Quantitive Method

1. In a linear programming problem, a valid objective function can be represented as
a. Max Z = 5xy
b. Max Z 5x2 + 2y2
c. Max 3x + 3y + 1/3z
d. Min (x1 + x2) / x3

2. 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 340 minutes, providing two additional machine hours will result in the:
a. same product mix, different total profit
b. different product mix, same total profit as before
c. same product mix, same total profit
d. different product mix, different total profit

3. The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He 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?
a. 0 L and 0 D
b. 0 L and 400 D
c. 200 L and 300 D
d. 400 L and 400 D

4. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of 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 regular cases and 0 diet cases, which resource is completely used up (at capacity)?
a. only time
b. only syrup
c. time and syrup
d. neither time nor syrup

5. Mallory 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. If the Mallory Furniture company decides to purchase 150 big shelves and no medium shelves, which of the two resources will be left over?
a. investment money only
b. storage space only
c. investment money and storage space
d. neither investment money nor storage space

6. Given the following linear programming problem:
Max Z = 15x + 20y
Subject to (1) 8x + 5y 40
(2) 4x + y 4
What is the maximum revenue at the optimal solution?
a. $120
b. $160
c. $185
d. $200

7. Given the following linear programming problem:
Min Z = 2x + 8y
Subject to (1) 8x + 4y 64
(2) 2x + 4y 32
(3) y 2
Determine the optimum values for x and y.
a. x = 2; y = 6
b. x = 6; y = 2
c. x = 12; y = 2
d. x = 2; y = 2
e. x = 6; y = 5

8. Which of the following could not be a linear programming problem constraint?
a. 3A + 21B > -2
b. 12A + 9B 3
c. 2A + 4B = 3
d. 4A + 7B + 2C + 9D 5

9. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of 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 optimal daily profit?
a. $220
b. $270
c. $320
d. $420
e. $520

10. Mallory 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 $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 objective function?
a. Z = $300B + $150M
b. Z = $300M + $150B
c. Z = $300B + $500M
d. Z = $500B + $300M

11. Mallory 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 storage space constraint?
a. 90B + 100M 18000
b. 100B + 90M ≤ 18000
c. 90B + 100M 18000
c. 500B + 300M 18000
d. 300B + 500M 18000

12. Mallory 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 weekly maximum profit?
a. $25000
b. $35000
c. $45000
d. $55000
e. $65000

13. For a linear programming problem, assume that a given resource has not been fully used. In other words, the slack value associated with the resource constraint is positive. We can conclude that the shadow price associated with that constraint:
a. will have a positive value
b. will have a negative value
c. will have a value of zero
d. could have a positive, negative or a value of zero. (no sign restrictions)

14. 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:
a. same product mix, different total profit
b. different product mix, same total profit as before
c. same product mix, same total profit
d. different product mix, different total profit

15. The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He 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 0 bottles of light beer and 400 bottles of dark beer, it will result in slack of
a. malt only
b. wheat only
c. both malt and wheat
d. neither malt nor wheat

16. Mallory 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. Which of the following is not a feasible purchase combination?
a. 0 big shelves and 200 medium shelves
b. 0 big shelves and 0 medium shelves
c. 150 big shelves and 0 medium shelves
d. 100 big shelves and 100 medium shelves

17. The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one "8 hour" shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of 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?
a. 90 R and 75 D
b. 135 R and 0 D
c. 0 R and 120 D
d. 75 R and 90 D
e. 50 R and 50 D

18. 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 two". The constraint for this requirement can be written as:
a. x2 .60
b. x2 .60 (x2 + x7 + x8)
c. .4x2 - .6x7 - .6x8 0
d. .4x2 - .6x7 - .6x8 0
e. -.4x2 + .6x7 + .6x8 0

19. The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 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 salt. 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?
a. flour only
b. salt only
c. herbs only
d. salt and flour
e. salt and herbs

20. 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,
a. X32 400
b. X21 + X22 + X23 400
c. X12 + X22 + X32 400
d. X12 + X22 + X32 400
e. X23 400

21. 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?
a. 2R + 4D 480
b. 2D + 4R 480
c. 2R + 3D 480
d. 3R + 2D 480
e. 3R + 4D 480

22. A croissant shop produces 2 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?
a. $380
b. $400
c. $420
d. $440
e. $480

23. 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.
a. x21 + x22 8000
b. x12 + x22 8000
c. x11 + x12 8000
d. x21 + x22 8000
e. x11 + x12 8000

24. 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 demand constraint for gasoline type 1.
a. x21 + x22 =11000
b. x12 + x22 =11000
c. x11 + x21 11000
d. x11 + x21 = 11000
e. x11 + x12 11000

25. 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 constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1.
a. x11 + x12 (.35)(x11 + x21)
b. x11 .35 (x11 + x21)
c. x11 .35 (x11 + x12)
d. -.65x11 + .35x21 0
e. .65x11 - .35x21 0

26. In a multi-period scheduling problem the production constraint usually takes the form of :
a. beginning inventory + demand - production = ending inventory
b. beginning inventory - demand + production = ending inventory
c. beginning inventory - ending inventory + demand = production
d. beginning inventory - production - ending inventory = demand
e. beginning inventory + demand + production = ending inventory

27. 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.

Machine Fixed cost to set up production runVariable cost per hose Capacity

1 750 1.25 6000
2 500 1.50 7500
3 1000 1.00 4000
4 300 2.00 5000

Write a constraint to ensure that if machine 4 is used, machine 1 will not be used.
a. y1 + y4 0
b. y1 + y4 = 0
c. y1 + y4 1
d. y1 + y4 0
e. y1 + y4 1

28. Max Z = 5x1 + 6x2
Subject to:17x1 + 8x2 136
3x1 + 4x2 36
x1, x2 0 and integer
What is the optimal solution?
a. x1= 6, x2 = 4, Z = 54
b. x1= 3, x2 = 6, Z = 51
c. x1= 2, x2 = 6, Z = 46
d. x1= 4, x2 = 6, Z = 56
e. x1= 0, x2 = 9 Z = 54

29. In a balanced transportation model where supply equals demand,
a. all constraints are equalities
b. none of the constraints are equalities
c. all constraints are inequalities
d. none of the constraints are inequalities

30. The linear programming model for a transportation problem has constraints for supply at each ________ and _________ at each destination.
a. destination / source
b. source / destination
c. demand / source
d. source / demand

31. The transportation method assumes that
a. the number of rows is equal to the number of columns
b. there must be at least 2 rows and at least 2 columns
c. a and b
d. the product of rows minus 1 and columns minus 1 should not be less than the number of completed cells

32. The transportation method is a linear programming technique where linearity can be found in the following way:
a. the cost of goods shipped from any source to any destination is a linear function of quantity shipped
b. the total cost associated with a given plan is a linear function of shipping costs
c. cell evaluations assume linear movements through the matrix
d. the cost of goods shipped from any source to any destination is a linear function of the cost per unit

33. Which of the following assumptions is not an assumption of the transportation model?
a. shipping costs per unit are constant
b. there is one transportation route between each source and destination
c. there is one transportation mode between each source and destination
d. actual total supply and actual total demand must be equal

34. __________ is not a bit of information needed in the transportation model.
a. Demand of the destinations
b. Unit shipping distances
c. Unit shipping costs
d. Capacity of the sources

35. Which of the following are assumptions or requirements of the transportation problem?
a. goods are the same, regardless of source
b. there must be multiple sources
c. shipping costs per unit do not vary with the quantity shipped
d. all of the above

36. Which of the following are assumptions or requirements of the transportation problem?
a. minimum quantities must be shipped
b. goods are the same, regardless of source
c. a and b
d. none of the above

37. Which of the following are assumptions or requirements of the transportation problem?
a. there must be multiple sources
b. there can only be one source
c. minimum quantities must be shipped
d. all of the above

38. Which of the following are assumptions or requirements of the transportation problem?
a. minimum quantities must be shipped
b. shipping costs per unit do not vary with the quantity shipped
c. there can only be one source
d. all of the above

39. In a transshipment problem, items may be transported
a. from destination to destination
b. from one transshipment point to another
c. directly from sources to destinations
d. all of the above

40. In a transshipment problem, items may be transported
a. from source to source
b. from one destination to another
c. from sources to transshipment points
d. from transshipment points to destinations
e. all of the above

41. An assignment problem is a special form of transportation problem where all supply and demand values equal
a. 0
b. 1
c. 2
d. greater than 1
e. none of the above

42. The transshipment model is an extension of the transportation model in which intermediate transshipment points are ___________ between the sources and destinations.
a. decreased
b. deleted
c. subtracted
d. added

43. Branches in a transshipment problem
a. indicate the direction of the flow
b. must connect every node to a transshipment node
c. represent the cost of shipments
d. all of the above

44. Constraints in a transshipment problem
a. include a variable for every branch
b. require the sum of the shipments out of an origin node to equal supply
c. correspond to branches
d. all of the above

45. The assignment problem constraint x31+x32+x33+x34 2 means
a. agent 2 can be assigned to 3 tasks
b. agent 3 can be assigned to 2 tasks
c. a mixture of agents 1, 2, 3 and 4 will be assigned to tasks
d. there is no feasible solution

46. 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. In order to maximize profit, how many big shelves (B) and how many medium shelves (M) should be purchased?
a. B = 90, M = 75
b. B = 135, M = 15
c. B = 150, M = 0
d. B = 0, M = 200
e. B = 100, M = 100