Forecasting in the fashion industry: a model for minimising supply-chain costs

ABSTRACT

Fashion is replaced every season and collections change rapidly, depending on certain events. There are only a few weeks between the fashion shows and the collections reaching their sale points. As the pattern of demand is seasonal, new items must be produced every season. Additionally, colours and patterns change rapidly, creating a need for producers and consumers to continually remain updated. This research study proposes a forecasting model that enhances the accuracy of fashion trend forecasting in the context of multiple variants of colour clothing. The model aims to maximise the firms’ profits, while minimising forecasting errors and reducing costs that result from excess production or, alternatively, from the loss of potential revenues due to low demand. In the proposed model, the expected profit was notably higher when the customers’ readiness to compromise was low or when only one type of product was in stock.

KEYWORDS:

• Fashion
• trend forecasting
• colour variants
• supply chain costs
• seasonal demand

1. Introduction

The global apparel industry is one of the most lucrative industries that experiences a significant annual growth rate. It is predicted to become worth US$2 trillion by 2023 (Euromonitor International, 2020). Part of the growth has been attributed to the emergence of new industry players, collectively referred to as ‘fast fashion retailers,’ which have substantially expanded since the turn of the century. Following the outbreak of the COVID-19 pandemic at the beginning of 2020, retail stores temporarily closed. Thus, COVID-19 was the catalyst for the majority of fashion retailers to quickly shift to operate via e-commerce in 2020, a change that has continued beyond the pandemic.

The fashion industry has undergone many changes since the early 1990s due to the strengthening of international trends in fashion production and the opening of new global markets. Consequently, the fashion industry, which was once characterised by standardised mass production, has expanded in the variety of products and increased the flexibility of supply chains (Wen, Choi, & Chung, 2019).

Following these changes in the fashion industry, prominent retailers, such as Zara and H&M, created fast fashion (Caro & Martínez-de-Albéniz, 2015). The development of fast fashion began in the early 1990s when the fashion market transitioned from two seasons to four, with most fast fashion retailers now producing more seasons. Fast fashion is not a long-term process but rather a variety of design and manufacturing processes that occur in response to rapid changes in demand. These changes follow the latest fashion trends and take on a competitive position based on a low-cost structure and shorter delivery times. Competition in the fashion market, similar to that in other consumer-product markets, is based on speed to market (Cachon & Swinney, 2011).

To maintain market share and profitability, the goal of retailers is to keep costs low while continuing the rapid response and flexibility of the supply chain. However, due to product development (i.e. design process), long lead times may exist despite these efforts, extending the time it takes for the product to reach the sale points. In many cases, outsourcing may result in significantly longer lead times (Cook & Yurchisin, 2016). Therefore, supply-chain flexibility and swift product design and development result in shorter lead times and, consequently, increase the company's profits.

Fast fashion retailers mostly sell products at affordable prices, therefore, prices at different retailers are usually within a similar range. In addition, fast fashion retailers combine supply-chain agility, quick responses, and constant product introductions to attract variety-seeking/fashion-conscious customers. Hence, most fast fashion retailers outperform other types of fashion retailers. In fact, fast fashion stores, such as H&M and Zara, have established themselves as recognised brands and grown to become the world’s largest apparel retailers.

The other necessary variables in companies’ production decisions are the quantities to be released to the market throughout the season, the bulk of which is usually produced in advance, and the demand by potential customers in terms of ‘design’ versus price (Niinimäki et al., 2020).

The contributions of this paper are as follows: (i) Derivation of closed form analytical solutions for optimal inventory levels; (ii) Modelling of the effect multiple variants of colour for clothing on the expected profit in contrast to the common approach with the order quantity set equal to the forecasted (expected) values and (iii) Different solutions for inventory levels are offered based on the ratio between the selling price and ordering cost.

2. Literature review

In the fashion industry, inventory management and forecasting are complex tasks due to the short life cycle of products and the high volatility of demand. Under the assumption that products offered in prior seasons cannot be resold or recycled in the following season, the retailer begins each new production cycle with a complete lack of inventory. Colour is one of the critical elements in fashion and is highly related to the inventory and production planning of fashion-apparel products. In the fashion industry, most clothes are presented in two main key colours (e.g. see Goyal, Levi, & Segev, 2016; Koren, Perlman, & Shnaiderman, 2022; Ovezmyradov & Kurata, 2019).

The supply chain should be based on the characteristics of the company's product demand. As mentioned, in the fashion world, products are characterised by high demand volatility and a short life cycle. The supply-chain structure of the fashion apparel industry is divided into four parts: raw materials, components, production and marketing (Guo, Choi, & Shen, 2020). Raw materials in industrial fashion are the base materials, such as cotton and wool, used to produce fibres and yarns. First, the raw materials are supplied to companies that manufacture the yarns and fabrics. At this stage, suppliers and manufacturers of raw materials and components face lower entry barriers into the market. Next, these materials are passed down the supply chain to the manufacturers to produce the final products. The manufacturers design and sew the clothes, which are then shipped to retailers for distribution to the points of sale. As the final products move down the supply chain, the difficulties of entering the market increase. It is also important to note that branding is significant for retailers. Therefore, branding requires a high initial investment in logistics, distribution, and marketing infrastructure, as well as in advertising, branding, brand reputation, and customer loyalty (Khoa, 2020).

Various complex factors, which are difficult to predict, affect sales in the fashion industry: weather, geopolitical conditions, cultural factors, marketing strategy, seasonal factors, and trends in the fashion market (Wen et al., 2019). However, the agile supply chain allows for quick changes throughout the chain’s various stages. This agility is necessary for a market characterised by volatile demand and a wide variety of products. Hence, the agile supply chain is suitable for companies in the fast fashion industry.

Some companies have hybrid supply chains, combining agility and lean management. For example, Zara's fast fashion business model contradicts most conventional views regarding the conduct of the supply chain (Essaber, Benmoussa, De Guio, & Dubois, 2021). The company chooses not to use the full output capability of its production lines. Instead, it manufactures and sells small series to ensure that every product sells out, produces half of the quantity in-house, manages most of its own logistical operations, encourages out-of-stock situations, and even leaves empty spaces in stores to create demand for an item. Therefore, certain risks, including unsold items, are eliminated in terms of their unreleased revenue and logistical management (Aftab, Yuanjian, Kabir, & Barua, 2018). This model is contrary to how most companies operate in the market, and gives Zara its unique character and profitability (Backs, Jahnke, Lüpke, Stücken, & Stummer, 2021).

To reduce costs and delivery times, the company's supply chain comprises a minimum number of points on the way to customers. Thus, the production and delivery of products are accomplished quickly, with an average delivery time of about three weeks from planning to arrival at the points of sale. The main advantage of this method is that the production agility prevents the need to anticipate trends and demands far in advance (Caro & Gallien, 2010).

Trend forecasting is significant both for matching the introduced collection to the market’s tastes and preferences and for adapting the supply of the produced goods to the demand. In terms of predicting consumer preferences and selecting items marketed to those preferences, items that are not sold during the season may be sold at the end of or after the season at a discount. Therefore, they are either sold at a loss to the retailer or not sold at all. Well-founded forecasts can prevent the costs associated with excess or lack of inventory. Additionally, forecasting is critical for enhancing production and procurement planning, lead times, and inventory management. Forecasting is a difficult task in the fashion industry due to the short life cycle of products and the high volatility of demand (Boone, Ganeshan, Jain, & Sanders, 2019; DuBreuil & Lu, 2020).

Under the assumption that the products offered in prior seasons cannot be resold or recycled in the following season, the retailer begins each new production cycle with no inventory. Consequently, for each season, fashion producers attempt to identify customers’ consumption habits and demands to increase the prospects of the success of their upcoming collections. As a result, they base their sales forecasts on the historical timelines of similar goods. It is also important to note that, despite the availability of historical data for similar products, no historical data exists for any of the new products. Previous empirical findings have supported the notion of brand loyalty, by which some consumers do not replace their intended purchase attitudes with other brands when their desired branded product is unavailable (Ovezmyradov & Kurata, 2019; Stathopoulou & Balabanis, 2016). These factors make demand forecasting more difficult compared to forecasting for existing products.

Most fashion-forecasting reports are produced by the milieu of designers and cultural experts who predict the attributes of upcoming collections based on cultural and societal developments. However, forecasting the success of new designs with the public provides only one part of the necessary information for producers. The reports rarely involve any insight from economists and engineers, despite the wealth of forecasting models and methodologies applied in statistical, demographic, and social analyses, as well as in economics and marketing research. Recent works have applied analytical and quantitative tools and techniques to assist in assessing the predictability of customer tastes and to decrease forecasting errors ex-ante (e.g. Zhou, Meng, Wang, & Xiaoxuan, 2020).

WGSN (https://www.wgsn.com) is a market leader in fashion-trend forecasting that forecasts trends in various markets, such as women's and children's clothing. Agencies such as Color Marketing Group (CMG; https://colormarketing.org/) and Color Association of the United States (CAUS; http://www.colorassociation.com) focus on colour forecasting, which are the design elements that have the most influence on fashion consumers. Experts from various fields, on behalf of the agencies, consider the current street fashion (i.e. grassroots fashion, most often seen in major urban centres) as well as the cultural and social elements of populations to predict future fashion trends. These experts may include sociologists, anthropologists, designers, and architects (DuBreuil & Lu, 2020).

Various methods for forecasting in the fashion industry are based on statistical methods, artificial neural networks, grey methods, and hybrid methods. Statistical methods to predict fashion trends include reliance on an average, moving average, weighted moving average, exponential smoothing, and regression. These methods are particularly suitable for forecasting collections with seasonal or minor yearly changes, such as winter suits and shirts in neutral colours. However, extreme values, such as the COVID-19 pandemic, are not considered (Ioannidis, Cripps, & Tanner, 2020; Lai & Westland, 2020).

In addition, novel methods are available for predictions, such as identifying statistically significant clickstream tracking on non-transactional websites to estimate ordering probability, as well as using Google Trends (https://trends.google.co.il/trends) to improve product forecasts (Boone, Ganeshan, Hicks, & Sanders, 2018). These methods are based primarily on reducing forecast error and do not include the cost components in the supply chain, as shown in the model developed within the framework of this study.

3. Model

The proposed model considered one retailer that sells two types of clothing products. Let iε{1,2} be the demand for product i, denoted by d_i, which was a random variable with a probability density function f_i(t_i) and cumulative distribution function F_i(t_i). The unit selling price and order cost of product i are p_i and c_i, respectively. Let x_i denote the inventory level of product i (i.e. a decision variable). If this level does not meet the demand of d_i, then each customer who does not receive their desired product may choose to buy the other one, if available, with a probability of δ_i,. Consequently, the expected profit from that customer would be δ_i(p_j-c_j) (j≠i). Alternatively, the consumer may decide not to buy the alternative product which may be related to brand loyalty, as mentioned earlier.

3.1. Calculation of the distributions of demand

The Autoregressive Integrated Moving Average (ARIMA) is a statistical technique that combines a variety of statistical methods, such as the integration of regression and automatically moving averages for predictions (Aburto & Weber, 2007; Box & Pierce, 1970). Although ARIMA models are based on the assumption that linear relationships between variables exist, they provide better clarity regarding the effects of independent variables on the resulting behaviour of fashion customers. Additionally, ARIMA models are based on a smaller volume of data, which describes market behaviour over shorter periods in various fields (Choi, Hui, & Yu, 2013).

Accordingly, the model hereinafter is based on the ARIMA statistical method to determine the function of the demand of fashion customers. The ARIMA method and the particular model, AR(1), were applied due to the need to carry out forecasts with a normally distributed demand for fashion goods and the more restricted time periods of fast fashion seasons (for similar practices in various industrial sectors, see Aviv, 2007; Hernandez-Matamoros, Fujita, Hayashi, & Perez-Meana, 2020; Nyoni, 2018; Satrio, Darmawan, Nadia, & Hanafiah, 2021). Nevertheless, other forecasting methods could also be used.

The demands were assumed to be normally distributed, satisfying the autoregressive processes of AR(1). For iε{1,2}, the demand in season n depended on that of the previous season, as follows: (1) $d_i^{\left(n\right)}=D_i+{\rho }_i{d}_i^{\left(n-1\right)}+{ϵ}_i^{\left(n\right)},$(1) where D_i and ρ_i were constant (such that |ρ_i|<1), and ϵ_i⁽ⁿ⁾ was the white noise, which was normally distributed with a mean of 0 and variance of ${\sigma }_{{ϵ}_i}^2$ (see also Aviv, 2007). Given a sample of d̅_i= [d_i⁽¹⁾, … , d_i⁽ⁿ⁻¹⁾], the coefficients D_i and ρ_i were calculated by the least squares method. The unconstrained mean of d_i⁽ⁿ⁾ was D_i/(1-ρ_i). Given the previous demand, d_i⁽ⁿ⁻¹⁾, the demand d_i⁽ⁿ⁾ was normally distributed with a mean of μ_i= D_i+ρ_i d_i⁽ⁿ⁻¹⁾ and variance of ${\sigma }_{{ϵ}_i}^2$.

Equipped with the AR(1) coefficients, the variance ${\sigma }_{{ϵ}_i}^2$ was estimated as follows. For every 2 ≤ k ≤ n-1, let d^_i^(k)  = D_i +ρ_id_i^(k−1) be the estimated value for the demand d_i^(k), then ϵ_i^(k)= d_i^(k)- d^_i^(k). Based on the sample [ϵ_i⁽²⁾, … , ϵ_i⁽ⁿ⁻¹⁾] of size n-2 as well as the known zero-mean of the white noise, the estimated value of the variance ${\sigma }_{{ϵ}_i}^2$ was ${\sigma }_i^2={\displaystyle \frac{1}{n-2}{\sum }_{k=2}^{n-1}({ϵ}_i^{\left(k\right)}{)}^2.}$

3.2. Setting of the optimal inventory levels

Given the inventory levels x₁ and x₂, there are four general cases. If x₁≥ d₁ and x₂≥ d₂, then each customer receives their preferred product, and the retailer's profit is (2) $\mathrm{\Pi }\left(x_1,x_2,d_1,d_2\right)=p_1{d}_1+p_2{d}_2-c_1{x}_1-c_2{x}_2.$(2)

If x₂< d₂ and x₁≥ d₁+δ₂(d₂-x₂), then the inventory of product 2 does not meet the demand for that product. On the other hand, the inventory level of product 1 satisfies both the original demand for that product and the demand among customers who do not receive product 2 and are ready to buy that product. Thus, the profit is (3) $\mathrm{\Pi }\left(x_1,x_2,d_1,d_2\right)=p_1\left[d_1+{\delta }_2\left(d_2-x_2\right)\right]+p_2{x}_2-c_1{x}_1-c_2{x}_2.$(3)

Similarly, if x₁< d₁ and x₂≥ d₂+δ₁(d₁-x₁), then (4) $\mathrm{\Pi }\left(x_1,x_2,d_1,d_2\right)=p_1{x}_1+p_2\left[d_2+{\delta }_1\left(d_1-x_1\right)\right]-c_1{x}_1-c_2{x}_2.$(4)

In all other cases, all the inventory is sold and the profit is equal to (5) $\mathrm{\Pi }\left(x_1,x_2,d_1,d_2\right)=p_1{x}_1+p_2{x}_2-c_1{x}_1-c_2{x}_2.$(5)

According to (2)–(5), the retailer's expected cost is (6) $\begin{array}{rl}& Z\left(x_1,x_2\right)=E\left[\mathrm{\Pi }\left(x_1,x_2,d_1,d_2\right)\right]\\ & ={\int }_0^{x_1}[{\int }_0^{x_2}\left(p_1{t}_1+p_2{t}_2\right)f_2\left(t_2\right)d{t}_2\\ & +{\int }_{x_2}^{x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}}\left(p_1\left[t_1+{\delta }_2\left(t_2-x_2\right)\right]+p_2{x}_2\right)f_2\left(t_2\right)d{t}_2\\ & +{\int }_{x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}}^{\mathrm{\infty }}\left(p_1{x}_1+p_2{x}_2\right)f_2\left(t_2\right)d{t}_2]f_1\left(t_1\right)d{t}_1\\ & +{\int }_{x_1}^{\mathrm{\infty }}[{\int }_0^{x_2-{\delta }_1\left(t_1-x_1\right)}\left(p_1{x}_1+p_2\left[t_2+{\delta }_1\left(t_1-x_1\right)\right]\right)\\ & f_2\left(t_2\right)d{t}_2+{\int }_{x_2-{\delta }_1\left(t_1-x_1\right)}^{\mathrm{\infty }}\left(p_1{x}_1+p_2{x}_2\right)f_2\left(t_2\right)d{t}_2]\\ & f_1\left(t_1\right)d{t}_1-c_1{x}_1-c_2{x}_2,\end{array}$(6) which satisfies the following lemma.

Lemma 3.1:

The objective function (6) is strictly concave in [0,∞)×[0,∞).

Proof:

function (6)'s first-order derivatives are: (7) $\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_1}=p_1\left[1-{\int }_0^{x_1}{F}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1\right]}\\ & -{\delta }_1{p}_2{\int }_{x_1}^{\mathrm{\infty }}{F}_2\left(x_2-{\delta }_1\left(t_1-x_1\right)\right)f_1\left(t_1\right)d{t}_1-c_1\end{array}$(7) and (8) $\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_2}=p_2\left[\begin{array}{c}1-F_1\left(x_1\right)F_2\left(x_2\right)\\ -{\int }_{x_1}^{\mathrm{\infty }}{F}_2\left(x_2-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1\end{array}\right]}\\ & -{\delta }_2{p}_1{\int }_0^{x_1}\left[F_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)-F_2\left(x_2\right)\right]\\ & f_1\left(t_1\right)d{t}_1-c_2.\end{array}$(8)

Let, $H_Z=\left(\begin{array}{cc}{\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_1^2}}& {\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}}\\ {\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}}& {\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_2^2}}\end{array}\right)$be the Hessian matrix of Z. The principal minors are (9) ${\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_1^2}=-{\displaystyle \frac{p_1}{{\delta }_2}{\int }_0^{x_1}{f}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1-{\delta }_1^2{p}_2{\int }_{x_1}^{\mathrm{\infty }}{f}_2\left(x_2-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1-(p_1+{\delta }_1{p}_2)F_2\left(x_2\right)f_1\left(x_1\right)<0}}$(9) and (10) ${\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_2^2}=-{\delta }_2{p}_1\underset{0}{\overset{x_1}{\int }}{f}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1-p_2\underset{x_1}{\overset{\mathrm{\infty }}{\int }}{f}_2\left(x_2-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1-(p_2+{\delta }_2{p}_1)F_1\left(x_1\right)f_2\left(x_2\right)<0.}$(10)

Also, the mixed derivative is equal to (11) ${\displaystyle \frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}=-p_1{\int }_0^{x_1}{f}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1-{\delta }_1{p}_2{\int }_{x_1}^{\mathrm{\infty }}{f}_2\left(x_2-{\delta }_1\left(t_1-x_1\right)\right)f_1\left(t_1\right)d{t}_1<0.}$(11)

Consequently, we obtain $\begin{array}{rl}& \left|H_Z\right|=\left({\delta }_1^2{\delta }_2+{\displaystyle \frac{1}{{\delta }_2}-2{\delta }_1}\right)p_1{p}_2\underset{0}{\overset{x_1}{\int }}{f}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1\\ & \cdot \underset{x_1}{\overset{\mathrm{\infty }}{\int }}{f}_2\left(x_2-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1\\ & +[{\displaystyle \frac{p_1}{{\delta }_2}(p_2+{\delta }_2{p}_1)F_1\left(x_1\right)f_2\left(x_2\right)}\\ & +{\delta }_2{p}_1(p_1+{\delta }_1{p}_2)F_2\left(x_2\right)f_1\left(x_1\right)]\\ & \underset{0}{\overset{x_1}{\int }}{f}_2\left(x_2+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1\\ & +[{\delta }_1^2{p}_2(p_2+{\delta }_2{p}_1)F_1\left(x_1\right)f_2\left(x_2\right)\\ & +p_2(p_1+{\delta }_1{p}_2)F_2\left(x_2\right)f_1\left(x_1\right)]\\ & \underset{x_1}{\overset{\mathrm{\infty }}{\int }}{f}_2\left(x_2-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1\\ & +(p_2+{\delta }_2{p}_1)F_1\left(x_1\right)f_2\left(x_2\right)(p_1+{\delta }_1{p}_2)F_2\left(x_2\right)f_1\left(x_1\right).\end{array}$

The first coefficient of |H_Z| is equal to (1-δ₁δ₂)²/ δ₂, which is non-negative. Therefore, (12) $\left|H_Z\right|>0.$(12)

According to (9)-(12), H_Z is negative-definite: that is, Z is strictly concave.

According to Lemma 1, the following problem is concave: (13) $maxZ\left(x_1,x_2\right)$(13) $s.t.x_1,x_2\ge 0.$

If we have (14) $p_1-c_1>{\delta }_1\left(p_2-c_2\right)$(14)

as well as (15) $p_2-c_2>{\delta }_2\left(p_1-c_1\right),$(15) then the retailer's profit from a customer is expected to be higher if that customer receives their preferred product, even if they are ready to compromise and purchase the other product. Consequently, the retailer should order products of both types to maximise their expected profits. In other cases, only products of one type are ordered. The solution for problem (13) is formulated in the following proposition.

Proposition 3.1:

While both (14) and (15) are met, the solution of (13), denoted by (x₁*,x₂*), satisfies the following equations: (16) $\begin{array}{rl}& {\int }_0^{x_1^{\ast }}{F}_2\left(x_2^{\ast }+{\displaystyle \frac{x_1^{\ast }-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1\\ & +{\displaystyle \frac{{\delta }_1{p}_2}{p_1}{\int }_{x_1^{\ast }}^{\mathrm{\infty }}{F}_2\left(x_2^{\ast }-{\delta }_1\left(t_1-x_1^{\ast }\right)\right)f_1\left(t_1\right)d{t}_1=1-{\displaystyle \frac{c_1}{p_1}}}\\ & {\displaystyle \frac{{\delta }_2{p}_1}{p_2}{\int }_0^{x_1^{\ast }}\left[F_2\left(x_2^{\ast }+{\displaystyle \frac{x_1^{\ast }-t_1}{{\delta }_2}}\right)-F_2\left(x_2^{\ast }\right)\right]}\\ & f_1\left(t_1\right)d{t}_1+F_1\left(x_1^{\ast }\right)F_2\left(x_2^{\ast }\right)\\ & +{\int }_{x_1^{\ast }}^{\mathrm{\infty }}{F}_2\left(x_2^{\ast }-{\delta }_1\left(t_1-x_1^{\ast }\right)\right)f_1\left(t_1\right)d{t}_1=1-{\displaystyle \frac{c_2}{p_2}}\end{array}$(16)

such that both x₁* and x₂* are positive. If only (14) is satisfied, then x₂* = 0, and x₁* satisfies (17) ${\int }_0^{x_1}{F}_2\left({\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1=1-{\displaystyle \frac{c_1}{p_1}.}$(17)

If only (15) is valid, then x₁* = 0, and x₂* satisfies (18) ${\int }_0^{\mathrm{\infty }}{F}_2\left(x_2^{\ast }-{\delta }_1{t}_1\right)f_1\left(t_1\right)d{t}_1=1-{\displaystyle \frac{c_2}{p_2}.}$(18)

Proof:

According to Karush-Kuhn-Tucker conditions, the optimal solution for (13) satisfies the following equations: (19) $\begin{array}{rl}& {\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_1}+{\lambda }_1=0}\\ & {\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_2}+{\lambda }_2=0}\\ & {\lambda }_1{x}_1=0\\ & {\lambda }_2{x}_2=0\end{array}$(19) such that (20) $x_1,x_2,{\lambda }_1,{\lambda }_2\ge 0.$(20)

According to Lemma 1, any solution for (19) that meets (20) is optimal. First, the case where one of conditions (14)–(15) is not satisfied will be considered.

Let λ₁= 0 and x₂= 0. Then, the first equation of (19) becomes ∂Z/∂x₁= 0, namely (see (7)), Equation (17), and the second becomes ∂Z/∂x₂+λ₂= 0, namely (see (8)) (21) ${\displaystyle \frac{{\delta }_2{p}_1}{p_2}{\int }_0^{x_1}{F}_2\left({\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1=1-{\displaystyle \frac{c_2}{p_2}+{\displaystyle \frac{{\lambda }_2}{p_2}.}}}$(21)

From (17) and (21) we find that ${\lambda }_2={\delta }_2\left(p_1-c_1\right)-\left(p_2-c_2\right).$

Specifically, λ₂ is positive while (15) is not met. Also, let x^₁ be the value that satisfies (17), according to the Intermediate Value Theorem, x^₁> 0. Then, the optimal value of x₁ is x₁* = x^₁. Hence, the solution (x^₁,0) meets (20) and is optimal.

Now, let x₁= 0 and λ₂= 0. The first and second equations of (19) respectively become (22) ${\delta }_1{p}_2{\int }_0^{\mathrm{\infty }}{F}_2\left(x_2^{\ast }-{\delta }_1{t}_1\right)f_1\left(t_1\right)d{t}_1=p_1-c_1+{\lambda }_1$(22) and Equation (18). Substituting (18) for (22) leads to ${\lambda }_1={\delta }_1\left(p_2-c_2\right)-\left(p_1-c_1\right).$

That is, λ₁ is positive under the contrary case of (14). The optimal value of x₂ becomes x₂* = x^₂, where x^₂ is defined as the value that satisfies (18). As x^₂ is positive, the solution (0, x^₂) satisfies (20) and is optimal.

Next, we assume that both (14) and (15) are met. Setting x₁= 0 or x₂= 0 respectively leads to λ₂< 0 or λ₁< 0. Therefore, we necessarily have λ₁= λ₂= 0. That is, any optimal solution sets both (7) and (8) to zero. According to (7), this defines the following functions: $G_0\left(x_1\right)={\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_1}\left(x_1,0\right)=p_1\left[1-{\int }_0^{x_1}{F}_2\left({\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1\right]-c_1}$and $G_{\mathrm{\infty }}\left(x_1\right)=\underset{x_2\to \mathrm{\infty }}{lim}{\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_1}\left(x_1,x_2\right)=\left(p_1-{\delta }_1{p}_2\right)\left(1-F_1\left(x_1\right)\right)-c_1.}$

Let x₁> 0. From (11), we find that for every x₂> 0, $G_{\mathrm{\infty }}\left(x_1\right)<{\displaystyle \frac{\mathrm{\partial }Z(x_1,x_2)}{\mathrm{\partial }{x}_1}\le G_0\left(x_1\right)}$. Thus, there exists x₂⁰(x₁) ≥ 0, such that ${\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_1}\left(x_1,x_2^0\left(x_1\right)\right)=0}$, if and only if (23) $G_{\mathrm{\infty }}\left(x_1\right)<0\le G_0\left(x_1\right)$(23)

Note that both G₀(x₁) and G_∞(x₁) are decreasing, and that G₀(x^₁) = 0. Therefore, the right inequality of (23) is valid if and only if x₁≤ x^₁. Regarding the left inequality of (23), under assumption (14), we may either have (24) ${\delta }_1\left(p_2-c_2\right)<p_1-c_1\le {\delta }_1{p}_2$(24)

or (25) $p_1-c_1>{\delta }_1{p}_2.$(25)

If (24) is satisfied, then G_∞(0) ≤ 0. Namely, every 0 ≤ x₁≤ x^₁ satisfies (23) and, according to (8), in that interval we define $\begin{array}{rl}& \mathrm{\Phi }\left(x_1\right)={\displaystyle \frac{\mathrm{\partial }Z}{\mathrm{\partial }{x}_2}\left(x_1,x_2^0\left(x_1\right)\right)=p_2}\\ & \left[1-F_1\left(x_1\right)F_2\left(x_2^0\right(x_1\left)\right)-\underset{x_1}{\overset{\mathrm{\infty }}{\int }}{F}_2\left(x_2^0\left(x_1\right)-{\delta }_1(t_1-x_1)\right)f_1\left(t_1\right)d{t}_1\right]\\ & -{\delta }_2{p}_1\underset{0}{\overset{x_1}{\int }}\left[F_2\left(x_2^0\left(x_1\right)+{\displaystyle \frac{x_1-t_1}{{\delta }_2}}\right)-F_2\left(x_2^0\right(x_1\left)\right)\right]f_1\left(t_1\right)d{t}_1-c_2.\end{array}$

Let x₁= 0. Then (0,x₂⁰(0)) sets (7) to zero, namely ${\int }_0^{\mathrm{\infty }}{F}_2\left({x^0}_2\left(0\right)-{\delta }_1{t}_1\right)f_1\left(t_1\right)d{t}_1={\displaystyle \frac{p_1-c_1}{{\delta }_1{p}_2}.}$

Therefore, under (14) we have (26) $\mathrm{\Phi }\left(0\right)=p_2\left[1-{\int }_0^{\mathrm{\infty }}{F}_2\left(x_2^0\left(0\right)-{\delta }_1{t}_1\right)f_1\left(t_1\right)d{t}_1\right]-c_2=p_2\left[1-{\displaystyle \frac{p_1-c_1}{{\delta }_1{p}_2}}\right]-c_2<p_2\left[1-{\displaystyle \frac{{\delta }_1(p_2-c_2)}{{\delta }_1{p}_2}}\right]-c_2=0.$(26)

Let x₁= x^₁. Then, x₂⁰(x^₁) = 0 and, under (15), (27) $\mathrm{\Phi }\left({\hat{x}}_1\right)=p_2-{\delta }_2{p}_1{\int }_0^{{\hat{x}}_1}{F}_2\left({\displaystyle \frac{{\hat{x}}_1-t_1}{{\delta }_2}}\right)f_1\left(t_1\right)d{t}_1-c_2=p_2-{\delta }_2\left(p_1-c_1\right)-c_2>0.$(27) According to (26) and (27), there exists 0 < X₁<x^₁, such that Φ(X₁) = 0. That is, (X₁,x₂⁰(X₁)) sets (8), as well as (7), to zero and, therefore, meets (16). This point solves (19), satisfying (20). Namely, the optimal solution is x₁* = X₁> 0 and x₂* = x₂⁰(X₁) > 0.

If condition (24) is replaced with (25), then G_∞(0) > 0, and there exists 0<x^^₁< x^₁, such that G_∞(x^^₁) = 0 and (23) is satisfied by x^^₁< x₁≤ x^₁. Then,$\underset{x_1\to {\widehat{\stackrel{ˆ}{x}}}_1^{+}}{lim}{x}_2^0\left(x_1\right)=\mathrm{\infty }$ and, therefore, (28) $\underset{x_1\to {\widehat{\stackrel{ˆ}{x}}}_1^{+}}{lim}\mathrm{\Phi }\left(x_1\right)=p_2\left[1-F_1\left({\widehat{\stackrel{ˆ}{x}}}_1^{+}\right)-{\int }_{{\widehat{\stackrel{ˆ}{x}}}_1}^{\mathrm{\infty }}{f}_1\left(t_1\right)d{t}_1\right]-c_2=-c_2<0.$(28) From (28) and (27), there exists x^^₁ < X₁<x^₁, such that Φ(X₁) = 0 and, as before, x₁* = X₁ and x₂* = x₂⁰(X₁).

4. Numerical example and sensitivity analysis

The optimal inventory levels, x₁ and x₂, as well as the retailer's expected profits were numerically considered regarding how they were affected by the model's parameters.

Example: Generally, the parameters were fixed as follows: (29) $p_1=100,p_2=70,c_1=40,c_2=25,{\mu }_1=500,{\mu }_2=300,{\sigma }_1=100,{\sigma }_2=60,{\delta }_1=0.4,{\delta }_2=\mathrm{0.6.}$(29)

First, the effect of the customers’ readiness to compromise on the results was considered. Figure 1 presents the optimal inventory levels as functions of δ₁ (1a) and δ₂ (1b).

Figure 1. Effect of δ₁ (a) and δ₂ (b) on the inventory levels.

Two graphs showing the values of the optimal inventory levels for each product as a function of δ1 (1a) and δ2 (1b) where the x-axis represents the value of δ and the y-axis represents the value of the optimal inventory level. In each graph, a red line indicates product 1, and a blue line indicates product 2.

As p₁-c₁=  60 and p₂-c₂= 45, (14) was necessarily satisfied, while (15) was met only if δ₂< 0.75; in particular, it was valid under δ₂= 0.6, according to (29). As mentioned in Figure 1a, the value of x₁ decreased in δ_1. Moreover, it was always greater than or equal to the demand d₁'s mean of 500 (note that p₁> 2c₁), and was equal to that value as δ₁ became close to 1 (i.e. all the customers who preferred product 1 were ready to buy product 2 when there was a lack of product 1). Since the net profit from product 1 was greater than that of product 2, it was preferable to order a quantity of inventory of product 1 which is greater than or equal to the mean of μ₁. The level of inventory of product 2 (x₂) increased in δ₁, and was lower than the mean of 300, while δ₁< 0.5, even though p₂> 2c₂. According to Figure 1b, the inventory levels, x₁ and x₂, were respectively increasing and decreasing in δ₂. Once δ₂ passed 0.75, condition (15) was not satisfied and, according to Proposition 1, x₂ vanished. Consequently, x₁ significantly grew to enable the sales of product 1 to almost all of the retailer's customers.

Figure 2 presents how the retailer's expected profit depended on δ₁ (2a) and δ₂ (2b). Increasing customers’ flexibility to buy their second-priority products increased the expected profit. However, this effect was negligible when δ₂< 0.75 (i.e. products of both types were ordered). Consider the following benchmark: the inventory levels were always equal to the expected values obtained from the forecasting process (see Subsection 3.1). That is, x_i was always equal to μ_i and the demands’ distributions, in particular their variances, were not taken into account. From Figure 2, it can be seen that the improvement in profits according to the proposed model, compared to the corresponding benchmark, was negligible if p_i> 2c_i, while both (14) and (15) were met (i.e. δ₂< 0.75).

Figure 2. Effect of δ₁ (a) and δ₂ (b) on the retailer's profits.

Two graphs showing the values of the expected profit as a function of δ1 (2a) and δ2 (2b) where the x-axis represents the value of δ and the y-axis represents the value of the expected profit. In each graph, a red line indicates our model, and a blue line indicates the benchmark.

The retailer's order costs (c₁ and c₂) were then increased, such that p_i< 2c_i. In particular, c₁= 70 and c₂= 50. The optimal inventory levels and the expected costs are presented in Figures 3 and 4, respectively. Under the higher order costs, constraint (15) was satisfied while δ₂< 2/3. In this case, the inventory levels were necessarily lower than or equal to the means of the demands. The effects of δ₁ and δ₂ in Figure 3 were more significant than in Figure 1 as the probabilities for shortages were higher and the inventory levels were lower. Moreover, the model may also lead to a noteworthy increase in the expected profits, compared to the corresponding benchmark case, if products of both types were ordered. In particular, the profit could grow in more than 7% if δ₁ was low (Figure 4a).

Figure 3. Effect of δ₁ (a) and δ₂ (b) on the inventory levels, p_i< 2c_i.

Two graphs showing the values of the optimal inventory levels for each product as a function of δ1 (3a) and δ2 (3b), under condition pi < 2ci, where the x-axis represents the value of δ and the y-axis represents the value of the optimal inventory level. In each graph, a red line indicates product 1, and a blue line indicates product 2.

Figure 4. Effect of δ₁ (a) and δ₂ (b) on the retailer's profits, p_i< 2c_i.

Two graphs showing the values of the expected profit as a function of δ1 (4a) and δ2 (4b), under condition pi < 2ci, where the x-axis represents the value of δ and the y-axis represents the value of the expected profit. In each graph, a red line indicates our model, and a blue line indicates the benchmark.

Next, the uncertainty level of the demands was considered regarding the effects on the inventory levels. If p_i> 2c_i, according to (29), then x₁ was necessarily greater than or equal to the mean μ_i (see Figure 5a). As the variances grew, the potential of high demands led to greater inventory of product 1. Although the demand may also be very low, the potential of very high demand defeated the risk of having very low demand. This situation was changed once p_i became lower than 2c_i, as shown in Figure 5b. The inventory levels were lower than or equal to the means of the demands and were reduced as the variances grew (e.g. x₂ may decrease by 28%). The risk of very low demands was now more significant than the potential for high demands.

Figure 5. Effect of uncertainty on the inventory levels when p_i> 2c_i (a) and p_i< 2c_i (b).

Two graphs showing the values of the optimal inventory levels for each product as a function of the uncertainty level, for pi > 2ci (5a) and pi < 2ci (5b) where the x-axis represents the value standard deviation and the y-axis represents the value of the optimal inventory level. In each graph, a red line indicates product 1, and a blue line indicates product 2.

5. Discussion

Forecasting ability is usually low in the fashion industry and demand volatility is high. Beheshti-Kashi et al. (2015) estimated that approximately 95% of a collection’s fashion products will be replaced the following season as new items are required every season. Due to the constantly changing consumer preferences and the lack of specific consumption history, forecasting for the next season can be challenging (Wen et al., 2019).

This paper presented a model of fashion forecasting and inventory management applied to the example of one retailer selling two types of clothing products. In the event of an inventory shortage, customers may compromise and buy the other product, if possible. Alternatively, they may choose to buy nothing. Assuming that the demands for the products are normally distributed, the autoregressive forecasting model was adopted to estimate not only the expected demands, but also their exact distribution. Consequently, the demands’ variances were also taken into account, such that the inventory levels were efficiently calculated, even if the linear correlation between the two following demands was low. The retailer's expected profit function was found to be concave and analytically formulated the non-trivial solution. In particular, conditions that led to the profitability of ordering inventory of both of the products were formulated.

The optimal inventory levels were numerically shown in relation to how they were affected by the customers’ readiness to compromise, the uncertainty of the demands, and the ratio of the retailer's selling price to order cost. In particular, the inventory levels were significantly affected by the readiness of the customers interested in buying the more profitable product to compromise. If the retailer's selling price was greater than the order cost, then the inventory level of the profitable product grew as the demand variance increased, due to the potential for very high demand. Otherwise, the inventory levels of both products were reduced as the uncertainty grew, due to the risk of very low demand. The expected profits, according to the proposed model, were compared to those obtained when the inventory levels were equal to the demand’s expected values (i.e. their exact distributions and, in particular, their variances were not taken into account). It was found that the expected profit corresponding to this model may be notably higher when the customers’ readiness to compromise was low or when only one type of product was in stock.

6. Conclusions

Three managerial insights are obtained. First, ordering inventory which is equal to or lower than the means of the demands is optimal when the retailer's order costs increase. Second, the expected profit increases when customers are more flexible with the purchase of their second-priority products. The third managerial insight is examined based on the comparison between the proposed model and the situation when the inventory levels are always equal to the expected values obtained from the forecasting process. When both types of products are ordered, the proposed model could also result in a notable increase in the expected profits.

It is important to note that there are some limitations to this research. First, the proposed model considered only two types of clothing products solved for a single period. It would be interesting to extend this research to include a multi-product model that is dynamically solved over a number of seasons. Second, this research focused on supply chain costs, unit selling price and the order cost of the product. Different supply chain costs, such as salvage value, should be further investigated.

Two directions for future research to extend the current results can be suggested. The first is an extension of the proposed model to include additional factors, such as n-products, other supply chain costs and several seasons. The second is to include a multi-echelon model that considers all of the supply chain members.

Disclosure statement

No potential conflict of interest was reported by the author(s).