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# 数学代写|线性规划代写Linear Programming代考|Comptroller as Pessimist

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## 数学代写|线性规划代写Linear Programming代考|Comptroller as Pessimist

In another office at the production facility sits an executive called the comptroller. The comptroller’s problem (among others) is to assign a value to the raw materials on hand. These values are needed for accounting and planning purposes to determine the cost of inventory. There are rules about how these values can be set. The most important such rule (and the only one relevant to our discussion) is the following:
The company must be willing to sell the raw materials should an outside firm offer to buy them at a price consistent with these values.
Let $w_i$ denote the assigned unit value of the $i$ th raw material, $i=1,2, \ldots, m$. That is, these are the numbers that the comptroller must determine. The lost opportunity cost of having $b_i$ units of raw material $i$ on hand is $b_i w_i$, and so the total lost opportunity cost is
$$\sum_{i=1}^m b_i w_i .$$
The comptroller’s goal is to minimize this lost opportunity cost (to make the financial statements look as good as possible). But again, there are constraints. First of all, each assigned unit value $w_i$ must be no less than the prevailing unit market value $\rho_i$, since if it were less an outsider would buy the company’s raw material at a price lower than $\rho_i$, contradicting the assumption that $\rho_i$ is the prevailing market price. That is,
$$w_i \geq \rho_i, \quad i=1,2, \ldots, m .$$
Similarly,
$$\sum_{i=1}^m w_i a_{i j} \geq \sigma_j, \quad j=1,2, \ldots, n$$

## 数学代写|线性规划代写Linear Programming代考|The Linear Programming Problem

In the two examples given above, there have been variables whose values are to be decided in some optimal fashion. These variables are referred to as decision variables. They are usually written as
$$x_j, \quad j=1,2, \ldots, n .$$
In linear programming, the objective is always to maximize or to minimize some linear function of these decision variables
$$\zeta=c_1 x_1+c_2 x_2+\cdots+c_n x_n .$$
This function is called the objective function. It often seems that real-world problems are most naturally formulated as minimizations (since real-world planners always seem to be pessimists), but when discussing mathematics it is usually nicer to work with maximization problems. Of course, converting from one to the other is trivial both from the modeler’s viewpoint (either minimize cost or maximize profit) and from the analyst’s viewpoint (either maximize $\zeta$ or minimize $-\zeta$ ). Since this book is primarily about the mathematics of linear programming, we shall usually consider our aim one of maximizing the objective function.

In addition to the objective function, the examples also had constraints. Some of these constraints were really simple, such as the requirement that some decision variable be nonnegative. Others were more involved. But in all cases the constraints consisted of either an equality or an inequality associated with some linear combination of the decision variables:
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n\left{\begin{array}{l} \leq \ = \ \geq \end{array}\right} b .$$

## 数学代写|线性规划代写Linear Programming代考|Comptroller as Pessimist

$$\sum_{i=1}^m b_i w_i .$$

$$w_i \geq \rho_i, \quad i=1,2, \ldots, m .$$

$$\sum_{i=1}^m w_i a_{i j} \geq \sigma_j, \quad j=1,2, \ldots, n$$

## 数学代写|线性规划代写Linear Programming代考|The Linear Programming Problem

$$x_j, \quad j=1,2, \ldots, n .$$

$$\zeta=c_1 x_1+c_2 x_2+\cdots+c_n x_n .$$

$$a_1 x_1+a_2 x_2+\cdots+a_n x_n\left{\begin{array}{l} \leq \ = \ \geq \end{array}\right} b .$$

## MATLAB代写

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