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Optimizing an Omnichannel Retail Strategy Considering Customer Segmentation

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Title: Optimizing an Omnichannel Retail Strategy Considering Customer Segmentation
Language: English
Authors: Shuangpeng Yang, Li Zhang (ORCID 0009-0008-5537-3660)
Source: Evaluation Review. 2025 49(5):814-850.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 37
Publication Date: 2025
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Retailing, Geographic Distribution, Models, Algorithms, Probability
DOI: 10.1177/0193841X251328710
ISSN: 0193-841X
1552-3926
Abstract: Unlike previous studies on fixed logistics nodes, this research explored how consumer distribution impacts store selection and inventory balance, integrating the "ship-from-store" strategy to increase fulfillment within multiperiod sales plans. Specifically, omnichannel retailers (O-tailer) must sequentially decide on inventory replenishment from suppliers to the distribution center (DC), allocation from the DC to stores, and which department will fulfill online orders. We introduce a multiperiod stochastic optimization model and solve it with a robust two-stage approach (RTA). In Stage 1, we use the K-means algorithm and silhouette coefficients to determine the optimal number of stores. In Stage 2, linear decision rule (LDR) are employed to decide on replenishment, allocation, and order fulfillment quantities. Numerical experiments show that RTA outperforms existing methods, achieving solutions with efficiency gaps of less than 10%, even when assumptions are not fully met. Additionally, the sensitivity analysis shows that variations in product prices, fulfillment costs, market share, and customer distribution consistently lead to greater profits with the "ship-from-store" strategy.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1482118
Database: ERIC
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  Value: <anid>AN0187567149;evr01oct.25;2025Aug29.06:59;v2.2.500</anid> <title id="AN0187567149-1">Optimizing an Omnichannel Retail Strategy Considering Customer Segmentation </title> <p>Unlike previous studies on fixed logistics nodes, this research explored how consumer distribution impacts store selection and inventory balance, integrating the ship-from-store strategy to increase fulfillment within multiperiod sales plans. Specifically, omnichannel retailers (O-tailer) must sequentially decide on inventory replenishment from suppliers to the distribution center (DC), allocation from the DC to stores, and which department will fulfill online orders. We introduce a multiperiod stochastic optimization model and solve it with a robust two-stage approach (RTA). In Stage 1, we use the K-means algorithm and silhouette coefficients to determine the optimal number of stores. In Stage 2, linear decision rule (LDR) are employed to decide on replenishment, allocation, and order fulfillment quantities. Numerical experiments show that RTA outperforms existing methods, achieving solutions with efficiency gaps of less than 10%, even when assumptions are not fully met. Additionally, the sensitivity analysis shows that variations in product prices, fulfillment costs, market share, and customer distribution consistently lead to greater profits with the ship-from-store strategy.</p> <p>Keywords: omnichannel; customer segmentation; ship-from-store; robust optimization</p> <hd id="AN0187567149-2">Introduction</hd> <p>The rise of omnichannel retail strategies allows retailers to leverage opportunities arising from technological advancements such as handheld devices. According to the Global E-commerce Sales Report,[<reflink idref="bib3" id="ref1">3</reflink>] the number of digital shoppers worldwide has reached 2.64 billion, comprising 33.3% of the global population, and is projected to increase to 2.77 billion by 2025. In response, traditional retailers such as Walmart and Macy's are adopting proactive strategies such as channel integration and inventory sharing. This facilitates a more convenient shopping experience for customers, allowing them to compare product prices, sizes, and more across various channels. The challenge lies in efficiently organizing product inventory within the network to meet diverse consumer demands and achieve seamless circulation between supply and demand. JD.com and Amazon have led the way by strategically placing multiple fulfillment centers near customer locations, ensuring <emph>same-day</emph> or <emph>next-day</emph> order fulfillment. In contrast, traditional <emph>brick-and-mortar</emph> retailers, constrained by high fixed costs, have fewer fulfillment centers. As a result, many retailers rely heavily on cross-channel fulfillment methods such as <emph>ship-from-store</emph> and buy online and pick up in-store to increase delivery speed for online customers ([<reflink idref="bib13" id="ref2">13</reflink>]).</p> <p>Specifically, the primary task for omnichannel retailers (o-tailer) to achieve efficient operations on both ends of product supply and demand is to construct a rational and efficient logistics network. In reality, this network typically comprises one or more large distribution centers (DCs) and multiple physical stores near different customer regions, providing customers with a sufficient product catalog to choose from. From a logistics perspective, the challenge for the o-tailer lies in balancing inventory allocation between DCs and physical stores at each time point to fulfill demand. Otherwise, it may result in lost orders or inventory backlog ([<reflink idref="bib16" id="ref3">16</reflink>]). Real-time sharing of inventory information among different nodes within the distribution network is crucial to address this issue ([<reflink idref="bib9" id="ref4">9</reflink>]; [<reflink idref="bib24" id="ref5">24</reflink>]). Specifically, a DC can fulfill orders for online individual customers, even if physical stores are out of stock, and replenish stock for them. Physical stores must be ready to handle in-store demand orders and process online customer orders at the end of each time period ([<reflink idref="bib1" id="ref6">1</reflink>]).</p> <p>The accompanying challenge is the selection of store locations. Naturally, the density and range of customer distribution are crucial influencing factors because they determine the specific customer base that a store serves, which in turn affects the retailer's order fulfillment costs, inventory allocation costs, and consumer purchasing efficiency. Once these two issues are addressed, that is, store locations and their service scopes are determined, o-tailer can typically implement a <emph>ship-from-store</emph> strategy to address the flexible order fulfillment issues mentioned earlier. Specifically, when processing online orders, if a customer's designated store is out of stock, the o-tailer can fulfill orders using inventory from nearby stores. This strategy has numerous benefits, such as optimizing inventory sharing within the network, increasing proximity to consumers, improving order processing efficiency, shortening delivery times, and reducing order losses. However, this also increases the complexity of inventory allocation for the o-tailer. Improper allocation can lead to stockouts and surplus issues, thereby increasing operational costs and causing customer attrition ([<reflink idref="bib10" id="ref7">10</reflink>]).</p> <p>This study aims to address the aforementioned issues, considering an o-tailer facing a known geographical distribution of customers. It divides the customer base into multiple demand regions and establishes specific stores for retail activities. The o-tailer sources its products from a single supplier, offering customers a diverse range of products through both online and offline channels, utilizing the <emph>ship-from-store</emph> strategy as a fulfillment option. The objective is to jointly optimize decisions on replenishment, allocation, and order fulfillment for the o-tailer across multiple sales periods in scenarios where demand cannot be obtained in advance, with the aim of maximizing profits. Apart from customer segmentation decisions, the o-tailer will also make four sequential decisions in each period. (i) At the beginning of each period, the o-tailer determines the replenishment quantity for each product at the DC, incurring replenishment costs. (ii) After the DC receives these products, the o-tailer determines the allocation quantity of each product to the stores, accounting for store inventory constraints and incurring allocation costs. (iii) During each period, in-store demands are met upon arrival, but if a store is out of stock, the demand is lost. (iv) At the end of each period, the o-tailer chooses either the DC or a store to fulfill the online orders, considering the inventory levels. If both the DC and all stores are out of stock, the order is lost.</p> <p>Considering reality, certain characteristics of an o-tailer's product distribution network pose significant challenges for modeling and problem solving, including the following. (i) <emph>Customer Segmentation</emph>: Customer service segmentation directly influences the subsequent joint decision execution of inventory allocation and order fulfillment selection, which is a critical prerequisite for multichannel retailers to gain profits. (ii) <emph>Proactive</emph> vs. <emph>Reactive Decision-making</emph>: Replenishing and allocating inventory require proactive decision-making before demand unfolds, whereas order fulfillment requires reactive decision-making after demand unfolds. The o-tailer must seamlessly integrate and coordinate these two types of decision-making. (iii) <emph>Limited Facility Capacity</emph>: In each period, the o-tailer must allocate inventory within a constrained logistics network to achieve the optimal balance between product receipt and retrievable quantity. (iv) <emph>Fulfillment Flexibility</emph>: The o-tailer offers three online order fulfillment options: DC, local stores within the region, and nearby stores. This necessitates the consideration of various factors, such as allocation costs, when making inventory distribution decisions. To address these challenges, our research makes the following contributions.</p> <hd id="AN0187567149-3">Theoretical Contributions</hd> <p>To the best of our knowledge, this is the first paper to incorporate customer segmentation into the joint optimization of multiperiod, multiproduct bulk sales planning for an o-tailer, aiming to optimize retail profits by making informed decisions regarding replenishment, allocation, and order fulfillment on the basis of segmented customer preferences. To achieve this goal, we developed a multiperiod mixed-integer stochastic optimization model that intricately integrates segmentation, replenishment allocation, and reactive fulfillment decisions while also considering facility capacity and fulfillment flexibility. By adjusting the hypothetical parameters in various scenarios, we offer valuable managerial insights for the o-tailer, contributing significantly to the literature on omnichannel retail.</p> <hd id="AN0187567149-4">Practical Contributions</hd> <p>We employ a two-stage robust method (RTA) framework to solve the model, expanding the applicability of the RTA in the field of omnichannel retail operations optimization. Specifically, in stage 1, on the basis of the geographical distribution characteristics of consumers, we utilize the K-means algorithm to partition them, determine the service customer groups for each store and select the optimal number of stores via the silhouette coefficient method. In stage 2, building upon the consumer segmentation obtained in stage 1, we utilize a linear decision rule (LDR) to dynamically adjust replenishment, allocation, and fulfillment quantities to adapt to real-time demand fluctuations. Numerical experiments show that, compared with the expected value of perfect information (EVPI) method, the RTA consistently generates high-quality and stable solutions with an efficiency gap of less than 10%, even under various scenarios deviating from the model's assumptions. Additionally, the <emph>ship-from-store</emph> strategy helps retailers achieve more profit than does fulfilling orders solely through stores or distribution centers.</p> <p>The structure of the paper is as follows: Section 2 reviews the relevant literature. Section 3 outlines the problem framework, constructs a multiperiod stochastic model, and describes the details involved. The RTA is then applied to solve the model in Section 4. Section 5 presents numerical experiments to assess the RTA's performance and explore omnichannel strategies' benefits for the o-tailer. Finally, Section 6 provides concluding remarks for the paper.</p> <hd id="AN0187567149-5">Literature Review</hd> <p>In this section, we review the relevant literature on four key topics: (i) <emph>omnichannel retailing</emph>, (ii) <emph>customer segmentation</emph> and <emph>order fulfillment</emph>, and (iii) <emph>methodologies</emph> for analyzing fulfillment problems in an omnichannel environment. The literature review provides the foundation for our proposed solution approach.</p> <hd id="AN0187567149-6">Omnichannel Retailing</hd> <p>The consumer demand for seamless integration across channels continues to fuel annual growth in online sales ([<reflink idref="bib16" id="ref8">16</reflink>]; [<reflink idref="bib32" id="ref9">32</reflink>]). Brick-and-mortar stores, once seen as supplements to the purchasing process, are now evolving into integral parts of the buying journey ([<reflink idref="bib31" id="ref10">31</reflink>]; [<reflink idref="bib34" id="ref11">34</reflink>]). Previous research has broadly categorized omnichannel operations into two models: buy online and pick up in-store (BOPS) and ship-from-store (SFS). While BOPS excels in product turnover ([<reflink idref="bib16" id="ref12">16</reflink>]), it lacks suitability for high-value items. Factors such as consumer search costs and cross-selling benefits influence its effectiveness ([<reflink idref="bib23" id="ref13">23</reflink>]; [<reflink idref="bib33" id="ref14">33</reflink>]). Conversely, the SFS model utilizes stores as stocking warehouses, reducing inventory pressure ([<reflink idref="bib14" id="ref15">14</reflink>]), enhancing order fulfillment, and improving profitability when customers face high waiting costs ([<reflink idref="bib15" id="ref16">15</reflink>]). Research also highlights retailers' focus on balancing customer visit frequency and waiting costs due to the varying operating costs of the two modes ([<reflink idref="bib19" id="ref17">19</reflink>]; [<reflink idref="bib36" id="ref18">36</reflink>]). Furthermore, warehouse location considerably impacts network capacity under the SFS model ([<reflink idref="bib27" id="ref19">27</reflink>]). Prior theoretical research primarily examined simplified models to assess o-tailer strategies, neglecting specific decision optimization at the logistics level. In contrast, our study emphasizes joint decision optimization within the distribution process, encompassing ordering, allocation, and order fulfillment, with the aim of offering practical insights to increase the efficiency of retailers.</p> <hd id="AN0187567149-7">Customer Segmentation and Order Fulfillment</hd> <p>As online retail has become increasingly prevalent, retailers often focus on optimizing costs across various segments of their logistics network, including channel selection ([<reflink idref="bib4" id="ref20">4</reflink>]; [<reflink idref="bib8" id="ref21">8</reflink>]) and inventory control ([<reflink idref="bib5" id="ref22">5</reflink>]; [<reflink idref="bib26" id="ref23">26</reflink>]). However, previous studies have typically assumed that physical retail stores are static, requiring no predecision-making ([<reflink idref="bib18" id="ref24">18</reflink>]; [<reflink idref="bib38" id="ref25">38</reflink>]). However, in reality, consumer distribution tends to cluster, necessitating a reasonable division of demand regions to determine inventory allocation and order fulfillment. Researchers have also explored factors such as high fulfillment costs ([<reflink idref="bib17" id="ref26">17</reflink>]), showrooms ([<reflink idref="bib11" id="ref27">11</reflink>]), fulfillment lead time ([<reflink idref="bib35" id="ref28">35</reflink>]), and channel switching costs ([<reflink idref="bib28" id="ref29">28</reflink>]) to predict customer demand and propose practical strategies. Nonetheless, these studies often overlook the specific optimization of retail bulk transaction order fulfillment within the logistics network, focusing solely on macrolevel strategic applicability. In contrast, our study integrates consumer segmentation with the "ship-from-store" strategy to assist retailers operationally by jointly optimizing across multiple products, periods, channels, and decision-making scenarios under demand uncertainty, offering a practical perspective on retail operations.</p> <hd id="AN0187567149-8">Methodologies</hd> <p>For customer segmentation, researchers often use the K-means algorithm of [<reflink idref="bib25" id="ref30">25</reflink>], which is known for its simplicity and low linear time complexity. However, it cannot directly determine the optimal number of clusters. To address this, [<reflink idref="bib29" id="ref31">29</reflink>] introduced the silhouette coefficient, which evaluates cluster quantity optimality by considering density correlations within and between clusters, overcoming this limitation. For handling uncertain parameters, robust optimization (RO) stands out as a prominent approach favored by researchers. Typically, dealing with uncertainty involves introducing a flexible space for uncertain factors, aimed at making the problem tractable ([<reflink idref="bib37" id="ref32">37</reflink>]). Many researchers tend to lean toward rule designs on the basis of the actual values of variables. For example, [<reflink idref="bib8" id="ref33">8</reflink>]'s study offers an effective method for avoiding interference from uncertain parameters. However, research by [<reflink idref="bib6" id="ref34">6</reflink>] and [<reflink idref="bib22" id="ref35">22</reflink>] suggested that these static rules may yield optimal results only in specific scenarios. In contrast, the linear decision rule (LDR) provides a more practical approach by transforming decision variables into affine functions of uncertain parameters for solving [<reflink idref="bib7" id="ref36">7</reflink>]; [<reflink idref="bib20" id="ref37">20</reflink>]; [<reflink idref="bib30" id="ref38">30</reflink>] extended the LDR further by handling uncertain parts through piecewise linear relationships. Additionally, unlike LDR, [<reflink idref="bib7" id="ref39">7</reflink>]; [<reflink idref="bib12" id="ref40">12</reflink>] introduced high-dimensional probability spaces to map uncertain components for solving. In summary, this study addresses a multiperiod stochastic model by first employing the K-means algorithm to partition consumers into regions and then using the silhouette coefficient method to determine the optimal number of stores. This method subsequently applies the LDR method to handle demand uncertainty and solve it. Ultimately, it provides retail businesses with operation plans tailored to real-world scenarios.</p> <hd id="AN0187567149-9">Problem Formulation</hd> <p>In this section, we introduce the customer segmentation and omnichannel distribution problems faced by an o-tailer. We first describe the basic problem setting. Then, we propose a consumer segmentation model. Finally, we present a stochastic optimization model that jointly determines consumer segmentation and omnichannel distribution decisions.</p> <hd id="AN0187567149-10">Basic Settings</hd> <p>Consider an o-tailer facing a DC and a demand population consisting of <emph>K</emph> customers. The o-tailer needs to partition them into <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> demand regions on the basis of customer geographical distribution and certain customer density constraints, noting the inability to predetermine them in advance. Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>=</mo><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></mrow><mo>}</mo></math> </ephtml> represent the set of demand regions. Then, the o-tailer needs to establish a unique physical store in each demand region; thus, the set of stores can naturally be represented by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>=</mo><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></mrow><mo>}</mo></math> </ephtml> . Notably, our model can be trivially extended to general cases with multiple or no stores in some zones. Subsequently, the o-tailer simultaneously sells <emph>I</emph> varieties of substitute products to customers both online and offline, with a sales cycle of <emph>T</emph>. Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>=</mo><mfenced close="}" open="{"><mrow><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>I</mi></mrow></mfenced></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>=</mo><mfenced close="}" open="{"><mrow><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>T</mi></mrow></mfenced></math> </ephtml> denote the sets of products and the sales horizon, respectively. Customers in region <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> purchase either from the online channel (<emph>e-commerce</emph> demand, denoted by <emph>e</emph>) or from store <emph>j</emph> (<emph>brick-and-mortar</emph> demand, denoted by <emph>b</emph>). Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><mrow><mi>e</mi><mo>,</mo><mi>b</mi></mrow><mo stretchy="false">}</mo></mrow></math> </ephtml> denote a collection of online and store channels. Importantly, because we cannot predefine the set of stores, we cannot know the number of customers served by each store in the demand regions. The DC supports both the online channel by shipping individual products to fulfill customer orders and the offline channel by shipping high-volume products to replenish store inventory. The stores also support online sales through <emph>ship-from-store</emph> implementation. This study assumes a multiperiod inventory allocation and fulfillment problem under random demand. Throughout the sales cycle, the o-tailer needs to make the following sequential decisions. (i) At the beginning of each period, the o-tailer must make replenishment and allocation decisions, which involve determining the quantities of each product to order from suppliers at the DC and the quantities of each product to allocate from the DC to each store. This process incurs unit replenishment and allocation costs. (ii) During each period, in-store demand is directly satisfied. (iii) At the end of each period, the o-tailer must make fulfillment decisions, determining which stores or DC fulfill specific online orders. This process incurs unit fulfillment costs. Specifically, in cases of stockouts when customers make purchases, we assume that the corresponding demand orders are lost.</p> <hd id="AN0187567149-11">Customer Segmentation</hd> <p>In reality, retailers often need to choose suitable store locations on the basis of factors such as the geographical distribution of their potential customers and their consumption preferences to control costs. In this study, we consider the former scenario, assuming a customer group set <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">K</mi></math> </ephtml> , where each customer <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="script">K</mi><mo>=</mo><mrow><mo stretchy="false">{</mo><mrow><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>K</mi></mrow><mo stretchy="false">}</mo></mrow></math> </ephtml> has a unique geographic coordinate <emph>B</emph><emph>k</emph>. Our goal is to partition them into compact demand areas <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> according to their geographical distribution while maximizing the density between customer groups within each demand area <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> to minimize delivery costs. Let <emph>S</emph><emph>j</emph> represent the customer group set for demand areas <emph>j</emph>, where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow><mi>c</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> </ephtml> represents the geographic location of corresponding store <emph>j</emph>, and binary variable <emph>Z</emph><emph>kj</emph> = 1 represents customers being served by stores <emph>j</emph>, and vice versa. For ease of representation, let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi mathvariant="bold">c</mi></mrow><mo>˜</mo></mover></mrow><mo>=</mo><mfenced close="}" open="{"><mrow><msub><mrow><mover accent="true"><mrow><mi>c</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></mrow></mfenced></math> </ephtml> denote the set of store locations to be determined. Furthermore, let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">z</mi><mfenced close=")" open="("><mrow><mover accent="true"><mrow><mi mathvariant="bold">c</mi></mrow><mo>˜</mo></mover></mrow></mfenced><mo>=</mo><mfenced close="}" open="{"><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><mfenced close=")" open="("><mrow><mover accent="true"><mrow><mi mathvariant="bold">c</mi></mrow><mo>˜</mo></mover></mrow></mfenced><mo>,</mo><mi>k</mi><mo>∈</mo><mi mathvariant="script">K</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow></mrow></mfenced></math> </ephtml> represent the customer partition set. Then, the following constraints exist: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi>min</mi><munder><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><mi mathvariant="script">K</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow></mrow></munder><mfenced close=")" open="("><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munder><munder><mrow><mo form="prefix" movablelimits="false">arg</mo><mo form="prefix" movablelimits="false">min</mo></mrow><mrow><mi>j</mi></mrow></munder><msup><mrow><mfenced close="‖" open="‖"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><msub><mrow><mover accent="true"><mrow><mi>c</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="right" /></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow></mrow></munder><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo></mtd><mtd columnalign="right"><mi>k</mi><mo>∈</mo><mi mathvariant="script">K</mi></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p>Equation (<reflink idref="bib1" id="ref41">1</reflink>) represents the optimal outcome pursued by consumer segmentation, namely, the maximum customer density across all regions. Constraints (<reflink idref="bib2" id="ref42">2</reflink>) indicate that a customer can be segmented into only one demand area.</p> <hd id="AN0187567149-12">Random Demand</hd> <p>Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>m</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> represent the random demand of channel <emph>m</emph> in zone <emph>j</emph> during period <emph>t</emph>, which is realized as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>d</mi></mrow><mrow><mi>m</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> at the end of period <emph>t</emph>. We assume that demands are exogenous and independent across zones but that they can exhibit channel correlation within the same zone. If the o-tailer cannot fulfill a specific product's online or in-store demand due to stockouts, we assume that this demand is lost. For simplicity, we use <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></math> </ephtml> to represent the collection of all demands from period 1 to period <emph>t</emph>, given by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup><mo>=</mo><mfenced close=")" open="("><mrow><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>m</mi><mi>j</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi></mrow></mfenced></math> </ephtml> . In our study, the LDR method can dynamically determine the current decision variables on the basis of historical data. For convenience, the decision variables mentioned in the subsequent text can be represented by the nonanticipative functions of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></math> </ephtml> .</p> <hd id="AN0187567149-13">Inventory Allocation</hd> <p>At the start of each period <emph>t</emph>, the o-tailer initiates the replenishment process. They order <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> </ephtml> units of product <emph>i</emph> from suppliers to the DC, updating the DC's inventory <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> </ephtml> . The o-tailer subsequently allocates <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> </ephtml> units of product <emph>i</emph> from the DC to store <emph>j</emph>, replenishing the store's inventory <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> </ephtml> . Store <emph>j</emph> has a capacity constraint <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> </ephtml> . These decisions are made after <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> is revealed and incur unit replenishment costs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> and allocation costs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> per unit. Importantly, the initial inventory levels are set to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> </ephtml> for the DC and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> </ephtml> for store <emph>j</emph>. Moreover, both the DC and the stores have holding costs, denoted as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> , respectively. The relevant constraints are as follows: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi></mrow></munder><mfenced close="]" open="["><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>,</mo><mo>∈</mo><msubsup><mrow><mi mathvariant="script">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p>Constraints (<reflink idref="bib3" id="ref43">3</reflink>) and (<reflink idref="bib4" id="ref44">4</reflink>) ensure that the initial inventory levels of the DC and all the stores are set to zero. Constraint (<reflink idref="bib5" id="ref45">5</reflink>) guarantees that at the start of each period <emph>t</emph>, after store <emph>j</emph> receives the allocated products, its inventory remains within the capacity limit. Constraint (<reflink idref="bib6" id="ref46">6</reflink>) restricts the feasible ranges of the corresponding variables. Here, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="script">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msubsup></math> </ephtml> represents a set of nonanticipative functions that map <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>M</mi><mo>×</mo><mi>I</mi><mo>×</mo><mi>J</mi><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msubsup></math> </ephtml> to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mo>+</mo></mrow></msub></math> </ephtml> .</p> <hd id="AN0187567149-14">Order Fulfillment</hd> <p>We assume that the <emph>in-store</emph> demand for product <emph>i</emph> in zone <emph>j</emph> is immediately met by store <emph>j</emph> upon its occurrence. For online demand, the o-tailer must determine whether to fulfill via the DC or the stores. We define <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></math> </ephtml> to represent the fulfillment orders for product <emph>i</emph> in zone <emph>j</emph>, fulfilled by the DC and store <emph>j</emph>′ at the end of period <emph>t</emph> and incurring unit fulfillment costs of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> , respectively. Note that the fulfillment order decisions are made after the revelation of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></math> </ephtml> . Orders that cannot be fulfilled are considered lost. Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mrow><mo>(</mo></mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>)</mo></mrow></mrow></math> </ephtml> denote the sales quantity of product <emph>i</emph> in zone <emph>j</emph> in period <emph>t</emph>. The corresponding constraints are as follows: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>≤</mo><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">eij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><mfenced close="]" open="["><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>+</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>−</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">bij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align" columnalign="left"><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>∈</mo><msubsup><mrow><mi mathvariant="script">R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="right" /><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>,</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>.</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p>Constraints (<reflink idref="bib7" id="ref47">7</reflink>) ensure that the sales quantities for product <emph>i</emph> in zone <emph>j</emph> from channel <emph>m</emph> do not exceed the corresponding demand. Constraint (<reflink idref="bib8" id="ref48">8</reflink>) states that the online sales quantity of product <emph>i</emph> in zone <emph>j</emph> equals the fulfillment quantities from both the DC and the stores. Constraints (<reflink idref="bib9" id="ref49">9</reflink>) and (<reflink idref="bib10" id="ref50">10</reflink>) indicate the inventory flow balance for the DC and stores, respectively. Constraint (<reflink idref="bib11" id="ref51">11</reflink>) restricts the feasible ranges of the variables involved.</p> <hd id="AN0187567149-15">Stochastic Optimization Model</hd> <p>Our goal is to maximize the total expected revenue by integrating distribution decisions across the omnichannel retail network. Let <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">g</mi><mrow><mrow><mo>(</mo></mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover><mrow><mo>)</mo></mrow></mrow><mo>=</mo><mfenced close="" open="("><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open=""><mrow><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>,</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mrow><mover accent="true"><mrow><mi mathvariant="script">J</mi></mrow><mo>→</mo></mover></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi></mrow></mfenced></math> </ephtml> denote a collection of all the adaptive decision variables. Given <bold>d</bold> and its corresponding result <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">g</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">d</mi></mrow></mfenced></math> </ephtml> , as well as <bold>c</bold> and its corresponding result <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">z</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">c</mi></mrow></mfenced></math> </ephtml> , the total revenue of the o-tailer for the entire sales period can be expressed as follows: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="array"><mtr><mtd columnalign="center"><mi mathvariant="normal">Ψ</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">g</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">d</mi></mrow></mfenced><mo>;</mo><mi mathvariant="bold">z</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">c</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi></mrow></munder><mo>[</mo><munder><mrow><munder accentunder="false"><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi></mrow></munder><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>sales revenue</mtext></mrow></munder><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>replenishment cost</mtext></mrow></munder><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>allocation cost</mtext></mrow></munder><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>online fulfillment cost</mtext></mrow></munder></mtd></mtr><mtr><mtd columnalign="center"><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>store fulfillment cost</mtext></mrow></munder><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><msubsup><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>DC holding cost</mtext></mrow></munder><mo>−</mo><munder><mrow><munder accentunder="false"><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mo>⏟</mo></munder></mrow><mrow><mtext>store holding cost</mtext></mrow></munder><mo>]</mo><mo>.</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>Next, we construct a stochastic optimization model P<subs>S</subs>. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="align-star" columnalign="left"><mtr><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mtext>P</mtext></mrow><mrow><mtext>S</mtext></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mi>max</mi><msub><mrow><mi mathvariant="double-struck">E</mi></mrow><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow></msub><mfenced close="]" open="["><mrow><mi mathvariant="normal">Ψ</mi><mfenced close=")" open="("><mrow><mi mathvariant="bold">g</mi><mfenced close=")" open="("><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow></mfenced><mo>;</mo><mi mathvariant="bold">z</mi><mfenced close=")" open="("><mrow><mover accent="true"><mrow><mi mathvariant="bold">c</mi></mrow><mo>˜</mo></mover></mrow></mfenced></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd columnalign="right"><mtext>s.t.</mtext></mtd><mtd columnalign="left"><mtext>Constraints </mtext><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mtext>–</mtext><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>When the locations of offline stores, customer segmentation, and randomly determined demand are obtained, by solving Problem P<subs>S</subs>, we can obtain the optimal replenishment, allocation, and fulfillment decisions, thereby maximizing the profit of the o-tailer. However, when dealing with the target expectations of solving Problem P<subs>S</subs>, we face two issues. (i) Store location selection and customer segmentation are typical NP-hard problems that cannot be directly solved through linear models. (ii) Traditional methods such as dynamic programming often assume specific demand distributions with known parameters. However, in reality, demand distribution information is often limited, meaning that current demand data cannot be directly obtained. Therefore, in this study, we consider a two-stage approach. First, we use the K-means algorithm to determine store locations and perform customer segmentation. Then, we adopt the linear decision rule (LDR) proposed by [<reflink idref="bib2" id="ref52">2</reflink>], combined with Lagrangian strong duality theory, to transform the stochastic problem P<subs>S</subs> into a linearly solvable problem.</p> <hd id="AN0187567149-16">Two-Stage Solving Approach</hd> <p>In this section, we propose a solution framework called the robust two-stage approach (RTA) to address Problem P<subs>S</subs> in the presence of demand distributional ambiguity and nonlinearity. Specifically, in stage 1, we utilize the K-means algorithm to partition consumers into different groups on the basis of geographical distance and ascertain the optimal number of stores via the silhouette coefficient method. In stage 2, we adaptively determine the replenishment, allocation, and fulfillment quantities by implementing the LDR approach.</p> <hd id="AN0187567149-17">Stage 1: Determining Customer Segmentation</hd> <p>We use the K-means algorithm to iteratively group customers who are geographically close together into the same service cluster <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>∈</mo><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> , with each cluster <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> </ephtml> corresponding to a setup store <emph>j</emph>. However, the limitation of the K-means algorithm lies in its inability to determine the optimal number of demand regions in advance. In this paper, following referenced studies by [<reflink idref="bib3" id="ref53">3</reflink>], the silhouette coefficient method is employed to measure the density between customers in demand regions, thereby determining the optimal number of demand regions/stores. Specifically, we use <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="normal">Γ</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub></math> </ephtml> to represent the silhouette coefficient when the number of stores is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> , with the calculation formula as follows: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="normal">Γ</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>K</mi></mrow></mfrac><munder><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><mi mathvariant="script">K</mi></mrow></munder><mfrac><mrow><msub><mrow><mi>α</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced><mo>−</mo><msub><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced></mrow><mrow><mi>max</mi><mfenced close="}" open="{"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced></mrow></mfenced></mrow></mfrac><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>In equation (<reflink idref="bib12" id="ref54">12</reflink>), <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>α</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mo form="prefix" movablelimits="false">∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msubsup><msub><mrow><mo form="prefix" movablelimits="false">∑</mo></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">K</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mi>k</mi></mrow></msub><mn>1</mn><mo>/</mo><msubsup><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow><mrow><mi mathvariant="italic">num</mi></mrow></msubsup><mo>−</mo><mn>1</mn><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math> </ephtml> represents the average distance between customer <emph>k</emph> and other customers within the current service range of the corresponding store, where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow><mrow><mi mathvariant="italic">num</mi></mrow></msubsup></math> </ephtml> represents the number of customers served by store <emph>j</emph>, and <emph>l</emph><emph>kk</emph>' represents the distance between customers <emph>k</emph> and <emph>k</emph>′. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mfenced close=")" open="("><mrow><mi>k</mi></mrow></mfenced><mo>=</mo><mi>min</mi><mfenced close="}" open="{"><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>k</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>k</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub></mrow></mfenced></mrow></mfenced></math> </ephtml> represents all shops where the average distance between customer <emph>k</emph> and the group of customers they serve is the smallest, and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>β</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>k</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mo form="prefix" movablelimits="false">∑</mo></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mn>1</mn><mo>/</mo><msubsup><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>j</mi></mrow><mrow><mi mathvariant="italic">num</mi></mrow></msubsup><msub><mrow><mi>l</mi></mrow><mrow><mi>k</mi><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math> </ephtml> . Obviously, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="normal">Γ</mi></mrow><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></msub><mo>∈</mo><mfenced close="]" open="["><mrow><mo>−</mo><mn>1,1</mn></mrow></mfenced></math> </ephtml> , and the closer its value is to 1, the more reasonable the partitioning is, which means that the selection of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> is more reasonable. On this basis, when the K-means algorithm is used, a range for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> is first specified, then the customer segmentation for each corresponding <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> is calculated, and finally, the best division of the demand areas is determined. The specific process is as shown in Algorithm 4.1:</p> <p>Algorithm 1 K-Means Clustering with Silhouette Coefficients</p> <p>Graph</p> <hd id="AN0187567149-18">Stage 2: Dealing With Demand Uncertainty</hd> <p>As described in Section 4.1, we obtained the set of customers for <emph>J</emph> stores and their corresponding service set <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>S</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></math> </ephtml> . At this point, the challenge for the o-tailer in implementing inventory allocation within the retail network lies mainly in the randomness of demand across different service sets. To address this, referencing the research of [<reflink idref="bib2" id="ref55">2</reflink>], we introduce a robust LDR method to address the uncertainty of demand distribution to optimize replenishment, allocation, and order fulfillment decisions in the problem.</p> <p>The key to applying robust optimization lies in constructing a set of uncertain parameters. To this end, we assume that at time <emph>t</emph>, the demand <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> for product <emph>i</emph> in service set <emph>S</emph><emph>j</emph> and channel <emph>m</emph> belongs to a set with a mean of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> and bounds <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msubsup><mrow><munder accentunder="false"><mrow><mi>d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced></math> </ephtml> . Furthermore, we construct the set <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>D</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>=</mo><mo>{</mo><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo stretchy="false">|</mo><msubsup><mrow><munder accentunder="false"><mrow><mi>d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>≤</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>≤</mo><msubsup><mrow><mover accent="true"><mrow><mi>d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow><mo>}</mo></math> </ephtml> , where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi></math> </ephtml> corresponds to all service sets. For ease of model representation, we further define <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>=</mo><mo>{</mo><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi></mrow><mo>}</mo></math> </ephtml> . Then, we employ the LDR method to approximate the optimal decision to minimize the optimality loss. Specifically, as shown in equation (<reflink idref="bib13" id="ref56">13</reflink>), we constrain the continuous decision variables involved in the model to be affine functions of the uncertain demand. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="array"><mtr><mtd columnalign="left"><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>,</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mo>∀</mo><msup><mrow><mi mathvariant="bold">d</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">D</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left" /></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>The coefficients <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> represent vectors of decision and dependent variables pertaining to the online and in-store demands of zone <emph>σ</emph> for product <emph>i</emph> in period <emph>τ</emph>, respectively. Using the LDR in equation (<reflink idref="bib13" id="ref57">13</reflink>), we can compute the coefficients <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></math> </ephtml> as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>˜</mo></mover></mrow><mrow><mi>t</mi></mrow></msup></math> </ephtml> is revealed. Next, we can compute the values of the variables involved in the model. This involves transforming the decision and dependent variables into the LDR coefficients, as shown in equation (<reflink idref="bib13" id="ref58">13</reflink>). With reference to Theorem 2 in the research by [<reflink idref="bib18" id="ref59">18</reflink>], we can obtain the optimal LDR coefficients for maximizing the expected total revenue through the following model Problem P<subs>LDR</subs>. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable class="eqnarray-star"><mtr><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mtext>P</mtext></mrow><mrow><mtext>LDR</mtext></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mspace class="nbsp" width="0.3333em" /><mspace class="nbsp" width="0.3333em" /><mi>max</mi><mspace class="nbsp" width="0.3333em" /><munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi></mrow></munder><mfenced close="" open="["><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>−</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced></mtd><mtd columnalign="left" /><mtd columnalign="left" /></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced></mtd><mtd columnalign="left" /></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>−</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced></mtd><mtd columnalign="left" /></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="("><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>]</mo></mtd><mtd columnalign="left" /></mtr><mtr><mtd columnalign="left"><munder><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi></mrow></munder><mfenced close="]" open="["><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced><mo>≤</mo><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo></mtd><mtd columnalign="left"><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≤</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mfenced close="" open="{"><mrow><mtable class="cases"><mtr><mtd columnalign="left"><mn>1</mn><mo>,</mo><mspace width="1em" /></mtd><mtd columnalign="left"><mtext>if</mtext><mspace class="nbsp" width="0.3333em" /><msup><mrow><mi>m</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>m</mi><mo>,</mo><mi>σ</mi><mo>=</mo><mi>j</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="left"><mn>0</mn><mo>,</mo><mspace width="1em" /></mtd><mtd columnalign="left"><mtext>otherwise</mtext><mo>;</mo></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">eij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">eij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><mfenced close=")" open="("><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">bij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">bij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi mathvariant="italic">bij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><munder><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">v</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>,</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">w</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>,</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>s</mi></mrow><mrow><mi mathvariant="italic">mij</mi></mrow><mrow><mi>t</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">s</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>m</mi><mo>∈</mo><mi mathvariant="script">M</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1,0</mn></mrow></msubsup><mo>−</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi></mrow></munder><munderover accent="true" accentunder="false"><mrow><mo>∑</mo></mrow><mrow><mi>τ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></munderover><mfenced close=")" open="("><mrow><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><mover accent="true"><mrow><mi mathvariant="bold">d</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><msubsup><mrow><munder accentunder="false"><mrow><mi mathvariant="bold">d</mi></mrow><mo accent="true">̲</mo></munder></mrow><mrow><mi>i</mi><mi>σ</mi></mrow><mrow><mi>τ</mi></mrow></msubsup></mrow></mfenced><mo>≥</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold">y</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd columnalign="left"><mi>i</mi><mo>∈</mo><mi mathvariant="script">I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi mathvariant="script">T</mi><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="script">J</mi><mo>,</mo><mi>τ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>≥</mo><mn>0</mn><mo>;</mo></mtd><mtd columnalign="left" /></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>s</mi><mi>m</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>q</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>u</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>v</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>w</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">μ</mi></mrow><mrow><mi>j</mi><mi>y</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>σ</mi><mi>τ</mi></mrow></msubsup><mo>≥</mo><mn>0</mn><mo>;</mo></mtd><mtd columnalign="left" /></mtr></mtable></math> </ephtml></p> <p>Graph</p> <hd id="AN0187567149-19">Numerical Experiments</hd> <p>In this section, we describe the simulation and evaluation of the performance of the RTA and explore the advantages of omnichannel strategies. First, we assess the clustering performance of the K-means algorithm with various consumer distributions, calculating silhouette coefficients to determine the optimal number of stores (Section 5.1). To evaluate the RTA approach comprehensively, we compare its solution quality and computational tractability with those of the deterministic approach (DET) and the expected value given perfect information approach (EVPI), considering different planning horizons and larger-size instances (Section 5.2). Additionally, we assess the robustness of the RTA across various demand distributions and deviations from specified uncertainty sets (Section 5.3). Furthermore, we conduct a sensitivity analysis (Section 5.4) to examine o-tailer response strategies in different retail environments. We also analyze the advantages of omnichannel operations under the <emph>ship-from-store</emph> strategy, considering different conditional assumptions (Section 5.5). Finally, we explore the benefits of flexible omnichannel fulfillment from a managerial perspective, focusing on its potential for driving profit growth for the o-tailer (Section 5.6).</p> <p>First, we assume that the range of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> in the K-means algorithm is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mi>K</mi><mo>/</mo><mn>400</mn><mo>,</mo><mi>K</mi><mo>/</mo><mn>600</mn></mrow></mfenced></math> </ephtml> and that each <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>J</mi></mrow><mo>→</mo></mover></mrow></math> </ephtml> value within this interval is approximately an integer. We take the customer distribution radius to be 50, and the sales horizon is <emph>T</emph> = 7. Second, we assume that the demand for product <emph>i</emph> over time <emph>t</emph> in each demand region <emph>j</emph> is related to the size of its customer base, with a range of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˘</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>/</mo><mn>10</mn><mo>−</mo><mn>10</mn><mo>,</mo><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˘</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>/</mo><mn>10</mn><mo>+</mo><mn>10</mn></mrow></mfenced></math> </ephtml> , where <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow><mi>S</mi></mrow><mo>˘</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> </ephtml> represents the number of customers that store <emph>j</emph> needs to serve. The online market share is represented by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>ϑ</mi></mrow><mrow><mi>o</mi><mi>n</mi></mrow></msub><mo>∈</mo><mfenced close="]" open="["><mrow><mn>0,1</mn></mrow></mfenced></math> </ephtml> , with a default value of 0.5 if not specified otherwise. Next, to ensure the reliability of the experimental data, we use a Monte Carlo simulation to select three beta distributions and generate 500 random demand samples, specifically, <emph>Beta</emph>(<reflink idref="bib1" id="ref60">1</reflink>, 1), <emph>Beta</emph>(<reflink idref="bib1" id="ref61">1</reflink>, 0.5) and <emph>Beta</emph>(0.5, 1). Finally, we solve the corresponding stochastic models for each scenario and calculate the average revenue for each sample group. Notably, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><mtext>i</mtext></mrow><mo stretchy="false">)</mo></mrow><mspace width="0.3333em" /><mspace width="0.3333em" /><msubsup><mrow><mi>f</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>></mo><msubsup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> </ephtml> , indicating higher costs for individual parcel shipping than for bulk inventory allocation. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><mtext>ii</mtext></mrow><mo stretchy="false">)</mo></mrow><mspace width="0.3333em" /><mspace width="0.3333em" /><msubsup><mrow><mi>F</mi></mrow><mrow><mi mathvariant="italic">ijj</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo><</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mo>,</mo><mspace width="0.3333em" /><mi>j</mi><mo>≠</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></math> </ephtml> , reflecting higher costs for shipping across different zones than for same-zone shipments. The parameter settings in Table 1 are randomly selected within specified ranges.</p> <p>Table 1. Parameter Setting.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left">Parameter</th><th align="center">Value</th><th align="center">Parameter</th><th align="center">Value</th><th align="center">Parameter</th><th align="center">Value</th><th align="center">Parameter</th><th align="center">Value</th></tr></thead><tbody valign="top"><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" xmlns=""><mrow><mn>15,20</mn></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left">0.3 × <italic>p</italic><italic>i</italic></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" xmlns=""><mrow><mn>1,2</mn></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>f</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo xmlns="">+</mo><mi xmlns="">ε</mi><mo xmlns="">,</mo><mi xmlns="">ε</mi><mo xmlns="">∈</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>1,3</mn></mrow></mfenced></math></p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>F</mi></mrow><mrow><mi mathvariant="italic">ijj</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" xmlns=""><mrow><mn>1,2</mn></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>j</mi><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced close=")" open="(" xmlns=""><mrow><mi>j</mi><mo>≠</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>F</mi></mrow><mrow><mi mathvariant="italic">ijj</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo xmlns="">+</mo><mi xmlns="">ε</mi><mo xmlns="">,</mo><mi xmlns="">ε</mi><mo xmlns="">∈</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>1,2</mn></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" xmlns=""><mrow><mn>0.3</mn><mo>,</mo><mn>0.6</mn></mrow></mfenced></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="[" xmlns=""><mrow><mn>0.6</mn><mo>,</mo><mn>0.9</mn></mrow></mfenced></math></p></td></tr></tbody></table> </ephtml> </p> <p>Additionally, the LDR method proposed in this paper requires taking certain measures to handle noninteger decisions when obtaining results. Specifically, the results in equation (<reflink idref="bib13" id="ref62">13</reflink>) are rounded to the nearest integer. However, this process necessitates adjustments in certain scenarios. For instance, if, after inventory allocation, the DC causes the total inventory quantity of a certain store to exceed capacity limits, we choose to reduce the quantity of the product with the highest allocation cost. Similarly, if the total number of orders for a product in a region exceeds its online demand, we opt to reduce the order quantity to the region from the department with the highest fulfillment cost at the DC or store. Furthermore, to better evaluate the performance of the algorithm proposed in the article, we simultaneously employ the deterministic method (DET) based on demand averages as proposed by [<reflink idref="bib21" id="ref63">21</reflink>] for comparative analysis. Through numerical experiments in various scenarios, the solution results of LDR and DET are juxtaposed with the solution method on the basis of the expected value given perfect information (EVPI). To achieve this, we define the effective gap formula as (<emph>Revenue</emph> − <emph>EVPI</emph>)/<emph>EVPI</emph> × 100% to assess the performance indicators of each method.</p> <p>When calculating the revenue for each set of samples via the RTA approach, we round nonintegral decisions from equation (<reflink idref="bib13" id="ref64">13</reflink>) to the nearest integers. Infeasibility issues may arise, and we handle them by adjusting the decisions. For example, if store <emph>j</emph> exceeds capacity after allocation, we reduce quantities from costly DCs. Similarly, if total fulfillment exceeds online demand in zone <emph>j</emph> for product <emph>i</emph>, we decrease fulfillment from the center or store with the highest cost. To better evaluate the performance of an RTA, we employ a deterministic (DET) approach based on the model proposed by [<reflink idref="bib21" id="ref65">21</reflink>], which solves a deterministic model using mean demand. Through experiments on various problem instances, we compare the tested approaches to the expected value given perfect information (EVPI). The EVPI is computed by solving optimal hindsight decisions and averaging the returns. The efficiency gap, (<emph>Revenue</emph> − <emph>EVPI</emph>)/<emph>EVPI</emph> × 100%, serves as the performance metric for each approach.</p> <hd id="AN0187567149-20">Customer Segmentation and Store Service Set Determination</hd> <p>In this section, we describe the partitioning of the customer population with a scale of <emph>K</emph> = 5000 using the K-means algorithm to assess the reasonableness of the division. As mentioned earlier, the range of customer region quantities is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo>∈</mo><mfenced close="]" open="["><mrow><mn>8,13</mn></mrow></mfenced></math> </ephtml> . The results are shown in Figure 1. When customer distributions vary, the demand areas ultimately divided by the K-means algorithm also exhibit certain differences. When <emph>Beta</emph>(<reflink idref="bib1" id="ref66">1</reflink>, 1), the customer distribution is relatively uniform. When <emph>Beta</emph>(<reflink idref="bib1" id="ref67">1</reflink>, 0.5), customers are significantly more concentrated in the upper right area, whereas when <emph>Beta</emph>(0.5, 1), they are more concentrated in the lower left area. Certainly, the K-means algorithm segmentation also exhibits characteristics of some stores with more customer services and others with less, which can be clearly seen from in Figure 1(c) and Figure 1(d). This reflects the great adjustability of the K-means algorithm, which can always partition reasonable demand areas on the basis of features such as the distance between customers, thereby facilitating retailers in making distribution decisions. On the other hand, the silhouette coefficients corresponding to the three distributions are shown in Figure 1(a)., showing that the optimal number of stores is 9 for <emph>Beta</emph>(<reflink idref="bib1" id="ref68">1</reflink>, 1), 8 for <emph>Beta</emph>(<reflink idref="bib1" id="ref69">1</reflink>, 0.5), and 12 for <emph>Beta</emph>(0.5, 1). The clustering of customers clearly has a certain effect on the selection of the final number of stores. Our proposed silhouette coefficient method can effectively address this impact and make the optimal division of the number of regions, thereby reducing the cost impact of serving customers over a wide range.</p> <p>Graph: Figure 1.Consumer segmentation and optimal store quantity determination.</p> <hd id="AN0187567149-21">Performance Evaluation</hd> <p>We evaluate the performance of RTA by comparing the solution quality with that of DET and EVPI, focusing on two aspects: (i) different sales horizons, and (ii) large-scale instances.</p> <hd id="AN0187567149-22">RTA Performance Across Different Planning Horizons</hd> <p>Then, we chose the omnichannel network with <emph>I</emph> = 10, <emph>J</emph> = 5, <emph>T</emph> = 3, 5, 7, 9 under different demand distributions: <emph>Beta</emph>(<reflink idref="bib1" id="ref70">1</reflink>, 1), <emph>Beta</emph>(<reflink idref="bib1" id="ref71">1</reflink>, 0.5), and <emph>Beta</emph>(0.5, 1). We solve these problems via the RTA, EVPI, and DET approaches and compare their solution quality. The results presented in Table 2 demonstrate the strong robustness of the RTA. As the sales horizon increases, the RTA can absorb more historical data, resulting in improved outcomes. For example, except for a marginal increase of only 0.6% in the cases of <emph>T</emph> = 7 and <emph>T</emph> = 9 under <emph>Beta</emph>(<reflink idref="bib1" id="ref72">1</reflink>, 0.5), the effective gap decreased from 5.8% to 3.45% and from 7.58% to 4.3% for the other two distributions. Furthermore, the maximum effective gap is less than 8%. In contrast, the DET method showed insufficient robustness, with a maximum effective gap of 9.37% and a minimum of 6.25%. Additionally, the computational time required by the RTA is well within a reasonable range, especially for an o-tailer who needs to operate a weekly sales plan.</p> <p>Table 2. Performance of Each Policy Across Different Planning Horizons.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2">Policy</th><th align="center" colspan="4"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">B</mi><mi xmlns="">e</mi><mi xmlns="">t</mi><mi xmlns="">a</mi><mfenced close=")" open="(" xmlns=""><mrow><mn>1,1</mn></mrow></mfenced></math></p></th><th align="center" colspan="4"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">B</mi><mi xmlns="">e</mi><mi xmlns="">t</mi><mi xmlns="">a</mi><mfenced close=")" open="(" xmlns=""><mrow><mn>1,0.5</mn></mrow></mfenced></math></p></th><th align="center" colspan="4"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">B</mi><mi xmlns="">e</mi><mi xmlns="">t</mi><mi xmlns="">a</mi><mfenced close=")" open="(" xmlns=""><mrow><mn>0.5</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math></p></th></tr><tr><th align="left"><italic>T</italic> = 3</th><th align="center">5</th><th align="center">7</th><th align="center">9</th><th align="center">3</th><th align="center">5</th><th align="center">7</th><th align="center">9</th><th align="center">3</th><th align="center">5</th><th align="center">7</th><th align="center">9</th></tr></thead><tbody valign="top"><tr><td align="left">EVPI</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left"> Rev(×104)</td><td align="char" char=".">63.05</td><td align="char" char=".">102.29</td><td align="char" char=".">143.92</td><td align="char" char=".">184.84</td><td align="char" char=".">66.1</td><td align="char" char=".">112.09</td><td align="char" char=".">152.89</td><td align="char" char=".">190.2</td><td align="char" char=".">55.56</td><td align="char" char=".">94.64</td><td align="char" char=".">135.13</td><td align="char" char=".">172.51</td></tr><tr><td align="left">DET</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left"> Rev(×104)</td><td align="char" char=".">58.49</td><td align="char" char=".">95.41</td><td align="char" char=".">134.14</td><td align="char" char=".">173.29</td><td align="char" char=".">59.9</td><td align="char" char=".">102.28</td><td align="char" char=".">139.54</td><td align="char" char=".">173.45</td><td align="char" char=".">50.49</td><td align="char" char=".">86.04</td><td align="char" char=".">122.81</td><td align="char" char=".">155.96</td></tr><tr><td align="left"> Gap(%)</td><td align="char" char=".">7.22</td><td align="char" char=".">6.74</td><td align="char" char=".">6.8</td><td align="char" char=".">6.25</td><td align="char" char=".">9.37</td><td align="char" char=".">8.75</td><td align="char" char=".">8.73</td><td align="char" char=".">8.81</td><td align="char" char=".">9.12</td><td align="char" char=".">9.09</td><td align="char" char=".">9.11</td><td align="char" char=".">9.59</td></tr><tr><td align="left">RTA</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left"> Rev(×104)</td><td align="char" char=".">59.36</td><td align="char" char=".">97.31</td><td align="char" char=".">138.04</td><td align="char" char=".">178.46</td><td align="char" char=".">63.24</td><td align="char" char=".">108.17</td><td align="char" char=".">148.6</td><td align="char" char=".">183.73</td><td align="char" char=".">51.34</td><td align="char" char=".">89.39</td><td align="char" char=".">129.21</td><td align="char" char=".">165.08</td></tr><tr><td align="left"> Gap(%)</td><td align="char" char=".">5.8</td><td align="char" char=".">4.87</td><td align="char" char=".">4.08</td><td align="char" char=".">3.45</td><td align="char" char=".">4.32</td><td align="char" char=".">3.5</td><td align="char" char=".">2.8</td><td align="char" char=".">3.4</td><td align="char" char=".">7.58</td><td align="char" char=".">5.55</td><td align="char" char=".">4.38</td><td align="char" char=".">4.3</td></tr><tr><td align="left"> Time(s)</td><td align="char" char=".">0.541</td><td align="char" char=".">1.652</td><td align="char" char=".">4.382</td><td align="char" char=".">7.643</td><td align="char" char=".">0.558</td><td align="char" char=".">1.632</td><td align="char" char=".">4.07</td><td align="char" char=".">9.055</td><td align="char" char=".">0.494</td><td align="char" char=".">1.651</td><td align="char" char=".">4.617</td><td align="char" char=".">10.625</td></tr></tbody></table> </ephtml> </p> <p>To further explore the performance of the RTA, we conducted tests using an omnichannel network with a scale of <emph>I</emph> = 10, <emph>K</emph> = 2000, <emph>T</emph> = 9 under <emph>Beta</emph>(<reflink idref="bib1" id="ref73">1</reflink>, 1) and summarized the respective cumulative revenue and the effective gap between the RTA and DET, as shown in Figure 2. As expected, with an extended sales period and observation of more historical data, the RTA gradually approaches the EVPI solution, resulting in a decrease in the effective gap from 12.51% at <emph>T</emph> = 1–3.51% at <emph>T</emph> = 9. In contrast, DET decreases from 8.6% at <emph>T</emph> = 1–6.86% at <emph>T</emph> = 9, reflecting the superior performance of the RTA as the sales period increases. Therefore, if an o-tailer can provide more historical sales data, the RTA can better assist them in achieving a profit, whereas DET cannot.</p> <p>Graph: Figure 2.Cumulative revenue and efficiency gaps across different planning horizons. (a) Cumulative revenue and (b) Efficiency gaps.</p> <hd id="AN0187567149-23">RTA's Performance in Larger Instances</hd> <p>We further tested the performance of the RTA under different demand distributions as the size of the distribution network increases. As shown in Table 3, the second column represents the instance size, which increases the network's scale by varying the number of products and customers to increase computational complexity. The RTA consistently performs well in the large-scale instances under the three demand distributions, with efficiency gaps all below 6%. Notably, for the <emph>Beta</emph>(<reflink idref="bib1" id="ref74">1</reflink>, 1) distribution with <emph>I</emph> = 100 and <emph>K</emph> = 2000, the efficiency gap is as small as 3.42%. This excellent performance of the RTA in large-scale instances is highly correlated with its ability to make adaptive decisions on the basis of historical observations. In other words, as the scale increases and more historical data are observed, the stability of the results improves. On the other hand, DET's performance stability is far inferior to that of the RTA. For instance, in the case of <emph>Beta</emph>(0.5, 1) with <emph>I</emph> = 60 and <emph>K</emph> = 6000, its efficiency gap reaches as high as 9.61%. Clearly, such a considerable gap is unacceptable for the o-tailer. In terms of computational time, an RTA requires a maximum of 10,000 seconds, which is acceptable for the o-tailer in formulating their weekly sales plans.</p> <p>Table 3. Performance of Each Policy for Larger-Size Instances.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2"><italic>Beta</italic>(<italic>κ</italic>, <italic>κ</italic>′)</th><th align="left" rowspan="2">(<italic>I</italic>, <italic>K</italic>)</th><th align="left" rowspan="2"><italic>J</italic></th><th align="center"><italic>EVPI</italic></th><th align="center" colspan="2"><italic>DET</italic></th><th align="center" colspan="3"><italic>LDR</italic></th></tr><tr><th align="center">Rev(×10<sup>5</sup>)</th><th align="center">Rev(×10<sup>5</sup>)</th><th align="center">Gap(%)</th><th align="center">Rev(×10<sup>5</sup>)</th><th align="center">Gap(%)</th><th align="center">Time(s)</th></tr></thead><tbody valign="top"><tr><td align="left"><italic>κ</italic> = 1</td><td align="char" char="(">(100, 2000)</td><td align="char" char=".">4</td><td align="char" char=".">141.27</td><td align="char" char=".">131.18</td><td align="char" char=".">7.14</td><td align="char" char=".">136.44</td><td align="char" char=".">3.42</td><td align="char" char=".">68.39</td></tr><tr><td align="left"><italic>κ</italic>′ = 1</td><td align="char" char="(">(80, 4000)</td><td align="char" char=".">6</td><td align="char" char=".">239.32</td><td align="char" char=".">226.21</td><td align="char" char=".">5.48</td><td align="char" char=".">226.26</td><td align="char" char=".">5.46</td><td align="char" char=".">256.27</td></tr><tr><td align="left" /><td align="char" char="(">(60, 6000)</td><td align="char" char=".">10</td><td align="char" char=".">263.47</td><td align="char" char=".">248.55</td><td align="char" char=".">5.66</td><td align="char" char=".">251.47</td><td align="char" char=".">4.55</td><td align="char" char=".">1365.75</td></tr><tr><td align="left" /><td align="char" char="(">(40, 8000)</td><td align="char" char=".">15</td><td align="char" char=".">228.6</td><td align="char" char=".">213.44</td><td align="char" char=".">6.63</td><td align="char" char=".">216.55</td><td align="char" char=".">5.27</td><td align="char" char=".">6912.91</td></tr><tr><td align="left" /><td align="char" char="(">(20, 10000)</td><td align="char" char=".">17</td><td align="char" char=".">145.48</td><td align="char" char=".">137.04</td><td align="char" char=".">5.8</td><td align="char" char=".">138.91</td><td align="char" char=".">4.52</td><td align="char" char=".">7467.23</td></tr><tr><td align="left"><italic>κ</italic> = 1</td><td align="char" char="(">(100, 2000)</td><td align="char" char=".">4</td><td align="char" char=".">150.17</td><td align="char" char=".">137.27</td><td align="char" char=".">8.59</td><td align="char" char=".">144.77</td><td align="char" char=".">3.6</td><td align="char" char=".">82.36</td></tr><tr><td align="left"><italic>κ</italic>′ = 0.5</td><td align="char" char="(">(80, 4000)</td><td align="char" char=".">6</td><td align="char" char=".">243.77</td><td align="char" char=".">226.19</td><td align="char" char=".">7.21</td><td align="char" char=".">231.38</td><td align="char" char=".">5.08</td><td align="char" char=".">256.27</td></tr><tr><td align="left" /><td align="char" char="(">(60, 6000)</td><td align="char" char=".">12</td><td align="char" char=".">285.34</td><td align="char" char=".">257.92</td><td align="char" char=".">9.61</td><td align="char" char=".">271.65</td><td align="char" char=".">4.79</td><td align="char" char=".">3460.23</td></tr><tr><td align="left" /><td align="char" char="(">(40, 8000)</td><td align="char" char=".">14</td><td align="char" char=".">244.79</td><td align="char" char=".">224.61</td><td align="char" char=".">8.24</td><td align="char" char=".">234.45</td><td align="char" char=".">4.22</td><td align="char" char=".">6791.06</td></tr><tr><td align="left" /><td align="char" char="(">(20, 10000)</td><td align="char" char=".">18</td><td align="char" char=".">158.79</td><td align="char" char=".">144.9</td><td align="char" char=".">8.75</td><td align="char" char=".">151.61</td><td align="char" char=".">4.52</td><td align="char" char=".">9978.3</td></tr><tr><td align="left"><italic>κ</italic> = 0.5</td><td align="char" char="(">(100, 2000)</td><td align="char" char=".">4</td><td align="char" char=".">136.76</td><td align="char" char=".">124.39</td><td align="char" char=".">9.04</td><td align="char" char=".">130.82</td><td align="char" char=".">4.34</td><td align="char" char=".">83.13</td></tr><tr><td align="left"><italic>κ</italic>′ = 1</td><td align="char" char="(">(80, 4000)</td><td align="char" char=".">6</td><td align="char" char=".">214.47</td><td align="char" char=".">198.42</td><td align="char" char=".">7.48</td><td align="char" char=".">204.39</td><td align="char" char=".">4.7</td><td align="char" char=".">258.44</td></tr><tr><td align="left" /><td align="char" char="(">(60, 6000)</td><td align="char" char=".">10</td><td align="char" char=".">243.16</td><td align="char" char=".">223.18</td><td align="char" char=".">8.22</td><td align="char" char=".">232.47</td><td align="char" char=".">4.4</td><td align="char" char=".">1338.96</td></tr><tr><td align="left" /><td align="char" char="(">(40, 8000)</td><td align="char" char=".">13</td><td align="char" char=".">219.84</td><td align="char" char=".">203.73</td><td align="char" char=".">7.33</td><td align="char" char=".">207.09</td><td align="char" char=".">5.8</td><td align="char" char=".">3253.29</td></tr><tr><td align="left" /><td align="char" char="(">(20, 10000)</td><td align="char" char=".">16</td><td align="char" char=".">135.3</td><td align="char" char=".">125.48</td><td align="char" char=".">7.26</td><td align="char" char=".">128.96</td><td align="char" char=".">4.69</td><td align="char" char=".">7046.22</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-24">Robustness Checking</hd> <p>We also assessed the robustness of the RTA by introducing parameter variations that deviate from the model assumptions, categorized into two groups: (i) modifying demand distributions, encompassing both symmetric and asymmetric distributions and (ii) broadening the spectrum of random demand distributions.</p> <hd id="AN0187567149-25">RTA Robustness across a Range of Demand Distributions</hd> <p>In this section, we discuss the analysis of the performance of the RTA across five different scales of omnichannel networks (refer to Column 2 of Table 4) under symmetric and asymmetric demand distributions (refer to the upper and lower parts of Table 4, respectively). Our findings reveal that the RTA exhibits remarkable resilience in both scenarios. Notably, the largest effective gap of 5.94% occurs only under the <emph>Beta</emph>(<reflink idref="bib2" id="ref75">2</reflink>, 5) distribution with <emph>I</emph> = 80, <emph>K</emph> = 4000, whereas in other instances, the gap remains below 5.2%, typically at approximately 3%. The lowest gap of 2.96% is observed under the <emph>Beta</emph>(<reflink idref="bib6" id="ref76">6</reflink>, 1) distribution with <emph>I</emph> = 100, <emph>K</emph> = 2000. Furthermore, we computed the coefficient of variation (CV) for all the omnichannel network scales (based on 500 samples), revealing robust solutions provided by the RTA, with all the CV values below 0.02%. These findings underscore the RTA's efficacy in driving profitability for o-tailer across diverse demand landscapes.</p> <p>Table 4. The RTA's Robustness Under a Range of Demand Distributions.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2" /><th align="center" rowspan="2">(<italic>I</italic>, <italic>K</italic>)</th><th align="center" colspan="6">Efficiency Gap(%)/CV(×10<sup>−3</sup>)</th></tr><tr><td align="left"><italic>Beta</italic>(1, 1)</td><td align="left"><italic>Beta</italic>(2, 2)</td><td align="left"><italic>Beta</italic>(3, 3)</td><td align="left"><italic>Beta</italic>(4, 4)</td><td align="left"><italic>Beta</italic>(5, 5)</td><td align="left"><italic>Beta</italic>(6, 6)</td></tr></thead><tbody valign="top"><tr><td align="left"> Symmetric</td><td align="left">(100, 2000)</td><td align="left">3.51/0.17</td><td align="left">3.71/0.14</td><td align="left">2.56/0.11</td><td align="left">4.52/0.1</td><td align="left">4.44/0.09</td><td align="left">3.17/0.1</td></tr><tr><td align="left" /><td align="left">(80, 4000)</td><td align="left">4.42/0.13</td><td align="left">4.55/0.1</td><td align="left">3.9/0.1</td><td align="left">4.22/0.12</td><td align="left">4.33/0.08</td><td align="left">4.54/0.06</td></tr><tr><td align="left" /><td align="left">(60, 6000)</td><td align="left">4.71/0.12</td><td align="left">4.42/0.1</td><td align="left">4.93/0.07</td><td align="left">5.45/0.09</td><td align="left">4.43/0.06</td><td align="left">4.51/0.06</td></tr><tr><td align="left" /><td align="left">(40, 8000)</td><td align="left">4.3/0.14</td><td align="left">4.57/0.1</td><td align="left">4.22/0.08</td><td align="left">5.44/0.07</td><td align="left">4.61/0.08</td><td align="left">4.57/0.07</td></tr><tr><td align="left" /><td align="left">(20, 10000)</td><td align="left">5.1/0.16</td><td align="left">4.84/0.12</td><td align="left">4.72/0.12</td><td align="left">4.25/0.09</td><td align="left">4.82/0.09</td><td align="left">4.95/0.15</td></tr><tr><td align="left" /><td align="center">(<italic>I</italic>, <italic>J</italic>)</td><td align="left"><italic>Beta</italic>(1, 6)</td><td align="left"><italic>Beta</italic>(2, 5)</td><td align="left"><italic>Beta</italic>(3, 4)</td><td align="left"><italic>Beta</italic>(4, 3)</td><td align="left"><italic>Beta</italic>(5, 2)</td><td align="left"><italic>Beta</italic>(6, 1)</td></tr><tr><td align="left"> Asymmetric</td><td align="left">(100, 2000)</td><td align="left">5.19/0.1</td><td align="left">4.56/0.12</td><td align="left">3.98/0.13</td><td align="left">4.55/0.08</td><td align="left">3.18/0.09</td><td align="left">2.96/0.07</td></tr><tr><td align="left" /><td align="left">(80, 4000)</td><td align="left">4.49/0.05</td><td align="left">5.94/0.14</td><td align="left">4.53/0.09</td><td align="left">4.39/0.07</td><td align="left">3.65/0.08</td><td align="left">4.81/0.05</td></tr><tr><td align="left" /><td align="left">(60, 6000)</td><td align="left">5.83/0.06</td><td align="left">5.54/0.08</td><td align="left">4.71/0.08</td><td align="left">4.28/0.08</td><td align="left">4.06/0.07</td><td align="left">4.11/0.05</td></tr><tr><td align="left" /><td align="left">(80, 8000)</td><td align="left">5.26/0.06</td><td align="left">4.95/0.07</td><td align="left">5.03/0.08</td><td align="left">4.86/0.08</td><td align="left">5.14/0.06</td><td align="left">4.89/0.05</td></tr><tr><td align="left" /><td align="left">(20, 10000)</td><td align="left">5.1/0.07</td><td align="left">4.89/0.09</td><td align="left">5.6/0.09</td><td align="left">4.63/0.09</td><td align="left">4.92/0.08</td><td align="left">5.15/0.08</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-26">RTA Robustness Under Demand Deviation</hd> <p>During promotional periods for o-tailer, customer demand tends to become more unpredictable. To address this, we explored adjusting the range of random demand intervals (refer to Row 2 of Table 5) to evaluate the RTA's resilience in such scenarios. The results in Table 5 indicate that despite a slight decrease in solution effectiveness with increasing demand range uncertainty, the RTA maintains overall stability, with a minimum effective gap of 3.75% and a maximum not exceeding 9.1%. This resilience is attributed to the RTA's adaptive adjustment of decision variables on the basis of historical data, ensuring o-tailer profitability. Moreover, the coefficient of variation (CV) index increases with network size expansion and widening random demand ranges. For example, when <emph>I</emph> = 20, <emph>K</emph> = 10,000, CV = 0.057%. This indicates that as the RTA addresses a broader range of random variables, it impacts the parameter calculations in LDR to some extent. Consequently, the coefficient of variation (CV) index increases with the expansion of the network scale and the widening of random demand ranges.</p> <p>Table 5. The LDR's Robustness Under Online Demand Deviation.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="4">(<italic>I</italic>, <italic>K</italic>)</th><th align="center" colspan="10">Online demand uncertainty set in the simulation</th></tr><tr><th align="left" colspan="2"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">I</mi><mo xmlns="">×</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>40,60</mn></mrow></mfenced></math></p></th><th align="center" colspan="2"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">I</mi><mo xmlns="">×</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>35,65</mn></mrow></mfenced></math></p></th><th align="center" colspan="2"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">I</mi><mo xmlns="">×</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>30,70</mn></mrow></mfenced></math></p></th><th align="center" colspan="2"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">I</mi><mo xmlns="">×</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>25,75</mn></mrow></mfenced></math></p></th><th align="center" colspan="2"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">I</mi><mo xmlns="">×</mo><mfenced close="]" open="[" xmlns=""><mrow><mn>20,80</mn></mrow></mfenced></math></p></th></tr><tr><th align="left">Gap</th><th align="center">CV</th><th align="center">Gap</th><th align="center">CV</th><th align="center">Gap</th><th align="center">CV</th><th align="center">Gap</th><th align="center">CV</th><th align="center">Gap</th><th align="center">CV</th></tr><tr><th align="left">(%)</th><th align="center">(×10<sup>−3</sup>)</th><th align="center">(%)</th><th align="center">(×10<sup>−3</sup>)</th><th align="center">(%)</th><th align="center">(×10<sup>−3</sup>)</th><th align="center">(%)</th><th align="center">(×10<sup>−3</sup>)</th><th align="center">(%)</th><th align="center">(×10<sup>−3</sup>)</th></tr></thead><tbody valign="top"><tr><td align="left">(100, 2000)</td><td align="char" char=".">3.65</td><td align="char" char=".">0.19</td><td align="char" char=".">5.19</td><td align="char" char=".">0.26</td><td align="char" char=".">4.4</td><td align="char" char=".">0.28</td><td align="char" char=".">7.75</td><td align="char" char=".">0.49</td><td align="char" char=".">9.01</td><td align="char" char=".">0.54</td></tr><tr><td align="left">(80, 4000)</td><td align="char" char=".">4.68</td><td align="char" char=".">0.13</td><td align="char" char=".">4.37</td><td align="char" char=".">0.18</td><td align="char" char=".">4.97</td><td align="char" char=".">0.25</td><td align="char" char=".">6.36</td><td align="char" char=".">0.29</td><td align="char" char=".">9.03</td><td align="char" char=".">0.48</td></tr><tr><td align="left">(60, 6000)</td><td align="char" char=".">4.46</td><td align="char" char=".">0.15</td><td align="char" char=".">5.37</td><td align="char" char=".">0.19</td><td align="char" char=".">7.38</td><td align="char" char=".">0.31</td><td align="char" char=".">6.65</td><td align="char" char=".">0.28</td><td align="char" char=".">7.90</td><td align="char" char=".">0.48</td></tr><tr><td align="left">(40, 8000)</td><td align="char" char=".">4.57</td><td align="char" char=".">0.15</td><td align="char" char=".">4.57</td><td align="char" char=".">0.13</td><td align="char" char=".">6.16</td><td align="char" char=".">0.29</td><td align="char" char=".">6.64</td><td align="char" char=".">0.36</td><td align="char" char=".">8.65</td><td align="char" char=".">0.43</td></tr><tr><td align="left">(20, 10000)</td><td align="char" char=".">5.08</td><td align="char" char=".">0.16</td><td align="char" char=".">5.01</td><td align="char" char=".">0.27</td><td align="char" char=".">5.81</td><td align="char" char=".">0.35</td><td align="char" char=".">6.64</td><td align="char" char=".">0.42</td><td align="char" char=".">8.5</td><td align="char" char=".">0.57</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-27">Sensitivity Analysis</hd> <p>In this section, we explore how the RTA enables o-tailer to consistently maximize profits by considering four key factors: unit product pricing, order fulfillment costs, online and offline market shares, and customer distribution patterns. By adjusting these parameters, tailored strategies can be developed to increase the adaptability of o-tailer in dynamic retail environments. Throughout our sensitivity analyses, we maintain a scale of <emph>I</emph> = 20, <emph>K</emph> = 2000, <emph>T</emph> = 7 for the omnichannel network, with a random distribution following <emph>Beta</emph>(<reflink idref="bib1" id="ref77">1</reflink>, 1).</p> <hd id="AN0187567149-28">Impact of Unit Product Pricing</hd> <p>We apply a discount factor <emph>ϑ</emph><emph>p</emph> (see the first column of Table 6) to the original prices as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi>ϑ</mi></mrow><mrow><mi>p</mi></mrow></msub></math> </ephtml> and then utilize the RTA to solve our problem. This allows us to examine the impact of price changes on the product distribution process. The first row of Table 6 represents the various discounts at different stages. The results indicate that as <emph>ϑ</emph><emph>p</emph> increases, except for the cases where <emph>ϑ</emph><emph>p</emph> =.6,.7, and 0.8, where the efficiency gap exceeds 5%, the values for the remaining scenarios are all less than 5%, demonstrating the operational viability of the RTA in ensuring retailer profits. The replenishment cost also increases (as we assume it to be a certain proportion of the price). The allocation and fulfillment costs fluctuate within a narrow range, confirming the stability of the RTA. Furthermore, the efficiency gap and inventory holding costs are typically proportional to the order loss rate, which is consistent with real-world observations. The computation time is also consistent within 30 seconds, which is acceptable for o-tailer.</p> <p>Table 6. Effect of Unit Product Pricing.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2"><italic>ϑ</italic><italic>p</italic></th><th align="center">Rev</th><th align="center">Replenishment</th><th align="center">Allocation</th><th align="center">Fulfillment</th><th align="center">Holding</th><th align="center">Gap</th><th align="center">Lost</th><th align="center">Time</th></tr><tr><th align="left">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(%)</th><th align="center">(%)</th><th align="center">(<italic>s</italic>)</th></tr></thead><tbody valign="top"><tr><td align="left">0.6</td><td align="char" char=".">148.24</td><td align="char" char=".">87.76</td><td align="char" char=".">27.85</td><td align="char" char=".">24.24</td><td align="char" char=".">3.28</td><td align="char" char=".">7.65</td><td align="char" char=".">4.27</td><td align="char" char=".">24.14</td></tr><tr><td align="left">0.7</td><td align="char" char=".">177.47</td><td align="char" char=".">105.4</td><td align="char" char=".">33.02</td><td align="char" char=".">28.58</td><td align="char" char=".">3.21</td><td align="char" char=".">6.82</td><td align="char" char=".">3.16</td><td align="char" char=".">26.07</td></tr><tr><td align="left">0.8</td><td align="char" char=".">215.39</td><td align="char" char=".">121.25</td><td align="char" char=".">31.34</td><td align="char" char=".">25.86</td><td align="char" char=".">3.35</td><td align="char" char=".">5.03</td><td align="char" char=".">2.05</td><td align="char" char=".">23.7</td></tr><tr><td align="left">0.9</td><td align="char" char=".">238.46</td><td align="char" char=".">136.44</td><td align="char" char=".">36.03</td><td align="char" char=".">31.74</td><td align="char" char=".">3.33</td><td align="char" char=".">4.94</td><td align="char" char=".">2.03</td><td align="char" char=".">24.05</td></tr><tr><td align="left">1</td><td align="char" char=".">272.24</td><td align="char" char=".">151.15</td><td align="char" char=".">31.83</td><td align="char" char=".">30.51</td><td align="char" char=".">3.53</td><td align="char" char=".">4.8</td><td align="char" char=".">1.13</td><td align="char" char=".">27.22</td></tr><tr><td align="left">1.1</td><td align="char" char=".">312.93</td><td align="char" char=".">169.78</td><td align="char" char=".">29.98</td><td align="char" char=".">33.27</td><td align="char" char=".">3.53</td><td align="char" char=".">4.21</td><td align="char" char=".">0.77</td><td align="char" char=".">25.65</td></tr><tr><td align="left">1.2</td><td align="char" char=".">363.44</td><td align="char" char=".">190.16</td><td align="char" char=".">29.05</td><td align="char" char=".">26.22</td><td align="char" char=".">3.55</td><td align="char" char=".">4.1</td><td align="char" char=".">0.21</td><td align="char" char=".">23.61</td></tr><tr><td align="left">1.3</td><td align="char" char=".">399.23</td><td align="char" char=".">208.77</td><td align="char" char=".">30.26</td><td align="char" char=".">28.81</td><td align="char" char=".">3.7</td><td align="char" char=".">3.48</td><td align="char" char=".">0.25</td><td align="char" char=".">25.66</td></tr><tr><td align="left">1.4</td><td align="char" char=".">403.12</td><td align="char" char=".">216.41</td><td align="char" char=".">38.29</td><td align="char" char=".">33.97</td><td align="char" char=".">3.65</td><td align="char" char=".">4.03</td><td align="char" char=".">0.42</td><td align="char" char=".">24.49</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-29">Impact of Order Fulfillment Costs</hd> <p>In this section, we explore the sensitivity of o-tailer profits to variations in the cost assumptions associated with order fulfillment from both the DC and stores. We achieve this by adjusting the costs via coefficients <emph>ϑ</emph><emph>DC</emph> and <emph>ϑ</emph><emph>store</emph>, as detailed in Columns 1 and 7 of Table 7, respectively. The findings presented in Table 7 underscore the efficacy of the RTA in enhancing o-tailer profits, irrespective of fluctuations in fulfillment costs. Notably, all the case studies maintain a negligible effective gap of less than 5.1%. Specifically, as both <emph>ϑ</emph><emph>DC</emph> and <emph>ϑ</emph><emph>store</emph> increase within the range of 0.7–1.3, o-tailer profits experience a gradual decline, aligning with expectations that higher fulfillment costs contribute to elevated operational expenses. Moreover, the fluctuations observed in order fulfillment costs as <emph>ϑ</emph><emph>DC</emph> and <emph>ϑ</emph><emph>store</emph> increase are attributed to the adaptive nature of the RTA, which optimizes the allocation of orders between the DC and stores to minimize costs. Furthermore, our analysis reveals that changes in the order loss rate primarily stem from variations in parameter <emph>ϑ</emph><emph>DC</emph>, with a relatively smaller impact observed for parameter <emph>ϑ</emph><emph>store</emph>. This trend can be attributed to our model's assumption that the number of stores exhibits superior adaptability to excess demand compared with the DC, thereby possessing enhanced regulatory capabilities. Consequently, the integration of the RTA with a <emph>ship-from-store</emph> strategy emerges as a viable approach for retailers to maximize profits by effectively managing the dynamic nature of order fulfillment costs.</p> <p>Table 7. Effects of Variations in Unit Order Fulfillment Costs.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" colspan="6">Effect of DC fulfillment cost</th><th align="center" colspan="6">Effect of store fulfillment cost</th></tr><tr><th align="left" rowspan="2">× <italic>ϑ</italic><italic>DC</italic></th><th align="center">Rev</th><th align="center">Fulfillment</th><th align="center">Gap</th><th align="center">Lost</th><th align="center">Time</th><th align="left" rowspan="2">× <italic>ϑ</italic><italic>store</italic></th><th align="center">Rev</th><th align="center">Fulfillment</th><th align="center">Gap</th><th align="center">Lost</th><th align="center">Time</th></tr><tr><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(%)</th><th align="center">(%)</th><th align="center">(<italic>s</italic>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(%)</th><th align="center">(%)</th><th align="center">(<italic>s</italic>)</th></tr></thead><tbody valign="top"><tr><td align="left">0.7</td><td align="char" char=".">296.5</td><td align="char" char=".">30.18</td><td align="char" char=".">4.00</td><td align="char" char=".">0.29</td><td align="char" char=".">25.516</td><td align="char" char=".">0.7</td><td align="char" char=".">296.61</td><td align="char" char=".">22.41</td><td align="char" char=".">4.74</td><td align="char" char=".">1.23</td><td align="char" char=".">26.012</td></tr><tr><td align="left">0.8</td><td align="char" char=".">293.98</td><td align="char" char=".">31.28</td><td align="char" char=".">4.45</td><td align="char" char=".">0.52</td><td align="char" char=".">23.79</td><td align="char" char=".">0.8</td><td align="char" char=".">294.35</td><td align="char" char=".">24.53</td><td align="char" char=".">4.85</td><td align="char" char=".">1.28</td><td align="char" char=".">26.213</td></tr><tr><td align="left">0.9</td><td align="char" char=".">292.34</td><td align="char" char=".">29.45</td><td align="char" char=".">4.75</td><td align="char" char=".">0.82</td><td align="char" char=".">27.223</td><td align="char" char=".">0.9</td><td align="char" char=".">292.56</td><td align="char" char=".">26.38</td><td align="char" char=".">4.93</td><td align="char" char=".">1.29</td><td align="char" char=".">27.186</td></tr><tr><td align="left">1.0</td><td align="char" char=".">290.9</td><td align="char" char=".">28.51</td><td align="char" char=".">4.99</td><td align="char" char=".">1.25</td><td align="char" char=".">25.548</td><td align="char" char=".">1</td><td align="char" char=".">290.9</td><td align="char" char=".">28.51</td><td align="char" char=".">4.99</td><td align="char" char=".">1.25</td><td align="char" char=".">25.444</td></tr><tr><td align="left">1.1</td><td align="char" char=".">290.17</td><td align="char" char=".">27.3</td><td align="char" char=".">5</td><td align="char" char=".">1.53</td><td align="char" char=".">25.324</td><td align="char" char=".">1.1</td><td align="char" char=".">289.48</td><td align="char" char=".">29.96</td><td align="char" char=".">5</td><td align="char" char=".">1.19</td><td align="char" char=".">23.567</td></tr><tr><td align="left">1.2</td><td align="char" char=".">290.16</td><td align="char" char=".">26.01</td><td align="char" char=".">4.83</td><td align="char" char=".">1.78</td><td align="char" char=".">26.528</td><td align="char" char=".">1.2</td><td align="char" char=".">287.91</td><td align="char" char=".">31.67</td><td align="char" char=".">5.06</td><td align="char" char=".">1.2</td><td align="char" char=".">24.16</td></tr><tr><td align="left">1.3</td><td align="char" char=".">289.9</td><td align="char" char=".">24.44</td><td align="char" char=".">4.92</td><td align="char" char=".">2.02</td><td align="char" char=".">25.312</td><td align="char" char=".">1.3</td><td align="char" char=".">286.55</td><td align="char" char=".">34.71</td><td align="char" char=".">5.05</td><td align="char" char=".">1.14</td><td align="char" char=".">23.421</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-30">Impact of Differences in Market Share</hd> <p>In this section, we introduce a market share parameter, <emph>ϑ</emph><emph>on</emph>, ranging from 10% to 90%, with a step size of 10% (see Column 1 of Table 8) to examine the impact of online market share on o-tailer operations. The results are presented in Table 8. As the online market scale increases, the o-tailer's profits decrease from 292.78 × 10<sups>3</sups> to 214.99 × 10<sups>3</sups>. This is because, in our problem assumption, in-store customers do not go through the fulfillment process, while an increase in online orders inevitably leads to higher fulfillment costs, as evident in Column 5 of Table 8, with an increase from 6.31 × 10<sups>3</sups> to 45.74 × 10<sups>3</sups>. The allocation cost gradually decreases as the number of online orders increases because, in our assumption, the total number of market orders is fixed. When the number of online orders increases, the number of in-store orders decreases, resulting in a decrease in the quantity of products allocated to the store. As a result, the allocation cost progressively decreases. On the other hand, online orders can be fulfilled through the DC, which does not require allocation to stores and therefore does not incur allocation costs. The replenishment costs remain stable, reflecting the constant total volume of orders. This also demonstrates the adaptability of the RTA, as it does not result in excessive or insufficient product orders when the scale of a single market increases. The holding cost of inventory is directly proportional to the order loss rate, aligning with reality, where a higher loss rate implies higher holding costs. The continuous increase in the latter is undoubtedly related to the fulfillment costs associated with online orders, where the RTA balances the gap between costs and benefits, thereby determining whether to fulfill orders.</p> <p>Table 8. Effect of Market Share.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2"><italic>ϑ</italic><italic>on</italic></th><th align="center">Rev</th><th align="center">Replenishment</th><th align="center">Allocation</th><th align="center">Fulfillment</th><th align="center">Holding</th><th align="center">Gap</th><th align="center">Lost</th><th align="center">Time</th></tr><tr><th align="left">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(%)</th><th align="center">(%)</th><th align="center">(<italic>s</italic>)</th></tr></thead><tbody valign="top"><tr><td align="left">0.1</td><td align="char" char=".">292.78</td><td align="char" char=".">151.36</td><td align="char" char=".">39</td><td align="char" char=".">6.31</td><td align="char" char=".">3.74</td><td align="char" char=".">4.94</td><td align="char" char=".">2.45</td><td align="char" char=".">24.481</td></tr><tr><td align="left">0.2</td><td align="char" char=".">289.61</td><td align="char" char=".">150.63</td><td align="char" char=".">36.91</td><td align="char" char=".">12.19</td><td align="char" char=".">3.2</td><td align="char" char=".">3.9</td><td align="char" char=".">1.72</td><td align="char" char=".">25.582</td></tr><tr><td align="left">0.3</td><td align="char" char=".">284.05</td><td align="char" char=".">150.05</td><td align="char" char=".">35.2</td><td align="char" char=".">17.30</td><td align="char" char=".">3.68</td><td align="char" char=".">5.08</td><td align="char" char=".">2.17</td><td align="char" char=".">23.671</td></tr><tr><td align="left">0.4</td><td align="char" char=".">278.02</td><td align="char" char=".">151.36</td><td align="char" char=".">33.59</td><td align="char" char=".">22.43</td><td align="char" char=".">4.43</td><td align="char" char=".">4.77</td><td align="char" char=".">3.46</td><td align="char" char=".">28.384</td></tr><tr><td align="left">0.5</td><td align="char" char=".">264.52</td><td align="char" char=".">150.73</td><td align="char" char=".">31.50</td><td align="char" char=".">26.89</td><td align="char" char=".">5.74</td><td align="char" char=".">5.51</td><td align="char" char=".">5.03</td><td align="char" char=".">25.291</td></tr><tr><td align="left">0.6</td><td align="char" char=".">253.2</td><td align="char" char=".">150.69</td><td align="char" char=".">29.98</td><td align="char" char=".">31.73</td><td align="char" char=".">6.8</td><td align="char" char=".">5.05</td><td align="char" char=".">6.38</td><td align="char" char=".">21.474</td></tr><tr><td align="left">0.7</td><td align="char" char=".">240.81</td><td align="char" char=".">150.58</td><td align="char" char=".">28.46</td><td align="char" char=".">36.47</td><td align="char" char=".">7.99</td><td align="char" char=".">4.38</td><td align="char" char=".">7.89</td><td align="char" char=".">24.46</td></tr><tr><td align="left">0.8</td><td align="char" char=".">229.99</td><td align="char" char=".">151.01</td><td align="char" char=".">26.39</td><td align="char" char=".">41.56</td><td align="char" char=".">9.33</td><td align="char" char=".">5.74</td><td align="char" char=".">9.4</td><td align="char" char=".">28.91</td></tr><tr><td align="left">0.9</td><td align="char" char=".">214.99</td><td align="char" char=".">150.62</td><td align="char" char=".">24.98</td><td align="char" char=".">45.74</td><td align="char" char=".">10.74</td><td align="char" char=".">5.16</td><td align="char" char=".">10.34</td><td align="char" char=".">26.383</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-31">Impact of Customer Distribution</hd> <p>Building on the earlier assumption, this section explores the impact of customer distribution on o-tailer operational profits and strategies by examining 12 different scenarios. The calculation results are shown in Table 9. There was minimal variation in replenishment costs because we assumed a constant total customer demand. Therefore, the quantity of products an o-tailer needs to order remains relatively constant, resulting in stable replenishment costs. Similar fluctuations were observed in allocation costs and fulfillment costs for similar reasons. However, inventory holding costs showed some variability. For example, inventory costs reached 6.66 × 10<sups>3</sups> and 6.61 × 10<sups>3</sups> in scenarios <emph>Beta</emph>(<reflink idref="bib1" id="ref78">1</reflink>, 6) and <emph>Beta</emph>(<reflink idref="bib2" id="ref79">2</reflink>, 5), respectively, whereas in scenario <emph>Beta</emph>(<reflink idref="bib1" id="ref80">1</reflink>, 1), they were 2.78 × 10<sups>3</sups>, almost double. The main reason for this may be the possibility of customer concentration under scenarios <emph>Beta</emph>(<reflink idref="bib1" id="ref81">1</reflink>, 6) and <emph>Beta</emph>(<reflink idref="bib2" id="ref82">2</reflink>, 5), as discussed in Section 5.1. In such scenarios, subsequent demand segmentation would show considerable disparities, with some demand areas containing far more customers than others do. Additionally, since our model assumes that demand uncertainty for each demand area is positively correlated with the number of customers it contains, more customers imply greater uncertainty for o-tailer. Consequently, the RTA may face more disruptions when inventory is allocated across the network, leading to excessive inventory allocation. However, despite the impact of excessive uncertainty, the RTA still maintains high-quality solutions with a maximum effective gap not exceeding 5.5%, demonstrating good robustness. There were also fluctuations in order loss rates, attributable to considerations similar to those of the previous three subsections, reflecting a trade-off between revenue and costs.</p> <p>Table 9. Effect of Customer Distribution.</p> <p>Graph</p> <p> <ephtml> <table><thead valign="top"><tr><th align="left" rowspan="2"><italic>Beta</italic>()</th><th align="center">Rev</th><th align="center">Replenishment</th><th align="center">Allocation</th><th align="center">Fulfillment</th><th align="center">Holding</th><th align="center">Gap</th><th align="center">Lost</th><th align="center">Time</th></tr><tr><th align="left">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(×10<sup>3</sup>)</th><th align="center">(%)</th><th align="center">(%)</th><th align="center">(<italic>s</italic>)</th></tr></thead><tbody valign="top"><tr><td align="left"><italic>Beta</italic>(1, 1)</td><td align="char" char=".">280.23</td><td align="char" char=".">150.28</td><td align="char" char=".">33.48</td><td align="char" char=".">31.45</td><td align="char" char=".">2.78</td><td align="char" char=".">3.89</td><td align="char" char=".">0.05</td><td align="char" char=".">29.107</td></tr><tr><td align="left"><italic>Beta</italic>(2, 2)</td><td align="char" char=".">277.43</td><td align="char" char=".">151.97</td><td align="char" char=".">33.51</td><td align="char" char=".">26.14</td><td align="char" char=".">2.85</td><td align="char" char=".">3.62</td><td align="char" char=".">0.35</td><td align="char" char=".">30.008</td></tr><tr><td align="left"><italic>Beta</italic>(3, 3)</td><td align="char" char=".">280.13</td><td align="char" char=".">148.72</td><td align="char" char=".">34.07</td><td align="char" char=".">29.97</td><td align="char" char=".">3.33</td><td align="char" char=".">3.3</td><td align="char" char=".">1.1</td><td align="char" char=".">24.469</td></tr><tr><td align="left"><italic>Beta</italic>(4, 4)</td><td align="char" char=".">269.95</td><td align="char" char=".">148.88</td><td align="char" char=".">42.74</td><td align="char" char=".">31.44</td><td align="char" char=".">2.94</td><td align="char" char=".">4.04</td><td align="char" char=".">2.21</td><td align="char" char=".">23.969</td></tr><tr><td align="left"><italic>Beta</italic>(5, 5)</td><td align="char" char=".">268.80</td><td align="char" char=".">146.61</td><td align="char" char=".">34.75</td><td align="char" char=".">31.3</td><td align="char" char=".">3.18</td><td align="char" char=".">3.58</td><td align="char" char=".">1.36</td><td align="char" char=".">24.58</td></tr><tr><td align="left"><italic>Beta</italic>(6, 6)</td><td align="char" char=".">280.28</td><td align="char" char=".">155.36</td><td align="char" char=".">35.09</td><td align="char" char=".">31.35</td><td align="char" char=".">3.7</td><td align="char" char=".">3.53</td><td align="char" char=".">0.4</td><td align="char" char=".">24.038</td></tr><tr><td align="left"><italic>Beta</italic>(1, 6)</td><td align="char" char=".">265.32</td><td align="char" char=".">147.19</td><td align="char" char=".">36.18</td><td align="char" char=".">34.62</td><td align="char" char=".">6.66</td><td align="char" char=".">4.26</td><td align="char" char=".">1.74</td><td align="char" char=".">24.587</td></tr><tr><td align="left"><italic>Beta</italic>(2, 5)</td><td align="char" char=".">265.05</td><td align="char" char=".">147.46</td><td align="char" char=".">36.82</td><td align="char" char=".">32.32</td><td align="char" char=".">6.61</td><td align="char" char=".">5.07</td><td align="char" char=".">3.51</td><td align="char" char=".">24.513</td></tr><tr><td align="left"><italic>Beta</italic>(3, 4)</td><td align="char" char=".">274.03</td><td align="char" char=".">147.11</td><td align="char" char=".">32.96</td><td align="char" char=".">32.87</td><td align="char" char=".">4.71</td><td align="char" char=".">3.61</td><td align="char" char=".">2.12</td><td align="char" char=".">24.12</td></tr><tr><td align="left"><italic>Beta</italic>(4, 3)</td><td align="char" char=".">270.5</td><td align="char" char=".">147.23</td><td align="char" char=".">30.82</td><td align="char" char=".">33.18</td><td align="char" char=".">5.37</td><td align="char" char=".">3.5</td><td align="char" char=".">1.09</td><td align="char" char=".">24.126</td></tr><tr><td align="left"><italic>Beta</italic>(5, 2)</td><td align="char" char=".">270.45</td><td align="char" char=".">145.77</td><td align="char" char=".">25.84</td><td align="char" char=".">33.71</td><td align="char" char=".">4.78</td><td align="char" char=".">3.5</td><td align="char" char=".">0.51</td><td align="char" char=".">24.284</td></tr><tr><td align="left"><italic>Beta</italic>(6, 1)</td><td align="char" char=".">267.59</td><td align="char" char=".">148.56</td><td align="char" char=".">32.24</td><td align="char" char=".">35.70</td><td align="char" char=".">3.52</td><td align="char" char=".">4.9</td><td align="char" char=".">1.44</td><td align="char" char=".">24.262</td></tr></tbody></table> </ephtml> </p> <hd id="AN0187567149-32">Advantage of Omnichannel Operations</hd> <p>This section describes the advantages of omnichannel operations under the <emph>ship-from-store</emph> strategy, considering various assumptions. Specifically, we compare the profits obtained from omnichannel fulfillment with those obtained solely from DC fulfillment or solely from store fulfillment. Note that these assumptions are mentioned in Sections 5.4.1 to 5.4.4. For this purpose, we introduce a formula (<reflink idref="bib14" id="ref83">14</reflink>) to compare the advantages of a single distribution center or store fulfillment versus omnichannel approaches. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Revenue improving</mtext><mo>=</mo><mfrac><mrow><mtext>Revenue omnichannel fulfillment</mtext><mo>−</mo><mtext>Revenue by DC/store fulfillment</mtext></mrow><mrow><mtext>Revenue omnichannel fulfillment</mtext></mrow></mfrac><mo>×</mo><mn>100</mn><mi>%</mi></math> </ephtml></p> <p>Graph</p> <p>The calculation results are shown in Figure 3. As expected, omnichannel retailers consistently lead in profit under all possible scenario assumptions. However, the results obtained solely from DC or store fulfillment vary. Specifically, the revenue improvement obtained solely from DC fulfillment gradually decreases with changes in price discounts and store fulfillment costs, indicating that profits gradually approach those of the omnichannel operations strategy. This rate increases with changes in DC fulfillment costs and market share, meaning that profits gradually diminish compared with those of the omnichannel operations strategy. In other scenarios, there is some fluctuation. For example, in Figure 3(e), when the customer distribution follows a <emph>Beta</emph>(2.5) distribution, the maximum profit increase rate approaches 7.5%, whereas when <emph>Beta</emph>(<reflink idref="bib6" id="ref84">6</reflink>, 1), it is close to 1%. Correspondingly, except for individual cases such as those shown in Figure 3(b), <emph>ϑ</emph><emph>DC</emph> ≤ 0.8, the revenue improvement corresponding to fulfillment solely from stores is inferior to that of DC fulfillment alone. Under other assumptions, however, fulfillment solely from stores is consistently superior to DC fulfillment alone. The reason, as analyzed earlier, lies in our assumption that the numerical advantage of stores aligns closely with the RTA to solving, enabling more realistic fulfillment department choices in retail and thus achieving profits closer to omnichannel operations. Overall, omnichannel operations under the <emph>ship-from-store</emph> strategy can maximize the advantages of flexible fulfillment, thereby helping retailers save costs and increase profits.</p> <p>Graph: Figure 3.Revenue improving under different condition assumptions. (a) Varied price discounts, (b) Varied DC fulfillment costs, (c) Varied store fulfillment costs, (d) Varied market share, and (e) Varied customer distributions.</p> <hd id="AN0187567149-33">Managerial Insights</hd> <p>In this section, we summarize some management insights derived from the preceding data experiments for optimizing omnichannel retail strategies.</p> <p>(i) When product prices are lower, adopting an omnichannel fulfillment strategy can help retailers achieve higher profits than can a single-channel fulfillment strategy. Additionally, in the fulfillment of online orders, employing the <emph>ship-from-store</emph> strategy can significantly increase profits compared with fulfilling online orders solely through distribution centers.</p> <p>(i) Faced with different unit fulfillment costs, omnichannel retailers can utilize the RTA to select departments with lower costs to fulfill online orders intelligently, ensuring profit maximization. Therefore, the <emph>ship-from-store</emph> strategy still has advantages over DC fulfillment alone.</p> <p>(i) Different channel proportions have a considerable effect on retailer profits, with larger online market shares typically leading to greater revenue growth through omnichannel fulfillment. Especially when there is high demand for online orders, implementing the <emph>ship-from-store</emph> strategy is more advantageous for omnichannel retailers.</p> <p>(i) When customer location distributions exhibit characteristics of concentration and dispersion, the <emph>ship-from-store</emph> strategy can better utilize inventory and cost advantages between stores for order fulfillment, achieving complementary advantages and striving for greater profits for retailers.</p> <p>These managerial insights suggest that for retailers facing complex and ever-changing consumer markets, choosing omnichannel operations under the <emph>ship-from-store</emph> strategy is a practical and viable way to realize potential profitability.</p> <hd id="AN0187567149-34">Conclusion</hd> <p>This article discusses the comprehensive issue of dividing customer demand areas on the basis of their distribution and setting up stores for customer order fulfillment. Similar to [<reflink idref="bib18" id="ref85">18</reflink>], this involves inventory joint replenishment, allocation, and fulfillment decisions within the operational network. Omnichannel retailers need to operate a limited retail network and sell multiple products across multiple planning periods. The decision-making process of omnichannel retailers follows a certain sequence, making four types of decisions in each period. (i) At the beginning of each period, omnichannel retailers determine the replenishment quantity of each product at the DC, incurring replenishment costs. (ii) After receiving these products, the DC determines how to allocate each product to various stores on the basis of inventory constraints and allocation costs. (iii) Within each period, they fulfill store demands, but if a store is out of stock, demand is lost. (iv) At the end of each period, omnichannel retailers choose between the DC or stores to fulfill online orders on the basis of inventory levels, considering fulfillment costs and pricing interactions. If both the DC and all the stores are out of stock, the orders are lost. The goal of omnichannel retailers is to maximize the expected total operational profit over the planning period.</p> <p>This problem involves several challenges: (i) the issue of regional demand division due to diverse customer location distributions, (ii) proactive and reactive decision-making, (iii) limited facility capacity, (iv) achieving flexibility, and (v) diversity in prices and costs. To address these challenges, we developed a multiperiod stochastic optimization model and used a combination of the K-means algorithm and LDR method for modeling and calculations. Similarly, this adaptive approach allows us to determine the variables involved in the model on the basis of evolving demands, making it well equipped to handle the uncertainty of the demand distribution.</p> <p>Empirical data show that even under complex conditions contradicting model assumptions, LDR consistently generates high-quality and stable solutions relying only on the demand distribution mean and bounds, with an efficiency gap of less than 10% compared with the expected value of perfect information (EVPI) method. Robustness analysis further confirms the consistent and reliable performance of the LDR method across different scenarios deviating from model assumptions. Sensitivity analysis explores the impacts of product prices, distribution center/store fulfillment costs, market share, and customer location distribution on omnichannel retail costs and revenue. The results demonstrate that the LDR method remains applicable and performs well. Furthermore, the fulfillment strategy analysis further confirms that adopting an omnichannel strategy with "ship-from-store" is beneficial for retailers to expand retail coverage and increase revenue.</p> <hd id="AN0187567149-35">ORCID iD</hd> <p>Li Zhang https://orcid.org/0009-0008-5537-3660</p> <ref id="AN0187567149-36"> <title> Note </title> <blist> <bibl id="bib1" idref="ref6" type="bt">1</bibl> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref42" type="bt">2</bibl> <bibtext> The author(s) received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref1" type="bt">3</bibl> <bibtext> https://<ulink href="http://www.shopify.com/blog/global-ecommerce-sales">www.shopify.com/blog/global-ecommerce-sales</ulink>.</bibtext> </blist> </ref> <ref id="AN0187567149-37"> <title> References </title> <blist> <bibtext> Acimovic J., Graves S. 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  Data: Optimizing an Omnichannel Retail Strategy Considering Customer Segmentation
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  Data: <searchLink fieldCode="AR" term="%22Shuangpeng+Yang%22">Shuangpeng Yang</searchLink><br /><searchLink fieldCode="AR" term="%22Li+Zhang%22">Li Zhang</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0008-5537-3660">0009-0008-5537-3660</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Evaluation+Review%22"><i>Evaluation Review</i></searchLink>. 2025 49(5):814-850.
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  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
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  Data: 37
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  Data: 2025
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  Data: Journal Articles<br />Reports - Descriptive
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  Data: <searchLink fieldCode="DE" term="%22Retailing%22">Retailing</searchLink><br /><searchLink fieldCode="DE" term="%22Geographic+Distribution%22">Geographic Distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Probability%22">Probability</searchLink>
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  Data: 10.1177/0193841X251328710
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  Data: 0193-841X<br />1552-3926
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  Data: Unlike previous studies on fixed logistics nodes, this research explored how consumer distribution impacts store selection and inventory balance, integrating the "ship-from-store" strategy to increase fulfillment within multiperiod sales plans. Specifically, omnichannel retailers (O-tailer) must sequentially decide on inventory replenishment from suppliers to the distribution center (DC), allocation from the DC to stores, and which department will fulfill online orders. We introduce a multiperiod stochastic optimization model and solve it with a robust two-stage approach (RTA). In Stage 1, we use the K-means algorithm and silhouette coefficients to determine the optimal number of stores. In Stage 2, linear decision rule (LDR) are employed to decide on replenishment, allocation, and order fulfillment quantities. Numerical experiments show that RTA outperforms existing methods, achieving solutions with efficiency gaps of less than 10%, even when assumptions are not fully met. Additionally, the sensitivity analysis shows that variations in product prices, fulfillment costs, market share, and customer distribution consistently lead to greater profits with the "ship-from-store" strategy.
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