Exam
Name___________________________________
1)
The region indicated in the diagram
is described by
1)
A)
y< 6
4x+ 4y–y> 0
x–y+ 2 0
B)
y< 6
3x +2y – 8 0
2x– 4y+ 5 > 0
C)
x< 6
3x+ 5y– 4 > 0
4x– 2y+ 9 0
D)
y 6
4x +5y – 5 < 0
x– 2y+ 8 0
E)
y< 6
x +y– 2 0
3x– 2y+ 12 < 0
2)
Maximize
Z =x1– 2x2+ 3x3
subject to
2x1+x2+2x310
x1–x2+x38
x1,x2,x30
2)
A)
10
B)
5
C)
15
D)
0
E)
20
3)
In a simplex table that gives an optimum solution, a zero indicator for a nonbasic variable suggests
3)
A)
there is a possibility for multiple optimum solutions.
B)
the problem has an unbounded solution.
C)
a degenerate BFS will occur.
4)
The dual of
Minimize
Z =x1+ 3x2
subject to
x1– 2x24
3x1+x21
x1, x20
is:
4)
A)
Maximize W= 4y1+y2 subject to y1+ 3y2 1; – 2y1+y2
3; y1, y2
0.
B)
Maximize W= 4y1+y2 subject to y1+ 3y2 1; – 2y1+y2
3; y1, y2
0.
C)
Maximize W=y1+ 3y2 subject to y1+ 3y2 4; – 2y1+y2
1; 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– 2y2 1; 3y1+y2 3; y1, y2,
0.
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)
When no quotients exist in a simplex table,
6)
A)
there is a possibility for multiple optimum solutions.
B)
the problem has an unbounded solution.
C)
a degenerate BFS will occur.
7)
When two quotients in a simplex table tie for being the smallest,
7)
A)
there is a possibility for multiple optimum solutions.
B)
the problem has an unbounded solution.
C)
a degenerate BFS will occur.
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)
x1
B)
s2
C)
s1
D)
Z
E)
x2
9)
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
9)
A)
10
B)
0
C)
6
D)
–1
E)
2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10)
Maximize
Z = 4x+y
subject to
–x+y2
3x+y18
x, y0.
10)
11)
Suppose a car dealer has showrooms in Atherton and Berkeley and warehouses in Concord
and Dublin. The cost of delivering a car is $60 from Concord to Atherton, $45 from
Concord to Berkeley, $50 from Dublin to Atheron, and $35 from Dublin to Berkely.
Suppose that the showroom in Atherton orders 7 cars and the showroom in Berkeley
orders 4 cars. Also suppose that the warehouse in Concord has 8 cars and the warehouse in
Dublin has 6 cars available. Find the best way to minimize cost and find the minimum cost.
(Hint: Let x1 be the number of cars delivered from Concord to Atherton and x2 be the
number of cars delivered from Concord to Berkeley. Thus 7 –x1 is the number of cars
delivered from Dublin to Atherton and 4 –x2 the number of cars shipped from Dublin to
Berkeley.)
11)
12)
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
12)
13)
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?
13)
14)
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.
14)
15)
Maximize
Z= 4x + 6y
subject to
x + y 3
y 5
x 4
x 0, y 0
15)
16)
Maximize
Z = 5x– 3y
subject to
2x–y
8
2x– 5y
0
x –y= – 2
x, y 0.
16)
17)
Sketch the region described by the inequalities
2x+y 1
x–y
1
17)
18)
Use the simplex method to minimize
Z = – x1+ 2x2
subject to
x1+x2 4
5x1+x2–12
2x1+ 5x2–14
3x1– 2x2 17
18)
19)
Sketch the region described by the inequalities.
x– 2y 4
y> – x– 2
19)
20)
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.)
20)
21)
Sketch the region described by the inequalities.
2x– 3y> 6
x + y 4
21)
22)
Sketch the region described by the inequalities.
x + y–2
y 5
–3x + y–3
22)
23)
Find the dual problem to the following: 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.
What is the number of days that each location needs to operate in order to fill the orders at
minimum cost.
Location 1 Location 2 Min. 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
23)
24)
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
24)
25)
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.
25)
26)
Use the simplex method to
maximize
Z =x1+ 4x2+x3
subject to
x1+x2+x3 6
x1–x2–2x3 2
x1, x2, x3
0 .
26)
27)
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.
27)
28)
Use the simplex method to
maximize
Z = 2x1– 3x2
subject to
x1+x25
x1+2x28
x1,x20
28)
29)
Use the dual and the simplex method to
minimize
Z = 4x1+ 5x2
subject to
x1–x24
2x1–x21
5x1+ 3x23
x1, x20.
29)
30)
Maximize
Z= 10x+ 15y
subject to
x+ 3y
15
4x + 3y
24
x 0, y 0
30)
31)
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?
31)
32)
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.
32)
33)
Use the simplex method to
maximize
Z = 30x+ 50y
subject to
2x+y
16
x+ 2y
11
x+ 3y
15
x, y
0
33)
34)
Use the simplex method to
minimize
Z = 4x1+x2
subject to
x1+x25
x1+2x28
x1,x20.
34)