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# 数学代写|运筹学代写Operations Research代考|MATH3202 The $(s, Q)$ Continuous Review Inventory Model

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## 数学代写|运筹学代写Operations 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.

## 数学代写|运筹学代写Operations Research代考|The Probability of Running out of Stock

By the basic result above, it is reasonable to approximate the probability that the system runs out of stock during the lead time by $\mathbb{P}\left(X_L>s\right)$. This leads to the approximation
$$\text { probability of running out of stock during the lead time } \approx \int_s^{\infty} f_L(x) d x$$

Fraction of the Demand Delivered Directly
To calculate this fraction, we use the fact that the stochastic process that describes the course of the economic stock is a so-called regenerative stochastic process. Probabilistically, as it were, this stochastic process starts over every time the economic inventory drops to the level $s$ and is then replenished to $s+Q$. The long-term behavior of the inventory process can be analyzed in terms of the behavior of the process between two consecutive regeneration points. For the regeneration periods, it is more convenient to choose the times when replenishment orders are received rather than the times when replenishment orders are placed. We define a cycle as the time interval between the moments when two consecutive replenishment orders come in. The further analysis is based on the following basic formula from the theory of regenerative stochastic processes: the long-term fraction of the demand that is not delivered directly from stock is equal to the following constant with probability 1:
$\mathbb{E}[$ demand per cycle that cannot be delivered directly]
$\mathbb{E}$ [total demand in a cycle]

## 数学代写|运筹学代写Operations Research代考|The $(s, Q)$ 持续审查 库存模刑

• 不断审車经济库存，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 $Q$ 每当经済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。

## 数学代写|运筹学代写Operations Research代考|The Probability of Running out of Stock

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

$\mathbb{E}[$ 无法直接交付的每个周期的需求]
$\mathbb{E}$ 【一个周期的总需求量】

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