 46

Nonbinding constraints will always have slack, which is the difference between the two sides of the inequality in the constraint equation.

Question 47

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon, with lines forming all sides.

Question 48

The optimal solution to any linear programming model is the:

A) maximum objective function line
B) minimum objective function line
C) corner point of a polygon
D) maximum or minimum of a parabola

Question 49

Linear programming problems can always be formulated algebraically, but not always on a spreadsheet.

Question 50

What is the equation of the line representing this constraint? ​

A)
B)
C)
D)

Question 51

The optimal solution to any linear programming model is a corner point of a polygon.

Question 52

A linear programming problem with _____decision variable(s) can be solved by a graphical solution method.

A) two
B) three
C) four
D) five

Question 53

In determining the optimal solution to a linear programming problem graphically, if the objective is to maximize the objective, we pull the objective function line down until it contacts the feasible region.

Question 54

Suppose an objective function has the equation: .
Then the slope of the objective function line is 2.

Question 55

There are two primary ways to formulate a linear programming problem: the traditional algebraic way and with spreadsheets.

Question 56

Suppose a constraint has this equation: Then the slope of the constraint line is -2.

Question 57

Suppose a company sells two different products, x and y, for net profits of $5 per unit and $10 per unit, respectively. The slope of the line representing the objective function is:

A) 0.5
B) -0.5
C) 2
D) -2

Question 58

Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant, which produces 5000 tons per month, and the Lakeview plant, which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants: Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:
What are the decision variables in this problem?

Question 59

The prototype linear programming problem is to select an optimal mix of products to produce to maximize profit. This type of problem is referred to as the:

A) product mix problem
B) production problem
C) product/process problem
D) product scheduling problem

Question 60

When formulating a linear programming spreadsheet model, we specify the constraints in a Solver dialog box, since Excel does not show the constraints directly.

Question 61

When the proportionality property of LP models is violated, we generally must use non-linear optimization.

Question 62

Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant, which produces 5000 tons per month, and the Lakeview plant, which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants: Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:
What are the constraints in this problem?

Question 63

The additivity property of linear programming implies that the contribution of any decision variable to the objective is of/on the levels of the other decision variables.

A) dependent
B) independent
C) conditional
D) the sum

Question 64

The additivity property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

Question 65

A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below: Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: The company's goal is to maximize the revenue obtained from selling units during the next two months.
What is the objective function in this problem?

Question 66

What is the objective function in this problem?

Question 67

It helps to ensure that Solver can find a solution to a linear programming problem if the model is well-scaled, that is, if all of the numbers are of roughly the same magnitude.

Question 68

​A decision support system is a user-friendly system where an end user can enter inputs to a model and see outputs, but need not be concerned with technical details.

Question 69

A marketing research professor is conducting a telephone survey and needs to contact at least 160 wives, 140 husbands, 110 single adult males, and 120 single adult females. It costs $2 to make a daytime call and $4 (because of higher labor costs) to make an evening call. The table shown below lists the expected results. For example, 10% of all daytime calls are answered by a single male, and 15% of all evening calls are answered by a single female. Because of a limited staff, at most half of all phone calls can be evening calls. Determine how to minimize the cost of completing the survey.
What are the decision variables in this problem?

Question 70

When there is a problem with Solver being able to find a solution, many times it is an indication of a(n):

A) older version of Excel^®
B) nonlinear programming problem
C) problem that cannot be solved using linear programming
D) mistake in the formulation of the problem

Question 71

What are the decision variables in this problem?

Question 72

In some cases, a linear programming problem can be formulated such that the objective can become infinitely large (for a maximization problem) or infinitely small (for a minimization problem). This type of problem is said to be:

A) infeasible
B) inconsistent
C) unbounded
D) redundant

Question 73

The proportionality property of LP models means that if the level of any activity is multiplied by a constant factor, then the contribution of this activity to the objective function, or to any of the constraints in which the activity is involved, is multiplied by the same factor.

Question 74

A company produces two products. Each product can be produced on either of two machines. The time (in hours) required to produce each product on each machine is shown below: Each month, 600 hours of time are available on each machine, and that customers are willing to buy up to the quantities of each product at the prices that are shown below: The company's goal is to maximize the revenue obtained from selling units during the next two months.
What are the constraints in this problem?

Question 75

What are the constraints in this problem?

Question 76

The divisibility property of linear programming means that a solution can have both:

A) integer and noninteger levels of an activity
B) linear and nonlinear relationships
C) positive and negative values
D) revenue and cost information in the model

Question 77

Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant, which produces 5000 tons per month, and the Lakeview plant, which can produce 7000 tons per month. Sinclair sells its polyethylene to three different GM auto plants: Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:
What is the optimal shipping plan? What are the total costs in that case?

Question 78

What is the objective function in this problem?

Question 79

Linear programming models have three important properties: _____.

A) optimality, additivity, and sensitivity
B) optimality, linearity, and divisibility
C) divisibility, linearity, and nonnegativity
D) proportionality, additivity, and divisibility

Question 80

Find an optimal solution to the problem, assuming that the company will not produce any units in either month that it cannot sell in that month. What is the production plan, and what is the total revenue?