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QUIZ 1 1 Probabilistic techniques assume that no uncertainty exists in model parameters. 2 Fixed cost is the difference between total cost and total variable cost. 3 Parameters are known, constant values that are usually coefficients of variables in equations. 4 In general, an increase in price increases the break even point if all costs are held constant. 5 A continuous random variable may assume only integer values within a given interval. 6 If events A and B are independent, then P(A|B) = P(B|A). 7 P(A | B) is the probability of event A, if we already know that event B has occurred. 8 The purpose of break]even analysis is to determine the number of units of a product to sell that will 9 If the price increases but fixed and variable costs do not change, the break even point 10 The indicator that results in total revenues being equal to total cost is called the 11 A bed and breakfast breaks even every month if they book 30 rooms over the course of a month. Their fixed cost is $4200 per month and the revenue they receive from each booked room is $180. What their variable cost per occupied room? 12 The expected value of the standard normal distribution is equal to 13 The area under the normal curve represents probability, and the total area under the curve sums to 14 In a binomial distribution, for each of n trials, the event 15 Administrators at a university are planning to offer a summer seminar. The costs of reserving a room, hiring an instructor, and bringing in the equipment amount to $3000. Suppose that it costs $25 per student for the administrators to provide the course materials. If we know that 20 people will attend, what price should be charged per person to break even? Note: please report the result as a whole number, rounding if necessary and omitting the decimal point. 16 Administrators at a university will charge students $158 to attend a seminar. It costs $2160 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $50 per student for the administrators to provide the course materials. How many students would have to register for the seminar for the university to break even? Note: please report the result as a whole number, omitting the decimal point. 17 A production run of toothpaste requires a fixed cost of $100,000. The variable cost per unit is $3.00. If 50,000 units of toothpaste will be sold during the next month, what sale price must be chosen in order to break even at the end of the month? Note: please report the result as a whole number, rounding if necessary and omitting the decimal point. 18 Wei is considering pursuing an MS in Information Systems degree. She has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Wei will be accepted by at least one of the two universities? {Express your answer as a percent. Round (if necessary) to the nearest whole percent and omit the decimal. For instance, 20.1% would be written as 20} 19 An automotive center keeps tracks of customer complaints received each week. The probability distribution for complaints can be represented as a table (shown below). The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints. xi 0 1 2 3 4 5 6 p(xi) .10 .15 .18 .20 .20 .10 .07 What is the average number of complaints received per week? Note: Please report your answer with two places to the right of the decimal, rounding if appropriate. 20 The variance of the standard normal distribution is equal to __________. QUIZ 2 1 Probability trees are used only to compute conditional probabilities. 2 If two events are not mutually exclusive, then P(A or B) = P(A) + P(B) 3 Seventy two percent of all observations fall within 1 standard deviation of the mean if the data is normally distributed. 4 The Hurwicz criterion is a compromise between the maximax and maximin criteria. 5 The maximin approach involves choosing the alternative with the highest or lowest payoff. 6 The minimin criterion is optimistic. 7 Both maximin and minimin criteria are optimistic. 8 A professor would like to utilize the normal distribution to assign grades such that 5% of students receive A’s. If the exam average is 62 with a standard deviation of 13, what grade should be the cutoff for an A? (Round your answer.) 9 The metropolitan airport commission is considering the establishment of limitations on noise pollution around a local airport. At the present time, the noise level per jet takeoff in one neighborhood near the airport is approximately normally distributed with a mean of 100 decibels and a standard deviation of 3 decibels. What is the probability that a randomly selected jet will generate a noise level of more than 105 decibels? 10 The chi]square test is a statistical test to see if an observed data fit a _________. 11 The maximin criterion results in the 12 A group of friends are planning a recreational outing and have constructed the following payoff table to help them decide which activity to engage in. Assume that the payoffs represent their level of enjoyment for each activity under the various weather conditions. Weather Cold Warm Rainy S1 S2 S3 Bike: A1 10 8 6 Hike: A2 14 15 2 Fish: A3 7 8 9 What is the conservative decision for this situation? 13 A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow. The maximax strategy is: 14 A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow. The maximin strategy is: 15 A brand of television has a lifetime that is normally distributed with a mean of 7 years and a standard deviation of 2.5 years. What is the probability that a randomly chosen TV will last more than 8 years? Note: Write your answers with two places after the decimal, rounding off as appropriate. 16 A life insurance company wants to update its actuarial tables. Assume that the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 71 years and a standard deviation of 3.5 years. What proportion of the plan participants are expected to see their 75th birthday? Note: Write your answers with two places after the decimal, rounding off as appropriate. 17 A group of friends are planning a recreational outing and have constructed the following payoff table to help them decide which activity to engage in. Assume that the payoffs represent their level of enjoyment for each activity under the various weather conditions. Weather Cold Warm Rainy S1 S2 S3 Bike: A1 10 8 6 Hike: A2 14 15 2 Fish: A3 7 8 9 If the probabilities of cold weather (S1), warm weather (S2), and rainy weather (S3) are 0.2, 0.4, and 0.4, respectively what is the EVPI for this situation? 18 A manager has developed a payoff table that indicates the profits associated with a set of alternatives under 2 possible states of nature. Alt S1 S2 1 10 2 2 ]2 8 3 8 5 What is the highest expected value? Assume that the probability of S2 is equal to 0.4. 19 Consider the following decision tree. What is the expected value for the best decision? Round your answer to the nearest whole number. 20 The quality control manager for ENTA Inc. must decide whether to accept (a1), further analyze (a2) or reject (a3) a lot of incoming material. Assume the following payoff table is available. Historical data indicates that there is 30% chance that the lot is poor quality (s1), 50 % chance that the lot is fair quality (s2) and 20% chance that the lot is good quality (s3). What is the numerical value of the minimax regret? Quiz 3 1 In a linear programming problem, all model parameters are assumed to be known with certainty. 2 A linear programming problem may have more than one set of solutions. 3 The following inequality represents a resource constraint for a maximization problem: X + Y . 20 4 In minimization LP problems the feasible region is always below the resource constraints. 5 If the objective function is parallel to a constraint, the constraint is infeasible. 6 If the objective function is parallel to a constraint, the constraint is infeasible. 7 A feasible solution violates at least one of the constraints. 8 Decision variables 9 The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z*. The equation for constraint DH is: 10 The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z*. Which of the following points are not feasible? 11 In a linear programming problem, the binding constraints for the optimal solution are: 5×1 + 3×2 . 30 2×1 + 5×2 . 20 Which of these objective functions will lead to the same optimal solution? 12 Which of the following could be a linear programming objective function? 13 The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the objective function? 14 Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the objective function? 15 The linear programming problem: MIN Z = 2×1 + 3×2 Subject to: x1 + 2×2 . 20 5×1 + x2 . 40 4×1 +6×2 . 60 x1 , x2 . 0 , 16 Which of the following statements is not true? 17 The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. For the production combination of 135 cases of regular and 0 cases of diet soft drink, which resources will not be completely used? 18 Consider the following minimization problem: Min z = x1 + 2×2 s.t. x1 + x2 . 300 2×1 + x2 . 400 2×1 + 5×2 . 750 x1, x2 . 0 Find the optimal solution. What is the value of the objective function at the optimal solution? Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty five) would be written 25 19 Solve the following graphically Max z = 3×1 +4×2 s.t. x1 + 2×2 . 16 2×1 + 3×2 . 18 x1 . 2 x2 . 10 x1, x2 . 0 Find the optimal solution. What is the value of the objective function at the optimal solution? Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty five) would be written 25 20 A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function. What would be the new slope of the objective function if multiple optimal solutions occurred along line segment AB? Write your answer in decimal notation. QUiz 4 1 In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities. 2 Fractional relationships between variables are permitted in the standard form of a linear program. 3 The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign 4 Product mix problems cannot have “greater than or equal to” (.) constraints. 5 In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories. 6 In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less]than]or equal]to constraint (.). 7 Balanced transportation problems have the following type of constraints: 8 When systematically formulating a linear program, the first step is 9 A systematic approach to model formulation is to first 10 In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct? 11 The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient. Ingredient Percent per pound in Feed A Percent per pound in Feed B Minimum daily requirement (pounds) 1 20 24 30 2 30 10 50 3 0 30 20 4 24 15 60 5 10 20 40 The constraint for ingredient 3 is: 12 In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint? 13 The following types of constraints are ones that might be found in linear programming formulations: 1. . 2. = 3. > 14 The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint? 15 Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1. 16 In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is 17 Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf blowers). The company wants to minimize the cost of transporting items between the facilities, taking into account the demand at the 3 different plants, and the supply at each manufacturing site. The table below shows the cost to ship one unit between each manufacturing facility and each plant, as well as the demand at each plant and the supply at each manufacturing facility. What is the demand constraint for plant B? 18 Compared to blending and product mix problems, transportation problems are unique because 19 Quickbrush Paint Company makes a profit of $2 per gallon on its oil]base paint and $3 per gallon on its water]base paint. Both paints contain two ingredients, A and B. The oil]base paint contains 90 percent A and 10 percent B, whereas the water]base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil]base and water]base paint to produce to maximize its total profit. How many gallons of water based paint should the Quickbrush make? Note: Please express your answer as a whole number, rounding the nearest whole number, if appropriate. 20 Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows: Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan? Express your answer with two places to the right of the decimal point. For instance, $9.32 (nine dollars and thirty]two cents) would be written as 9.32 QUIZ 5 1 In a mixed integer model, some solution values for decision variables are integer and others are only 0 or 1. 2 If we are solving a 0]1 integer programming problem with three decision variables, the constraint x1 + x2 + x3 . 3 is a mutually exclusive constraint. 3 Rounding non]integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem. 4 The solution to the LP relaxation of a maximization integer linear program provides an upper bound for the value of the objective function. 5 A conditional constraint specifies the conditions under which variables are integers or real variables. 6 If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate constraints in an integer program. 7 Max Z = 5×1 + 6×2 Subject to: 17×1 + 8×2 . 136 3×1 + 4×2 . 36 x1, x2 . 0 and integer What is the optimal solution? 8 Assume that we are using 0]1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise. The constraint (x1 + x2 + x3 + x4 . 2) means that __________ out of the 4 projects must be selected. 9 If we are solving a 0]1 integer programming problem, the constraint x1 + x2 . 1 is a __________ constraint. 10 In a __________ integer model, some solution values for decision variables are integers and others can be non]integer. 11 If we are solving a 0]1 integer programming problem, the constraint x1 . x2 is a __________ constraint. 12 In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation? 13 You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint(s) for the second restriction 14 You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, the constraint for the first restriction is 15 The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. Write the constraint that indicates they can purchase no more than 3 machines. 16 In a 0]1 integer programming model, if the constraint x1]x2 = 0, it means when project 1 is selected, project 2 __________ be selected. 17 If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is 18 If we are solving a 0]1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint. 19 Consider the following integer linear programming problem Max Z = 3×1 + 2×2 Subject to: 3×1 + 5×2 . 30 5×1 + 2×2 . 28 x1 . 8 x1 ,x2 . 0 and integer Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty]five) would be written 25 20 Consider the following integer linear programming problem Max Z = 3×1 + 2×2 Subject to: 3×1 + 5×2 . 30 4×1 + 2×2 . 28 x1 . 8 x1 , x2 . 0 and integer Find the optimal solution. What is the value of the objective function at the optimal solution. Note: The answer will be an integer. Please give your answer as an integer without any decimal point. For example, 25.0 (twenty]five) would be written 25

QUIZ 1

1

Probabilistic techniques assume that no uncertainty exists in model parameters.

2

Fixed cost is the difference between total cost and total variable cost.

3

Parameters are known, constant values that are usually coefficients of variables in equations.

4

In general, an increase in price increases the break even point if all costs are held constant.

5

A continuous random variable may assume only integer values within a given interval.

6

If events A and B are independent, then P(A|B) = P(B|A).

7

P(A | B) is the probability of event A, if we already know that event B has occurred.

8

The purpose of break]even analysis is to determine the number of units of a product to sell that

will

9

If the price increases but fixed and variable costs do not change, the break even point

10

The indicator that results in total revenues being equal to total cost is called the

11

A bed and breakfast breaks even every month if they book 30 rooms over the course of a

month. Their fixed cost is $4200 per month and the revenue they receive from each booked

room is $180. What their variable cost per occupied room?

12

The expected value of the standard normal distribution is equal to

13

The area under the normal curve represents probability, and the total area under the curve

sums to

14

In a binomial distribution, for each of n trials, the event

15

Administrators at a university are planning to offer a summer seminar. The costs of reserving a

room, hiring an instructor, and bringing in the equipment amount to $3000.

Suppose that it costs $25 per student for the administrators to provide the course materials. If

we know that 20 people will attend, what price should be charged per person to break

even? Note: please report the result as a whole number, rounding if necessary and omitting

the decimal point.

16

Administrators at a university will charge students $158 to attend a seminar. It costs $2160 to

reserve a room, hire an instructor, and bring in the equipment. Assume it costs $50 per student

for the administrators to provide the course materials. How many students would have to

register for the seminar for the university to break even? Note: please report the result as a

whole number, omitting the decimal point.

17

A production run of toothpaste requires a fixed cost of $100,000. The variable cost per unit is

$3.00. If 50,000 units of toothpaste will be sold during the next month, what sale price must be

chosen in order to break even at the end of the month? Note: please report the result as a

whole number, rounding if necessary and omitting the decimal point.

18

Wei is considering pursuing an MS in Information Systems degree. She has applied to two

different universities. The acceptance rate for applicants with similar qualifications is 20% for

University X and 45% for University Y. What is the probability that Wei will be accepted by at

least one of the two universities? {Express your answer as a percent. Round (if necessary) to the

nearest whole percent and omit the decimal. For instance, 20.1% would be written as 20}

19

An automotive center keeps tracks of customer complaints received each week. The probability

distribution for complaints can be represented as a table (shown below). The random variable

xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.

xi 0 1 2 3 4 5 6

p(xi) .10 .15 .18 .20 .20 .10 .07

What is the average number of complaints received per week? Note: Please report your

answer with two places to the right of the decimal, rounding if appropriate.

20

The variance of the standard normal distribution is equal to __________.

QUIZ 2

1

Probability trees are used only to compute conditional probabilities.

2

If two events are not mutually exclusive, then P(A or B) = P(A) + P(B)

3

Seventy two percent of all observations fall within 1 standard deviation of the mean if the data

is normally distributed.

4

The Hurwicz criterion is a compromise between the maximax and maximin criteria.

5

The maximin approach involves choosing the alternative with the highest or lowest payoff.

6

The minimin criterion is optimistic.

7

Both maximin and minimin criteria are optimistic.

8

A professor would like to utilize the normal distribution to assign grades such that 5% of

students receive A’s. If the exam average is 62 with a standard deviation of 13, what grade

should be the cutoff for an A? (Round your answer.)

9

The metropolitan airport commission is considering the establishment of limitations on noise

pollution around a local airport. At the present time, the noise level per jet takeoff in one

neighborhood near the airport is approximately normally distributed with a mean of 100

decibels and a standard deviation of 3 decibels. What is the probability that a randomly

selected jet will generate a noise level of more than 105 decibels?

10

The chi]square test is a statistical test to see if an observed data fit a _________.

11

The maximin criterion results in the

12

A group of friends are planning a recreational outing and have constructed the following payoff

table to help them decide which activity to engage in. Assume that the payoffs represent their

level of enjoyment for each activity under the various weather conditions.

Weather

Cold Warm Rainy

S1 S2 S3

Bike: A1 10 8 6

Hike: A2 14 15 2

Fish: A3 7 8 9

What is the conservative decision for this situation?

13

A business owner is trying to decide whether to buy, rent, or lease office space and has

constructed the following payoff table based on whether business is brisk or slow.

The maximax strategy is:

14

A business owner is trying to decide whether to buy, rent, or lease office space and has

constructed the following payoff table based on whether business is brisk or slow.

The maximin strategy is:

15

A brand of television has a lifetime that is normally distributed with a mean of 7 years and a

standard deviation of 2.5 years. What is the probability that a randomly chosen TV will last

more than 8 years? Note: Write your answers with two places after the decimal, rounding off as

appropriate.

16

A life insurance company wants to update its actuarial tables. Assume that the probability

distribution of the lifetimes of the participants is approximately a normal distribution with a

mean of 71 years and a standard deviation of 3.5 years. What proportion of the plan

participants are expected to see their 75th birthday? Note: Write your answers with two places

after the decimal, rounding off as appropriate.

17

A group of friends are planning a recreational outing and have constructed the following payoff

table to help them decide which activity to engage in. Assume that the payoffs represent their

level of enjoyment for each activity under the various weather conditions.

Weather

Cold Warm Rainy

S1 S2 S3

Bike: A1 10 8 6

Hike: A2 14 15 2

Fish: A3 7 8 9

If the probabilities of cold weather (S1), warm weather (S2), and rainy weather (S3) are 0.2, 0.4,

and 0.4, respectively what is the EVPI for this situation?

18

A manager has developed a payoff table that indicates the profits associated with a set of

alternatives under 2 possible states of nature.

Alt S1 S2

1 10 2

2 ]2 8

3 8 5

What is the highest expected value? Assume that the probability of S2 is equal to 0.4.

19

Consider the following decision tree.

What is the expected value for the best decision? Round your answer to the nearest whole

number.

20

The quality control manager for ENTA Inc. must decide whether to accept (a1), further analyze

(a2) or reject (a3) a lot of incoming material. Assume the following payoff table is available.

Historical data indicates that there is 30% chance that the lot is poor quality (s1), 50 % chance

that the lot is fair quality (s2) and 20% chance that the lot is good quality (s3).

What is the numerical value of the minimax regret?

Quiz 3

1

In a linear programming problem, all model parameters are assumed to be known with

certainty.

2

A linear programming problem may have more than one set of solutions.

3

The following inequality represents a resource constraint for a maximization problem:

X + Y . 20

4

In minimization LP problems the feasible region is always below the resource constraints.

5

If the objective function is parallel to a constraint, the constraint is infeasible.

6

If the objective function is parallel to a constraint, the constraint is infeasible.

7

A feasible solution violates at least one of the constraints.

8

Decision variables

9

The following is a graph of a linear programming problem. The feasible solution space is shaded,

and the optimal solution is at the point labeled Z*.

The equation for constraint DH is:

10

The following is a graph of a linear programming problem. The feasible solution space is shaded,

and the optimal solution is at the point labeled Z*.

Which of the following points are not feasible?

11

In a linear programming problem, the binding constraints for the optimal solution are:

5×1 + 3×2 . 30

2×1 + 5×2 . 20

Which of these objective functions will lead to the same optimal solution?

12

Which of the following could be a linear programming objective function?

13

The production manager for the Coory soft drink company is considering the production of 2

kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8

hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To

produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4

minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for

diet soft drink are $2.00 per case. What is the objective function?

14

Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big

shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs

$300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves

this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big

shelf is $300 and for each medium shelf is $150. What is the objective function?

15

The linear programming problem:

MIN Z = 2×1 + 3×2

Subject to: x1 + 2×2 . 20

5×1 + x2 . 40

4×1 +6×2 . 60

x1 , x2 . 0 ,

16

Which of the following statements is not true?

17

The production manager for the Coory soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours

= 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To

produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4

minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for

diet soft drink are $2.00 per case. For the production combination of 135 cases of regular and 0

cases of diet soft drink, which resources will not be completely used?

18

Consider the following minimization problem:

Min z = x1 + 2×2

s.t. x1 + x2 . 300

2×1 + x2 . 400

2×1 + 5×2 . 750

x1, x2 . 0

Find the optimal solution. What is the value of the objective function at the optimal

solution? Note: The answer will be an integer. Please give your answer as an integer without

any decimal point. For example, 25.0 (twenty five) would be written 25

19

Solve the following graphically

Max z = 3×1 +4×2

s.t. x1 + 2×2 . 16

2×1 + 3×2 . 18

x1 . 2

x2 . 10

x1, x2 . 0

Find the optimal solution. What is the value of the objective function at the optimal

solution? Note: The answer will be an integer. Please give your answer as an integer without

any decimal point. For example, 25.0 (twenty five) would be written 25

20

A graphical representation of a linear program is shown below. The shaded area represents the

feasible region, and the dashed line in the middle is the slope of the objective function.

What would be the new slope of the objective function if multiple optimal solutions occurred

along line segment AB? Write your answer in decimal notation.

QUiz 4

1

In a balanced transportation model, supply equals demand such that all constraints can be

treated as equalities.

2

Fractional relationships between variables are permitted in the standard form of a linear

program.

3

The standard form for the computer solution of a linear programming problem requires all

variables to be to the right and all numerical values to be to the left of the inequality or equality

sign

4

Product mix problems cannot have “greater than or equal to” (.) constraints.

5

In formulating a typical diet problem using a linear programming model, we would expect most

of the constraints to be related to calories.

6

In a transportation problem, a demand constraint (the amount of product demanded at a given

destination) is a less]than]or equal]to constraint (.).

7

Balanced transportation problems have the following type of constraints:

8

When systematically formulating a linear program, the first step is

9

A systematic approach to model formulation is to first

10

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2,

an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates

that stock 1 must not account for more than 35% of the number of shares purchased. Which

constraint is correct?

11

The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef

feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential

ingredients are contained in the feed, shown in the table below. The table also shows the

minimum daily requirements of each ingredient.

Ingredient

Percent per pound

in Feed A

Percent per pound

in Feed B

Minimum daily

requirement

(pounds)

1 20 24 30

2 30 10 50

3 0 30 20

4 24 15 60

5 10 20 40

The constraint for ingredient 3 is:

12

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2,

an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to

$50,000 to invest. The stockbroker suggests limiting the investments so that no more than

$10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed

350, whichever is more restrictive. How would this be formulated as a linear programming

constraint?

13

The following types of constraints are ones that might be found in linear programming

formulations:

1. .

2. =

3. >

14

The production manager for the Softy soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480

minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a

regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3

gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink

are $2.00 per case. What is the time constraint?

15

Let xij = gallons of component i used in gasoline j. Assume that we have two components and

two types of gasoline. There are 8,000 gallons of component 1 available, and the demand

gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint

for component 1.

16

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2,

an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to

$50,000 to invest. The expected returns on investment of the three stocks are 6%, 8%, and

11%. An appropriate objective function is

17

Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be

shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf

blowers). The company wants to minimize the cost of transporting items between the facilities,

taking into account the demand at the 3 different plants, and the supply at each manufacturing

site. The table below shows the cost to ship one unit between each manufacturing facility and

each plant, as well as the demand at each plant and the supply at each manufacturing facility.

What is the demand constraint for plant B?

18

Compared to blending and product mix problems, transportation problems are unique because

19

Quickbrush Paint Company makes a profit of $2 per gallon on its oil]base paint and $3 per

gallon on its water]base paint. Both paints contain two ingredients, A and B. The oil]base paint

contains 90 percent A and 10 percent B, whereas the water]base paint contains 30 percent A

and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of

ingredient B in inventory and cannot obtain more at this time. The company wishes to use

linear programming to determine the appropriate mix of oil]base and water]base paint to

produce to maximize its total profit. How many gallons of water based paint should the

Quickbrush make? Note: Please express your answer as a whole number, rounding the nearest

whole number, if appropriate.

20

Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality

of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by

mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the

two cat foods are as follows:

Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3

ounces of fat per day. What is the cost of this plan? Express your answer with two places to the

right of the decimal point. For instance, $9.32 (nine dollars and thirty]two cents) would be

written as 9.32

QUIZ 5

1

In a mixed integer model, some solution values for decision variables are integer and others are

only 0 or 1.

2

If we are solving a 0]1 integer programming problem with three decision variables, the

constraint x1 + x2 + x3 . 3 is a mutually exclusive constraint.

3

Rounding non]integer solution values up to the nearest integer value will result in an infeasible

solution to an integer linear programming problem.

4

The solution to the LP relaxation of a maximization integer linear program provides an upper

bound for the value of the objective function.

5

A conditional constraint specifies the conditions under which variables are integers or real

variables.

6

If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3

separate constraints in an integer program.

7

Max Z = 5×1 + 6×2

Subject to: 17×1 + 8×2 . 136

3×1 + 4×2 . 36

x1, x2 . 0 and integer

What is the optimal solution?

8

Assume that we are using 0]1 integer programming model to solve a capital budgeting problem

and xj = 1 if project j is selected and xj = 0, otherwise.

The constraint (x1 + x2 + x3 + x4 . 2) means that __________ out of the 4 projects must be

selected.

9

If we are solving a 0]1 integer programming problem, the constraint x1 + x2 . 1 is a __________

constraint.

10

In a __________ integer model, some solution values for decision variables are integers and

others can be non]integer.

11

If we are solving a 0]1 integer programming problem, the constraint x1 . x2 is a

__________ constraint.

12

In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot

be selected. Which of the alternatives listed below correctly models this situation?

13

You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate

each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:

Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.

Restriction 2. Evaluating sites S2 or

S4 will prevent you from assessing site S5.

Restriction 3. Of all the sites, at least 3 should be assessed.

Assuming that Si is a binary variable, write the constraint(s) for the second restriction

14

You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate

each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:

Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.

Restriction 2. Evaluating sites S2 or

S4 will prevent you from assessing site S5.

Restriction 3. Of all the sites, at least 3 should be assessed.

Assuming that Si is a binary variable, the constraint for the first restriction is

15

The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff

has 4 different machines that can produce this kind of hose. Because these machines are from

different manufacturers and use differing technologies, their specifications are not the same.

Write the constraint that indicates they can purchase no more than 3 machines.

16

In a 0]1 integer programming model, if the constraint x1]x2 = 0, it means when project 1 is

selected, project 2 __________ be selected.

17

If the solution values of a linear program are rounded in order to obtain an integer solution, the

solution is

18

If we are solving a 0]1 integer programming problem, the constraint x1 + x2 = 1 is a __________

constraint.

19

Consider the following integer linear programming problem

Max Z = 3×1 + 2×2

Subject to: 3×1 + 5×2 . 30

5×1 + 2×2 . 28

x1 . 8

x1 ,x2 . 0 and integer

Find the optimal solution. What is the value of the objective function at the optimal

solution. Note: The answer will be an integer. Please give your answer as an integer without any

decimal point. For example, 25.0 (twenty]five) would be written 25

20

Consider the following integer linear programming problem

Max Z = 3×1 + 2×2

Subject to: 3×1 + 5×2 . 30

4×1 + 2×2 . 28

x1 . 8

x1 , x2 . 0 and integer

Find the optimal solution. What is the value of the objective function at the optimal

solution. Note: The answer will be an integer. Please give your answer as an integer without any

decimal point. For example, 25.0 (twenty]five) would be written 25

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