数学代写|运筹学作业代写operational research代考|MATH2730

2023年3月30日

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数学代写|运筹学作业代写operational research代考|Stochastic Inventory Models

In this section, we consider stochastic inventory models in which the stock must be managed over a very long period of time. We assume that the model’s parameters do not change significantly during this time period. We consider the situation with stochastic demand and positive lead times. In this situation, it is not possible to prevent the product from being sold out. A shortage occurs when the demand during the lead time exceeds the stock on the shelves at the reorder time. The two basic questions we want to answer are:

1. When should one order? (reorder point)
2. How much should one order? (order quantity)

The reorder point is expressed in units of product. If the inventory level has dropped to this reorder point, new stock is ordered. An important term is that of safety stock. The safety stock level determines the service delivered to the customers. In the situation where the inventory can be replenished at any time, the safety stock level is the difference between the reorder point and the expected demand during the lead time (in stochastic inventory models with periodic inventory review, the concept of safety stock is slightly more subtle). An increase in the safety stock level decreases the probability of the product being sold out but increases the average inventory level. In inventory management, one typically seeks a balance between the service to customers and the holding and ordering costs.

In the following subsections, we discuss, among other things, the $(s, Q)$ inventory model with continuous inventory review and the $(R, S)$ inventory model with periodic inventory review. Before we discuss these inventory models, which are widely used in practice, we introduce a number of basic concepts. In stochastic inventory models, one needs to indicate what happens to the demand that occurs while the system is out of stock. We distinguish two cases:

Back-ordering. The demand that occurs when the system is out of stock is delivered at a later time when sufficient stock is available again.
No back-ordering. The demand that occurs when the system is out of stock is lost.

数学代写|运筹学作业代写operational research代考|The (s, Q) Continuous Review Inventory Model

The widely used $(s, Q)$ inventory model is applicable in the following situation:

• The economic inventory is continuously reviewed and inventory can be reordered at any time.
• The individual demand transactions are so small that the inventory level can be seen as a continuous variable.
• A replenishment order of quantity $Q$ is placed whenever the economic inventory decreases to the reorder point $s$.
• The lead time of an order is a positive constant $L .^2$
• The demanded quantities in disjoint time intervals can be treated as independent random variables.

These assumptions represent, in one way or another, approximations of reality. Nevertheless, the model and the heuristic solution have proved to be extremely useful in practice. In the heuristic analysis, it is not necessary to specify the stochastic demand process completely; instead, it is sufficient to know the probability distribution of the random variable
$X_L=$ total demand during the lead time.
We introduce the following notation:
$f_L(x)=$ probability density of the demand during the lead time,
$\mu_L=$ expected value of the demand during the lead time,
$\sigma_L=$ standard deviation of the demand during the lead time.
In fact, we will only need $\mu_L$ and $\sigma_L$ for the heuristic solution. In practice, $\mu_L$ and $\sigma_L$ are calculated based on collected data on the demand. Suppose that $\mu_1$ and $\sigma_1$ are the expected value and standard deviation of the demand during one week. If the lead time is fixed and equal to $L$ weeks, then we have
$$\mu_L=L \mu_1 \quad \text { and } \quad \sigma_L=\sqrt{L} \sigma_1 \text {. }$$

运筹学代考

数学代写|运筹学作业代写operational research代考|Stochastic Inventory Models

1. 应该什么时候订购？（订货点）
2. 一个应该订多少？（订单数量）

数学代写|运筹学作业代写operational research代考|The (s, Q) Continuous Review Inventory Model

• 不断审查经济库存，可以随时重新订购库存。
• 单个需求交易非常小，库存水平可以看作是一个 连续变量。
• 数量补货单 $Q$ 每当经济库存减少到再订货点时放 置s.
• 订单的提前期是一个正常数 $L .^2$
• 不相交时间间隔内的需求量可以视为独立的随机 变量。
这些假设以某种方式代表现实的近似值。尽管如此，该 模型和启发式解决方案已被证明在实践中非常有用。在 启发式分析中，不需要完全指定随机需求过程; 相反， 知道随机变量的概率分布就足够了 $X_L=$ 交货期内的总需求。
我们引入以下符号:
$f_L(x)=$ 提前期内需求的概率密度，
$\mu_L=$ 提前期内需求的预期值，
$\sigma_L=$ 提前期内需求的标准偏差。
其实我们只需要 $\mu_L$ 和 $\sigma_L$ 对于启发式解决方案。在实践 中， $\mu_L$ 和 $\sigma_L$ 是根据收集到的需求数据计算得出的。假 设 $\mu_1$ 和 $\sigma_1$ 是一周内需求的期望值和标准差。如果提前期 是固定的并且等于 $L$ 周，那么我们有
$$\mu_L=L \mu_1 \quad \text { and } \quad \sigma_L=\sqrt{L} \sigma_1$$

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