ch 2 app stat
What is the equation for the constraint AB? 3X + 12Y ≥ 15 12X +3Y ≥ 36 X + Y ≥ 15 X + 4Y ≥ 12
12X +3Y ≥ 36
What is the proper interpretation of the shadow price for Facebook? Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$9 Facebook 4 20 4 229.3333333 4 Every additional dollar spent on Facebook advertising gains 20 customers as long as the number of postings does not exceed 233. If they spend $4, then can reach 20 customers. Every additional dollar spent on Facebook advertising gains 20 customers as long as the number of postings does not exceed 229. If they spend less than $20, they can reach 229.333 customers.
Every additional dollar spent on Facebook advertising gains 20 customers as long as the number of postings does not exceed 233.
Coory soft drink $3 per case regular $2 per case diet what is the objective function?
Max $3r + $2d
In a linear programming problem, a valid objective function can be represented as: Max Z 5x2 + 2y2 Min (x1 + x2) / x3 Max Z = 5xy Max 3x + 3y + 1/3 z
Max 3x + 3y + 1/3 z
For a maximization problem, the shadow price measures the ________ in the value of the optimal solution, per unit increase for a given ________. decrease, resource decrease, parameter improvement, resource increase, parameter
improvement, resource
Billy has decided that he can raise the price on the Curious t-shirt by 10% without losing sales. If he raises the price, his profits will: increase by $125. decrease by 10%. increase by $2.50. increase by 10%.
increase by $125.
Greg, a young entrepreneur, has developed an aggressive business plan and is presenting his profit projections on the popular show Shark Tank in hopes of securing some venture capital. He concludes his presentation with an LP model of his planned product mix, and is convinced he will seal the deal by demonstrating that his profits are limitless since his LP model is unbounded. What should the sharks tell him? "Limitless profits are possible only in maximization models, and we want you to minimize profits." "Limitless profits are possible only in minimization models, and we want you to maximize profits." "Unlimited profits aren't possible. You must have made a mistake in your LP model." "Limitless profits sound fantastic, here's a blank check."
"Unlimited profits aren't possible. You must have made a mistake in your LP model."
Billy's accountant made an error, and the budget has been reduced from $3000 to $2500. Billy's profit will go down by: $0. $1650. $625. $1350.
$0.
Aunt Anastasia feels that her prices are too low, particularly for her eggs. How much would her profit have to increase on the eggs before it is profitable for her to make and sell eggs? $1.50 $0.50 $2.50 $1.00
$1.00
Max Z = 15x + 20y s.t. 8x + 5y ≤ 40 4x + y ≥ 4 What is the maximum revenue at the optimal solution? $185 $160 $200 $120
$160
What is the increase in revenue if Taco Loco purchases 20 pounds of cheese for $1 and uses it optimally? $29.00 $158.18 $0 $9.09
$29.00
If one of Billy's machines breaks down, it usually results in about 6 hours of downtime. When this happens, Billy's profits are reduced by: $15. $18. $35. $25.
$35.
A croissant shop produces two products: bear claws (B) and almond-filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond-filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each. What is the optimal daily profit? $380 $400 $440 $420
$380
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? $55,000 $65,000 $35,000 $45,000
$45,000
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? $700 $800 $1000 $900
$800
The constraint for ingredient 3 is: .3B ≥ 20. .3B = 20. .3B≤ 20. .5A + .75B = 20.
.3B ≥ 20.
Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York Stock Exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in "stock 2". The constraint for this requirement can be written as: x2 ≥ .60. .4x2 - .6x7 - .6x8 ≤ 0. x2 ≥ .60 (x2 + x7 + x8). .4x2 - .6x7 - .6x8 ≥ 0.
.4x2 - .6x7 - .6x8 ≤ 0.
Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1. .65x11 - .35x21 ≤ 0 x11 + .35(x11 + x12) -.65x11 + .35x21 ≤ 0 x11 + x12 (.35)(x11 + x21)
.65x11 - .35x21 ≤ 0
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? .9x + .1y ≤ 10,000 .3x + .7y ≤ 10,000 .9A + .1B ≤ 10,000 .9x + .3y ≤ 10,000
.9x + .3y ≤ 10,000
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? 1000L and 0V 0L and 0V 0L and 1000V 0L and 1200V
0L and 1200V
generate $17,000 for the ship. How should the requirement that the bar setups should change at least every other day but no more than twice per day? 1 Bar ≥ 7 1 Bar ≤ 28 1 Bar ≥ 7 1 Bar ≤ 28 7 ≤ 1 Bar ≤ 28
1 Bar ≥ 7 1 Bar ≤ 28
The optimal quantity of the three products and resulting revenue for Taco Loco is: 14 Z, 13 Y, and 17 X for $9.81. 1.45 Z, 8.36 Y, and 0 Z for $129.09. 10.22 beef, 5.33 cheese, and 28.73 beans for $147.27. 28 beef, 80 cheese, and 39.27 beans for $147.27.
1.45 Z, 8.36 Y, and 0 Z for $129.09.
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. Which of the following is not a feasible purchase combination? 150 big shelves and 0 medium shelves 100 big shelves and 0 medium shelves 100 big shelves and 100 medium shelves 100 big shelves and 82 medium shelves
100 big shelves and 100 medium shelves
In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3, which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. An appropriate part of the model would be: MAX 50(15)X1 + 50 (47.25)X2 + 50 (110)X3. MAX 15X1 + 47.25X2 + 110X3. X1 + X2 +X3 ≤ 50,000. 15X1 + 47.25X2 + 110 X3 ≤ 50,000.
15X1 + 47.25X2 + 110 X3 ≤ 50,000.
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? 12.22 to 28 pounds 27.55 to 28.45 pounds 17.78 to 30 pounds 26 to 38.22 pounds
17.78 to 30 pounds
Which of the following could not be a linear programming problem constraint? 1A + 2B ≤ 3 1A + 2B = 3 1A + 2B ≥ 3 1A + 2B ≠ 3
1A + 2B ≠ 3
In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 Which of these objective functions will lead to the same optimal solution? 2x1 + 1x2 25x1 + 15x2 80x1 + 60x2 7x1 + 8x2
25x1 + 15x2
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 1E+30 39.27 0
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? 3L + 2V ≤ 4800 1L + 2V ≤ 4800 2L + 3V ≤ 4800 6L + 8V ≤ 4800
2L + 3V ≤ 4800
The assistant manager checks the cooler one fine Monday morning and sees that they have 400 ounces of cheese, 150 ounces of meat, 400 ounces of beans and 250 tortillas on hand. What is an appropriate constraint for this fast food vignette? 2N + 4B + 4Q + 3E ≤ 400 4C + 1M + 0B + 4T ≤ 4 250T + 400B + 150M + 400C 2.75N + 4B + 2Q + 3E
2N + 4B + 4Q + 3E ≤ 400
Coory soft drink 2 minutes regular 4 minutes diet what is the time constraint?
2r + 4d≤ 480
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? 200 L and 300 D 0 L and 400 D 0 L and 0 D 300 L and 200 D
300 L and 200 D
How many decision variables are in the LP formulation? 1 2 3 4
4
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? Final Reduced Objective Cell Name Value Cost Coefficient $C$2 Newspaper 0 -1000 5000 $D$2 Radio 13.76 0 3000 $E$2 Twitter 0 -800 700 $F$2 Facebook 4 0 200 8,900 17.76 42,080 cannot be determined from the sensitivity report
42,080
In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 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? 47.25X2 ≤ 10,000 X2 + X3 ≤ 350 47.25X2 ≤ 10,000 47.25 X2 + 110X3 ≤ 350 10,000 X2 ≤ 350X2 + 350X3 X2 ≤ 10000 X2 + X3 ≤ 350
47.25X2 ≤ 10,000 X2 + X3 ≤ 350
What is the equation for constraint EF? 4X + 8Y ≥ 64 16X + 8Y ≥ 32 16X + 8Y ≥ 24 4X + 8Y ≥ 12
4X + 8Y ≥ 64
Which of these is an appropriate constraint for this scenario? 5,000N + 3,000R + 700T + 200F ≤ 3,500 T ≤ 2,800 N + R + T + F ≥ 36 500N + 250R + 125T + 15F ≤ 3,500
500N + 250R + 125T + 15F ≤ 3,500
The poultry farmer would like to generate 80 ounces of fertilizer per week while feeding a maximum of 160 ounces of layer pellets each week. Which of these constraints is correct? 5L + 4C + 8B ≤ 160 3L + 2C + 4C ≤ 80 5L + 4C + 8B ≥ 160 5L + 4C +8B ≥ 80
5L + 4C + 8B ≤ 160
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. Which of the following is not a feasible production combination? 75R and 90D 40R and 100D 135R and 0D 90R and 75D
75R and 90D
generate $17,000 for the ship. What is the appropriate constraint for the budget? 900 Bar + 1500 Food + 5,000 Excursion ≤ 150,000 1 Bar + 1 Food + 1 Excursion ≤ 150,000 7 Bar + 12 Food + 5 Excursion ≤ 150,000 1,500 Bar + 5,000 Food + 17,000 Excursion ≤ 150,000
900 Bar + 1500 Food + 5,000 Excursion ≤ 150,000
Balanced transportation problems have which of the following type of constraints? ≥ < ≤ =
=
Which of the following statements about infeasible problems is best? At least one of the possible solutions violates all of the constraints. All of the possible solutions violate at least one constraint. All of the possible solutions violate all of the constraints. At least one of the possible solutions violates at least one of the constraints.
All of the possible solutions violate at least one constraint.
Which of the following could not be a linear programming problem constraint? A - B ≤ 3 -A + B ≤ -3 A - B ≤ -3 A + B ≤ -3
A + B ≤ -3
Line segment GH represents the objective function. Which constraint has surplus? CD AB EF none of the constraints has surplus
AB
Which of the following statements is not true? A feasible solution point does not have to lie on the boundary of the feasible solution. An optimal solution satisfies all constraints. An infeasible solution violates all constraints. A feasible solution satisfies all constraints.
An infeasible solution violates all constraints.
Which of these statements is best? An infeasible problem has unbounded solutions. An infeasible problem is also unbounded. An unbounded problem is also infeasible. An unbounded problem has feasible solutions.
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 B = 90, M = 75 B = 100, M = 100 B = 0, M = 200
B = 150, M = 0
Which line is represented by the equation 2X + Y ≥ 8? AJ BF CG DH
BF
Aunt Anastasia can obtain an additional 10 hours of kiln capacity free of charge from a friend. If she did this, how would her profits be affected? Profits would decrease by $25. Profits would decrease by $6.25. Profit would increase by $25. Cannot tell from the information provided.
Cannot tell from the information provided.
The poultry farmer has in mind the following levels for each of his metrics of interest: a plumage score greater than 75, fertilizer production greater than 80 ounces per week, temperament less than 100 decibels, and an appetite less than 160 ounces of layer pellets per week. When he runs his linear programming model, he discovers that his flock will consist entirely of Leghorn birds. He can picture the sad little faces of his children when he tells them that there will be no variety of birds gracing their front lawn this summer. Help him avoid the embarrassment by selecting a constraint that will ensure that there is some variety in his flock. C + B ≥ 5 5L + 4C + 8B ≥ 160 5L + 4C + 8B ≤ 160 L + C + B ≥ 5
C + B ≥ 5
generate $17,000 for the ship. Here's a lovely portion of the sensitivity report for the constraints. Which of these conclusions is reasonable? Captain Stubing should exhaust his Budget. There is no Bar or Food being used. Captain Stubing is scheduling more Excursions than he should be. The value for Restaurant will rise by $1,660 for each additional cuisine required. Question 5 0 / 1.34 points
Captain Stubing should exhaust his Budget.
Which of the following constraints has a surplus greater than 0? AJ BF DH CG
DH
What is the appropriate constraint for the requirement that there should be at least one different bar setup for every different type of food? Food + Bar ≥ 0 Bar - Food ≤ 0 Food - Bar ≤ 0 Bar + Food ≤ 0
Food - Bar ≤ 0
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.) Which of these statements about the sensitivity report for the constraints is best? Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$5 Budget 3500 12 3500 2560 3440 The advertising campaign needs another $2,560 to reach the maximum number of customers. Only $940 of the advertising budget is being spent on this campaign. It is possible to reach enough customers by spending only $60. For every $1 increase in the budget, the ad campaign can reach twelve more customers.
For every $1 increase in the budget, the ad campaign can reach twelve more customers.
Which of the following points is not feasible? A B G H
G
generate $17,000 for the ship. Here's a portion of the sensitivity analysis for the constraints. Which of these statements is best? Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$10 Bar/Food 0 1560 0 5 16 $E$11 Budget 150000 3.4 150000 1E+30 96200 $E$6 Excursions 24.24 0 5 19.24 1E+30 $E$7 Restaurant 12 -1660 12 16 5 If the restaurant constraint increased to 13, the optimal mix will result in increased revenues of $1,660. If the budget rises to $150,001, then the optimal mix will result in an increased revenue of $3.40. If the excursions rise to 6, then the optimal mix will result in increased revenue of $19.24. If the bar/food constraint rises to 1560, then the revenue will rise by a factor of 16.
If the budget rises to $150,001, then the optimal mix will result in an increased revenue of $3.40.
How should the entry for the Newspaper decision variable be interpreted? Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$2 Newspaper 0 -1000 5000 1000 1E+30 It is possible to reach an almost infinite fewer number (10 to the 30th power) of customers by newspaper than by any other method. The director should lower the newspaper advertising expense by $1,000 to reach the objective of 5,000 customers contacted. 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. If the director increased newspaper advertisement by 1,000, he would spend $1000 less than with the current campaign.
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.
Using this snippet of the sensitivity report for variable cells, ? Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$13 A 9 0 5 1E+30 1.25 $C$13 B 0 -2 6 2 1E+30 $D$13 C 0 -2 1 2 1E+30 The final value of this problem is 9. Items B and C are the only elements in the final model. Insisting that one additional unit of B, be included in the model will reduce the profit by $2. Item A can drop in value all the way down to 1.25 before it harms the result.
Insisting that one additional unit of B, be included in the model will reduce the profit by $2.
generate $17,000 for the ship. Here's the sensitivity report for the decision variables. Which of these conclusions is correct? Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$2 Bar 12 0 1500 1560 1E+30 $C$2 Cuisine 12 0 5000 1660 1E+30 $D$2 Excursion 24.24 0 17000 1E+30 3458.333333 It doesn't matter how much money they make on excursions, the optimal answer will continue to be 24.24 of them. If the cost of the Bar variable is increased by more than 0, then more of it will be used. If the coefficient for the Cuisine variable rises to 1660, then more of it will be used. If the cost of the Bar variable is reduced by more than 0, then more of it will be used.
It doesn't matter how much money they make on excursions, the optimal answer will continue to be 24.24 of them.
The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is: MAX .06X1 +.08X2 +.11X3. MAX .06(15)X1 +.08(47.25)X2 +.11(110)X3. MAX (1/.06)X1 +.(1/08)X2 + (1/.11)X3. MAX 15X1 + 47.25X2 +.110X3.
MAX .06(15)X1 +.08(47.25)X2 +.11(110)X3.
generate $17,000 for the ship. What should Captain Stubing's objective function be? Max Z = Bar + Food + Excursion Max Z = 1500 Bar + 5000 Food + 17000 Excursion Max Z = 5 Excursions + 12 Cuisine + 7 Bar + 150000 Budget Max Z = 900 Bar + 1500 Food + 5000 Excursion
Max Z = 1500 Bar + 5000 Food + 17000 Excursion
The assistant manager checks the cooler one fine Monday morning and sees that they have 400 ounces of cheese, 150 ounces of meat, 400 ounces of beans and 250 tortillas on hand. What is an appropriate objective function for this fast food vignette? Max Z = 400C + 150M + 400B + 250T Max Z = 2.75N + 4B + 2Q + 3E Min Z = 400C + 150M + 400B + 250T Min Z = 2.75N + 4B + 2Q + 3E
Max Z = 2.75N + 4B + 2Q + 3E
The communications director's market research revealed the following: Medium Exposure Cost Newspaper 5,000 $500 Radio 3,000 $250 Twitter 700 $25 Facebook 200 $15 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 $25 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 Max Z = 500N + 250R + 125T + 15F Min Z = 5,000N + 3,000R + 700T + 200F Min Z = 500N + 250R + 125T + 15F
Max Z = 5,000N + 3,000R + 700T + 200F
Temperament is actually measured by the average volume of cackling, clucking, and crowing and is measured in decibels per bird. Appetite is measured in ounces of layer pellets per week consumed by each of the breeds, while fertilizer is measured as the output in ounces per week. Which of the following is an appropriate objective function? Min Z = 5L + 2C + 4.5B Min Z = 10L + 10C + 10B Max Z = 5L + 2C + 4.5B Max Z = 10L + 10C + 10B
Max Z = 5L + 2C + 4.5B
The assistant manager checks the cooler one fine Monday morning and sees that they have 400 ounces of cheese, 150 ounces of meat, 400 ounces of beans and 250 tortillas on hand. Which of these is a decision variable for the LP formulation of this problem? cheese 400 Nacholupa $2.75
Nacholupa
Which of the statements about this portion of the sensitivity report is best? Cheese costs more than meat. There is both cheese and meat left over if the optimal product mix is produced. One additional pound of meat should be purchased if it can be acquired for $2 or less. The shadow price of cheese can rise to 16.8167 before the right hand side changes.
One additional pound of meat should be purchased if it can be acquired for $2 or less.
Aunt Anastasia's available hours for paint and seal have fallen from 80 hours to 60 hours because of other commitments. How will this affect her profits? Profits will increase by $20. Profits will increase by $30. Profits will decrease by $20. Profits will not change.
Profits will not change.
Suppose the charitable organization contacted Aunt Anastasia and told her that they had underestimated the amount of rabbits they needed. Instead of 25 rabbits, they need 35. How would this affect Aunt Anastasia's profits? Profits would increase by $5. Profits would increase by $2.50. Profits would decrease by $5. Profits would decrease by $2.50.
Profits would decrease by $5.
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 = 40, D= 100, Z = $320 R = 75, D = 90, Z = $405 R = 90, D = 75, Z = $420 R = 135, D = 0, Z = $405
R = 90, D = 75, Z = $420
________ is used to analyze changes in model parameters. Optimal solution Sensitivity analysis Feasible solution A slack variable
Sensitivity analysis
Using this snippet of the sensitivity report for constraints, which of these conclusions is best? Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$15 One 45 1 45 1.666666667 45 $E$16 Two 54 0 56 1E+30 2 $E$17 Three 27 0 67 1E+30 40 $E$18 Four 72 0 78 1E+30 6 Adding two units of One will increase the objective function value by two. None of items Two, Three or Four is being used. The most valuable resource is Four. Taking away four units of One will lower the objective function value by four.
Taking away four units of One will lower the objective function value by four.
When the poultry farmer runs his linear programming model, he discovers that his flock will consist entirely of Leghorn birds. He studies his sensitivity report (copied below) and decides to write a constraint that requires two Cochin chickens to be selected. If the current optimal mix of breeds results in 160 eggs per week, which of the following statements is best? Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease $B$1 Leghorn 32 0 5 1E+30 2.1875 $C$1 Cochin 0 -2 2 3 1E+30 $D$1 Buff Orpington 0 -3.5 4.5 3.5 1E+30 The new weekly egg output will be 158. The new weekly egg output will be 162. The new weekly egg output will be 169. The new weekly egg output will be 153.
The new weekly egg output will be 153.
The poultry farmer has in mind the following levels for each of his metrics of interest: a plumage score greater than 75, fertilizer production greater than 80 ounces per week, temperament less than 100 decibels, and an appetite less than 160 ounces of layer pellets per week. Use this portion of the sensitivity report to evaluate the following statements: Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease $B$1 Leghorn 32 0 5 1E+30 2.1875 $C$1 Cochin 0 -2 2 3 1E+30 $D$1 Buff Orpington 0 -3.5 4.5 3.5 1E+30 The plumage score will be less than 40. Layer pellets will be consumed at the rate of 96 ounces per week. The fertilizer generated will be 160 ounces per week. The noise level around the farm should be over 60 decibels.
The noise level around the farm should be over 60 decibels.
Based on the variable cells sensitivity report, what conclusion is best? Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$2 Bar 12 0 1500 1560 1E+30 $C$2 Cuisine 12 0 5000 1660 1E+30 $D$2 Excursion 24.24 0 17000 1E+30 3458.333333 The value of Excursion can be infinite, since 1E+30 is 10 to the 30th power. The optimal value for the objective function is $490,080 The values of Bar, Cuisine, and Excursion are all equal to zero. The values of Bar and Cuisine can be negative since they have allowable decreases of 1E+30.
The optimal value for the objective function is $490,080
Which of the following is not a typical characteristic of a linear programming problem? Restrictions exist. The problem has an objective. A choice among alternatives is required. The problem can be solved graphically.
The problem can be solved graphically.
The assistant manager checks the cooler one fine Monday morning and sees that they have 400 ounces of cheese, 150 ounces of meat, 400 ounces of beans and 250 tortillas on hand. Which of these statements about the sensitivity report is best? Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $D$4 Burritaco 0 -0.125 4 0.125 1E+30 $C$4 Nacholupa 55 0 2.75 0.25 0.114 If the Nacholupa has a cost reduction of more than 0, none will be made. The company can make an additional 0.25 Nacholupas if they want to with the leftover ingredients. There are no Burritacos being made. The company can make up to 1E+30 Burritacos.
There are no Burritacos being made.
Consider the optimization problem represented by this graph. Which of the following statements is best? This is a maximization problem with no feasible solution. This is a minimization problem with no feasible solution. This is a minimization problem with a feasible solution. This is a maximization problem with a feasible solution.
This is a minimization problem with a feasible solution.
Consider the optimization problem represented by this graph. Line GH represents the objective function. Which of the following statements is best? All points along GH are optimal. This is a single optimal solution. All points on lines AB, CD and DE that touch the shaded region are optimal.
This is a single optimal solution.
The equation for constraint DH is: X + 2Y ≥ 8. 2X + Y ≥ 8. 8X + 4Y ≥ 32. 4X + 8Y ≥ 32.
X + 2Y ≥ 8.
In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3, which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct? X1 = 0.35(X1 + X2 + X3) X1 = 0.35 (50000) X1 ≤ 0.35 X1 ≤ 0.35(X1 + X2 + X3)
X1 ≤ 0.35(X1 + X2 + X3)
If Xab = the production of product a in period b, then to indicate that the limit on production of the company's 3 products in period 2 is 400, we write: X21 + X22 + X23 ≤ 400. X12 + X22 + X32 ≤ 400. X12 + X22 + X32 ≥ 400. X32 ≤ 400.
X12 + X22 + X32 ≤ 400.
Which of the following could be a linear programming objective function? Z = 1A + 2B / C + 3D Z = 1A + 2B + 3C + 4D Z = 1A + 2BC + 3D Z = 1A + 2B2 + 3D
Z = 1A + 2B + 3C + 4D
A plant manager is attempting to determine the production schedule of various products to maximize profit. Assume that a machine hour constraint is binding. If the original amount of machine hours available is 200 minutes., and the range of feasibility is from 130 minutes to 300 minutes, providing two additional machine hours will result in: the same product mix, different total profit. a different product mix, same total profit as before. a different product mix, different total profit. the same product mix, same total profit.
a different product mix, different total profit.
How would multiple optimal solutions typically appear on a graphical solution? a plane a line a cube a point
a line
In the formulation of a ≥ constraint: a slack variable is added. a slack variable is subtracted. a surplus variable is subtracted. a surplus variable is added.
a surplus variable is subtracted.
An intern sets up a linear program to optimize the use of paper products in the men's washroom. The system of equations he develops is: Max 2T + 3S + 4ST s.t 3T + 6S ≤ 40 10T + 10S ≤ 66 10T + 15S ≤ 99 His mentor studies the model, frowns, and admonishes the intern for violating which of the following properties of linear programming models? additivity divisibility proportionality certainty
additivity
Which of the following special cases does not require reformulation of the problem in order to obtain a solution? alternate optimality infeasibility unboundedness Each one of these cases requires reformulation.
alternate optimality
In a multiperiod scheduling problem, the production constraint usually takes the form of: beginning inventory - demand + production = ending inventory. beginning inventory - ending inventory + demand = production. beginning inventory + demand - production = ending inventory. beginning inventory + demand + production = ending inventory.
beginning inventory - demand + production = ending inventory.
Aunt Anastasia is planning for next spring, and she is considering making only two products. Based on the results from the linear program, which two products would you recommend that she make? baskets and rabbits eggs and rabbits baskets and eggs She should continue to make all three.
baskets and rabbits
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. buy at least 126 pounds of cheese for $5.33 or less. refuse to buy any cheese. buy less than 80 pounds of cheese for $1.45 per pound.
buy 46 pounds or less of cheese for $1.45 or less.
The ________ property of linear programming models indicates that the values of all the model parameters are known and are assumed to be constant. divisibility additive certainty proportionality
certainty
The optimal solution of a minimization problem is at the extreme point ________ the origin. exactly at parallel to farthest from closest to
closest to
The type of linear program that compares services to indicate which one is less productive or inefficient is called: blending. product mix. data envelopment analysis. marketing.
data envelopment analysis.
A systematic approach to model formulation is to first: define decision variables. construct the objective function . determine the right hand side of each constraint. develop each constraint separately.
define decision variables.
The region that satisfies all of the constraints in a graphical linear programming problem is called the: feasible solution space. region of optimality. optimal solution space. region of non-negativity.
feasible solution space.
Multiple optimal solutions provide ________ flexibility to the decision maker. greater greater or equal less or equal less
greater
The constraint AJ: does not contain feasible points. is a binding constraint. contains the optimal solution. has no surplus.
has no surplus.
When systematically formulating a linear program, the first step is to: construct the objective function. formulate the constraints. identify the parameter values. identify the decision variables.
identify the decision variables.
Without satisfying the non-negativity constraint, a solution that satisfies all the other constraints of a linear programming problem is called: feasible. optimal. semi-feasible. infeasible.
infeasible.
The optimal solution to a linear programming model that has been solved using the graphical approach: must be below and on the left side of all constraint lines. is typically at some corner of the feasible region. must be above and the right of all constraint lines. is typically located at the origin.
is typically at some corner of the feasible region.
Which additional resources would you recommend that Aunt Anastasia try to obtain?
kiln
If Billy could acquire more of any resource, which would it be? buyers machining time money labor time
machining time
Decision variables: measure the objective function. measure the values of each constraint. always exist for each constraint. measure how much or how many items to produce, purchase, hire, etc.
measure how much or how many items to produce, purchase, hire, etc.
This linear programming problem is a(n): minimization problem. irregular problem. maximization problem. cannot tell from the information given
minimization problem.
Consider the following maximization problem. MAX z = x + 2y s.t. 2x + 3y ≤ 6 5x + 6y ≤ 30 y ≥ 1 The optimal solution: occurs where x = 4.67 and y = 1.11. occurs where x = 0 and y = 2. results in an objective function value of 12. occurs where x = 6 and y = 0.
occurs where x = 0 and y = 2.
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 only syrup time and syrup neither time nor syrup
only time
Multiple optimal solutions can occur when the objective function is ________ a constraint line. unequal to parallel to perpendicular to equal to
parallel to
Sensitivity analysis is the analysis of the effect of ________ changes on the ________. price, company cost, production constraint, parameter parameter, optimal solution
parameter, optimal solution
The first step in solving a graphical linear programming model is to: solve simultaneous equations at each corner point to find the solution values at each point. plot the model constraints as equations on the graph and indicate the feasible solution area. plot the objective function and move this line out from the origin to locate the optimal solution point. determine which constraints are binding.
plot the model constraints as equations on the graph and indicate the feasible solution area.
The ________ of linear programming models indicates that the rate of change or slope of the objective function or a constraint is constant. certainty divisibility proportionality additive
proportionality
An optimization problem that has multiple optimal solutions: means that the surplus for a third constraint cannot be calculated. is reflected by the entire feasible region being optimal means that there are actually no optimal solutions. provides the decision-maker with increased flexibility.
provides the decision-maker with increased flexibility.
MAX z = 5x + 3y s.t. x - y ≤ 6 x ≤ 1 The optimal solution: results in an objective function value of 5. is infeasible. occurs where x = 0 and y = 1.
results in an objective function value of 5.
Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ij = inventory quantity in periodj, and dj = demand quantity in period j. Correct formulation of the demand constraint for a multiperiod scheduling problem is: rj + oj + i1 - i2 ≥ dj. rj - oj - i1 + i2 ≥ dj. rj + oj + i1 - i2 ≤ dj. rj + oj + i2 - i1 ≥ dj.
rj + oj + i2 - i1 ≥ dj.
For the production combination of 800 bags of lime and 600 bags of vinegar, which of the three resources is (are) not completely used? herbs only flour only salt only salt and flour
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, same total profit. different product mix, same total profit as before. same product mix, different total profit. different product mix, different total profit.
same product mix, different total profit.
If the feasible region for a linear programming problem is unbounded, then the solution to the corresponding linear programming problem is ________ unbounded. sometimes never always There is not enough information to complete this statement.
sometimes
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. If the furniture company purchases no big shelves and 200 medium shelves, which of the two resources will be completely used (at capacity)? investment money only storage space only investment money and storage space neither investment money nor storage space
storage space only
A slack variable is
the amount by which the left side of a constraint is smaller than the right
Consider the optimization problem represented by this graph. The objective function is represented by line GH. Where is the optimal solution? the intersection of lines CD and EF the intersection of lines AB and EF the intersection of lines AB and CD the upper right corner of the shaded region
the intersection of lines CD and EF
A shadow price reflects which of the following in a maximization problem? the marginal gain in the objective that would be realized by subtracting one unit of a resource the marginal gain in the objective that would be realized by adding one unit of a resource the marginal cost of adding additional resources the marginal gain of selling one more unit
the marginal gain in the objective that would be realized by adding one unit of a resource
Hot dog manufacturer minimize cost:
the minimum cost hot dog has 1300 milligrams more sodium that required
In order for an optimization problem to have multiple optimal solutions: two or more of the constraints must not have intersection points. the objective function and one constraint must have the same slope. two or more of the constraints must have the same slope. the objective function and one constraint must have the same y-intercept.
the objective function and one constraint must have the same slope.
Compared to blending and product mix problems, transportation problems are unique because: they maximize profit. they contain fewer variables. the constraints are all equality constraints with no "≤" or "≥" constraints. the solution values are always integers.
the solution values are always integers.
The theoretical limit on the number of constraints that can be handled by a linear programming problem is: 3. 2. 4. unlimited.
unlimited.
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: malt only. wheat only. both malt and wheat. neither malt nor wheat.
wheat only.
For a linear programming problem, assume that a given resource has not been fully used. We can conclude that the shadow price associated with that constraint: will have a positive value. will have a value of zero. will have a negative value. could have a positive, negative or a value of zero. (no sign restrictions).
will have a value of zero.
Given the following linear programming problem: Max Z = 15x + 20y s.t. 8x + 5y ≤ 40 4x + y ≥ 4 What would be the values of x and y that will maximize revenue? x = 1; y = 0 x = 0; y = 8 x = 5; y = 0 x = 0; y = 1
x = 0; y = 8
For the constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y ≤ 42 (2) x + 3y ≥ 6 x = 1; y = 2 x = 2; y = 5 x = 2; y = 1 x = 2; y = 3
x = 1; y = 2
Use the constraints given below and determine which of the following points is feasible. (1) 14x + 6y ≤ 42 (2) x - y ≤ 3 x = 1; y = 4 x = 3; y = 0.5 x = 2; y = 4 x = 2; y = 8
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 = 5; y = 0 x = 6; y = 0 x = 0; y = 5 x = 0; y = 3
x = 5; y = 0
Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 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. x11 + x12 ≤ 8000 x21 + x22 ≤ 8000 x12 + x22 ≥ 8000 x21 + x22 ≥ 8000
x11 + x12 ≤ 8000
Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the demand constraint for gasoline type 1. x11 + x21 ≤ 11000 x11 + x21 = 11000 x21 + x22 = 11000 x12 + x22 = 11000
x11 + x21 = 11000
For a resource constraint, either its slack value must be ________ or its shadow price must be ________. negative, negative negative, zero zero, negative zero, zero
zero, zero