Deck 16: Linear Programming

Question 1

A firm produces shampoo (S) and conditioner (C). The profit contribution of shampoo is $3 per unit, while conditioner's contribution is $6 per unit. What is the slope, (C/S) of the firm's objective function?

A) -18
B) -2
C) -6
D) -.5
E) 0

Answer 1

D

Question 2

In a linear programming problem, multiple optimal solutions are possible if:

A) the contour of objective function touches one corner of the feasible region.
B) there are more than two nonbinding constraints.
C) the objective function is non-linear.
D) the slope of the objective function equals the slope of a binding constraint.
E) the feasible region is unbounded.

Answer 2

D

Question 3

In a linear programming problem, the objective function:

A) formulates the target in terms of the relevant decision variables.
B) restricts the values of decision variables.
C) shows the feasible region.
D) defines each resource constraint.
E) None of these are correct.

Answer 3

A

Question 4

To identify the feasible region, one must graph:

A) all non-binding constraints.
B) the objective function contours.
C) all binding constraints.
D) the complete set of simultaneous equations.
E) the capacities of all relevant resources.

Question 5

A firm is pondering the introduction of a new good with a profit contribution of $70 per unit. Each unit of the good uses 2 units of input A and 3 units of input B. The shadow price of A is $10, and the shadow price of input B is $15. Which of the following statements is true?

A) The firm should produce the new good.
B) The opportunity cost of producing the new good is $75.
C) The firm should not produce the new good.
D) Producing the new good will earn a profit of $10.
E) Producing the new good will earn a loss of $5.

Question 6

In an LP problem the inequalities 2X + Y  800 and X + 2Y  700, hold as binding constraints. The optimal solution is:

A) X = 300 and Y = 200.
B) X = 200 and Y = 300.
C) X = 100 and Y = 300.
D) X = 400 and Y = 0.
E) X = 200 and Y = 400.

Question 7

The combination of decision variables optimizing a linear programming problem, occurs at:

A) a point where the objective function contour touches the feasible region.
B) a point in the interior of the feasible region.
C) a point outside the feasible region.
D) a point where the constraint functions intersects the objective function.
E) a point which satisfies at least one nonbinding constraint.

Question 8

The optimal solution in a linear programming problem:

A) is determined by solving all constraints as equalities.
B) always exists.
C) typically occurs in the interior of the feasible region.
D) occurs at a corner of the feasible region.
E) always involves at least one nonbinding constraint.

Question 9

In an LP problem, the feasible region contains values of the decision variables that:

A) optimize the value of the objective function.
B) are finite in magnitude.
C) satisfy all the nonbinding constraints
D) satisfy all binding constraints.
E) are nonnegative.

Question 10

Linear programming is useful for solving optimization problems involving:

A) both linear and nonlinear constraints.
B) both linear and nonlinear objectives.
C) linear objectives and nonlinear constraints.
D) linear objectives and linear constraints.
E) a small number of decision variables.

Question 11

In a linear programming problem, there are 3 binding constraints. Constraint A has slope -1.5, constraint B has slope -0.5, and constraint C has slope -0.2. The objective function's slope is -1.2. Where would the optimal solution lie?

A) At the intersection of constraints A and B
B) At the intersection of constraints B and C
C) At the intersection of constraint A and the horizontal axis
D) At the intersection of constraint C and the vertical axis
E) Anywhere along constraint A

Question 12

A shadow price measures the:

A) impact of all nonbinding constraints on the objective function simultaneously.
B) market price of the resource in question.
C) total profit earned from the firm's given decisions.
D) change in the value of the objective function associated with a unit change in the resource.
E) loss incurred by not operating at the optimal point of the feasible region.

Question 13

Which of the following statements concerning sensitivity analysis is incorrect?

A) It tracks changes in the objective function with respect to per unit change in one or more coefficients.
B) It tracks changes in the constraint function when the amount of a resource is altered.
C) It's used to calculate the shadow prices of resources.
D) It's used to derive the new optimal point of operation, when the objective function, or the constraint functions, or both change.
E) It's used to solve the appropriate set of simultaneous equations.

Question 14

If the shadow price of a given resource is $100 and the cost of expanding the capacity of that resource is $80 per unit, then the expansion of the capacity will:

A) earn positive profits.
B) earn zero profit, but the firm will stay in business.
C) temporarily earn negative profits.
D) not be a rational decision.
E) not affect the profit level.

Question 15

In a linear programming problem, the goal to be maximized (or minimized) is referred to as the:

A) max-min function.
B) constraint function.
C) objective function.
D) production function.
E) algebraic function.

Question 16

A small machine shop produces steel shafts and metal plates. The production process depends on labor, milling machine hours, and lathe machine hours. The firm tries to identify how many metal shafts and plates should be produced per week in order to maximize profit. Which, among the following, are the decision variables?

A) Profit contributions of shafts and plates
B) Maximum weekly labor hours available
C) Milling and lathe machine hours available per week
D) Quantities of shafts and plates produced per week
E) The total profit earned by the shop

Question 17

In a linear programming problem involving two decision variables, a change in one of their coefficients in the objective function will:

A) always change the optimal solution.
B) change the feasible region.
C) never change the optimal solution because the other coefficient is still fixed.
D) change the optimal solution only if the slope of the objective function's contour changes significantly.
E) necessarily change all shadow prices.

Question 18

In an LP problem, the goal is to maximize the objective function S + 3T, subject to the binding constraints S + T  700 and S + 2T  1,000. The optimal solution is:

A) S = 400 and T = 300.
B) S = 200 and T = 600.
C) S = 700 and T = 0.
D) S = 0 and T = 500
E) S = 500 and T = 200

Question 19

In a linear programming problem, the inequalities, X + 2Y ≤ 12 and 3X + 4Y  28, hold as binding constraints. The problem's optimal solution would be:

A) X = 4, and Y = -4.
B) X = 4, and Y = 8.
C) X = 4, and Y = 4.
D) X = 8, and Y = 4.
E) X = 12, Y = 0.

Question 20

In an LP problem, the goal is to maximize the objective function 2S + 3T, subject to the binding constraints S + T  700 and S + 2T  1,000. The optimal solution is:

A) S = 400 and T = 300.
B) S = 200 and T = 600.
C) S = 700 and T = 0.
D) S = 0 and T = 500
E) S = 500 and T = 200

Question 21

Why are computer solutions necessary to solve most linear programming problems?

Question 22

Determine the feasible region given the following constraints:
4x + 3y $\le$ 120
x $\le$ 20
y $\ge$ 10
x, y $\ge$ 0

Question 23

Most of the large-scale linear programming problems are solved using:

A) the graphical approach.
B) spreadsheet-based linear programming computer programs packages.
C) standard differential calculus techniques.
D) a combination of algebraic and geometric techniques.
E) matrix algebra.

Question 24

The shadow price of a nonbinding constraint is:

A) negative in value
B) zero.
C) a value between zero and one.
D) proportionate to the amount of profit
E) equal to the profit per unit of the decision variable concerned.

Question 25

A firm is maximizing profit by producing goods X and Y, using resources A and B. The firm is fully utilizing its supply of resource A, while a surplus of resource B is available. The profit contributions from per units of goods X and Y are $5 and $4 respectively. The firm is considering expansion of its supply of resource A (at a cost of $8 per unit). Increasing A by one unit would allow the firm to produce 3 additional units of X, while producing 1 fewer units of Y. Should the firm expand its supply of A? Explain.

Question 26

A new product should be introduced if its profit contribution:

A) exceeds the shadow price of each resource needed to produce it.
B) is less than the opportunity cost of producing it.
C) exceeds the total value of the resources used, valued at their respective shadow prices.
D) is greater than zero.
E) is greater than the profit contributions of the other goods produced by the firm.

Question 27

A manufacturer of nutritional products is formulating a new liquid vitamin supplement. A bottle of the new product must contain at least 30 units of vitamin B and 50 units of vitamin C. A unit of vegetable extract (V) contains 1.2 units of vitamin B and .8 units of vitamin C. A unit of fruit extract (F) contains .25 units of vitamin B and 1.8 units of vitamin C. The cost of vegetable extract is $.05 per unit and fruit extract costs $.06 per unit. The firm's goal is to minimize its cost per bottle. Formulate a linear programming problem for the firm.

Question 28

Graph and solve the following linear programming problem:
Maximize Z = 100x + 50y
Subject to: 10x + 10y $\le$ 50
y $\le$ 3, x, y $\ge$ 0

Question 29

Given, MB = Marginal benefit and MC = Marginal cost. Then, for a positive decision variable in the optimal solution, which of the following relations holds?

A) MB = MC = 0
B) MB - MC > 0
C) MB - MC = 0
D) MB - MC < 0
E) The relationship between MB and MC is indeterminate.

Question 30

A manufacturer of leather goods produces two models of briefcases - the Executive (E) and the Student (S). Each unit of the E requires 1 square yard of leather, 2.5 hours of labor, and 1 hour of machine time. The S requires .75 square yard of leather, 2 hours of labor, and .5 hours of machine time. Each unit of E contributes $8 of profit while S contributes $5. The manufacturer has 500 square yards of leather available per week, 400 labor hours, and 180 machine hours. Formulate as a linear programming problem. The basic objective is to maximize profit.

Question 31

An investor wishes to maximize the return on her portfolio while also maintaining certain liquidity and risk standards. The alternatives and their corresponding returns are:
$\begin{array} { l r } \text { Alternative } & \text { Return } \\\text { Municipal Bonds (M) } & 6.2 \% \\\text { Certificates of Deposit (S) } & 5.1 \% \\\text { Treasury Bills (T) } & 6.9 \% \\\text { AA Bonds (B) } & 10.5 \%\end{array}$
The investor wishes to have at least 25% of the portfolio in Treasury Bills, no more than 20% in AA bonds; no more than 15% in Certificates of Deposit, and no more than 10% in municipal bonds. Formulate a linear programming problem for the investor seeking to maximize the expected return of a $200,000 portfolio.

Question 32

A firm is mulling over its optimal mix of print advertising. Magazine ads (M) cost $2,500 each, reach an estimated 20,000 consumers, and generate about $7,500 revenue per ad. Newspaper ads (N) cost $1,500 each, reach an estimated 15,000 consumers, and generate about $6,000 revenue per ad. The advertising budget is $75,000 and management wishes to reach at least 600,000 consumers in total. In addition, the firm wants to have at least twice as many magazine ads as newspaper ads. Formulate the firm's linear programming problem.

Question 33

The optimal combination of decision variables of a linear programming problem is x = 20 and y = 30. The given resource constraints are: 5x + 2y  150 and y  15. The shadow price of the second constraint is:

A) 1.5.
B) )67.
C) Zero.
D) 10.
E) There is insufficient information to provide and answer.

Question 34

An investor seeks to create a portfolio from three types of securities: Treasury Bills (T), Corporate Paper (C), and Junk Bonds (J). The table lists the expected rates of return for the asset types and their average risk (on a 1-5 scale, where '5' denotes maximum risk). The investor seeks portfolio allocations (T + C + J = 1) that will maximize the expected return on her portfolio, while maintaining an overall risk of no more than '3'.
Security Return (%) Risk
Treasury Bills 2.5 2.0
Corporate Papers 5.0 2.5
Junk Bonds 8.0 5.0
(a) Formulate the investor's linear programming problem.

Question 35

A furniture manufacturer produces three types of chairs. Model A requires 1.2 hours of labor and .25 hour of machine time. Model B requires 1 hour of labor and .2 hour of machine time. Model C requires .8 hours of labor and .15 hours of machine time. There are 600 labor hours and 200 machine hours available per period. The manufacturer seeks to maximize profit and has determined that profit contributions of each unit of A, B, and C are $20, $15, and $12, respectively. Formulate as a linear program.

Question 36

Determine the feasible region and the optimal corner of the following linear programming problem:
Minimize: Z = 3x + 5y
Subject to: 2x + 5y $\ge$ 21
x + y $\ge$ 6
x $\ge$ 2 and y $\ge$ 0

Question 37

A discount appliance company is planning to advertise extensively before opening a new store. It has allocated $240,000 for advertising. The objective of the company is to maximize the total number of customers exposed to the firm's ads. Formulate a linear programming problem for the company given the information provide and find the optimal solution.
$\begin{array}{l}\begin{array}{cccc}\text { TYPE OF }&\text {COST PER AD } & \text { CUSTOMERS (1000s) } &\text { MAXIMUM }\\ \text { ADVERTISING}& (\$ 1000 \mathrm{~s})&\text { EXPOSED TO EACH AD}&\text { NUMBER OF ADS}\\ \hline \text { Radio }(\mathrm{R}) & 6 & 20 & 20 \\\text { Television }(\mathrm{T}) & 16 & 40 & 15\end{array}\end{array}$

Question 38

Determine the feasible region for the following linear programming problem:
Maximize Z = 10x + 8y
Subject to: 2x + 3y $\le$ 11
5x + 2y $\le$ 11; x, y $\ge$ 0

Question 39

A furniture manufacturer produces two types of tables. Table A sells for $430 and Table B for $300 per unit. Both types require 10 hours of labor. The hardwood requirement of Table A is $120 of per unit and that of Table B is $ per unit. The cost of labor is $10 an hour, and 400 labor hours are available per week. The firm's available supply of hardwood is $3,600 per week. Formulate, graph and solve the firm's linear programming problem.

Question 40

A firm can profitably introduce a new activity if and only if:

A) the activity's direct benefit exceeds its opportunity cost.
B))the activity's direct benefit equals its opportunity cost.
C) all the constraints become nonbinding.
D) the slope of the resource constraint equals the slope of the objective function.
E) the shadow price of any one of the resources becomes zero.

Question 41

A firm produces tires by utilizing machine-hours and labor-hours. It has the choice of producing through three separate processes using different combinations of inputs. The optimization can be done by undertaking a process singly or in combination. The combination matrix is provided below:
Process 1(X₁) Process 2(X₂) Process 3(X₃)
Machine-hours 1 2 3
Labor-hours 3 2 1
The firm can rent a machine at a price $10 and hire a labor at a wage $15. The firm needs to produce a minimum target of 50 tires per day.
(a) Formulate and solve a linear programming problem which will minimize the firm's daily cost (C).

Question 42

What are shadow prices and why are they important? Explain.

Question 43

A beverage producer produces cola at two bottling plants and distributes it to 4 major cities in the Southeast. The table below shows the bottling plant capacities, the number of cases to be shipped to each city, and the transport costs ($ per case) between plants and cities. The producer's objective is to meet its delivery requirements while minimizing its total transport costs.
(a) Formulate the firm's linear programming problem.

Question 44

The manager of an appliance store wishes to obtain survey responses from a sample of at least 900 households regarding their future-purchase plans. The cost of mail surveys (M) is $1 each, while the cost of telephone surveys (T) is $3 per call. Response rates are 60% for telephone calls and 30% for mail questionnaires. To assure sufficient response accuracy, the manager insists on gathering at least three times as many actual telephone responses as mail responses. The manager's objective is to minimize the cost of the survey (C), while meeting the stipulated goals. Formulate and solve the firm's linear programming problem.

Question 45

A producer of two types of fine chocolate bars utilizes four basic ingredients: milk, sugar, cocoa, and almonds. The milk bar (M) requires 8 ounces of milk, 2 ounces of sugar, and 3 ounces of cocoa. The almond bar (A) requires 5 ounces of milk, 1.5 ounces of sugar, 2.5 ounces of cocoa, and 2 ounces of almonds. The profit contribution of each bar is $.50. The daily availability of the ingredients is limited up to 5,000 ounces of milk, 1,200 ounces of sugar, 2,000 ounces of cocoa, and 1,000 ounces of almonds.
(a) Formulate and solve the producer's linear programming problem.