小白科研筆記 | Newsvendor Model

08-14

4 人贊了文章

這學期在一門講實證研究的課上看了兩篇文章，剛好最近期末要交這兩篇文章的總結和反思，就順便寫在這裡好了。今天先寫第一篇，主要是基於報童模型（Newsvendor model），通過將限制不斷放鬆得到一個更為柔性和普適的模型。

由於作業要用英文交上去，所以關於文章的回顧都是英文的，不過本人英語水平一般，有不當之處歡迎指正。

Review of Structural Estimation of the Newsvendor Model

1. Summary of the work

1.1 Motivation

For hospitals, they face a dilemma of reserving operating room time. They have to balance the costs of allocating too much time, which typically means idle time and low efficiency, with the costs of allocating too little time, which typically means crash between surgery and overtime costs. This is similar to the newsvendor problem that a newsboy needs to decide how many newspapers that he needs to book in advance for the next day. Therefore, the authors apply newsvendor model to the healthcare system and make it more flexible by adjusting the structure of it.

1.2 Model Construction

1.2.1 Basic newsvendor model

Id like to start with the basic newsvendor model as a foundation. Imagine a scenario where a newsboy needs to determine the optimal amount of inventory $Q^*$ based on the demand distribution $F(D)$ , price $p$ and cost $c$ . There are two ways to find the $Q^*$ .

For the first solution, we can assume that the newsboy has booked $Q-1$ newspapers, then the question is transformed into whether he should order the $Q$ th newspaper. The expected revenue for the $Q$ th newspaper will be $(p-c)Pr(Dgeq Q)$ , which is equal to $(p-c)[1-F(Q)]$ , and the expected cost will be $cPr(Dleq Q)$ , which is equal to $cF(Q)$ . As the value of $Q$ grows, the expected revenue will decrease and the expected cost will increase, which finally leads to a corner case, $(p-c)[1-F(Q)]=cF(Q)$ . Therefore, $F(Q^*)=frac{p-c}{p}$ .

Theres also another way to think about it. The overage cost ( $Dleq Q$ ) for the newsboy is $c_o=c$ , and the underage cost ( $Dgeq Q$ ) is $c_u=p-c$ . Then the objective function is to $min {c_o(Q-D)^++c_u(D-Q)^+ }$ , and the solution is $F(Q^*)=frac{c_u}{c_u+c_o}=frac{1}{1+gamma}, gamma=frac{c_o}{c_u}$ .

1.2.2 Application to reserving operating room time

As stated above, to apply newsvendor model to healthcare system and find the optimal reserving time, we need to know the distribution of demand $D$ and value of $gamma$ . The data available to us contains the reservation decision $Q$ , characteristics of each case, and actual duration for each of the surgery $D$ .

Since the time needed for different surgery varies a lot, we can assume that for surgery type $i$ , the cumulative distribution function of its demand is $F(D_i; heta_i), heta_i=h(x_i,eta)$ . And the function of $gamma$ is $gamma_i=g(z_i,alpha)$ .

1.3 Estimation

1.3.1 Demand specification

In healthcare field, the demand is usually treated as log-normally distributed, that is $ln(D_i)sim N(mu_i, sigma^2), mu_i=eta x_i+varepsilon_i$ , which can be estimated through MLE or OLS.

1.3.2 $gamma$ specification

Since $gamma$ is non-negative, we can assume its function is $log(gamma_i)=alpha z_i+xi_i$ . However, the true value of $gamma$ can never be observed, so we need to find ways to approximate it.

If the decision maker behaves optimally, then $gamma_i=frac{1}{F(Q_i; heta_i)}-1$ (Model N1).

If the decision maker is limited rational, then we can add some noise to the reservation decision, that is, $Q_i=Q_i^*+v_i$ . Then $Q_i^*=F^{-1}left[ frac{1}{1+exp(alpha z_i)}; h(x_i, eta) ight]$ , $Q_i=Q^*(x_i, z_i, hateta, alpha)+v_i$ , which can be estimated through non-linear LS (Model N2).

1.3.3 Forecasting error

However, when comparing the estimation results with extant literature, the authors find there existing a gap due to doctors tendency to overestimate their own ability and underestimate the length of time.

To take this into consideration, authors assume the doctors behave based on their perceived demand distribution, $log(D_i^p)sim N(mu_i +b_i, sigma^2), b_i=varphi w_i+epsilon_i$ . Then the original solution can be transformed into $Phi(frac{ln(Q_i)-(b_i+mu_i)}{sigma})=frac{1}{1+gamma_i}$ , that is, $ln(Q_i)=mu_i+b_i+sigmaPhi^{-1}(frac{1}{1+gamma_i})$ . However, $varphi$ and $alpha$ can not be estimated simultaneously. So they are estimated step by step.

Based on related literatures, we can assign $frac{4}{7}$ to $gamma$ and then estimate $varphi$ (Model B1).

Or we can adjust $b_i$ to make $frac{E(D_i^p)}{E(D_i)}=0.8$ and then estimate $alpha$ (Model B2).

2. Pros and Cons

From my understanding, the contribution of this paper lies in that it generalizes the original newsvendor model and makes it more flexible by successfully accommodating the heterogeneity of demand distribution and $gamma$ .

But when taking the forecasting error into account, Im still curious about the way of just inserting the value from literatures directly. Maybe this is not the best way of doing it especially for some industries these values may still be unknown. But is there a better way or is it even possible to estimate the value of $gamma$ and $alpha$ simultaneously by simulation or some other way?

References

[1] Olivares M, Terwiesch C, Cassorla L. Structural estimation of the newsvendor model: an application to reserving operating room time[J]. Management Science, 2008, 54(1): 41-55.