Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Maximize
Z =x1– 2x2+ 3x3
subject to
2x1+x2+2x310
x1–x2+x38
x1,x2,x30
1)
A)
0
B)
10
C)
5
D)
20
E)
15
2)
The region indicated in the diagram
is described by
2)
A)
y< 6
3x +2y – 8 0
2x– 4y+ 5 > 0
B)
x< 6
3x+ 5y– 4 > 0
4x– 2y+ 9 0
C)
y 6
4x +5y – 5 < 0
x– 2y+ 8 0
1
D)
y< 6
x +y– 2 0
3x– 2y+ 12 < 0
E)
y< 6
4x+ 4y–y> 0
x–y+ 2 0
3)
When no quotients exist in a simplex table,
3)
A)
the problem has an unbounded solution.
B)
a degenerate BFS will occur.
C)
there is a possibility for multiple optimum solutions.
4)
When two quotients in a simplex table tie for being the smallest,
4)
A)
the problem has an unbounded solution.
B)
a degenerate BFS will occur.
C)
there is a possibility for multiple optimum solutions.
5)
The region indicated in the diagram
is described by
5)
A)
y 2x
x + y > 1
B)
y 2x
x + y > 1
C)
y 2x
x + y
1
D)
y 2x
x + y
1
E)
y< 2x
x + y > 1
6)
The dual of
Minimize
Z =x1+ 3x2
subject to
x1– 2x24
3x1+x21
x1, x20
is:
6)
A)
Maximize W=y1+ 3y2 subject to y1+ 3y2 4; – 2y1+y2
1; y1, y2
0.
B)
Maximize W= 4y1+y2 subject to y1– 2y2 1; 3y1+y2 3; y1, y2,
0.
C)
Maximize W= 4y1+y2 subject to y1+ 3y2 1; – 2y1+y2
3; y1, y2
0.
D)
Maximize W=y1+ 3y2 subject to y1– 2y2 4; 3y1+y2 1; y1, y2 0.
E)
Maximize W= 4y1+y2 subject to y1+ 3y2 1; – 2y1+y2
3; y1, y2
0.
7)
In the initial simplex table below, find the pivot entry.
x1 x2s1s2Z
s1
s2
Z
–1 2 1 0 0 8
10 6 0 1 0 12
–3–8 0 0 1 0
7)
A)
–1
B)
10
C)
2
D)
0
E)
6
8)
In the initial simplex table below, find the departing variable.
x1x2 s1s2 Z
s1
s2
Z
–1 2 1 0 0 8
10 6 0 1 0 12
–3–8 0 0 1 0
8)
A)
Z
B)
x1
C)
s2
D)
x2
E)
s1
9)
In a simplex table that gives an optimum solution, a zero indicator for a nonbasic variable suggests
9)
A)
the problem has an unbounded solution.
B)
a degenerate BFS will occur.
C)
there is a possibility for multiple optimum solutions.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10)
A store sells two types of calculators. In order to cover overhead, it must sell at least 40
calculators total per week, and in order to satisfy distribution requirements, it must sell at
least twice as many of type II as type I. Write a system of inequalities to describe the
situation. Let x be the number of type I that it sells in a week and y be the number of type II
that it sells in a week. Find the region described by the system of linear inequalities.
10)
11)
Use the simplex method to solve the following problem: United Blimpo Co. produces two
types of exercise devices ,regular and heavy–duty, each of which requires in its
manufacture the use of two machines, A and B. A regular model requires the use of
machine A for 2 hours and machine B for 3 hours. A heavy–duty model requires the use of
machine A for 3 hours and machine B for 3 hours. Machine A can be used at most 18 hours
a day and machine B can be used at most 21 hours a day. If the profits on the regular and
heavy–duty models are $20 and $15, respectively, and United Blimpo Co. can sell all it
produces, how many of each model should be produced per day in order to realize
maximum profit? What is the maximum profit per day?
11)
12)
Find the dual problem to the following: The What If Company has $30,000 for the
purchase of material to make three types of gadgets. The company has allocated a total of
1200 hours of assembly time and 180 hours of packaging time for the gadgets. The
following table gives the cost per gadget, the number of hours per gadget, and the profit per
gadget for each type.
Type 1 Type 2 Type 3
Cost/gadget $300 $300 $400
Assembly Hours/gadget 15 15 10
Packaging Hours/gadget 2 2 3
Profit $150 $ 250 $200
What is the number of gadgets of each type the company should produce to maximize
profit?
12)
13)
Sketch the region described by the inequalities.
x– 2y 4
y> – x– 2
13)
14)
A company has two different locations to assemble three different models of PCs. The table
below summarizes the daily production capacity, the minimum number of each type
needed, and the daily operating costs for each location. Find the number of days that each
location needs to operate in order to fill the orders at minimum cost.
Location 1 Location 2 Minimum Number
Model 1 60/day 60/day 2400
Model 2 40/day 80/day 2000
Model 3 60/day 40/day 1800
Weekly Cost $16,000 $12,000
14)
15)
A car rental company has $540,000 to purchase up to 25 new cars of two different models.
One model costs $18,000 each and the other model costs $24,000 each. Write a system of
linear inequalities to describe the situation. Let x represent the first model and y represent
the second. Find the region described by the system of linear inequalities.
15)
16)
The XYZ Corporation produces two models of home computers, the Alpha model and the
Beta model. Let x be the number of Alpha models and y the number of Beta models
produced at the San Antonio factory per week. If the factory can produce at the most 100
Beta models in a week, write an inequality to describe this situation. Describe the region
for the inequality. What additional inequalities can you add to the situation?
16)
17)
Use the simplex method to
maximize
Z = 2x1– 3x2
subject to
x1+x25
x1+2x28
x1,x20
17)
18)
A manufacturer produces two products, product A and product B. Both products require
processing on Machines I and II. The number of hours needed to produce one unit is given
by the following chart:
Machine I Machine II
Product A 3 hrs 2 hrs
Product B 4 hrs 6 hrs
Machine I is available for at most 1150 hours and Machine II is available for at most 1100
hours. If the profit made on product A is $15 / unit and the profit made on product B is
$30 / unit. Find the production level that will maximize profit and find the Maximum
profit.
18)
19)
Sketch the region described by the following system of inequalities:
x> 1
y
2x– 3
2y+x< 10
19)
20)
Use the corner–point technique to
maximize
Z =x+ 2y
subject to
yx+ 3
x+ 2y
24
x, y
0.
Also determine the values of x and y at which the maximum value occurs.
20)
21)
A manufacturer produces two products, product A and product B. Both products require
processing on Machines I and II. The number of hours needed to produce one unit is given
by the following chart:
Machine I Machine II
Product A 2 hrs 3 hrs
Product B 1 hrs 4 hrs
Machine I is available for at most 1000 hours and Machine II is available for at most 2500
hours. If the profit made on product A is $20 / unit and the profit made on product B is
$25 / unit. Find the production level that will maximize profit and find the Maximum
profit.
21)
22)
A chair company produces two models of chairs, the Sequoia and the Saratoga. The
Sequoia model takes 3 hours to assemble and 1
2 hour to paint. The Saratoga model takes 2
hours to assemble and 1 hour to paint.The maximum number of hours available to
assemble is 24 per day and the maximum number of hours available to paint is 8 per day.
(a) If the company earns a profit of $20 per Sequoia model and $30 per Saratoga model,
find the number of models produced per day in order to maximize profit.
(b) If the company earns a profit of $30 per Sequoia model and $15 per Saratoga model,
find the number of models produced per day in order to maximize profit.
(c) Suppose the company decides to upgrade the two models so it takes an additional 2
hours to detail the Sequoia and 2 hours to detail the Saratoga. The maximum number of
hours available to detail is 18 per day. If the company earns a profit of $45 per Sequoia
model and $35 per Saratoga model, find the number of models produced per day in order
to maximize profit.
(d) Suppose the company decides to upgrade the two models so it takes an additional 2
hours to detail the Sequoia and 2 hours to detail the Saratoga. The maximum number of
hours available to detail is 18 per day. If the company earns a profit of $30 per Sequoia
model and $40 per Saratoga model, find the number of models produced per day in order
to maximize profit.
22)
23)
A company manufactures two models of inline skates, Alpha and Beta, at two different
manufacturing plants. The maximum output at plant I is 1000 per month, while the
maximum output at plant II is 1200 per month. Due to contractual obligations, the number
of Alpha models produced at plant I must exceed the number of Beta models produced at
plant I by at least 100. The profit per Alpha and Beta model manufactured at plant I is $50
and $80, respectively, while the profit per Alpha and Beta model manufactured at plant II
is $60 and $70, respectively. This month, the company received an order for 900 Alpha and
1000 Beta models. Find how many of each model should be produced at each plant in order
to satisfy the order and maximize the profit. (Hint: Let x1 represent the number of Alpha
models and x2 represent the number of Beta models manufactured at plant I.)
23)
24)
Give the dual of:
Maximize
Z =x1+ 3x2+ 4x3
subject to
3x1+x2–x34
4x1–x2+2x3–1
x1, x2, x30.
24)
25)
Use the simplex method to
maximize
Z = 30x+ 50y
subject to
2x+y
16
x+ 2y
11
x+ 3y
15
x, y
0
25)
26)
Suppose a TV dealer has stores A and B and warehouses C and D. The cost of shipping a
TV is $18 from C to A, $9 from C to B, $24 from D to A, and $15 from D to B. Suppose that
store A orders 25 TV sets and store B orders 30 TV sets. Also suppose that warehouse C has
40 TV sets and warehouse D has 45 TV sets available. Find the best way to minimize cost
and find the minimum cost (Hint: Let x1 be the number of TV sets shipped from C to A
and x2 be the number of TV sets shipped from C to B. Thus, 25 –x1 is the number of TV
sets shipped from D to A and 30 –x2 is the number of TV sets shipped from D to B.)
26)
27)
Maximize
Z = 4x+y
subject to
–x+y2
3x+y18
x, y0.
27)
28)
Use the simplex method to
minimize
Z = 4x1+x2
subject to
x1+x25
x1+2x28
x1,x20.
28)
29)
Sketch the region described by the inequalities.
2x– 3y> 6
x + y 4
29)
30)
A person decides to take two different dietary supplements. Each supplement contains two
essential ingredients, A and B, for which there are minimum daily requirements, and each
contains a third ingredient, C, which needs to be minimized. Find the amount of each
supplement that the person should take each day in order to satisfy the requirements for A
and B while minimizing C. Also find the amount of C the person takes each day.
Supplement 1 Supplement 2 Daily Requirement
A20 mg/oz 6 mg/oz 98 mg
B8 mg/oz 16 mg/oz 80 mg
C2 mg/oz 6 mg/oz
30)
31)
To make some extra money, you make two types of picture frames, type A and type B, for
sale. You have an initial start–up expense of $75. The production cost for type A is $3.60
per frame, and the production cost for type B is $5.20 per frame. The price for type A is
$6.00 per frame and the price for type B is $10.00 per frame. Let x be the number of type A
and y be the number of type B produced and sold. Write an inequality describing revenue
less than cost. Solve the inequality and describe the region. Also, describe what this means
in terms of frames.
31)
32)
Sketch the region described by the inequalities.
x + y–2
y 5
–3x + y–3
32)