Solutions Manual For
Quantitative Methods An Introduction for Business Management 1st Edition by Paolo
Brandimarte
Chapter 1-14 With Appendix [A B C]
Contents
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal
law, corporate law, human rights
1 Quantitative Methods: Should We Bother? 1
1.1 Solutions 1
1.2 Computational supplement: How to solve the optimal mix
problem 3
2 Calculus 7
2.1 Solutions 7
3 Linear Algebra 15
3.1 Solutions 15
4 Descriptive Statistics: On the Way to Elementary Probability 25
4.1 Solutions 25
5 Probability Theories 29
5.1 Solutions 29
5.2 Additional problems 30
5.3 Solutions of additional problems 31
6 Discrete Random Variables 33
6.1 Solutions 33
7 Continuous Random Variables 37
7.1 Solutions 37
8 Dependence, Correlation, and Conditional Expectation 43
8.1 Solutions 43
9 Inferential Statistics 47
9.1 Solutions 47
10 Simple Linear Regression 63
10.1 Solutions 63
11 Time Series Models 69
11.1 Solutions 69
12 Deterministic Decision Models 75
12.1 Solutions 75
, 13 Decision Making Under Risk 91
13.1 Solutions 91
14 Advanced Regression Models 99
14.1 Solutions 99
Appendix A R – A software tool for statistics 103
Appendix B Introduction to MATLAB 105
B.1 Working with vectors and matrices in the MATLAB
environment 105
B.2 MATLAB graphics 113
B.3 Solving equations and computing integrals 114
B.4 Statistics in MATLAB 116
B.5 Using MATLAB to solve linear and quadratic programming
problems 118
Appendix C Introduction to AMPL 121
C.1 Running optimization models in AMPL 122
C.2 Mean-variance efficient portfolios in AMPL 123
C.3 The knapsack model in AMPL 125
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal
law, corporate law, human rights
, 1
Quantitative Methods: Should We Bother?
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal law, corporate law,
human rights
1.1 SOLUTIONS
Problem 1.1 We consider the strategy of trying Plan A first and then Plan B; a more
complete solution approach should rely on the decision tree framework of Chapter 13 (see
Problem 13.1).
Imagine that we are at the end of year 1, and say that the first movie has been a success.
If we try Plan B for the second movie, we invest 4000 now, and at the end of the second year
we will make 6600 with probability 0.5+ α and 0 with probability 1 (0.5+ α) = 0.5 α.
The expected NPV for this part of the strategy is
6600
NPVh = × (0.5+ α) — 4000 = 6000 × α — 1000;
1.1
note that here we are discounting the cash flow at the end of year 2 back to the end of year
1. We go on with Plan B only if this NPV is positive, i.e.,
1000
α≥ = 0.1667.
6000
On the other branch of the tree, the first movie has been a flop. The second movie, if we
adopt plan B, yields the following NPV (discounted back to the end of year 1):
6600
× (0.5 — α) — 4000 = —6000 × α — 5000.
NPVf =
1.1
This will always be negative for α ≥ 0, which makes sense: With 50–50 probabilities, the
bet is not quite promising, and the situation does not improve after a first flop if this makes
the odds even less favorable, and we dismiss the possibility of producing the second movie.
Let us step back to the root node (beginning of year 1), where we apply Plan A. There,
the expected NPV is
4400 + NPVh 0
— 2500 + 0.5 × + 0.5 ×
1.1 1.1
= —954.5455 + α × 2727.273,
1
, 2 QUANTITATIVE METHODS: SHOULD WE BOTHER?
which is positive if
954.5455
α≥
= 0.35.
2727.273
This condition is more stringent that the previous one. Thus, the conditional probability of
a second hit after the first one should not be less than 0.85 (which also implies that the the
conditional probability of a hit after a first flop is 0.15).
Problem 1.2 Rather than extending the little numerical example of Chapter 1, let us
state the model in general form (also see Chapter 12):
N
max (pi — ci)xi,
i=1
N
s.t. rimxi ≤ Rm, m = 1, . .., M,
i=1
0 ≤ xi ≤ di, i = 1, . .. , N,
where:
• Items are indexed by i = 1, . .., N
• Resources are indexed by m = 1, . .., M
• di, pi, and ci are demand, selling price, and production cost, respectively, for item i
rim is the unit requirement of resource m for item i, and Rm is the total availability
of resource m
In this model, we have a single decision variable, xi, representing what we produce and
sell. If we introduce the possibility of third-party production, we can no longer identify
production and sales. We need to change decision variables as follows:
• xi is what we produce
• yi is what we buy principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include
constitutional law, criminal law, corporate law, human rights
We could also introduce a variable zi to denote what we sell, but since zi = xi + yi, we can
1
avoid this. Let us denote by gi > ci the cost of purchasing item i from the third-party
supplier. The model is now
N
max [(pi — ci)xi + (pi — gi)yi]
i=1
N
s.t. rimxi ≤ Rm m = 1, . .., M
i=1
xi + yi ≤ di i = 1, . .., N
x i , yi ≥ 0 i = 1, . .., N
1However, in multiperiod problems involving inventory holding we do need such a variable; see Chapter 12.
Quantitative Methods An Introduction for Business Management 1st Edition by Paolo
Brandimarte
Chapter 1-14 With Appendix [A B C]
Contents
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal
law, corporate law, human rights
1 Quantitative Methods: Should We Bother? 1
1.1 Solutions 1
1.2 Computational supplement: How to solve the optimal mix
problem 3
2 Calculus 7
2.1 Solutions 7
3 Linear Algebra 15
3.1 Solutions 15
4 Descriptive Statistics: On the Way to Elementary Probability 25
4.1 Solutions 25
5 Probability Theories 29
5.1 Solutions 29
5.2 Additional problems 30
5.3 Solutions of additional problems 31
6 Discrete Random Variables 33
6.1 Solutions 33
7 Continuous Random Variables 37
7.1 Solutions 37
8 Dependence, Correlation, and Conditional Expectation 43
8.1 Solutions 43
9 Inferential Statistics 47
9.1 Solutions 47
10 Simple Linear Regression 63
10.1 Solutions 63
11 Time Series Models 69
11.1 Solutions 69
12 Deterministic Decision Models 75
12.1 Solutions 75
, 13 Decision Making Under Risk 91
13.1 Solutions 91
14 Advanced Regression Models 99
14.1 Solutions 99
Appendix A R – A software tool for statistics 103
Appendix B Introduction to MATLAB 105
B.1 Working with vectors and matrices in the MATLAB
environment 105
B.2 MATLAB graphics 113
B.3 Solving equations and computing integrals 114
B.4 Statistics in MATLAB 116
B.5 Using MATLAB to solve linear and quadratic programming
problems 118
Appendix C Introduction to AMPL 121
C.1 Running optimization models in AMPL 122
C.2 Mean-variance efficient portfolios in AMPL 123
C.3 The knapsack model in AMPL 125
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal
law, corporate law, human rights
, 1
Quantitative Methods: Should We Bother?
principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include constitutional law, criminal law, corporate law,
human rights
1.1 SOLUTIONS
Problem 1.1 We consider the strategy of trying Plan A first and then Plan B; a more
complete solution approach should rely on the decision tree framework of Chapter 13 (see
Problem 13.1).
Imagine that we are at the end of year 1, and say that the first movie has been a success.
If we try Plan B for the second movie, we invest 4000 now, and at the end of the second year
we will make 6600 with probability 0.5+ α and 0 with probability 1 (0.5+ α) = 0.5 α.
The expected NPV for this part of the strategy is
6600
NPVh = × (0.5+ α) — 4000 = 6000 × α — 1000;
1.1
note that here we are discounting the cash flow at the end of year 2 back to the end of year
1. We go on with Plan B only if this NPV is positive, i.e.,
1000
α≥ = 0.1667.
6000
On the other branch of the tree, the first movie has been a flop. The second movie, if we
adopt plan B, yields the following NPV (discounted back to the end of year 1):
6600
× (0.5 — α) — 4000 = —6000 × α — 5000.
NPVf =
1.1
This will always be negative for α ≥ 0, which makes sense: With 50–50 probabilities, the
bet is not quite promising, and the situation does not improve after a first flop if this makes
the odds even less favorable, and we dismiss the possibility of producing the second movie.
Let us step back to the root node (beginning of year 1), where we apply Plan A. There,
the expected NPV is
4400 + NPVh 0
— 2500 + 0.5 × + 0.5 ×
1.1 1.1
= —954.5455 + α × 2727.273,
1
, 2 QUANTITATIVE METHODS: SHOULD WE BOTHER?
which is positive if
954.5455
α≥
= 0.35.
2727.273
This condition is more stringent that the previous one. Thus, the conditional probability of
a second hit after the first one should not be less than 0.85 (which also implies that the the
conditional probability of a hit after a first flop is 0.15).
Problem 1.2 Rather than extending the little numerical example of Chapter 1, let us
state the model in general form (also see Chapter 12):
N
max (pi — ci)xi,
i=1
N
s.t. rimxi ≤ Rm, m = 1, . .., M,
i=1
0 ≤ xi ≤ di, i = 1, . .. , N,
where:
• Items are indexed by i = 1, . .., N
• Resources are indexed by m = 1, . .., M
• di, pi, and ci are demand, selling price, and production cost, respectively, for item i
rim is the unit requirement of resource m for item i, and Rm is the total availability
of resource m
In this model, we have a single decision variable, xi, representing what we produce and
sell. If we introduce the possibility of third-party production, we can no longer identify
production and sales. We need to change decision variables as follows:
• xi is what we produce
• yi is what we buy principles, case law, legal frameworks, and ethical issues. It prepares students for legal practice, with specializations that may include
constitutional law, criminal law, corporate law, human rights
We could also introduce a variable zi to denote what we sell, but since zi = xi + yi, we can
1
avoid this. Let us denote by gi > ci the cost of purchasing item i from the third-party
supplier. The model is now
N
max [(pi — ci)xi + (pi — gi)yi]
i=1
N
s.t. rimxi ≤ Rm m = 1, . .., M
i=1
xi + yi ≤ di i = 1, . .., N
x i , yi ≥ 0 i = 1, . .., N
1However, in multiperiod problems involving inventory holding we do need such a variable; see Chapter 12.
