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# 数学代写|线性规划代写Linear Programming代考|MS-E2121 Managing a Production Facility

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## 数学代写|线性规划代写Linear Programming代考|Managing a Production Facility

Consider a production facility for a manufacturing company. The facility is capable of producing a variety of products that, for simplicity, we shall enumerate as $1,2, \ldots, n$. These products are constructed/manufactured/produced out of certain raw materials. Let us assume that there are $m$ different raw materials, which again we shall simply enumerate as $1,2, \ldots, m$. The decisions involved in managing/operating this facility are complicated and arise dynamically as market conditions evolve around it. However, to describe a simple, fairly realistic optimization problem, we shall consider a particular snapshot of the dynamic evolution. At this specific point in time, the facility has, for each raw material $i=1,2, \ldots, m$, a known amount, say $b_i$, on hand. Furthermore, each raw material has at this moment in time a known unit market value. We shall denote the unit value of the $i$ th raw material by $\rho_i$.

In addition, each product is made from known amounts of the various raw materials. That is, producing one unit of product $j$ requires a certain known amount, say $a_{i j}$ units, of raw material $i$. Also, the $j$ th final product can be sold at the known prevailing market price of $\sigma_j$ dollars per unit.
Throughout this section we make an important assumption:
The production facility is small relative to the market as a whole and therefore cannot through its actions alter the prevailing market value of its raw materials, nor can it affect the prevailing market price for its products.
We shall consider two optimization problems related to the efficient operation of this facility (later, in Chapter 5, we shall see that these two problems are in fact closely related to each other).

## 数学代写|线性规划代写Linear Programming代考|Production Manager as Optimist

The first problem we wish to consider is the one faced by the company’s production manager. It is the problem of how to use the raw materials on hand. Let us assume that she decides to produce $x_j$ units of the $j$ th product, $j=1,2, \ldots, n$. The revenue associated with the production of one unit of product $j$ is $\sigma_j$. But there is also a cost of raw materials that must be considered. The cost of producing one unit of product $j$ is $\sum_{i=1}^m \rho_i a_{i j}$. Therefore, the net revenue associated with the production of one unit is the difference between the revenue and the cost. Since the net revenue plays an important role in our model, we introduce notation for it by setting
$$c_j=\sigma_j-\sum_{i=1}^m \rho_i a_{i j}, \quad j=1,2, \ldots, n .$$
Now, the net revenue corresponding to the production of $x_j$ units of product $j$ is simply $c_j x_j$, and the total net revenue is
$$\sum_{j=1}^n c_j x_j$$

## 数学代写|线性规划代写Linear Programming代考|Production Manager as Optimist

$$c_j=\sigma_j-\sum_{i=1}^m \rho_i a_{i j}, \quad j=1,2, \ldots, n .$$

$$\sum_{j=1}^n c_j x_j$$

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