Posted on Categories:Operations Research, 数学代写, 运筹学

# 数学代写|运筹学代写Operations Research代考|STAT360 Model Formulation

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## 数学代写|运筹学代写Operations Research代考|Model Formulation

We consider a firm that receives recoverable product from the market. The firm can manufacture new products and recover the value of a used product or return through remanufacturing with dismantling for parts. The firm provides product at a constant demand rate of $d$ items per time unit. Product consists of two parts, denoted as part 1 and part 2. Each part is manufactured separately and placed in inventory (SS1-serviceable stock inventory for part 1, SS2 – serviceable stock inventory for part 2), then two parts are assembled with the cost $c_A$ and are sold in a market. Products are returned to the firm according the rate $\beta$, other products are immediately disposed of at the rate $\alpha=1-\beta$. The dismantling operation costs $c_D$. Returned product is dismantled for parts, any part is inspected whether it is usable or not, and then is placed in inventory (RS1-inventory for returned stock of part 1 , RS2-inventory for returned stock of part 2). Part 1 is not usable at the rate $q_1$ and should be remanufactured, the rest $\beta_1-q_1$ are as good as new and directly reused, part 2 isn’t usable at the rate $q_2$. Figure 1 represents the integrated closed-loop supply chain inventory system. The sequence of production activities is the following: in any time cycle $[0, T]$ demand for part 1 and part 2 is satisfied firstly through usable parts, then through remanufacturing of used parts and at last manufacturing of new parts. All activities in the model are supposed to be instantaneous and lot-for-lot. The production activities of each part are evaluated on separate production lines (Fig. 2).
Assumptions
This paper assumes:
(1) production and recovery are instantaneous,
(2) remanufactured items are as good as new,
(3) demand is known, constant and independent,
(5) the product consists of two parts
(6) no shortages are allowed,
(7) unlimited storage, and
(8) infinite planning horizon.

## 数学代写|运筹学代写Operations Research代考|Solution of the Model

Instead of solving the problem (9) the function $L(m, n)$ can be minimized subject to $m_j \geq 1, n_i \geq 1$, i.e., the following two-dimensional nonlinear integer optimization problem is relevant
$$\begin{gathered} \min {(m, n)} L(m, n)=\min {(m, n)}\left(P+\sum_{j=1}^l R_j m_j+\sum_{i=1}^k S_i n_i\right) \cdot\left(h_1+\sum_{j=1}^l \frac{h_2^j}{m_j}+\sum_{i=1}^k \frac{h_3^i}{n_i}\right), \ m=\left(m_1, m_2, \ldots, m_l\right), n=\left(n_1, n_2, \ldots, n_k\right) \ m_j, n_i \in{1,2, \ldots} \end{gathered}$$
For the solution of the problem (10), consider the following two-dimensional nonlinear integer optimization problem
\begin{aligned} & \min {\left(x_1, x_2, \ldots, x_n\right)} K\left(x_1, x_2, \ldots, x_n\right)=\min {\left(x_1, x_2, \ldots, x_n\right)}\left(b_0+\sum_{i=1}^i b_i x_i\right) \cdot\left(a_0+\sum_{i=1}^n \frac{a_i}{x_i}\right), \ & x_i \in{1,2, \ldots}, i=1,2, \ldots n . \end{aligned}
First, let us consider the following continuous auxiliary problem:
\begin{aligned} & \min {\left(x_1, x_2, \ldots, x_n\right)} K\left(x_1, x_2, \ldots, x_n\right)=\min {\left(x_1, x_2, \ldots, x_n\right)}\left(b_0+\sum_{i=1}^i b_i x_i\right) \cdot\left(a_0+\sum_{i=1}^n \frac{a_i}{x_i}\right) \ & x_i \geq 1, i=1,2, \ldots, n \end{aligned}

## 数学代可|运营管理学代写运营研究代考|模型的提出

(1) 生产和回收是瞬时的。
(2)再制造的物品和新的一样好。
(3) 需求是已知的、恒定的和独立的。
(4) 准备时间为零。
(5) 产品由两部分组成
(6)不允许出现短缺。
(7) 无限储存，以及
(8)无限的计划范围。

## 数学代写|运筹学代写运营研究代考|模型的解决方法

$$\L(m, n)=\min (m, n)\left(P+sum_{j=1}^l R_j m_j+\sum_{i=1}^k S_i n_i\right) \cdot\left(h_1+sum_{j=1}^l\frac{h_2^j}{m_j}+\sum_{i=1}^k \frac{h_3^i}{n_i}\right) 。m=left(m_1, m_2, \ldots, m_l\right), n=left(n_1, n_2, \ldots, n_k\right) m_j, n_i\in 1,2, .$$

$$K\left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)\left(b_0+sum_{i=1}^i b_i x_i\right) cdot\left(a_0+sum_{i=1}^n frac{a_i}{x_i}\right), \quad x_i \in 1,2, \ldots, i=1,2, \ldots n 。$$

$$K\left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)=\min \left(x_1, x_2, \ldots, x_n\right)\left(b_0+sum_{i=1}^i b_i x_i\right) cdot\left(a_0+sum_{i=1}^n \frac{a_i}{x_i}\right) \quad x_i \geq 1, i=1, 2, \ldots, n$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。