Obtaining Conservative Estimates of Integrated Profitability for a Single-Period Product in an Own-Branding-and-Manufacturing Enterprise with Multiple Owned Channels

Abstract

The achievable capacity index (ACI) is a simple and efficient approach for estimating the profitability of newsboy-type products, wherein profitability is defined as the probability of achieving the target profit by optimizing the order quantity. At present, the ACI is applicable to single retail stores (i.e., single demand) but not to multiple sales channels (i.e., multiple demand). This paper presents an integrated achievable capacity index (IACI) by which to measure the aggregate profitability of multiple mutually independent channels under normally distributed demand. An unbiased IACI estimator is also developed, to which is applied the Taylor expansion to approximate its sampling distribution, wherein the sizes, means, and variances of demand differ in each channel. Furthermore, overestimates due to sampling error are avoided by deriving the lower confidence bound for the IACI. This paper also provides generic tables to aid managers seeking conservative estimates of profitability. The applicability of the proposed scheme is demonstrated numerically using a real-world example involving an own-branding-and-manufacturing (OBM) enterprise with multiple owned channels.

1. Introduction

Inventory management involves the formulation of policies aimed at minimizing costs and/or maximizing profits. This can be achieved by optimizing the quantity of goods ordered or manufactured or by determining the optimal stock level or reorder point. The single-period problem (referred to as the newsboy/newsvendor problem) describes an inventory management situation involving products with a restricted shelf life (e.g., daily newspapers, weekly magazines, fresh foods, and deteriorating chemicals). The newsboy problem assumes that the order quantity can only be decided at the beginning of the period, and replenishment is unavailable during the sales period. Furthermore, the vendor may face costs associated with scrap or shortages resulting from uncertain demand. Essentially, the newsboy problem requires the vendor to obtain an accurate estimate of actual demand (based on historical demand) for use in determining the order quantity with the best performance in satisfying a designated objective function, such as minimizing the expected cost, maximizing the expected profit, or maximizing the probability of achieving a target profit.

A number of recent studies have sought to use marketing scenarios to enhance the applicability of newsboy models to practical scenarios. For current marketing scenarios, carbon emissions reduction schemes, including carbon tax and cap-and-trade regulations, have been the most environmentally friendly approaches to reducing carbon emissions in the last several decades. Bai and Chen [1] and Xu et al. [2] considered carbon tax and cap-and-trade regulations in their newsboy models aimed at reducing carbon emissions. Bai et al. [3] established a newsboy model based on a remanufacturing system with a cap-and-trade policy. Qu et al. [4] explored newsboy models under cap-and-trade regulations in which the carbon footprint was calculated for the production, sales, residual disposal, and return warranty service. With the development of information and communication technologies, e-commerce has become an available sales model in addition to the bricks-and-clicks business. Hovelaque et al. [5] compared the advantages of three shipping models (i.e., store picking, dedicated warehouse picking, and drop shipping) in the context of a newsboy policy based on an e-commerce model. Note that the return ratio is generally higher for online shopping than for bricks-and-clicks business models. Ma et al. [6] considered a newsboy problem with drop shipping and resalable returns, wherein both bricks-and-clicks and online shipping businesses were available. Ma and Jemai [7] further analyzed two policies aimed at optimizing the rationing of store inventory, including a threshold policy and a fixed-portion policy. Kamanchi et al. [8] explored a scenario that involved optimizing the supply chain of an e-commerce fashion vendor.

In a competitive market environment, inventory management involves a series of complex issues beyond the simple determination of how much to order. For example, in situations with limited shelf space, managers must select from among multiple homogeneous products to find those with the highest profitability (the issue of what/which to order). Consequently, the market profitability of candidate products must be assessed before ordering products. Some literature pertaining to the newsboy problem has realized this issue and further used the estimated result of the objective function to present the market performance for a newsboy-type product, such as the maximum expected profit or the maximum probability of achieving a target profit. Kevork [9] presented an estimator for the maximum expected profit and derived its statistical properties using Monte Carlo simulations for small and large samples. Halkos and Kevork [10,11] applied interval estimation to determine the maximum expected profit in cases of normally-, exponentially-, and Rayleigh-distributed demand. Sundar et al. [12] explored the lower bound of interval estimation for the maximum expected profit under various distributions with various parameters. Su and Pearn [13] defined the maximum probability of achieving a target profit as the profitability of a newsboy-type product with a normally distributed demand. To overcome difficulties in obtaining the statistical properties of profitability, they developed the achievable capacity index (ACI), which can also be used to measure profitability in situations where the relevant costs and prices remain constant. Note that the degree of profitability is positively correlated with the ACI value. Since the formulation of ACI is more simplified, it is possible to derive the statistical properties of the ACI estimator. Consequently, some practical applications can be explored by performing statistical inference of the ACI. For example, Su et al. [14] considered a conservative profitability evaluation to prevent overestimates of profitability due to sampling error. To address this issue, Su et al. [14] derived the lower confidence bound of the ACI (LCBA). Su et al. [15] further efficiently obtained the value of LCBA using four parametric bootstrap methods.

2. Problem Description

The ACI can be used to measure the profitability of a single-period product based on a single sales channel (i.e., single demand); however, it is inapplicable to scenarios involving multiple sales channels (i.e., multiple demand), such as own-branding-and-manufacturing (OBM) enterprises. For the most part, OBM enterprises adjust their output based on the aggregate profitability of products in all channels. As a matter of convenience, OBM managers tend to use the total revenue of all channels as an indication of product profitability; however, this approach does not reflect the impact of costs due to surplus or shortage. Furthermore, OMB managers lack reliable methods by which to make statistical inferences pertaining to profits. In the current study, we sought to develop a tool by which to assess aggregate profitability under multiple demands for OBM enterprises.

OBM enterprises generally own manufactories and sales channels and own or rent warehouses. To safeguard brand image, the sale of branded products is generally limited to owned channels (e.g., flagship stores, direct-selling stores, franchise stores, and counters) or contracted to designated retailers. It is important to consider that the demand patterns for a given product tend to vary from one channel to the next due to regional differences in the customer base and sales model. However, allowing the transfer of stocks among channels makes it possible to minimize out-of-stock and over-stock situations, thereby decreasing operational losses due to uncertain demand. Figure 1 presents a schematic diagram illustrating the logistics, flow of information and flow of revenue in a general OBM enterprise. From Figure 1, since the OBM system uses regional production and distribution to reduce transportation costs, warehouse A distributes inventory to channels 1–3, and warehouse B distributes inventory to channels 4–5. Each channel is in a different region, and the customer demand patterns in each region are also different and independent of each other. Although out-of-stock and over-stock events may occur in each channel due to the uncertainty in the demand of each channel, the transfer of stocks among channels is allowed to reduce the cost caused by both events.

Consequently, the above-mentioned scenarios must be taken into account when structuring the aggregate profitability with multiple owned channels. On the other hand, it is also necessary to establish an integrated ACI (IACI) to simplify the expression of aggregate profitability. Note that our focus on normally distributed demand in all channels means that the aggregate profitability and IACI depend on the mean and variance of the distributions. To deal with situations where these parameters are unknown, we use statistical inference to estimate the true IACI based on historical demand data from all channels. We also derive conservative IACI estimates by enhancing the robustness of sampling error resulting from inconsistencies in sample size and population. The achievements of this study are as follows:

(1)

We addressed OBM enterprises with multiple owned channels, in which the demand level in each channel is arbitrary, mutually exclusive, and unknown. The transfer of products among channels is allowed to decrease operational losses due to uncertain demand.

(2)

Based on Su and Pearn. [13], we further established aggregate profitability with normally distributed demand in all channels and defined an IACI index to simplify the measurement of integrated profitability. We also sought to derive the statistical properties of the IACI estimator, despite inconsistencies in the sample size and variations in the populations among the various channels.

(3)

According to Su et al. [14], we also derived the lower confidence bound for the IACI (LCBIA) to minimize the risk of profitability overestimation due to sampling error. We also provide generic tables to inform the decisions of managers seeking conservative estimates of profitability.

Figure 2 presents a simple diagram pertaining to the application scenario, development, and procedure of the proposed model, as well as a comparison between the existing literature and the presented paper. The remainder of this article is organized as follows. Section 3 presents the methods used to determine the aggregate profitability using the proposed IACI. Section 4 derives the statistical properties of the IACI estimator for use in obtaining conservative estimates of profitability. Section 5 illustrates the applicability of the proposed scheme using numerical simulations of a real-world example involving an OBM enterprise with multiple owned channels. Concluding remarks and directions for future research are outlined in Section 6.

3. Integrated Profitability and Proposed IACI

3.1. Integrated Profitability

Integrated profitability is defined as the probability of achieving the target profit ($k_t$) under the optimal order quantity for each location. Assume that among $h$ channels, $s$ channels are out of stock, and $h-s$ channels meet the demand or possess surplus stocks at the end of the sales period. We formulate the integrated profit per period ($z_t$) as follows:

$$z_t=\left\{\begin{array}{l}p{\displaystyle {\sum }_{i=1}^h{d}_i}-c{\displaystyle {\sum }_{i=1}^h{Q}_i}-c_d\left[{\displaystyle {\sum }_{i=s+1}^h(Q_i-d_i)}-{\displaystyle {\sum }_{i=1}^s(d_i-Q_i)}\right],{\displaystyle {\sum }_{i=s+1}^h(Q_i-d_i)}\ge {\displaystyle {\sum }_{i=1}^s(d_i-Q_i),}\\ p{\displaystyle {\sum }_{i=1}^h{Q}_i}-c{\displaystyle {\sum }_{i=1}^h{Q}_i}-c_s\left[{\displaystyle {\sum }_{i=1}^s(d_i-Q_i)-{\displaystyle {\sum }_{i=s+1}^h(Q_i-d_i)}}\right],{\displaystyle {\sum }_{i=1}^s(d_i-Q_i)}>{\displaystyle {\sum }_{i=s+1}^h(Q_i-d_i),}\end{array}\right.$$

(1)

where $Q_i$ and $d_i$ respectively indicate the order quantity and demand per period for each channel, selling price ($p$), manufacturing/purchasing cost ($c$), shortage cost ($c_s$) and disposal cost due to surplus ($c_d$) are given. Note that the upper and lower lines in Equation (1) respectively indicate the overall excess and shortage where mutual transfers are allowed. Equation (1) can be simplified as follows:

$$z_t=\left\{\begin{array}{l}(c_p+c_e)d_t-c_e{Q}_t,Q_t\ge d_t\\ (c_p+c_s)Q_t-c_s{d}_t,Q_t<d_t\end{array}\right.$$

(2)

where $c_p=p-c$ is the net profit for a given product, $c_e=c_d+c$ is the excess cost, $Q_t={\displaystyle {\sum }_{i=1}^h{Q}_i}$ is the total order quantity per period for all channels (or the total quantity manufactured per period by the OBM firm), and $d_t={\displaystyle {\sum }_{i=1}^h{d}_i}$ is the total demand per period for all channels. According to Equation (2), the aggregate profit can be met by the target profit (i.e., $z_t\ge k_t$) as long as $LA{L}_t(Q_t)<d_t<UA{L}_t(Q_t)$, wherein $UA{L}_t(Q_t)=[-k_t+(c_p+c_s)Q_t]/c_s$ and $LA{L}_t(Q_t)=(k_t+c_e{Q}_t)/(c_p+c_e)$ respectively indicate the upper and lower achievable limits, depending on the total order quantity. In this paper, we consider situations in which the demands among channels are independent and all obey normal distributions with unknown parameters; i.e., $d_i~N({\mu }_i,{\sigma }_i^2)$, where $i=1,2,\dots ,h$. Note that because $d_t~N({\displaystyle {\sum }_{i=1}^h{\mu }_i},{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2})$, the probability of achieving the target profit can be derived as follows:

$$\begin{array}{ll}\hfill \mathrm{Pr}(z_t\ge k_t)& =\mathrm{Pr}(LA{L}_t(Q_t)\le d_t\le UA{L}_t(Q_t))\\ & =\mathsf{\Phi }\left(\frac{UA{L}_t(Q_t)-{\displaystyle {\sum }_{i=1}^h{\mu }_i}}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}}\right)-\mathsf{\Phi }\left(\frac{LA{L}_t(Q_t)-{\displaystyle {\sum }_{i=1}^h{\mu }_i}}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}}\right),\end{array}$$

(3)

where $\mathsf{\Phi }(\cdot )$ is the cumulative distribution function of the standard normal distribution. Obviously, the above probability also depends on the total order quantity. By solving $d\mathrm{Pr}(z_t\ge k_t)/d{Q}_t=0$, we obtain a positive extreme point of the total order quantity in Equation (3), as follows:

$$\begin{gathered} Q_t^{*}=T_t+\frac{c_s(c_p+c_e)(c_p{\displaystyle {\sum }_{i=1}^h{\mu }_i}-k_t)}{c_p[c_p(c_p+c_e+c_s)+2c_e{c}_s]}+\left\{{\left[\frac{c_s(c_p+c_e)(c_p{\displaystyle {\sum }_{i=1}^h{\mu }_i}-k_t)}{c_p[c_p(c_p+c_e+c_s)+2c_e{c}_s]}\right]}^2\right. \\ {\left.+\frac{2c_s^2{(c_p+c_e)}^2\ln [1+c_p(c_p+c_e+c_s)/c_s{c}_e]{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}{c_p(c_p+c_e+c_s)[c_p(c_p+c_e+c_s)+2c_e{c}_s]}\right\}}^{1/2}, \end{gathered}$$

(4)

where $\omega =\ln [1+c_p(c_p+c_e+c_s)/c_s{c}_e]$. The sufficient condition of the extreme point for maximizing $\mathrm{Pr}(z_t\ge k_t)$ is negative, as follows:

$$\begin{gathered} \frac{d^2\mathrm{Pr}(z_t\ge k_t)}{d{Q}_t^2}{|}_{Q_t=Q_t^{*}}=-\frac{(c_p+c_s)\exp \left\{-\frac{1}{2}{\left[\frac{UA{L}_t(Q_t^{*})-{\mu }_t}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}\right]}^2\right\}}{\sqrt{2\pi }{({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2})}^3{c}_s^2(c_p+c_e)} \\ \times \left\{\frac{\left[UA{L}_t(Q_t^{*})-LA{L}_t(Q_t^{*})\right][c_p(c_p+c_e+c_s)+2c_e{c}_s]}{2}\right. \\ \left.+\frac{c_p(c_p+c_e+c_s)\omega {\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}{UA{L}_t(Q_t^{*})-LA{L}_t(Q_t^{*})}\right\}<0 \end{gathered}$$

(5)

The extreme point represents the ideal total order quantity (i.e., the maximum probability of meeting the target profit), thereby indicating the optimal choice. By substituting the optimal order quantity into Equation (3), we can obtain the aggregate profitability of a newsboy-type product (denoted by ${\mathsf{\Omega }}_t$), as follows:

$${\mathsf{\Omega }}_t=\mathrm{Pr}(z_t>k_t\left|Q_t=Q_t^{*}\right.)=\mathsf{\Phi }\left\{A+\frac{\omega }{2A}\right\}-\mathsf{\Phi }\left\{-A+\frac{\omega }{2A}\right\},$$

(6)

where

$$A=A_0\times \frac{{\displaystyle {\sum }_{i=1}^h{\mu }_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}}+\sqrt{{\left[A_0\times \frac{{\displaystyle {\sum }_{i=1}^h{\mu }_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}}\right]}^2+A_0\omega },$$

(7)

$A_0=c_p(c_p+c_e+c_s)/[2c_p(c_p+c_e+c_s)+4c_e{c}_s]$ and $T_t=k_t/c_p$ indicates the target demand; i.e., the minimum overall demand required to achieve the target profit (disregarding shortages and excess costs).

3.2. Development of IACI

As shown in Equations (6) and (7), the parameters pertaining to demand can be assembled into the following ratio (denoted by $I_A^T$):

$$I_A^T=\frac{{\displaystyle {\sum }_{i=1}^h{\mu }_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}}.$$

(8)

Thus, integrated profitability can be rearranged as a function of $I_A^T$ (denoted by ${\mathsf{\Omega }}_t(I_A^T)$) simply by substituting the above equation into Equation (7), as follows: $A=A_0{I}_A^T+{[{(A_0{I}_A^T)}^2+A_0\omega ]}^{1/2}$. In formulating $I_A^T$, the numerator indicates the difference between the mean total demand and target demand, whereas the denominator is the root of the total variances. Intuitively, we can assume that the value of $I_A^T$ is positively correlated with integrated profitability as long as the parameters unrelated to demand remain constant. In this study, this ratio is referred to as the integrated achievable capacity index (IACI). To confirm the correlation mentioned above, we take the first-order derivative of ${\mathsf{\Omega }}_t(I_A^T)$ with respect to $I_A^T$ based on the Leibniz integral rule, to obtain the following:

$$\frac{d{\mathsf{\Omega }}_t(I_A^T)}{d{I}_A^T}=\frac{A_{0}A}{\sqrt{2\pi \left(A_0{I}_A^{T2}+A_0\omega \right)}}\left[1+e^{\omega }+\frac{\omega \left(e^{\omega }-1\right)}{2A^2}\right]e^{-\frac{1}{2}{\left(A+\frac{1}{2A}\right)}^2}>0.$$

(9)

Thus, profitability ${\mathsf{\Omega }}_T(I_A^T)$ is a strictly increasing function of $I_A^T$. Thus, when the price and the associated costs are fixed, it is possible to use $I_A^T$ to measure the integrated profitability of a single-period product.

4. Conservative Estimation of IACI

In many practical applications, it is simply not feasible to obtain population parameters or the value of $I_A^T$. This necessitates the collection of historical demand data from all channels to estimate the true $I_A^T$. In the current study, we explored situations in which every channel differed from all other channels in terms of sample size and the means and variances of demand. If a sample of size $n_i$ in channel $i$ is given as $\{x_{i1},x_{i2},\dots x_{i{n}_i}\}$ where $i=1,2,\dots ,h$, then we consider the natural estimator of $I_A^T$ (denoted by ${\widehat{I}}_A^T$). The formulation of ${\widehat{I}}_A^T$ is obtained by replacing the population parameters with their estimators, as follows:

$${\widehat{I}}_A^T=\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}},$$

(10)

where ${\overline{x}}_i={\displaystyle {\sum }_{j=1}^{n_i}{x}_{ij}}/n_i$ and $s_i^2={\displaystyle {\sum }_{j=1}^{n_i}{(x_{ij}-{\overline{x}}_i)}^2}/(n_i-1)$.

4.1. Approximated Sampling Distribution

Under the assumption that ${\overline{x}}_i~N\left({\mu }_i,{\sigma }_i^2/n_i\right)$, the sampling distribution of the sum of the sample means can be derived as ${\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}~N\left({\displaystyle {\sum }_{i=1}^h{\mu }_i},{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2/n_i}\right)$. Consequently, the natural estimator ${\widehat{I}}_A^T$ can be represented as

$${\widehat{I}}_A^T=\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}}=\frac{\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-{\displaystyle {\sum }_{i=1}^h{\mu }_i}}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}/n_i}}+\frac{{\displaystyle {\sum }_{i=1}^h{\mu }_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}/n_i}}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}/n_i}}}=\frac{Z+\frac{{\displaystyle {\sum }_{i=1}^h{\mu }_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}/n_i}}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}/n_i}}},$$

(11)

where $Z$ is a standard normal random variable. Note that in Equation (11), the denominator cannot be presented as a chi-square random variable due to the unequal variances in demand. Moreover, the numerator cannot be simplified in the form of $I_A^T$.

The above issues make it difficult to derive the sampling distribution of ${\widehat{I}}_A^T$. We thus applied the Taylor expansion to approximate the sampling distribution of ${\widehat{I}}_A^T$. Let $\theta =(\mu ,{\sigma }^2)=({\mu }_1,{\mu }_2,\dots ,{\mu }_h,{\sigma }_1^2,{\sigma }_2^2,\dots ,{\sigma }_h^2)$ be the unknown population parameters and $\widehat{\theta }=(\overline{x},s^2)=({\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_h,s_1^2,s_2^2,\dots ,s_h^2)$ be the corresponding estimators. Regarding ${\widehat{I}}_A^T$ as a function of $\widehat{\theta }$ and applying the Taylor expansion at point $\theta$, we obtain the following:

$${\widehat{I}}_A^T=f\left(\widehat{\theta }\right)\approx f\left(\theta \right)+{\left.\frac{\partial f^{\mathrm{T}}}{\partial \overline{x}}\right|}_{\widehat{\theta }=\theta }\left(\overline{x}-\mu \right)+{\left.\frac{\partial f^{\mathrm{T}}}{\partial s^2}\right|}_{\widehat{\theta }=\theta }\left(s^2-{\sigma }^2\right),$$

(12)

where the derivatives are

$${\left.\frac{\partial f}{\partial {\overline{x}}_i}\right|}_{\widehat{\theta }=\theta }={\left.\frac{\partial }{\partial {\overline{x}}_i}\left[\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}}\right]\right|}_{\widehat{\theta }=\theta }={\left.{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^{-\frac{1}{2}}\right|}_{\widehat{\theta }=\theta }={\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-\frac{1}{2}},$$

(13)

and

$${\left.\frac{\partial f}{\partial s_i^2}\right|}_{\widehat{\theta }=\theta }=\frac{\partial }{\partial s_i^2}{\left.\left[\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-T_t}{\sqrt{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}}\right]\right|}_{\widehat{\theta }=\theta }={\left.\left[-\frac{{\displaystyle {\sum }_{i=1}^h{\overline{x}}_i}-T_t}{2{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^{\frac{3}{2}}}\right]\right|}_{\widehat{\theta }=\theta }=-\frac{I_A^T}{2}{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-1}.$$

(14)

Let $e=\left[1,1,\dots ,1\right]$ denote the identity row vector. The derivation in Equation (12) implies that

$$\begin{array}{ll}{\widehat{I}}_A^T& \approx I_A^T+{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-\frac{1}{2}}\left[1,1,\dots ,1\right]\left(\overline{x}-\mu \right)-\frac{I_A^T}{2}{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-1}\left[1,1,\dots ,1\right]\left(s^2-{\sigma }^2\right)\\ & =I_A^T+{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-\frac{1}{2}}{\displaystyle {\sum }_{i=1}^h\left({\overline{x}}_i-{\mu }_i\right)}-\frac{I_A^T}{2}{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-1}{\displaystyle {\sum }_{i=1}^h\left(s_i^2-{\sigma }_i^2\right)}.\end{array}$$

(15)

Using Equation (15), the expected value of ${\widehat{I}}_A^T$ can be approximated as follows:

$$E\left({\widehat{I}}_A^T\right)\approx I_A^T+{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-\frac{1}{2}}{\displaystyle {\sum }_{i=1}^h\left[E\left({\overline{x}}_i\right)-{\mu }_i\right]}-\frac{I_A^T}{2}{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-1}{\displaystyle {\sum }_{i=1}^h\left[E\left(s_i^2\right)-{\sigma }_i^2\right]}=I_A^T.$$

(16)

These results indicate that ${\widehat{I}}_A^T$ is an asymptotic unbiased estimator.

In Equation (15), the estimator ${\widehat{I}}_A^T$ includes two terms that are associated with random variables, ${\displaystyle {\sum }_{i=1}^h\left({\overline{x}}_i-{\mu }_i\right)}$ and ${\displaystyle {\sum }_{i=1}^h\left(s_i^2-{\sigma }_i^2\right)}$. Since ${\overline{x}}_i~N\left({\mu }_i,{\sigma }_i^2/n_i\right)$, we have ${\displaystyle {\sum }_{i=1}^h\left({\overline{x}}_i-{\mu }_i\right)}~N\left(0,{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2/n_i}\right)$. Considering the random variables $s_i^2-{\sigma }_i^2$ in Equation (15), we have $s_i^2~{\sigma }_i^2{\chi }_{n_i-1}^2/\left(n_i-1\right)$, $E\left(s_i^2\right)={\sigma }_i^2$, and $Var\left(s_i^2\right)=2{\sigma }_i^4{\left(n_i-1\right)}^{-1}$, where ${\chi }_{n_i-1}^2$ is a chi-square random variable with $n_i-1$ degrees of freedom, where $i=1,2,\dots ,h$. According to Central Limit Theorem, the statistic $s_i^2-{\sigma }_i^2$ approaches normal distribution; i.e., $s_i^2-{\sigma }_i^2~N\left(0,2{\sigma }_i^4/(n_i-1)\right)$. Thus, we obtain the following: ${\displaystyle {\sum }_{i=1}^h\left(s_i^2-{\sigma }_i^2\right)}~N\left(0,2{\displaystyle {\sum }_{i=1}^h{\sigma }_i^4/(n_i-1)}\right)$. Based on the above results, the sampling distribution of ${\widehat{I}}_A^T$ can be approximated as follows:

$${\widehat{I}}_A^T~I_A^T+{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-\frac{1}{2}}N\left(0,{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}\right)-\frac{I_A^T}{2}{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^{-1}N\left(0,2{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}\right)=N\left(I_A^T,\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}\right).$$

(17)

4.2. LCB on IACI

Based on the approximated sampling distribution of ${\widehat{I}}_A^T$, we would derive the lower confidence bound for IACI (LCBIA) to express the conservative integrated profitability, which is denoted as $I_A^{TLCB}$. For a given Type-I error $\alpha$, we establish the following equation according to the definition of the confidence interval:

$$\begin{array}{ll}1-\alpha & =\mathrm{Pr}(I_A^T\ge I_A^{TLCB})\\ & =\mathrm{Pr}\left(\frac{{\widehat{I}}_A^T-I_A^T}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}}\le \frac{{\widehat{I}}_A^T-I_A^{TLCB}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}}\right)\\ & =\mathrm{Pr}\left(Z\le \frac{{\widehat{I}}_A^T-I_A^{TLCB}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}}\right)=\mathsf{\Phi }\left(\frac{{\widehat{I}}_A^T-I_A^{TLCB}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}}\right).\end{array}$$

(18)

Thus, we obtain

$$z_{1-\alpha }=\frac{{\widehat{I}}_A^T-I_A^{TLCB}}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}}.$$

(19)

The above equation can be rearranged to obtain a closed form of LCBIA, as follows:

$$I_A^{TLCB}={\widehat{I}}_A^T-z_{1-\alpha }\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}}+\frac{I_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{{\sigma }_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{\sigma }_i^2}\right)}^2}}.$$

(20)

In this equation, the population parameters pertaining to demand in each channel are unknown. Thus, when the sample size is large enough (roughly > 30), we can replace these parameters with their estimators as follows:

$$I_A^{TLCB}={\widehat{I}}_A^T-z_{1-\alpha }\sqrt{\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}+\frac{{\widehat{I}}_A^{T2}{\displaystyle {\sum }_{i=1}^h\frac{s_i^4}{n_i-1}}}{2{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^2}}.$$

(21)

Below, we consider situations with equal and unequal sample sizes in each channel in accordance with the above formulation of LCBIA.

4.2.1. Equal Sample Sizes

If $n_1=n_2=\dots =n_h=n$, the formulation of LCBIA can be written as follows:

$$I_A^{TLCB}={\widehat{I}}_A^T-z_{1-\alpha }\sqrt{\frac{1}{n}+\frac{{\widehat{I}}_A^{T2}}{2(n-1)}\xi },$$

(22)

where the term with a root sign indicates the sampling error and $0<\xi ={\displaystyle {\sum }_{j=1}^h{s}_j^4}/{\left({\displaystyle {\sum }_{j=1}^h{s}_j^2}\right)}^2<1$ is a ratio indicating variations in demand.

Table 1. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.1$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6920	0.7904	0.8887	0.9868	1.0848	1.1827	1.2804	1.3780	1.4755	1.5729	1.6701
40		0.7333	0.8320	0.9305	1.0289	1.1272	1.2253	1.3234	1.4213	1.5191	1.6169	1.7145
50		0.7615	0.8603	0.9590	1.0576	1.1560	1.2544	1.3527	1.4508	1.5489	1.6469	1.7447
60		0.7823	0.8812	0.9800	1.0787	1.1773	1.2758	1.3743	1.4726	1.5708	1.6690	1.7671
70		0.7985	0.8975	0.9963	1.0951	1.1939	1.2925	1.3910	1.4895	1.5879	1.6862	1.7844
80		0.8115	0.9106	1.0095	1.1084	1.2072	1.3059	1.4045	1.5031	1.6016	1.7000	1.7983
90		0.8223	0.9214	1.0204	1.1194	1.2182	1.3170	1.4157	1.5144	1.6130	1.7115	1.8099
100		0.8314	0.9306	1.0296	1.1286	1.2276	1.3264	1.4252	1.5239	1.6226	1.7211	1.8197
110		0.8393	0.9385	1.0376	1.1366	1.2356	1.3345	1.4333	1.5321	1.6308	1.7295	1.8281
120		0.8461	0.9453	1.0445	1.1436	1.2426	1.3416	1.4404	1.5393	1.6380	1.7368	1.8354
130		0.8521	0.9514	1.0506	1.1497	1.2488	1.3478	1.4467	1.5456	1.6444	1.7432	1.8419
140		0.8575	0.9568	1.0560	1.1552	1.2543	1.3533	1.4523	1.5512	1.6501	1.7489	1.8476
150		0.8624	0.9617	1.0609	1.1601	1.2592	1.3583	1.4573	1.5563	1.6552	1.7540	1.8528
160		0.8667	0.9661	1.0653	1.1645	1.2637	1.3628	1.4618	1.5608	1.6598	1.7586	1.8575
170		0.8707	0.9701	1.0694	1.1686	1.2678	1.3669	1.4660	1.5650	1.6640	1.7629	1.8617
180		0.8744	0.9737	1.0730	1.1723	1.2715	1.3707	1.4697	1.5688	1.6678	1.7667	1.8656
190		0.8777	0.9771	1.0764	1.1757	1.2749	1.3741	1.4732	1.5723	1.6713	1.7703	1.8692
200		0.8808	0.9802	1.0796	1.1789	1.2781	1.3773	1.4764	1.5755	1.6746	1.7736	1.8725

,

Table 2. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.2$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6845	0.7815	0.8781	0.9745	1.0707	1.1666	1.2623	1.3577	1.4530	1.5481	1.6429
40		0.7269	0.8243	0.9214	1.0183	1.1150	1.2115	1.3078	1.4039	1.4998	1.5956	1.6912
50		0.7558	0.8534	0.9509	1.0481	1.1452	1.2421	1.3388	1.4353	1.5317	1.6279	1.7240
60		0.7771	0.8750	0.9726	1.0701	1.1674	1.2646	1.3616	1.4585	1.5552	1.6517	1.7481
70		0.7937	0.8917	0.9895	1.0872	1.1847	1.2821	1.3793	1.4764	1.5734	1.6702	1.7669
80		0.8070	0.9052	1.0031	1.1010	1.1987	1.2962	1.3936	1.4909	1.5881	1.6851	1.7820
90		0.8181	0.9163	1.0144	1.1124	1.2102	1.3079	1.4055	1.5029	1.6002	1.6974	1.7945
100		0.8274	0.9258	1.0240	1.1220	1.2200	1.3178	1.4155	1.5130	1.6105	1.7079	1.8051
110		0.8354	0.9339	1.0322	1.1303	1.2284	1.3263	1.4241	1.5218	1.6193	1.7168	1.8142
120		0.8425	0.9409	1.0393	1.1376	1.2357	1.3337	1.4316	1.5294	1.6270	1.7246	1.8221
130		0.8486	0.9472	1.0456	1.1439	1.2421	1.3402	1.4382	1.5361	1.6338	1.7315	1.8291
140		0.8542	0.9528	1.0512	1.1496	1.2479	1.3460	1.4441	1.5420	1.6399	1.7377	1.8353
150		0.8591	0.9578	1.0563	1.1547	1.2530	1.3513	1.4494	1.5474	1.6453	1.7432	1.8409
160		0.8636	0.9623	1.0609	1.1593	1.2577	1.3560	1.4542	1.5523	1.6503	1.7482	1.8460
170		0.8677	0.9664	1.0650	1.1635	1.2620	1.3603	1.4585	1.5567	1.6547	1.7527	1.8506
180		0.8714	0.9702	1.0688	1.1674	1.2659	1.3642	1.4625	1.5607	1.6588	1.7569	1.8548
190		0.8748	0.9736	1.0723	1.1709	1.2694	1.3679	1.4662	1.5644	1.6626	1.7607	1.8587
200		0.8780	0.9768	1.0756	1.1742	1.2728	1.3712	1.4696	1.5679	1.6661	1.7642	1.8623

,

Table 3. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.3$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6772	0.7727	0.8678	0.9626	1.0571	1.1512	1.2450	1.3386	1.4319	1.5249	1.6177
40		0.7206	0.8168	0.9126	1.0081	1.1033	1.1983	1.2930	1.3874	1.4816	1.5756	1.6695
50		0.7502	0.8468	0.9430	1.0390	1.1348	1.2303	1.3256	1.4206	1.5155	1.6102	1.7046
60		0.7720	0.8689	0.9655	1.0618	1.1580	1.2539	1.3496	1.4451	1.5404	1.6356	1.7305
70		0.7890	0.8861	0.9829	1.0796	1.1760	1.2722	1.3683	1.4641	1.5598	1.6553	1.7506
80		0.8026	0.8999	0.9970	1.0938	1.1905	1.2870	1.3833	1.4794	1.5754	1.6712	1.7668
90		0.8139	0.9114	1.0086	1.1057	1.2025	1.2992	1.3957	1.4921	1.5883	1.6843	1.7802
100		0.8235	0.9211	1.0185	1.1157	1.2127	1.3095	1.4062	1.5028	1.5992	1.6954	1.7915
110		0.8317	0.9294	1.0269	1.1242	1.2214	1.3184	1.4153	1.5120	1.6085	1.7050	1.8013
120		0.8389	0.9367	1.0343	1.1317	1.2290	1.3262	1.4231	1.5200	1.6167	1.7133	1.8098
130		0.8452	0.9431	1.0408	1.1384	1.2358	1.3330	1.4301	1.5271	1.6239	1.7206	1.8173
140		0.8509	0.9488	1.0466	1.1442	1.2417	1.3391	1.4363	1.5334	1.6303	1.7272	1.8239
150		0.8559	0.9539	1.0518	1.1495	1.2471	1.3445	1.4419	1.5390	1.6361	1.7331	1.8299
160		0.8605	0.9586	1.0565	1.1543	1.2520	1.3495	1.4469	1.5442	1.6413	1.7384	1.8353
170		0.8647	0.9628	1.0608	1.1587	1.2564	1.3540	1.4515	1.5488	1.6461	1.7432	1.8402
180		0.8685	0.9667	1.0647	1.1627	1.2604	1.3581	1.4557	1.5531	1.6504	1.7476	1.8448
190		0.8720	0.9702	1.0683	1.1663	1.2642	1.3619	1.4595	1.5570	1.6544	1.7517	1.8489
200		0.8752	0.9735	1.0717	1.1697	1.2676	1.3654	1.4631	1.5606	1.6581	1.7555	1.8527

,

Table 4. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.4$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6701	0.7642	0.8579	0.9511	1.0440	1.1365	1.2286	1.3204	1.4119	1.5031	1.5940
40		0.7145	0.8094	0.9040	0.9982	1.0921	1.1856	1.2788	1.3718	1.4645	1.5569	1.6491
50		0.7447	0.8402	0.9354	1.0302	1.1248	1.2190	1.3130	1.4067	1.5002	1.5934	1.6865
60		0.7671	0.8630	0.9585	1.0538	1.1489	1.2436	1.3381	1.4324	1.5265	1.6204	1.7140
70		0.7844	0.8806	0.9765	1.0722	1.1676	1.2627	1.3577	1.4524	1.5469	1.6412	1.7354
80		0.7983	0.8948	0.9910	1.0869	1.1826	1.2781	1.3734	1.4685	1.5633	1.6580	1.7526
90		0.8099	0.9066	1.0030	1.0992	1.1951	1.2909	1.3864	1.4818	1.5769	1.6719	1.7668
100		0.8197	0.9165	1.0131	1.1095	1.2057	1.3016	1.3974	1.4930	1.5884	1.6837	1.7788
110		0.8281	0.9251	1.0218	1.1184	1.2147	1.3109	1.4069	1.5027	1.5983	1.6938	1.7892
120		0.8354	0.9325	1.0294	1.1261	1.2226	1.3190	1.4151	1.5111	1.6069	1.7026	1.7982
130		0.8419	0.9391	1.0361	1.1330	1.2296	1.3261	1.4224	1.5185	1.6145	1.7104	1.8061
140		0.8476	0.9450	1.0421	1.1391	1.2358	1.3324	1.4289	1.5251	1.6213	1.7173	1.8132
150		0.8528	0.9502	1.0475	1.1445	1.2414	1.3381	1.4347	1.5311	1.6274	1.7235	1.8195
160		0.8575	0.9550	1.0523	1.1495	1.2464	1.3433	1.4399	1.5365	1.6329	1.7291	1.8253
170		0.8617	0.9593	1.0567	1.1540	1.2510	1.3480	1.4447	1.5414	1.6379	1.7342	1.8305
180		0.8656	0.9633	1.0608	1.1581	1.2552	1.3522	1.4491	1.5458	1.6424	1.7389	1.8353
190		0.8692	0.9669	1.0645	1.1619	1.2591	1.3562	1.4531	1.5500	1.6467	1.7432	1.8397
200		0.8725	0.9703	1.0679	1.1654	1.2627	1.3598	1.4569	1.5538	1.6505	1.7472	1.8438

,

Table 5. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.5$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6631	0.7559	0.8482	0.9400	1.0314	1.1223	1.2128	1.3030	1.3929	1.4824	1.5717
40		0.7085	0.8023	0.8957	0.9886	1.0812	1.1734	1.2653	1.3568	1.4481	1.5391	1.6298
50		0.7394	0.8339	0.9280	1.0217	1.1151	1.2082	1.3009	1.3934	1.4856	1.5776	1.6694
60		0.7622	0.8572	0.9518	1.0461	1.1401	1.2338	1.3272	1.4203	1.5132	1.6059	1.6984
70		0.7799	0.8753	0.9703	1.0650	1.1595	1.2536	1.3475	1.4412	1.5346	1.6279	1.7210
80		0.7941	0.8898	0.9852	1.0803	1.1751	1.2696	1.3639	1.4580	1.5519	1.6456	1.7391
90		0.8059	0.9019	0.9975	1.0929	1.1880	1.2828	1.3775	1.4719	1.5662	1.6602	1.7541
100		0.8159	0.9121	1.0079	1.1035	1.1989	1.2940	1.3889	1.4837	1.5782	1.6726	1.7668
110		0.8245	0.9208	1.0169	1.1127	1.2083	1.3036	1.3988	1.4938	1.5886	1.6832	1.7777
120		0.8320	0.9285	1.0247	1.1207	1.2165	1.3120	1.4074	1.5026	1.5976	1.6925	1.7872
130		0.8386	0.9352	1.0316	1.1277	1.2237	1.3194	1.4150	1.5104	1.6056	1.7007	1.7956
140		0.8445	0.9412	1.0377	1.1340	1.2301	1.3260	1.4217	1.5173	1.6127	1.7079	1.8030
150		0.8497	0.9466	1.0432	1.1397	1.2359	1.3319	1.4278	1.5235	1.6190	1.7145	1.8098
160		0.8545	0.9515	1.0482	1.1448	1.2411	1.3373	1.4333	1.5291	1.6248	1.7204	1.8158
170		0.8589	0.9559	1.0528	1.1494	1.2459	1.3421	1.4383	1.5342	1.6301	1.7257	1.8213
180		0.8629	0.9600	1.0569	1.1537	1.2502	1.3466	1.4428	1.5389	1.6349	1.7307	1.8264
190		0.8665	0.9637	1.0607	1.1576	1.2542	1.3507	1.4470	1.5432	1.6393	1.7352	1.8310
200		0.8699	0.9672	1.0643	1.1612	1.2579	1.3545	1.4509	1.5472	1.6433	1.7394	1.8353

,

Table 6. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.6$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6562	0.7478	0.8388	0.9292	1.0192	1.1086	1.1977	1.2864	1.3747	1.4627	1.5504
40		0.7026	0.7953	0.8876	0.9794	1.0707	1.1617	1.2523	1.3425	1.4325	1.5222	1.6116
50		0.7342	0.8277	0.9208	1.0135	1.1058	1.1977	1.2893	1.3807	1.4717	1.5625	1.6531
60		0.7574	0.8515	0.9452	1.0386	1.1316	1.2242	1.3166	1.4087	1.5006	1.5922	1.6836
70		0.7755	0.8700	0.9642	1.0581	1.1516	1.2448	1.3378	1.4305	1.5229	1.6152	1.7072
80		0.7900	0.8849	0.9795	1.0738	1.1677	1.2614	1.3548	1.4480	1.5409	1.6337	1.7263
90		0.8021	0.8973	0.9922	1.0868	1.1811	1.2751	1.3689	1.4625	1.5558	1.6490	1.7420
100		0.8122	0.9077	1.0029	1.0977	1.1923	1.2867	1.3808	1.4747	1.5684	1.6620	1.7554
110		0.8210	0.9167	1.0121	1.1072	1.2020	1.2967	1.3911	1.4853	1.5793	1.6731	1.7668
120		0.8286	0.9245	1.0201	1.1154	1.2105	1.3053	1.4000	1.4944	1.5887	1.6828	1.7768
130		0.8354	0.9314	1.0272	1.1227	1.2179	1.3130	1.4079	1.5025	1.5970	1.6914	1.7856
140		0.8414	0.9375	1.0335	1.1291	1.2246	1.3198	1.4149	1.5097	1.6044	1.6990	1.7934
150		0.8468	0.9431	1.0391	1.1349	1.2305	1.3259	1.4212	1.5162	1.6111	1.7058	1.8004
160		0.8516	0.9481	1.0442	1.1402	1.2359	1.3315	1.4269	1.5221	1.6171	1.7120	1.8068
170		0.8561	0.9526	1.0489	1.1450	1.2409	1.3365	1.4320	1.5274	1.6226	1.7176	1.8126
180		0.8601	0.9568	1.0532	1.1494	1.2453	1.3412	1.4368	1.5323	1.6276	1.7228	1.8179
190		0.8639	0.9606	1.0571	1.1534	1.2495	1.3454	1.4411	1.5367	1.6322	1.7275	1.8227
200		0.8673	0.9641	1.0607	1.1571	1.2533	1.3493	1.4452	1.5409	1.6365	1.7319	1.8273

,

Table 7. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.7$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6495	0.7399	0.8296	0.9187	1.0073	1.0955	1.1831	1.2704	1.3573	1.4439	1.5301
40		0.6968	0.7885	0.8797	0.9703	1.0605	1.1503	1.2397	1.3288	1.4175	1.5059	1.5941
50		0.7290	0.8216	0.9137	1.0054	1.0967	1.1876	1.2782	1.3684	1.4583	1.5480	1.6375
60		0.7527	0.8460	0.9388	1.0313	1.1233	1.2150	1.3064	1.3975	1.4884	1.5790	1.6694
70		0.7711	0.8649	0.9583	1.0513	1.1440	1.2363	1.3284	1.4202	1.5117	1.6030	1.6941
80		0.7860	0.8802	0.9740	1.0675	1.1606	1.2534	1.3460	1.4384	1.5305	1.6223	1.7141
90		0.7983	0.8928	0.9870	1.0808	1.1744	1.2676	1.3606	1.4534	1.5460	1.6383	1.7305
100		0.8086	0.9035	0.9979	1.0921	1.1860	1.2796	1.3730	1.4661	1.5591	1.6518	1.7444
110		0.8176	0.9126	1.0074	1.1018	1.1960	1.2899	1.3836	1.4771	1.5703	1.6634	1.7564
120		0.8253	0.9206	1.0156	1.1103	1.2047	1.2989	1.3928	1.4866	1.5802	1.6736	1.7668
130		0.8322	0.9277	1.0228	1.1177	1.2124	1.3068	1.4010	1.4950	1.5888	1.6825	1.7760
140		0.8383	0.9340	1.0293	1.1244	1.2192	1.3138	1.4083	1.5025	1.5965	1.6904	1.7842
150		0.8438	0.9396	1.0351	1.1304	1.2254	1.3202	1.4148	1.5092	1.6035	1.6976	1.7915
160		0.8488	0.9447	1.0404	1.1358	1.2309	1.3259	1.4207	1.5153	1.6097	1.7040	1.7982
170		0.8533	0.9494	1.0451	1.1407	1.2360	1.3311	1.4260	1.5208	1.6154	1.7099	1.8042
180		0.8574	0.9536	1.0495	1.1452	1.2406	1.3359	1.4310	1.5259	1.6206	1.7153	1.8098
190		0.8613	0.9575	1.0535	1.1493	1.2449	1.3403	1.4355	1.5305	1.6254	1.7202	1.8148
200		0.8648	0.9611	1.0572	1.1531	1.2488	1.3443	1.4397	1.5348	1.6299	1.7248	1.8196

,

Table 8. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.8$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6429	0.7321	0.8206	0.9085	0.9959	1.0827	1.1690	1.2550	1.3405	1.4258	1.5107
40		0.6912	0.7819	0.8720	0.9616	1.0507	1.1393	1.2276	1.3155	1.4031	1.4904	1.5773
50		0.7240	0.8157	0.9069	0.9976	1.0879	1.1778	1.2674	1.3566	1.4455	1.5342	1.6226
60		0.7481	0.8406	0.9326	1.0242	1.1153	1.2061	1.2966	1.3868	1.4767	1.5664	1.6558
70		0.7669	0.8599	0.9525	1.0447	1.1366	1.2281	1.3193	1.4102	1.5009	1.5913	1.6816
80		0.7820	0.8755	0.9686	1.0613	1.1537	1.2458	1.3375	1.4291	1.5204	1.6114	1.7023
90		0.7945	0.8884	0.9819	1.0750	1.1678	1.2604	1.3526	1.4447	1.5364	1.6280	1.7195
100		0.8051	0.8993	0.9931	1.0866	1.1798	1.2727	1.3654	1.4578	1.5501	1.6421	1.7340
110		0.8142	0.9087	1.0028	1.0966	1.1901	1.2834	1.3764	1.4692	1.5617	1.6542	1.7464
120		0.8221	0.9168	1.0112	1.1053	1.1991	1.2926	1.3859	1.4790	1.5719	1.6647	1.7573
130		0.8291	0.9240	1.0186	1.1129	1.2070	1.3008	1.3944	1.4877	1.5809	1.6740	1.7668
140		0.8353	0.9305	1.0253	1.1198	1.2140	1.3081	1.4019	1.4955	1.5889	1.6822	1.7753
150		0.8409	0.9362	1.0312	1.1259	1.2204	1.3146	1.4086	1.5024	1.5961	1.6896	1.7830
160		0.8460	0.9414	1.0366	1.1314	1.2261	1.3205	1.4147	1.5087	1.6026	1.6963	1.7899
170		0.8506	0.9462	1.0415	1.1365	1.2313	1.3259	1.4203	1.5145	1.6085	1.7024	1.7962
180		0.8548	0.9505	1.0459	1.1411	1.2360	1.3308	1.4253	1.5197	1.6139	1.7080	1.8020
190		0.8587	0.9545	1.0500	1.1453	1.2404	1.3353	1.4300	1.5245	1.6189	1.7132	1.8073
200		0.8623	0.9582	1.0539	1.1493	1.2445	1.3395	1.4343	1.5290	1.6235	1.7179	1.8122

and

Table 9. LCBIA for various values of $n$ and ${\widehat{I}}_A^T$ as $\xi =0.9$.

	${\widehat{\mathit{I}}}_{\mathit{A}}^{\mathit{T}}$	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	19	2.0
n
30		0.6365	0.7245	0.8119	0.8986	0.9847	1.0703	1.1554	1.2401	1.3244	1.4083	1.4919
40		0.6856	0.7753	0.8645	0.9530	1.0411	1.1287	1.2159	1.3027	1.3892	1.4753	1.5612
50		0.7190	0.8099	0.9002	0.9900	1.0794	1.1683	1.2569	1.3452	1.4331	1.5208	1.6082
60		0.7436	0.8353	0.9265	1.0172	1.1075	1.1975	1.2871	1.3764	1.4654	1.5542	1.6427
70		0.7627	0.8550	0.9469	1.0383	1.1294	1.2201	1.3105	1.4006	1.4905	1.5801	1.6695
80		0.7781	0.8709	0.9633	1.0553	1.1470	1.2383	1.3293	1.4201	1.5106	1.6009	1.6910
90		0.7909	0.8841	0.9769	1.0694	1.1615	1.2533	1.3449	1.4362	1.5273	1.6181	1.7088
100		0.8016	0.8952	0.9884	1.0813	1.1738	1.2661	1.3581	1.4498	1.5413	1.6327	1.7239
110		0.8109	0.9048	0.9983	1.0915	1.1844	1.2770	1.3694	1.4615	1.5535	1.6452	1.7368
120		0.8190	0.9131	1.0069	1.1004	1.1936	1.2865	1.3792	1.4717	1.5640	1.6561	1.7481
130		0.8261	0.9205	1.0145	1.1083	1.2017	1.2949	1.3879	1.4807	1.5733	1.6657	1.7580
140		0.8324	0.9270	1.0213	1.1153	1.2090	1.3024	1.3957	1.4887	1.5816	1.6743	1.7668
150		0.8381	0.9329	1.0274	1.1215	1.2155	1.3092	1.4026	1.4959	1.5890	1.6820	1.7748
160		0.8433	0.9382	1.0329	1.1272	1.2213	1.3152	1.4089	1.5024	1.5957	1.6889	1.7820
170		0.8480	0.9431	1.0379	1.1324	1.2267	1.3208	1.4146	1.5083	1.6019	1.6953	1.7885
180		0.8522	0.9475	1.0424	1.1371	1.2316	1.3258	1.4199	1.5138	1.6075	1.7010	1.7945
190		0.8562	0.9516	1.0467	1.1415	1.2361	1.3305	1.4247	1.5187	1.6126	1.7064	1.8000
200		0.8598	0.9553	1.0505	1.1455	1.2403	1.3348	1.4291	1.5233	1.6174	1.7113	1.8051

present the LCBIA values for ${\widehat{I}}_A^T=1.0(0.1)2.0$ and $n=30(10)200$ by implementing R software (version 1.1.419; R Foundation for Statistical Computing, Vienna, Austria), where $\xi =0.1(0.1)0.9$ under a 95% confidence level (i.e., $\alpha =0.05$). Note that the proposed formulation can be applied to any decision-making situation that requires a conservative estimate of profitability simply by calculating the values of ${\widehat{I}}_A^T$ and $\xi$ (based on historical demand in all channels) and then looking up the LCBIA values in the corresponding table. For example, when ${\widehat{I}}_A^T=1.5$ and $\xi =0.6$ for a given sample size of $n=50$, the corresponding LCBIA can be found in Table 6 (i.e., $I_A^{TLCB}=1.1977$). If the required profitability (denoted by $I_A^{TR}$) of a given product is 1.2 (under a 95% confidence level), then the result (i.e., $I_A^{TLCB}=1.1977<I_A^{TR}=1.2$) indicates that the profitability of that product is insufficient, such that it should be discontinued or actively promoted. Figure 3 presents three-dimensional LCBIA graphs for various ${\widehat{I}}_A^T$, $\xi$, and $n$ as a function of the designated confidence level ($\alpha =0.1,0.05,0.025,0.01$). Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 and Figure 3 illustrate various trends in the LCBIA, which could be considered managerial insights to guide decision-making, including the following:

(1)

The LCBIA value increases (approaching ${\widehat{I}}_A^T$) with an increase in the sample size as long as the values of ${\widehat{I}}_A^T$, $\xi$, and $\alpha$ remain unchanged. This implies that the evaluation does not have to be particularly conservative due to a reduction in sampling error resulting from an increase in the size of the sample data. Note, however, that when the sample size increases, the value of ${\widehat{I}}_A^T$ may vary, such that the above results do not necessarily hold in practice. Thus, it is necessary to consider the sample size as well as changes in ${\widehat{I}}_A^T$.

(2)

The LCBIA value increases with an increase in the estimator ${\widehat{I}}_A^T$, as long as the values of $n$, $\xi$, and $\alpha$ remain unchanged. This correlation can also be observed in Equation (22). Note that the sampling error is positively correlated with the estimator.

(3)

Decreasing the risk of Type-I error will lead to a decrease in the value of LCBIA, as long as the values of ${\widehat{I}}_A^T$, $\xi$, and $n$ remain unchanged. This implies that the estimates of profitability become increasingly conservative as the confidence level increases.

(4)

The LCBIA value increases with a decrease in the value of $\xi$, as long as the values of ${\widehat{I}}_A^T$, $\alpha$, and $n$ remain unchanged. Based on the formulation of $\xi$, we know that the denominator is always greater than the numerator. Thus, as the value of $\xi$ decreases, any increase in the number of channels will lead to a greater disparity between the denominator and numerator. Figure 4 presents simulation results of $\xi$ for specific ranges of standard deviation (i.e., $s=1~10$, $s=10~20$, $s=20~30$, $s=30~40$, and $s=40~50$) and channel numbers. Note that we generated 10,000 sets of standard deviations for $h=5(1)25$. For each $h$, we calculated the value of $\xi$ for each set and then obtained the average of $\xi$ for all sets. As shown in Figure 4, when $h\ge 17$, the $\xi$ values simulated for all the specified standard deviation ranges were less than 0.1. This observation opens the door to establishing a preliminary approach for the estimation of LCBIA. When $h\ge 17$, we can disregard the calculation of $\xi$ and directly set $\xi =0.1$ to obtain the most conservative estimate of profitability by looking at the LCBIA values in Table 1. We compared this with the most conservative estimate and $I_A^{TR}$ as a preliminary evaluation to make decision-making more efficient.

4.2.2. Unequal Sample Sizes

Note that Equation (21) cannot be simplified when $n_1$, $n_2$,…, and $n_h$ are unequal. In this situation, it is possible only to calculate the estimate (i.e., ${\widehat{I}}_A^T$) and three summation terms (i.e., ${\displaystyle {\sum }_{i=1}^h{s}_i^2{n_i}^{-1}}$, ${\displaystyle {\sum }_{i=1}^h{s}_i^2}$, and ${\displaystyle {\sum }_{i=1}^h{s}_i^4{(n_i-1)}^{-1}}$) separately, which are then substituted into Equation (21) to obtain the LCBIA value. This complex calculation procedure is inconvenient and inefficient for decision-making. To simplify the process, it is possible to take the worst- and best-case profitability scenarios as the preliminary results. If we define $n_{\max }=\max \{n_1,n_2,\dots ,n_h\}$ and $n_{\min }=\min \{n_1,n_2,\dots ,n_h\}$ as the maximum and minimum sample sizes in all channels, we obtain the following inequalities:

$$\frac{1}{n_{\max }}=\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^2}{n_{\max }}}}{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}<\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^2}{n_i}}}{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}<\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^2}{n_{\min }}}}{{\displaystyle {\sum }_{i=1}^h{s}_i^2}}=\frac{1}{n_{\min }}$$

(23)

and

$$\frac{\xi }{n_{\max }-1}=\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^4}{n_{\max }-1}}}{{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^2}<\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^4}{n_i-1}}}{{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^2}<\frac{{\displaystyle {\sum }_{i=1}^h\frac{s_i^4}{n_{\min }-1}}}{{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^2}=\frac{\xi }{n_{\min }-1}.$$

(24)

From Equation (21), it is easy to see that the LCBIA value is negatively correlated with the values of ${\displaystyle {\sum }_{i=1}^h{s}_i^2{n}_i^{-1}}/{\displaystyle {\sum }_{i=1}^h{s}_i^2}$ and ${\displaystyle {\sum }_{i=1}^h{s}_i^4{(n_i-1)}^{-1}/{\left({\displaystyle {\sum }_{i=1}^h{s}_i^2}\right)}^2}$. Thus, the accuracy of the LCBIA value within interval $I_A^{TLCB}\in (I_A^{TLCB(w)},I_A^{TLCB(b)})$, where

$$I_A^{TLCB(w)}={\widehat{I}}_A^T-z_{1-\alpha }\sqrt{\frac{1}{n_{\min }}+\frac{{\widehat{I}}_A^{T2}}{2(n_{\min }-1)}\xi }$$

(25)

and

$$I_A^{TLCB(b)}={\widehat{I}}_A^T-z_{1-\alpha }\sqrt{\frac{1}{n_{\max }}+\frac{{\widehat{I}}_A^{T2}}{2(n_{\max }-1)}\xi }$$

(26)

are defined as the worst- and best cases of LCBIA. Note that the formulas for Equations (25) and (26) are similar to Equation (21), such that the values of $I_A^{TLCB(w)}$ and $I_A^{TLCB(b)}$ are easily obtained simply by looking them up in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and

Table 10. Historical data collected from 18 channels (C: Channel and B: batch).

	C	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
B
1		46	39	43	39	27	23	21	27	26	18	31	30	31	12	18	20	22	30
2		35	38	36	41	28	22	28	36	25	17	26	32	39	12	17	20	27	29
3		41	37	33	36	25	21	21	25	23	19	25	32	38	9	14	24	28	31
4		46	35	40	44	26	27	22	27	22	20	28	33	37	11	20	20	29	35
5		43	30	37	35	22	24	22	24	20	18	24	27	27	7	21	23	32	35
6		36	35	38	44	22	26	29	23	19	18	21	30	31	13	21	21	25	37
7		35	35	41	42	24	20	29	32	22	17	30	26	33	13	23	25	28	26
8		43	35	38	38	32	21	31	29	28	17	29	32	30	8	16	24	30	30
9		36	39	38	38	28	22	25	27	24	16	30	30	27	13	18	17	24	38
10		41	39	41	35	27	26	26	27	26	13	25	27	34	12	20	26	26	32
11		44	31	45	33	23	24	28	24	27	18	22	29	30	11	16	26	33	35
12		36	37	40	42	20	24	29	20	15	18	24	32	31	10	19	25	25	29
13		38	36	40	37	29	22	25	34	17	19	25	37	36	13	23	23	24	34
14		35	37	33	42	21	24	26	25	20	19	31	28	39	11	19	23	26	32
15		41	41	35	31	26	26	18	26	24	13	27	30	31	11	23	25	28	33
16		41	41	36	43	29	22	25	29	25	17	35	29	37	14	24	23	29	30
17		44	34	38	37	28	22	20	28	20	18	34	29	24	12	22	15	26	32
18		43	35	41	40	25	17	24	24	8	20	31	32	30	9	17	25	24	30
19		44	30	38	43	26	24	28	19	17	19	25	28	31	7	19	26	28	30
20		37	40	40	44	16	20	27	25	16	21	32	34	32	11	22	18	28	36
21		40	43	38	44	25	25	22	30	22	17	27	31	30	10	22	24	26	35
22		40	41	38	36	23	22	29	24	12	15	33	35	31	11	21	21	26	29
23		42	40	42	38	23	24	21	26	23	17	29	29	31	13	23	21	35	40
24		43	33	39	39	25	24	29	26	21	15	28	28	36	10	20	23	27	37
25		43	28	35	42	28	22	25	27	18	18	30	32	33	12	18	21	23	34
26		37	39	38	40	26	22	21	28	13	20	26	27	29	13	18	21	25	30
27		42	24	40	44	20	21	23	28	15	18	26	23	27	12	19	25	26	33
28		47	32	40	37	26	18	27	25	10	15	33	30	24	14	15	22	30	33
29		35	29	43	42	21	23	28	32	24	18	26	28	35	13	22	24	22	34
30		40	40	31	41	26	19	21	28	19	20	30	28	36	9	26	21	26	29
31			39	36	42	29	26	26	25	24	17	26	29	38	10			25	32
32			41	41	42	29	23	25	30	26	19	38	24	21				29	36
33			38	39	41	24	24		21	15	17	31		40				23	37
34			32	40	46	26	23		28	29	16	21		38				24	31
35			39	45	40	18	23		25	25	20			31				24	35
36			33	40	35	25	22		30	19	23			32				28	32
37			38	40	45	31	26		28	19	21			29				26	36
38			32	38	32	22	25		25	27				34				22	30
39			33	47		17	23		27	14				32				30	35
40			37	39		23	26		27	21				27				26	29
41			34	38		26			28	24				35				26	30
42			34	36		20			33	15				33				26	28
43			36	42		28			29	22				26				30	26
44			40	40		21								32				31	38
45			33	35		29								37				26	37
46			30	40														28
47			35	41														30
48			33	39														23
49			34
50			30
K-S		0.710	0.621	0.252	0.391	0.490	0.430	0.658	0.467	0.717	0.362	0.700	0.759	0.830	0.45	0.893	0.480	0.138	0.332
A-D		0.05	0.234	0.05	0.100	0.211	0.090	0.065	0.124	0.242	0.098	0.572	0.351	0.291	0.051	0.673	0.068	0.093	0.099

. It is possible to make a preliminary decision using the values of $I_A^{TLCB(w)}$ and $I_A^{TLCB(b)}$ without calculating the accuracy value of $I_A^{TLCB}$. For example, if $I_A^{TLCB(b)}<I_A^{TR}$, then we can conclude that the designated profitability is unmet. Conversely, if $I_A^{TLCB(w)}>I_A^{TR}$, then the designated profitability is met. Note however that using this method, it is not possible to obtain absolute results if $I_A^{TLCB(w)}<I_A^{TR}<I_A^{TLCB(b)}$. It is possible only to confirm the results by calculating the accuracy of LCBIA. Note that the difference between $I_A^{TLCB(b)}$ and $I_A^{TLCB(w)}$ is positively correlated with the difference between $n_{\max }$ and $n_{\min }$. This implies that when the values of $n_{\max }$ and $n_{\min }$ are nearly equal, the proposed method could theoretically be used to obtain absolute results for decision-making. The results listed above are summarized in Figure 5 in the form of an illustration showing the decision-making process based on conservative estimates of integrated profitability in scenarios involving samples of equal or unequal sizes. Essentially, the user needs only to calculate the value for ${\widehat{I}}_A^T$ and then look up the corresponding value in a table.

5. Practical Applicability

In the following section, we demonstrate the applicability of the proposed method by applying it to an OBM enterprise engaged in the manufacture and sale of own-brand bedding (e.g., pillows, mattresses, and cushions). In this scenario, most of the sales channels are department stores and flagship stores, thereby allowing the transfer of products among channels to decrease operational losses due to uncertain demand. Note that the color of the foam used in these products tends to darken over time. This in no way impedes the function of the product; however, it reduces the visual appeal. A basic requirement of bedding is cleanliness, such that in seeking to preserve its reputation, the enterprise must remove expired (i.e., darkened) stock from the shelves, thereby incurring the initial cost of manufacturing and shipping, as well as additional costs for disposal. Shortages incur additional shipping costs and delay delivery. Thus, bedding in this scenario can be regarded as a newsboy-type product.

In the following, we explore a real-world case involving a change in the sales model. The growing popularity of e-commerce has prompted many brands to adopt sales models that include virtual stores; however, there are inherent differences between the operating models used for virtual and physical stores, particularly in terms of pricing and promotion. Sales staff generally receive complaints whenever customers perceive a discrepancy between the online price and the price in physical stores, thereby necessitating the differentiation of products as physical-exclusive and virtual-exclusive. It has also been reported that enterprises tend to transfer unprofitable products from physical stores to virtual stores. As a result, many OBM enterprises must deal with the issue of integrated profitability evaluation.

Our focus in this study was on determining whether a particular pillow should be listed as a virtual-exclusive item. We sought to achieve this by measuring the integrated profitability of the pillow using the IACI index. This involved collecting historical demand data from all channels in order to estimate the true value of IACI based on the fact that the demand parameters for all channels are unknown. Table 10 lists the historical demand data for the pillow from $h=18$ channels. This OBM enterprise does not adopt unified replenishment among the channels; therefore, the replenishment batches and demand sample sizes among channels are inconsistent. Table 10 also lists the p-values obtained using R software (v1.1.419) based on the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) normality tests. The fact that all p-values exceeded 0.05 confirmed that there was no significant violation of normal distribution.

The parameter values for the target item (pillow) are as follows:

• Selling price: $p=3300$;
• Manufacturing cost: $c=2100$;
• Net profit: $c_p=p-c=1200$;
• Total target profit per period: $k_t=\mathrm{564,000}$;
• Total target demand per period: $T_t=k_t/c_p=470$;
• Shortage cost: $c_s=250$;
• Disposal cost: $c_d=150$;
• Excess cost: $c_e=c_d+c=2250$.

Note that the above-mentioned historical demand data, selling price, and costs have been slightly modified to prevent the leaking of sensitive data. As shown in Table 10, the sizes of the demand samples in the various channels were not identical. The maximum and minimum sample sizes were $n_{\max }=n_1=50$ and $n_{\min }=n_2=30$, respectively. The OBM manager should transfer this pillow to virtual stores when the $I_A^T$ value drops to below $I_A^{TR}=1.25$ (i.e., integrated profitability is ${\mathsf{\Omega }}_t(1.25)=0.7919$).

In the following, we discuss this issue from the perspective of conservative profitability based on samples of equal and unequal sizes:

(1)

Case involving samples of unequal size: The black text in Table 10 refers to the case where all samples in all channels are of equal size ($n_1=n_2=\dots =n_{18}=n_{\min }=30$). Based on the partial historical demand data, we first calculate the IACI estimate at ${\widehat{I}}_A^T=1.6931$. The fact that ${\widehat{I}}_A^T=1.6931>I_A^{TR}=1.25$ means that this pillow can still be sold profitably in physical stores. Note, however, that this finding is the result of point estimation, which introduces the risk that IACI was overestimated due to sampling error. Thus, we also obtain the LCBIA value in order to derive a conservative estimate. Note that this case involves a large number of channels and the ${\widehat{I}}_A^T$ estimate is relatively small (close to 1.6), which means that the preliminary estimation method is applicable using the following parameters: $\xi =0.1$ and ${\widehat{I}}_A^T=1.6$. Based on the LCBIA value in Table 1, we can obtain the LCBIA value for ${\widehat{I}}_A^T=1.6$ and $n=30$ ($I_A^{TLCB}=1.2804$). This conservative estimate $I_A^{TLCB}=1.2804>I_A^{TR}=1.25$ also indicates that this pillow could still be sold profitably in physical stores with 95% confidence. Finally, we compute the exact estimate for LCBIA, where the exact ratio of variation is $\xi =0.068$. By substituting $\xi =0.068$, ${\widehat{I}}_A^T=1.6931$, $\alpha =0.05$, and $n=30$ into Equation (22), we obtain the exact LCBIA value ($I_A^{TLCB}=1.3780$), which does not deviate from the previous suggestions.

(2)

Case involving samples of equal size: As shown in Table 10, the maximum and minimum sample sizes for all channels were $n_{\max }=50$ and $n_{\min }=30$, respectively. Note that $h\ge 17$, such that $\xi =0.1$. Based on LCBIA values from Table 1 where $n=30$, $n=50$, and ${\widehat{I}}_A^T=1.6$, we obtain $I_A^{TLCB(w)}=1.3780$ and $I_A^{TLCB(b)}=1.4508$. The fact that $I_A^{TLCB(w)}=1.3780>I_A^{TR}=1.2$ indicates that this pillow can still be sold profitably in physical stores with 95% confidence.

6. Conclusions

This article explores the issue of assessing the conservative profitability of single-period products in an OBM enterprise with multiple owned channels, where products can be transferred among channels to avoid operational losses due to unknown demand (i.e., the sizes, means, and variances of demand differ in each channel). We structured the integrated profitability of a single-period product in all channels with normally distributed demand and defined an IACI index by which to derive the integrated profitability in a simple and straightforward manner. To deal with inter-channel variations in sample size and population, we applied the Taylor expansion to approximate the statistical properties of the IACI estimator for use in deriving the lower confidence bound on the IACI as a conservative indicator of profitability. We also tabulated the LCBIA values for samples of various sizes at the 95% confidence level. Note that the LCBIA can be used to prevent the overestimation of profits due to sampling error, and the proposed tables are universally applicable to any OBM enterprise requiring a conservative estimate of profitability. We also developed an efficient decision-making procedure that allows OBM managers to determine whether the aggregate profitability of a given product meets the minimum profitability requirements simply by looking up a single value in a table. In previous literature, some authors have also determined the optimal order quantities in a newsboy problem with multiple locations (demands), such as Şen and Zhang [16], Cherikh [17], and Lin et al. [18]. This paper not only determines “how much to manufacture” but also “what/which to manufacture” in the OBM enterprise with multiple channels (demands). The applicability of the proposed method is demonstrated using a real-world case involving the sale of household items in an OBM enterprise that includes virtual and physical channels.

This work could be extended in several directions: (1) researchers could consider other approximated estimation approaches by which to estimate IACI and compare their performance; (2) researchers could study IACI estimates based on non-normal distributions; (3) researchers could develop an LCBIA based on Bayesian estimation rather than frequentist estimation for situations where a priori demand-related information is available; (4) researchers could perform sensitivity analysis for parameters involving decision transitions, such as confidence level, target profit, related costs, and selling price; and (5) researchers could explore nonparametric bootstrap methods for situations in which the demand pattern is unknown; (6) This paper has some limitations of the research. For example, we assume that the demands of each channel are mutually exclusive. However, this assumption may not be met in some scenarios involving special sale models. The proposed model may not accurately measure the integrated profitability when the demands of each or partial channel are dependent. Therefore, it is worth developing an IACI that includes correlation factors (e.g., correlation coefficients) of demand between channels.