Deck 4: Linear Programming: an Algebraic Approach

Question 1

Solve the linear programming problem by the simplex method.

A) x = 0, y = 0, z = 3, u = 1, v = 0, P = 49
B) x = 0, y = 3, z = 1, u = 0, v = 0, P = 27
C) x = 0, y = 0, z = 0, u = 3, v = 1, P = 44
D) x = 3, y = 1, z = 0, u = 0, v = 0, P = 43

Answer 1

x = 0, y = 3, z = 1, u = 0, v = 0, P = 27

Question 2

Check that the given simplex tableau is in final form. Find the solution to the associated regular linear programming problem.

A) x = 0, y = , z = 0, u = , v = 0, w = 0, P =
B) x = , y = , z = 0, u = 0 , v = 0, w = 0, P =
C) x = 0, y = , z = , u = 0 , v = 0, w = , P = 0
D) x = 0, y = , z = 0, u = , v = 0, w = 0, P =

Answer 2

x = 0, y = , z = 0, u = , v = 0, w = 0, P =

Question 3

Solve the linear programming problem by the simplex method.

A) x = 0, y = 8, u = 16, v = 10, w = 0, P = 40
B) x = 8, y = 8, u = 0, v = 10, w = 0, P = 64
C) x = 8, y = 0, u = 0, v = 2, w = 16, P = 64
D) x = 6, y = 6, u = 0, v = 0, w = 0, P = 78

Answer 3

x = 6, y = 6, u = 0, v = 0, w = 0, P = 78

Question 4

Solve the linear programming problem by the simplex method.

A) x = 24, y = 0, z = 24, t = 0, u = 11, v = 21, w = 32, P = 134
B) x = 0, y = 24, z = 0, t = 0, u = 11, v = 21, w = 32, P = 168
C) x = 11, y = 24, z = 0, t = 0, u = 0, v = 21, w = 32, P = 179
D) x = 11, y = 0, z = 0, t = 24, u = 11, v = 21, w = 32, P = 96

Question 5

Solve the linear programming problem by the simplex method.

A) x = , y = , P =
B) x = , y = , P =
C) x = , y = , P =
D) x = , y = , P =
E) x = , y = , P =

Question 6

Solve the linear programming problem by the simplex method.

A) x = 1, y = 1, z = 1, u = 0, v = 0, w = 0, P = 66
B) x = 1, y = 0, z = 1, u = 1, v = 0, w = 0, P = 109
C) x = 1, y = 1, z = 1, u = 0, v = 0, w = 0, P = 112
D) x = 0, y = 1, z = 0, u = 1, v = 1, w = 0, P = 97

Question 7

Check that the given simplex tableau is not in final form. Find the pivot element to be used in the next iteration of the simplex method.

A) pivot element is , lying in the second row, second column.
B) pivot element is , lying in the first row, second column.
C) pivot element is 0, lying in the second row, third column.
D) pivot element is 0, lying in the first row, first column.

Question 8

Ashley has earmarked at most $350,000 for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and-income fund. The money market fund has a rate of return of 7%/year, the international equity fund has a rate of return of 12%/year, and the growth-and-income fund has a rate of return of 16%/year. Ashley has stipulated that no more than 25% of her total portfolio should be in the growth-and-income fund and that no more than 50% of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?

A) $57,500 in the money market fund, $205,000 in the international equity fund, $57,500 in the growth-and-income fund; maximum return $37,825
B) $87,500 in the money market fund, $175,000 in the international equity fund, $175,000 in the growth-and-income fund; maximum return $55,125
C) $87,500 in the money market fund, $175,000 in the international equity fund, $87,500 in the growth-and-income fund; maximum return $41,125
D) $205,000 in the money market fund, $57,500 in the international equity fund, $205,000 in the growth-and-income fund; maximum return $54,050

Question 9

Check that the given simplex tableau is in final form. Find the solution to the associated regular linear programming problem.

A) x = , y= , u = , v = , P =
B) x = , y= , u = , v = , P =
C) x = , y= , u = 0, v = 0, P =
D) x = , y= , u = 0, v = 0, P =

Question 10

Solve the linear programming problem by the simplex method.

A) x = 0, y = 4, u = 0, v = 1, P = 32
B) x = 1, y = 0, u = 0, v = 0, P = 30
C) x = 1, y = 3, u = 0, v = 0, P = 30
D) x = 2.5, y = 0, u = 1.5, v = 0, P = 15

Question 11

Find the pivot element to be used in the next iteration of the simplex method.

A) Pivot element is 3, lying in first row, second column
B) Pivot element is 0, lying in second row, second column
C) Pivot element is 1, lying in third row, second column
D) Pivot element is , lying in first row, first column
E) Pivot element is 1, lying in fourth row, second column

Question 12

A furniture company is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and C - are to be marketed. Each model A desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model B desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model C desk requires hr, hr, and hr for fabrication, assembly, and finishing, respectively. The profit on each model A desk is $34, the profit on each model B desk is $37, and the profit on each model C desk is $28. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is 279 hr, 181 hr, and 166 hr, respectively. To maximize company's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?

A) 60 units of model A, 76 units of model B, 60 units of model C; maximum profit $6,532; 0 hr left over in the fabrication department, 0 hr left over in the assembly department, 0 hr left over in the finishing department
B) 60 units of model A, 60 units of model B, 76 units of model C; maximum profit $6,487; 0 hr left over in the fabrication department, 3.5 hr left over in the assembly department, 0 hr left over in the finishing department
C) 76 units of model A, 60 units of model B, 60 units of model C; maximum profit $8,932; 1 hr left over in the fabrication department, 0 hr left over in the assembly department, 2 hr left over in the finishing department
D) 60 units of model A, 0 units of model B, 76 units of model C; maximum profit $12,850; 0 hr left over in the fabrication department, 3 hr left over in the assembly department, 12 hr left over in the finishing department

Question 13

As part of a campaign to promote its annual clearance sale, a company decided to buy television-advertising time on a TV station. Company's television-advertising budget is $102,000. Morning time costs $3,000/minute, afternoon time costs $1,000/minute, and evening (prime) time costs $12,000/minute. Because of previous commitments, the station cannot offer the company more than 7 min of prime time or more than a total of 25 min of advertising time over the 4 wk in which the commercials are to be run. The station estimates that morning commercials are seen by 210,000 people, afternoon commercials are seen by 105,000 people, and evening commercials are seen by 630,000 people. How much morning, afternoon, and evening advertising time should the company buy to maximize exposure of its commercials? What is the maximum number of viewers?

A) 22 min of morning advertising time, 0 min of afternoon advertising time, 3 min of evening advertising time; 6.51 million viewers
B) 10 min of morning advertising time, 15 min of afternoon advertising time, 0 min of evening advertising time; 4.4 million viewers
C) 16 min of morning advertising time, 7 min of afternoon advertising time, 2 min of evening advertising time; 5 million viewers
D) 15 min of morning advertising time, 3 min of afternoon advertising time, 7 min of evening advertising time; 5.67 million viewers

Question 14

Solve the linear programming problem by the simplex method.

A) x = 13, y = 4, z = 0, u = 73, v = 0, w = 0, P = 129
B) x = 4, y = 0, z = 13, u = 73, v = 0, w = 0, P = 250
C) x = 0, y = 13, z = 0, u = 73, v = 0, w = 0, P = 134
D) x = 0, y = 0, z = 4, u = 73, v = 0, w = 0, P = 76

Question 15

Determine wheather the given simplex table is in the final form. If so, find the solution to the associated regular linear programming problem.

A) x = ; y = ; u = 0; v = 0; P =
B) Not in final form.
C) x = ; y = ; u = 0; v = 0; P =
D) x = ; y = ; u = 0; v = 0; P =
E) x = ; y = ; u = 0; v = 0; P =

Question 16

A farmer has 150 acres of land suitable for cultivating crops A and B. The cost of cultivating crop A is $40/acre, whereas that of crop B is $60/acre. The farmer has a maximum of $7,400 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3,300 labor-hours available. If he expects to make a profit of $150/acre on crop A and $200/acre on crop B, how many acres of each crop should he plant in order to maximize his profit? What is the largest profit the farmer can realize? Are there any resources left over?

A) 65 acres of crop A, 80 acres of crop B; maximum profit $25750; $0 left over, 5 acres of land left over, 0 labor-hours left over
B) 150 acres of crop A, 80 acres of crop B; maximum profit $51,000; $2,200 left over, 0 acres of land left over, 0 labor-hours left over
C) 90 acres of crop A, 60 acres of crop B; maximum profit $51,000; $200 left over, 0 acres of land left over, 0 labor-hours left over
D) 150 acres of crop A, 0 acres of crop B; maximum profit $22,500; $2,200 left over, 0 acres of land left over, 300 labor-hours left over

Question 17

National Business Machines Corporation manufactures two models of fax machines: A and B. Each model A costs $150 to make, and each model B costs $105. The profits are $45 for each model A and $30 for each model B fax machine. If the total number of fax machines demanded each month does not exceed 2,500 and the company has earmarked no more than $600,000/month for manufacturing costs, find how many units of each model National should make each month in order to maximize its monthly profits. What is the largest monthly profit the company can make?

A) 0 units of model A, 2,500 units of model B; $225,000
B) 2,500 units of model A, 0 units of model B; $112,500
C) 1,250 units of model A, 1,250 units of model B; $225,000
D) 0 units of model A, 2,500 units of model B; $75,000

Question 18

Check that the given simplex tableau is not in final form. Find the pivot element to be used in the next iteration of the simplex method.

A) pivot element is 5, lying in the third row, third column.
B) pivot element is 0, lying in the first row, third column.
C) pivot element is 5, lying in the third row, second column.
D) pivot element is 2, lying in the first row, second column.

Question 19

Solve the linear programming problem by the simplex method.

A) x = , y = , u = 0, v = 1, P =
B) x = , y = , u = 1, v = 0, P =
C) x = , y = , u = 1, v = 0, P =
D) x = , y = , u = 1, v = 0, P =
E) x = , y = , u = 0, v = 1, P =

Question 20

Find the pivot element to be used in the next iteration of the simplex method.

A) Pivot element , lying in first row, second column
B) Pivot element - , lying in first row, fourth column
C) Pivot element , lying in second row, second column
D) Pivot element 0, lying in second row, second column
E) Pivot element 1, lying in first row, third column

Question 21

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows: For the first year's production, a sum of $7,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 198,000 hr; and the amount of labor for on-site work is to be less than or equal to 207,000 labor-hours. Determine how many houses of each type Boise should produce (market research has confirmed that there should be no problems with sales) to maximize its profit from this new venture.

A) 300 houses of Standard Model, 200 houses of Deluxe Model, 200 houses of Luxury Model.
B) 300 houses of Standard Model, 300 houses of Deluxe Model, 300 houses of Luxury Model.
C) 200 houses of Standard Model, 300 houses of Deluxe Model, 200 houses of Luxury Model.
D) 200 houses of Standard Model, 200 houses of Deluxe Model, 300 houses of Luxury Model.
E) 200 houses of Standard Model, 200 houses of Deluxe Model, 200 houses of Luxury Model.

Question 22

Consider the linear programming problem. Maximize subject to Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 23

A financier plans to invest up to $7 million in three projects. She estimates that project A will yield a return of 5% on her investment, project B will yield a return of 10% on her investment, and project C will yield a return of 20% on her investment. Because of the risks associated with the investments, she decided to put not more than 20% of her total investment in project C. She also decided that her investments in projects B and C should not exceed 60% of her total investment. Finally, she decided that her investment in project A should be at least 60% of her total investment. How much should the financier invest in each project if she wishes to maximize the total returns on her investments? What is the maximum amount she can expect to make from her investments? ​

A) Project A: $1,350,000, project B: $1,350,000, and project C: $2,900,000; maximum profit: $690000
B) Project A: $2,850,000, project B: $2,900,000, and project C: $1,350,000; maximum profit: $230,000
C) Project A: $1,400,000, project B: $2,800,000, and project C:$2,800,000; maximum profit: $910,000
D) Project A: $2,800,000, project B: $2,800,000, and project C: $1,400,000; maximum profit: $700,000

Question 24

Solve the linear programming problem by the simplex method.

Question 25

Find the pivot element to be used in the next iteration of the simplex method.

A) Pivot element is 2, lying in the third row, first column.
B) Pivot element is 1, lying in the second row, first column.
C) Pivot element is 8, lying in the third row, first column.
D) Pivot element is 0, lying in the second row, sixth column.
E) Pivot element is 2, lying in the first row, second column.

Question 26

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 27

A pharmaceutical company produces three kinds of cold formulas: I, II, and III. It takes 5 hr to produce 1,000 bottles of formula I, 4 hr to produce 1,000 bottles of formula II, and 3 hr to produce 1,000 bottles of formula III. The profits for each 1,000 bottles of formula I, formula II, and formula III are $240, $170, and $210, respectively. Suppose, for a certain production run, there are enough ingredients on hand to make at most 7,000 bottles of formula I, 13,000 bottles of formula II, and 6,000 bottles of formula III. Furthermore, suppose the time for the production run is limited to a maximum of 90 hr. How many bottles of each formula should be produced in this production run so that the profit is maximized? What is the maximum profit realizable by the company? Are there any resources left over?

A) 6,000 bottles of formula I, 7,000 bottles of formula II, 9,250 bottles of formula III; maximum profit: $8,631.15; Yes, ingredients for 3,750 bottles of formula III
B) 9,250 bottles of formula I, 6,000 bottles of formula II, 7,000 bottles of formula III; maximum profit: $8,524.95; Yes, ingredients for 3,750 bottles of formula I
C) 7,000 bottles of formula I, 9,250 bottles of formula II, 6,000 bottles of formula III; maximum profit: $4,512.50; Yes, ingredients for 3,750 bottles of formula II
D) 16,250 bottles of formula I, no bottles of formula II, 9,250 bottles of formula III; maximum profit: $6,921.35; Yes, 3 hr of the time for the production run

Question 28

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 29

Solve the linear programming problem by the simplex method.

Question 30

A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of product A and a profit of $5/unit of product B. To manufacture 1 unit of product A requires 7 min on machine I and 9 min on machine II. To manufacture 1 unit of product B requires 9 min on machine I and 3 min on machine II. There are 6 hr of machine time available on machine I and 4 hr of machine time available on machine II in each work shift. What is the largest profit the company can realize?

A) $192
B) $202
C) $209
D) $204
E) $188

Question 31

Consider the linear programming problem. ​
Sketch the feasible set for the linear programming problem.

A)
B)
C)
D) ​
E)

Question 32

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 33

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 34

Consider the linear programming problem. Maximize subject to Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 35

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 36

Solve the linear programming problem by the simplex method.
​

Question 37

Find the solution to the associated regular linear programming problem.

A) x = , y = 0, z = , u = 0, v = , w = 0, P =
B) x = 0, y = , z = , u = 0, v = , w = 0, P =
C) x = , y = 0, z = , u = 0, v = , w = 0, P =
D) x = , y = 0, z = , u = 0, v = , w = 0, P =
E) x = 0, y = , z = , u = 0, v = , w = 0, P =

Question 38

Solve the linear programming problem by the simplex method.
​

Question 39

Maximize subject to 3x + 10y + 4z ≤ 120
5x + 3y + 8z ≤ 9
8x + 10y + 3z ≤ 104
X ≥ 0, y ≥ 0, z ≥ 0

A) x = 0, y = 0, z = 0, u = 92, v = 0, w = 74, P = 3
B) x = 0, y = 0, z = 0, u = 92, v = 0, w = 76, P = 15
C) x = 0, y = 0, z = 3, u = 90, v = 0, w = 76, P = 3
D) x = 0, y = 3, z = 0, u = 90, v = 0, w = 74, P = 21
E) x = 0, y = 3, z = 0, u = 90, v = 0, w = 74, P = 15

Question 40

Consider the linear programming problem. ​
Sketch the feasible set for the linear programming problem.

A)
B)
C)
D)
E)

Question 41

Solve the linear programming problem by the simplex method. Minimize subject to

A)
B)
C)
D)

Question 42

An oil company operates two refineries in a certain city. Refinery I has an output of 200, 100, and 100 barrels of low-, medium-, and high-grade oil per day, respectively. Refinery II has an output of 100, 200, and 600 barrels of low-, medium-, and high-grade oil per day, respectively. The company wishes to produce at least 1,000, 1,400, and 3,000 barrels of low-, medium-, and high-grade oil to fill an order. If it costs $500/day to operate refinery I and $700/day to operate refinery II, determine how many days each refinery should be operated to meet the requirements of the order at minimum cost to the company. What is the minimum cost?

A) refinery I for 0 days; refinery II for 10 days; $5,300
B) refinery I for 6 days; refinery II for 4 days; $5,400
C) refinery I for 2 days; refinery II for 6 days; $5,200
D) refinery I for 3 days; refinery II for 5 days; $5,200

Question 43

Solve the linear programming problem by the simplex method.
​

Question 44

As part of a campaign to promote its annual clearance sale, a company decided to buy television-advertising time on a TV station. Company's television-advertising budget is $102,000. Morning time costs $3,000/minute, afternoon time costs $1,000/minute, and evening (prime) time costs $12,000/minute. Because of previous commitments, the station cannot offer the company more than 7 min of prime time or more than a total of 25 min of advertising time over the 3 wk in which the commercials are to be run. The station estimates that morning commercials are seen by 160,000 people, afternoon commercials are seen by 80,000 people, and evening commercials are seen by 480,000 people. How much morning, afternoon, and evening advertising time should the company buy to maximize exposure of its commercials?
__________ min of morning advertising time,
__________ min of afternoon advertising time,
__________ min of evening advertising time
What is the maximum number of viewers? Round your answer to two decimal places, if necessary.
__________ million viewers

Question 45

Solve the linear programming problem by the simplex method. Minimize subject to

A)
B)
C)
D)
E)

Question 46

A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of product A and a profit of $5/unit of product B. To manufacture 1 unit of product A requires 7 min on machine I and 9 min on machine II. To manufacture 1 unit of product B requires 9 min on machine I and 3 min on machine II. There are 6 hr of machine time available on machine I and 4 hr of machine time available on machine II in each work shift.
How many units of each product should be produced in each shift to maximize the company's profit?
A = __________ B = __________
What is the largest profit the company can realize?
$ __________
Is there any time left unused on the machines? Answer yes or no.
__________

Question 47

Solve the linear programming problem by the simplex method. Minimize subject to

A)
B)
C)
D)
E)

Question 48

Everest Deluxe World Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in newspaper I is seen by 70,000 group A customers, 40,000 group B customers, and 20,000 group C customers. Each advertisement in newspaper II is seen by 10,000 group A, 20,000 group B, and 40,000 group C customers. Each advertisement in newspaper I costs $800, and each advertisement in newspaper II costs $700. Everest would like their advertisements to be read by at least 2 million people from group A, 1.4 million people from group B, and 1 million people from group C. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost? What is the minimum cost?

A) 25 in newspaper I; 20 in newspaper II; $33,000
B) 28 in newspaper I; 12 in newspaper II; $33,000
C) 30 in newspaper I; 10 in newspaper II; $31,000
D) 26 in newspaper I; 18 in newspaper II; $32,000

Question 49

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 50 deluxe cabins and 190 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 140 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 340 deluxe and 800 standard cabins for their 15-day cruise in May. It costs $44,000 to operate a type-A vessel and $54,000 to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

A) 3 type-A vessels; 2 type-B vessels; $240,000
B) 0 type-A vessels; 6 type-B vessels; $324,000
C) 2 type-A vessels; 3 type-B vessels; $250,000
D) 7 type-A vessels; 0 type-B vessels; $308,000

Question 50

Solve the given minimization problem by the simplex method. Minimize subject to

A)
B)
C)
D)
E)

Question 51

A financier plans to invest up to $7 million in three projects. She estimates that project A will yield a return of 10% on her investment, project B will yield a return of 20% on her investment, and project C will yield a return of 25% on her investment. Because of the risks associated with the investments, she decided to put not more than 20% of her total investment in project C. She also decided that her investments in projects B and C should not exceed 60% of her total investment. Finally, she decided that her investment in project A should be at least 60% of her total investment. How much should the financier invest in each project if she wishes to maximize the total returns on her investments?
​
Project A: $ __________, project B: $ __________, and project C: $ __________
​
What is the maximum amount she can expect to make from her investments?
​
$ __________

Question 52

A pharmaceutical company produces three kinds of cold formulas: I, II, and III. It takes 2 hr to produce 1,000 bottles of formula I, 4 hr to produce 1,000 bottles of formula II, and 3 hr to produce 1,000 bottles of formula III. The profits for each 1,000 bottles of formula I, formula II, and formula III are $220, $130, and $190, respectively. Suppose, for a certain production run, there are enough ingredients on hand to make at most 7,000 bottles of formula I, 17,000 bottles of formula II, and 9,000 bottles of formula III. Furthermore, suppose the time for the production run is limited to a maximum of 100 hr. How many bottles of each formula should be produced in this production run so that the profit is maximized?
​
__________ bottles of formula I,
​
__________ bottles of formula II,
​
__________ bottles of formula III
​
What is the maximum profit realizable by the company? $ __________
​
Are there any resources left over?
​
ingredients for __________ bottles of formula I,
​
ingredients for __________ bottles of formula II,
​
ingredients for __________ bottles of formula III,
​
__________ hr of the time for the production run

Question 53

Solve the given minimization problem by the simplex method. Minimize subject to

A)
B)
C)
D)

Question 54

A furniture company is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and C - are to be marketed. Each model A desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model B desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model C desk requires hr, hr, and hr for fabrication, assembly, and finishing, respectively. The profit on each model A desk is $39, the profit on each model B desk is $42, and the profit on each model C desk is $31. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is 265 hr, 176 hr, and 160 hr, respectively. To maximize company's profit, how many desks of each model should be made in the month?
__________ units of model A,
__________ units of model B,
__________ units of model C
What is the largest profit the company can realize? $ __________
Are there any resources left over?
__________ hr left over in the fabrication department,
__________ hr left over in the assembly department,
__________ hr left over in the finishing department

Question 55

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows:
For the first year's production, a sum of $7,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 198,000 hr; and the amount of labor for on-site work is to be less than or equal to 207,000 labor-hours. Determine how many houses of each type Boise should produce (market research has confirmed that there should be no problems with sales) to maximize its profit from this new venture.
__________ houses of Standard Model.
__________ houses of Deluxe Model.
__________ houses of Luxury Model.

Question 56

A farmer has 150 acres of land suitable for cultivating crops A and B. The cost of cultivating crop A is $40/acre, whereas that of crop B is $60/acre. The farmer has a maximum of $7,400 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3,300 labor-hours available. If he expects to make a profit of $170/acre on crop A and $180/acre on crop B, how many acres of each crop should he plant in order to maximize his profit?
__________ acres of crop A, __________ acres of crop B
What is the largest profit the farmer can realize? $__________
Are there any resources left over?
$__________ left over, __________ acres of land left over, __________ labor-hours left over

Question 57

National Business Machines Corporation manufactures two models of fax machines: A and B. Each model A costs $110 to make, and each model B costs $130. The profits are $30 for each model A and $40 for each model B fax machine. If the total number of fax machines demanded each month does not exceed 2,500 and the company has earmarked no more than $600,000/month for manufacturing costs, find how many units of each model National should make each month in order to maximize its monthly profits.
__________ units of model A, __________ units of model B
What is the largest monthly profit the company can make? $ __________

Question 58

Solve the linear programming problem by the simplex method. Minimize subject to

A)
B)
C)
D)
E)

Question 59

Ashley has earmarked at most $200,000 for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and-income fund. The money market fund has a rate of return of 6%/year, the international equity fund has a rate of return of 9%/year, and the growth-and-income fund has a rate of return of 16%/year. Ashley has stipulated that no more than 25% of her total portfolio should be in the growth-and-income fund and that no more than 50% of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund?
$ __________ in the money market fund, $ __________ in the international equity fund,
$ __________ in the growth-and-income fund
What is the maximum return? $ __________

Question 60

Solve the linear programming problem by the simplex method. Minimize subject to

A)
B)
C)
D)

Question 61

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 0, y = 5, P = 10
B) x = 2, y = 0, P = 2
C) x = 0, y = 0, P = 11
D) x = 5, y = 3, P = 11

Question 62

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) 34
B) 34
C) 34
D) 34
E) 34

Question 63

The optimal value attained by the objective function of the primal problem may be different from that attained by the objective function of the dual problem.

A) True
B) False

Question 64

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 0, y = 6, z = 6, P = 12
B) x = 6, y = 6, z = 6, P = 12
C) x = 6, y = 6, z = 0, P = 12
D) x = 0 , y = 0, z = 6, P = 6

Question 65

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 0, y = 3, C = 3
B) x = 6, y = 0, C = - 24
C) x = 0, y = 0, C = 0
D) x = 5, y = , C = - 24

Question 66

Solve the primal problem.

A)
B)
C)
D)
E)

Question 67

Solve the primal problem.

A)
B)
C)
D)
E)

Question 68

Solve the given minimization problem by the simplex method.

Question 69

Use the technique developed in this section to solve the minimization problem.

A)
B)
C)
D)
E)

Question 70

Rewrite the given linear programming problem as a maximization problem with constraints involving inequalities of the form (with the exception of the inequalities ).

A) Maximize
Subject to
B) Maximize
Subject to
​
C) Maximize
Subject to
​
D) Maximize
Subject to
​

Question 71

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 10, y = 0, C = - 40
B) x = 5, y = 5, C = - 15
C) x = 0, y = 0, C = 0
D) x = 0, y = 25, C = - 175

Question 72

Use the technique developed in this section to solve the minimization problem.

A)
B)
C)
D)
E)

Question 73

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 3, y = 2, P = 7
B) x = 1, y = 0, P = 1
C) x = 0, y = 0, C = 0
D) x = 0, y = 3, P = 2

Question 74

Use the technique developed in this section to solve the minimization problem.
__________; __________; __________; __________ __________; __________; __________

Question 75

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 5, y = 10, C = 25
B) x = 0, y = 5, C = 3
C) x = 0, y = 0, C = 0
D) x = 10, y = 0, C = 10

Question 76

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A) x = 1, y = 3, P = 9
B) x = 0, y = 2, P = 2
C) x = 0, y = 0, C = 0
D) x = 6, y = 0, P = 36

Question 77

The owner of the Health JuiceBar wishes to prepare a low-calorie fruit juice with a high vitamin A and vitamin C content by blending orange juice and pink grapefruit juice. Each glass of the blended juice is to contain at least 1,610 International Units (IU) of vitamin A and 194 IU of vitamin C. One ounce of orange juice contains 70 IU of vitamin A, 14 IU of vitamin C, and 13 calories; each ounce of pink grapefruit juice contains 140 IU of vitamin A, 12 IU of vitamin C, and 12 calories. How many ounces of each juice should a glass of the blend contain if it is to meet the minimum vitamin requirements while containing a minimum number of calories?
__________ ounces of orange juice
__________ ounces of pink grapefruit juice
__________ calories

Question 78

The owner of the Health JuiceBar wishes to prepare a low-calorie fruit juice with a high vitamin-A and -C content by blending orange juice and pink grapefruit juice. Each glass of the blended juice is to contain at least 1,260 International Units (IU) of vitamin A and 184 IU of vitamin C. One ounce of orange juice contains 70 IU of vitamin A, 18 IU of vitamin C, and 13 calories; each ounce of pink grapefruit juice contains 140 IU of vitamin A, 16 IU of vitamin C, and 12 calories. How many ounces of each juice should a glass of the blend contain if it is to meet the minimum vitamin requirements while containing a minimum number of calories?

A) 4 oz of orange juice;
6 oz of pink grapefruit juice;
168 calories
B) 7 oz of orange juice;
4 oz of pink grapefruit juice;
139 calories
C) 2 oz of orange juice;
6 oz of pink grapefruit juice;
132 calories
D) 2 oz of orange juice;
7 oz of pink grapefruit juice;
148 calories
E) 4 oz of orange juice;
7 oz of pink grapefruit juice;
136 calories

Question 79

Solve the linear programming problem by the simplex method.

Question 80

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If a standard minimization linear programming problem has a unique solution, then so does the corresponding maximization problem with objective function , where is the objective function for the minimization problem.