Deck 17: Linear Programming

Question 1

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

A)always change the optimal solution since the slope of the objective function changes.
B)typically change the feasible region till a stage,beyond which the constraint functions become nonbinding.
C)never change the optimal solution as the other coefficient is still fixed.
D)change the optimal solution only if the slope of the objective function's contour changes significantly.
E)their shadow prices although the constraints are not affected directly.

Answer 1

D

Question 2

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

Answer 2

A

Question 3

Constrained optimization problems form the core of a distinct managerial field.This managerial field is known as _____.

A)decision theory
B)research analysis
C)operations research
D)market analysis
E)constraint analysis

Answer 3

C

Question 4

A firm produces shaving cream and razor.The profit contribution of shaving cream is $2 per unit and that of razor is $4 per unit.The firm wishes to maximize total profit under certain constraints.What would be the slope (-razor/shaving cream,where represents the change in the variables)of the firm's objective function?

A)-4
B)-2
C)0.5
D)-0.5
E)0

Question 5

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)imposes the non-negativity condition on the decision variables

Question 6

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)does not depend on all the constraints

Question 7

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 = Y = 8.

Question 8

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 9

What must be done in order to identify the feasible region in a decision-variable plane?

A)Graph has to be plotted for all non-binding constraints.
B)Objective function contours are to be plotted.
C)Graph has to be plotted for all constraints with strict equality.
D)Appropriate equations are to be solved simultaneously.
E)One of the constraint functions is to made binding.

Question 10

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 11

A firm is pondering over 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 12

The linear programming method resembles the optimization problem.What are the common features that they incorporate?

A)Decision variables and objectives
B)Conditional variables and objectives
C)Constraints and decision variables
D)Target variables and constraints
E)Predictor variables and objectives

Question 13

If constraints in a linear programing problem take the form 2X + Y  800 and X + 2Y  700.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 14

In a linear programming problem,maximize the objective function 2S + 3T,subject to the binding constraints S + T  700 and S + 2T  1,000.The optimal combination of decision variables for this optimization is _____.

A)S = 400 and T = 300
B)S = 200 and T = 600
C)S = 700 and T = 1000
D)S = 0 and T = 1700
E)S = 1700 and T = 0

Question 15

A shadow price measures:

A)the impact of all nonbinding constraints on the objective function simultaneously.
B)the market price of the resource in question.
C)the revenue earnings from the possible sale of goods.
D)the change in the value of the objective function associated with a unit change in the resource..
E)the amount of loss incurred by not operating at the optimal point of the feasible region.

Question 16

Linear programming is useful for solving optimization problems involving _____.

A)both linear and nonlinear constraints
B)linear constraints
C)linear objective function but nonlinear constraints
D)nonlinear objective function but linear constraints
E)conventional calculus

Question 17

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)any point within the feasible region.
C)any point outside the feasible region.
D)any point where the constraint functions intersects the objective function.
E)a point which will satisfy at least one of the constraints.

Question 18

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

A)the contour of objective function does not meet to that of any constraints
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

Question 19

The feasible region in a linear programming problem contains values of the decision variable that:

A)optimize the value of the objective function.
B)are non-negative and finite in magnitude.
C)satisfy only the nonbinding constraints
D)satisfy all relevant constraints.
E)are nonnegative.

Question 20

Which of the following concerning sensitivity analysis is incorrect?

A)It is used to study changes in the objective function with respect to per unit change in one or more coefficients.
B)It is used to observe changes in the constraint function when the amount of an available resource is altered.
C)It is used to calculate the shadow prices of the decision variables.
D)It is used to derive the new optimal point of operation,when the objective function,or the constraint functions,or both change.
E)It is used to identify the correct set of simultaneous equations.

Question 21

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)0.67
C)Zero
D)10
E)0.1

Question 22

A furniture manufacturer produces three types of chairs.Model A requires 1.2 hours of labor and 0.25 hour of machine time.Model B requires 1 hour of labor and 0.2 hour of machine time.Model C requires 0.8 hours of labor and 0.15 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 23

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)MB - MC ≠ 0

Question 24

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 25

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 matrix given below and find the optimal solution.
TYPE OF COST PER AD CUSTOMERS (1000s)MAXIMUM
ADVERTISING ($1000s)EXPOSED TO EACH AD NUMBER OF ADS
Radio (R)6 20 20
Television (T)16 40 15
Direct Mail (M)1 5 40
Newspaper (N 5 10 10________

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

An investor wishes to maximize the return on her portfolio and also maintain certain liquidity and risk standards.The alternatives and their corresponding returns are:
Alternative Return
Municipal Bonds (M)6.2%
Certificates of Deposit (S)5.1%
Treasury Bills (T) 6.9%
AA Bonds (B)10.5%
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 28

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 29

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)the matrix algebra.

Question 30

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

Question 31

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 32

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 0.75 square yard of leather,2 hours of labor,and 0.5 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 33

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 34

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.Newspaper ads (N)cost $1,500 each,reach an estimated 15,000 consumers,and generate about $6,000 revenue.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 35

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 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 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 0.8 units of vitamin C.A unit of fruit extract (F)contains 0.25 units of vitamin B and 1.8 units of vitamin C.The cost of vegetable extract is $0.05 per unit and fruit extract costs $0.06 per unit.The firm's goal is to minimize its cost per bottle.Formulate a linear programming problem for the firm.

Question 38

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 39

Why are computer solutions necessary,now days,to solve linear programming problems?

Question 40

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 41

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

Question 42

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 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

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.

Question 45

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.