CBAD 292 Exam 4

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:

-0.5

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:

3x + 5y <_ 100

An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below. Medium Cost Per Ad # Reached Exposure Quality TV 500 10000 30 Radio 200 3000 40 Newspaper 400 5000 25 If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000, you will develop the model that will maximize the number reached and achieve an exposure quality if at least 1000. Let T = the number of TV adsLet R = the number of radio adsLet N = the number of newspaper ads (C) Please select the constraints for this decision problem.

500T + 200R + 400N ≤ 10000

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

60x + 50y

If a transportation problem has four origins and five destinations, the LP formulation of the problem will have

9 constraints

Which of the following is not true regarding an LP model of the assignment problem?

All constraints are of the ≥ form.

A 12-month rolling planning horizon is a single model where the decision in the first period is implemented.

False

A company makes two products, A and B. A sells for $100 and B sells for $90. The variable production costs are $30 per unit for A and $25 for B. The company's objective could be written as: MAX 190x1 − 55x2.

False

A decision maker would be wise to not deviate from the optimal solution found by an LP model because it is the best solution.

False

A feasible solution does not have to satisfy any constraints as long as it is logical.

False

A multiple choice constraint involves selecting k out of n alternatives, where k ≥ 2.

False

A transportation problem with 3 sources and 4 destinations will have 7 decision variables.

False

A transportation problem with 3 sources and 4 destinations will have 7 variables in the objective function.

False

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

False

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

False

Double-subscript notation for decision variables should be avoided unless the number of decision variables exceeds nine.

False

Generally, the optimal solution to an integer linear program is less sensitive to the constraint coefficients than is a linear program.

False

If a real-world problem is correctly formulated, it is not possible to have alternative optimal solutions.

False

If an LP problem is not correctly formulated, the computer software will indicate it is infeasible when trying to solve it.

False

If the LP relaxation of an integer program has a feasible solution, then the integer program has a feasible solution.

False

In a capacitated transshipment problem, some or all of the transfer points are subject to capacity restrictions.

False

In a transportation problem with total supply equal to total demand, if there are four origins and seven destinations, and there is a unique optimal solution, the optimal solution will utilize 11 shipping routes.

False

It is improper to combine manufacturing costs and overtime costs in the same objective function.

False

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

False

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

False

Production constraints frequently take the form:beginning inventory + sales − production = ending inventory

False

Revenue management methodology was originally developed for the banking industry.

False

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

False

The media selection model presented in the textbook involves maximizing the number of potential customers reached subject to a minimum total exposure quality rating.

False

The primary limitation of linear programming's applicability is the requirement that all decision variables be nonnegative.

False

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

False

The solution to the LP Relaxation of a minimization problem will always be less than or equal to the value of the integer program minimization problem.

False

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

False

Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the right-hand side is acceptable since both are a measure of time.

False

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.

False

When the number of agents exceeds the number of tasks in an assignment problem, one or more dummy tasks must be introduced in the LP formulation or else the LP will not have a feasible solution.

False

Let M be the number of units to make and B be the number of units to buy. If it costs $2 to make a unit and $3 to buy a unit and 4000 units are needed, the objective function is

Min 2M + 3B

Canning Transport is to move goods from three factories to three distribution centers. Information about the move is given below. Give the network model and the linear programming model for this problem. Source Supply Destination Demand A 200 X 50 B 100 Y 125 C 150 Z 125 ​ Shipping costs are: ​ Destination Source X Y Z A 3 2 5 B 9 10 -- C 5 6 4 ​ (Source B cannot ship to destination Z) Canning Transport is to move goods from three factories to three distribution... We define the amount of goods shipped from a factory to a distribution center in the following table. ​ Destination Source 1 2 3 A XA1 XA2 XA3 B XB1 XB2 -- C XC1 XC2 XC3 ​ (Source B cannot ship to destination Z) (B) Please provide the objective function

Min 3XA1 + 2XA2 + 5XA3 + 9XB1 + 10XB2 + 5XC1 + 6XC2 + 4XC3

Use the table from above:

Min 9X1A+5X1B+4X1C+2X1D+12X2A+6X2B+3X2C+5X2D+11X3A+6X3B+5X3C+7X3D

Which of the following is not true regarding the linear programming formulation of a transportation problem?

The number of constraints is (number of origins) x (number of destinations).

A company makes two products from steel; one requires 2 tons of steel and the other requires 3 tons. There are 100 tons of steel available daily. A constraint on daily production could be written as: 2x1 + 3x2 ≤ 100.

True

A feasible solution is a solution that satisfies all of the constraints.

True

A marketing research firm must determine how many daytime interviews (D) and evening interviews (E) to conduct. At least 40% of the interviews must be in the evening. A correct modeling of this constraint is: -0.4D + 0.6E > 0.

True

A transshipment constraint must contain a variable for every arc entering or leaving the node.

True

A transshipment problem is a generalization of the transportation problem in which certain nodes are neither supply nodes nor destination nodes.

True

Compared to the problems in the textbook, real-world problems generally require more variables and constraints.

True

Flow in a transportation network is limited to one direction.

True

If a solution to an LP problem satisfies all of the constraints, then it must be feasible.

True

If a transportation problem has four origins and five destinations, the LP formulation of the problem will have nine constraints.

True

If an LP model has an unbounded solution, then we must have made a mistake - either we have made an input error or we omitted one or more constraints.

True

If the optimal solution to the LP relaxation problem is integer, it is the optimal solution to the integer linear program.

True

If x1 + x2 ≤ 500y1 and y1 is 0 - 1, then if y1 is 0, x1 and x2 will be 0.

True

In a model involving fixed costs, the 0 - 1 variable guarantees that the capacity is not available unless the cost has been incurred.

True

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.

True

In general, rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values.

True

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.

True

In the general assignment problem, one agent can be assigned to several tasks.

True

Infeasibility refers to the situation in which there are no feasible solutions to the LP model.

True

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.

True

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.

True

Media selection problems can maximize exposure quality and use number of customers reached as a constraint, or maximize the number of customers reached and use exposure quality as a constraint.

True

Multiple choice constraints involve binary variables.

True

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

True

Portfolio selection problems should acknowledge both risk and return.

True

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

True

Some linear programming problems have a special structure that guarantees the variables will have integer values.

True

The LP Relaxation contains the objective function and constraints of the IP problem, but drops all integer restrictions.

True

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.

True

The assignment problem is a special case of the transportation problem in which all supply and demand values equal one.

True

The capacitated transportation problem includes constraints which reflect limited capacity on a route.

True

The classic assignment problem can be modeled as a 0-1 integer program.

True

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

True

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

True

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.

True

The value, such as profit, to be optimized in an optimization model is the objective.

True

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

True

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

True

Transshipment problem allows shipments both in and out of some nodes while transportation problems do not.

True

When a route in a transportation problem is unacceptable, the corresponding variable can be removed from the LP formulation.

True

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

True

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

True

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

True

Whenever total supply is less than total demand in a transportation problem, the LP model does not determine how the unsatisfied demand is handled.

True

​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.

True

​A rolling planning horizon is a multiperiod model where only the decision in the first period is implemented, and then a new multiperiod model is solved in succeeding periods.

True

​Data collection for large-scale LP models can be more time-consuming than either the formulation of the model or the development of the computer solution.

True

​Integer linear programs are harder to solve than linear programs.

True

​Most practical applications of integer linear programming involve only 0 -1 integer variables.

True

​The marketing research model presented in the textbook involves minimizing total interview cost subject to interview quota guidelines.

True

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

Two

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: Product Machine A Machine B (hours) (hours) Chemical X 2 3 Chemical Y 4 2 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. We let x be the amount of chemical X to produce and y be the amount of chemical Y to produce. What are the decision variables in this problem?

X & Y

As part of the settlement for a class action lawsuit, Hoxworth Corporation must provide sufficient cash to make the following annual payments (in thousands of dollars). The annual payments must be made at the beginning of each year. The judge will approve an amount that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to savings (at 4% annually) and government securities, at prices and rates currently quoted in The Wall Street Journal. Hoxworth wants to develop a plan for making the annual payments by investing in the following securities (par value 5 $1000). Funds not invested in these securities will be placed in savings. Assume that interest is paid annually. The plan will be submitted to the judge and, if ap- proved, Hoxworth will be required to pay a trustee the amount that will be required to fund the plan. a. Use linear programming to find the minimum cash settlement necessary to fund the annual payments Let F = total funds required to meet the six years of payments G1 = units of government security 1 G2 = units of government security 2 S1 = investment in savings at the beginning of year 1 S2 = investment in savings at the beginning of year 2 S3 = investment in savings at the beginning of year 3 S4 = investment in savings at the beginning of year 4 S5 = investment in savings at the beginning of year 5 Note: All decision variables are expressed in thousands of dollars (C) Please select the correct constraints

a. 1.05125G2 + 1.04S4 - S5 = 315 b. F - 1.055G1 - 1.000G2 - S1 = 190 d. All decision variables are non-negative. e. 1.0675G1 + .05125G2 + 1.04S3 - S4 = 285 g. 1.04S5 = 460 h. .0675G1 + .05125G2 +1.04S1 - S2 = 215 i. .0675G1 + .05125G2 + 1.04S2 - S3 = 240

Use the above problem: (C) Please select the constraints.

a. X1A + X2A + X3A + X4A = 1 b. X2A + X2B + X2C + X2D ≤ 1 c. X1C + X2C + X3C + X4C = 1 d. X1D + X2D + X3D + X4D = 1 e. X4A + X4B + X4C + X4D ≤ 1 f. X1B + X2B + X3B + X4B = 1 g. X1A + X1B + X1C + X1D ≤ 1 h. X 3A + X3B + X3C + X3D ≤ 1

The cost of completing a task by a worker is shown in the following table. Task Person A B C D 1 9 5 4 2 2 12 6 3 5 3 11 6 5 7 Let X1A denote whether we assign person 1 to task A. If we assign person 1 to task A, X1A = 1. If we do not assign person 1 to task A, X1A = 0. ​ Task Person A B C D 1 X1A X1B X1C X1D 2 X2A X2B X2C X2D 3 X3A X3B X3C X3D (A) What are the decision variables?

a. X1D, X2D, X3B b. X1C, X2A, X3A c. X1B, X2C, X3D d. X1A, X2B, X3C

The assignment problem constraint x31 + x32 + x33 + x34 ≤ 2 means

agent 3 can be assigned to 2 tasks.

In an all-integer linear program,

all variables must be integer.

Rounding the solution of an LP Relaxation to the nearest integer values provides

an integer solution that might be neither feasible nor optimal.

All optimization problems have:

an objective function and decision variables

In a transshipment problem, shipments

can occur between any two nodes.

Conditions that must be satisfied in an optimization model are:​

constraints

If the acceptance of project A is conditional on the acceptance of project B, and vice versa, the appropriate constraint to use is a

corequisite constraint.

The optimal solution to any linear programming model is the:

corner point of a polygon

The term nonnegativity refers to the condition in which the:

decision variables cannot be less than zero

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

decision variables.

The difference between the transportation and assignment problems is that

each supply and demand value is 1 in the assignment problem

Consider the following linear programming problem: Maximize: 2x1 + 4x2 Subject to: The above linear programming problem:

exhibits infeasibility

Consider the following linear programming problem: Maximize: 2x1 + 2x2 Subject to: The above linear programming problem:

exhibits unboundedness

Rounded solutions to linear programs must be evaluated for

feasibility and optimality.

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

feasible region

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

formulating the problem, invoking Solver, and sensitivity analysis

The shortest-route problem finds the shortest-route

from the source to any other node.

Consider the following linear programming problem: Maximize: 5x1 + 5x2 Subject to: The above linear programming problem:

has more than one optimal solution

Consider the following linear programming problem: Minimize: 5x1 + 6x2 Subject to: The above linear programming problem:

has only one optimal solution

Media selection problems usually determine

how many times to use each media source.

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

hyperplanes

Constraints in a transshipment problem

include a variable for every arc.

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.

independent

Arcs in a transshipment problem

indicate the direction of the flow.

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

integer and noninteger levels of an activity

Every linear programming problem involves optimizing a:

linear function subject to several linear constraints

To study consumer characteristics, attitudes, and preferences, a company would engage in

marketing research.

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

mathematical programming models

The objective of the transportation problem is to

minimize the cost of shipping products from several origins to several destinations.

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

mistake in the formulation of the problem

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

nonnegative

In an optimization model, there can only be one:

objective function

The graph of a problem that requires x1 and x2 to be integer has a feasible region

of dots.

In the general linear programming model of the assignment problem,

one agent is assigned to one and only one task.

Most practical applications of integer linear programming involve

only 0-1 integer variables and not ordinary integer variables.

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:

product mix problem

Linear programming models have three important properties: _____.

proportionality, additivity, and divisibility

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:

shadow price

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

simplex method

The parts of a network that represent the origins are

the nodes

The problem which deals with the distribution of goods from several sources to several destinations is the

transportation problem

The assignment problem is a special case of the

transportation problem.

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:

unbounded

Which of the following is the most useful contribution of integer programming?

using 0-1 variables for modeling flexibility

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

value of the objective function

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: Product Machine A Machine B (hours) (hours) Chemical X 2 3 Chemical Y 4 2 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. Choose algebraic expressions for all of the constraints in this problem.

x <= 16 y <= 18 2x + 4y <= 80 3x + 2y <= 60 y >= 0 x>= 0

Let x1 , x2 , and x3 be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1). Which answer below indicates that at least two of the projects must be done?

x1 + x2 + x3 ≥ 2

In a model, x1 ≥ 0 and integer, x2 ≥ 0, and x3 = 0, 1. Which solution would not be feasible?

x1 = 2, x2 = 3, x3 = .578

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:

x>_60, y>_80

The number of units shipped from origin i to destination j is represented by

xij.

What is the equation of the line representing this constraint? 20x + 10y<_1000

y<_-2x+100

​A mutual fund manager must decide how much money to invest in Atlantic Oil (A) and how much to invest in Pacific Oil (P). At least 60% of the money invested in the two oil companies must be in Pacific Oil. A correct modeling of this constraint is

​-0.6A + 0.4P > 0.

​In a production scheduling LP, the demand requirement constraint for a time period takes the form

​beginning inventory + production - ending inventory = demand.

​The production scheduling problem modeled in the textbook involves capacity constraints on all of the following types of resources except

​material.

​Assuming W1, W2 and W3 are 0 -1 integer variables, the constraint W1 + W2 + W3 < 1 is often called a

​mutually exclusive constraint.

Which of the following is not a characteristic of assignment problems?​

​the signs of constraints are always <

Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, then

−.5A + .5B − .5C ≤ 0