A new generalized newsvendor model with random demand

Newsvendor problem is one of the most extensively discussed problems in the inventory management literature due to its applicability in different fields (Silver et al., 1998). This inventory problem relies on offsetting the shortage and leftover cost in order to obtain optimal order quantity. In the standard newsvendor problem, each of the shortage and leftover costs are assumed to be proportional to the loss amount, i.e linear in the order quantity and demand. Also, the demand is assumed to be a random variable with known probability distributions. However, there may be situations where the loss is more severe than the usual linear one. Further in real life situations, the demand distribution is often unknown. In this paper, we generalize the classical newsvendor problem with non-linear losses and study the estimation of optimal order quantity subject to random demand with unknown parameters.

In classical newsvendor problem, we consider a newsvendor selling a perishable commodity procured from a single supplier. Further let the demand be a random variable. Newsvendor takes one time decision on how much quantity she should order from supplier. The newsvendor faces leftover cost if the demand is lower than the inventory, as a penalty for ordering too much. Similarly, shortage cost is faced if the demand is higher than the inventory, as a penalty for ordering too less. The shortage (leftover) cost is computed in terms of currency units as the product of per unit shortage cost and the loss amount, i.e difference between demand and inventory. Thus the loss is linear in difference between demand and inventory. To determine the optimal order quantity, the newsvendor has to minimize the total cost or maximize the total profit. This classical version of the newsvendor problem has been generalized, application wise, in many directions since its inception. Veinott Jr (1965) generalized the newsvendor model for multiple time periods. Extension of newsvendor problem for multiple products is being studied extensively in the literature (e.g., see Chernonog and Goldberg, 2018, and the references therein). We refer to a recent review by (Qin et al., 2011) for extensions of newsvendor problem that adds dimensions like marketing efforts, buyer’s risk appetite etc.

In many real life situations, the notion of loss may be higher than the quantity of lost units. For example, chemotherapy drugs are critical for administering it to a patient on the scheduled days. Shortage of the drug of that day would result in breaking the cycle of treatment. Hence the loss is not merely the quantity but more than that. Similarly, in the case of excess inventory, not only the excess amount of the drug but its disposal method also contributes to the notion of loss. This is due to the chance of vast environmental and microbial hazards that may be created through improper disposal, like creation anti-biotic resistant bacteria or super-bugs. Thus the losses due to excess and shortage could be thought of as non-linear.  Newsvendor problem with nonlinear losses, however, remains not much addressed until recently.

Chandra and Mukherjee (2005) have considered optimization of different risk alternatives to expected cost function, like cost volatility and Value-at-Risk, which results in non-linear objective functions. Parlar and Rempala (1992) considered the periodic review inventory problem and derived the solution for a quadratic loss function for both shortage and leftover. Gerchak and Wang (1997) proposed a newsvendor model using power type loss function for asset allocation. Their work assumes asymmetric losses with linear leftover but power type shortage loss with details only up to quadratic loss. In this paper we consider a generalization of the newsvendor problem in the line of Parlar and Rempala (1992) and Gerchak and Wang (1997), i.e. power type losses. In particular, we consider the shortage and excess losses as general power function of same degree. Determination of optimal order quantity from such a generalized newsvendor problem still remains unexplored to the best of our knowledge. Since the power of excess and shortage loss are same, we refer to this problem as symmetric generalized (SyGen) newsvendor problem.

An interesting observation that can be made from the newsvendor literature is that majority of related work considers a completely specified demand distribution, whereas in reality, it is seldom known. In such cases, the optimal order quantity needs to be estimated. Dvoretzky et al. (1952) first addressed the estimation problem in classical newsvendor setup using Wald’s decision theoretic approach. Scarf (1959); Hayes (1969); Fukuda (1960) considered estimation problems in inventory control using maximum likelihood estimation (MLE) under different parametric demand distributions. Conrad (1976) estimated the demand and hence the optimal order quantity for Poisson distribution. Nahmias (1994) estimated the optimal order quantity for Normal demand with unknown parameters and Agrawal and Smith (1996) estimated the order quantity for negative binomial demand. Sok (1980) in his master’s thesis, presented estimators of the optimal order quantity based on order statistics for parametric distributions including uniform and exponential. Rossi et al. (2014) has given bounds on the optimal order quantity using confidence interval for parametric demand distributions.

On the other hand, non-parametric estimation of optimal order quantity in a classical newsvendor problem is comparatively recent. For example, Bookbinder and Lordahl (1989) considered bootstrap estimator of the optimal order quantity and Pal (1996) discussed construction of asymptotic confidence interval of the cost using bootstrapping. Another important non-parametric data-driven approach is the sampling average approximation (SAA) (Kleywegt et al., 2001). In this method, the expected cost is replaced by the sample average of the corresponding objective function and then optimized. Levi et al. (2015) provides bounds of the relative bias of estimated optimal cost using SAA based on full sample data. Bertsimas and Thiele (2005) ranks the objective functions evaluated at each demand sample data and shows that the trimmed mean of the ordered objective functions leads to a convex problem ensuring robust and tractable solution.

In this paper we consider estimation of the optimal order quantity for parametric demand distributions under SyGen newsvendor set-up. In particular, we consider two specific demand distributions, viz. uniform and exponential. We present here the optimal order quantity and its estimators using (1) full data in a random sample and (2) order statistic from a full sample. We investigate the conditions for existence of the estimator of the optimal order quantity. Further, we do a simulation study to gauge the performance of different estimators in terms of bias and mean square error (MSE) for different shortage to excess cost ratio, degree of loss-importance and sample size.

Rest of this paper is organized as follows. Section 2. describes determination of optimal order quantities for uniform and exponential demand distribution using full sample and order statistics, along with the existence condition for the same, wherever required. In section 3, we investigate estimation of the optimal order quantity using full sample and order statistics. Results of simulation study is presented in section 4. Section 5 is the concluding section with discussion on the future problems.

In a single period classical newvendor problem, the vendor has to order the inventory before observing the demand so that the excess and shortage costs are balanced out. Let us denote the demand by a positive random variable $X$ with cumulative distribution function (CDF) $F_{\theta}(x),\;\theta\in\Theta$ and probability density function $f_{\theta}(x)$. We further assume that the first moment of $X$ exists finitely. Suppose $Q\in\mathbb{R}^{+}$ is the inventory level at the beginning of period. Then, shortage loss is defined as $(X-Q)^{+}$ and the excess loss is defined as $(Q-X)^{+}$, where $d^{+}=max(d,0)$. Let $C_{s}$ and $C_{e}$ denote the per unit shortage and excess costs (constant) respectively and the corresponding costs are defined as $C_{s}(X-Q)^{+}$ and $C_{e}(Q-X)^{+}$. Thus the total cost can be written as a piece-wise linear function in the following manner:

The optimal order quantity ($Q^{*}$) is obtained by minimizing the expected total cost, $E[\chi]$. The analytical solution in this problem is given by $Q^{*}=F_{\theta}^{-1}\left(\frac{C_{s}}{C_{e}+C_{s}}\right)$, that is the $\gamma^{th}$ quantile of the demand distribution ($\gamma=\frac{C_{s}}{C_{s}+C_{e}}$).

We propose the following extension the classical newsvendor problem replacing the piece-wise linear loss functions by piece-wise power losses of same degree. Thus, the generalized cost function is given by

The powers of losses on both sides of $Q$ on the support of $X$ being same we would refer to this problem as symmetric generalized (SyGen) newsvendor problem.

The expected total cost in SyGen newsvendort problem is given by,

In the subsequent sections, we show that the expected cost admits minimum for uniform and exponential distributions.

In order to do so, the first order condition (FOC) is given by

It is difficult to provide further insight without having more information on $f_{\theta}(x)$. In the following section, we assume two choices for the demand distribution - (i) Uniform and (ii) Exponential.

In this section we consider the problem of estimating the optimal order quantity in SyGen newsvendor setup, based on a random sample of fixed size on demand, when the parameters of the two demand distributions discussed above are unknown.

In this section we present simulation studies for exponential demand in order to estimate the optimum order quantities from the estimating equations discussed above. Let us consider standard exponential demand distribution (exp($1$)) and observe performances of the estimated optimal order quantities corresponding to different estimating equations over different values of $m$ and ${\alpha_{m}}$. In this simulation we draw samples of size $n$, ($=10,50,100,500,1000,5000,10000$). For a given sample size $n$, the estimated optimal quantities are determined from each of the estimating equations described in the previous section. This procedure is repeated $1000$ times. We compute the bias in the estimated optimal order quantities by the average of ($\hat{Q}^{*}-Q^{*}$) over these $1000$ repetitions, where $Q^{*}$ is true optimal order quantity and $\hat{Q}^{*}$ is its estimate. The tables below report bias and MSE of different estimators of the optimal order quantity.

Full sample bias and MSE of $\hat{Q}_{1}^{*}$ are reported in tables 2a-2c, respectively and the same for $\hat{Q}_{2}^{*}$ are given in tables 3a-3c. It can be observed from the figures that the bias and MSEs of these two estimators of $Q^{*}$ are comparable. Also, none of the two estimators uniformly outperforms the other in terms of absolute bias or MSE across the given degrees of importance of loss, $m$ , in either small or large sample cases.

Comparing the bias and MSE of the estimators of $Q^{*}$ based on $2^{nd}$ order statistic (viz. $\hat{Q}_{(1)}^{*}$ and $\hat{Q}_{(2)}^{*}$) given in tables Tab.4a-4c and Tab.5a-5c respectively, similar observations as in the full sample case, can be made. However, in this case, the margin in bias and/or MSE for certain $(\alpha_{1},m,n)$ are much larger. For example, MSE of $\hat{Q}_{(1)}^{*}$ is quite smaller than that of $\hat{Q}_{(2)}^{*}$ for $m=50$ over all the considered sample sizes, when $\alpha_{1}=2$, whereas similar margins could be observed favoring $\hat{Q}_{(2)}^{*}$ in case of $\alpha_{1}=1$ and $m=10$, for all sample sizes except 100 and 1000. Thus, neither of the two estimators outperform the other.

Contributions of our study in this paper are 2-fold. First we have proposed a generalization of the standard news vendor problem assuming random demand and higher degree of shortage and excess loss. In particular, we have developed a symmetric generalized news vendor cost structure using power losses of same degree for both of shortage and excess inventory. We have presented the method to determine the optimal order quantity. In particular, we have presented the analytical expression of the optimal order quantity for uniform demand. In case of exponential demand, determination of optimal order quantity requires finding zeroes of a transcendental equation. We have proven the existence of real roots of the equation, ensuring that there exists a realistic solution to the proposed general model. Our second contribution is to provide different estimators of the optimal order quantity. We have provided estimators of the optimal order quantity using (i) full sample and (ii) broken sample data on demand. Whereas, analytical form of the estimator and its properties are easy to verify for uniform demand distribution, it is difficult for exponential distribution. We have presented a simulation study to compare different estimators. In this paper, we have presented estimators based on full sample as well as broken sample like single order statistic. Finally, we have provided a simulation study to gauge the performance of the proposed estimators in terms of bias and MSE.

A natural extension of this work could be to consider asymmetric shortage and excess losses. However, asymmetric power type shortage and excess losses would result in different dimensions of shortage and excess costs making them incomparable. Unpublished manuscript by Baraiya and Mukhoti (2019) proposes an inventory model for such a case.

$3^{rd}$ author remains deeply indebted to Late Prof. S Sengupta and Prof. Bikas K Sinha for their intriguing discussions and guidance on broken sample estimation. Research of $1^{st}$ author is funded by Indian Institute of Management Indore. The authors acknowledge gratefully the valuable comments by the participants of the SMDTDS-2020 conference organized by IAPQR at Kolkata.