### 统计代写|决策与风险作业代写decision and risk代考|Output Analysis for Transient Simulations

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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
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|决策与风险作业代写decision and risk代考|A Simple Inventory Model

The simple model that we introduce below is based on the model proposed in Muñoz and Muñoz (2011) to forecast the demand of items with sporadic demand, and is inspired in the ideas of Kalchsmidth et al. (2006), where they suggest the use of forecasting techniques that take into account not only the time series, but also the structure of the process that generates the demand (non-systematic variability). In what follows we will refer to this model as Model $1 .$

Let us suppose that a seller uses a $(Q, R$ ) policy (see, e.g., Nahmias 2013 ) to order the supply of a certain product, i.e., when the inventory level reaches the reorder point $(R)$, a quantity $Q$ is ordered. If an order is placed at time $t=0$ and $L$ is the delay, then the demand for the product during the delay is
$$W_{1}=\left{\begin{array}{cc} \sum_{i=1}^{N(L)} U_{i}, & N(L)>0 \ 0, & \text { otherwise } \end{array}\right.$$
where, for $t \geq 0, N(t)$ is the number of clients that arrived up to time t, and for $i=1,2, \ldots, U_{i}$ is the demand for client $i$. We assume that $U_{1}, U_{2}, \ldots$ are i.i.d. random variables and are also independent of the stochastic process ${N(t): t \geq 0}$. In order to obtain analytical results for some of our performance measures, we also assume that ${N(t): t \geq 0}$ is a Poisson process with rate $\Theta_{0}$.

Under our assumptions, we can define the following parameters that represent important properties for the policy $(Q, R)$ (for details on the derivation of the analytical expressions see Muñoz et al. 2013).

The expected demand is an important measure to forecast demand $W_{1}$, and is defined by
$$\mu_{W}=E\left[W_{1}\right]=\Theta_{0} L \mu_{U}$$
where $E[X]$ denotes the expected value of a random variable $X$, and $\mu_{U}=E\left[U_{1}\right]$.
The variance of demand $W_{1}$ is an important measure for the magnitude of the uncertainty on the forecast of demand $W_{1}$, and is defined by
$$\sigma_{W}^{2}=E\left[W_{1}^{2}\right]-E\left[W_{1}\right]^{2}=\Theta_{0} L\left(\mu_{U}^{2}+\sigma_{U}^{2}\right)$$
where $\mu_{U}^{2}=E\left[U_{1}\right]^{2}, \sigma_{U}^{2}=E\left[U_{1}^{2}\right]-\mu_{U^{*}}^{2}$
Given a value $R$ for the reorder point, an important risk measure is the probability of no stockout (called type-l service level), and is defined by
$$\alpha_{1}(R)=P\left[W_{1} \leq R\right]=E\left[I\left(W_{1} \leq R\right)\right]$$
where, for any event $A, I(A)$ denotes the indicator random variable that takes a value of 1 if event $A$ occurs and zero otherwise.

Given a value $0<\alpha<1$, the type-1 reorder point is a value for the reorder point that provides an approximate type-1 service level of $\alpha$, and is defined by
$$r_{1}(\alpha)=\inf \left{R \geq 0: \alpha_{1}(R) \geq \alpha\right}$$
where $\alpha_{1}(R)$ is defined in (6.5).
Similarly, given a value $R$ for the reorder point, another important risk measure is the proportion of demand that is met from stock (called type-2 service level or fill

rate), and is defined by
$$\alpha_{2}(R)=1-\frac{E\left[\left(W_{1}-R\right) I\left(W_{1}>R\right)\right]}{Q}$$
and given a value $0<\alpha<1$, the type-2 reorder point is a value for the reorder point that provides an approximate type- 2 service level of $\alpha$, and is defined by
$$r_{2}(\alpha)=\inf \left{R \geq 0: \alpha_{2}(R) \geq \alpha\right}$$
where $\alpha_{2}(R)$ is defined in $(6.7)$.
In the next sections, we show how to estimate the measures of risk defined by Eqs. (6.3) through (6.6) from the output of a simulation, and remark that Model 1 , as defined in (6.1), is a very simple model just to verify (using simulation) that we are proposing valid estimation procedures that may be applied to a complex model, for which we would not have analytical solutions.

## 统计代写|决策与风险作业代写decision and risk代考|Properties of a Good Estimator

In order to discuss the main properties that a good simulation-based estimator must satisfy, we use the concept of weak converge of random variables. We say that a sequence of random variables $X_{1}, X_{2}, \ldots$ converge weakly to a random variable $X$ (and denote $X_{m} \Rightarrow X$, as $m \rightarrow \infty$ ), if $\lim {m \rightarrow \infty} F{X_{m}}(x)=F_{X}(x)$, at any point $x$ where $F$ is continuous, where $F_{X_{m}}$ and $F_{X}$ denote the c.d.f. of $X_{m}$ and $X$, respectively (see, e.g., Chung 2000 ). Note that $X$ can also be a constant $(M)$, on which case $X$ is simply the random variable that takes the value of $M$ with probability 1 .

A first property that a good estimator must satisfy is consistency. We say that the estimator $T\left(F_{m}\right)$, where $F_{m}$ is defined in (6.1) is consistent if
$$T\left(F_{m}\right) \Rightarrow T(F)$$
as $m \rightarrow \infty$. Note that consistency means that the estimator $T\left(F_{m}\right)$ approaches the parameter $T(F)$ as the sample size $m$ increases, and this is a required property since we do not want the estimator to converge to a different value (or not to converge at all).

In order to assess the accuracy of a consistent estimator $T\left(F_{m}\right)$, we usually verify if a Central Limit Theorem (CLT) in the form of
$$\frac{\sqrt{m}\left(T\left(F_{m}\right)-T(F)\right)}{\sigma_{T}} \Rightarrow N(0,1)$$
as $m \rightarrow \infty$ is satisfied, on which case we may also look for a consistent estimator $\hat{\sigma}^{2}(m)$ for the asymptotic variance $\sigma_{T}^{2}$, so that if follows from $(6.10)$ and standard

properties of weak convergence (see, e.g., Serfling 2009) that
$$\frac{\sqrt{m}\left(T\left(F_{m}\right)-T(F)\right)}{\hat{\sigma}(m)} \Rightarrow N(0,1),$$
as $m \rightarrow \infty$, where $N(0,1)$ denotes a random variable distributed as Normal with mean 0 and variance 1 . It is worth mentioning that the CLT of (6.10) implies consistency of $T\left(F_{m}\right)$, as defined in (6.9).

A CLT in the form of $(6.11)$ is sufficient to assess the accuracy of the point estimator $T\left(F_{m}\right)$, since (6.11) implies that
$$\lim {m \rightarrow \infty} P\left[\left|T\left(F{m}\right)-T(F)\right| \leq z_{\beta} \frac{\hat{\sigma}(m)}{\sqrt{m}}\right]=1-\beta$$
where $z_{\beta}$ denotes the $(1-\beta / 2)$-quantile of a $N(0,1)$, which is sufficient to establish an asymptotically valid $(1-\beta) 100 \%$ confidence interval (CI) for $T(F)$ with halfwidth
$$H W_{T}=\frac{t_{(m-1, \beta)} \hat{\sigma}(m)}{\sqrt{m}}$$
where $t_{(m-1, \beta)}$ denotes the $(1-\beta / 2)$-quantile of a Student-t distribution with $(\mathrm{m}-$ 1) degrees of freedom. A halfwidth in the form of $(6.12)$ is the typical measure used in simulation software to assess the accuracy of $T\left(F_{m}\right)$ for the estimation of a parameter $T(F)$. Note that we are using a Student-t distribution to have a wider CI when the value of $m$ is small, and the CI is still asymptotically valid since $t_{(m-1, \beta)} \rightarrow z_{\beta}$, as $m \rightarrow \infty$. Note also from (6.12) that, to lower the value of a halfwidth (i.e., improve the accuracy of $T\left(F_{m}\right)$ ), we need to increase the sample size $m$, so that the halfwidth will be reduced approximately by half if we multiply the sample size $m$ by 4 .

## 统计代写|决策与风险作业代写decision and risk代考|Estimation of Expected Values

For the estimation of the expected value $T(F)=E\left[W_{1}\right]$, the point estimator $T\left(F_{m}\right)$ becomes the sample average
$$\bar{W}(m)=\frac{\sum_{i=1}^{m} W_{i}}{m},$$
and it is well-known from the classical CLT that the CLT (6.10) is satisfied for $T\left(F_{m}\right)=\bar{W}(m)$, and $\sigma_{T}^{2}=E\left[W_{1}^{2}\right]-E\left[W_{1}\right]^{2}$. Moreover, since $W_{1}, W_{2}, \ldots, W_{m}$ are i.i.d., it is well known that a consistent (and unbiased) estimator for $\sigma_{T}^{2}$ is

$$S_{W}^{2}(m)=\frac{\sum_{i=1}^{m}\left(W_{i}-\bar{W}(m)\right)^{2}}{m-1}$$
so that, it follows from (6.11) that an asymptotically valid $(1-\beta) 100 \%$ halfwidth for the expected value $\mu_{W}=E\left[W_{1}\right]$ is given by
$$H W_{\mu_{W}}=\frac{t_{(m-1 . \beta)} S_{W}(m)}{\sqrt{m}},$$
where $S_{W}(m)$ is defined in (6.14).
Thus, Eq. (6.13) can be used to compute a point estimator for the expected value in a transient simulation, and Eq. (6.15) allows us to compute an assessment of the accuracy of the point estimator. Note that Eqs. (6.13) and (6.15) can be applied not only to the estimation of the expected demand (6.3) in Model 1 but also for the estimation of a type-1 service level defined in (6.5) or a type- 2 service level defined in (6.7), since the service levels are also expectations, to be more precise, we can take $W_{1 i}=I\left[W_{i} \leq R\right]$ for parameter $(6.5)$ and $W_{2 i}=1-\left(W_{i}-R\right) I\left[W_{i}>R\right] / Q$ for parameter $(6.7), i=1, \ldots, m$. A $\mathrm{C}++$ code for the estimation of these risk measures using simulation output was compiled to produce a library and below we report numerical examples using this code.

## 统计代写|决策与风险作业代写decision and risk代考|A Simple Inventory Model

$$W_{1}=\left{∑一世=1ñ(大号)ü一世,ñ(大号)>0 0, 除此以外 \对。$$

μ在=和[在1]=θ0大号μü

σ在2=和[在12]−和[在1]2=θ0大号(μü2+σü2)

r_{1}(\alpha)=\inf \left{R \geq 0: \alpha_{1}(R) \geq \alpha\right}r_{1}(\alpha)=\inf \left{R \geq 0: \alpha_{1}(R) \geq \alpha\right}

r_{2}(\alpha)=\inf \left{R \geq 0: \alpha_{2}(R) \geq \alpha\right}r_{2}(\alpha)=\inf \left{R \geq 0: \alpha_{2}(R) \geq \alpha\right}

## 统计代写|决策与风险作业代写decision and risk代考|Properties of a Good Estimator

H在吨=吨(米−1,b)σ^(米)米

## 统计代写|决策与风险作业代写decision and risk代考|Estimation of Expected Values

H在μ在=吨(米−1.b)小号在(米)米,

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