# 数学代写|运筹学作业代写operational research代考|KMA355

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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
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## 数学代写|运筹学作业代写operational research代考|Cost Minimization With Penalty Costs

Suppose that instead of a service requirement, penalty costs are considered for demand that cannot be delivered directly from stock. Assume that in addition to the fixed ordering cost $K$ for a replenishment order and holding cost $h=v r$ per unit of inventory per time unit, a penalty cost of $b$ is incurred for each unit of product that is delivered late. A heuristic solution for how to choose $s$ and $Q$ to minimize the total cost is as follows:

Step 1. Determine the order quantity $Q$ from the $\mathrm{EOQ}$ formula $Q^=\sqrt{\frac{2 K \mu_1}{v r}}$, where $\mu_1$ is the average demand per unit of time. Step 2. Determine the reorder point $s$ using $$F_L(s)=1-\frac{v r Q^}{b \mu_1}$$
assuming $\operatorname{vr} Q^* / b \mu_1<1$, where $F_L(x)$ is the probability distribution function of the total demand during the lead time. If the demand during the lead time is normally distributed, use $s=\mu_L+k \sigma_L$ to simplify this formula to
$$\Phi(k)=1-\frac{v r Q^*}{b \mu_1}$$
In a way similar to that done for the news vendor problem, the formula for the reorder point $s$ can be found using marginal analysis. Suppose that for a given value of the order quantity $Q$, the reorder point increases from $s$ to $s+\Delta$ with $\Delta$ small. The average increase of the holding cost per unit of time is then approximately equal to $v r \Delta$. What is the average saving on the penalty cost per unit of time? In every cycle in which the total demand during the lead time is greater than $s$, approximately $b \Delta$ is saved in penalty costs. The fraction of the cycle for which the demand during the lead time is greater than $s$ is equal to $1-F_L(s)$. The average number of cycles per unit of time is $\mu_1 / Q$ in the back-order model (because the average demand per cycle is $Q$ ). This means that an increase of the reorder point from $s$ to $s+\Delta$ leads to an average decrease in the penalty cost of about
$$\frac{b \Delta \mu_1}{Q}\left[1-F_L(s)\right]$$
per unit of time. As a function of $s$, this decrease itself decreases as $s$ increases and therefore at some point becomes less than the average increase of $v r \Delta$ in the holding cost. This suggests that one should choose the reorder point $s$ according to
$$\frac{b \Delta \mu_1}{Q}\left[1-F_L(s)\right]=v r \Delta$$

## 数学代写|运筹学作业代写operational research代考|The (s, Q) Model with Lost Sales

An exact analysis of the inventory model with lost sales is even more difficult than that of the back-order model. However, the heuristic analysis in the previous subsection requires only minor adjustments for the model with lost sales. The basic result concerning the net inventory right before the replenishment arrives was crucial in the analysis of the back-order model. What is the corresponding result for the model with lost sales? It is tempting to say that the net inventory right before the replenishment arrives is exactly equal to $\left(s-X_L\right)^{+}$. However, this need not hold if other replenishment orders were outstanding when the relevant replenishment order was placed (verify!). Nevertheless, it is reasonable to take $\left(s-X_L\right)^{+}$as an approximation for the net inventory right before a replenishment arrives, especially if we assume that $s$ and $Q$ are such that lost sales do not occur too often.

The probability of running out of stock during the lead time of an order is again approximated by $P\left(X_L>s\right)$, so
$$\text { probability of running out of stock during the lead time } \approx \int_s^{\infty} f_L(x) d x \text {. }$$
A subtler argument is required for the fraction of sales that are lost. The starting point is again formula (6.12). The numerator is approximated by
$\mathbb{E}[$ amount of lost sales per cycle $] \approx \mathbb{E}\left[\left(X_L-s\right)^{+}\right]$.
To obtain the numerator of $(6.12)$, we note that
$\mathbb{E}[$ total demand per cycle $]=\mathbb{E}[$ amount of lost sales per cycle]
$+\mathbb{E}[$ amount of delivered sales per cycle].

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Cost Minimization With Penalty Costs

F_L(s)=1-Ifrac $\left{v \mathrm{v} \mathrm{Q}^{\wedge}\right}\left{b \backslash m \mathrm{~b}_{-} 1\right}$

$$\Phi(k)=1-\frac{v r Q^*}{b \mu_1}$$

$$\frac{b \Delta \mu_1}{Q}\left[1-F_L(s)\right]$$

$$\frac{b \Delta \mu_1}{Q}\left[1-F_L(s)\right]=v r \Delta$$

## 数学代写|运筹学作业代写operational research代考|The (s, Q) Model with Lost Sales

probability of running out of stock during tl
\begin{aligned} & \mathbb{E}[] \approx \mathbb{E}\left[\left(X_L-s\right)^{+}\right] \ & (6.12) \ & \mathbb{E}[]=\mathbb{E}[ \end{aligned}
$+\mathbb{E}[$ 每个周期交付的销售额]。

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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。

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