# 数学代写|运筹学作业代写operational research代考|MATH4202

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## 数学代写|运筹学作业代写operational research代考|The (R, S) Periodic Review Inventory Model

In the $(s, Q)$ inventory model from Sections 6.4 .1 and 6.4 .2 , a replenishment order could be placed at any time. In this subsection and the next, we consider two important inventory policies for stochastic inventory models in which the inventory can only be replenished periodically (for example, at the beginning of every week). In the $(R, S)$ inventory model, the stock is replenished at every inventory review. The assumptions for this inventory model are as follows:

• The inventory position is reviewed every $R$ period, where $R$ is a given positive integer.
• At every inventory review, the economic inventory is replenished to the level $S$, where $S$ is positive.
• The lead time of a replenishment order is $L$ periods, where $L$ is a given nonnegative integer.
• The sizes of the demand for the product in time periods $t=1,2, \ldots$ are independent random variables that have the same probability density $f_1(x)$ with expected value $\mu_1$ and standard deviation $\sigma_1$.

In the analysis of the $(R, S)$ periodic review model, we restrict ourselves to determining the long-term fraction of the demand that is delivered directly from stock. The analysis runs parallel to the analysis for the $(s, Q)$ continuous review model. First, we introduce some notation. We define the random variable $T_k$ as
$T_k=$ total demand in $k$ consecutive periods.
We denote the probability density of $T_k$ by $f_k(x)$ and the expected value and standard deviation of $T_k$ by $\mu_k$ and $\sigma_k$. As a result of the assumption that the sizes of the demand are independent, we have
$$\mu_k=k \mu_1 \text { and } \sigma_k=\sqrt{k} \sigma_1 \text { for } k \geq 1$$

## 数学代写|运筹学作业代写operational research代考|The (R, s, S) Inventory Model

The difference between the $(R, s, S)$ model and the $(R, S)$ model is that a replenishment order is not necessarily placed at every inventory review. In the $(R, s, S)$ inventory model, a replenishment order to bring the economic inventory up to $S$ is only placed if the economic inventory is less than $s$ at inventory review; otherwise, nothing is ordered. We assume $0 \leq s<S$. In real-world applications, the value of $S-s$ is often based on cost considerations (for example, $S-s$ is chosen according to the EOQ formula). Now, suppose that $S-s$ is given and that the aim is to choose the reorder point $s$ such that the fraction of demand delivered directly from stock is at least $\beta$ for a given value of $\beta$. In case $S-s \geq 1.5 \mu_R$ and $\beta \geq 0.9$, a simple approximation formula for the reorder point $s$ can be given if the demand per period is normally distributed. In this situation, one can show that for the back-order model, the reorder point $s$ can be approximated by
$$s=\mu_{R+L}+k \sigma_{R+L}$$
where $k$ is the solution to the equation
$$\sigma_{R+L}^2 J(k)=(1-\beta) 2 \mu_R\left{S-s+\frac{\sigma_R^2+\mu_R^2}{2 \mu_R}\right} .$$
Here, $J(k)=(1 / \sqrt{2 \pi}) \int_k^{\infty}(x-k)^2 e^{-\frac{1}{2} x^2} d x$. This function is called the normal quadratic loss function. ${ }^4$ For the inventory model with lost sales, equation (6.25) is adjusted by replacing $1-\beta$ with $(1-\beta) / \beta$. In the derivation of the heuristic solution, an important role is played by the probability distribution of the quantity by which the economic inventory is lower than $s$ when a replenishment order is necessary. Assuming that $S-s$ is sufficiently large in relation to $\mu_R$, we have that a good approximation to this probability distribution does not depend on $S-s$ and is given by the famous balance distribution $\frac{1}{\mu_R} \int_0^x\left(1-F_R(y)\right) d y$ from renewal theory, where $F_R(x)$ is the probability distribution function of the demand in the time between two inventory reviews.

In the case that the demand per period is not normally distributed but we do have $\sigma_{R+L} / \mu_{R+L} \leq 0.5$, the requested reorder point $s$ can also be approximated well by using the equation above with the normal quadratic loss function $J(k)$.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|The (R, S) Periodic Review Inventory Model

• 库存状况每 $R$ 期间，其中 $R$ 是给定的正整数。
• 在每次库存审查时，经济库存被补充到水平 $S$ ， 在哪里 $S$ 是积极的。
• 补货订单的提前期为 $L$ 期间，其中 $L$ 是给定的非负 整数。
• 时间段内对产品的需求大小 $t=1,2, \ldots$. 是具有 相同概率密度的独立随机变量 $f_1(x)$ 具有预期价 值 $\mu_1$ 和标准差 $\sigma_1$.
在分析的 $(R, S)$ 定期审查模型，我们限制自己确定直接 从库存交付的需求的长期部分。该分析与对 $(s, Q)$ 持续 审查模型。首先，我们介绍一些符号。我们定义随机变 量 $T_k$ 作为
$T_k=$ 总需求 $k$ 连续的时期。
我们表示概率密度 $T_k$ 经过 $f_k(x)$ 以及期望值和标准差 $T_k$ 经过 $\mu_k$ 和 $\sigma_k$. 由于假设需求的大小是独立的，我们有
$$\mu_k=k \mu_1 \text { and } \sigma_k=\sqrt{k} \sigma_1 \text { for } k \geq 1$$

## 数学代写|运筹学作业代写operational research代考|The (R, s, S) Inventory Model

$$s=\mu_{R+L}+k \sigma_{R+L}$$

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