# Joint Pricing and Inventory Control Decisions for Fashion Apparel with Considering Fashion Level and Partial Backlogging

Authors: Lili Zhou; Qi Chen; Qi Xu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Apparel inventories have been crawling up for decades. The fashion level of clothing drops over time, resulting in a continuous decline in market value, a rapid reduction in demand, and unsold inventory. Poor inventory management weighs down the apparel industry. Inventory planning decisions are crucial for a fashion apparel retailer. When the good is highly perishable, the retailer may need to backlog demand in order to market the good at a reasonable price and reduce inventory costs. This study formulates a finite horizon inventory model for fashion apparel under deteriorating fashion level and partial backlogging conditions. This model jointly optimizes the selling price, length of inventory-holding period, and length of shortage period by maximizing the average profit. The conditions that characterize the optimal solution are established and an algorithm is proposed to search for the optimal solutions. A numerical example is provided to illustrate the algorithm and the solution procedure. Furthermore, sensitivity analysis of the key system parameters is carried out and some interesting managerial insights are presented. Findings clearly suggest that the presence of shortage has got an affirmative effect on ordering policy of the fashion apparel retailer.


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## Body

## 1. Introduction

With the rapid development of the world economy and the advent of the era of globalization, people’s lifestyles have undergone great changes. In addition to comfort, they are also pursuing the individuality and fashion sense of clothing. According to the statistics from American Fashion [1], in recent years, the global apparel market has reached 3 trillion US dollars, accounting for about 2% of the global gross domestic product (GDP). Taking the American sports and leisure and lifestyle brand Nike as an example, the company is the world’s second largest fashion brand. The company’s 2017 revenue reached US $344.6 billion, and its market value was close to US $105 billion. Fast fashion is an industrial practice that is widely applied in fashion retailing. The central idea is to offer a continuous stream of new merchandise to the market which reflects the latest fashion trends and helps capture the hottest designs in the current market. For example, famous international fashion retail brands such as H&M, Top Shop, and Zara have all implemented fast fashion strategies [2]. Fashion clothing sales are influenced not only by price but also by fashion level, which depends on elements such as style, color, and material. Fashion products differ significantly from other types of commercial products. In general, the higher the fashion level of clothing products, the higher their market value. As time goes on, the fashion level of the clothing drops, resulting in a continuous decline in market value; market demand may thus fall off rapidly and result in an accumulation of unsold inventory. In other words, fashion apparel decreases in value over the selling season. The latest data show that excess inventory is a major problem in the fashion industry. For example, H&M announced at the end of March 2018 that the total value of its accumulated unsold apparel was US $4 billion. In 2016, the total inventory of 79 quoted textile and apparel companies in China amounted to US $81 billion [3]. Inventory planning decisions are crucial for supply chain managers in the fashion industry. Having short product lifecycle and highly volatile demand, fashion items require extra attention for the initial ordering before selling season starts, as well as the subsequent replenishment decisions during the selling season [4, 5]. Bowers and Agarwal [6] proposed a dynamic hierarchical model for apparel production planning and scheduling in which they considered all processes from daily scheduling and short-term production planning to long-term decisions to better match the demand with a minimal total cost. Fisher et al. [7] studied replenishment order quantity to minimize the cost of lost sales and obsolete inventory in a two-stage model. They identified the optimal reorder time and quantified the benefit of lead time reduction. Raman and Kim [8] quantified the impact of inventory-holding cost in the fashion supply chain based on data from Northco, a school uniform manufacturer in Northeastern USA. However, the impact of the fashion level of clothing products on pricing and replenishment strategies of apparel retailer has been largely neglected in prior studies [6–8]. Due to the time-varying characteristics of fashion clothing, implementing a joint pricing and replenishment strategy has obvious practical significance for managing inventory replenishment and improving net profits. Chen et al. [3] analyzed the joint optimal dynamic pricing strategy and replenishment cycle for fashion apparel considering the effect of fashion level. The decay of fashion over time leads to a decrease in clothing sales demand, which makes the storage time of products longer and ultimately leads to an increase in inventory costs. In reality, clothing retailers can reduce inventory costs by delaying in meeting demand of consumers, that is, allowing the products to be temporarily out of stock or partly out of stock to delay meeting consumers’ needs. When it is economic to backlog demand, the reseller can plan for periods of shortage during which demand can be partially back-ordered [9]. When the good is highly perishable, the retailer may need to backlog demand in order to market the good at a reasonable price. Apart from the direct effects of fashion level and demand rate on changes in inventory levels, apparel inventory will inevitably suffer from a shortage. In some inventory systems, such as fashionable items, the length of the waiting time for next replenishment would determine whether the backlogging will be accepted or not. Therefore, the backlogging rate is variable and is dependent on the waiting time for the next replenishment [10]. To enhance the exploratory power and the connection between research and practice, we develop our theoretical framework learning from the study by Chen et al. [3] on inventory management for fashion apparel. Chen et al. [3] used a dynamic pricing model to examine the effect of fashion level on firm performance. We draw upon inventory planning decisions theory as a response to the call for an increased focus on the partial backlogging and contribute to the broader application of inventory planning decisions theory in inventory management for fashion apparel. In addition, this study also contributes to the deterioration items literature by simultaneously considering the fashion level and partial backlogging.Therefore, this paper aims to focus on the pricing and replenishment strategy problems for the fashion apparel retailer with considering fashion level and partial backlogging. Based on this research background, the following questions arise: (1) How does the fashion level of apparel affect product demand? (2) How can we determine the optimal price, the optimal length of inventory-holding period, and the optimal length of shortage period to maximize the apparel retailer’s average profit during the selling period? (3) How the inventory control parameters, such as the deteriorating rate of the fashion level, the backlogging resistance, the shortage cost, and inventory-holding cost, influence apparel retailer’s performance? In our inventory model, the shortage is allowed, and the partial backlogging rate is sensitive to the waiting time. Firstly, we prove that, for any given replenishment cycle, the average profit is strictly concave in selling price and, for any given selling price, the average profit function is strictly pseudoconcave in replenishment cycle; that is, there exists a unique and optimal solution to the problem. Next, an algorithm is proposed to obtain the optimal solution of the problem. Finally, the sensitivity analyses of the parameters are conducted to demonstrate the effectiveness of the proposed model. The research results show that when the purchasing cost, the shortage cost, and the holding of fashion apparel increase, the average profit of system will decrease. The optimal average profit and order quantity are highly sensitive to the deterioration rate of fashion level and the backordering resistance.The rest of this paper is organized as follows. InSection 2, we briefly review the related literature. In Section 3, we outline the notations and assumptions used to address the problem. Section 4 contributes to the formulation of the model. We obtain theoretical results and provide an iterative algorithm in Section 5. Numerical examples, sensitivity analysis, and some managerial insights are provided in Section 6. Finally, Section 7 offers a conclusion and discusses the implications for management.


## 2. Literature Review

To date, the literature on fashion inventory management has focused on quick response strategies. In the fashion industry, more and more companies, such as Uniqlo, Zara, Topshop, and Primark, are adopting such strategies [11, 12]. Quick response strategies were first proposed in the 1980s in the US fashion industry [13]. Fashion retailers can postpone order placement until close to the beginning of the selling season to obtain more accurate market information about the sales of particular types of fashion items (i.e., style trends). This allows them to adjust their initial inventory forecast, so that supply can better match demand. Other related studies include that by Aviv et al. [14], where they studied the potential benefits of response pricing for seasonal fashion merchandise sellers. Cachon and Swinney [15] discuss the quick response mediated fast fashion program in the apparel industry. Chow et al. [16] study the impacts of imposing a minimum order quantity on a two-echelon supply chain implementing the quick response strategies. Choi [17] studied a sourcing decision with the quick response strategy under an environmental sustainability practice. Chan et al. [18] determined the number of observations to take under quick response for apparel inventory control. Choi [19] constructed a formal analytical fashion supply chain model based on the Bayesian information updating to examine the effects of quick response. Cosgun et al. [20] addressed the simultaneous determination of markdown prices and optimal initial inventory levels under the cross-price effects in a random demand setting for multiproduct groups. All of these studies proposed using quick response strategies to deal with volatile market demand by reducing order lead time to more accurately determine the appropriate quantity to order.Inventory planning decisions are crucial for fashion apparel. As such apparel has a short product lifecycle and is associated with highly volatile demand, initial orders before the selling season begins require careful attention [4, 5], and the same applies for replenishment decisions during the selling season. Due to the complexity of fashion inventory management, very few studies have focused on fashion inventory management from a pricing and replenishment policy perspective. Deterioration items exist commonly in the real business environment. As stated in previous literature [21, 22], deterioration refers to spoilage, decay, pilferage, decay, evaporation, obsolescence, or damage such that the items cannot maintain the original value or utility. Most of the physical goods undergo decay or deterioration over time, such as medicine, volatile liquids, fashion items, and others [23]. For the property of the deterioration items, the research on pricing and replenishment strategy for the fashion apparel retailer can be referred to the research on inventory and pricing decisions of perishable products [24–26]. For example, Wu et al. [24] formulated an inventory model with deteriorating items and price-sensitive demand and obtained the retailer’s optimal selling price and the length of replenishment cycle. Avinadav et al. [25] used a price- and time-dependent demand function to develop a mathematical model that calculates the optimal price and replenishment period of perishable items. Herbon et al. [26] formulated an inventory model with deteriorating items and price-sensitive demand and obtained the retailer’s optimal dynamic selling price and the length of replenishment cycle. In the above-mentioned inventory models, the shortage of products has been neglected. The shortage setting should be considered to face the various practical requirements.Nevertheless, few research efforts have explicitly considered the inventory decisions of fashion apparel from the perspective of pricing and replenishment cycles. Fisher et al. [27] studied how replenishment order quantity may minimize the cost of lost sales, backorders, and obsolete inventory in a two-stage model. Kogan and Spiegel [28] developed a model with an initial inventory level and addressed the problem of maximizing profits from the production, storage, and sale of fashion goods. In their proposed inventory model, the selling price remains unchanged throughout the sales period. Such a price strategy ignores the fashion level of the apparel, which declines over time. Tsao [29] developed an analytical model of inventory control for deteriorating goods and fashion goods under trade credit and partial backlogging conditions. In that study, two-phase pricing and inventory decisions were considered and the optimal replenishment strategy was identified. The above-mentioned works of literature have not considered the influence of fashion level on pricing and inventory decisions. Chen et al. [3] used a dynamic pricing model to examine the effect of fashion level on firm performance. From the aspect of modeling, the most closely related studies to this paper are those by Tsao [29] and Chen et al. [3]. However, unlike these studies, we incorporate the fashion level and shortage into the optimization model. This paper derives a pricing and replenishment policy for fashion apparel and adopts a price- and time-dependent demand function to model the finite time horizon inventory. The major objective of this policy is to simultaneously determine the optimal price, optimal length of inventory-holding period, and optimal length of shortage period to maximize average profit. We also establish the conditions that characterize the optimal solutions and propose an algorithm to search for optimal solutions. To be specific, a systematic comparison between this paper and other related papers is given in Table 1 to show the novelty and advantage of this paper.Table 1
A comparison of the present work with related previous works.
ReferencesQuick responseInventory planning decisionsFashion levelShortage[14–19]YesNoNoNo[24–26]NoYesNoNo[3]NoYesYesNo[29]NoYesNoYesOur paperNoYesYesYes


## 3. The Model

### 3.1. Problem Formulation

First, we describe the problem as follows. We model an apparel retailer that sells fashionable clothing products. The demand for fashionable clothing is determined by the selling pricep and fashion level of the clothing ωt during the selling period. The replenishment problem for a single fashion item is considered. The inventory system evolves as follows: Q units of items arrive at the inventory system at the beginning of each cycle. The inventory level It decreases due to consumers’ demand during the replenishment cycle. During the interval 0≤t≤t1, the inventory level decreases due to demand. A shortage occurs due to demand and partial backlogging during the time interval t1≤t≤t1+t2. The length of replenishment cycle T equals the length of inventory-holding period t1 plus the length of shortage period t2. The backlogged demand during the shortage period is S. Thus, the retailer’s ordering quantity for one replenishment cycle is Q=I0+S. The selling price, length of inventory-holding period, and length of shortage period are decision variables. The major objective of this policy is to simultaneously determine the selling price, the length of inventory-holding period, and the length of shortage period to maximize average profit.


### 3.2. Assumptions

The mathematical model in this paper is developed on the basis of the following assumptions.Assumption 1.
The demand function is nonnegative and explicitly depends separately on both the price and fashion level. In practice, the more fashionable a garment, the higher its market value, increasing consumers’ incentive to buy it [3]. That is, the higher the fashion level is, the larger the demand will be, according to the following formula:(1)Dp,t=dp+ωt,0≤t≤t1,dp,t1≤t≤t1+t2.
Without loss of generality, let the functiondp be linear; that is, dp=a−bp. a is the fixed term of demand determined by the market size and b represents the sensitivity of demand with respect to price. The demand function of paper is based on several previous studies of perishable products. In those studies, the demand function was usually assumed to have a linear form. This assumption has been widely adopted in the perishable inventory management literature, such as Li et al. [22], Zhang et al. [23], and Herbon et al. [26].Assumption 2.
The fashion level at timet is given by(2)ωt=ω0e−ηt.
Equation (2) is motivated by the model presented in Chen et al.’s work [3] and is proposed based on the following consideration. The fashion level function is strictly decreasing with time t; dω/dt<0. Therefore, we can deduce that the demand is decreasing with time t, so the assumption of the demand function in (1) is reasonable.Assumption 3.
Shortages are allowed. We adopt the notation used in Abad [9] where the unsatisﬁed demand is backlogged, and the fraction of shortage back-ordered is(3)Bτ=k0e−k1τ.where 0<k0<1 is the backlogging intensity, τ>0 is the waiting time up to the next replenishment, and k1>0 is the backlogging resistance. Note that Bτ proportions of the consumers who arrive during the shortage period are willing to wait for the next replenishment resulting in the backlogged demand, whereas the remaining 1−Bτ proportions of the consumers are lost, resulting in lost sales [22]. From formula (3), it can be seen that ∂Bτ/∂τ<0 and ∂2Bτ/∂τ2>0, and the partial backlogging decreases rapidly with the increase of waiting time, indicating that customers are becoming impatient, which is consistent with the reality.


## 3.1. Problem Formulation

First, we describe the problem as follows. We model an apparel retailer that sells fashionable clothing products. The demand for fashionable clothing is determined by the selling pricep and fashion level of the clothing ωt during the selling period. The replenishment problem for a single fashion item is considered. The inventory system evolves as follows: Q units of items arrive at the inventory system at the beginning of each cycle. The inventory level It decreases due to consumers’ demand during the replenishment cycle. During the interval 0≤t≤t1, the inventory level decreases due to demand. A shortage occurs due to demand and partial backlogging during the time interval t1≤t≤t1+t2. The length of replenishment cycle T equals the length of inventory-holding period t1 plus the length of shortage period t2. The backlogged demand during the shortage period is S. Thus, the retailer’s ordering quantity for one replenishment cycle is Q=I0+S. The selling price, length of inventory-holding period, and length of shortage period are decision variables. The major objective of this policy is to simultaneously determine the selling price, the length of inventory-holding period, and the length of shortage period to maximize average profit.


## 3.2. Assumptions

The mathematical model in this paper is developed on the basis of the following assumptions.Assumption 1.
The demand function is nonnegative and explicitly depends separately on both the price and fashion level. In practice, the more fashionable a garment, the higher its market value, increasing consumers’ incentive to buy it [3]. That is, the higher the fashion level is, the larger the demand will be, according to the following formula:(1)Dp,t=dp+ωt,0≤t≤t1,dp,t1≤t≤t1+t2.
Without loss of generality, let the functiondp be linear; that is, dp=a−bp. a is the fixed term of demand determined by the market size and b represents the sensitivity of demand with respect to price. The demand function of paper is based on several previous studies of perishable products. In those studies, the demand function was usually assumed to have a linear form. This assumption has been widely adopted in the perishable inventory management literature, such as Li et al. [22], Zhang et al. [23], and Herbon et al. [26].Assumption 2.
The fashion level at timet is given by(2)ωt=ω0e−ηt.
Equation (2) is motivated by the model presented in Chen et al.’s work [3] and is proposed based on the following consideration. The fashion level function is strictly decreasing with time t; dω/dt<0. Therefore, we can deduce that the demand is decreasing with time t, so the assumption of the demand function in (1) is reasonable.Assumption 3.
Shortages are allowed. We adopt the notation used in Abad [9] where the unsatisﬁed demand is backlogged, and the fraction of shortage back-ordered is(3)Bτ=k0e−k1τ.where 0<k0<1 is the backlogging intensity, τ>0 is the waiting time up to the next replenishment, and k1>0 is the backlogging resistance. Note that Bτ proportions of the consumers who arrive during the shortage period are willing to wait for the next replenishment resulting in the backlogged demand, whereas the remaining 1−Bτ proportions of the consumers are lost, resulting in lost sales [22]. From formula (3), it can be seen that ∂Bτ/∂τ<0 and ∂2Bτ/∂τ2>0, and the partial backlogging decreases rapidly with the increase of waiting time, indicating that customers are becoming impatient, which is consistent with the reality.


## 4. Model Formulation

From Figure1, we can see that the inventory level It decreases due to consumers’ demand during the time interval 0≤t≤t1. The differential equation that represents inventory status is given by(4)dI1tdt=−dp−ωt.Figure 1
The apparel inventory level.With the conditionIt1=0, solving formula (4) yields I1t=∫tt1dp+ωudu. Therefore, the initial inventory level is then given by I0=I10=∫0t1dp+ωudu. During the second interval t1≤t≤t1+t2, shortage occurred, and the demand is partially backlogged according to the fraction BT−t. That is, the inventory level at time t is governed by the following differential equation:(5)dI2tdt=−dpBt1+t2−t.With the conditionIt1=0, solving (5) yields I2t=−dp∫t1+t2−tt2Bτdτ. If we put t=T into the above formula, the maximum amount of demand backlogging will be obtained as follows: S=dpMt2, where Mt2=∫0t2Bτdτ. Order quantity per cycle is the total of S and I0; that is,(6)Q=∫0t1dp+ωudu+dpMt2.Thus, the inventory level during the replenishment cycle is described as follows:(7)It=∫tt1dp+ωudu,0≤t≤t1,−dp∫t1+t2−tt2Bτdτ,t1≤t≤t1+t2.Now, we can obtain inventory costs and sales revenue per cycle which consist of the following five elements: fixed order costA, sales revenue SR=pQ, purchase cost PC=cQ, inventory cost HC=h∫0λI1tdt, and shortage cost SC=sdp∫0t2τBτdτ, where p is the selling price per unit, c is the purchasing cost per unit, h is inventory-holding cost per unit, and s is the shortage cost per unit.Therefore, the total profit (denoted byΠt1,t2,p) during the time interval 0,T is given by(8)Πt1,t2,p=SR−PC−HC−SC−OC−A=p−c∫0t1dp+ωudu+dpMt2−h∫0t1∫tt1dp+ωududt−sdp∫0t2τBτdτ.The average profit functions are given by(9)πt1,t2,p=Πt1,t2,pt1+t2.


## 5. Results and Algorithm

The retailer’s objective is to maximizeπt1,t2,p by jointly determining t1,t2,p. In this section, we will explore the mathematical properties of the model to assist in proposing an algorithm to search for the optimal solution.The necessary conditions for the average profit (9) to be maximized are(10)∂πt1,t2,p∂p=1t1+t2∂Π∂p=0,(11)∂πt1,t2,p∂t1=∂Π/∂t1t1+t2−Πt1+t22=0,(12)∂πt1,t2,p∂t2=∂Π/∂t2t1+t2−Πt1+t22=0.Proposition 1.
For any givent1 and t2, the average profit πt1,t2,p is strictly concave in p, and there exists a unique global maximum point p∗∈c,∞ that satisfies the first-order necessary condition ∂π/∂p=0.Proof.
From equation (8), we get ∂2Π/∂p2=−2bt1+Mt2<0. According to equation (10), we conclude that ∂2π/∂p2=1/t1+t2∂2Π/∂p2<0; therefore, π is strictly concave in p. In addition, we have limp⟶+∞∂π/∂p<0 and limp⟶c∂π/∂p>0. Therefore, there exists a unique maximum point p∗∈c,∞ to satisfy ∂πt1,t2,p/∂p=0.
The length of shortage periodt2 falls in the range of 0,p−c/s. For any given selling price p, the objective function (9) is equivalent to the following programming problem:(13)maxπt1,t2|p,s.t.t1≥0,p−cs−t2≥0.
Note that formula (13) represents fractional programs. A fractional program is defined as the nonlinear program sup Φx/Γx, x∈U=x∈X:hx≤0, where Γx>0. If Φx>0 is concave, Γx and hx are convex; then the problem is a concave fractional program. For a differentiable and strictly concave fractional program, there is at most one maximum, and a solution to the Kuhn-Tucker conditions is a maximum. We refer readers to Li et al.’s work [22] for the details on the definition and properties of fractional program.Proposition 2.
For any givenp, the average profit πt1,t2|p is strictly pseudoconcave in t1,t2.Proof.
From equation (9), it is easy to get ∂2Π/∂t12<0, ∂2Π/∂t1∂t2=0, and ∂2Π/∂t22=dpp−c−st2B′t2−sBt2. Since 0≤Bt2≤1 and B′t2≤0, we get ∂2Π/∂t22<0. Let H denote Hessian matrix of Π; then we have H=∂2Π/∂t12∂2Π/∂t22−∂2Π/∂t1∂t2∂2Π/∂t1∂t2>0. Therefore, H is negative definite, and Πt1,t2,p is strictly concave in t1,t2. Also, the denominator t1+t2 is a linear function. Hence, πt1,t2|p is strictly pseudoconcave in t1,t2.
Next, to simplify the expression of the model, let(14)Δ=p−cω0g0+A+dpp−cg0−Mg0+s∫0g0τBτdτ,where g0=t2|Gt1=0,t2=0. We also define the roots of ∂πt1,t2,p/∂t1=0 and ∂πt1,t2,p/∂t2=0 as t1+ and t2+, respectively.Proposition 3.
For any givenp∈c,∞, πt1,t2|p has a global maximum point t1∗,t2∗, which is determined by the following rules:(1)
IfΔ≤0, then t1∗,t2∗=0,g0(2)
IfΔ≥0, then t1∗,t2∗=t1+,t2+Proof.
Letλ1 and λ2 be two Lagrange Multipliers. Kuhn-Tucker condition of equation (13) is(15)∂πt1,t2,p∂t1+λ1=0,∂πt1,t2,p∂t2−λ2=0,λ1t1=0,λ2p−cs−t1=0,λ1,λ2≥0.
To solve the above equations, we can consider the three following cases:Case (i). Ifλ2≠0, then t2=p−c/s and ∂π/∂t2<0, which contradicts the constraint of λ2≥0. Therefore, this scenario will not occur.Case (ii). Ifλ1=0 and λ2=0, then ∂πt1,t2|p/∂t1=0 and ∂πt1,t2|p/∂t2=0.Case (iii). Ifλ1≠0 and λ2=0, then t1=0, ∂πt1,t2|p/∂t2=0, and λ1=−∂πt1,t2|p/∂t1.
Next, we prove that case (ii) and case (iii) are feasible. Equations (11) and (12) are equivalent to(16)∂Π∂t1−t1+t2=Π,(17)∂Π∂t2−t1+t2=Π.
From equations (16) and (17), we get(18)∂Π∂t1−∂Π∂t2=0.
We defineGt1,t2=∂Π/∂t1−∂Π/∂t2. Since ∂Gt1,t2/∂t2>0, Gt1,t2 increases as t2 increases. Moreover limt2⟶p−c/sGt1,t2=∂Π/∂t1>0 and limt2⟶0+Gt1,t2=c−p−ht1ωt1−ht1dp<0. Thus, there exists a unique t2∈0,p−c/s such that Gt1,t2=0. Therefore, t2 can be regarded as the function of t1. Let t2=gt1 and substitute it into equation (16); then we get ∂Π/∂t1t1+gt1−Π=0. Let Ft1=∂Π/∂t1t1+gt1−Π; then we obtain the first derivative of Ft1 with respect to t1: dFt1/dt1=∂2Π/∂t12t1+gt1<0. In addition, limt1⟶0+∂Π/∂t1=p−cdp+ω0>0 and limt1⟶∞∂Π/∂t1<0. Thus, there exists a unique t¯1∈0,+∞ such that ∂Π/∂t1|t1=t¯1=0. Therefore, Ft1 is strictly decreasing in t1∈0,t¯1, and we get limt1⟶t¯Ft1=−Π<0. Moreover, limλ⟶0+Ft1=Δ.(a)
IfΔ≥0, then limt1⟶0+Ft1>0. By Intermediate Value Theorem, there exists a unique t1+∈0,t¯1 satisfying equation (16). Thus, Case (ii) is feasible, and t1∗=t1+.(b)
IfΔ≤0, then limλ⟶0+Ft1<0. Therefore, ∂π/∂t1<0, and then λ1=−∂π/∂t1>0, which means that Case (iii) is feasible and t1∗=0. Subsequently, we get t2∗ by substituting t1∗ into equation (18). From Proposition 2, the local maximum point t1∗,t2∗ is the global maximum point.
Not surprisingly, similar to the works of Li et al. [22] and Tsao [29], it is very difficult to solve the nonlinear equations (10)–(12). Based on the preceding propositions, we use a Gauss-Newton iterative method and propose the following algorithm to obtain the optimal solution:Step 1: Seti=1, and initialize the value of p=pi>c.Step 2: CalculateΔpi. According to Proposition 3, execute one of the following substeps:(1)
IfΔpi≥0, then t1i=t1+ and t2i=t2+.(2)
IfΔpi≤0, then t1i=0 and t2i=g0.Step 3: Solve∂πλi,μi,p/∂p=0 and mark the solution as p=pi+1. If the difference between pi and pi+1 is small enough, such as pi+1−pi<10−6, the optimal solution is t1∗,t2∗,p∗=t1i,t2i,pi+1. Otherwise, set i=i+1 and go back to Step 2.Step 4: We can obtainQ∗ using (6) and π∗ using (9), respectively.


## 6. Numerical Analysis

In this section, we conduct numerical analyses to gain managerial insights. The basic parameter values are chosen mainly based on the work of Chen et al. [3] and satisfy the assumptions mentioned in the Model Formulation section. The parameters are designed as follows: market size a=1000, price sensitivity parameter b=20, purchasing cost per unit c=40, inventory-holding cost per unit h=2, shortage cost per unit s=3, k0=0.9, k1=0.01, the deteriorating rate of the fashion level η=0.02, initial fashion level ω0=100 , and fixed cost per order A=400. Then the corresponding optimal solutions are given as follows: the optimal price p∗=47.4156, the optimal length of inventory-holding period t1∗=2.6312, the optimal length of shortage period t2∗=0.7487, the optimal order quantity Q∗=452.6448, and the optimal average profit π∗t1,t2,p=890.806. Next, we examine the impacts of the related parameters on the optimal solutions and obtain some useful insights.


### 6.1. Impact of the Deteriorating Rate of the Fashion Levelη

The optimal selling pricep∗, optimal length of inventory-holding period t1∗, optimal length of shortage period t2∗, and optimal average profit π∗ decrease in η (Table 2). Table 2 shows the effect of deterioration rate η of the fashion level. The smaller the deterioration rate, the slower the lowering of fashion level because the fashion level largely depends on the elements of style and material, all of which affect the deterioration rate. Therefore, the larger the deterioration rate, the faster the attenuation of fashion products. The faster attenuation of fashion level leads to the shorter sales cycle. Then the market demand will be correspondingly reduced, resulting in the lower average profit. It can be seen from Table 2 that the change of the deterioration rate has a great influence on average profit but little influence on optimal pricing and replenishment period. The results show that when the deterioration rate of fashion apparel increases, the fashion products retailer should pay attention to reducing the selling price appropriately, reducing the inventory-holding period and out of stock period, and reducing the quantity of orders as much as possible, so as to improve the average profit.Table 2
Impact of the deteriorating rate of the fashion level.
ηp∗t1∗t2∗π∗0.0247.412.630.74878900.0347.402.610.74878850.0447.392.590.74868840.0547.372.570.74858830.0647.362.560.74828810.0747.352.540.74788790.0847.342.520.74758770.0947.322.500.7470873


### 6.2. Impact of the Backlogging Resistancek1

As can be seen from Table3, the optimal selling price p∗ and optimal length of inventory-holding period t1∗ increase in k1, whereas the optimal length of shortage period t2∗ and optimal average profit π∗ decrease in k1. The larger the backlogging resistance is, the more reluctant the customer is to wait. In this case, the inventory-holding time should be appropriately extended, the shortage time should be shortened, and the total order quantity should be correspondingly reduced. It can be seen from Table 3 that the change of the backlogging resistance has a great influence on average profit but little influence on optimal pricing and replenishment period. Therefore, for customer groups with different backlogging resistance, the corresponding pricing and replenishment strategies will also be very different, which needs attention of clothing retailers.Table 3
Impact of the backlogging resistance.
k1p∗t1∗t2∗π∗0.01047.412.610.74878900.01547.892.620.72958860.02048.072.640.71348810.02548.402.650.69528770.03048.912.680.6790873


### 6.3. Impact of the Shortage Costs

As can be seen from Table4, the optimal selling price p∗ and optimal length of inventory-holding period t1∗ increase in s, whereas the optimal length of shortage period t2∗ and optimal average profit π∗ decrease in s. The optimal average profit is very sensitive to the change of shortage cost. When the shortage cost increases, the clothing retailer should reduce the order quantity in order to obtain higher profit. Furthermore, the results of Tables 2–4 show that the optimal average profit π∗ increases with the optimal length of shortage period t2∗. It suggests that the presence of shortage has got an affirmative effect on fashion apparel retailer ordering policy.Table 4
Impact of the shortage cost.
sp∗t1∗t2∗π∗2.047.222.580.84359012.547.312.610.77198933.047.412.630.74878913.547.522.670.73768864.047.622.680.7180879


### 6.4. Sensitivity Analysis of Other Parametersa,b,c,h

We now study the effects of changes in the values of the parametersa, b, c, and h on the optimal selling price p∗, order quantity Q∗, and average profit π∗. The sensitivity analysis is performed by changing each value of the parameters by −60%, −40%, −20%, 0%, +20%, +40%, and +60%, taking one parameter at a time and keeping the remaining parameter values unchanged. The computational results are shown in Figure 2.Figure 2
Effect of changes in values of parametersa, b, c, and h on the model variables.
(a)(b)(c)(d)The sensitivity analysis shown in Figure2 indicates the following observations.The optimal selling pricep∗ decreases in the values of parameters b, c, and h. Moreover, p∗ is lowly positive sensitive to changes in parameters c and h, whereas p∗ is a highly positive sensitive to change in c. It is reasonable that the purchase cost has a great and positive effect upon the optimal price.When the values of parametersb, c, and h increase, the optimal order quantity Q∗ will decrease. Q∗ is moderately sensitive to changes in parameters c and h, whereas Q∗ is highly sensitive to changes in a and b.When the values of parametersb, c, and h increase, the optimal average profit π∗ decreases. π∗ is moderately sensitive to changes in parameters c and h, whereas Q∗ is highly sensitive to changes in a and b. These results indicate that increases in cost have a negative effect on the average profit.


## 6.1. Impact of the Deteriorating Rate of the Fashion Levelη

The optimal selling pricep∗, optimal length of inventory-holding period t1∗, optimal length of shortage period t2∗, and optimal average profit π∗ decrease in η (Table 2). Table 2 shows the effect of deterioration rate η of the fashion level. The smaller the deterioration rate, the slower the lowering of fashion level because the fashion level largely depends on the elements of style and material, all of which affect the deterioration rate. Therefore, the larger the deterioration rate, the faster the attenuation of fashion products. The faster attenuation of fashion level leads to the shorter sales cycle. Then the market demand will be correspondingly reduced, resulting in the lower average profit. It can be seen from Table 2 that the change of the deterioration rate has a great influence on average profit but little influence on optimal pricing and replenishment period. The results show that when the deterioration rate of fashion apparel increases, the fashion products retailer should pay attention to reducing the selling price appropriately, reducing the inventory-holding period and out of stock period, and reducing the quantity of orders as much as possible, so as to improve the average profit.Table 2
Impact of the deteriorating rate of the fashion level.
ηp∗t1∗t2∗π∗0.0247.412.630.74878900.0347.402.610.74878850.0447.392.590.74868840.0547.372.570.74858830.0647.362.560.74828810.0747.352.540.74788790.0847.342.520.74758770.0947.322.500.7470873


## 6.2. Impact of the Backlogging Resistancek1

As can be seen from Table3, the optimal selling price p∗ and optimal length of inventory-holding period t1∗ increase in k1, whereas the optimal length of shortage period t2∗ and optimal average profit π∗ decrease in k1. The larger the backlogging resistance is, the more reluctant the customer is to wait. In this case, the inventory-holding time should be appropriately extended, the shortage time should be shortened, and the total order quantity should be correspondingly reduced. It can be seen from Table 3 that the change of the backlogging resistance has a great influence on average profit but little influence on optimal pricing and replenishment period. Therefore, for customer groups with different backlogging resistance, the corresponding pricing and replenishment strategies will also be very different, which needs attention of clothing retailers.Table 3
Impact of the backlogging resistance.
k1p∗t1∗t2∗π∗0.01047.412.610.74878900.01547.892.620.72958860.02048.072.640.71348810.02548.402.650.69528770.03048.912.680.6790873


## 6.3. Impact of the Shortage Costs

As can be seen from Table4, the optimal selling price p∗ and optimal length of inventory-holding period t1∗ increase in s, whereas the optimal length of shortage period t2∗ and optimal average profit π∗ decrease in s. The optimal average profit is very sensitive to the change of shortage cost. When the shortage cost increases, the clothing retailer should reduce the order quantity in order to obtain higher profit. Furthermore, the results of Tables 2–4 show that the optimal average profit π∗ increases with the optimal length of shortage period t2∗. It suggests that the presence of shortage has got an affirmative effect on fashion apparel retailer ordering policy.Table 4
Impact of the shortage cost.
sp∗t1∗t2∗π∗2.047.222.580.84359012.547.312.610.77198933.047.412.630.74878913.547.522.670.73768864.047.622.680.7180879


## 6.4. Sensitivity Analysis of Other Parametersa,b,c,h

We now study the effects of changes in the values of the parametersa, b, c, and h on the optimal selling price p∗, order quantity Q∗, and average profit π∗. The sensitivity analysis is performed by changing each value of the parameters by −60%, −40%, −20%, 0%, +20%, +40%, and +60%, taking one parameter at a time and keeping the remaining parameter values unchanged. The computational results are shown in Figure 2.Figure 2
Effect of changes in values of parametersa, b, c, and h on the model variables.
(a)(b)(c)(d)The sensitivity analysis shown in Figure2 indicates the following observations.The optimal selling pricep∗ decreases in the values of parameters b, c, and h. Moreover, p∗ is lowly positive sensitive to changes in parameters c and h, whereas p∗ is a highly positive sensitive to change in c. It is reasonable that the purchase cost has a great and positive effect upon the optimal price.When the values of parametersb, c, and h increase, the optimal order quantity Q∗ will decrease. Q∗ is moderately sensitive to changes in parameters c and h, whereas Q∗ is highly sensitive to changes in a and b.When the values of parametersb, c, and h increase, the optimal average profit π∗ decreases. π∗ is moderately sensitive to changes in parameters c and h, whereas Q∗ is highly sensitive to changes in a and b. These results indicate that increases in cost have a negative effect on the average profit.


## 7. Conclusions

The fashion level of clothing drops over time, resulting in a continuous decline in market value, a rapid reduction in demand, and unsold inventory. In reality, clothing retailers can reduce inventory costs by delaying in meeting demand of consumers, that is, allowing the products to be temporarily out of stock or partly out of stock to delay meeting consumers’ needs. Thus, an inventory model is proposed for deteriorating fashion level apparel under shortage policy. In the model, the shortage is allowed, and the partial backlogging rate is sensitive to the waiting time. Firstly, we prove that, for any given replenishment cycle, the average profit is strictly concave in selling price and, for any given selling price, the average profit function is strictly pseudoconcave in replenishment cycle; that is, there exists a unique and optimal solution to the problem. Next, an algorithm is proposed to obtain the optimal price, optimal length of inventory-holding period, and optimal length of shortage period, so as to maximize the profit. Finally, the sensitivity analyses of the parameters are conducted to demonstrate the effectiveness of the proposed model. The research results show that when the purchasing cost, the shortage cost, and the holding of fashion apparel increase, the average profit of system will decrease. The optimal average profit and order quantity are highly sensitive to the deterioration rate of fashion level and the backordering resistance. The presence of shortage has got an affirmative effect on ordering policy of the fashion apparel retailer.In this paper, we mainly focus on the impact of fashion level and products’ shortage on the operational policy. Although we take the pricing decision into account in the model, the attention on the pricing policy is limited, since it is not affected by fashion level and time. In fact, dynamic pricing is a powerful tool to manage the demand changing over time products. Dynamic pricing is a price discrimination based on the time of purchase by customers. It can reduce the fluctuation of demand to a certain extent and increase the company’s income. Therefore, the problem of dynamic pricing strategy and replenishment cycle for fashion apparel by considering the effect of fashion level and products’ shortage would be worth exploring in the future. In addition, this paper considers the deterministic situation, thus considering the stochastic situation, such as stochastic demand, which can be another future research direction.



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*Source: 7092981-2021-01-05.xml*