Data-Driven Order Consolidation with Vehicle Routing Optimization

Abstract

This study compares time-based and quantity-based consolidation strategies within the Vehicle Routing Problem (VRP) framework to optimize supplier profitability and logistical efficiency. The time-based model consolidates deliveries at fixed intervals, offering predictable routes, reduced customer wait times, and cost efficiency in stable markets. Conversely, the quantity-based model dynamically adjusts delivery volumes to meet fluctuating demand, providing flexibility in dynamic environments but potentially increasing long-term costs due to logistical complexity. Using a mixed-integer linear programming (MILP) model, sensitivity analyses, and scenario-based experiments, the study demonstrates that the time-based model excels in stable conditions, while the quantity-based model performs better in highly variable demand scenarios. These findings provide actionable insights for selecting consolidation strategies that optimize delivery operations and enhance supply chain performance based on market dynamics.

1. Introduction

The global e-commerce market has seen unprecedented growth, reshaping supply chain operations worldwide. This expansion, driven by advancements in digital technology and increasing internet accessibility, has made online shopping more accessible and convenient for consumers [1,2,3]. As a result, businesses are investing heavily in logistics, warehousing, and transportation infrastructure to meet rising consumer demands while also adapting to the increasing complexity of global supply chains [3,4,5]. To remain competitive in this dynamic environment, firms must transition toward more agile and resilient supply chain models [6].

Effective inventory management has become a cornerstone of maintaining competitiveness in this expanding market. Optimized inventory control reduces costs, enhances service levels, and provides businesses with a critical edge in the crowded e-commerce sector [7,8]. By employing advanced techniques, firms can minimize holding costs, improve order fulfillment rates, and better align inventory levels with market fluctuations. The integration of real-time analytics and inventory tracking further allows companies to respond swiftly to changing consumer demands, thereby improving operational efficiency [9,10,11]. As supply chains grow in scale and complexity, effective inventory management remains vital for sustaining success in the global market [12,13].

Shipping and delivery integration play a central role in achieving economic efficiencies through economies of scale, which are crucial for maintaining competitiveness. Consolidating delivery operations reduces fuel consumption, labor costs, and per-unit transportation expenses while enhancing coordination among supply chain partners [3,4]. This method enhances operational efficiency while facilitating collaborative initiatives, thereby contributing to cost optimization and improved profitability [14]. Integrated delivery systems thus represent a key strategy for driving economic performance and sustaining competitiveness in the fast-evolving market.

Shipment integration strategies can be categorized into two main approaches: time-based and quantity-based integration. Time-based shipment integration focuses on aligning deliveries with specific timeframes to ensure just-in-time delivery, thereby reducing lead times and improving service levels [7]. This is particularly advantageous in industries with high demand variability, as it minimizes inventory holding costs and ensures faster responses to consumer needs [15]. Conversely, quantity-based shipment integration emphasizes shipment consolidation to achieve economies of scale. By maximizing vehicle load capacity, businesses can significantly reduce per-unit shipping costs while contributing to environmental sustainability through fewer trips and lower emissions [16,17]. Each strategy offers unique benefits, making both essential for optimizing logistics operations and addressing diverse supply chain demands [18].

The Vehicle Routing Problem (VRP) has emerged as a fundamental tool for optimizing delivery logistics by determining the most efficient routes for fleets to deliver goods across multiple locations [17]. Over time, VRP research has incorporated a wide array of constraints—such as time windows, capacity limits, and service levels—to enhance delivery efficiency and effectiveness. Time-dependent VRPs (TD-VRPs) prioritize timely deliveries within specific windows, improving customer satisfaction, while Capacitated VRPs (CVRPs) focus on optimizing vehicle utilization to transport maximum quantities of goods with minimal distance traveled [19,20]. The integration of time- and quantity-based considerations into VRP models offers comprehensive solutions that address both cost efficiency and customer satisfaction [21,22].

Delivering integrated orders based on time and quantity through optimized routes significantly enhances supply chain performance. Time-based constraints in VRP models ensure timely deliveries, which are critical for customer satisfaction and loyalty, while quantity-based constraints maximize vehicle capacity utilization, reducing transportation costs and environmental impact [15,23]. Hybrid VRP models that combine these approaches cater to the dynamic demands of modern logistics, offering innovative solutions to complex challenges [24]. These strategies underscore the importance of leveraging efficient routing systems to strengthen supply chain resilience and competitiveness [16,25].

This study presents a systematic mathematical framework that synthesizes the time-based model and quantity-based model within the context of the Vehicle Routing Problem (VRP). This model incorporates both time- and quantity-based considerations to optimize logistics decision-making. By combining these frameworks, we create a versatile tool that addresses diverse delivery scenarios while balancing efficiency and sustainability. Validation using real-world company data demonstrates the model’s practical applicability and robustness, highlighting its potential to enhance supply chain performance.

2. Literature Review

2.1. The Importance of Shipment Integration and Delivery Routing

As the importance of shipment integration in supply chain management continues to grow, selecting appropriate shipment integration strategies has become a crucial element for achieving efficient supply chain operations [26,27]. Shipment integration refers to the strategy of consolidating accumulated orders from buyers to enhance the efficiency of deliveries, thereby realizing economies of scale in transportation [28,29]. This approach not only reduces redundant shipments but also optimizes the utilization of resources, contributing to cost efficiency and operational sustainability.

To deliver consolidated orders effectively to customers, it is essential to derive optimal delivery routes. Such optimization offers significant advantages, including the reduction of transportation costs, shorter delivery times, and minimized environmental impact [29]. Moreover, efficient routing ensures timely and reliable deliveries, which are critical for enhancing customer satisfaction and building long-term competitive advantages.

With the increasing complexity of supply chain networks, simultaneously integrating shipment strategies and routing optimization holds great promise for addressing dynamic market demands. This combined approach enables businesses to enhance supply chain efficiency while maintaining adaptability to fluctuating operational requirements [21,30]. Additionally, integrating these strategies supports broader organizational goals, such as sustainability and responsiveness, which are becoming increasingly vital in global supply chain management [24,31].

2.2. Research on Shipment Integration

Shipment integration policies, particularly time-based and quantity-based approaches, are widely applied in both academic research and industry practices, offering unique characteristics and advantages [23,31]. Time-based shipment integration involves delivering accumulated orders at regular intervals, making it well-suited for scenarios with high-order frequencies or predictable demand cycles. In contrast, quantity-based policies trigger shipments once a predefined order quantity is reached, which is particularly beneficial for irregular demand patterns or situations requiring bulk deliveries [32].

Optimal delivery cycles and order quantities for time-based and quantity-based policies have been extensively studied, identifying conditions for maximum efficiency [27,33]. The cost-effectiveness of time-based, quantity-based, and hybrid policies under Poisson-distributed order arrivals has also been examined, providing insights into their optimal implementation [29,34]. Time-based shipment integration policies within a single-inventory system were investigated to derive optimal delivery cycles and pricing strategies for improved operational outcomes [30]. Additionally, time-based and quantity-based policies were compared in a single-item inventory system, incorporating order cancellations to identify optimal delivery cycles and quantities under various conditions [31].

While time-based and quantity-based policies offer distinct advantages and can be effectively applied in diverse operational contexts, existing research has primarily focused on determining their optimal parameters independently. This limitation underscores the need to explore integrated approaches that combine these policies with efficient delivery routing strategies. Such integration could significantly enhance supply chain performance by reducing logistics costs, improving service quality, and providing a comprehensive framework to address the challenges of dynamic supply chain environments.

2.3. VRP and Shipment Consolidation Strategies

The Vehicle Routing Problem (VRP) is one of the most critical challenges in logistics and supply chain management, aiming to enhance operational efficiency and minimize costs [35,36]. Efficient vehicle routing improves logistics performance by reducing transportation costs, streamlining delivery operations, and promoting sustainability [30]. In recent years, VRP research has increasingly focused on addressing the complexities of modern logistics networks through innovative methodologies [23,25].

The last-mile delivery problem has been combined with time-based shipment consolidation policies, demonstrating that integrated approaches outperform traditional first-in, first-out methods [32]. Cost efficiency in last-mile logistics can be significantly improved by integrating intermediate delivery routes for vehicles [36]. Stochastic models for VRP with probabilistic demand and time windows have been developed to enhance on-time delivery rates and overall transportation efficiency [37]. Shipment consolidation and dispatching problems have also been addressed to minimize total costs and travel distances, with proposed mathematical models comparing direct delivery and transshipment-based routing strategies [38].

This study aims to integrate time-based and quantity-based consolidation policies within the framework of the Vehicle Routing Problem (VRP) to determine the optimal policy for maximizing supplier profitability under diverse logistics and market conditions. By examining how each consolidation policy functions within the VRP framework and identifying scenarios where each approach achieves superior performance, this research seeks to provide actionable insights. The study highlights the potential of combining time-based and quantity-based shipment consolidation strategies with VRP optimization to simultaneously enhance operational efficiency and profitability. The findings will offer practical guidance for logistics companies in selecting and implementing consolidation strategies tailored to their specific operational requirements and constraints.

3. Integrated Model Formulation

3.1. Problem Description

This study addresses the problem of integrating time-based consolidation policies and quantity-based consolidation policies within the framework of the Vehicle Routing Problem (VRP). These policies are analyzed to determine their respective impacts on cost efficiency, delivery performance, and overall profitability. To facilitate this comparison, the study examines the operational and strategic characteristics of time-based consolidation and quantity-based consolidation under the following several fundamental assumptions (the parameters and decision variables are listed in Table 1):

1. Customer demand is influenced by either time or quantity fluctuations. In the time-based model, demand is sensitive to price and lead time changes, whereas in the quantity-based model, demand remains constant concerning lead time since deliveries are quantity-based.

2. Customer demand, location, and price are assumed to be predetermined and known.

3. It is assumed that all customers have uniform sensitivity to lead time and price variations.

4. The market scenario considered involves a single product offered by a solitary supplier.

3.2. Time-Based Mathematical Model

In the time-based consolidation policy, the framework is designed to systematically address demand fluctuations arising from variations in time, price, and lead time. This approach incorporates a comprehensive suite of functions, including the demand function, total demand function, revenue function, cost function, and the expected long-term profit function, to optimize operational efficiency and profitability.

The demand for customer $i$ at time t, denoted ${\lambda }_{t,i}$ incorporates the initial demand ${\lambda }_{0,i}$ incorporates adjustments for customer preference-dependent demand increases, price sensitivity, and lead-time sensitivity as follows in Equation (1). This approach aligns with current advances in integrating demand management strategies within vehicle routing frameworks, highlighting the critical role of demand sensitivity in optimizing logistics operations and improving service efficiency [37,38].

(1)${\lambda }_{t,i}={\lambda }_{0,i}\times (1+{\theta }_i\times T)-{\delta }_p\ast p_i-{\delta }_L\ast T \forall i \in C \backslash \left\{0\right\}$

In this equation, T represents the delivery cycle time, ${\theta }_i$ is the demand growth specific to the customer attributed to time. The parameters ${\delta }_p$ and ${\delta }_L$ capture the price sensitivities and lead time sensitivity, respectively, while $p_i$ is the price for customer $i$. In the time-based model, ${\delta }_p$ quantifies the degree to which changes in product pricing influence customer demand, while ${\delta }_L$ measures the impact of delivery delays on demand levels. Both parameters are critical in adjusting the based demand (${\lambda }_0$) to account for variation arising from real-world market dynamics. For this study ${\delta }_p$ and ${\delta }_L$ are assumed to be uniform across all customers to represent generalized market behavior and simplify the numerical experiments. This formulation effectively captures demand shifts driven by customer preferences, pricing strategies, and delivery lead times. To calculate the total demand for customer $i$ over the cycle T, denoted as $Q_{T,i}$ we integrate ${\lambda }_{t,i}$ across the cycle T, as shown in Equations (2) and (3) as follows:

(2)$Q_{T,i}={\int }_0^T{\lambda }_{t,i} dt$

(3)$Q_T=\sum _{i\in C\backslash 0}{Q}_{T,i}$

Within the time-based model, costs are effectively organized into four major categories: inventory holding costs, order processing costs, delivery costs, and customer waiting costs. These components are crucial for optimizing supply chain operations and sustaining high levels of customer satisfaction.

1. Inventory holding costs: These costs are related to maintaining adequate stock to meet customer demand. Increasing inventory levels lead to higher holding costs, intricately linked to customer service quality. The expected inventory holding cost is calculated as Equation (4) as follows:

   (4)$\mathrm{Expected} \mathrm{inventory} \mathrm{holding} \mathrm{cost}: E\left[{IC}_T\right]=h\times {\sum }_{i\in C\backslash 0}{\lambda }_{t,i}\times T$

2. Order processing costs: Representing fixed costs incurred per cycle, these cover administrative and logistical expenses essential for operational efficiency. The expected inventory dispatch cost is calculated as Equation (5) as follows:

   (5)$\mathrm{Expected} \mathrm{inventory} \mathrm{dispatch} \mathrm{cost}: E\left[{HC}_T\right]={\displaystyle \frac{F}{T}}$

3. Delivery costs: These costs are determined by factors such as customer location, travel distance, and vehicle routes, playing a vital role in logistics efficiency. They can be minimized through route optimization. The expected delivery cost is expressed as Equation (6) as follows:

   (6)$\mathrm{Expected} \mathrm{delivery} \mathrm{cost}: E\left[{DC}_T\right]=C_d\ast {\sum }_{i\in C\backslash 0}{\sum }_{j\in C, j\ne i}{d}_{i,j}\times x_{i,j,k}\times {\lambda }_{t,i}$

4. Customer waiting costs: Directly impacting customer satisfaction, these costs depend on the waiting time for order delivery and can be minimized through optimal cycle time and route planning and calculated as Equation (7) as follows:

   (7)$\mathrm{Expected} \mathrm{customer} \mathrm{waiting} \mathrm{cost}: E\left[{WC}_T\right]=w\times {\sum }_{i\in C}(T_{process}+{\displaystyle \frac{d_{i,j}}{V}}+{\displaystyle \frac{{\lambda }_{t,i}}{Q_k}})$

The cumulative costs in the time-based model are strategically optimized to minimize the cycle time T, thereby maximizing the supplier’s profit. By integrating these cost components, suppliers can effectively manage trade-offs between cost efficiency and service quality, ultimately enhancing supply chain performance. This structured approach emphasizes the critical balance between cost management and revenue generation, as highlighted in contemporary research on supply chain cost optimization and service quality improvement [37,38]. Specifically, the costs incurred in the time-based model are detailed in Equations (4)–(7), which outline the individual cost components, while the supplier’s revenue and total costs are defined in Equations (8) and (9), respectively. The final expected long-term profit for the time-based model, integrating these components, is expressed in Equation (10) as follows:

(8)$\mathrm{Total} \mathrm{profit}: E\left[{TR}_T\right]={\sum }_{i\in C\backslash 0}{p}_i\times Q_{T,i}$

(9)$\mathrm{Total} \mathrm{cost}: E\left[{TC}_T\right]=E\left[{IC}_T\right]+E\left[{HC}_T\right]+E\left[{DC}_T\right]+E\left[{WC}_T\right]$

(10)$\mathrm{Long} \mathrm{term} \mathrm{profit} \mathrm{for} \mathrm{the} \mathrm{time}-\mathrm{based} \mathrm{model}: E\left[\Pi \left(T\right)\right] =E\left[{TR}_T\right]-E\left[{TC}_T\right]$

This formulation highlights the interconnectedness of revenue, cost efficiency, and service quality in determining the long-term profitability of the time-based model, making it a robust framework for optimizing supply chain operations.

To extend the utility of this formulation, the next step involves formalizing the mathematical framework that enables operational optimization. The time-based model aims to maximize the supplier’s long-term profit by minimizing cumulative costs across key components, such as inventory holding, order processing, delivery, and customer waiting. To accomplish this, a Mixed Integer Linear Programming (MILP) formulation is proposed, integrating the defined cost functions and operational constraints. The MILP formulation employs an objective function that maximizes the supplier’s profit, defined as the difference between total revenue and cumulative costs. This rigorously structured framework serves as a practical tool for optimizing supply chain performance in diverse operational scenarios.

(11)$\mathrm{Objective} \mathrm{function}: maxE\left[\Pi \left(T\right)\right]$

Subjective to:

(12)$\sum _{j\in C, j\ne i}\sum _{k\in K}{x}_{i,j,k}=1 \forall i \in C \backslash \{0\}$

(13)$\sum _{j\in C\left\{0\right\}, j\ne 0}\sum _{k\in K}{x}_{0,j,k}=1 \forall k \in K$

(14)$\sum _{i\in C\backslash \left\{0\right\}, i\ne 0}\sum _{k\in K}{x}_{i,0,k}=1 \forall k \in K$

(15)$\sum _{j\in C, j\ne i} x_{i,j,k}=\sum _{j\in C, j\ne i} x_{j,i,k} \forall i \in C$

(16)$\sum _{i\in C, i\ne 0}{d}_{i,j} x_{i,j,k} \le {dist}_{max} \forall k \in K$

(17)$\sum _{i\in C, i\ne 0}{q}_{i,k} \le Q_k \forall k \in K$

(18)$q_i={\lambda }_{t,i}\forall i\ne C\left\{0\right\}$

(19)$u_i-u_j+\left|\mathrm{C}\right|x_{i,j,k}\le \left|\mathrm{C}\right|-1\forall i,j\in C, i\ne j, i\ne 0, j\ne 0$

(20)$x_{i,j,k} \in \left\{\mathrm{0,1}\right\}, \forall i,j\in C, k\in K$

(21)$u_i\ge 1, u_0=0, \forall i\in C$

The constraints in this formulation are meticulously designed to ensure the feasibility and practicality of vehicle routing within the optimization model. Constraint (12) guarantees that each customer is visited precisely once by a single vehicle, thereby guaranteeing complete service without redundancy. Constraint (13) mandates that every vehicle departs from the depot at the beginning of its route, while Constraint (14) enforces that all vehicles return to the depot upon completing their deliveries, ensuring the circularity and completeness of all routes. To maintain route continuity, Constraint (15) requires that a vehicle entering a customer’s location must also depart from the same location. Constraint (16) imposes a restriction on the total distance traveled by each vehicle, ensuring it does not exceed the maximum allowable distance, which accounts for operational limitations such as fuel capacity or time constraints. Constraint (17) ensures that the total quantity delivered by each vehicle remains within its designated capacity, preserving the logistical feasibility of the operations. Furthermore, Constraint (18) aligns the delivery quantities with the specific demand of each customer, ensuring accurate and reliable service. To prevent infeasible routing scenarios, Constraint (19) eliminates sub-tours by enforcing continuity and validity in the routes. Constraint (20) specifies that the route selection variables must be binary, explicitly indicating whether a particular route is part of the solution. Lastly, Constraint (21) establishes the order of visits to customers, setting the depot as the starting point and eliminating sub-tours by maintaining a logical sequence of customer visits. Together, these constraints form a comprehensive framework that ensures the model’s adherence to operational realities, facilitating effective vehicle routing while optimizing logistical efficiency.

3.3. Quantity-Based Mathematical Model

The quantity-based mathematical model is constructed to examine demand sensitivity in relation to quantity and price, providing a comprehensive framework that includes demand, total demand, revenue, cost, and expected long-term profit functions. These functions are tailored to address the dynamics inherent in quantity-based delivery systems.

In the quantity-based model, the demand function, total demand function, revenue function, cost function, and expected long-term profit function are defined to capture demand sensitivity to quantity and price. The demand for customer $i$ when the delivered quantity Q is reached, denoted ${\lambda }_{q,i}$ incorporates the initial demand ${\lambda }_{0,i}$ with adjustments for quantity-based demand increase and price sensitivity as follows (Equation (22)):

(22)${\lambda }_{q,i}={\lambda }_{0,i}\ast (1+{\theta }_i\ast {\displaystyle \frac{Q_{used}}{Q_k}})-{\delta }_p\ast p_i \forall i \in C \backslash \{0\}$

Here, ${\theta }_i$ represents the demand growth rated specific to customer $i$, attributed to increasing delivery quantities, while $Q_k$ denotes the maximum vehicle capacity. The parameter ${\delta }_p$ adjusts price sensitivity to effectively capture changes in demand influenced by product pricing. In the quantity-based model, the lead-time sensitivity parameter ${\delta }_L$ is not explicitly included, as delivery cycles are dynamically adjusted based on accumulated demand rather than fixed time intervals. This dynamic approach minimizes the influence of lead time on demand, positioning ${\delta }_P$ as the primary factor for reflecting customer preferences and pricing strategies the primary factor for reflecting customer preferences and pricing strategies. This formulation effectively models how demand evolves in response to delivery quantities and pricing, providing a robust framework for addressing the complexities of quantity-based delivery systems. By incorporating customer preferences and price sensitivity, this function reflects the dynamic shifts in demand characteristic of quantity-based delivery models. The total demand for customer $i$ over the delivery cycle $Q_{used}$, denoted as $Q_{Q,i},$ is computed as follows (Equations (23) and (24)):

(23)$Q_{Q,i}={\lambda }_{q,i}\times Q_{used}$

(24)$Q_Q=\sum _{i\in C}{Q}_{Q,i}$

In the quantity-based model, cost components and revenue functions are structured around the delivered quantity $Q_{used}$, diverging from the traditional fixed-time cycles of the time-based model to optimize delivery quantities for maximizing long-term profitability. While the primary cost factors—inventory holding costs, order processing costs, delivery costs, and customer waiting costs—are consistent with those in the time-based model, they are driven by $Q_{used}$ rather than time T. This distinction reflects a strategic alignment with modern supply chain management objectives, emphasizing both cost efficiency and enhanced customer satisfaction.

The optimization of cumulative costs within the quantity-based model focuses on determining the optimal delivery quantity $Q_{used}$ to maximize supplier profitability. Recent studies [1,8,39] highlight the importance of integrating quantity-based demand management with advanced logistics strategies, demonstrating how such approaches can significantly improve cost efficiency and service quality. These findings reinforce the relevance of quantity-driven models, such as the quantity-based model, in addressing the complexities of modern supply chains.

Consequently, the quantity-based model’s cumulative costs are explicitly designed to optimize $Q_{used}$, thereby enhancing supplier profitability. The specific cost equations and revenue functions underlying the quantity-based model are detailed in Equations (25)–(31), providing a comprehensive framework for evaluating and optimizing supply chain performance.

(25)$\mathrm{Expected} \mathrm{inventory} \mathrm{holding} \mathrm{cost}: E\left[{IC}_Q\right]=h\times {\sum }_{i\in C\backslash \left\{0\right\}}{\lambda }_{Q,i}\times Q_{used}$

(26)$\mathrm{Expected} \mathrm{inventory} \mathrm{dispatch} \mathrm{cost}: E\left[{HC}_Q\right]={\displaystyle \frac{F}{Q_{used}}}$

(27)$\mathrm{Expected} \mathrm{delivery} \mathrm{cost}: E\left[{DC}_Q\right]=C_d\ast {\sum }_{i\in C\backslash \left\{0\right\}}{\sum }_{j\in C, j\ne i}{d}_{i,j}\times x_{i,j,k}\times {\lambda }_{Q,i}$

(28)$\mathrm{Expected} \mathrm{customer} \mathrm{waiting} \mathrm{Cost}:E\left[{WC}_Q\right]=w\times {\sum }_{i\in C}(T_{process}+{\displaystyle \frac{d_{i,j}}{V}}+{\displaystyle \frac{{\lambda }_{q,i}}{Q_k}})$

(29)$\mathrm{Total} \mathrm{profit}: E\left[{TR}_Q\right]={\sum }_{i\in C\backslash \left\{0\right\}}{p}_i\times Q_{Q,i}$

(30)$\mathrm{Total} \mathrm{cost}: E\left[{TC}_Q\right]=E\left[{IC}_Q\right]+E\left[{HC}_Q\right]+E\left[{DC}_Q\right]+E\left[{WC}_Q\right]$

(31)$\mathrm{Long} \mathrm{term} \mathrm{profit} \mathrm{for} \mathrm{the} \mathrm{quantity}\text{-}\mathrm{based} \mathrm{model}: E\left[\Pi \left(Q\right)\right] =E\left[{TR}_Q\right]-E\left[{TC}_Q\right]$

The goal of the quantity-based model is to maximize the supplier’s long-term profit by minimizing cumulative costs associated with inventory holding, order processing, delivery, and customer waiting. This leads to an optimal quantity $Q_{used}$ that aligns with customer demand and logistical efficiency.

The constraints for the quantity-based model are similar to those in the time-based model but adapted for quantity-based decision-making. The MILP model includes the following elements to optimize $Q_{used}:$

(32)$\mathrm{Objective} \mathrm{function}: maxE\left[\Pi \left(Q\right)\right]$

Subjective to:

(33)$\sum _{j\in C, j\ne i}\sum _{k\in K}{x}_{i,j,k}=1 \forall i \in C \backslash \{0\}$

(34)$\sum _{j\in C\left\{0\right\}, j\ne 0}\sum _{k\in K}{x}_{0,j,k}=1 \forall k \in K$

(35)$\sum _{i\in C\backslash \left\{0\right\}, i\ne 0}\sum _{k\in K}{x}_{i,0,k}=1 \forall k \in K$

(36)$\sum _{j\in C, j\ne i} x_{i,j,k}=\sum _{j\in C, j\ne i} x_{j,i,k} \forall i \in C$

(37)$\sum _{i\in C, i\ne 0}{d}_{i,j} x_{i,j,k} \le {dist}_{max} \forall k \in K$

(38)$\sum _{i\in C, i\ne 0}{q}_{i,k} \le Q_k \forall k \in K$

(39)$q_i={\lambda }_{q,i}\forall i\ne C\left\{0\right\}$

(40)$u_i-u_j+\left|\mathrm{C}\right|x_{i,j,k}\le \left|\mathrm{C}\right|-1\forall i,j\in C, i\ne j, i\ne 0, j\ne 0$

(41)$x_{i,j,k} \in \left\{\mathrm{0,1}\right\}, \forall i,j\in C, k\in K$

(42)$u_i\ge 1, u_0=0, \forall i\in C$

The constraints in this formulation are carefully designed to ensure feasible and efficient vehicle routing for the quantity-based (quantity-based model) consolidation strategy within the optimization model. Constraint (33) ensures that each customer is served exactly once by a single vehicle, preventing duplicate visits or missed deliveries. Constraint (34) guarantees that every vehicle departs from the central depot at the start of its route, establishing a clear starting point for all operations. Constraint (35) enforces that each vehicle must return to the depot upon completing its deliveries, ensuring the routes are circular and complete. Constraint (36) maintains route continuity by ensuring that if a vehicle arrives at a customer location, it must also depart from that location. Constraint (37) restricts the total travel distance of each vehicle so that it does not exceed the allowable operational limit, accounting for fuel capacity or time constraints. Constraint (38) ensures that the total quantity of goods delivered by a vehicle does not exceed its designated capacity, maintaining logistical feasibility. Constraint (39) aligns the quantity delivered to each customer with their specific demand, ensuring that customer requirements are accurately fulfilled. Constraint (40) eliminates any potential for infeasible routing scenarios, such as sub-tours, by enforcing logical and continuous routes. Constraint (41) requires that route selection variables are binary, clearly defining whether a particular route between two locations is part of the final solution. Constraint (42) establishes the order in which customers are visited, starting from the depot and maintaining a logical and practical sequence for deliveries. These constraints collectively ensure that the quantity-based model achieves efficient delivery operations while meeting customer demands, respecting vehicle capacity limits, and adhering to operational restrictions. By addressing practical challenges in vehicle routing, this model optimizes the overall performance and reliability of the delivery system.

4. Numerical Experiments

4.1. Numerical Experiment Setup Using Real-World Industry Data

This study conducts numerical experiments utilizing real-world data collected from a Korean dental inventory management company from 2020 to June 2022. The company plays a pivotal role in supplying various dental inventory items nationwide, including dental masks, alcohol swabs, and implants. By leveraging detailed data encompassing demand, pricing, order cycles, and holding costs for each inventory item, this research aims to rigorously evaluate and analyze the proposed models. Such data facilitates the accurate simulation of complex scenarios inherent in dental inventory management and contributes to assessing the models’ responsiveness to industry-specific variables. Figure 1 illustrates the monthly demand patterns observed for dental inventory during the study period, while Figure 2 presents the pricing of dental inventory items. Initial customer demand was determined using a weighted average of demand across companies, and the base price for inventory items was set at the 50th percentile value, which is 13,000 KRW.

The principal objective of this study is to formulate optimal delivery routes and inventory management strategies through the tune-based model and quantity-based model, focusing on maximizing supplier profitability while minimizing total costs. The experimental analysis aims to determine whether the models can efficiently meet customer demand while optimizing expenses related to delivery, inventory, and customer waiting time. By utilizing actual data, this study enhances the practical relevance of its findings, aiming to effectively address the operational challenges faced within the dental inventory sector.

Through these numerical experiments, the study addresses the practical complexities faced by dental inventory companies, particularly in achieving stable demand fulfillment and efficient inventory turnover. The findings aim to improve operational efficiency in inventory management by providing strategic insights that contribute to cost reduction and enhanced customer service standards. Given the critical need for a reliable inventory system in the dental industry, the insights derived from this study are expected to significantly contribute to optimizing operations within dental inventory management.

To support a fair and accurate comparison between time-based and quantity-based consolidation models, a synchronization constraint was introduced to align their delivery cycle timings. The time-based model relies on fixed delivery intervals, whereas the quantity-based model bases deliveries on accumulated demand. Therefore, a synchronization constraint was established to ensure that the delivery cycles of both models remain closely aligned within a permissible tolerance level $\epsilon$ as follows:

$|T-Q_{used}| \le \epsilon .$

Here, $\epsilon$ represents the synchronization tolerance designed to minimize timing differences and maintain consistency in delivery cycle timing. In this study, $\epsilon$ was deliberately set to a very small value to ensure that the permissible differences in delivery cycles between the time-based model and quantity-based model remain negligible. This approach allows for the accommodation of minor variations inherent to each model’s characteristics while maintaining the overall alignment and consistency of delivery schedules.

The value of $Q_{used}$ was derived to approximate the delivery interval based on the quantity-based model’s demand fulfillment process. Specifically, $Q_{used}$ s is calculated as the total cumulative demand across all customers during the designated cycle period, divided by the amount scheduled for delivery in each cycle based on anticipated demand as follows:

$Q_{used}={\displaystyle \frac{\mathrm{c}\mathrm{u}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{f} \mathrm{a}\mathrm{l}\mathrm{l} \mathrm{c}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{s} \mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}{\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{d} \mathrm{d}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y} \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{y} \mathrm{p}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}}$

By calculating $Q_{used}$ in this manner, the delivery cycle timing in the quantity-based model aligns closely with that of the time-based model, providing a consistent basis for comparing the operational efficiency and cost-effectiveness of both models. The synchronization constraint minimizes discrepancies caused by differing delivery intervals, ensuring that any observed differences in performance are attributed to the unique characteristics of each model rather than variations in timing. This approach provides a robust foundation for evaluating the strategic implications of each model within the context of dental inventory management.

4.2. Optimal Solution Derivation and Experimental Results

Section 4.2 is dedicated to deriving concrete findings by leveraging experimental data collected in prior sections to assess the effectiveness of the proposed optimization models. This study employs both the time-based model and the quantity-based model to achieve efficient inventory management and optimize delivery routes for a dental inventory company, integrating diverse parameters within an optimization framework. In the experimental design, a randomized combination of the primary parameters and variable settings, as delineated in Table 2, was utilized across 100 iterations to facilitate a comprehensive comparative analysis of the model’s performance. By aggregating the results of these iterations, the study aims to identify strategies for profit maximization and cost minimization, thereby assessing the practical applicability of the models. The parameters presented in Table 2 were derived from empirical data, thereby enhancing the precision and credibility of the obtained results.

The results presented in Table 3 and Figure 3 provide a clear comparison of the performance differences between the time-based model and quantity-based model across various consolidation cycles. Table 3 highlights the maximum profit achieved by each model, consistently demonstrating the superior performance of the time-based model over the quantity-based model. The time-based model exhibits increasingly favorable outcomes as the consolidation cycle lengthens, with a profit difference of 260,938 KRW in Cycle 5 widening to 795,514 KRW in Cycle 10. This trend underscores the operational efficiency and cost-minimization capabilities of the time-based model’s fixed-cycle delivery strategy, which effectively balances supply and demand over time.

The box plot analysis in Figure 3 offers deeper insights into the profit distribution of the two models across consolidation cycles. The time-based model not only demonstrates higher average profits compared to the quantity-based model but also exhibits a significantly broader profit distribution range. This indicates the time-based model’s ability to maintain greater profitability across various cycles while accommodating variability and exceptions that may arise. For instance, the time-based model consistently achieves higher average profits across all consolidation cycles, with the profit disparity becoming increasingly pronounced as the cycle duration extends. Additionally, the time-based model’s broader interquartile range, as depicted in the box plots, reflects its structural robustness in managing profit variability effectively.

In contrast, the quantity-based model exhibits a relatively narrower profit distribution, reflecting its sensitivity to fluctuations in individual customer demand. This narrower range suggests potential limitations in the quantity-based model’s operational efficiency. Moreover, the quantity-based model demonstrates a decline in profitability as the consolidation cycle lengthens, highlighting its reduced adaptability to operational uncertainties over extended cycles. Nevertheless, the quantity-based model shows potential advantages in shorter consolidation cycles or dynamic market environments characterized by frequent and unpredictable demand changes.

In summary, the time-based model effectively leverages its fixed-cycle delivery strategy to achieve higher average profits and a broader profit distribution range, offering both stability and profitability in consolidated delivery operations. Conversely, the quantity-based model’s flexible approach enables it to adapt to demand fluctuations but may result in lower average profits and narrower profit ranges, particularly over extended cycles. The findings from the box plot and table analysis confirm that the time-based model is better suited for stable markets where predictability and operational efficiency are paramount, while the quantity-based model may excel in dynamic markets requiring responsiveness to rapid demand changes.

In particular, the strategic advantages of the time-based model are evident in a fixed consolidation cycle, demonstrating that a fixed-cycle delivery strategy leads to favorable outcomes in terms of demand forecasting and cost optimization. These findings suggest that in a fixed-cycle system where regular deliveries are scheduled, a stable delivery approach like that of the time-based model can yield superior performance.

The experimental results reveal not only differences in profitability but also distinct patterns in optimal vehicle routing between the time-based model and quantity-based model, highlighting the unique operational characteristics of each approach. Figure 4 visualizes the optimal vehicle routing patterns derived from the time-based model and quantity-based model across varying consolidation cycles, clearly illustrating the strategic and operational differences between the two models.

The time-based model adopts a fixed-cycle delivery strategy that generates consistent and predictable routing patterns, often sequential in nature, with customers visited in a stable order during each cycle. This fixed-cycle approach enables the time-based model to regularly align supply and demand, minimizing transportation costs, reducing customer wait times, and lowering inventory holding costs. Such a structured methodology ensures operational stability and cost efficiency, ultimately leading to higher long-term profitability. Figure 4 demonstrates the time-based model’s ability to maintain clear and repetitive routing structures across cycles, minimizing operational variability and effectively maximizing supplier profitability over time.

In contrast, the quantity-based model employs a quantity-based delivery strategy, dynamically adjusting routes based on fluctuations in customer demand. While this flexibility allows the quantity-based model to address demand variability within specific cycles, it often results in irregular routing patterns that deviate from sequential or circular structures. The adaptability of the quantity-based model facilitates responsiveness to immediate changes in customer requirements but also introduces inefficiencies such as increased travel distances and more complex routing configurations. Figure 4 visually captures the non-circular and irregular routing patterns of the quantity-based model, emphasizing its flexibility while simultaneously highlighting the challenges it faces in maintaining long-term cost efficiency.

The observations from Figure 4 succinctly summarize the key differences between the time-based model and the quantity-based model, underscoring the strategic advantages of each approach. The time-based model’s predictable routing patterns ensure stability and cost-effectiveness, making it ideal for environments with regular demand patterns. Conversely, the quantity-based model’s flexible routing approach is better suited for dynamic markets characterized by frequent and significant demand fluctuations, though it may result in higher operational complexity and costs. Specifically, the time-based model focuses on reducing transportation and operational costs through its structured routing framework, while the quantity-based model, despite its adaptability, risks reduced cost efficiency as consolidation cycles lengthen.

In conclusion, the time-based model’s fixed-cycle delivery strategy offers long-term stability and cost efficiency by stabilizing customer wait times and reducing inventory holding costs. On the other hand, the quantity-based model’s quantity-based approach demonstrates flexibility in route adjustments to meet immediate demand fluctuations but may incur increased travel distances and higher operational costs, making it more suitable for short-term demand management rather than extended consolidation cycles. Figure 4 effectively highlights these distinctions, providing a clear visual comparison of the operational strategies and efficiencies of the two models.

4.3. Sensitivity Analysis for Evaluating the Performance of Time-Based Model and Quantity-Based Model

In Section 4.3, a sensitivity analysis was conducted to evaluate the performance of the time-based model and quantity-based models under varying price sensitivity (${\delta }_p$) conditions. This analysis holds significant managerial implications, as price sensitivity directly influences demand elasticity, profitability, and operational decision-making. Understanding how each model responds to changes in the price sensitivity coefficient provides valuable insights for designing integrated strategies tailored to specific market dynamics.

From an inventory management and operational strategy perspective, such analysis is critical for identifying which model delivers superior profitability and efficiency under varying conditions. In this study, lead time sensitivity (${\delta }_L$) was held constant while ${\delta }_p$ was systematically varied across a range of values to analyze the trade-offs between the two models. This approach allows for an evaluation of the balance between stability and flexibility, enabling decision-makers to select the most suitable model based on objectives such as profit maximization, cost minimization, or long-term stability in volatile markets.

Furthermore, the robustness of the results was ensured through 100 iterative simulations for each condition, providing reliable and actionable insights for strategic planning in diverse business environments.

Table 4 presents clear profitability trends across varying consolidation cycles, price sensitivity $\left({\delta }_p\right)$, and lead time sensitivity $\left({\delta }_L\right)$. The time-based model consistently demonstrates superior average profitability compared to the quantity-based model, particularly under conditions of high price sensitivity and extended lead times. These results highlight the resilience of the time-based model in managing market environments where price-induced demand fluctuations and delivery delays pose significant challenges to operational stability. However, under specific conditions, such as ${\delta }_p=0.3$ and ${\delta }_L=0.4$ the quantity-based model, it outperforms the time-based model, reflecting its adaptability to dynamic demand scenarios.

From a managerial perspective, the time-based model provides significant strategic advantages in mitigating demand variability caused by price sensitivity. As ${\delta }_p$ increases, both models exhibit declining profitability; however, the decline is more gradual for the time-based model due to its fixed delivery cycles. For instance, under a 10-day consolidation cycle with ${\delta }_p=0.3$, the time-based model achieves an average profit of 657,888 KRW, whereas the quantity-based model achieves a slightly higher profit of 776,064 KRW. This indicates that the time-based model is generally more robust in stabilizing demand and sustaining profitability in highly volatile markets, although the quantity-based model’s flexibility may occasionally prove advantageous under specific market conditions.

In contrast, the quantity-based model demonstrates heightened sensitivity to price fluctuations, as its adaptive nature necessitates frequent adjustments to align with dynamic demand, often resulting in increased operational costs. While this flexibility is beneficial in low ${\delta }_p$ markets, it undermines profitability in high ${\delta }_p$ scenarios. Managers operating in price-sensitive environments can leverage the time-based model’s predictable and consolidated delivery approach to mitigate demand instability and maintain consistent profit margins. Conversely, the quantity-based model may be more suitable in markets requiring rapid responsiveness to immediate demand changes.

The length of the consolidation cycle further amplifies the profitability differences between the two models. The time-based model achieves greater economies of scale with extended cycles, effectively lowering per-unit distribution costs. For instance, under a 10-day consolidation cycle with ${\delta }_p=0.1$, the time-based model achieves a maximum profit of 3,679,687 KRW, compared to the quantity-based model’s 2,326,707 KRW. These findings underscore the time-based model’s suitability for cost-sensitive markets that prioritize bulk shipments and predictable demand patterns. In contrast, while the quantity-based model benefits modestly from extended cycles, its frequent adjustments to accommodate demand variations limit its ability to fully capitalize on the cost efficiencies associated with longer consolidation periods.

The analysis of lead time sensitivity (${\delta }_L)$ provides further critical insights. As ${\delta }_L$ increases, the time-based model demonstrates superior profitability by mitigating the adverse effects of prolonged lead times on demand. For example, under a 9-day consolidation cycle with ${\delta }_L=0.4$, the time-based model achieves a profit of 1,791,332 KRW, significantly surpassing the quantity-based model’s 1,355,818 KRW. This indicates that the time-based model’s fixed delivery intervals act as a stabilizing mechanism, reducing demand variability even under high ${\delta }_L$ conditions. Conversely, the quantity-based model’s flexibility, while advantageous in other scenarios, can exacerbate demand instability in high ${\delta }_L$ environments, thereby further diminishing its profitability.

In conclusion, the sensitivity analysis highlights the time-based model’s robustness and efficiency in achieving consistent profitability under challenging market conditions. Its fixed-cycle framework optimally balances supply and demand, rendering it ideal for markets with high price and lead time sensitivity. Conversely, the quantity-based model, while highly flexible, is more suitable for dynamic markets requiring rapid responsiveness to demand fluctuations, albeit with trade-offs in cost efficiency. By aligning model selection with specific market characteristics, logistics managers can enhance supply chain resilience, optimize operational efficiency, and achieve sustainable profitability across diverse operational contexts.

4.4. Scenario Experimentation Based on Number of Buyers

This section conducts scenario-based experiments by setting up three cases with varying numbers of customers to compare the performance of the time-based model and quantity-based models. Specifically, cases with 4, 6, and 10 customers were analyzed to evaluate how each model’s cost efficiency, profitability, and route optimization adapt to different customer counts.

With fewer customers, logistics cost savings are expected to stem from shorter routes and smaller shipment volumes. Conversely, as the number of customers increases, the complexity of route optimization intensifies, potentially incurring additional logistics costs that significantly influence each model’s profitability structure. From a managerial perspective, this experiment seeks to assess how changes in customer volume impact the complexity and efficiency of logistics operations and identify the optimal model selection strategy under various market conditions. The design also aims to evaluate the interplay between economies of scale, shipment frequency, and service quality, providing practical insights into balancing these key factors for operational success.

For each customer scenario, 100 repeated trials were conducted to identify the superior model. In each trial, optimal routes were determined for both the time-based model and quantity-based models by identifying the routes yielding maximum profit. This iterative approach ensures that logistics managers are provided with reliable and consistent operational data to inform decision-making. It also highlights how each model optimizes performance to achieve higher cost efficiency and profitability as customer volume changes.

The findings of the scenario analysis are summarized in Table 5, which outlines the profitability and cost-efficiency of both models under varying customer counts. Additionally, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 visually represent the route distributions and profit outcomes for both models, offering critical insights into their operational dynamics across different customer configurations. As customer numbers increase, the fixed-cycle approach of the time-based model demonstrates superior stability and cost-saving potential even in complex routing situations, whereas the quantity-based model exhibits greater adaptability in smaller markets with higher demand variability.

Through this scenario analysis, the study provides strategic guidance for logistics and supply chain managers in selecting the optimal model based on customer count. Specifically, it evaluates whether the time-based model’s fixed-cycle approach facilitates economies of scale and reduced routing complexity in larger customer networks or whether the quantity-based model’s flexibility and adaptability prove more advantageous in markets with fewer customers or higher demand fluctuations. By systematically analyzing the differential performance of both models across varying customer counts, this section establishes a comprehensive understanding of the preferred model under diverse operational scenarios.

Table 5 builds on this analysis by demonstrating that for a customer base of four, the time-based model consistently outperforms the quantity-based model across all consolidation cycles, achieving higher profitability in every scenario. For instance, at a 5-day consolidation cycle, the time-based model records an average profit of 414,314 KRW, surpassing the quantity-based model’s 357,684 KRW. As the consolidation cycle lengthens, the profitability gap becomes more pronounced; at a 10-day consolidation cycle, the time-based model achieves a profit of 1,128,763 KRW, approximately 27% higher than the quantity-based model’s profit of 887,905 KRW. This consistent performance underscores the time-based model’s capacity to leverage economies of scale and maintain profitability even in small-scale operational scenarios.

Figure 6 further supports these findings by illustrating that the optimal delivery routes for both the time-based and quantity-based models remain identical across all consolidation cycles, as the same geospatial optimization is applied to minimize travel distance. Each route involves departing from the depot, visiting all four customers (C1 to C4), and returning to the depot, ensuring logistical efficiency. This consistency in routing confirms that the performance differences between the models are primarily driven by operational strategies rather than routing efficiency. The time-based model’s fixed-cycle approach minimizes cost variability and promotes consistent, predictable operations, while the quantity-based model, despite utilizing the same routes, incurs elevated operational costs due to frequent adjustments required by dynamic demand fluctuations, which adversely impact its profitability.

The box plots in Figure 6 provide additional evidence by showing that the time-based model maintains a more stable profit distribution with narrower variability compared to the quantity-based model. This stability highlights the time-based model’s effectiveness in managing costs and delivering predictable outcomes, even in scenarios with a small customer base. In such cases, the time-based model emerges as the strategically optimal choice, leveraging its structured approach to maximize economies of scale and reduce operational complexity. Its fixed-cycle delivery strategy minimizes cost variability, ensuring operational stability and reliability. In contrast, while the quantity-based model offers greater flexibility, its demand-driven adjustments result in higher operational costs and diminished profitability, particularly during shorter consolidation cycles. Therefore, managers focusing on small-scale markets should prioritize the time-based model to achieve cost efficiency, predictable performance, and simplified routing strategies, aligning with broader strategic objectives in logistical planning.

Table 5 and Figure 7 highlight the comparative profitability of the time-based and quantity-based models for a customer base of six, demonstrating that the time-based model consistently outperforms the quantity-based model across all consolidation cycles. At a 5-day consolidation cycle, the time-based model achieves an average profit of 699,731 KRW, exceeding the quantity-based model’s 638,375 KRW. As the consolidation cycle lengthens, this profitability gap becomes more pronounced, with the time-based model recording a profit of 1,766,324 KRW at a 10-day consolidation cycle. This is approximately 37% higher than the quantity-based model’s 1,292,699 KRW.

Figure 8 illustrates that the optimal delivery routes for both models remain identical across all consolidation cycles, as both utilize the same geospatial optimization to minimize travel distance. Each route involves departing from the depot, visiting all six customers (C1 to C6), and returning to the depot, emphasizing that performance differences are attributed to the models’ distinct operational strategies rather than routing efficiency. The time-based model’s fixed-cycle approach ensures predictable cost structures and operational stability, whereas the quantity-based model, despite using the same route, incurs higher operational costs due to frequent dynamic adjustments based on fluctuating demand. For a moderate customer base of six, the time-based model emerges as the preferred choice, leveraging economies of scale and its structured approach to minimize variability in costs and maximize profitability. Its fixed-cycle delivery strategy provides stability, making it particularly suitable for logistics operations requiring consistency. In contrast, the quantity-based model, while offering greater flexibility, demonstrates reduced profitability due to the higher operational costs associated with its dynamic adjustments. Managers in markets with moderate customer bases should prioritize the time-based model for its ability to effectively balance cost efficiency and predictability, ensuring sustained profitability and reduced operational complexity.

Table 5 and Figure 9 reveal that for a customer base of ten, the time-based model consistently outperforms the quantity-based model across all consolidation cycles, achieving higher profitability in every scenario. At a 5-day consolidation cycle, the time-based model achieves an average profit of 1,081,505 KRW, surpassing the quantity-based model’s 860,876 KRW. As the consolidation cycle lengthens, the profitability gap becomes more pronounced, with the time-based model recording 2,421,069 KRW at a 10-day cycle, approximately 20% higher than the quantity-based model’s 2,020,624 KRW. These results underscore the ability of the time-based model to manage increasing customer complexity while maintaining profitability, showcasing its operational scalability.

The box plots in Figure 9 further validate the stability of the time-based model’s profit distribution. The time-based model consistently demonstrates a narrower range of variability and higher mean profits across all cycles, emphasizing its ability to provide stable and predictable outcomes, even with a larger customer base. Conversely, the quantity-based model exhibits a wider profit range, reflecting increased sensitivity to dynamic demand fluctuations and frequent operational adjustments as customer numbers grow.

Figure 10 illustrates the optimal delivery routes for both models across consolidation cycles for ten customers. As the customer base expands, routing complexity significantly increases, involving multiple intersections and occasional backtracking to optimize travel distance. Both models apply identical geospatial optimization to minimize inefficiencies, visiting all ten customers before returning to the depot. This consistency in routing efficiency highlights that the performance differences between the models arise from their distinct operational strategies rather than differences in routing optimization. The time-based model’s fixed-cycle approach ensures predictable cost structures, mitigating variability and delivering stable performance even in high-complexity scenarios. In contrast, the quantity-based model’s dynamic adjustments to fluctuating demand lead to higher operational costs, ultimately reducing profitability in such conditions.

From a managerial perspective, the time-based model emerges as the superior choice for larger customer bases, as its structured approach effectively manages complexity while maintaining scalability, profitability, and cost stability. Its ability to minimize operational variability makes it ideal for logistics operations requiring consistent and reliable performance. On the other hand, while the quantity-based model offers flexibility, it becomes increasingly inefficient with larger customer bases due to the higher costs associated with frequent adjustments. Logistics managers should prioritize the time-based model for networks with a larger number of customers, leveraging its scalability and predictable outcomes to optimize profitability and operational efficiency. For smaller, dynamic markets, where responsiveness outweighs cost stability, the quantity-based model may remain a viable option. This analysis emphasizes the critical importance of aligning model selection with customer base size and market characteristics to achieve optimal logistical performance and profitability.

4.5. Observations and Decision-Making

In this study, we analyzed the impact of key sensitivity parameters, specifically price sensitivity (${\delta }_P$) and lead time sensitivity (${\delta }_L$), as well as the effect of varying customer counts on the profitability and cost efficiency of the time-based model and quantity-based models. By comparing the performance of both models across diverse scenarios, we identified their strengths and weaknesses and derived strategic insights that could guide decision-making in logistics and supply chain management.

The sensitivity analysis results demonstrate that as price sensitivity (${\delta }_P$) and lead time sensitivity (${\delta }_L$) increase, both models experience a decline in profitability. However, the time-based model exhibits a more gradual decline in profitability compared to the quantity-based model, indicating greater stability. For example, under high price sensitivity (${\delta }_P$ = 0.3), the quantity-based model’s profitability decreases sharply, whereas the time-based model maintains a relatively consistent profit structure. This suggests that the time-based model’s fixed-cycle approach effectively dampens the impact of demand fluctuations caused by price volatility, allowing it to sustain profitability even in highly uncertain markets. In contrast, the quantity-based model, which is more responsive to demand changes, shows heightened variability in profitability as it adjusts to frequent changes in price and lead time.

As lead time sensitivity increases, the quantity-based model experiences greater fluctuations in route optimization and delivery schedules, which leads to added complexity and rising costs. Conversely, the time-based model leverages its fixed-cycle strategy to maintain predictable deliveries, ensuring a more stable profit structure even when lead time sensitivity is high. These findings indicate that in markets with significant fluctuations, the time-based model’s resilience to price and lead time volatility is advantageous, providing a steady framework for cost control and profitability.

The analysis further explored how varying customer counts—specifically scenarios with 4, 6, and 10 customers—affect the cost efficiency and route optimization of each model. With a small customer base (4 customers), both models achieve relatively high profitability, although the time-based model demonstrates a slight edge. This advantage arises from the time-based model’s ability to maintain balanced demand and predictable routes, allowing effective cost savings even with fewer customers.

With a moderate customer count (6 customers), the difference in profitability and cost efficiency between the models becomes more pronounced. The time-based model continues to excel by maintaining an efficient and predictable route structure, while the quantity-based model incurs additional costs due to its flexible route adjustments to meet changing demand. In high customer count scenarios (10 customers), the time-based model’s fixed-cycle approach demonstrates significant benefits over the quantity-based model, which faces increased complexity in managing route optimization effectively. The quantity-based model’s flexible routing in response to demand fluctuations results in longer travel distances and increased waiting times, leading to higher operational costs. Conversely, the time-based model’s consistent route structure enables substantial cost savings and higher profitability.

The visualizations of route patterns confirm these findings, showing that while the quantity-based model dynamically adjusts routes based on individual customer demands, the time-based model maintains a stable, circular route pattern. This stability in routing allows the time-based model to achieve a predictable cost structure and effective time management, further supporting its cost efficiency and profitability, particularly in scenarios with a larger customer base.

These insights lead to several strategic recommendations for logistics and supply chain managers. In environments with a limited customer base, the quantity-based model’s flexible response to demand changes can achieve efficient route optimization, making it suitable for small-scale operations. For larger customer bases, the time-based model’s fixed-cycle strategy offers greater predictability in routing, cost reductions, and stability, maximizing efficiency in complex delivery settings. In high-volatility markets with significant price and lead time sensitivities, the time-based model’s fixed-cycle approach helps stabilize profitability, providing a robust structure to handle market fluctuations. However, in rapidly changing demand environments where frequent adjustments are required, the quantity-based model’s ability to dynamically respond to each order’s demand variations may prove advantageous.

In conclusion, this study provides practical insights by comparing the effectiveness of the T and quantity-based models under various customer count and market sensitivity scenarios. The findings suggest that the time-based model is generally better suited for high customer volumes and volatile markets, where stability and predictable routing are essential, while the quantity-based model is preferable in environments with smaller customer bases or highly dynamic demand patterns. This comparison offers valuable guidance for selecting the optimal model based on specific operational needs and market conditions.

5. Conclusions

This study systematically evaluated the performance of the time-based and quantity-based models under various supply chain scenarios, focusing on their adaptability to differing market conditions and operational environments. By analyzing key variables such as price sensitivity (${\delta }_P$), lead time sensitivity (${\delta }_L$), and customer count, this research provided a robust comparative assessment of cost efficiency, profitability, and route optimization outcomes. Additionally, sensitivity analyses and experiments with varying customer counts (4, 6, and 10 customers) were conducted to assess the models’ performance across different operational scales and market conditions.

The sensitivity analysis revealed that as price sensitivity (${\delta }_P$) and lead time sensitivity (${\delta }_L$) increased, the time-based model demonstrated greater resilience, maintaining stable profitability compared to the quantity-based model. This stability was particularly evident under high sensitivity conditions, where the fixed delivery cycles of the time-based model mitigated the impact of fluctuating demand, ensuring predictable cost structures. Conversely, the quantity-based model, while agile in responding to demand fluctuations, exhibited significant declines in profitability as operational adjustments became more frequent and costly under increased sensitivity levels.

Experiments with varying customer counts further highlighted the scalability of the time-based model. For smaller customer bases (4 customers), the time-based model leveraged economies of scale and consistent delivery cycles to achieve higher profitability. As customer numbers increased (6 and 10 customers), the model’s structured approach effectively managed the growing complexity of route optimization, maintaining cost efficiency and stable profitability. On the other hand, the quantity-based model struggled with higher operational costs as customer numbers grew, making it less effective in larger-scale scenarios.

From a managerial perspective, these findings provide actionable insights for optimizing supply chain performance by aligning model selection with market dynamics and operational scales. In large-scale networks with stable demand, the time-based model should be prioritized for its ability to enhance cost efficiency, maintain predictable operations, and leverage economies of scale. Furthermore, its contribution to environmental sustainability—achieved through reduced logistics inefficiencies and integrated delivery strategies—positions it as a strategic choice for organizations aiming to align profitability with sustainability objectives. In contrast, the quantity-based model remains suitable for smaller or dynamic markets requiring rapid responsiveness to fluctuating demand, albeit with trade-offs in cost stability.

For expanding supply chains, the time-based model offers a structured framework for managing increasing customer complexity while minimizing variability and ensuring consistent performance across growing networks. These insights emphasize the importance of tailoring supply chain strategies to specific market characteristics, enabling managers to streamline operations, optimize profitability, and build resilience against market volatility. Managers operating in high-sensitivity or large-scale environments should prioritize the time-based model for its scalability and operational robustness. Conversely, the quantity-based model is better suited for smaller-scale or volatile markets, where adaptability and agility outweigh cost predictability.

Despite its contributions, this study acknowledges several limitations that highlight opportunities for future research. While customer demand was dynamically modeled, more complex demand patterns, such as multivariate-dependent or non-linear demand, were not incorporated. This may limit the applicability of the findings to real-world scenarios characterized by unpredictable fluctuations. Additionally, the analysis was constrained to fixed customer counts (4, 6, and 10 buyers), which may not adequately capture the diversity of customer compositions and market environments encountered in practice. Moreover, the study primarily focused on cost efficiency and profitability metrics without explicitly addressing other critical dimensions, such as customer satisfaction or service quality.

Future research should address these limitations by incorporating advanced optimization techniques, such as heuristic and metaheuristic algorithms, to efficiently solve large-scale supply chain problems. Expanding the scope to include diverse customer distributions, geographical constraints, and product categories would improve the generalizability of the findings. Furthermore, integrating customer satisfaction and service quality metrics into the evaluation framework would align supply chain strategies with customer-centric objectives, ensuring a more holistic analysis. These advancements would enhance the practical relevance of this research, providing innovative solutions for modern logistics challenges while fostering alignment between operational efficiency, customer satisfaction, and sustainability goals.

Author Contributions

C.Y. performed data curation, funding acquisition, investigation, methodology, mathematical model development, validation, and the writing of the original manuscript; C.L. contributed to the conceptualization, funding, supervision, and editing of the manuscript. Y.L. contributed to the conceptualization, investigation, project administration, mathematical model development, validation, and writing of the original manuscript. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author/s.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Monthly average demand for dental supplies.

Figure 2. Price distribution of dental inventory.

Figure 3. Distribution of objective function for each model according to consolidation cycle.

Figure 4. Optimal delivery routes demonstrated by each model across consolidation cycles.

Figure 5. The objective function for each model distributed according to the consolidation cycle (4 customers).

Figure 6. Optimal Delivery Routes Demonstrated by Each Model Across Consolidation Cycles (4 customers).

Figure 7. Distribution of objective function for each model according to consolidation cycle (6 customers).

Figure 8. Optimal delivery routes demonstrated by each model across consolidation cycles (6 customers).

Figure 9. Distribution of objective function for each model according to consolidation cycle (10 customers).

Figure 10. Optimal delivery routes demonstrated by each model across consolidation cycles (10 customers).

Table 1

Parameters and decision variables.

Parameters and Decision Variables	Description
C	Set of customers (Including Depot)
C∖{0}	Set of customers (Excluding Depot)
K	Set of vehicles
k	Index for vehicle
$i,j$	Index for buyer
$d_{ij}$	$\mathrm{Distance} \mathrm{between} \mathrm{customer} i$ $\mathrm{and} \mathrm{customer} j$
T	Time-based integrated cycle
$Q_{used}$	Integrated cycle set in the quantity-based model (quantity-based model)
${\lambda }_{0,i}$	$\mathrm{Initial} \mathrm{demand} \mathrm{of} \mathrm{customer} i$
${\lambda }_{t,i}$	$\mathrm{Demand} \mathrm{for} \mathrm{customer} i$ in the time-based cycle
${\lambda }_{q,i}$	$\mathrm{Demand} \mathrm{for} \mathrm{customer} i$ in the quantity-based cycle
$Q_{T,i}$	$\mathrm{Demand} \mathrm{of} \mathrm{customer} i$ during cycle T
$Q_T$	Total demand of customer during cycle T
$Q_{q,i}$	$\mathrm{Demand} \mathrm{of} \mathrm{customer} i$ during cycle Q
$Q_Q$	Total demand of customer during cycle Q
$p_i$	$\mathrm{Product} \mathrm{price} \mathrm{for} \mathrm{customer} i$
${\theta }_i$	$\mathrm{Demand} \mathrm{growth} \mathrm{rate} \mathrm{due} \mathrm{to} \mathrm{preference} \mathrm{for} \mathrm{customer} i$
$q_i$	$\mathrm{Quantity} \mathrm{of} \mathrm{product} \mathrm{delivered} \mathrm{to} \mathrm{customer} i$
${\mu }_i$	$\mathrm{MTZ} \mathrm{variable} \mathrm{representing} \mathrm{visit} \mathrm{order} \mathrm{for} \mathrm{customer} i$
${\delta }_p$	Sensitivity to product price
${\delta }_L$	Sensitivity to lead time
$Q_k$	Maximum transport capacity of vehicle k
v	Vehicle travel speed
$\epsilon$	Permissible tolerance for aligning delivery cycles of time-based model and quantity-based model
${dist}_{max}$	Maximum travel distance for vehicles
$T_{process}$	Fixed order processing time
F	Fixed order processing cost
H	Inventory holding cost
$C_d$	Unit delivery cost
w	Unit cost of customer waiting
$E\left[\Pi \right(T\left)\right]$	Long term profit for the time-based model
$E\left[{TR}_T\right]$	Total profit for the time-based model
$E\left[{TC}_T\right]$	Total cost for the time-based model
$E\left[{IC}_T\right]$	Expected inventory holding cost in the time-based model
$E\left[{HC}_T\right]$	Expected inventory dispatch cost in the time-based model
$E\left[{DC}_T\right]$	Expected inventory delivery cost in the time-based model
$E\left[{WC}_T\right]$	Expected inventory waiting cost in the time-based model
$E\left[\Pi \left(Q\right)\right]$	Long term profit for the quantity-based model
$E\left[{TR}_Q\right]$	Total profit for the quantity-based model
$E\left[{TC}_Q\right]$	Total cost for the quantity-based model
$E\left[{IC}_Q\right]$	Expected inventory holding cost in the quantity-based model
$E\left[{HC}_Q\right]$	Expected inventory dispatch cost in the quantity-based model
$E\left[{DC}_Q\right]$	Expected inventory delivery cost in the quantity-based model
$E\left[{WC}_Q\right]$	Expected inventory waiting cost in the quantity-based model
$x_{i,j,k}$	$\mathrm{Binary} \mathrm{variable} \mathrm{indication} \mathrm{vehicle} \mathrm{movement} \mathrm{from} \mathrm{customer} i$ $\mathrm{to} j$

Table 2

Parameter settings in the numerical experiments.

Parameters	Value/Range
$p_i$	13,000 KRW (Empirical Data)
${\lambda }_{0,i}$	Random (5~8)
${\theta }_i$	Random (0.05~0.1)
${\delta }_p$	[0.1, 0.2, 0.3]
${\delta }_L$	[0.2, 0.3, 0.4]
$T$	[5, 6, 7, 8, 9, 10] day
$v$	60 km/h
$Q_k$	400
${dist}_{max}$	600 km
$T_{process}$	0.5 day
$F$	[50,000, 100,000, 150,000] KRW
$H$	[5,000, 10,000, 15,000] KRW
$C_d$	[100, 200, 300] KRW
$w$	[30, 40, 50] KRW

Table 3

The maximized profit at a given consolidation cycle.

Consolidation Cycle	Time-Based Model	Quantity-Based Model	Optimal Model
5	995,287	734,349	Time-based model
6	1,228,166	927,887	Time-based model
7	1,480,687	1,090,199	Time-based model
8	1,782,087	1,235,946	Time-based model
9	1,972,155	1,452,392	Time-based model
10	2,373,131	1,577,617	Time-based model

Table 4

Sensitivity analysis results for T and quantity-based models by consolidation cycle (average profit in KRW).

ConsolidationCycle	Price Sensitivity${\mathit{\delta }}_{\mathit{P}}$	Lead TimeSensitivity${\mathit{\delta }}_{\mathit{L}}$	Time-Based Model Profit	Quantity-Based Model Profit	Optimal Model
5	0.1	0.2	1,468,632	1,167,747	Time-based model
	0.1	0.3	1,383,569	1,113,872	Time-based model
	0.1	0.4	1,220,222	1,149,387	Time-based model
	0.2	0.2	1,105,814	747,987	Time-based model
	0.2	0.3	971,224	766,452	Time-based model
	0.2	0.4	827,275	746,919	Time-based model
	0.3	0.2	695,602	371,101	Time-based model
	0.3	0.3	569,468.	377,038	Time-based model
	0.3	0.4	295,785	398,941	Quantity-based model
6	0.1	0.2	1,856,363	1,388,619	Time-based model
	0.1	0.3	1,710,499	1,387,216	Time-based model
	0.1	0.4	1,504,005	1,402,794	Time-based model
	0.2	0.2	1,431,566	932,174	Time-based model
	0.2	0.3	1,161,809	924,989	Time-based model
	0.2	0.4	1,079,020	913,700	Time-based model
	0.3	0.2	932,522	471,524	Time-based model
	0.3	0.3	751,082	472,952	Time-based model
	0.3	0.4	265,571	479,289	Quantity-based model
7	0.1	0.2	2,314,449	1,630,362	Time-based model
	0.1	0.3	2,063,042	1,590,348	Time-based model
	0.1	0.4	1,785,279	1,597,133	Time-based model
	0.2	0.2	1,739,510	1,095,233	Time-based model
	0.2	0.3	1,466,044	1,094,391	Time-based model
	0.2	0.4	1,278,618	1,066,949	Time-based model
	0.3	0.2	1,192,158	589,211	Time-based model
	0.3	0.3	877,319	558,472	Time-based model
	0.3	0.4	364,961	577,613	Quantity-based model
8	0.1	0.2	2,703,205	1,875,054	Time-based model
	0.1	0.3	2,337,812	1,862,678	Time-based model
	0.1	0.4	2,034,267	1,867,548	Time-based model
	0.2	0.2	2,094,386	1,249,122	Time-based model
	0.2	0.3	1,715,505	1,262,719	Time-based model
	0.2	0.4	1,402,041	1,279,788	Time-based model
	0.3	0.2	1,483,057	604,466	Time-based model
	0.3	0.3	1,090,839	619,780	Time-based model
	0.3	0.4	447,343	606,549	Quantity-based model
9	0.1	0.2	3,198,219	2,075,533	Time-based model
	0.1	0.3	2,763,826	2,103,471	Time-based model
	0.1	0.4	2,463,451	2,056,291	Time-based model
	0.2	0.2	2,466,440	1,448,146	Time-based model
	0.2	0.3	2,034,711	1,430,963	Time-based model
	0.2	0.4	1,791,332	1,355,818	Time-based model
	0.3	0.2	1,719,139	689,743	Time-based model
	0.3	0.3	1,241,670	711,252	Time-based model
	0.3	0.4	606,105	725,080	Quantity-based model
10	0.1	0.2	3,679,687	2,326,707	Time-based model
	0.1	0.3	3,161,790	2,348,265	Time-based model
	0.1	0.4	2,700,088	2,305,240	Time-based model
	0.2	0.2	2,823,009	1,594,306	Time-based model
	0.2	0.3	2,351,947	1,572,971	Time-based model
	0.2	0.4	1,874,859	1,597,827	Time-based model
	0.3	0.2	2,073,701	814,164	Time-based model
	0.3	0.3	1,445,218	783,167	Time-based model
	0.3	0.4	657,888	776,064	Quantity-based model

Table 5

Optimized profit across consolidation cycles for different customer scenarios.

Number of Buyer	Consolidation Cycle	Time-Based Model	Quantity-Based Model	Optimal Model
4	5	414,314	357,684	Time-based model
	6	537,491	446,988	Time-based model
	7	710,791	505,695	Time-based model
	8	849,709	604,161	Time-based model
	9	990,447	718,610	Time-based model
	10	1,128,763	887,905	Time-based model
6	5	699,731	638,375	Time-based model
	6	815,812	797,034	Time-based model
	7	1,000,427	804,245	Time-based model
	8	1,216,009	965,818	Time-based model
	9	1,450,308	1,131,575	Time-based model
	10	1,766,324	1,292,699	Time-based model
10	5	1,081,505	860,876	Time-based model
	6	1,460,760	1,127,467	Time-based model
	7	1,773,298	1,149,688	Time-based model
	8	1,975,233	1,652,826	Time-based model
	9	2,391,455	1,878,409	Time-based model
	10	2,421,069	2,020,624	Time-based model

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