Introduction

Exercise 1.1

Let \(x_i\) denote the number of hours a rolling mill spends on product \(i\). Define \(p_i\) as the profit and \(r_i\) as the hourly production rate for product \(i\). The machine can only be used for \(t = 40\) hours this week and the quantity of each product must not exceed \(b_i\).

\[\begin{split}\begin{aligned} \text{maximize} \quad \sum_i x_i r_i p_i &\\ \text{subject to} \quad \sum_i x_i &= t\\ x_i r_i &\leq b_i\\ x_i &\geq 0 \end{aligned}\end{split}\]

From inspection, bands will bring in an hourly profit of \(200 \times 25 = $5000\) while coils will yield \(140 \times 30 = $4200\). Thus the optimal solution should be to produce as many bands as possible and then roll out coils.

Exercise 1.2

Let \(x_{ij}\) denote the number of tickets sold for flight \(i\) with fare class \(j\). Define \(f_{ij}\) as the ticket price and \(d_{ij}\) as the maximum number of potential customers. Each leg of the flight cannot exceed \(p = 30\) passengers and the number of tickets made available cannot exceed the forecasted maximum demand.

\[\begin{split}\begin{aligned} \text{maximize} \quad \sum_i \sum_j x_{ij} f_{ij} &\\ \text{subject to} \quad \sum_j x_{1j} + x_{3j} &\leq p\\ \sum_j x_{2j} + x_{3j} &\leq p\\ 0 \leq x_{ij} &\leq d_{ij} \end{aligned}\end{split}\]

Exercise 1.3

Let \(b_j\) denote the event of asserting that the distribution is \(p\) when in fact it is \(q\).

Let \(c_j\) denote the event of asserting that the distribution is \(p\) when in fact it is \(p\).

The latter parts of this exercise can be interpreted as applying Boole’s inequality:

\[\mathbb{P}\left( \bigcup_j b_j \right) \leq \sum_j \mathbb{P}(b_j) \qquad \land \qquad \mathbb{P}\left( \bigcup_j c_j \right) \leq \sum_j \mathbb{P}(c_j)\]

where \(\mathbb{P}(b_j) = x_j q_j\) and \(\mathbb{P}(c_j) = x_j p_j\). The probability of each individual event is defined as such since the act of asserting which distribution an observation came from is independent of that observation’s generative process.

\[\begin{split}\begin{aligned} \text{maximize} \quad \sum_j p_j x_j &\\ \text{subject to } \quad \sum_j x_j q_j &\leq \beta\\ \sum_j x_j &= 1\\ 0 \leq x_j &\leq 1 \end{aligned}\end{split}\]