Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1)
A set S in Rn is convex if, for each p and q in S, the line segment between p and q lies in S.
[This line segment is the set of points of the form (1 – t)p+ tq for 0 t 1. ]
Let F be the feasible set of all solutions x of a linear programming problem Axb with x
0. Assume that F is nonempty. Show that F is a convex set in Rn.
[Hint: Consider points p and q in F and t such that 0 t 1. Show that (1 – t)p+ tq is in F.]
1)
2)
A store makes two different types of smoothies by blending different fruit juices. Each
bottle of Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of
pineapple juice, and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue
smoothie contains 5 fluid ounces of orange juice, 6 fluid ounces of pineapple juice, and 4
fluid ounces of blueberry juice. The store has 500 fluid ounces of orange juice, 360 fluid
ounces of pineapple juice, and 250 fluid ounces of blueberry juice available to put into its
smoothies. The store makes a profit of $1.50 on each bottle of Orange Daze and $1 on each
bottle of Pineapple Blue. To determine the maximum profit, the simplex method can be
used and the final tableau is:
x1x2 x3x4x5 M
1 0 3
20 –1
8 0 0 30
0 1 –1
10 1
4 0 0 40
0 0 1
10 –3
4 1 0 30
0 0 1
81
16 0 1 85
Give an interpretation to the fact that the marginal value of blueberry juice is zero. Refer in
your explanation to the value in the final tableau of the slack variable x5.
2)
3)
The set S of all x that satisfy the inequality ax b is a ray in R1. Show that the set S is
convex.
[Hint: A set S in Rn is convex if, for each p and q in S, the line segment between p and q
lies in S. This line segment is the set of points of the form (1 – t)p+ tq for 0 t 1. ]
3)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value of the strategy.
4)
Let M be the matrix game having payoff matrix 0 1 –2–2
3 0 2 0
–1 3 0 –2.
Find (x) when x=
0
2
3
1
3
.
4)
A)
1
B)
–2
3
C)
5
3
D)
4
3
Solve by using the simplex method.
5)
Minimize 8x1+7x2
subject to 3x1–4x212
2x1+x220
and x1 0, x2 0
5)
A)
Minimum =988
11 when x1=92
11 and x2=36
11
B)
Minimum = – 988
11 when x1=92
11 and x2=36
11
C)
Minimum =988
11 when x1=32
11 and x2=36
11
D)
Minimum =140 when x1= 0 and x2=20
2
Provide an appropriate response.
6)
A linear programming problem has objective function 4x and constraints
x a
–x –b
x 0
where a and b are positive numbers and b > a.
Does a maximum exist for the objective function? If not, why not? Does a minimum exist? If not,
why not?
6)
A)
Maximum exists. Minimum does not exist as the objective function may be arbitrarily small
(unbounded).
B)
Neither maximum nor minimum exist as the feasible set is the empty set.
C)
Maximum does not exist as the objective function may be arbitrarily large (unbounded).
Minimum does exist.
D)
Maximum and minimum both exist.
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
7)
Minimize 5x1+x2–3x3
subject to x1+6x2–2x322
–6x2+6x335
and x1 0, x2 0, x3 0
7)
A)
b =–22
–35 , c =5
1
–3, and A =–1–6 2
0 6 –6
B)
b =22
35 , c =5
1
–3, and A =1 6 –2
0–6 6
C)
b =–22
–35 , c =–5
–1
3, and A =–1–6 2
0 6 –6
D)
b =22
35 , c =–5
–1
3, and A =1 6 –2
0–6 6
3
The primal problem and the final tableau in the solution of the primal problem are given. Use the final tableau to solve
the dual problem.
8)
Primal problem :
Maximize 7x1+10x2
subject to x1+ 2x212
5x1+4x235
and x1 0, x2 0
Final tableau in solution to primal problem:
x1x2x3x4M
0 1 5
6–1
6 0 25
6
1 0 –2
31
3 0 11
3
0 0 11
32
3 1 202
3
8)
A)
Minimum = 0 when y1= 0 and y2= 0
B)
Minimum =202
3 when y1=11
3 and y2=2
3
C)
Maximum =202
3 when y1=11
3 and y2=2
3
D)
Minimum =202
3 when y1=25
6 and y2=11
6
Provide an appropriate response.
9)
Anne and Michael are playing a game in which each player has a choice of two colors: green or
blue. The payoff matrix with Anne as the row player is given below:
gr bl
green
blue a b
c d
Using the same payoffs for Anne and Michael, write the matrix that shows the winnings with
Michael as the row player.
9)
A)
gr bl
green
blue a c
b d
B)
gr bl
green
blue –a–b
–c–d
C)
gr bl
green
blue d c
b a
D)
gr bl
green
blue –a–c
–b–d
Find the value of the matrix game.
10)
Let M be the matrix game having payoff matrix 1–1
–2 4 .
10)
A)
1
4
B)
1
2
C)
2
9
D)
3
4
Find all saddle points for the matrix game.
11)
9 3 –5
–6 3 7
11)
A)
Entry a21
B)
Entry a11
C)
Entry a13
D)
No saddle points
The primal problem and the final tableau in the solution of the primal problem are given. Use the final tableau to solve
the dual problem.
12)
Primal problem:
Maximize 3x1+ 4x2+ 2x3
subject to x1+x3 16
2x2+x3 20
3x1+6x2 36
and x1 0, x2 0, x3 0
Final tableau in solution to primal problem:
x1x2 x3x4x5x6 M
1 0 0 1
2–1
21
6 0 4
0 0 1 1
21
2–1
6 0 12
0 1 0 –1
41
41
12 0 4
0 0 0 3
21
21
2 1 52
12)
A)
Minimum = 576 when y1= 4, y2= 4, and y3= 12
B)
Minimum = 52 when y1= 4, y2= 4, and y3= 12
C)
Minimum = 52 when y1=3
2, y2=1
2, and y3=1
2
D)
Minimum =15
2 when y1=3
2, y2=1
2, and y3=1
2
Find the expected payoff.
13)
Let M be the matrix game having payoff matrix 0 2 1 –1
–1 0 2–1
0 1 0 –1.
Find E(x, y) when x=
1
4
1
2
1
4
and y=
1
4
0
1
2
1
4
13)
A)
1
2
B)
1
8
C)
5
16
D)
1
4
Solve the problem.
14)
A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The
average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100.
The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They
must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many
counselors and how many aides should the camp hire to minimize cost?
14)
A)
18 counselors and 12 aides
B)
27 counselors and 18 aides
C)
12 counselors and 18 aides
D)
35 counselors and 10 aides
Provide an appropriate response.
15)
Given the primal problem P below:
Maximize f(x) =cxT
subject to Ax b
x 0
where c is a vector in
n, b is a vector in
n, and A is an m × n matrix, how many constraints and
how many variables are in the dual problem?
15)
A)
m constraints, m variables
B)
n constraints, m variables
C)
m constraints, n variables
D)
n constraints, n variables
Solve the problem.
16)
Alan wants to invest a total of $21,000 in mutual funds and a certificate of deposit (CD). He wants
to invest no more in mutual funds than half the amount he invests in the CD. His expected return
on mutual funds is 10% and on the CD is 4%. How much money should Alan invest in each area in
order to have the largest return on his investments? What is his maximum one–year return?
16)
A)
Maximum one–year return is $840 when he invests $0 in mutual funds and $21,000 in the CD.
B)
Maximum one–year return is $2100 when he invests $21,000 in mutual funds and $0 in the
CD.
C)
Maximum one–year return is $1680 when he invests $14,000 in mutual funds and $7000 in the
CD.
D)
Maximum one–year return is $1260 when he invests $7000 in mutual funds and $14,000 in the
CD.
Determine whether the statement is true or false.
17)
In an initial simplex tableau, if the augmented column above the horizontal line contains a
negative number, then the objective function is unbounded and no optimal solution exists.
17)
A)
True
B)
False
Find all saddle points for the matrix game.
18)
–8–4
6–5
18)
A)
Entry a11
B)
Entry a12
C)
Entry a22
D)
No saddle point
Set up the initial simplex tableau for the given linear programming problem.
19)
Minimize 8x1+4x2
subject to 4x1– x213
3x1+2x213
and x1 0, x2 0
19)
A)
x1x2 x3x4 M
4–1 1 0 0 13
3 2 0 1 0 13
–8–4 0 0 1 0
B)
x1x2 x3x4 M
4–1 1 0 0 13
3 2 0 1 0 13
8 4 0 0 1 0
C)
x1x2 x3x4 M
4–1 1 0 0 13
–3–2 0 1 0 –13
8 4 0 0 1 0
D)
x1x2 x3x4 M
4–1 1 0 0 13
–3–2 0 1 0 –13
–8–4 0 0 1 0
Solve the problem.
20)
An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide
transportation for a minimum of 420 first class, 720 tourist class, and 1450 economy class
passengers. For a certain trip, airplane P1 costs $11,000 to operate and can accommodate 23 first
class, 52 tourist class, and 92 economy class passengers. Airplane P2 costs $8500 to operate and can
accommodate 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? Set this up as a linear
programming problem in the following form:
Minimize cTx subject to Ax b and x 0. Do not find the solution.
20)
A)
Let x1 be the number of airplanes of type P1 and x2 the number of airplanes of type P2
Then b =0
0
0, x =x1
x2, c =11,000
8500 , and A =23 18 420
52 30 720
92 44 1450
B)
Let x1 be the number of airplanes of type P1 and x2 the number of airplanes of type P2
Then b =11,000
8500 , x =x1
x2, c =420
720
1450 , and A =23 18
52 30
92 44
C)
Let x1 be the number of airplanes of type P1 and x2 the number of airplanes of type P2
Then b =420
720
1450 , x =x1
x2, c =11,000
8500 , and A =23 52 92
18 30 44
D)
Let x1 be the number of airplanes of type P1 and x2 the number of airplanes of type P2
Then b =420
720
1450 , x =x1
x2, c =11,000
8500 , and A =23 18
52 30
92 44
Solve by using the simplex method.
21)
Minimize 3x1+ 3x2+ 2x3
subject to –2x1+x3–12
2x1+x2 20
–x1+x2+ 2x3 24
and x1 0, x2 0, x3 0
21)
A)
Minimum = 36 when x1= 6, x2= 6, and x3= 0
B)
Minimum = 40 when x1= 6, x2= 6, and x3= 2
C)
Minimum = 42 when x1= 6, x2= 8, and x3= 0
D)
Minimum = 44 when x1= 4, x2= 8, and x3= 4
Identify the basic feasible solution corresponding to the given simplex tableau.
22)
x1x2 x3x4 M
1 –1
2 0 2 0 11
0 –3 1 2
3 0 5
0 2 0 3 1 77
22)
A)
x1=5, x2= 0, x3=11, x4= 0, M =77
B)
x1= 1, x2= – 1
2, x3= 0, x4=2, M = 0
C)
x1= 0, x2=2, x3= 0, x4= 3, M = 1
D)
x1=11, x2= 0, x3=5, x4= 0, M =77
Explanation:
Solve the linear programming problem.
23)
Minimize 3x1+2x2
subject to x1–3x2–6
–2x1+x2–6
and x1 0, x2 0
23)
A)
Unbounded
B)
Minimum =4 when x1= 0 and x2= 2
C)
Infeasible
D)
Minimum =108
5 when x1=24
5 and x2=18
5
Explanation:
10
Explanation:
Find all saddle points for the matrix game.
24)
–7 3
1 8
24)
A)
Entry a12 and entry a21
B)
Entry a12
C)
Entry a21
D)
No saddle points
Find the value of the matrix game.
25)
Let M be the matrix game having payoff matrix 6–1 6
–3 5 2
–8 3 1 .
25)
A)
19
5
B)
9
7
C)
27
10
D)
9
5
Solve the problem.
26)
A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it
can send a convoy up the river road or it can send a convoy overland through the jungle. On a
given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river
and the guerrillas are there, the convoy will have to turn back and 4 army soldiers will be lost. If
the convoy goes overland and encounters the guerrillas, 1
4 of the supplies will get through, but 5
army soldiers will be lost. Each day a supply convoy travels one of the roads, and if the guerrillas
are watching the other road, the convoy gets through with no losses. What is the optimal strategy
for the guerrillas if they want to inflict maximum losses on the army?
26)
A)
5
9 river, 4
9 land
B)
4
9 river, 5
9 land
C)
1
2 river, 1
2 land
D)
0 river, 1 land
Find the value of the strategy.
27)
Let M be the matrix game having payoff matrix 0 1 –2–3
1 0 2 0
–2 1 0 –3.
Find (y) when y=
1
4
1
4
1
2
0
.
27)
A)
5
4
B)
–1
4
C)
–3
4
D)
–9
4
Solve the problem.
28)
A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it
can send a convoy up the river road or it can send a convoy overland through the jungle. On a
given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river
and the guerrillas are there, the convoy will have to turn back and 6 army soldiers will be lost. If
the convoy goes overland and encounters the guerrillas, 3
4 of the supplies will get through, but 7
army soldiers will be lost. Each day a supply convoy travels one of the roads, and if the guerrillas
are watching the other road, the convoy gets through with no losses. What is the optimal strategy
for the army if it wants to maximize the amount of supplies it gets to its troops?
28)
A)
0 river, 1 land
B)
4
5 river, 1
5 land
C)
1
5 river, 4
5 land
D)
1
2 river, 1
2 land
Provide an appropriate response.
29)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. To determine the maximum profit,
the simplex method can be used and the final tableau is:
x1x2 x3x4x5 M
1 0 3
20 –1
8 0 0 30
0 1 –1
10 1
4 0 0 40
0 0 1
10 –3
4 1 0 30
0 0 1
81
16 0 1 85
If the store could obtain 10 fluid ounces extra of one of the juices, which type of juice should they
get? Why?
29)
A)
It makes no difference which juice they get – the increase in profit would be the same for any
of them.
B)
Orange juice, since it has the highest marginal value of 1
8.
C)
Pineapple juice, since it has the highest marginal value of 40.
D)
Blueberry juice; in the optimal solution the value of the slack variable x5 corresponding to the
blueberry juice constraint is 30, while the values of the other slack variables are zero.
Solve by using the simplex method.
30)
Minimize 4x1+ 5x2
subject to 2x1– 4x2 10
2x1+x2 15
and x1 0, x2 0
30)
A)
Minimum = 33 when x1= 7 and x2= 1
B)
Minimum = 30 when x1=15
2 and x2= 0
C)
Minimum = 20 when x1= 5 and x2= 0
D)
Minimum = 75 when x1= 0 and x2= 15
13
Find the value of the strategy.
31)
Let M be the matrix game having payoff matrix 1–3 5
–3 4 –3
2–2 0 .
Find (y) when y=
1
4
1
4
1
2
.
31)
A)
–3
4
B)
0
C)
–5
4
D)
2
The primal problem and the final tableau in the solution of the primal problem are given. Use the final tableau to solve
the dual problem.
32)
Primal problem:
Maximize 6x1+ 7x2
subject to 2x1+ 3x2 12
2x1+x2 8
and x1 0, x2 0
Final tableau in solution to primal problem:
x1x2x3x4 M
0 1 1
2–1
2 0 2
1 0 –1
43
4 0 3
0 0 2 1 1 32
32)
A)
Maximum = 32 when y1= 2 and y2= 1
B)
Minimum = 32 when y1= 2 and y2= 1
C)
Minimum = 32 when y1= 2 and y2= 3
D)
Minimum = 0 when y1= 0 and y2= 0
14
Solve the problem.
33)
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 $40 and on an SST
ring is $35?
33)
A)
12 VIP and 12 SST
B)
14 VIP and 10 SST
C)
18 VIP and 6 SST
D)
16 VIP and 8 SST
34)
Alan wants to invest a total of $13,000 in mutual funds, CDs, and a high yield savings account. He
wants to invest no more in mutual funds than half the amount he invests in CDs. He also wants the
amount in savings to be at least twice the sum of his CDs and mutual funds. His expected return on
mutual funds is 8%, on the CDs is 5%, and on savings is 3%. How much money should Alan invest
in each area in order to have the largest return on his investments? Set this up as a linear
programming problem in the following form: Maximize cTx subject to Ax b and x 0. Do not find
the solution.
34)
A)
Let x1 be the amount invested in mutual funds, x2 the amount in CDs, and x3 the amount in
savings.
Then b =13,000
0.5
2 , x =x1
x2
x3
, c =0.08
0.05
0.03 , and A =1 1 1
2–1 0
2 2 –1
B)
Let x1 be the amount invested in mutual funds, x2 the amount in CDs, and x3 the amount in
savings.
Then b =13,000
0
0 , x =x1
x2
x3
, c =0.08
0.05
0.03 , and A =1 1 1
2–1 0
2 2 –1
C)
Let x1 be the amount invested in mutual funds, x2 the amount in CDs, and x3 the amount in
savings.
Then b =13,000
0
0 , x =x1
x2
x3
, c =0.08
0.05
0.03 , and A =1 2 2
1–1 2
1 0 –1
D)
Let x1 be the amount invested in mutual funds, x2 the amount in CDs, and x3 the amount in
savings.
Then b =0
0
0, x =x1
x2
x3
, c =0.08
0.05
0.03 , and A =1 1 1
0 2 –1
2 2 –1
Find the value of the strategy.
35)
Let M be the matrix game having payoff matrix 0–2 2
3 4 –2
–2 1 0 .
Find (y) when y=
1
3
1
2
1
6
.
35)
A)
–2
3
B)
8
3
C)
–1
3
D)
–1
6
Provide an appropriate response.
36)
Determine whether the statement below is true or false. If it is false, give an explanation.
Given a canonical linear programming problem, if f(x) is greater than or equal to f(x) for all vectors
x in the feasible set, then x must be an optimal solution.
36)
A)
False. x itself may not be in the feasible set.
B)
False. The objective function may be unbounded.
C)
False. The problem may ask for the objective function to be minimized.
D)
True
37)
In a matrix game with payoff matrix A, how can you find the value (y) of a strategy y to player C?
37)
A)
(y) is the minimum of the inner product of y with each of the rows of A.
B)
(y) is the maximum of the inner product of y with each of the rows of A.
C)
(y) is the minimum of the inner product of y with each of the columns of A.
D)
(y) is the maximum of the inner product of y with each of the columns of A.
Solve the minimization problem by using the simplex method to solve the dual problem.
38)
An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide
transportation for a minimum of 400 first class, 800 tourist class, and 1500 economy class
passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first
class, 50 tourist class, and 100 economy class passengers. Airplane P2 costs $8000 to operate and
can accommodate 20 first class, 30 tourist class, and 50 economy class passengers. How many of
each type of airplane should be used in order to minimize the operating cost? What is the minimum
operating cost?
38)
A)
Minimum cost is $180,000 when they use 10 of airplane P1 and 10 of airplane P2.
B)
Minimum cost is $178,000 when they use 9 of airplane P1 and 11 of airplane P2.
C)
Minimum cost is $182,000 when they use 11 of airplane P1 and 9 of airplane P2.
D)
Minimum cost is $186,000 when they use 9 of airplane P1 and 12 of airplane P2.
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
39)
x1x2 x3x4 M
6 1 1 0 0 8
–22 0 –4 1 0 12
47 0 9 0 1 0
39)
A)
Not optimal
B)
Optimal
Identify the basic feasible solution corresponding to the given simplex tableau.
40)
x1x2 x3x4 M
12 6 1 0 0 29
2 5 0 1 0 32
–19 –17 0 0 1 0
40)
A)
x1= –19, x2= –17, x3= 0, x4= 0, M = 1
B)
x1= –19, x2= –17, x3=29, x4=32, M = 0
C)
x1= 0, x2= 0, x3=29, x4=32, M = 0
D)
x1=12, x2=6, x3= 1, x4= 0, M = 0
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
41)
x1x2x3x4 x5 M
1
3 0 1 2 0 0 16
–1
2 1 0 5 0 0 3
1
3 0 0 4 1 0 13
–5 0 0 8 0 1 0
41)
A)
Optimal
B)
Not optimal
Use the simplex method to solve the linear programming problem.
42)
A furniture company makes two different types of lamp stand. Each lamp stand A requires 16
minutes for sanding, 48 minutes for assembly, and 6 minutes for packaging. Each lamp stand B
requires 8 minutes for sanding, 32 minutes for assembly, and 8 minutes for packaging. The total
number of minutes available each day in each department are as follows: for sanding 3360 minutes,
for assembly 9600 minutes, and for packaging 2000 minutes. The profit on each lamp stand A is $30
and the profit on each lamp stand B is $22. How many of each type of lamp stand should the
company make per day to maximize their profit? What is the maximum profit?
42)
A)
Maximum profit is $7336 when they make 136 of lamp stand A and 148 of lamp stand B
B)
Maximum profit is $6000 when they make 200 of lamp stand A and 0 of lamp stand B
C)
Maximum profit is $6380 when they make 66 of lamp stand A and 200 of lamp stand B
D)
Maximum profit is $6164 when they make 138 of lamp stand A and 92 of lamp stand B
Solve by using the simplex method.
43)
Maximize 50x1+ 35x2
subject to 3x1+x2 24
x1+x2 16
2x1+3x2 30
and x1 0, x2 0
43)
A)
Maximum = 510 when x1= 6 and x2= 6
B)
Maximum = 620 when x1= 4 and x2= 12
C)
Maximum = 350 when x1= 0 and x2= 10
D)
Maximum = 400 when x1= 8 and x2= 0
18
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
44)
Maximize x1–2x2+6x3
subject to 2x1+x2–3x320
6x1+3x2–x3=28
and x1 0, x2 0, x3 0
44)
A)
b =20
28
–28 , c =1
–2
6, and A =2 1 –3
6 3 –1
–6–3 1
B)
b =20
28
–28 , c =1
–2
6, and A =2 6 –6
1 3 –3
–3–1 1
C)
b =20
28 , c =1
–2
6, and A =2 6
1 3
–3–1
D)
b =20
28 , c =1
–2
6, and A =2 1 –3
6 3 –1
Determine whether the statement is true or false.
45)
You wish to solve the following linear programming problem:
Minimize 2x1+x2
subject to x1–x2 4
x1+ 2x2 10
and x1 0, x2 0
This can be solved by the simplex method by solving the equivalent problem:
Maximize –2x1–x2
subject to x1–x2 4
x1+ 2x2–10
and x1 0, x2 0
45)
A)
True
B)
False
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
46)
x1x2 x3x4 M
12 11 1 0 0 43
3 5 0 1 0 34
–7–8 0 0 1 0
46)
A)
x2, row 2
B)
x2, row 1
C)
x1, row 1
D)
x1, row 2
A simplex tableau is given. Compute the next tableau.
47)
x1x2x3x4 M
4 1 0 4 0 16
7 0 1 7 0 31
–10 0 0 9 1 0
47)
A)
x1x2x3x4 M
11
4 0 1 0 4
0–7
4–6 0 –7 3
05
2 0 19 1 40
B)
x1x2x3x4 M
11
4 0 1 0 4
0–7
4 1 0 0 3
0–5
2 0 19 1 28
C)
x1x2x3x4 M
1 0 0 0 0 4
0–7
4 1 0 0 3
05
2 0 19 1 40
D)
x1x2x3x4 M
11
4 0 1 0 4
0–7
4 1 0 0 3
05
2 0 19 1 40
Solve the problem.
48)
A furniture company makes two different types of lamp stands. Each lamp stand A requires 12
minutes for sanding, 20 minutes for assembly, and 7 minutes for packaging. Each lamp stand B
requires 14 minutes for sanding, 23 minutes for assembly, and 7 minutes for packaging. The total
number of minutes available each day in each department are as follows: for sanding 4000 minutes,
for assembly 8000 minutes, and for packaging 2000 minutes. The profit on each lamp stand A is $17
and the profit on each lamp stand B is $27. How many of each type of lamp stand should the
company make per day to maximize their profit? Set this up as a linear programming problem in
the following form: Maximize cTx subject to Ax b and x 0. Do not find the solution.
48)
A)
Let x1 be the number of lamp stands of type A made per day and x2 the number of lamp
stands of type B made per day.
Then b =4000
8000
2000 , x =x1
x2, c =17
27 , and A =12 20 7
14 23 7
B)
Let x1 be the number of lamp stands of type A made per day and x2 the number of lamp
stands of type B made per day.
Then b =0
0
0, x =x1
x2, c =17
27 , and A =12 14 4000
20 23 8000
7 7 2000
C)
Let x1 be the number of lamp stands of type A made per day and x2 the number of lamp
stands of type B made per day.
Then b =17
27 , x =x1
x2, c =4000
8000
2000 , and A =12 14
20 23
7 7
D)
Let x1 be the number of lamp stands of type A made per day and x2 the number of lamp
stands of type B made per day.
Then b =4000
8000
2000 , x =x1
x2, c =17
27 , and A =12 14
20 23
7 7
Find the optimal row or column strategy of the matrix game.
49)
Let M be the matrix game having payoff matrix 3–3 3
–2 4 1
–4 3 0 .
Find the optimal column strategy.
49)
A)
^
y=
7
12
5
12
0
B)
^
y=
5
12
7
12
0
C)
^
y=
5
12
0
7
12
D)
^
y=
1
2
1
2
0
Use the simplex method to solve the linear programming problem.
50)
Alan wants to invest a total of $18,000 in mutual funds and a certificate of deposit (CD). He wants
to invest no more in mutual funds than half the amount he invests in the CD. His expected return
on mutual funds is 9% and on the CD is 5%. How much money should Alan invest in each area in
order to have the largest return on his investments? What is his maximum one–year return?
50)
A)
Maximum one–year return is $4140 when he invests $12,000 in mutual funds and $6000 in the
CD.
B)
Maximum one–year return is $1140 when he invests $6000 in mutual funds and $12,000 in the
CD.
C)
Maximum one–year return is $1620 when he invests $18,000 in mutual funds and $0 in the
CD.
D)
Maximum one–year return is $900 when he invests $0 in mutual funds and $18,000 in the CD.
Provide an appropriate response.
51)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. To determine the maximum profit,
the simplex method can be used and the final tableau is:
x1x2 x3x4x5 M
51)
22
1 0 3
20 –1
8 0 0 30
0 1 –1
10 1
4 0 0 40
0 0 1
10 –3
4 1 0 30
0 0 1
81
16 0 1 85
Give an interpretation to the number 1
8 in the bottom row.
A)
1
8 is the value of the slack variable x3 in the optimal solution. The slack variable x3
corresponds to the orange juice constraint. Since x3 is greater than zero, this means that in the
optimal solution, not all the available orange juice is used. Thus the marginal value of the
orange juice is zero.
B)
1
8 represents the number of Pineapple Blue smoothies they should make to maximize profit.
(In practice this would be rounded to 0).
C)
1
8 is the marginal value of the pineapple juice. If the amount of pineapple juice available were
increased by one fluid ounce, the profit would increase by 1
8 dollars.
D)
1
8 is the marginal value of the orange juice. If the amount of orange juice available were
increased by one fluid ounce, the profit would increase by 1
8 dollars.
52)
If the payoff matrix of a matrix game contains a saddle point, what is the optimal strategy for the
row player?
52)
A)
Always choose the row with the smallest minimum.
B)
Always choose the row with the largest minimum.
C)
Always choose the row with the largest maximum.
D)
Always choose the row with the smallest maximum.
Solve the problem.
53)
Player R has two cards: a red 2 and a black 8. Player C has three cards: a red 3, a black 5, and a
black 10. They each show one of their cards. If the cards are the same color, C receives the larger of
the two numbers. If the cards are of different colors, R receives the sum of the two numbers. The
payoff matrix is :
r3 b5 b10
r2
b8 –3 7 12
11 –8–10
Find the optimal strategy for player R.
53)
A)
^
x=
10
29
19
29
B)
^
x=1
0
C)
^
x=
2
3
1
3
D)
^
x=
19
29
10
29
Provide an appropriate response.
54)
A linear programming problem has objective function 3x1+ 4x2 and constraints
ax1+ bx2 10
cx1+ dx2 20
x1 0, x2 0
where a, b, c, and d are positive numbers.
Does a maximum exist for the objective function? If not, why not? Does a minimum exist? If not,
why not?
54)
A)
Neither maximum nor minimum exist as the problem is unbounded.
B)
Maximum does exist. Minimum does not exist as the objective function may be arbitrarily
small (unbounded).
C)
Neither maximum nor minimum exist as the feasible set is the empty set.
D)
Maximum does not exist as the objective function may be arbitrarily large (unbounded).
Minimum does exist.
24
Find the value of the strategy.
55)
Let M be the matrix game having payoff matrix 0–3 4
–2 5 3
–2 1 0 .
Find (x) when x=
1
3
1
2
1
6
.
55)
A)
–7
6
B)
5
3
C)
17
6
D)
–4
3
Identify the basic feasible solution corresponding to the given simplex tableau.
56)
x1x2x3x4 M
1
6 0 –2 1 0 7
2 1 2
3 0 0 8
–1 0 4 0 1 64
56)
A)
x1=1
6, x2= 0, x3= –2, x4= 1, M = 0
B)
x1= –1, x2= 0, x3=4, x4= 0, M = 1
C)
x1= 0, x2=8, x3= 0, x4=7, M =64
D)
x1= 0, x2=7, x3= 0, x4=8, M =64
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
57)
x1x2x3x4x5 M
0 0 1 –1–1 0 2
0 1 0 19
23
2 0 19
1 0 0 9 3 0 30
0 0 0 60 18 1 180
57)
A)
Not optimal
B)
Optimal
Provide an appropriate response.
58)
You have decided to solve a matrix game by using linear programming. You add 3 to each entry of
the payoff matrix A to obtain a matrix B =a b c
d e f with only positive entries. You find the
optimal strategy for column player C by solving the linear programming problem:
Maximize y1+y2+y3
subject to ay1+ by2+ cy3 1
dy1+ ey2+ fy3 1
and y1 0, y2 0, y3 0
The initial simplex tableau is
y1y2y3y4 y5 M
a b c 1 0 0 1
d e f 0 1 0 1
–1–1–1 0 0 1 0
and suppose that the final tableau is
y1y2y3y4y5 M
g 0 1 h i 0 q
j 1 0 k l 0 r
m 0 0 n p 1 v
where g, h, i, j, k, and l are nonzero numbers and m, n, p, q, r, and v are positive numbers.
What is the optimal strategy for player R?
58)
A)
^
x=0
n
p
B)
^
x=
0
r
r+q
q
r+q
C)
^
x=
n
n+p
p
n+p
D)
^
x=n
p
Solve the problem.
59)
Irene wants to invest $36,000 in stocks, bonds, and gold coins. She knows that her rate of return will
depend on the economic climate of the country, which is difficult to predict. After some research
and analysis, she determines the annual profit in dollars that she would expect per hundred
dollars on each type of investment, depending on whether the economy is strong, stable, or weak.
Strong Stable Weak
Stocks 3 1 –2
Bonds 1 2 0
Gold –1 1 3
How should Irene invest her money in order to maximize her profit regardless of what the
economy does? In other words, consider the problem as a matrix game in which Irene is playing
against “fate”.
59)
A)
Irene should invest $15,000 in stocks and $21,000 in gold.
B)
Irene should invest $12,000 in stocks, $8000 in bonds, and $16,000 in gold.
C)
Irene should invest $16,000 in stocks and $20,000 in gold.
D)
Irene should invest $14,000 in stocks, $8000 in bonds, and $18,000 in gold.
Find the value of the strategy.
60)
Let M be the matrix game having payoff matrix 0–2 4
–1 1 3
2–2 0 .
Find (x) when x=
1
4
1
4
1
2
.
60)
A)
–1
2
B)
7
4
C)
3
4
D)
–5
4
Solve the minimization problem by using the simplex method to solve the dual problem.
61)
Minimize 6x1+ 8x2
subject to x1+ 2x2 6
2x1+x2 8
and x1 0, x2 0
61)
A)
Minimum = 36 when x1= 6 and x2= 0
B)
Minimum =86
3 when x1= 3 and x2=4
3
C)
Minimum = 64 when x1= 0 and x2= 8
D)
Minimum =92
3 when x1=10
3 and x2=4
3
Determine whether the statement is true or false.
62)
Consider the simplex method for a canonical linear programming problem in which each entry in
the vector b is positive. If the variable x3 is to be brought into solution then row i will be chosen as
the pivot row if the ratio bi
ai3 is the largest among all ratios for which ai3 > 0.
62)
A)
True
B)
False
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
63)
Maximize 2x1+x2+5x3
subject to x1+5x2–2x327
–2x2+6x320
and x1 0, x2 0, x3 0
63)
A)
b =27
20 , c =2
1
5, and A =1 0
5–2
–2 6
B)
b =2
1
5, c =27
20 , and A =1 5 –2
0–2 6
C)
b =27
20 , c =2
1
5, and A =1 5 –2
0–2 6
D)
b =27
20 , c =2
1
5, and A =2 1 5
1 5 –2
0–2 6
28
Provide an appropriate response.
64)
You have decided to solve a matrix game by using linear programming. You add 3 to each entry of
the payoff matrix A to obtain a matrix B =a b c
d e f with only positive entries. You find the
optimal strategy for column player C by solving the linear programming problem:
Maximize y1+y2+y3
subject to ay1+ by2+ cy3 1
dy1+ ey2+ fy3 1
and y1 0, y2 0, y3 0
The initial simplex tableau is
y1y2y3y4 y5 M
a b c 1 0 0 1
d e f 0 1 0 1
–1–1–1 0 0 1 0
and suppose that the final tableau is
y1y2y3y4y5 M
g 0 1 h i 0 q
j 1 0 k l 0 r
m 0 0 n p 1 v
where g, h, i, j, k, and l are nonzero numbers and m, n, p, q, r, and v are positive numbers.
What is the value of the original game with payoff matrix A?
64)
A)
1
v– 3
B)
v – 3
C)
1
v
D)
v
Determine whether the statement is true or false.
65)
Consider the simplex method for a canonical linear programming problem. If x is an extreme point
of the feasible set and if the objective function cannot be decreased by moving along any of the
edges of the feasible set which join at x, then x is an optimal solution.
65)
A)
True
B)
False
A simplex tableau is given. Compute the next tableau.
66)
x1x2x3x4 M
212 1 0 0 24
2 5 0 1 0 15
–9–4 0 0 1 0
66)
A)
x1x2x3x4 M
0 7 1 –1 0 9
1 5
2 0 1
2 0 15
2
0 37
2 0 9
2 1 135
2
B)
x1x2x3x4 M
0 7–1–1 0 9
1 5
2 0 1
2 0 15
2
0 37
299
2 1 135
2
C)
x1x2x3x4 M
0 7 1 –1 0 9
1 0 0 0 0 15
2
0 37
2 0 9
2 1 135
2
D)
x1x2x3x4 M
0 7 1 –1 0 9
1 5
2 0 1
2 0 15
2
0 –37
20–1 1 15
Solve the minimization problem by using the simplex method to solve the dual problem.
67)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. To determine the maximum profit,
the simplex method can be used and the final tableau is:
x1x2 x3x4x5 M
1 0 3
20 –1
8 0 0 30
0 1 –1
10 1
4 0 0 40
0 0 1
10 –3
4 1 0 30
0 0 1
81
16 0 1 85
What is the marginal value of blueberry juice?
67)
A)
1
8
B)
0
C)
1
16
D)
30
Determine whether the statement is true or false.
68)
In the simplex method, a basic variable is a variable which has a coefficient of 1 and which occurs
in only one equation.
68)
A)
True
B)
False
A simplex tableau is given. Compute the next tableau.
69)
x1x2x3x4 M
3 1 1 0 0 15
12 2 0 1 0 38
–9–11 0 0 1 0
69)
A)
x1x2x3x4 M
3 1 1 0 0 15
10 0 –2 1 0 8
–15 0 –2 0 1 30
B)
x1x2x3x4 M
0 1 1 0 0 15
6 0 –2 1 0 8
–15 0 –2 0 1 30
C)
x1x2x3x4 M
3 1 1 0 0 15
10 0 –2 1 0 8
24 0 11 11 1 165
D)
x1x2x3x4 M
3 1 1 0 0 15
6 0 –2 1 0 8
24 0 11 0 1 165
Determine whether the statement is true or false.
70)
Suppose a system contains three inequalities and three unknowns x1, x2, and x3. If the simplex
method results in an optimal solution in which the slack variable x6 is zero, this means that at the
optimal values of x1, x2, and x3, the third inequality is an equality.
70)
A)
True
B)
False
Solve the problem.
71)
A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it
can send a convoy up the river road or it can send a convoy overland through the jungle. On a
given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river
and the guerrillas are there, the convoy will have to turn back and 4 army soldiers will be lost. If
the convoy goes overland and encounters the guerrillas, 2
3 of the supplies will get through, but 6
army soldiers will be lost. Each day a supply convoy travels one of the roads, and if the guerrillas
are watching the other road, the convoy gets through with no losses. What is the optimal strategy
for the army if it wants to minimize its casualties?
71)
A)
2
5 river, 3
5 land
B)
3
5 river, 2
5 land
C)
1 river, 0 land
D)
1
2 river, 1
2 land
Find the value of the matrix game.
72)
Let M be the matrix game having payoff matrix 2 2 –1 6
–3 2 3 3 .
72)
A)
3
11
B)
1
3
C)
1
2
D)
1
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
73)
x1x2 x3x4 M
12 7 1 0 0 46
411 0 1 0 26
–11 –5 0 0 1 0
73)
A)
x2, row 2
B)
x1, row 2
C)
x1, row 1
D)
x2, row 1
State the dual of the linear programming problem.
74)
Maximize 4x1+7x2
subject to 4x1+5x245
3x1+x218
x1+4x234
and x1 0, x2 0
74)
A)
Minimize 45y1+18y2
subject to 4y1+5y24
3y1+y27
y1+4y234
and y1 0, y2 0
B)
Minimize 45y1+18y2+34y3
subject to 4y1+3y2+y34
5y1+y2+4y37
and y1 0, y2 0, y3 0
C)
Minimize 45y1+18y2+34y3
subject to 4y1+3y2+y34
5y1+y2+4y37
and y1 0, y2 0, y3 0
D)
Minimize 4y1+7y2
subject to 4y1+5y245
3y1+y218
y1+4y234
and y1 0, y2 0
33
Provide an appropriate response.
75)
In a matrix game with payoff matrix A, how can you find the value (x) of a strategy x to row
player R?
75)
A)
(x) is the maximum of the inner product of x with each of the rows of A.
B)
(x) is the minimum of the inner product of x with each of the columns of A.
C)
(x) is the maximum of the inner product of x with each of the columns of A.
D)
(x) is the minimum of the inner product of x with each of the rows of A.
Solve the minimization problem by using the simplex method to solve the dual problem.
76)
Minimize 10x1+30x2
subject to x1+ 5x27
2x1+4x28
and x1 0, x2 0
76)
A)
Minimum =70 when x1=7 and x2=0
B)
Minimum =50 when x1=10
3 and x2=5
3
C)
Minimum =50 when x1=2 and x2=1
D)
Minimum =42 when x1=0 and x2=7
5
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
77)
x1x2 x3x4 M
12 1 1 0 0 13
8 2 0 1 0 40
–3–7 0 0 1 0
77)
A)
Not optimal
B)
Optimal
Determine whether the statement is true or false.
78)
In a matrix game, the value (y) of a particular strategy y to player C is equal to the minimum of
the inner product of y with each of the rows of the payoff matrix A.
78)
A)
True
B)
False
Set up the initial simplex tableau for the given linear programming problem.
79)
Maximize 23x1+28x2+21x3
subject to 9x1+7x2+4x3 30
3x1+12x2+9x3 40
and x1 0, x2 0, x3 0
79)
A)
x1x2x3x4 x5
9 7 4 1 0 30
312 9 0 1 40
–23 –28 –21 0 0 0
B)
x1x2x3x4 x5 M
9 7 4 1 0 0 30
312 9 0 1 0 40
23 28 21 0 0 1 0
C)
x1x2x3x4 x5 M
9 7 4 1 0 0 30
312 9 0 1 0 40
–23 –28 –21 0 0 1 0
D)
x1x2x3x4 x5
9 7 4 1 0 30
312 9 0 1 40
23 28 21 0 0 0
Solve the linear programming problem.
80)
Maximize 6x1+ 7x2
subject to 2x1+ 3x2 12
2x1+x2 8
and x1 0, x2 0
80)
A)
maximum = 39 when x1= 3 and x2= 3
B)
maximum = 32 when x1= 3 and x2= 2
C)
maximum = 28 when x1= 0 and x2= 4
D)
maximum = 38 when x1= 4 and x2= 2
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
81)
x1x2 x3x4 M
12 12 1 3 0 21
7 7 0 6 0 35
–3–3 0 10 1 0
81)
A)
x2, row 1
B)
x1, row 1
C)
x2, row 2
D)
x1, row 2
Write the payoff matrix for the game.
82)
Players R and C each show 1, 2, or 3 fingers. If the total number N of fingers shown is even, then R
pays N dollars to C. If N is odd, C pays N dollars to R.
82)
A)
1 2 3
1
2
3
1 –2 1
–1 2 –3
3–2 3
B)
1 2 3
1
2
3
–2 3 –4
3 –4 5
–4 5 –6
C)
1 2 3
1
2
3
2 –3 4
–3 4 –5
4–5 6
D)
1 2 3
1
2
3
–1 2 –1
1 –2 3
–3 2 –3
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
83)
Minimize x1–5x2+3x3
subject to 5x1+x2+6x331
4x1–4x328
and x1 0, x2 0, x3 0
83)
A)
b =31
28 , c =1
–5
3, and A =5 1 6
–4 0 4
B)
b =31
28 , c =1
–5
3, and A =5 1 6
4 0 –4
C)
b =31
–28 , c =–1
5
–3, and A =5 1 6
4 0 –4
D)
b =31
–28 , c =–1
5
–3, and A =5 1 6
–4 0 4
36
Solve the minimization problem by using the simplex method to solve the dual problem.
84)
Minimize 24x1+ 30x2
subject to x1 1
2x1+ 3x2 4
x1+ 4x2 3
and x1 0, x2 0
84)
A)
Minimum =111
2 when x1= 2 and x2=1
4
B)
Minimum = 39 when x1= 1 and x2=1
2
C)
Minimum = 44 when x1= 1 and x2=2
3
D)
Minimum = 46 when x1=3
2 and x2=1
3
Determine whether the statement is true or false.
85)
In the simplex method for a canonical linear programming problem, if all entries in the bottom row
to the left of the vertical line are nonnegative, then the solution is optimal.
85)
A)
True
B)
False
86)
In a matrix game, if row s is dominant to some other row in payoff matrix A, then row s will not be
used in some optimal strategy for row player R.
86)
A)
True
B)
False
Find all saddle points for the matrix game.
87)
8 4 7
–2–4–8
87)
A)
Entry a23
B)
Entry a12
C)
Entry a12 and entry a23
D)
No saddle points
Identify the basic feasible solution corresponding to the given simplex tableau.
88)
x1x2 x3x4 M
0 1 1
2–2 0 12
1 0 –3–4
3 0 8
0 0 2 3 1 112
88)
A)
x1= 0, x2= 0, x3=2, x4= 3, M =112
B)
x1= 0, x2= 1, x3=1
2, x4= –2, M = 0
C)
x1=12, x2=8, x3= 0, x4= 0, M =112
D)
x1=8, x2=12, x3= 0, x4= 0, M =112
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
89)
Minimize x1–4x2+5x3
subject to 2x1–4x326
4x1–x2=21
and x1 0, x2 0, x3 0
89)
A)
b =–26
21
–21 , c =1
–4
5, and A =–2 0 4
4–1 0
–4 1 0
B)
b =–26
21
–21 , c =–1
4
–5, and A =–2 0 4
4–1 0
–4 1 0
C)
b =–26
21
–21 , c =–1
4
–5, and A =2 0 –4
4–1 0
–4 1 0
D)
b =–26
21 , c =1
–4
5, and A =–2 0 4
4–1 0
Provide an appropriate response.
90)
Determine whether the statement below is true or false. If it is false, give an explanation.
If the feasible set for a canonical linear programming problem is not empty, there must be at least
one optimal solution.
90)
A)
False. The objective function may be unbounded.
B)
False. The objective function may have the same value for more than one extreme point of the
feasible set.
C)
False. The feasible set may contain only one point.
D)
True
Determine whether the statement is true or false.
91)
In the simplex method, a basic solution is infeasible if one of the slack variables is negative.
91)
A)
True
B)
False
Find the optimal row or column strategy of the matrix game.
92)
Let M be the matrix game having payoff matrix 5–3 5
–1 6 3
–4 1 2 .
Find the optimal row strategy.
92)
A)
^
x=
3
5
2
5
0
B)
^
x=
3
5
0
2
5
C)
^
x=
7
15
8
15
0
D)
^
x=
7
15
0
8
15
Provide an appropriate response.
93)
Given the primal and dual problems below:
Primal:
Maximize f(x) =cxT
subject to Ax b
x 0
where c is a vector in
n and A is an m × n matrix
Dual:
Minimize g(y) =bTy
subject to ATy c
y 0
where b is a vector in
m
If x is an optimal solution to the primal problem and y is an optimal solution to the dual problem,
then which of the following statements is (are) true?
A: f(x) = g(y)
B: f(x) = f(y)
C: g(x) = g(y)
D: g(x) = f(y)
93)
A)
A and D
B)
B and C
C)
A only
D)
All of them.
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
94)
x1x2 x3x4 M
10 8 1 0 0 44
3 5 0 1 0 25
–3–4 0 0 1 0
94)
A)
x1, row 2
B)
x2, row 1
C)
x2, row 2
D)
x1, row 1
40
Set up the initial simplex tableau for the given linear programming problem.
95)
Maximize 19x1+16x2
subject to 8x1+3x238
11x1+11x237
x1+12x224
and x1 0, x2 0
95)
A)
x1x2 x3x4 x5 M
8 3 1 0 0 0 –38
11 11 0 1 0 0 –37
112 0 0 1 0 –24
–19 –16 0 0 0 1 0
B)
x1x2 x3x4 x5
11 3 1 0 0 38
11 11 0 1 0 37
1 12 0 0 1 24
19 16 0 0 0 0
C)
x1x2 x3x4 x5 M
8 3 1 0 0 0 38
11 11 0 1 0 0 37
112 0 0 1 0 24
19 16 0 0 0 1 0
D)
x1x2 x3x4 x5 M
8 3 1 0 0 0 38
11 11 0 1 0 0 37
112 0 0 1 0 24
–19 –16 0 0 0 1 0
Find the optimal row or column strategy of the matrix game.
96)
Let M be the matrix game having payoff matrix 4–2
–1 1 .
Find the optimal column strategy.
96)
A)
^
y=
1
4
3
4
B)
^
y=
5
8
3
8
C)
^
y=
3
4
1
4
D)
^
y=
3
8
5
8
Use linear programming to solve the matrix game with the given payoff matrix.
97)
A =–1 2 1
1 –1 0
97)
A)
^ ^
x=
2
5
3
5
, y=
3
5
2
5
0
, value =1
5
B)
^ ^
x=
2
11
3
11
, y=
3
11
2
11
0
, value =5
11
C)
^ ^
x=
3
5
2
5
, y=
3
11
0
2
11
, value =11
5
D)
^ ^
x=
1
4
3
4
, y=
3
4
1
4
0
, value =1
4
Solve by using the simplex method.
98)
Maximize 6x1+ 7x2
subject to 2x1+ 3x2 12
2x1+x2 8
and x1 0, x2 0
98)
A)
Maximum = 28 when x1= 0 and x2= 4
B)
Maximum = 32 when x1= 3 and x2= 2
C)
Maximum = 39 when x1= 3 and x2= 3
D)
Maximum = 38 when x1= 4 and x2= 2
Determine whether the statement is true or false.
99)
If the payoff matrix of a matrix game contains a saddle point, the optimal strategy for each player
will be a pure strategy.
99)
A)
True
B)
False
Solve the minimization problem by using the simplex method to solve the dual problem.
100)
Minimize 24x1+ 16x2+ 30x3
subject to 3x1+x2+ 2x3 50
x1+x2+ 3x3 35
and x1 0, x2 0, x3 0
100)
A)
Minimum = 510 when x1=80
7, x2= 0, and x3=55
7
B)
Minimum =3810
7 when x1= 10, x2=30
7, and x3=55
7
C)
Minimum = 696 when x1= 17, x2= 18, and x3= 0
D)
Minimum = 498 when x1= 12, x2= 0, and x3= 7
Find the expected payoff.
101)
Let M be the matrix game having payoff matrix –1 0 1
2 0 5
–2 1 0 .
Find E(x, y) when x=
1
4
1
2
1
4
and y=
1
3
1
6
1
2
.
101)
A)
5
3
B)
4
3
C)
3
2
D)
13
8
Find all saddle points for the matrix game.
102)
2–5–6
4 4 9
–3 2 3
102)
A)
Entry a21 and entry a22
B)
Entry a21 and entry a13
C)
Entry a21
D)
No saddle points
Solve by using the simplex method.
103)
Maximize 3x1+ 4x2+ 2x3
subject to x1+x3 16
2x2+x3 20
3x1+6x2 36
and x1 0, x2 0, x3 0
103)
A)
Maximum = 58 when x1= 6, x2= 5, and x3= 10
B)
Maximum = 64 when x1= 8, x2= 6, and x3= 8
C)
Maximum = 52 when x1= 4, x2= 4, and x3= 12
D)
Maximum = 48 when x1= 4, x2= 4, and x3= 10
Provide an appropriate response.
104)
Given the primal linear programming problem below:
Maximize f(x) =cxT
subject to Ax b
x 0
If a slack variable is not in an optimal solution, then what can be said about the marginal value of
the item corresponding to its equation?
104)
A)
There is not enough information to say anything about the marginal value.
B)
The marginal value is greater than zero.
C)
The marginal value is less than zero.
D)
The marginal value is zero.
Use linear programming to solve the matrix game with the given payoff matrix.
105)
A =–3 4
–2 2
2 –1
105)
A)
^ ^
x=
3
10
0
7
10
, y=
1
2
1
2
, value =1
2
B)
^ ^
x=
2
5
0
3
5
, y=
1
2
1
2
, value =1
2
C)
^ ^
x=
3
10
0
7
10
, y=
1
2
1
2
, value =2
9
D)
^ ^
x=
1
15
0
7
45
, y=
1
9
1
9
, value =2
9
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
106)
x1x2 x3x4 M
1–9 8 0 0 34
0 7 12 1 0 38
0–6 3 0 1 0
106)
A)
x2, row 1
B)
x1, row 1
C)
x2, row 2
D)
x1, row 2
Find the expected payoff.
107)
Let M be the matrix game having payoff matrix 0 3–3–2
1 0 1 0
–2 3 0 –3.
Find E(x, y) when x=
0
2
3
1
3
and y=
1
4
1
4
1
2
0
107)
A)
3
4
B)
7
12
C)
1
4
D)
1
3
Write the payoff matrix for the game.
108)
Player R has a supply of nickels, dimes, and quarters. Player R chooses one of the coins, and player
C must guess which coin R has chosen. If the guess is correct, C takes the coin. If the guess is
incorrect, C gives R an amount equal to R’s chosen coin.
108)
A)
n d q
n
d
q
5 –5–5
–10 10 –10
–25 –25 25
B)
n d q
n
d
q
–5 10 25
5 –10 25
5 10 –25
C)
n d q
n
d
q
5 –10 –25
–5 10 –25
–5–10 25
D)
n d q
n
d
q
–5 5 5
10 –10 10
25 25 –25
Solve the linear programming problem.
109)
Minimize 6x1+ 8x2
subject to 2x1+ 4x2 12
2x1+x2 8
and x1 0, x2 0
109)
A)
Minimum = 64 when x1= 0 and x2= 8
B)
Minimum =92
3 when x1=10
3 and x2=4
3
C)
Minimum = 36 when x1= 6 and x2= 0
D)
Minimum =86
3 when x1= 3 and x2=4
3
Determine whether the statement is true or false.
110)
You wish to solve the following linear programming problem:
Minimize 2x1+x2
subject to x1–x2 4
x1+ 2x2 10
and x1 0, x2 0
This can be solved by the simplex method by solving the equivalent problem:
Maximize 2x1+x2
subject to x1–x2 4
–x1– 2x2–10
and x1 0, x2 0
110)
A)
True
B)
False
Find the optimal row or column strategy of the matrix game.
111)
Let M be the matrix game having payoff matrix 6–4
–3 1 .
Find the optimal row strategy.
111)
A)
^
x=
9
14
5
14
B)
^
x=
5
7
2
7
C)
^
x=
2
7
5
7
D)
^
x=
5
14
9
14
Solve by using the simplex method.
112)
Maximize 2x1+ 5x2
subject to 3x1+ 2x2 6
–2x1+4x2 8
and x1 0, x2 0
112)
A)
Maximum = 4 when x1= 2 and x2= 0
B)
Maximum = 15 when x1= 0 and x2= 3
C)
Maximum =49
4 when x1=1
2 and x2=9
4
D)
Maximum =27
2 when x1=1
2 and x2=5
2
Solve the linear programming problem.
113)
Minimize 4x1+ 5x2
subject to 2x1– 4x2 10
2x1+x2 15
and x1 0, x2 0
113)
A)
Minimum = 75 when x1= 0 and x2= 15
B)
Minimum = 33 when x1= 7 and x2= 1
C)
Minimum = 20 when x1= 5 and x2= 0
D)
Minimum = 30 when x1=15
2 and x2= 0
Write the payoff matrix for the game.
114)
Player R has two cards: a red 2 and a black 7. Player C has three cards: a red 4, a black 6, and a
black 9. They each show one of their cards. If the cards are the same color, C receives the larger of
the two numbers. If the cards are of different colors, R receives the sum of the two numbers.
114)
A)
r4 b6 b9
r2
b7 4–8–11
–11 7 9
B)
r4 b6 b9
r2
b7 6–6–9
–713 16
C)
r4 b6 b9
r2
b7 –6 6 9
7–13 –16
D)
r4 b6 b9
r2
b7 –4 8 11
11 –7–9
Solve the minimization problem by using the simplex method to solve the dual problem.
115)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. To determine the maximum profit,
the simplex method can be used and the final tableau is:
x1x2 x3x4x5 M
1 0 3
20 –1
8 0 0 30
0 1 –1
10 1
4 0 0 40
0 0 1
10 –3
4 1 0 30
0 0 1
81
16 0 1 85
What is the marginal value of pineapple juice?
115)
A)
1
16
B)
1
8
C)
30
D)
0
116)
Daniel decides to feed his cat a combination of two foods: Max Cat and Mighty Cat. He wants his
cat to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d)
contained in one bag of each food are shown in the chart, together with the total amounts needed.
a b c d
Max Cat 4 2 1 4
Mighty Cat 6 2 2 3
Needed 54 36 20 60
The costs per bag are $40 for Max Cat and $36 for Mighty Cat. How many bags of each food should
be blended to meet the nutritional requirements at the lowest cost? What is the minimum cost?
116)
A)
Minimum cost = $624 when he blends 12 bags of Max Cat and 4 bags of Mighty Cat.
B)
Minimum cost = $720 when he blends 0 bags of Max Cat and 20 bags of Mighty Cat.
C)
Minimum cost = $712 when he blends 16 bags of Max Cat and 2 bags of Mighty Cat.
D)
Minimum cost = $672 when he blends 6 bags of Max Cat and 12 bags of Mighty Cat.
A simplex tableau is given. Compute the next tableau.
117)
x1x2x3x4x5 M
1
3 0 1 2 0 0 15
–1
2 1 0 3 0 0 2
1
3 0 0 4 1 0 13
–8 0 0 5 0 1 0
117)
A)
x1x2x3x4x5 M
0 0 1 –2–1 0 2
0 1 0 93
2 0 43
2
1 0 0 12 3 0 39
0 0 0 9 1 1 5
B)
x1x2x3x4x5 M
0 0 1 –2–1 0 2
0 1 0 – 3 3
2 0 –35
2
1 0 0 12 3 0 39
0 0 0 101 24 1 312
C)
x1x2x3x4x5 M
0 0 1 –2–1 0 2
0 1 0 93
2 0 43
2
1 0 0 12 3 0 39
0 0 0 101 24 1 312
D)
x1x2x3x4x5 M
0 0 1 –2–1 0 2
0 1 0 93
2 0 43
2
1 0 0 4 1 0 13
0 0 0 101 24 1 312
Solve the linear programming problem.
118)
Minimize 4x1+ 5x2
subject to 5x1+4x2 20
4x1+x2 12
x1+3x2 9
and x1 0, x2 0
118)
A)
Minimum =228
11 when x1=27
11 and x2=24
11
B)
Minimum =212
11 when x1=28
11 and x2=20
11
C)
Minimum =221
11 when x1=24
11 and x2=25
11
D)
Minimum = 36 when x1= 9 and x2= 0
50
State the dual of the linear programming problem.
119)
Maximize 9x1+6x2
subject to x1+2x211
6x1+5x249
and x1 0, x2 0
119)
A)
Minimize 9y1+6y2
subject to y1+6y211
2y1+5y249
and y1 0, y2 0
B)
Minimize 11y1+49y2
subject to y1+6y29
2y1+5y26
and y1 0, y2 0
C)
Minimize 11y1+49y2
subject to y1+2y29
6y1+5y26
and y1 0, y2 0
D)
Minimize 11y1+49y2
subject to y1+6y29
2y1+5y26
and y1 0, y2 0
Solve the problem.
120)
Daniel decides to feed his cat a combination of two foods: Max Cat and Mighty Cat. He wants his
cat to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d)
contained in one bag of each food are shown in the chart, together with the total amounts needed.
a b c d
Max Cat 4 3 1 6
Mighty Cat 5 2 2 3
Needed 54 37 20 60
The costs per bag are $40 for Max Cat and $35 for Mighty Cat. How many bags of each food should
be blended to meet the nutritional requirements at the lowest cost? What is the minimum cost?
120)
A)
Minimum cost = $610 when he blends 3 bags of Max Cat and 14 bags of Mighty Cat.
B)
Minimum cost = $700 when he blends 0 bags of Max Cat and 20 bags of Mighty Cat.
C)
Minimum cost = $500 when he blends 20
3 bags of Max Cat and 20
3 bags of Mighty Cat.
D)
Minimum cost = $541.25 when he blends 17
2 bags of Max Cat and 23
4 bags of Mighty Cat.
Provide an appropriate response.
121)
You have decided to solve a matrix game by using linear programming. You add 3 to each entry of
the payoff matrix A to obtain a matrix B =a b c
d e f with only positive entries. You find the
optimal strategy for column player C by solving the linear programming problem:
Maximize y1+y2+y3
subject to ay1+ by2+ cy3 1
dy1+ ey2+ fy3 1
and y1 0, y2 0, y3 0
The initial simplex tableau is
y1y2y3y4 y5 M
a b c 1 0 0 1
d e f 0 1 0 1
–1–1–1 0 0 1 0
and suppose that the final tableau is
y1y2y3y4y5 M
g 0 1 h i 0 q
j 1 0 k l 0 r
m 0 0 n p 1 v
where g, h, i, j, k, and l are nonzero numbers and m, n, p, q, r, and v are positive numbers.
What is the optimal strategy for player C?
121)
A)
^
y=0
r
q
B)
^
y=
n
n+p
p
n+p
C)
^
y=m
0
0
D)
^
y=
0
r
r+q
q
r+q
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
122)
x1x2x3x4 M
5 1 0 10 0 35
2 0 1 5 0 26
–6 0 0 7 1 0
122)
A)
Optimal
B)
C)
Not optimal
Solve the problem.
123)
Player R has two cards: a red 2 and a black 8. Player C has three cards: a red 3, a black 5, and a
black 10. They each show one of their cards. If the cards are the same color, C receives the larger of
the two numbers. If the cards are of different colors, R receives the sum of the two numbers. The
payoff matrix is :
r3 b5 b10
r2
b8 –3 7 12
11 –8–10
Find the value of the game.
123)
A)
53
29
B)
10
29
C)
1
2
D)
19
29
Identify the basic feasible solution corresponding to the given simplex tableau.
124)
x1x2x3x4 x5 M
6 6 7 1 0 0 44
44 42 7 0 1 0 42
–19 –17 –30 0 0 1 0
124)
A)
x1=6, x2=6, x3=7, x4=44, x5=42, M = 0
B)
x1=6, x2=6, x3=7, x4= 1, x5= 0, M = 0
C)
x1= –19, x2= –17, x3= –30, x4= 0, x5= 0, M = 1
D)
x1= 0, x2= 0, x3= 0, x4=44, x5=42, M = 0
Find the optimal row or column strategy of the matrix game.
125)
Let M be the matrix game having payoff matrix 1 1 3 –1 4
1–2–1 1 –2
2–1 0 3 –2.
Find the optimal row strategy.
125)
A)
^
x=
1
2
1
6
1
3
B)
^
x=
1
3
0
2
3
C)
^
x=
2
3
0
1
3
D)
^
x=
1
2
0
1
2
Solve the problem.
126)
A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it
can send a convoy up the river road or it can send a convoy overland through the jungle. On a
given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river
and the guerrillas are there, the convoy will have to turn back and 5 army soldiers will be lost. If
the convoy goes overland and encounters the guerrillas, 1
3 of the supplies will get through, but 8
army soldiers will be lost. Each day a supply convoy travels one of the roads, and if the guerrillas
are watching the other road, the convoy gets through with no losses. If the army chooses the
optimal strategy to maximize the amount of supplies it gets to its troops and the guerrillas choose
the optimal strategy to prevent the most supplies from getting through, then what portion of the
supplies will get through?
126)
A)
3
5
B)
1
2
C)
2
5
D)
3
8
127)
An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide
transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class
passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first
class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and
can accommodate 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?
127)
A)
7 P1 planes and 11 P2 planes
B)
20 P1 planes and 0 P2 planes
C)
9 P1 planes and 13 P2 planes
D)
11 P1 planes and 7 P2 planes
Solve the linear programming problem.
128)
Maximize 2x1+ 5x2
subject to 3x1+ 2x2 6
–2x1+4x2 8
and x1 0, x2 0
128)
A)
Maximum =49
4 when x1=1
2 and x2=9
4
B)
Maximum =27
2 when x1=1
2 and x2=5
2
C)
Maximum = 4 when x1= 2 and x2= 0
D)
Maximum = 15 when x1= 0 and x2= 3
Write the payoff matrix for the game.
129)
Each player has a supply of pennies, nickels, and dimes. At a given signal, both players display one
coin. If the total number of cents N is even, then R pays N cents to C. If N is odd, then C pays N
cents to R.
129)
A)
p n d
p
n
d
–1–1 10
–5–5 10
1 5 –10
B)
p n d
p
n
d
2 6 –11
6 10 –15
–11 –15 20
C)
p n d
p
n
d
–2–6 11
–6–10 15
11 15 –20
D)
p n d
p
n
d
1 1 –10
5 5 –10
–1–5 10
Determine whether the statement is true or false.
130)
You wish to solve the following linear programming problem:
Minimize 2x1+x2
subject to x1–x2 4
x1+ 2x2 10
and x1 0, x2 0
This can be solved by the simplex method by solving the equivalent problem:
Maximize –2x1–x2
subject to x1–x2 4
–x1– 2x2–10
and x1 0, x2 0
130)
A)
True
B)
False
Provide an appropriate response.
131)
A matrix game has payoff matrix
a11 a12 a13
a21 a22 a23
in which entry a21 is a saddle point. What are the optimal strategies for player R and player C?
131)
A)
^ ^
x=0
1, y=1
0
0
B)
^ ^
x=
1
2
1
2
, y=
1
3
1
3
1
3
C)
^ ^
x=0
1, y=0
0
1
D)
^ ^
x=1
0, y=1
0
0
56
132)
The optimal strategy for a 2 × n matrix game can be found as follows: obtain n linear functions by
finding the inner product of x(t) =1 – t
t with each of the columns of the payoff matrix A. Graph
the n linear functions on a t–z coordinate system. Then (x(t)) is the minimum value of the n linear
functions which will be seen on the graph as a polygonal path. The z–coordinate of any point on
this path is the minimum of the corresponding z coordinates of points on the n lines. The highest
point on the path (x(t)) is M. Suppose that M has coordinates (a, b). What information is given by
these coordinates?
132)
A)
The optimal strategy for R is 1 – a
a and the optimal strategy for C is 1 – b
b .
B)
The optimal strategy for R is a
1 – a and the value of the game for R is b.
C)
The optimal strategy for R is 1 – a
a and the value of the game for R is b.
D)
The optimal strategy for C is a
1 – a and the value of the game for C is b.
Determine whether the statement is true or false.
133)
If the payoff matrix of a matrix game contains a saddle point, the optimal strategy for the row
player will be to always choose the row with the largest minimum while the optimal strategy for
the column player will be to always choose the column with the smallest maximum.
133)
A)
True
B)
False
Solve the problem.
134)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. How many bottles of each smoothie
should the store make to maximize its profit? What is the maximum profit?
134)
A)
Maximum profit is $85 when the store makes 30 bottles of Orange Daze and 40 bottles of
Pineapple Blue.
B)
Maximum profit is $80 when the store makes 40 bottles of Orange Daze and 20 bottles of
Pineapple Blue.
C)
Maximum profit is $87.50 when the store makes 35 bottles of Orange Daze and 35 bottles of
Pineapple Blue.
D)
Maximum profit is $75 when the store makes 50 bottles of Orange Daze and 0 bottles of
Pineapple Blue.
Determine whether the statement is true or false.
135)
The value C of a matrix game to player C is the maximum of the values of the various possible
strategies for C.
135)
A)
True
B)
False
Explanation:
Solve the linear programming problem.
136)
Maximize 4x1+6x2
subject to x1–5x2–10
–2x1+x2–6
and x1 0, x2 0
136)
A)
Maximum =316
9 when x1=40
9 and x2=26
9
B)
Unbounded
C)
Infeasible
D)
Maximum =12 when x1= 0 and x2= 2
Explanation:
58
Explanation:
A simplex tableau is given. Compute the next tableau.
137)
x1x2 x3x4 M
1 –10 10 0 0 24
0 2 4 1 0 12
0 –8 8 0 1 0
137)
A)
x1x2 x3x4 M
1 0 30 5 0 84
0 1 21
2 0 6
0 0 24 4 1 48
B)
x1x2 x3x4 M
1 0 30 5 0 84
0 10 20 5 0 60
0 0 24 4 1 48
C)
x1x2 x3x4 M
1 0 30 5 0 84
0 1 21
2 0 6
0 0 24 5 1 60
D)
x1x2 x3x4 M
1 0 26 4 0 72
0 1 21
2 0 6
0 0 24 4 1 48
Solve the problem.
138)
Each player has a supply of nickels, dimes, and quarters. At a given signal, both players display
one coin. If the displayed coins are not the same, then the player showing the higher valued coin
gets to keep both. If they are both nickels or dimes, then player R keeps both; but if they are both
quarters, then player C keeps both. Find the optimal strategy for player C.
n d q
n
d
q
5–5–5
5 10 –10
5 10 –25
138)
A)
^
y=0
0
1
B)
^
y=0
1
0
C)
^
y=
0
1
2
1
2
D)
^
y=
0
1
3
2
3
Find all saddle points for the matrix game.
139)
0 3 –1 4
0 3 –1–8
1 9 1 6
139)
A)
Entry a13 and entry a31
B)
Entry a31
C)
Entry a31 and entry a33
D)
No saddle points
Solve the minimization problem by using the simplex method to solve the dual problem.
140)
Minimize 12x1+ 8x2
subject to x1+x2 3
3x1+x2 7
and x1 0, x2 0
140)
A)
Minimum = 36 when x1= 3 and x2= 0
B)
Minimum = 24 when x1= 0 and x2= 3
C)
Minimum = 56 when x1= 0 and x2= 7
D)
Minimum = 32 when x1= 2 and x2= 1
For the given simplex tableau, determine which variable should be brought into the solution and which row to use as a
pivot.
141)
x1x2x3x4 x5 M
2
3 0 1 3 0 0 16
–1
2 1 0 12 0 0 2
2
3 0 0 4 1 0 14
–8 0 0 8 0 1 0
141)
A)
x1, row 2
B)
x4, row 2
C)
x1, row 1
D)
x1, row 3
Set up the initial simplex tableau for the given linear programming problem.
142)
Maximize 34x1+33x2
subject to 2x1+6x237
4x1+12x233
and x1 0, x2 0
142)
A)
x1x2 x3x4 M
2 6 1 0 0 37
412 0 1 0 33
34 33 0 0 1 0
B)
x1x2 x3x4 M
2 6 1 0 0 37
412 0 1 0 33
–34 –33 0 0 1 0
C)
x1x2 x3x4 M
2 6 1 0 0 37
412 0 1 0 33
–34 –33 0 0 0 0
D)
x1x2 x3x4x5 M
2 6 1 0 1 0 37
412 0 1 1 0 33
–34 –33 0 0 0 1 0
Solve the minimization problem by using the simplex method to solve the dual problem.
143)
Minimize 4x1+ 5x2
subject to 5x1+4x2 20
4x1+x2 12
x1+3x2 9
and x1 0, x2 0
143)
A)
Minimum =228
11 when x1=27
11 and x2=24
11
B)
Minimum =221
11 when x1=24
11 and x2=25
11
C)
Minimum =212
11 when x1=28
11 and x2=20
11
D)
Minimum =41
2 when x1= 2 and x2=5
2
61
State the dual of the linear programming problem.
144)
Maximize 4x1+3x2+3x3
subject to 4x1+3x239
5x1+x320
4x1+x2+2x345
and x1 0, x2 0, x3 0
144)
A)
Minimize 4y1+3y2+3y3
subject to 4y1+5y2+4y339
3y1+y320
y2+2y345
and y1 0, y2 0, y3 0
B)
Minimize 39y1+20y2+45y3
subject to 4y1+5y2+4y34
3y1+y33
y2+2y33
and y1 0, y2 0, y3 0
C)
Minimize 39y1+20y2+45y3
subject to 4y1+3y24
5y1+y33
4y1+y2+2y33
and y1 0, y2 0, y3 0
D)
Minimize 39y1+20y2+45y3
subject to 4y1+5y2+4y34
3y1+y33
y2+2y33
and y1 0, y2 0, y3 0
Find the optimal row or column strategy of the matrix game.
145)
Let M be the matrix game having payoff matrix 1 1 3 –1 4
1–2–1 1 –2
2–1 0 3 –2.
Find the optimal column strategy.
145)
A)
^
y=
0
1
3
0
1
3
1
3
B)
^
y=
0
1
2
0
1
3
1
6
C)
^
y=
0
2
3
0
1
3
0
D)
^
y=
0
1
3
0
2
3
0
Determine whether the statement is true or false.
146)
In the simplex method for a canonical linear programming problem, if a solution is not optimal
then the variable which should be brought into the solution is the variable xk for which the entry in
the bottom row is as negative as possible.
146)
A)
True
B)
False
Answer:
A
Explanation:
A)
B)
Find all saddle points for the matrix game.
147)
7–1–7
6 2 –5
–1 6 –7
147)
A)
Entry a23
B)
Entry a13
C)
Entry a23 and entry a33
D)
No saddle points
Write the payoff matrix for the game.
148)
Each player has a supply of nickels, dimes, and quarters. At a given signal, both players display
one coin. If the displayed coins are not the same, then the player showing the higher valued coin
gets to keep both. If they are both nickels or dimes, then player R keeps both; but if they are both
quarters, then player C keeps both.
148)
A)
n d q
n
d
q
–5–5–5
5 –10 –10
5 10 25
B)
n d q
n
d
q
5–5–5
5 10 –10
5 10 –25
C)
n d q
n
d
q
–5–10 –25
10 –10 –25
25 25 25
D)
n d q
n
d
q
5 –10 –25
10 10 –25
25 25 –25
Answer:
B
Explanation:
A)
B)
Answer:
A
Explanation:
A)
B)
Provide an appropriate response.
149)
Determine whether the statement below is true or false. If it is false, give an explanation.
Given a canonical linear programming problem with an optimal solution, if the vector x is an
extreme point of the feasible set, then x must be an optimal solution.
149)
A)
False. Some extreme point is an optimal solution but not every extreme point is an optimal
solution.
B)
False. x may not be in the feasible set.
C)
True
D)
False. Not every optimal solution is an extreme point.
The primal problem and the final tableau in the solution of the primal problem are given. Use the final tableau to solve
the dual problem.
150)
Primal problem:
Maximize 50x1+ 35x2
subject to 3x1+x2 24
x1+x2 16
2x1+ 3x2 30
and x1 0, x2 0
Final tableau in solution to primal problem:
x1x2 x3x4x5 M
1 0 3
7 0 –1
7 0 6
0 0 –1
7 1 –2
7 0 4
0 1 –2
7 0 3
7 0 6
0 0 80
7 0 55
7 1 510
150)
A)
Minimum = 510 when y1= 6, y2= 6
B)
Minimum = 0 when y1= 0, y2= 0
C)
Minimum = 510 when y1=80
7, y2= 0, and y3=55
7
D)
Minimum = 510 when y1= 6, y2= 4, and y3= 6
Find the expected payoff.
151)
Let M be the matrix game having payoff matrix –4 3 3
0–1 3
–2 1 0 .
Find E(x, y) when x=
1
3
1
2
1
6
and y=
1
4
1
4
1
2
.
151)
A)
3
8
B)
1
C)
25
24
D)
7
8
Provide an appropriate response.
152)
^
The optimal strategy for a 2 × 5 matrix game can be found as follows: obtain 5 linear functions by
finding the inner product of x(t) =1 – t
t with each of the columns of the payoff matrix A. Graph
the 5 linear functions on a t–z coordinate system. Then (x(t)) is the minimum value of the 5 linear
functions which will be seen on the graph as a polygonal path. The z–coordinate of any point on
this path is the minimum of the corresponding z coordinates of points on the 5 lines. The highest
point on the path (x(t)) is M. Suppose that only the lines corresponding to columns 1 and 4 of
matrix A pass through the point M. What can be said about the optimal column strategy y ?
152)
A)
^
y=
c1
0
0
c4
0
where c1+c4= 1
B)
^
y=
0
0
0
1
0
C)
^
y=
1
2
0
0
1
2
0
D)
^
y=
0
c2
c3
0
c5
where c2+c3+c5= 1
Determine whether the statement is true or false.
153)
Suppose that you are using the simplex method to solve a canonical linear programming problem
in which each entry in the vector b is positive. If you obtain a tableau which contains just one
negative entry in the bottom row and no positive entry aik above it, then the objective function is
unbounded and no optimal solution exists.
153)
A)
True
B)
False
Use linear programming to solve the matrix game with the given payoff matrix.
154)
A = 3 1 –2
1 2 0
–1 1 3
154)
A)
^ ^
x=
5
11
0
6
11
, y=
6
11
0
5
11
, value =9
34
B)
^ ^
x=
5
9
0
4
9
, y=
4
9
0
5
9
, value =34
9
C)
^ ^
x=
4
9
0
5
9
, y=
5
9
0
4
9
, value =7
9
D)
^ ^
x=
4
34
0
5
34
, y=
5
34
0
4
34
, value =9
34
Provide an appropriate response.
155)
If the payoff matrix of a matrix game contains a saddle point, what is the optimal strategy for the
column player?
155)
A)
Always choose the column with the smallest maximum.
B)
Always choose the column with the largest minimum.
C)
Always choose the column with the largest maximum.
D)
Always choose the column with the smallest minimum.
Solve the minimization problem by using the simplex method to solve the dual problem.
156)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue.
What is the marginal value of orange juice?
[Hint: use the simplex method to determine the maximum profit and use the final tableau to
determine the marginal value.]
156)
A)
0
B)
1
8
C)
30
D)
1
16
Find the optimal row or column strategy of the matrix game.
157)
Let M be the matrix game having payoff matrix 4 4 –410
0 4 5 3 .
Find the optimal row strategy.
157)
A)
^
x=
4
13
9
13
B)
^
x=
9
13
4
13
C)
^
x=
5
13
8
13
D)
^
x=
8
13
5
13
Solve by using the simplex method.
158)
Maximize 7x1+ 8x2
subject to x1+ 2x214
5x1+4x245
and x1 0, x2 0
158)
A)
Maximum =63 when x1=9 and x2= 0
B)
Maximum =275
3 when x1=17
3 and x2=13
2
C)
Maximum =73 when x1=17
3 and x2=25
6
D)
Maximum =56 when x1= 0 and x2=7
Identify the basic feasible solution corresponding to the given simplex tableau.
159)
x1x2 x3x4 x5 M
9 4 1 0 0 0 40
7 4 0 1 0 0 43
112 0 0 1 0 20
–26 –34 0 0 0 1 0
159)
A)
x1=9, x2=4, x3=40, x4=43, x5=20, M = 0
B)
x1= 0, x2= 0, x3=40, x4=43, x5=20, M = 0
C)
x1= –26, x2= –34, x3= 0, x4= 0, x5= 0, M = 1
D)
x1=9, x2=4, x3= 1, x4= 0, x5= 0, M = 0
Provide an appropriate response.
160)
A linear programming problem has objective function 6x and constraints
x a
–x –b
where a and b are positive numbers.
Does a maximum exist for the objective function? If not, why not? Does a minimum exist? If not,
why not?
160)
A)
Maximum does not exist as the objective function may be arbitrarily large (unbounded).
Minimum does exist.
B)
Maximum and minimum both exist.
C)
Neither maximum nor minimum exist as the feasible set is the empty set.
D)
Maximum exists. Minimum does not exist as the objective function may be arbitrarily small
(unbounded).
Solve the problem.
161)
A furniture company makes two different types of lamp stand. Each lamp stand A requires 20
minutes for sanding, 48 minutes for assembly, and 6 minutes for packaging. Each lamp stand B
requires 9 minutes for sanding, 32 minutes for assembly, and 8 minutes for packaging. The total
number of minutes available each day in each department are as follows: for sanding 3600 minutes,
for assembly 9600 minutes, and for packaging 2000 minutes. The profit on each lamp stand A is $30
and the profit on each lamp stand B is $22. How many of each type of lamp stand should the
company make per day to maximize their profit? What is the maximum profit?
161)
A)
Maximum profit is $5400 when they make 180 of lamp stand A and 0 of lamp stand B.
B)
Maximum profit is $6164 when they make 138 of lamp stand A and 92 of lamp stand B.
C)
Maximum profit is $6850 when they make 100 of lamp stand A and 175 of lamp stand B.
D)
Maximum profit is $6380 when they make 66 of lamp stand A and 200 of lamp stand B.
Use the simplex method to solve the linear programming problem.
162)
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 $40 and on an SST
ring is $35?
162)
A)
18 VIP and 6 SST
B)
16 VIP and 8 SST
C)
14 VIP and 10 SST
D)
12 VIP and 12 SST
Provide an appropriate response.
163)
In certain situations, a matrix game can be reduced to a smaller game by deleting certain rows
and/or columns from the payoff matrix. The optimal strategy for the reduced game will then
determine the optimal strategy for the original game. In what circumstances may a row or column
be deleted from the payoff matrix?
163)
A)
A row may be deleted if it is recessive to some other row and a column may be deleted if it is
dominant to some other column.
B)
A row may be deleted if it is dominant to some other row and a column may be deleted if it is
recessive to some other column.
C)
A row may be deleted if it is dominant to some other row and a column may be deleted if it is
dominant to some other column.
D)
A row may be deleted if it is recessive to some other row and a column may be deleted if it is
recessive to some other column.
The primal problem and the final tableau in the solution of the primal problem are given. Use the final tableau to solve
the dual problem.
164)
Primal problem:
Maximize x1+ 4x2+ 3x3
subject to x1+2x2+x3 24
3x2+ 4x3 30
and x1 0, x2 0, x3 0
Final tableau in solution to primal problem:
x1x2 x3x4x5 M
1 0 –5
3 1 –2
3 0 4
0 1 4
3 0 1
3 0 10
0 0 2
3 1 2
3 1 44
164)
A)
Minimum = 44 when y1= 1 and y2=2
3
B)
Maximum = 44 when y1= 0, y2= 0, and y3=2
3
C)
Minimum = 44 when y1=2
3, y2= 1, and y3=2
3
D)
Minimum = 44 when y1= 4 and y2= 10
Write the payoff matrix for the game.
165)
In the traditional Japanese children’s game janken (or “stone, scissors, paper”), at a given signal,
each of two players shows either no fingers (stone), two fingers (scissors), or all five fingers (paper).
Stone beats scissors, scissors beats paper, and paper beats stone. In the case of a tie, there is no
payoff. In the case of a win, the winner collects 30 yen.
165)
A)
st sc p
st
sc
p
30 30 –30
–30 30 30
30 –30 30
B)
st sc p
st
sc
p
0 30 –30
–30 0 30
30 –30 0
C)
st sc p
st
sc
p
0 –30 30
30 0 –30
–30 30 0
D)
st sc p
st
sc
p
0 30 –30
30 0 –30
30 –30 0
Find the optimal row or column strategy of the matrix game.
166)
Let M be the matrix game having payoff matrix 1 1 –2 4
–1 4 6 3 .
Find the optimal column strategy.
166)
A)
^
y=
4
5
0
1
5
0
B)
^
y=
7
10
3
10
0
0
C)
^
y=
7
10
0
3
10
0
D)
^
y=
4
5
1
5
0
0
Provide an appropriate response.
167)
A linear programming problem has objective function 3x1+ 4x2 and constraints
ax1+ bx2 10
cx1+ dx2 20
x1 0, x2 0
where a, b, c, and d are positive numbers.
Does a maximum exist for the objective function? If not, why not? Does a minimum exist? If not,
why not?
167)
A)
Neither maximum nor minimum exist as the feasible set is the empty set.
B)
Maximum exists. Minimum does not exist as the objective function may be arbitrarily small
(unbounded).
C)
Maximum does not exist as the objective function may be arbitrarily large (unbounded).
Minimum does exist.
D)
Maximum and minimum both exist.
Find the value of the matrix game.
168)
Let M be the matrix game having payoff matrix 1 1 3 –1 4
1–2–1 1 –2
2–1 0 3 –2.
168)
A)
2
3
B)
1
3
C)
1
2
D)
1
4
Solve the linear programming problem.
169)
Maximize 50x1+ 35x2
subject to 3x1+x2 24
x1+x2 16
2x1+3x2 30
and x1 0, x2 0
169)
A)
Maximum = 400 when x1= 8 and x2= 0
B)
Maximum = 620 when x1= 4 and x2= 12
C)
Maximum = 510 when x1= 6 and x2= 6
D)
Maximum = 350 when x1= 0 and x2= 10
72
Determine whether the basic feasible solution corresponding to the given tableau is optimal.
170)
x1x2x3x4 M
048 1 –2 0 4
1 1 0 1
2 0 4
0 7 0 6 1 48
170)
A)
Not optimal
B)
Optimal
Determine whether the statement is true or false.
171)
Suppose a system contains three inequalities and three unknowns x1, x2, and x3. If the simplex
method results in an optimal solution in which the slack variable x4 is greater than zero, this means
that at the optimal values of x1, x2, and x3, the first inequality is an equality.
171)
A)
True
B)
False
State the dual of the linear programming problem.
172)
Maximize 6x1+4x2+4x3
subject to 3x1+2x245
3x1+x316
4x2+x316
and x1 0, x2 0, x3 0
172)
A)
Minimize 6y1+4y2+4y3
subject to 3y1+3y245
2y1+4y316
y2+y316
and y1 0, y2 0, y3 0
B)
Minimize 45y1+16y2+16y3
subject to 3y1+3y26
2y1+4y34
y2+y34
and y1 0, y2 0, y3 0
C)
Minimize 45y1+16y2+16y3
subject to 3y1+3y26
2y1+4y34
y2+y34
and y1 0, y2 0, y3 0
D)
Minimize 45y1+16y2+16y3
subject to 3y1+2y26
3y1+4y34
y2+y34
and y1 0, y2 0, y3 0
73
Solve the linear programming problem.
173)
Maximize 6x1+7x2
subject to x1–x24
–x1+5x2–5
and x1 0, x2 0
173)
A)
Unbounded
B)
Maximum =83
4 when x1=15
4 and x2= – 1
4
C)
Maximum =24 when x1=4 and x2= 0
D)
Infeasible
74
Explanation:
Solve the problem.
174)
Daniel decides to feed his cat a combination of two foods: Max Cat and Mighty Cat. He wants his
cat to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d)
contained in one bag of each food are shown in the chart, together with the total amounts needed.
a b c d
Max Cat 1 2 2 2
Mighty Cat 3 1 2 1
Needed 20 26 21 22
The costs per bag are $42 for Max Cat and $40 for Mighty Cat. How many bags of each food should
be blended to meet the nutritional requirements at the lowest cost? Set this up as a linear
programming problem in the following form:
Minimize cTx subject to Ax b and x 0. Do not find the solution.
174)
A)
Let x1 be the number of bags of Max Cat and x2 the number of bags of Mighty Cat.
Then b =
20
26
21
22
, x =x1
x2, c =42
40 , and A =
1 3
2 1
2 2
2 1
B)
Let x1 be the number of bags of Max Cat and x2 the number of bags of Mighty Cat.
Then b =
0
0
0
0
, x =x1
x2, c =42
40 , and A =
1 3 20
2 1 26
2 2 21
2 1 22
C)
Let x1 be the number of bags of Max Cat and x2 the number of bags of Mighty Cat.
Then b =
20
26
21
22
, x =x1
x2, c =42
40 , and A =1 2 2 2
3 1 2 1
D)
Let x1 be the number of bags of Max Cat and x2 the number of bags of Mighty Cat.
Then b =42
40 , x =x1
x2, c =
20
26
21
22
, and A =
1 3
2 1
2 2
2 1
Solve the minimization problem by using the simplex method to solve the dual problem.
175)
Minimize 16x1+ 20x2+ 36x3
subject to x1+3x3 3
2x2+ 6x3 4
x1+x2 2
and x1 0, x2 0, x3 0
175)
A)
Minimum =152
3 when x1=2, x2=1
3, and x3=1
3
B)
Minimum =172
3 when x1=2, x2=2
3, and x3=1
3
C)
Minimum = 60 when x1= 1, x2= 1, and x3=2
3
D)
Minimum = 52 when x1=3
2, x2=1
2, and x3=1
2
State the dual of the linear programming problem.
176)
Maximize 5x1+8x2+7x3
subject to 5x1+x315
5x1+4x2+x349
and x1 0, x2 0, x3 0
176)
A)
Minimize 5y1+8y2+7y3
subject to 5y1+y315
5y1+4y2+y349
and y1 0, y2 0, y3 0
B)
Minimize 15y1+49y2
subject to 5y1+5y25
4y28
y1+y27
and y1 0, y2 0
C)
Minimize 15y1+49y2
subject to 5y1+5y25
4y28
y1+y27
and y1 0, y2 0
D)
Minimize 5y1+8y2
subject to 5y1+5y215
4y249
y1+y27
and y1 0, y2 0
76
Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form:
maximize cTx subject to Ax b and x
0. Do not find the solution.
177)
Maximize x1–5x2+3x3
subject to 6x1+x2+2x336
4x1–5x332
and x1 0, x2 0, x3 0
177)
A)
b =36
–32 , c =1
–5
3, and A =6 1 2
–4 0 5
B)
b =1
–5
3, c =36
–32 , and A =6 1 2
–4 0 5
C)
b =36
32 , c =1
–5
3, and A =6 1 2
4 0 –5
D)
b =36
–32 , c =1
–5
3, and A =6 4
1 0
2–5
Use the simplex method to solve the linear programming problem.
178)
A store makes two different types of smoothies by blending different fruit juices. Each bottle of
Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice,
and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid
ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The
store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces
of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each
bottle of Orange Daze and $1 on each bottle of Pineapple Blue. How many bottles of each smoothie
should the store make to maximize its profit? What is the maximum profit?
178)
A)
Maximum profit is $87.50 when the store makes 35 bottles of Orange Daze and 35 bottles of
Pineapple Blue.
B)
Maximum profit is $85 when the store makes 30 bottles of Orange Daze and 40 bottles of
Pineapple Blue.
C)
Maximum profit is $75 when the store makes 50 bottles of Orange Daze and 0 bottles of
Pineapple Blue.
D)
Maximum profit is $80 when the store makes 40 bottles of Orange Daze and 20 bottles of
Pineapple Blue.
77
Answer Key
Testname: C9
Answer Key
Testname: C9
79
Answer Key
Testname: C9
Answer Key
Testname: C9