##### Mktg 372 supplement B

Suppose that a firm makes gears (G) and axles (A). Each gear requires 15 minutes of labor and each axle requires 30 minutes of labor. One thousand hours of labor are available. How should the labor constraint(s) be written?

.25G + .5A ≤ 1000

If a firm is using a linear program to determine production amounts of its chairs and tables, which of the following constraints must be in the model?

non-negativity

What does LP stand for?

Linear Program

Sausage and Cheese Ltd. prepares three gift packages containing sausages and cheeses. The "Tasters," "Succulent," and "Gourmet" gift packages contain 3 sausages and 6 cheeses, 5 sausages and 4 cheeses, and 6 sausages and 5 cheeses, respectively. There are 2500 sausages and 3000 cheeses available for packing, and demand is unlimited. Profits are $2.50, $3.50, and $4.00 for the "Tasters," "Succulent," and "Gourmet" gift packages, respectively. The goal is to maximize profits. Let T, S, and G represent the number of gift packages produced of type "Tasters," "Succulent," and "Gourmet," respectively. What is the objective function for this linear program?

Max 2.5T + 3.5S + 4G

Which of the following statements is not correct?

A Linear Program as described in Supplement B can have more than one objective function if it has only one constraint.

What is the optimal solution (X, Y, Z) to the following linear
program?

Max 2X + 4Y + 6Z

Subject to

Z ≤ 0

X + Y +
Z ≤ 20

X, Y, Z ≥ 0

(0, 20, 0)

Consider the following constraints from a two-variable Linear
Program.

(1) X ≥ 0

(2) Y ≥ 0

(3) X + Y ≤ 50

If
constraints (2) and (3) are binding, what is the optimal solution (X, Y)?

(50, 0)

Consider the following three functions:

g(x, y) = 4x -
9y2

h(x, y, z) = 13x + y + 3z -6

i(y, z) = z - y

Which
of the following is true regarding the linearity of the functions?

h and i are linear, but g is not linear

George Dantzig developed the ___________________ in 1947 to solve Linear Programs.

Simplex Method

When was the Simplex Method developed?

1947

A constraint in Excel Solver consists of what three pieces of information?

Cell Reference, relationship operator, and Constraint

In the "Solver Options" box of Excel Solver, what should be checked to ensure that all decision variables are ≥ 0?

Assume Non-Negative

In the "Solver Options" box of Excel Solver, what should be checked to ensure that the Simplex Method is used to solve the model?

Assume Linear Model

In the "Solver Parameters" box of Excel Solver, what is clicked to actually solve the problem?

"SOLVE"

Three reports are available when Solver has successfully found an optimal solution. These are _______.

Answer Report, Sensitivity Report, and Limits Report

The ________________________ for a constraint is the amount the optimal objective value will change if the right-hand-side of the constraint is increased by one unit.

Shadow Price

One use of the Answer report is:

as a debugging tool.

In Linear Programming models, what do you want to do with the objective?

maximize or minimize

A mathematical model in which one is trying to maximize or minimize some quantity while satisfying a set of constraints is a(n):

constrained optimization problem

Human intelligence is not needed in which of the following steps of solving optimization problems?

solving the problem

Once you've written the algebraic formulation of the problem, the next setup involved in Solving Optimization Problems is:

set up the Solver settings

A ____________________ contains explicit definitions of the decision variables, an algebraic expression of the objective function, and algebraic statements of the constraints.

formulation

When formulating optimization problems, which of the following represent the typical sequence?

(1) diagram, (2) text-based formulation, (3) algebraic formulation

A diagram of the situation can help _______ the problem as well as be a(n) ______ _______ tool.

structure, valuable communication

The algebraic formulation of an optimization problem must state what three things?

decision variables, objective function, and constraints

What are the allowable constraint relationship types in optimization problems?

=, ≤, ≥

If C represents the number of chairs produced, which of the following is a proper non-negativity constraint?

C ≥ 0

Which type of constraint does not allow the solution for a decision variable of an optimization problem to be less than zero?

non-negativity

Testing of the LP model should include __________ and _____________.

base case, test values

How can the following Linear Program be characterized?

Min X +
Y

Subject to

X ≤ 20

Y ≤ 5

X + Y ≥ 40

X, Y ≥ 0

bounded and infeasible

Consider the following three functions:

f(x) = 6x2

g(x, y)
=4x - 3y + 19

h(x, y) = 3xy

Which of the following is true
regarding the linearity of the functions?

g(x, y) is linear, but f(x) and h(x, y) are not linear

Sausage and Cheese Ltd. prepares three gift packages containing sausages and cheeses. The "Tasters," "Succulent," and "Gourmet" gift packages contain 3 sausages and 6 cheeses, 5 sausages and 4 cheeses, and 6 sausages and 5 cheeses, respectively. There are 2500 sausages and 3000 cheeses available for packing, and demand is unlimited. Profits are $2.50, $3.50, and $4.00 for the "Tasters," "Succulent," and "Gourmet" gift packages, respectively. The goal is to maximize profits. Let T, S, and G represent the number of gift packages produced of type "Tasters," "Succulent," and "Gourmet," respectively. What is the constraint describing the sausage capacity?

3T + 5S + 6G ≤ 2500

Dane's aircraft muffler manufacturers have 1500 linear feet of steel on hand to manufacture the three top selling muffler sets. Super mufflers (S) provide $285 profit and common (C) mufflers' profit margin is $310, while the deluxe (D) muffler set provides a $400 profit margin. It costs Dane $310, $295, and $400 to build each muffler set, respectively. What is the objective function of Dane's aircraft muffler manufacturing?

Max 285S + 310C + 400D

How can the following Linear Program be characterized?

Min X +
Y

Subject to

X ≥ 20

Y ≥ -5

X + Y ≤ 23

bounded and feasible

Capital Co. is considering five different projects. Define Xi as a binary (0-1) variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1,2,3,4,5. Which of the following represents the constraint(s) stating that projects 2, 3, and 4 cannot all be undertaken simultaneously?

X2 + X3 + X4 ≤ 2

Which of the following is not one of the steps in setting up the Solver optimization problem?

specify the constraints

Solver provides many options for the solution process. For LPs, the two most commonly used are:

assume linear model, assume non-negative

Consider a mathematical program where Xi represents the amount produced of item i (i = 1,2,3,4), and you want the total amount produced over all four items to equal either 100, 120, 140, or 200. If you define qi as binary (0-1) variables (i = 1,2,3,4) and add the constraint q1 + q2 + q3 + q4 = 1, what other constraint do you need to add to the program?

X1 + X2 + X3 + X4 = 100q1 + 120q2 + 140q3 + 200q4

Consider the mathematical program below. Which of the following
choices represents an upper bound to the problem?

Max 10 -
X2

Subject to

X ≥ 3

X = 0

Consider the Linear Program below. Which of the choices represents
the best (tightest) lower bound?

Max 2X + Y

Subject
to

X + Y ≤ 10

X, Y ≥ 0

(5, 0)

How can the following Linear Program be characterized?

Min X +
2Y

Subject to

X ≤ 20

Y ≤ 5

X, Y ≥ -40

bounded and feasible

How can the following Linear Program be characterized?

Min X +
2Y

Subject to

X ≤ 20

X, Y ≥ -40

unbounded and feasible

Consider the following constraints from a two-variable Linear
Program.

(1) X ≥ 0

(2) Y ≥ 0

(3) X + Y ≤ 20

(4) 2X
+ 5Y ≤ 70

If constraints (3) and (4) are binding, what is the
optimal solution (X, Y)?

(10, 10)

Consider the following constraints from a two-variable Linear
Program.

(1) X ≥ 0

(2) Y ≥ 0

(3) 10X + 4Y ≤
110

(4) 5X - Y ≤ 40

If constraints (3) and (4) are binding,
what is the optimal solution (X, Y)?

(9, 5)

For an optimization problem a(an) __________________ violates at least one of the constraints.

infeasible solution

If the solution to an optimization problem violates two constraints but satisfies three, it is a(an) ________.

infeasible solution

The two primary Excel tools for diagnosing problems in models are ___________________.

Error Checking and Formula Auditing

Which Excel tool provides solutions to Linear Programs?

Solver

What in Excel Solver corresponds to the objective function in the algebraic model?

Target Cell

What in Excel Solver corresponds to the decision variables in the algebraic model?

Changing Cells

In the Excel Solver "Add Constraint" box, what two additional choices are available under the relationship operator list besides ≤, ≥, and =?

int and bin

Constraints at their limits at the optimal solution of a Linear Program, that is, with the left-hand-side value equal to the right-hand-side value, are called _________________________.

binding constraints

Constraints that are not at their limits at the optimal solution of a Linear Program, that is, with the left-hand-side value not equal to the right-hand-side value, are called _________________________.

non-binding constraints

To retain model flexibility while using Solver you must:

use only cell references

At the optimal solution of a Linear Program, the difference between the right-hand-side value and the left-hand-side value of a constraint is the ____________________.

slack

The Answer Report Target Cell, Adjustable Cell, and Constraint sections all include:

original value, final value

What do you need to do before using Solver?

have a working, flexible spreadsheet model

Consider the following three functions:

g(x, y) = 4x - 3y +
21

h(x, y, z) = 13x2 + y + 3z

i(z) = z

Which of the
following is true regarding the linearity of the functions?

g and i are linear, but h is not linear

Consider the following two functions:

g(x, y) =4x - 3y +
21

h(x, y) = 13xy

Which of the following is true regarding
the linearity of the functions?

g(x, y) is linear, but h(x, y) is not linear

How can the following Linear Program be characterized?

Max X +
Y

Subject to

X ≤ 34

X, Y ≥ 0

unbounded and feasible

In Linear Programming models, over what quantities do you have control?

decision variables

What is the optimal solution to the following linear
program?

Max 2X + Y

Subject to

2X + 2Y ≥ 40

X + Y
≤ 10

X ≥ 0

Y ≥ 0

the program is infeasible

Which of the following statements is correct?

Given a Linear Program with a maximization objective, the optimal objective function value may decrease if a ≥ constraint is added to the program.

The text-based formulation of an optimization problem should state what three things?

decision variables, objective function, and constraints