Mathematical Problems in Engineering

Volume 2019, Article ID 7219326, 21 pages

https://doi.org/10.1155/2019/7219326

## Estimation of Jointly Normally Distributed Demand for Cross-Selling Items in Inventory Systems with Lost Sales

School of Economics and Management, Beihang University, Beijing 100083, China

Correspondence should be addressed to Hai-Tao Zheng; nc.ude.aaub@oatiahgnehz

Received 1 November 2018; Revised 29 May 2019; Accepted 17 June 2019; Published 16 July 2019

Academic Editor: Giuseppe D'Aniello

Copyright © 2019 Ren-Qian Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Demand estimation is often confronted with incomplete information of censored demand because of lost sales. Many estimators have been proposed to deal with lost sales when estimating the parameters of demand distribution. This study introduces the cross-selling effect into estimations, where two items are cross-sold because of the positive externality in a newsvendor-type inventory system. We propose an approach to estimate the parameters of a jointly normally distributed demand for two cross-selling items based on an iterative framework considering lost sales. Computational results based on more than two million numerical examples show that our estimator achieves high precision. Compared with the point estimations without lost sales, all the relative errors of the estimations of demand expectation, standard deviation, and correlation coefficient are no larger than 2% on average if the sample size is no smaller than 800. In particular, for demand expectation, the error is smaller than 1% if the comprehensive censoring level is no larger than four standard deviations (implying a -level of safety stock for each item), even if the sample size decreases to 50. This implies that the demand estimator should be competent in modern inventory systems that are rich in data.

#### 1. Introduction

In newsvendor-type inventory models, it is frequently assumed that the demand distribution for items is known or can be estimated based on sales data [1, 2]. Estimating the demand directly based on sales data implies that the sales quantity must be exactly equal to the realistic demand amount. However, this is not the case in practice [3]. The order quantity of an item is finite; thus, the inventory is limited but demand is often uncertain. It is very difficult to meet an uncertain demand based on a limited inventory. As a result, stock-outs emerge. Studies show that more than 85% of customers choose to give up the products when the items are out of stock [4], and therefore, the sales amount is the censored information of demand. When sales data is considered, demand might be underestimated. To remedy this flaw, demand estimation methods for censored sales data have been proposed [5, 6].

Another factor that should be considered in inventory management is the presence of externalities, including positive and negative ones [7]. As a type of positive externalities, cross-selling has not been considered in demand estimation. Cross-selling means that “a customer who has purchased a particular product may also be willing to purchase a related product” [8], and this implies that items are possibly purchased together owing to their unknown interior associations [9].

This study considers the cross-selling effect between two items to establish a new method to estimate the jointly (bivariate) normally distributed demand for two cross-selling items based on demand information from sales data that is censored by inventory order quantities. In our study, the sales data of any one item are not only censored because of lost sales caused by its own stock-out, but also affected by another associated item because of cross-selling. The core task is to recover the original demand distributions from the incomplete data that is a mixture of censored demand and cross sales. The proposed estimation is entirely based on the sales data, which is easily implemented in a modern business scenario. To the best of our knowledge, this is the first time that the demand estimation for cross-selling items with unobservable lost sales is proposed. Our work will consolidate the inventory model with positive externalities, as in the models of Netessine and Zhang [7] and Zhang et al. [10], and it can be used to provide basic information for inventory decision making.

The remainder of the paper is organized as follows. Section 2 reviews the related work. Section 3 includes assumptions and notations. Section 4 presents the demand estimation method and provides estimators. Section 5 illustrates the method by a numerical example, and Section 6 conducts numerical experiments to evaluate the performance of the estimators based on numerical examples. The paper is concluded in Section 7.

#### 2. Literature Review

Many studies have focused on single-variable demand estimation considering censored data based on parametric methods where the type of random distribution is given. For censored data of a normal demand, Fisher [11] developed estimators for the mean and standard deviation by maximizing the likelihood function, where two specifically predetermined tables are needed. Cohen [12] introduced new formulae to improve Fisher's method, based on which the special tables were excluded. Later, Gupta [13] investigated the problem and presented a better linear unbiased estimator. Nahmias [5] developed a parametric methodology for estimating the normal demand and validated the approach based on censored samples. The method was further extended to the case of a negative binomial distribution [6]. Berk et al. [14] estimated the demand with censored data of negative binomial, gamma, Poisson, and normal distributions by Bayesian updating, adopting an approximate posterior distribution that matches the first two moments of the exact posterior. Heeseab [15] provided a Bayesian method for demand estimation considering unobserved lost sales, which depends on the prior information to estimate the posterior distribution. Jain et al. [16] modeled the cumulative demand arrivals during a period as a stochastic process with an unknown parameter. The parameter is updated in a Bayesian fashion by the observation of sales and stock-out events, which is further simplified by introducing stock-out timing observation.

Without a given type of demand distribution, nonparametric demand estimation methods are adopted. Beutel and Minner [17] proposed demand forecasting by solving a linear programming (LP) problem with fully observed demand, which does not need to know the random distribution. The method was further extended to deal with the censored observations of demand by Sachs and Minner [18]. They both assume that the demand is linearly dependent on multiple exogenous variables. For the nonlinear case, an intelligent forecasting algorithm based on artificial neural network (ANN) and conventional regression has been adopted. The method was applied for forecasting gasoline demand [19].

The above estimators are focused on the case of a single stochastic variable and did not consider consumption externalities. For multiple variables, Dempster et al. [20] proposed the expectation-maximization (EM) algorithm for the case of incomplete data, which, in contrast to the previous approaches, is an iterative method. Dahiya and Korwar [21] provided a parameter estimation method for the likelihood equation in the bivariate normal case. They assumed that the missing data were from only one variable and no consumption externalities existed. Adamids and Loukas [22] considered the missing values and proposed bivariate Poisson estimations for a two-item inventory based on the EM method without externalities. Regarding the time-varying demand, multivariate auto regressive integrated moving average (ARIMA) models were used to forecast the demand for perishable goods [23], but consumption externalities were not considered either.

Consumption externality means that the demand/consumption for one item affects the demand/consumption for other items, and there are two types of externality in inventory models: negative and positive [7]. Besanko et al. [24] presented a logit demand estimation based on the consumer choice model of substitutable items with negative externalities, where the endogenous price is the Nash equilibrium of wholesale and retail in a market. Anupindi et al. [25] and Smith and Agrawal [26] developed demand estimators considering stock-outs and multi-item substitutions, assuming Poisson arrivals of demand. Kök and Fisher [27] presented an EM-based method for estimating the parameters of substitution behavior and demand for products, which was used for optimizing the assortment planning of a retailer. Conlon and Mortimer [28] estimated the multinomial distribution demand under incomplete product availability by maximizing a likelihood function based on an EM algorithm. They assumed that the customer behavior can be described based on the discrete choice model of substitutable items. For multi-store multi-product substitution, Wan et al. [29] built demand estimation models considering that consumers may substitute a product for the stocked out item in the same store or switching to a neighboring store, where the parameters were estimated by the Markov chain Monte Carlo algorithm in a Bayesian manner.

It is noticeable that the extant inventory models and demand estimations with externalities are mainly focused on substitutable items, where choice happens because of substitution, and the customer behavior can be modeled by the consumer choice theory. However, this is not the case for cross-selling scenarios, where positive externalities exist between complementary items and no choice happens even if some items are stocked out. For comparing our study with previous works, Table 1 classifies the above-reviewed related literature into three groups.