统计代写|运筹学作业代写operational research代考|Integer Linear Programming

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|运筹学作业代写operational research代考|Integer Linear Programming

Integer linear programming, or integer programming (IP), has been widely adopted as a method of modeling because some variables are not continuous but are integers in many cases in real life. Actually, IP is a subset of LP, with an additional constraint that some or all decision variables are restricted to integral values, depending on the type of IP. The general maximization-type IP model can be formulated as shown in Model 1.2.1.

Model1.2.1 Standard maximization-type integer linear programming model Maximize $z=\sum_{j=1}^{n} c_{j} x_{j}$
subject to
$$\sum_{j=1}^{n} a_{i j} x_{i} \leq b_{i} \quad \text { for all } i$$
All $x_{j} \geq 0$ and integer

The general minimization-type IP model can be formulated as shown in Model 1.2.2.

Model 1.2.2 Standard minimization-type integer linear programming model
Minimize $z=\sum_{j=1}^{n} c_{j} x_{j}$
subject to
\begin{aligned} &\sum_{j=1}^{n} a_{i j} x_{j} \leq b_{i} \quad \text { for all } i \ &\sum_{j=1}^{n} a_{i j} x_{j}=b_{i} \quad \text { for all } i \end{aligned}
All $x_{j} \geq 0$ and $x_{1}$ integer
Models 1.2.1 and 1.2.2 are almost the same as Models 1.1.1 and 1.1.2, respectively, except that there are integrality requirements in Models $1.2 .1$ and 1.2.2. Generally, there are three types of IP:

1. Pure integer linear programming is used if all variables must be integral, as is the case with Model 1.2.1.
2. Mixed integer linear programming (MILP) is used if only some of the variables must be integers, as is the case with Model 1.2.2.
3. Binary integer linear programming is used if all the variables must be either 0 or 1 .
Unlike LP with the simplex method, a good IP algorithm for a very wide class of IP problems has not been developed (Williams 1999). Different algorithms are good with different types of problem. Generally, IP algorithms are based on exploiting the tremendous computational success of LP. Thus, before applying an IP algorithm, the integer restriction on the problem should be relaxed first to form an LP model. Starting from the continuous optimum point obtained from the LP model, integer constraints are incorporated repeatedly to modify the LP solution space in a manner that will eventually render the optimum extreme point, satisfying the integer requirements.

统计代写|运筹学作业代写operational research代考|Branch-and-Bound Algorithm

In practice, the branch-and-bound $(B \mathcal{E} B)$ algorithm is widely used for solving IP models, especially MILP models (Williams 1999). The idea of the B\&B

algorithm is to perform the enumeration efficiently so that not all combinations of decision variables must be examined. Sometimes, the terms implicit enumeration, tree search, and strategic partitioning are used, depending on the implementation of the algorithm (Jensen and Bard 2003).

The B\&B algorithm starts with solving an IP model as an LP model by relaxing the integrality conditions. In cases in which the resultant LP solution or the continuous optimum is an integer, this solution will also be the integer optimum. Otherwise, the B\&B algorithm sets up lower and upper bounds for the optimal solution. The branching strategy repetitively decreases the upper bound and increases the lower bound. The process terminates, provided that the processing list is empty (Castillo et al. 2002).

统计代写|运筹学作业代写operational research代考|Goal Programming

Model 1.3.1 Standard goal programming model
$$\text { Minimize } z=\sum_{i}\left(d_{i}^{+}+d_{i}^{-}\right)$$
subject to
$$\begin{gathered} \sum_{j} a_{i \pi}^{i j} x_{j} \leq b_{i} \quad \text { for all } i \ \sum_{j} a_{i j} x_{j}-d_{i}^{+}+d_{i}^{-}=b_{i} \quad \text { for all } i \end{gathered}$$
All $x_{j}=0$ or $1 ; d_{i}^{+}$and $d_{i}^{-} \geq 0$
In this GP model, $a_{i j}$ is the coefficient, whereas $b_{i}$ is the right-side value. $d_{i}^{+}$and $d_{i}^{-}$are overachievement and underachievement of goal $i$, respectively. The decision variable of the GP model is denoted as $x_{\dot{r}}$ Objective function 1.3.1 minimizes the total deviations from the goals, while subject to system constraint set 1.3.2 and resource constraint set 1.3.3. Because all the objective function and constraint sets are in the linear form, it belongs to the LP type. In addition, decision variables (i.e., $x_{j}$ ) are binary, and deviation variables (i.e., $d_{i}^{+}$and $d_{i}^{-}$) are continuous. Therefore, it is regarded as the mixed IP model. In the next two sections, two algorithms for solving GP models are discussed, the weights method and the preemptive method. The common point of both methods is that they convert multiple goals into a single objective function.

Goal programming (GP), invented by Charnes and Cooper (1961), is very similar to the LP model except that multiple goals are considered at the same time. Deviation variables (i.e., $d_{1}^{+}, d_{1}^{-}, d_{2}^{+}, d_{2}^{-}, \ldots, d_{n}^{+}, d_{n}^{-}$) are included in each goal equation to represent the possible deviations from goals. Deviation variables with positive signs refer to overachievement, which means that deviations are greater than the target value; those with negative signs indicate underachievement, which means that deviations are less than the target value. The objective function of a GP is to minimize deviations from desired goals. For each goal, there are three possible alternatives of incorporating deviation variables in the objective function. If both overachievement and underachievement of a goal are not desirable, then both $d_{i}^{+}$and $d_{i}^{-}$are included in the objective function. If overachievement of a goal is regarded as unsatisfactory, then only $d_{i}^{+}$is included in the objective function. If underachievement of a goal is regarded as unsatisfactory, then only $d_{i}^{-}$is included in the objective function. The general GP model in the form of MILP can be formulated as shown in Model 1.3.1.

统计代写|运筹学作业代写operational research代考|Integer Linear Programming

Model1.2.1 标准最大化型整数线性规划模型Maximize和=∑j=1nCjXj

∑j=1n一种一世jX一世≤b一世 对全部 一世

Minimize和=∑j=1nCjXj

∑j=1n一种一世jXj≤b一世 对全部 一世 ∑j=1n一种一世jXj=b一世 对全部 一世

1. 如果所有变量都必须是整数，则使用纯整数线性规划，如模型 1.2.1 的情况。
2. 如果只有一些变量必须是整数，则使用混合整数线性规划 (MILP)，如模型 1.2.2 的情况。
3. 如果所有变量必须为 0 或 1 ，则使用二进制整数线性规划。
与采用单纯形法的 LP 不同，尚未开发出针对非常广泛的 IP 问题的良好 IP 算法（Williams 1999）。不同的算法适用于不同类型的问题。通常，IP 算法基于利用 LP 的巨大计算成功。因此，在应用 IP 算法之前，应首先放宽对问题的整数限制，形成 LP 模型。从LP模型得到的连续最优点出发，反复加入整数约束，以最终呈现最优极值的方式修改LP解空间，满足整数要求。

统计代写|运筹学作业代写operational research代考|Branch-and-Bound Algorithm

B\&B 算法首先通过放宽完整性条件将 IP 模型求解为 LP 模型。在得到的 LP 解或连续最优解是整数的情况下，该解也将是整数最优解。否则，B\&B 算法为最优解设置下限和上限。分支策略反复降低上限并增加下限。如果处理列表为空，则该过程终止（Castillo 等人，2002）。

统计代写|运筹学作业代写operational research代考|Goal Programming

最小化 和=∑一世(d一世++d一世−)

∑j一种一世圆周率一世jXj≤b一世 对全部 一世 ∑j一种一世jXj−d一世++d一世−=b一世 对全部 一世

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