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In the above, we have always assumed that the supplier’s lead time is deterministic. A closer look at the heuristic analysis makes it clear that the analysis also holds if the lead time is stochastic, provided that the probability that two or more outstanding replenishment orders overlap is negligible. The results found earlier then also apply to the case of a stochastic lead time. In particular, the formulas (6.17), (6.18), and (6.20) remain valid when $\mu_L$ and $\sigma_L$ are taken equal to
$$\mu_L=\mathbb{E}[L] \mu_1 \quad \text { and } \quad \sigma_L=\sqrt{\mathbb{E}[L] \sigma_1^2+\sigma^2(L) \mu_1^2}$$
where $\mathbb{E}[L]$ and $\sigma(L)$ are the expected value and the standard deviation of the stochastic lead time $L$. Variability in the lead time has a great influence on the safety stock. Smaller safety stocks are sufficient if the variability of the supplier’s lead time is reduced. As an illustration, consider, once again, the bicycle wholesaler. Suppose that the factory’s lead time is not exactly 3 weeks, but 2 or 4 weeks, each with probability $\frac{1}{2}$. What is the reorder point $s$ subject to the service requirement of $99 \%$ of the demand being delivered directly from stock? For $\mu_L$ and $\sigma_L$, we find the values (the unit of time is one year)
$$\mu_L=\frac{3}{52} \times 2600=150, \sigma_L=\sqrt{\frac{3}{52} \times 40000+\frac{1}{(52)^2} \times(2600)^2}=52.257$$
Next, the safety factor $k$ is solved from $52.257 \times I(k)=0.01 \times 198$. This gives $k=2.288$, so $s=150+2.288 \times 52.257 \approx 277$. The reorder point $s$ therefore increases from 215 to 270 if the lead time is not 3 weeks but 2 or 4 weeks, each with probability $\frac{1}{2}$. The safety stock increases by 55 bikes, and therefore nearly doubles.

## 数学代写|运筹学代写Operations 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
$$\text { probability of running out of stock during the lead time } \approx \int_s^{\infty} f_L(x) d x .$$
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
$$\mathbb{E}[\text { amount of lost sales per cycle }] \approx \mathbb{E}\left[\left(X_L-s\right)^{+}\right] .$$
To obtain the numerator of $(6.12)$, we note that
\begin{aligned} \mathbb{E}[\text { total demand per cycle }]= & \mathbb{E}[\text { amount of lost sales per cycle }] \ & +\mathbb{E}[\text { amount of delivered sales per cycle }] . \end{aligned}

## 运筹学代写

$$\mu_L=\mathbb{E}[L] \mu_1 \quad \text { and } \quad \sigma_L=\sqrt{\mathbb{E}[L] \sigma_1^2+\sigma^2(L) \mu_1^2}$$

$$\mu_L=\frac{3}{52} \times 2600=150, \sigma_L=\sqrt{\frac{3}{52} \times 40000+\frac{1}{(52)^2} \times(2600)^2}=52.257$$

## 数学代写|运筹学代写Operations Research代考|销售损失的模型

$\left(s-X_L\right)^{+}$作为补货到达之前净库存的近似值，特别是如果我们假设 $s$ 和 $Q$ 这样就不会经常发生销

probability of running out of stock during the lead time $\approx \int_s^{\infty} f_L(x) d x$.

$\mathbb{E}[$ amount of lost sales per cycle $] \approx \mathbb{E}\left[\left(X_L-s\right)^{+}\right]$.

$\mathbb{E}[$ total demand per cycle $]=\mathbb{E}[$ amount of lost sales per cycle $] \quad+\mathbb{E}[$ amount of delivered sales per cycle $]$.

## 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。