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# 数学代写|运筹学代写Operations Research代考|MATH3202 A Distribution Problem

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## 数学代写|运筹学代写Operations Research代考|A Distribution Problem

Problem statement. For the distribution problem in Section 1.2.3, we now consider the situation that extra storage capacity is needed. The first possibility is to open a new depot $D_4$ with a capacity of 50 thousand tons. It is also possible to expand the capacity of depot $D_2$ from 40 thousand tons to 65 thousand tons. The monthly depreciation amounts to 10 thousand euros for the new depot and 3 thousand euros for the capacity expansion of depot $D_2$. The distribution cost from the factory to the new depot $D_4$ is 5 euros per ton. The distribution costs from the new depot $D_4$ to the customers $C_1, C_2, C_3$, and $C_4$ are, respectively, 7, 5, 8, and 10 euros per ton. How should the depot capacity be increased, and what is the optimal distribution schedule in the new situation to minimize the total cost?
Formulation. In order to answer the questions, the LP model from Section 1.2.2 needs to be extended to an integer programming model. This extension requires the following binary variables:
$$\delta_1=\left{\begin{array}{l} 1 \text { if the new depot } D_4 \text { is built } \ 0 \text { otherwise } \end{array}\right.$$
and
$$\delta_2= \begin{cases}1 & \text { if the capacity of depot } D_2 \text { is expanded } \ 0 & \text { otherwise }\end{cases}$$

## 数学代写|运筹学代写Operations Research代考|A Timetabling Problem

Problem statement. A bus company needs the following numbers of drivers every week:

Each bus driver works five consecutive days a week and then has two days off. The salary is 100 euros for each weekday, 115 euros for Saturdays, and 125 euros for Sundays. What work schedule minimizes the weekly salary costs?

Formulation. The seven days of the week are numbered $1, \ldots, 7$, where day 1 $=$ Monday, day 2 = Tuesday, and so on. We use the following decision variables:
$x_i=$ number of bus drivers beginning work on day $i$
for $i=1, \ldots, 7$. We do not take the number of bus drivers working on day $i$ as a decision variable because that variable does not give us any information about the remaining number of days the bus drivers still have to work. The chosen decision variables $x_i$ immediately allow us to specify the objective function and constraints. The number of bus drivers working on day 1 is equal to $x_1+x_7+x_6+x_5+x_4$. This number must be at least 25 , which results in the inequality $x_1+x_7+x_6+x_5+x_4 \geq 25$. The constraints for the other days are found likewise. The aim is to minimize the weekly salary costs. The total salary of a bus driver who starts working on day 7 , for example, is $125+4 \times 100=525$ euros, which contributes $525 x_7$ to the objective function. The complete formulation of the integer programming model and the computer solution are given in Figure 2.4.

## 数学代写|运筹学代写Operations Research代考|A Distribution Problem

$x_i=$在第i天开始工作的巴士司机人数

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