quant methods in business ch. 1-4 study guide

Given the following linear program that maximizes revenue: Max Z = 15 x + 20 y s.t. 8 x + 5 y ≤ 40 4 x + y ≥ 4 What is the maximum revenue at the optimal solution?

$160

A university is planning a seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume 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?

$175

#15 Taco Loco on pt2 What is the increase in revenue if Taco Loco purchases 20 pounds of cheese for $1 and uses it optimally?

$29.00

The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the optimal weekly profit?

$800

Variable cost does include:

-material handling and freight. -raw materials and resources. -direct labor and packaging.

# 16 on pt 2 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. The constraint for ingredient 3 is:

.3B ≥ 20.

Quickbrush Paint Company is developing a linear program to determine the optimal quantities of ingredient A and ingredient B to blend together to make oil-base and water-base paint. 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. Assuming that x represents the number of gallons of oil-base paint, and y represents the gallons of water-base paint, which constraint is correctly represents the constraint on ingredient A?

.9 x + .3 y ≤ 10,000

The owner of Chips etc. produces two kinds of chips: lime (L) and vinegar (V). He has a limited amount of the three ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of lime chips are $0.40, and for a bag of vinegar chips $0.50. Which of the following is not a feasible production combination?

0L and 1200V

#13 Taco Loco on pt2 Taco Loco is unsure whether the amount of beef that their computer thinks is in inventory is correct. What is the range in values for beef inventory that would not affect the optimal product mix?

17.78 to 30 pounds

Administrators at a university will charge students $150 to attend a seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 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?

24

#14 Taco Loco on pt2 How many pounds of beans will Taco Loco have left over if they produce the optimal quantity of products X, Y, and Z?

28.73

The owner of Chips etc. produces two kinds of chips: lime (L) and vinegar (V). He has a limited amount of the three ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of lime chips are $0.40, and for a bag of vinegar chips $0.50. What is the constraint for salt?

2L + 3V ≤ 4800

The production manager for the Coory soft drink company is considering the production of two kinds of soft drinks: regular (R) and diet(D). Two of the limited 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?

2R + 4D ≤ 480

The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. Which of the following is not a feasible solution?

300 L and 200 D

Company A has a fixed cost of 75,000 and a variable cost of 15. Company B's fixed cost is 90,000 and variable cost is 11. At what point is Yowzah indifferent between the two bidders?

3750

#3. on pt 2 Atwitter A portion of the variable cells section of the sensitivity report in Excel appears in the table below. How many potential customers will be reached by the optimal advertising campaign?

42,080

Cully Furniture buys two 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 $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the maximum profit?

45,000

#2. on pt 2 Atwitter Which of these is an appropriate constraint for this scenario?

500N + 250R + 125T + 15F ≤ 3,500

Which of the following statements about infeasible problems is best?

All of the possible solutions violate at least one constraint.

Which of these statements is best?

An unbounded problem has feasible solutions.

Cully Furniture buys two 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 $75,000 to invest in shelves this week, and the warehouse has 18,000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. In order to maximize profit, how many big shelves (B) and how many medium shelves (M) should be purchased?

B = 150, M = 0

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 line is represented by the equation 2X + Y ≥ 8?

BF

#5. on pt 2 Atwitter What is the proper interpretation of the shadow price for Facebook?

Every additional dollar spent on Facebook advertising gains 20 customers as long as the number of postings does not exceed 233

________ are generally independent of the volume of units produced and sold.

Fixed costs

#4. on pt 2 Atwitter Which of these statements about the sensitivity report for the constraints is best?

For every $1 increase in the budget, the ad campaign can reach twelve more customers.

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 is not feasible?

G

#6. on pt 2 Atwitter How should the entry for the Newspaper decision variable be interpreted?

If the director were forced to purchase a newspaper advertisement, he would reach 1000 fewer customers than would be reached using the optimal advertising campaign.

In a linear programming problem, a valid objective function can be represented as:

Max 3 x + 3 y + 1/3 z

# 1. on pt. 2 AtwitterThe college director of global activities was hopeful that the print ads in the school newspaper and 30 second spots on the campus radio station would spur some interest in the array of study tour and study abroad options he had secured. The communications director for the college had other ideas; she favored a social media campaign consisting of tweets and facebook postings. "This is the most ridiculous thing I ever heard of," he whined to the dean. The communications director's market research revealed the following: The advertising budget is $3500, but there is no requirement that all the money be spent. The newspaper has only four issues before the end of the semester, but the radio is a 24/7 operation and has two dozen 30 second slots available. Facebook postings must be alternated with the rest of the mindless drivel posted on the college page; thus there is space for only three postings before the end of the semester. Twitter is complicated by the 140 character requirement. The communications director feels she needs five tweets to convey a single message about tours and semesters abroad, so for one message, the cost would be $125 for each of the five components of the single ad. Due to thumb fatigue, she feels that she has only 2800 characters left in her thumbs before the end of the semester. (A side note - During the intersession period, she plans to embark on a strict regimen of thumb yoga to prepare for the coming semester.) What is an appropriate objective function for this scenario?

Max Z = 5,000N + 3,000R + 700T + 200F

The production manager for the Coory soft drink company is considering the production of two 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 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 are the optimal daily production quantities of each product and the optimal daily profit?

R = 90, D = 75, Z = $420

A hot dog manufacturer wishes to minimize the cost in dollars of producing a low-cost niched product while meeting the dietary guidelines for protein and sodium. Once the model has been run, the surplus variable in the sodium constraint has a value of 1300 milligrams. The best interpretation of this outcome is:

The minimum cost hot dog has 1300 milligrams more sodium than required.

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:

X + 2Y ≥ 8.

Which of the following is an equation or an inequality that expresses a resource restriction in a mathematical model?

a constraint

How would multiple optimal solutions typically appear on a graphical solution?

a line

In the formulation of a ≥ constraint:

a surplus variable is subtracted.

In a multiperiod scheduling problem, the production constraint usually takes the form of:

beginning inventory - demand + production = ending inventory.

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

break-even point.

#12 Taco Loco on pt2 Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. (Note, all units are in pounds, not ounces like we used on our in-class worksheet.) Max R = 14Z + 13Y + 17Xsubject to:Beef 2Z + 3Y + 4X ≤ 28Cheese 9Z + 8Y + 11X ≤ 80Beans 4Z + 4Y + 2X ≤ 68X,Y,Z ≥ 0The sensitivity report from the computer model reads as follows: The local cheese vendor offers to sell Taco Loco 200 pounds of cheese for these three products. Taco Loco should:

buy 46 pounds or less of cheese for $1.45 or less.

If fixed costs decrease, but variable cost and price remain the same, the break-even point:

decreases.

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

decreases.

Sensitivity analysis

determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations.

A model is a functional relationship that includes:

equations. variables. parameters.

The region that satisfies all of the constraints in a graphical linear programming problem is called the:

feasible solution space.

When systematically formulating a linear program, the first step is to:

identify the decision variables.

For a maximization problem, the shadow price measures the ________ in the value of the optimal solution, per unit increase for a given ________.

improvement, resource

A slack variable:

is the amount by which the left side of a ≤ constraint is smaller than the right side.

The optimal solution to a linear programming model that has been solved using the graphical approach:

is typically at some corner of the feasible region.

Consider the following linear program:

is unbounded.

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*. This linear programming problem is a(n):

minimization problem.

The production manager for the Coory soft drink company is considering the production of two 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 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. For the production combination of 135 cases of regular and 0 cases of diet soft drink, which resources will not be completely used?

only time

Multiple optimal solutions can occur when the objective function is ________ a constraint line.

parallel to

Standard form

requires all variables in the constraint equations to appear as sums or differences on the left of the inequality/equality and all numeric values to be on the right-hand side.

The purpose of break-even analysis is to determine the number of units of a product to sell that will:

result in zero profit.

The owner of Chips etc. produces two kinds of chips: lime (L) and vinegar (V). He has a limited amount of the three ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of lime chips are $0.40, and for a bag of vinegar chips $0.50. For the production combination of 800 bags of lime and 600 bags of vinegar, which of the three resources is (are) not completely used?

salt only

For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs. and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:

same product mix, different total profit.

Variable cost does not include:

staff and management salaries.

At the break-even point:

total revenue equals total cost.

The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. The manager can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week, respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. If the production manager decides to produce of 400 bottles of light beer and 0 bottles of dark beer, it will result in slack of:

wheat only.

Use the constraints given below and determine which of the following points is feasible. (1) 14 x + 6 y ≤ 42 (2) x - y ≤ 3

x = 1; y = 4

What combination of x and y is a feasible solution that minimizes the value of the objective function? Min Z = 3x + 15y (1) 2x + 4y ≥ 12 (2) 5x + 2y ≥10

x = 6; y = 0