Exam
Name___________________________________
Use the indicated region of feasible solutions to find the
maximum and minimum values of the given objective
function.
Solve the problem.
3) The Acme Class Ring Company designs and
sells two types of rings: the VIP and the SST.
They can produce up to 24 rings each day
using up to 60 total man-hours of labor. It
takes 3 man-hours to make one VIP ring and 2
man-hours to make one SST ring. How many
of each type of ring should be made daily to
maximize the company's profit, if the profit on
a VIP ring is $30 and on an SST ring is $40?
1) z = 12x - 22y
y
(0, 6)
(1.2, 5)
4) Zach is planning to invest up to $50,000 in
corporate and municipal bonds. The least he
will invest in corporate bonds is $6000 and he
does not want to invest more than $29,000 in
corporate bonds. He also does not want to
invest more than $28,174 in municipal bonds.
The interest is 8.5% on corporate bonds and
6.8% on municipal bonds. This is simple
interest for one year. What is the maximum
value of his investment after one year?
(5, 0)
x
Use graphical methods to solve the linear programming
problem.
2) Maximize
z = 8x + 12y
subject to:
40x + 80y ≤ 560
5) An airline with two types of airplanes, P1 and
6x + 8y ≤ 72
x≥0
P2 , has contracted with a tour group to
y≥ 0
provide transportation for a minimum of 400
first class, 900 tourist class, and 1500 economy
class passengers. For a certain trip, airplane P1
y
costs $10,000 to operate and can accomodate 20
first class, 50 tourist class, and 110 economy
class passengers. Airplane P2 costs $8500 to
10
10
-10
operate and can accomodate 18 first class, 30
tourist class, and 44 economy class passengers.
How many of each type of airplane should be
used in order to minimize the operating cost?
x
Pivot once about the circled element in the simplex
tableau, and read the solution from the result.
-10
6)
1
Write the solutions that can be read from the simplex
tableau.
7)
13)
8)
14)
15)
9)
16)
Introduce slack variables as necessary, and write the
initial simplex tableau for the problem.
10) Find x1 ≥ 0 and x2 ≥ 0 such that
17)
5x1 + 10x2 ≤ 105
10x1 + 15x2 ≤ 136
and z = 2x1 + 5x2 is maximized.
x1
1
0
0
x2 x 3
4 1
1 3
0 0
s1 s2
0 5
1 2
0 4
z
x1
1
3
3
2
x2 s1
0 1
0 0
1 0
0 0
s2 s3
3 0
1 1
2 0
3 0
z
x1
0
0
1
0
x2 s1 s2 s3 z
3 0 1 1 0
4 1 0 1 0
5 0 0 1 0
-3 0 0 1 1
x1
2
1
4
x2 x 3 s 1 s 2 z
x1
3
1
-3
x2 x 3
4 0
5 1
4 0
0
1
0
3
2
1
1
4
8
1
0
0
s1 s2
3 1
7 0
1 0
0 4
0 3
1 2
0 9
0 16
0 20
1 21
16
18
17
10
0 10
0 8
1 16
z
0 14
0 21
1 20
A manufacturing company wants to maximize profits on
products A, B, and C. The profit margin is $3 for A, $6 for
B, and $15 for C. The production requirements and
departmental capacities are as follows:
Department Production requirement Departmental capacity
by product (hours)
(Total hours)
A B C
Assembling
2 3 2
30,000
Painting
1 2 2
38,000
Finishing
2 3 1
28,000
11) Find x1 ≥ 0 and x2 ≥ 0 such that
2x1 + 5x2 ≤ 7
3x1 + 3x2 ≤ 18
and z = 4x1 + x2 is maximized.
18) What is the maximum profit if the capacity of
the painting department changes to 40,000
hours?
12) Find x1 ≥ 0 and x2 ≥ 0 such that
x1 + x2 ≤ 64
3x1 + x2 ≤ 171
19) What is the maximum profit if the profit
margin on A changes to $7.00?
and z = 2x1 + x2 is maximized.
20) What is the constraint for the finishing
department?
2
The initial tableau of a linear programming problem is
given. Use the simplex method to solve the problem.
21)
x1 x2
2 2
1 1
2 4
-2 -5
x3 s1
6 1
2 0
3 0
-3 0
s2
0
1
0
0
s3
0
0
1
0
Each day Larry needs at least 10 units of vitamin A, 12
units of vitamin B, and 20 units of vitamin C. Pill #1
contains 4 units of A and 3 of B. Pill #2 contains 1 unit of
A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B,
and 5 of C.
z
0 8
0 3
0 20
1 0
28) Pill #1 costs 10 cents, pill #2 costs 7 cents, and
pill #3 costs 3 cents. Larry wants to minimize
cost. What is the constraint inequality for
vitamin A?
Use the simplex method to solve the linear programming
problem.
22) Maximize
z = 2x1 + 5x2 + 3x3
subject to:
2x1 + x2 + 3x3 ≤ 9
4x1 + 3x2 + 5x3 ≤ 12
with
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
23) Maximize
2x1 + x2 + 5x3 + 6x4 ≤ 25
with
5x1 + 3x2 + 4x3 + x4 ≤ 60
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
subject to:
State the dual problem. Use y1 , y2 , y3 and y4 as the
variables. Given: y1 ≥ 0, y2 ≥ 0, y3 ≥ 0, and y4 ≥ 0.
z = x1 + 3x2 + x3 + 2x4
subject to:
24) Maximize
29) Pill #1 costs 9 cents, pill #2 costs 9 cents, and
pill #3 costs 1 cent. How many of each pill
must Larry take to minimize his cost?
30) Maximize
subject to:
9x1 + 3x2 ≤ 190
x1 + x2 ≤ 18
2x1 + 6x2 ≤ 85
x1 ≥ 0, x2 ≥ 0
z = x1 + 2x2 + 4x3 + 6x4
x1 + 2x2 + 3x3 + x4 ≤ 100
3x1 + x2 + 2x3 + x4 ≤ 75
with
z = 5x1 + 7x2
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
31) Minimize
w = 2x1 + 3x2 + x3
subject to:
x1 + 3x2 + 2x3 ≥ 31
2x1 + 4x2 + 3x3 ≥ 67
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
A toy making company has at least 300 squares of felt, 700
oz of stuffing, and 230 ft of trim to make dogs and
dinosaurs. A dog uses 1 square of felt, 4 oz of stuffing, and
1 ft of trim. A dinosaur uses 2 squares of felt, 3 oz of
stuffing, and 1 ft of trim.
32) Minimize
subject to:
w = 6x1 + 2x2 + 3x3 + 2x4
4x1 + 5x2 + 4x3 + 3x4 ≥ 45
4x1 + 6x2 + 5x3 + 11x4 ≥ 55
25) It costs the company $1.65 to make each dog
and $1.52 for each dinosaur. What is the
company's minimum cost?
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
33) Maximize
subject to:
26) It costs the company $1.72 to make each dog
and $1.88 for each dinosaur. The company
wants to minimize its costs. What are the
coefficients of the objective function?
z = x1 + 2x2 + 3x3
6x1 + 3x2 + x3 ≤ 16
4x1 + 7x2 + 3x3 ≤ 21
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
27) It costs the company $1.62 to make each dog
and $1.83 for each dinosaur. The company
wants to minimize its costs. What are the
coefficients of the constraint inequality for
trim?
34) Minimize
w = 6x1 + 3x2
subject to:
3x1 + 2x2 ≥ 28
2x1 + 5x2 ≥ 48
x1 ≥ 0, x2 ≥ 0
3
35) Maximize
subject to:
z = 3x1 + 2x2
x1 + x2 ≤ 28
2x1 + x2 ≤ 22
x1 ≥ 0, x2 ≥ 0
Use the simplex method to solve the linear programming
problem.
36) Minimize
subject to:
w = y1 + 4y2
4y1 + 3y2 ≥ 75
3y1 + 5y2 ≥ 93
y1 ≥ 0, y2 ≥ 0
37) Minimize
subject to:
w = 5y1 + 3y2
2y1 + 3y2 ≥ 9
2y1 + y2 ≥ 11
y1 ≥ 0, y2 ≥ 0
4
Answer Key
Testname: 1324-SIMP-PT
1)
2)
3)
4)
5)
Maximum of 60; minimum of -132
Maximum of 100 when x = 8 and y = 3
0 VIP and 24 SST
$53,893
14 P1 planes and 7 P2 planes
32) Maximize
subject to:
33) Minimize
subject to:
8) x3 = 5, s2 = 1, s3 = -27, z = 15; x1 , x2 , s1 = 0
9) x3 = 14, s2 = -6, s3 = 4, z = 42; x1 , x2 , s1 = 0
10)
34) Maximize
105
136
0
subject to:
35) Minimize
7
18
0
subject to:
16) s2 = 10, x2 = 8, z = 16; x1 , x3 , s1 = 0
17) x1, x2, s1 = 0, x3 = 21, s2 = 14, z = 20
$225,000
$225,000
2A + 3B + C ≤ 28,000
Maximum at 15 for x2 = 3, s1 = 2, s3 = 8
22) Maximum is 20 when x1 = 0, x2 = 4, x3 = 0
23) Maximum is 60 when x1 = 0, x2 = 20, x3 = 0, x4 = 0
24) Maximum is 450 when x1 = 0, x2 = 0, x3 = 0, x4 = 75
$317
1.72, 1.88
1, 1
4P1 + P2 + 10P3 ≥ 10
29) P1 = 0, P2 = 0, P3 = 12
30) Minimize
w = 190y1 + 18y2 + 85y3
subject to: 9y1 + y2 + 2y3 ≥ 5
3y1 + y2 + 6y3 ≥ 7
31) Maximize
subject to:
6y1 + 4y2 ≥ 1
3y1 + 7y2 ≥ 2
y1 + 3y2 ≥ 3
z = 28y1 + 48y2
3y1 + 2y2 ≤ 6
2y1 + 5y2 ≤ 3
w = 28y1 + 22y2
y1 + 2y2 ≥ 3
y1 + y2 ≥ 2
37) 27.5 when y1 = 5.5 and y2 = 0
14) s1 = 9, s3 = 16, x2 = 20, z = 21; x1 , s2 = 0
15) s2 = 16, s1 = 18, x1 = 17, z = 10; x2 , s3 = 0
25)
26)
27)
28)
w = 16y1 + 21y2
36) 31 when y1 = 31 and y2 = 0
1
1 1 0 0
64
3
1 0 1 0 171
0
-2 -1 0 0 1
13) x1 = 4, s1 = 3, z = 2; x2 , x3 , s2 = 0
18)
19)
20)
21)
4y1 + 4y2 ≤ 6
5y1 + 6y2 ≤ 2
4y1 + 5y2 ≤ 3
3y1 + 11y2 ≤ 2
6) x2 = 2, s2 = 4, s3 = 2, z = 12; x1, x3, s1 = 0
7) x3 = 5, x2 = -7, z = -10; x1 , s1 , s2 = 0
x1 x2 s1 s2 z
5 10 1 0 0
10 15 0 1 0
-2 -5 0 0 1
11) x1 x2 s1 s2 z
2
5 1 0 0
3
3 0 1 0
-4 -1 0 0 1
12) x1 x2 s1 s2 z
z = 45y1 + 55y2
z = 31y1 + 67y2
y1 + 2y2 ≤ 2
3y1 + 4y2 ≤ 3
2y1 + 3y2 ≤ 1
5