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# 数学代写|整数优化代写Integer Programming代考|MGO336 The Linear Integer Programming (LIP) Model and some preliminaries

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## 数学代写|整数优化代写Integer Programming代考|The Linear Integer Programming (LIP) Model and some preliminaries

A standard linear integer programming model is given by: Maximize
$$Z=\sum_j c_j x_j$$
Such that
$$\sum_i \sum_j a_{i j} x_j \leq b_i$$
Where $a_{i j}, b_i, c_j$ are constants, $x_j \geq 0$ and integer.
$$i=1,2, \ldots, m \text { and } j=1,2, \ldots, n$$
Earlier approaches solved the relaxed linear integer model as a LP and modified the LP optimal solution to an IP optimal solution in different ways, see Kumar et al. (2010). Instead of solving the relaxed integer linear model, an alternative possibility is to determine some very useful additional constraints, which can be in the form of:
(i) original variable sum constraint,
(ii) slack and excess variable sum constraint, and
(iii) slack and excess variable limits.
These approaches are discussed below.
The original variable sum constraint can be in the form of equation (1.2).
$$x_1+x_2+\ldots+x_n-\phi_c=0 .$$
Where $\phi_c$ is not necessarily integer.
The excess and or slack variable sum constraint can be in the form of equation (1.3)
$$s_1+s_2+\ldots+s_m-\lambda=0$$
Where $\lambda$ is an integer.

## 数学代写|整数优化代写Integer Programming代考|The concept of segments

From Figure 1.1, the segments are in descending order i.e.
$$Z_0>Z_1>Z_2>Z_3>\ldots>Z_k>Z_{k+1} \text {. }$$

The idea is to search segment 1 , then segment 2 in that order until the optimal solution is found in the $k^{\text {th }}$ segment. It is easier to search a segment at a time than the whole feasible region.
1.2.1 Selection of the segment interval
There is need to select segment intervals so that the sizes of these segments are approximately equal. This may pose a challenge, and in this chapter, we present a technique to select the segment intervals. Suitable shapes that can be used to approximate the intervals are triangle, cone, pyramid etc. In this chapter a triangle shape is selected for its simplicity.
From Figure 1.1, the segments are in descending order i.e.
$$Z_0>Z_1>Z_2>Z_3>\ldots>Z_{k-1}>Z_k>Z_{k+1}$$
The idea is to search segment 1 , then segment 2 in that order until the optimal solution is found. Suppose the optimal solution is obtained in the $k^{\text {th }}$ segment, which becomes the required optimal solution. It is easier to search a segment at a time than the whole feasible region.

## 数学代写|整数优化代写INTEGER PROGRAMMING代考|THE LINEAR INTEGER PROGRAMMING $L I P$

$$Z=\sum_j c_j x_j$$

$$\sum_i \sum_j a_{i j} x_j \leq b_i$$

$$i=1,2, \ldots, m \text { and } j=1,2, \ldots, n$$

$i$ 原始变量总和约束，
$i i$ 松弛和过剩变量总和约束，以及
$i i i$ 松弛和过度的可变限制。

$$x_1+x_2+\ldots+x_n-\phi_c=0 .$$

$$s_1+s_2+\ldots+s_m-\lambda=0$$

## 数学代写|整数优化代写INTEGER PROGRAMMING代考|THE CONCEPT OF SEGMENTS

$$Z_0>Z_1>Z_2>Z_3>\ldots>Z_k>Z_{k+1} .$$

1.2.1段间隔

$$Z_0>Z_1>Z_2>Z_3>\ldots>Z_{k-1}>Z_k>Z_{k+1}$$

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