An Analysis of the Impact of Service Consistency on the Vehicle Routing Problem with Time Windows

Abstract

This paper proposes a mixed-integer linear programming (MILP) model for the consistent vehicle routing problem with time windows (ConVRPTW), motivated by the need to enhance customer satisfaction in the last-mile logistics industry. The problem involves designing a set of consistent routes for a group of customers whose demand fluctuates from one period to another within a specified planning horizon. The objective is to minimize the cost associated with the number of vehicles used and the total travel time while satisfying service consistency constraints. A tabu search (TS) metaheuristic is implemented to solve the proposed model, and its performance is compared with the optimal solutions. Instances of 10 and 20 customers are designed based on the well-known Solomon instances for VRPTW. Finally, the solutions obtained from the ConVRPTW model are compared with those of a traditional VRPTW model using some key performance indicators (KPIs) to measure the contribution of service consistency to route design.

1. Introduction

This paper addresses the vehicle routing problem with service consistency and time windows (ConVRPTW). This variant of the classic vehicle routing problem (VRP) aims to incorporate service consistency into designing a multi-period routing plan, thereby improving customer satisfaction. The problem is characterized by customers with known demands that must be served by a fleet of vehicles with homogeneous capacity. A solution of the ConVRPTW must ensure that each customer is served by the same driver throughout the planning horizon. Furthermore, each customer defines a fixed time window for service, maintaining this range constant throughout the planning horizon. Therefore, maintaining service consistency is a challenging task because customer demand varies from one period to another.

Service consistency is a highly valued characteristic for customers, consisting of providing a service by the same familiar driver at roughly the same time each day [1]. Service consistency can be described in three dimensions: (i) the driver, i.e., ensuring that the same driver always serves a customer; (ii) time, ensuring that a customer is served in similar time ranges throughout a planning horizon; (iii) quantity, ensuring stable inventory levels, addressing variability in demand and minimizing stockouts. In particular, this paper considers the driver and time dimensions, and we propose a set of key performance indicators (KPIs) to evaluate the impact of service consistency on route design in the context of last-mile logistics.

KPIs help monitor and understand a system’s performance, enabling the assessment of whether it is operating within its desired parameters. They are used to support strategic, tactical, and operational decision-making. In distribution logistics, it is essential to identify the key elements to measure in order to define KPIs for effective route planning. These elements are directly related to vehicle performance in a route plan, where emphasis is placed on the duration of the delivery system, resource utilization, or the number of customers served.

The contributions of this study are threefold. First, a mixed-integer linear programming (MILP) formulation is proposed for ConVRPTW, with the objective of simultaneously minimizing the vehicle fleet size and total travel time. This approach is essential in last-mile logistics, since fleet reduction has a direct impact on fixed operating costs, while reducing travel time impacts operational efficiency and service quality. The combination of both objectives enables strategic and operational decisions in designing a routing plan. Secondly, a tabu search metaheuristic is developed to solve the proposed model, capable of reaching optimal solutions for instances with ten and twenty customers within a five-day planning horizon. Finally, a set of KPIs is proposed to evaluate the operational performance of the solutions obtained through the ConVRPTW model, which is compared with the solutions of a standard vehicle routing model with time windows (VRPTW). The solutions are evaluated in a scenario with and without service consistency. In last-mile logistics, these indicators are essential for measuring the operational performance of a routing plan. Most studies only report indicators associated with a cost-based objective function, limiting a more detailed analysis of the solution’s performance.

This paper is organized as follows. Section 2 presents a literature review on service consistency in last-mile logistics. Section 3 introduces the proposed mathematical model for ConVRPTW. Section 4 describes the proposed KPIs and their importance in evaluating the operational performance of a routing plan. Section 5 describes the tabu search (TS) algorithm for solving the proposed model. Section 6 discusses the computational experiments performed and the performance analysis of the obtained solutions. Finally, Section 7 presents the main conclusions.

2. Literature Review

Much of the work on the vehicle routing problem focuses on minimizing total route costs, often based on the total distance traveled or total travel time [2]. An approach based only on cost has the advantage of generating an economically efficient solution for companies. However, key aspects such as customer and driver satisfaction are also relevant [3]. Service consistency increases customer satisfaction and driver productivity [1]. By assigning the same driver to a set of customers, drivers gain knowledge about the route and delivery area, enabling better management of potential setbacks and reducing the risk of delivery delays. Additionally, customers and drivers can establish more effective delivery schedules and protocols, thereby improving service quality. Traditionally, two dimensions of consistency have been studied in the literature: driver consistency and time consistency.

Driver consistency is characterized by assigning a single driver to serve the same group of customers over a specific time horizon. The advantage of this dimension lies in the driver’s knowledge of the assigned route and possible parking areas, streamlining the delivery system and making it more productive. Furthermore, the customer and driver can establish delivery protocols, which increases customer satisfaction. Cases in which this dimension has been addressed include [2,4,5,6,7,8,9,10,11,12,13,14,15].

Time consistency seeks to serve customers within similar time ranges over a planning horizon. The advantage of this dimension is that customers know when their orders will arrive, allowing them to coordinate with a recipient, streamline the delivery system, and reduce waiting times for both drivers and customers. Cases in which this dimension has been addressed are [5,7,9,10,11,14,15,16]. In the literature, this dimension has also been addressed, considering the time windows defined by customers. Applications of time consistency through the formulation of time windows can be seen in [2,4,6,12,13,17].

In small instances, the use of commercial solvers such as CPLEX and Gurobi has traditionally been used to solve the ConVRP [4,5,7,9,10,12,13,15,17]. To solve medium-sized instances, strategies such as branch-and-cut algorithms [2,18], column generation algorithms [9], and e-constraint algorithms [6] have been proposed. These algorithms share the commonality that, although they guarantee optimality, they require a high computational time, especially in applied study cases, where the search for solutions using these methods becomes impractical.

Metaheuristic algorithms are widely used to solve the ConVRP in large instances, motivated by real-world cases. These algorithms do not guarantee optimality, but they enable the attainment of high-quality solutions in considerably less time than exact algorithms. In the literature, some metaheuristic algorithms have been applied, such as the large neighborhood search (LNS) [2,8,10], adaptive large neighborhood search (ALNS) [5,9,19], multi-directional local search (MDLS) [11], multi-directional local neighborhood search (MDLNS) [6], ad-hoc algorithm [12], multiperiod route elimination heuristic (MREH) algorithm [4], tabu search (TS) [16], adaptive tabu search (ATS) [13,15], variable neighborhood search (VNS) [14], and record-to-record (RTR) algorithm [7,15].

Finally, it is worth noting that the works described above do not provide an in-depth examination of the operational performance of the reported solutions.

Table 1. KPIs proposed in the literature for the analysis of a routing plan.

Paper	Number of Vehicles	Travel Time	Routing Cost	Travel Distance	Others Indicators
[2]			✔
[17]	✔	✔		✔
[8]		✔
[9]	✔	✔			Waiting time
[7]	✔	✔			Customers receiving consistent service
[5]	✔	✔			Difference in arrival times
[1]	✔	✔			Number of customers served–Savings time
[10]	✔	✔
[6]	✔	✔			Maximum difference in arrival times
[4]	✔	✔			Number of expelled customers–Active clients per day
[11]	✔	✔
[12]	✔	✔	✔		Total demand by route
[18]	✔		✔
[13]	✔		✔		Benefit
[16]	✔	✔	✔	✔
[14]		✔	✔
[15]	✔	✔	✔
Our study	✔	✔			Waiting time–Capacity used per vehicle–Coefficient of variation in route length, vehicle capacity, and customers served per vehicle

summarizes the main indicators used in the literature to evaluate the performance of the reported solutions. In particular, the most commonly used performance indicators are the number of vehicles and the total travel time. Other performance indicators are the cost and total distance traveled. These indicators are directly related to the economic performance of a solution. Thus, it is observed that most of the work is characterized by obtaining an economically profitable but potentially inefficient routing plan in operational aspects. On the other hand, some works propose indicators that help understand how a routing plan behaves in terms of workload, such as the number of customers to be served [4,5], the maximum difference in arrival times [6], the number of customers receiving consistent service [7], the number of orders [7], the waiting time [9], and the amount of demand satisfied per route [12]. Therefore, it is evident that a more detailed study of the operational performance of a routing plan makes a significant contribution to the literature, facilitating a more precise understanding of the operational impact of considering service consistency.

3. Mathematical Model

This section introduces a mixed integer linear programming (MILP) formulation for the ConVRPTW.

Table 2. ConVRTPW notation.

Sets
N	Set of nodes: customers and depot
A	Set of arcs
K	Set of vehicles
D	Set of days in the planning horizon
Parameters
$t_{ij}$	Travel time in the arc $(i,j)\in A$
T	Working day hours
$q_{id}$	Customer demand i on day d, $i\in N\setminus \left\{0\right\},d\in D$
Q	Maximum capacity of each vehicle
$w_{id}$	1 if customer i requires service on day d, 0 otherwise, $i\in N\setminus \left\{0\right\},d\in D$
$s_i$	Customer service time i, $i\in N\setminus \left\{0\right\}$
$l_i$	Low time window to serve the customer i, $i\in N\setminus \left\{0\right\}$
$u_i$	Upper time window to serve the customer i, $i\in N\setminus \left\{0\right\}$
T	Maximum travel time for each vehicle
Decision Variables
$x_{ijkd}$	1 if vehicle k uses arc $(i,j)$ on day d, 0 otherwise, $(i,j)\in A,k\in K,d\in D$
$y_{ikd}$	1 if customer i is assigned to vehicle k on day d, 0 otherwise, $i\in N\setminus \left\{0\right\},k\in K,d\in D$
$z_k$	1 if vehicle k is assigned on at least one day within the planning horizon, 0 otherwise, $k\in K$
$v_{kd}$	1 if vehicle k is assigned on day d, 0 otherwise, $k\in K,d\in D$
$a_{id}$	Arrival time to customer i on the day d, $i\in N\setminus \left\{0\right\},d\in D$

summarizes the parameters and decision variables described. Let $N=\{0,1,\dots ,\widehat{n}\}$ denote the set of nodes, where node $\left\{0\right\}$ represents the depot, and nodes $\{1,\dots ,\widehat{n}\}$ represent the customers. The parameter $\widehat{n}$ specifies the total number of customers. The set $A=\left\{\right(i,j)\mid i,j\in N,i\ne j\}$ describes the arcs. The set $D=\{1,\dots ,\widehat{d}\}$ defines the days in the planning horizon, where $\widehat{d}$ is the total number of days. The fleet consists of vehicles with homogeneous capacity Q. These vehicles are represented by $K=\{1,\dots ,\widehat{k}\}$, where $\widehat{k}$ is the total number of vehicles. Each vehicle $k\in K$ can perform at most one trip per day $d\in D$, and each driver is exclusively assigned to a single vehicle.

The parameters $t_{ij}$ represent the travel time to cross the arc $(i,j)\in A$. $q_{id}$ denotes the delivery demand of customer $i\in N\setminus \left\{0\right\}$ on a day $d\in D$, such that $q_{id}\le Q$$\forall i\in N,d\in D$. The matrix $w_{id}$ indicates whether a customer requires attention on any day within the planning horizon, being equal to 1 if customer $i\in N\setminus \left\{0\right\}$ requires attention on day $d\in D$, and 0 otherwise. To incorporate consistency over time into the mathematical model, restrictions associated with time windows are used. In this context, the service time parameter $s_i$ is defined for each customer $i\in N\setminus \left\{0\right\}$. The parameters $l_i$ and $u_i$ represent the lower and upper time limits for delivery start time, respectively, which are the time windows of the customer $i\in N$. T represents the working day’s hours, that is to say, the maximum time each vehicle has to make all deliveries and return to the depot.

The ConVRPTW model uses the following decision variables. The binary variables $x_{ijkd}$ take the value 1 if vehicle $k\in K$ uses arc $(i,j)\in A$ on day $d\in D$, and 0 otherwise. The binary variables $y_{ikd}$ take the value 1 if customer $i\in N\setminus \left\{0\right\}$ is served by vehicle $k\in K$ on day $d\in D$, and 0 otherwise. The binary variables $v_{kd}$ take the value 1 if vehicle $k\in K$ is used on the day $d\in D$ and 0 otherwise. The binary variables $z_k$ take the value 1 if vehicle $k\in K$ is used on at least one day within the planning horizon, and 0 otherwise. The continuous variables $a_{id}$ represent the arrival time to start service for customer $i\in N$ on the day $d\in D$.

$$\begin{array}{cc}\hfill & Min\overline{c}\sum _{k\in K}{z}_k+\sum _{d\in D}\sum _{k\in K}\sum _{(i,j)\in A}{t}_{ij}{x}_{ijkd},\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill \mathit{s}.\mathit{t}:& \sum _{i\in N/i\ne j}{x}_{ijkd}=\sum _{i\in N/i\ne j}{x}_{jikd}\hfill & \hfill & \forall j\in N\setminus \left\{0\right\},k\in K,d\in D,\hfill \end{array}$$

(2)

$$\begin{array}{cccc}\hfill & \sum _{i\in N/i\ne j}{x}_{jikd}=y_{jkd}\hfill & \hfill & \forall j\in N\setminus \left\{0\right\},k\in K,d\in D,\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill & \sum _{j\in N\setminus \left\{0\right\}}{x}_{0jkd}=\sum _{i\in N\setminus \left\{0\right\}}{x}_{i0kd}\hfill & \hfill & \forall k\in K,d\in D,\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill & \sum _{i\in N\setminus \left\{0\right\}}{x}_{i0kd}=v_{kd}\hfill & \hfill & \forall k\in K,d\in D,\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill & \sum _{i\in N\setminus \left\{0\right\}}{q}_{id}{y}_{ikd}\le Q\hfill & \hfill & \forall k\in K,d\in D,\hfill \end{array}$$

(6)

$$\begin{array}{cccc}\hfill & v_{kd}\ge y_{ikd}\hfill & \hfill & \forall i\in N\setminus \left\{0\right\},k\in K,d\in D,\hfill \end{array}$$

(7)

$$\begin{array}{cccc}\hfill & z_k\ge v_{kd}\hfill & \hfill & \forall k\in K,d\in D,\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill & \sum _{k\in K}{y}_{ikd}=w_{id}\hfill & \hfill & \forall i\in N\setminus \left\{0\right\},d\in D,\hfill \end{array}$$

(9)

$$\begin{array}{cccc}\hfill & w_{i\alpha }+w_{i\beta }-2\le y_{ik\alpha }-y_{ik\beta }\hfill & \hfill & \forall i\in N\setminus \left\{0\right\},k\in K,\alpha ,\beta \in D,\alpha \ne \beta ,\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill & a_{id}+x_{ijkd}(s_i+t_{ij})-(1-x_{ijkd})T\le a_{jd}\hfill & \hfill & \forall i\in N,j\in N\setminus \left\{0\right\},k\in K,d\in D,i\ne j,\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill & a_{id}+s_i+t_{i0}\le T\hfill & \hfill & \forall i\in N\setminus \left\{0\right\},d\in D,\hfill \end{array}$$

(12)

$$\begin{array}{cccc}\hfill & l_i{w}_{id}\le a_{id}\le u_i{w}_{id}\hfill & \hfill & \forall i\in N,d\in D,\hfill \end{array}$$

(13)

$$\begin{array}{cccc}\hfill & x_{ijkd}\in \left\{0,1\right\}\hfill & \hfill & \forall (i,j)\in A,k\in K,d\in D,\hfill \end{array}$$

(14)

$$\begin{array}{cccc}\hfill & y_{ikd}\in \left\{0,1\right\}\hfill & \hfill & \forall i\in N\setminus \left\{0\right\},k\in K,d\in D,\hfill \end{array}$$

(15)

$$\begin{array}{cccc}\hfill & z_k\in \left\{0,1\right\}\hfill & \hfill & \forall k\in K,\hfill \end{array}$$

(16)

$$\begin{array}{cccc}\hfill & a_{id}\ge 0\hfill & \hfill & \forall i\in N,d\in D.\hfill \end{array}$$

(17)

The objective function (1) minimizes the number of vehicles and the total travel time, where the parameter $\overline{c}$ represents a weight, which is large enough to give greater importance to the minimization of the number of vehicles, since the size of the fleet has a significant impact on the operating costs of a routing plan. Equations (2)–(5) are typical vehicle flow restrictions. Equation (6) ensures that vehicle capacity is not exceeded. Equation (7) assigns the vehicles to be used daily. Equation (8) defines the global vehicle activation in the planning horizon; that is, it accounts for the maximum number of vehicles used daily. Equation (9) guarantees that each customer receives service on the required day. Equation (10) is the constraint that integrates driver consistency, where if customer i requires service on days $\alpha$ and $\beta$, then $w_{i,\alpha }+w_{i,\beta }-2=0$, guaranteeing driver consistency by fixing the assignment of a customer in the same vehicle throughout the planning horizon; that is, $y_{ik\alpha }=y_{ik\beta }$. Equations (11) and (12) represent the arrival time constraints for serving each customer, where, if $x_{ijkd}=1$, the arrival time at customer $j\in N\setminus \left\{0\right\}$ on day $d\in D$ is calculated as $a_{jd}\ge a_{id}+s_i+t_{ij}$, considering the arrival time at the previous customer, the service time, and the travel time to customer j. On the other hand, if $x_{ijkd}=0$, this implies that there is no direct connection between customers i and j on day d, and the arrival time at customer j is set as $a_{jd}\ge -T$, where T represents a sufficiently large value that invalidates this restriction for such a case. Equation (13) represents the time window constraints. Finally, Equations (14)–(17) guarantee the integrality and non-negativity constraints of the decision variables, respectively.

4. Key Performance Indicators (KPIs) for the ConVRPTW

The KPIs address the need to evaluate the performance of a routing plan in a real-world environment, where operational efficiency and load balance are key factors for the sustainability and effectiveness of the delivery system. In practice, it is essential not only to minimize the costs, distance, time, or number of vehicles but also to equitably distribute the workload between vehicles and drivers, avoiding resource misallocation. A set of KPIs is proposed to measure the operational performance of the solutions obtained using the ConVRPTW model.

4.1. Daily Waiting Time per Vehicle

The daily wait time per vehicle ($W{T}_{kd}$) measures the total daily time each vehicle must wait before visiting a customer. These waits occur when a vehicle arrives to serve customer i before its lower time window $l_i$ and must wait to begin service. This indicator helps quantify idle time, defined as follows:

$$\begin{array}{ccc}W{T}_{kd}=\sum _{i\in N}max(a_{id}-l_i,0)\ast y_{ikd}\hfill & \hfill & \hfill \forall k\in K,d\in D.\end{array}$$

(18)

The $W{T}_{kd}$ can identify inefficiencies in route planning, such as long periods of inactivity, which can indicate opportunities to improve visit sequencing and reduce downtime. Furthermore, when combined with each vehicle’s operating time, this indicator can determine the total length of the workday. In this sense, two vehicles may have similar operating times but significant differences in their waiting times, which can result in unequal workdays for drivers. This workload inequity affects system efficiency, creates scheduling difficulties, and impacts worker satisfaction and well-being.

4.2. Daily Capacity Used per Vehicle

The daily capacity utilization per vehicle ($C{U}_{kd}$) measures the load used by each vehicle on each day of operation. This performance indicator considers the capacity used on each vehicle to meet the demand of assigned customers over a time horizon. This indicator enables the determination of each vehicle’s idle capacity and the identification of any load imbalances between vehicles. It is defined as follows:

$$\begin{array}{ccc}C{U}_{kd}={\displaystyle \frac{{\sum }_{i\in N}{q}_{id}\ast y_{ikd}}{Q}}\hfill & \hfill & \hfill \forall k\in K,d\in D.\end{array}$$

(19)

The $C{U}_{kd}$ offers opportunities for enhancing resource allocation in route planning. Although a classical mathematical model guarantees feasible solutions regarding vehicle capacity, it is possible to strategically redistribute this capacity to improve operational efficiency. For example, operating some vehicles at maximum capacity constantly is not always ideal, as disruptive events such as accidents or thefts could result in significant resource losses. This would require reallocating deliveries, increasing delivery times, causing missed customer deadlines, and even leaving orders unfulfilled due to a lack of available capacity. Furthermore, a vehicle operating at maximum capacity can imply greater logistical and physical effort for the driver, which can affect performance and lead to unfair workload distribution.

4.3. Coefficient of Variation in Route Length

The coefficient of variation in route length ($CVR{L}_d$) measures the percentage variation in the length of all routes on a particular day. This indicator helps us to understand the extent of the difference between the lengths of the routes. The higher the value of this indicator, the greater the imbalance in route length; that is, at least one route is much longer than the rest. It is defined as follows:

• $ML{R}_d$ is the average length of routes per day;
• $VL{R}_d$ is the variance in route length per day.

$$\begin{array}{ccc}ML{R}_d={\displaystyle \frac{{\sum }_{k\in K}{\sum }_{(i,j)\in A}{t}_{ij}\ast x_{ijkd}}{{\sum }_{k\in K}{v}_{kd}}}\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(20)

$$\begin{array}{ccc}VL{R}_d=\sum _{k\in K}{\left[\sum _{(i,j)\in A}{t}_{ij}\ast x_{ijkd}-ML{R}_d\right]}^2\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(21)

$$\begin{array}{ccc}CVR{L}_d={\displaystyle \frac{\sqrt{\frac{VL{R}_d}{{\sum }_{k\in K}{v}_{kd}}}}{ML{R}_d}}\hfill & \hfill & \hfill \forall d\in D.\end{array}$$

(22)

The $CVL{R}_d$ enables the assessment of the fairness of operational load distribution based on drivers’ work hours. A significant imbalance in route length can lead to inequalities in work hours, which in turn affect drivers’ performance, fatigue, and overall well-being. This can lead to internal conflicts, team discontent, logistical mismatches, and inefficient resource allocation.

4.4. Coefficient of Variation in Vehicle Capacity

The coefficient of variation in vehicle capacity ($CVV{C}_d$) measures the percentage change in the demand met by the vehicles on a particular day. This indicator helps understand the difference between the quantity of products each vehicle supplies. The higher the value of this indicator, the greater the imbalance in each vehicle’s load, with some vehicles responsible for transporting a greater quantity of products than others. It is defined as follows:

• $MCa{p}_d$ is the average capacity used per day;
• $VCa{p}_d$ is the variance in capacity used per day.

$$\begin{array}{ccc}MCa{p}_d={\displaystyle \frac{{\sum }_{i\in N}{q}_{id}}{{\sum }_{k\in K}{v}_{kd}}}\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(23)

$$\begin{array}{ccc}VCa{p}_d=\sum _{k\in K}{\left[\sum _{i\in N}{q}_{id}\ast y_{ikd}-MCa{p}_d\right]}^2\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(24)

$$\begin{array}{ccc}CVV{C}_d={\displaystyle \frac{\sqrt{\frac{VCa{p}_d}{{\sum }_{k\in K}{v}_{kd}}}}{MCa{p}_d}}\hfill & \hfill & \hfill \forall d\in D.\end{array}$$

(25)

The $CVV{C}_d$ measures the fairness of operational load distribution based on the capacity used of the vehicle fleet. Unlike the $C{U}_{kd}$ indicator, this coefficient measures the variability in load allocation among all vehicles in the fleet, providing a broader view of the balance in resource utilization. A disproportionate distribution can lead to the overloading of certain vehicles, increasing the risk of unforeseen events typical in a delivery system, such as last-minute changes in demand, where a customer requests a new batch of products, or unexpected events along the route, including breakdowns, collisions, or vehicle theft. Maintaining an appropriate balance in capacity used is crucial for ensuring operational flexibility, mitigating risks, and enhancing efficiency in responding to contingencies.

4.5. Coefficient of Variation in Customers per Vehicle

The coefficient of variation in customers per vehicle ($CVC{V}_d$) measures the percentage variation in the number of customers served by vehicles on a particular day. This indicator helps to understand the difference in the number of customers each vehicle serves. The higher the value of this indicator, the greater the imbalance in the number of customers served per vehicle. It is defined as follows:

• $MCu{s}_d$ is the average number of customers served per day;
• $VCu{s}_d$ is the variance in customers served per day.

$$\begin{array}{ccc}MCu{s}_d={\displaystyle \frac{{\sum }_{k\in K}{\sum }_{i\in N}{y}_{ikd}}{{\sum }_{k\in K}{v}_{kd}}}\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(26)

$$\begin{array}{ccc}VCu{s}_d=\sum _{k\in K}{\left[\sum _{i\in N}{y}_{ikd}-MCu{s}_d\right]}^2\hfill & \hfill & \hfill \forall d\in D,\end{array}$$

(27)

$$\begin{array}{ccc}CVC{V}_d={\displaystyle \frac{\sqrt{\frac{VCu{s}_d}{{\sum }_{k\in K}{v}_{kd}}}}{MCu{s}_d}}\hfill & \hfill & \hfill \forall d\in D.\end{array}$$

(28)

The $CVC{V}_d$ measures the fairness of operational load distribution based on the number of customers assigned to each vehicle. An unequal distribution of customers served per vehicle can lead to operational overloads. From a resource distribution perspective, a balanced allocation of customers within the fleet supports dynamic allocation strategies. This is especially helpful in contexts where demand is sporadic, as last-minute orders may arise. Balanced allocation enables more efficient load redistribution and reduces operational overloads. As a result, it improves the capacity to respond to disruptive events.

The indicators described above help assess the impact of planning on operational stability. They ensure that implemented solutions are not only efficient in terms of costs and travel times but also promote equity in the distribution of effort among resources. This equity is crucial for ensuring long-term sustainability and service quality.

It is essential to note that these indicators are supplementary tools to support decision-making, rather than providing a definitive response. The relevance and use of each indicator depend on the specific objectives and requirements of each delivery system. These indicators may also conflict, meaning some may be more relevant than others in certain contexts. For example, there is a conflict between the coefficient of variation in customers per vehicle ($CVC{V}_d$) and the coefficient of variation in vehicle capacity ($CVV{C}_d$). An equitable customer assignment based on $CVC{V}_d$ can create imbalances in vehicle loads, affecting $CVV{C}_d$, especially when some customers have higher demands. Assigning based on $CVC{V}_d$ or $CVV{C}_d$ can worsen the $CVR{L}_d$ or $WT$ indicator. When optimizing a routing plan, it is crucial to evaluate operational priorities and find a balance between these indicators. This ensures efficient and equitable performance of the delivery system.

It is worth emphasizing that the proposed KPIs are not directly integrated into the optimization model but are instead applied ex post to evaluate and compare the solutions obtained. This design choice enables fair benchmarking between the ConVRPTW and the classical VRPTW, as incorporating service consistency or workload balance measures directly into the optimization would alter the objective function and compromise comparability. Therefore, the indicators were carefully selected to capture dimensions that are not traditionally measured in the literature, such as workload fairness, idle times, and equitable capacity utilization, providing a broader and more transparent assessment of the operational implications of enforcing service consistency. This broader assessment lays the foundation for rethinking how we evaluate routing strategies beyond traditional metrics.

In summary, the proposed KPIs broaden the traditional evaluation of ConVRPTW solutions beyond cost and efficiency criteria by incorporating equity and sustainability as fundamental dimensions of operational performance. While previous studies have mainly focused on minimizing fleet size, distance, or travel time, less attention has been given to fairness in workload distribution and the long-term sustainability of delivery systems. By explicitly addressing these aspects, the proposed indicators provide a more comprehensive framework for assessing routing plans, supporting solutions that are not only efficient but also equitable and sustainable.

5. Solution Framework

A tabu search (TS) algorithm is proposed to solve the ConVRPTW. This metaheuristic was proposed in [20] and has been successfully applied to a wide variety of combinatorial optimization problems [21]. The TS algorithm iteratively improves the quality of an initial solution by exploring different neighborhood structures. Unlike other techniques such as GRASP [22], VNS [23], and LNS [24], TS allows a solution to be temporarily worsened in order to explore different neighborhood spaces, thereby escaping local optima.

A key element of TS is the tabu list, which stores changes made to the solution to avoid cycles and facilitate the exploration of new search neighborhoods. The size of this list, known as tabu tenure, defines the number of stored moves. This list stores the most recent moves made by the algorithm, which are categorized as tabu and are disabled. However, an aspiration criterion allows tabu moves to be made as long as this move represents an improvement on the current solution.

The design of the algorithm to solve the ConVRPTW is described in detail below, considering (i) the algorithm for constructing an initial solution and (ii) the tabu search algorithm.

5.1. Initial Solution

To construct a feasible initial solution, a constructive heuristic based on the sequential insertion procedure is proposed. Sequential insertion takes as input a set of nodes N, a set D of days in the planning horizon, and a customer insertion prioritization criterion. We use the earliest time (ET) criterion, which consists of ordering customer insertion according to the closest lower time window $l_i,\forall i\in N\setminus \left\{0\right\}$. The output of this heuristic is a set of feasible routes S.

The algorithm begins by organizing the set of customers $N\setminus \left\{0\right\}$ into a list V according to the prioritization criterion $ET$. The heuristic initializes a route k and sequentially inserts customers according to the order of the list V. The process continues until all customers are assigned to some route. The feasibility of inserting a customer at the end of the current route k is evaluated at each iteration. If the insertion is feasible according to the ConVRPTW constraints, the customer is included in the solution S and removed from the list V. Otherwise, the next customer on the list is evaluated. If, after traversing the entire list V, there are still unassigned clients, the current route k is closed, and a new route $k+1$ is opened, repeating until all clients are inserted into a route. This heuristic is presented in Algorithm 1.

Algorithm 1: Sequential insertion

5.2. Tabu Search Algorithm

The tabu search is described in Algorithm 2. It receives as input a set of nodes N, a set D of days in the planning horizon, and the prioritization criterion for customer insertion ET. The output of the TS is a set of improved feasible routes, denoted as $S^{\ast }$.

Algorithm 2: Tabu search algorithm

The TS algorithm requires an initial feasible solution generated using Algorithm 1 described in the previous subsection. The TS iteratively seeks to improve its quality from this initial solution until it meets the stopping criterion, which is defined as the maximum computation time. It utilizes classical local operators to generate the $\nu$-neighborhood structures during the search process. We consider inter-route local operators: Relocate, Exchange, 2-Exchange, and intra-route local operators: Relocate, 2-opt, OR-opt. These operators have been adapted to ConVRPTW in order to satisfy capacity constraints (4), time windows constraints (11), and driver consistency constraints (8) in a multi-period approach. For more details on how these operators work, see [25].

An intra-route intensification procedure was considered in the design of the algorithm, which seeks to improve the quality of the routes in the solution $\overline{S}$ by the sequential application of the intra-route operators: Relocate, 2-opt, and OR-opt. These operators are applied iteratively until no improvement is possible.

A refactibilization procedure is designed using the inter-route operators Relocate and Exchange. Unlike their standard functionality, these operators do not discard solutions initially identified as infeasible. Instead, a refactibilization subroutine is incorporated to modify these solutions and transform them into feasible ones. Another process added to the TS implementation is the ejection chain operator. This operator is designed to diversify the search and escape from local optima by randomly removing customers from a solution $\overline{S}$ and then reinserting them into the same route in a random order. For more details on this operator, see [25].

The neighborhoods $\nu$ generated by the local operators are grouped into the set $\mathsf{\Phi }$, which covers the search space explored by the TS algorithm. To determine the activation sequence of local operators, a counter called times without improvement (TWI) is used. This counter increments by one each time, where a new solution is not better than the incumbent solution; otherwise, it is updated to zero.

$$TWI=\left\{\begin{array}{cc}TWI+1\hfill & \mathrm{if}f\left(S\right)\ge f^{\ast },\hfill \\ 0\hfill & \mathrm{if}f\left(S\right)<f^{\ast }.\hfill \end{array}\right.$$

(29)

Local operators are activated at each iteration depending on the values reached by the TWI. Some local operators are activated in all iterations. In contrast, the Exchange, 2-Exchange, Ejection Chain and Refactibilization operators are activated only when the VSM reaches the values of $\alpha$, $\beta$, $\gamma$, and $\delta$, respectively. The sequence of operators used is detailed in

Table 3. Activation mechanism of local operators.

Procedure	Activation Criteria
Intensification	In the first iteration, the intra-route intensification process is applied to improve the initial solution. This step is performed only in the first iteration.
Relocate inter-route	Triggered when $TWI\ge 0$, to prioritize route minimization.
Exchange	Triggered when $TWI\ge \alpha$, to diversify the search for solutions.
2-Exchange	Triggered when $TWI\ge \beta$, to diversify the search for solutions.
Intra-route relocate	Triggered when $TWI\ge 0$, to prioritize minimizing total travel time.
2-opt	Activated when $TWI\ge 0$, to prioritize minimizing total travel time.
OR-opt	Triggered when $TWI\ge 0$, to prioritize minimizing total travel time.
Ejection Chain	Triggered if $TWI\ge \gamma$, to diversify the search.
Refactibilization	When $TWI\ge \delta$, the refactibilization operators (Relocate inter-route and Exchange) are activated.

.

6. Computational Results

The proposed algorithm was implemented in Python 3.12 using the NumPy 1.26 library. The algorithm was tested with instance sets of ten and twenty clients for validation. These instances were generated from the data of the instances proposed in [26] for the VRPTW problem and were optimally solved using Gurobi 11.0.3, running on an Intel Core i9-14900K processor.

6.1. Instance Characteristics

The instances used are based on the 56 instances proposed in [26], adapted for implementation in ConVRPTW. These instances include customers with variable demands distributed over a 5-day planning horizon, with a $70\%$ probability that a customer requires service on a specific day. They are classified into six groups based on customer distribution: classes C1 and C2 distribute customers in clusters; classes R1 and R2 distribute customers randomly; classes RC1 and RC2 distribute a mix of random and clustered customers. Furthermore, instances in classes C1, R1, and RC1 have narrow time windows, while classes C2, R2, and RC2 have wider time windows. Vehicle capacity also varies: 100 units for classes C1, R1, and RC1, and greater capacities of 350 for C2, and 500 for R2 and RC2. Based on these specifications, there are small instances with 10 and 20 customers. Instances with 10 customers maintain the originally defined vehicle capacities. Instances with 20 clients, on the other hand, double the vehicle capacity.

6.2. Parameter Settings

This section defines the parameters used in the Gurobi solver and TS algorithm. For the solutions obtained using Gurobi 11.0.3, the computation time was limited to 1 h per instance. The instances with 10 clients all reached the optimum before the time limit. In this case, the TS algorithm stops automatically upon finding the optimal solution or reaching the established time limit.

In instances with 20 clients, some failed to reach the optimum within 1 h of computation using Gurobi. For these instances, the results of the TS algorithm are analyzed considering the maximum computation times proposed by [27], specifically 1, 10, and 60 min. The solutions generated using TS are compared with those obtained by Gurobi, allowing for an evaluation of both the quality of the solutions obtained and the performance of the algorithms.

A value of $\overline{c}=100,000$ is set, ensuring that the increase of one unit in the vehicle fleet is penalized more than any increase in the total time traveled.

The TS algorithm parameters are set as follows: for inter-route operators (Relocate, Exchange, 2-Exchange) a tabu tenure of 25 is defined, while for intra-route operators (Relocate, 2-opt, OR-opt) the value is 625. In addition, the local operator activation parameters are set as $\alpha =2, \beta =2, \gamma =250, \delta =5$. These values are selected after preliminary testing to ensure good performance in all the instances evaluated.

6.3. Analysis of Results

We evaluate the performance of the solution obtained using the TS algorithm by comparing it with the optimal solution obtained by solving the ConVRPTW model with the Gurobi solver in the instances with 10 and 20 customers. We report the results for total computing time (computing time), the objective function ($OF$), and the $GAP$ of the solutions. In

Table 4. Results for the set of 10 customer instances.

	Gurobi			TS
Instances	Computing Time	OF	GAP (%)	Computing Time	OF	GAP (%)
C101	0.56	200,425	0	0.44	200,425	0
C102	0.31	100,244	0	0.00	100,244	0
C103	1.82	200,345	0	0.13	200,345	0
C104	1.70	200,354	0	0.45	200,354	0
C105	0.52	200,411	0	0.28	200,411	0
C106	0.31	200,337	0	0.07	200,337	0
C107	0.68	200,374	0	0.20	200,374	0
C108	1.14	200,374	0	0.70	200,374	0
C109	1.28	200,384	0	10.81	200,384	0
C201	0.07	100,801	0	0.00	100,801	0
C202	0.27	100,507	0	0.00	100,507	0
C203	0.30	100,427	0	0.00	100,427	0
C204	0.45	100,489	0	0.00	100,489	0
C205	0.18	100,709	0	0.00	100,709	0
C206	0.31	100,641	0	0.00	100,641	0
C207	0.24	100,726	0	0.00	100,726	0
C208	0.30	100,695	0	0.00	100,695	0
R101	0.10	400,818	0	0.01	400,818	0
R102	6.92	300,927	0	0.10	300,927	0
R103	1.13	200,792	0	0.07	200,792	0
R104	3.36	200,873	0	0.05	200,873	0
R105	0.36	301,042	0	0.04	301,042	0
R106	0.63	200,795	0	0.07	200,795	0
R107	2.70	200,955	0	0.15	200,955	0
R108	1.54	200,756	0	0.26	200,756	0
R109	1.21	201,001	0	1.49	201,001	0
R110	1.67	200,835	0	0.69	200,835	0
R111	2.02	200,941	0	0.05	200,941	0
R112	12.69	200,785	0	2.19	200,785	0
R201	0.13	101,022	0	0.00	101,022	0
R202	1.92	100,788	0	0.00	100,788	0
R203	0.53	100,783	0	0.00	100,783	0
R204	0.31	100,677	0	0.49	100,677	0
R205	0.22	100,860	0	0.00	100,860	0
R206	0.39	100,774	0	0.10	100,774	0
R207	0.28	100,667	0	0.06	100,667	0
R208	0.28	100,729	0	0.15	100,729	0
R209	0.24	100,734	0	0.00	100,734	0
R210	0.67	100,808	0	0.00	100,808	0
R211	0.38	100,774	0	0.28	100,774	0
RC101	1.42	301,155	0	0.17	301,155	0
RC102	2.62	301,079	0	1.27	301,079	0
RC103	3.03	200,987	0	0.38	200,987	0
RC104	414.15	301,086	0	32.84	301,086	0
RC105	6.51	301,076	0	0.37	301,076	0
RC106	2.61	301,092	0	0.42	301,092	0
RC107	2.64	300,994	0	0.35	300,994	0
RC108	2.80	300,981	0	0.18	300,981	0
RC201	0.16	100,758	0	0.00	100,758	0
RC202	1.44	100,774	0	0.07	100,774	0
RC203	1.37	100,685	0	0.55	100,685	0
RC204	0.54	100,658	0	0.00	100,658	0
RC205	0.73	100,748	0	0.13	100,748	0
RC206	0.43	100,797	0	0.00	100,797	0
RC207	0.42	100,684	0	0.77	100,684	0
RC208	0.81	100,624	0	0.00	100,624	0
Mean	8.78	170,384.95	0	1.01	170,384.95	0

and

Table 5. Results for the set of 20 customer instances in which the Gurobi solver reached the optimal solutions.

	Gurobi			Tabu Search
Instances	Computing Time	OF	GAP (%)	Computing Time	OF	GAP (%)
C101	1.40	300,821	0	0.00	300,821	0
C105	4.86	200,761	0	1.48	200,761	0
C106	2.37	200,762	0	1.07	200,762	0
C107	52.88	300,794	0	1.58	300,794	0
C108	24.64	200,736	0	0.66	200,736	0
C201	0.76	100,892	0	0.00	100,892	0
C202	5.09	100,839	0	3.93	100,839	0
C203	225.55	100,917	0	65.54	100,917	0
C204	29.44	100,751	0	330.02	100,751	0
C205	3.57	100,979	0	0.00	100,979	0
C206	9.03	101,061	0	1.07	101,061	0
C207	8.54	100,855	0	0.00	100,855	0
C208	7.18	100,956	0	0.00	100,956	0
R101	42.43	601,934	0	1.70	601,934	0
R105	545.49	401,672	0	2.96	401,672	0
R106	2592.61	401,640	0	1559.41	401,640	0
R201	426.58	201,673	0	783.58	201,673	0
R202	21.53	101,696	0	0.34	101,696	0
R203	9.66	101,025	0	0.39	101,025	0
R204	32.76	101,192	0	35.99	101,192	0
R205	13.72	101,356	0	3.15	101,356	0
R206	119.98	101,354	0	8.34	101,354	0
R207	46.73	101,167	0	61.29	101,167	0
R208	12.47	101,161	0	10.09	101,161	0
R209	12.82	101,338	0	9.06	101,338	0
R210	23.24	101,318	0	1.38	101,318	0
RC101	37.15	401,501	0	1.35	401,501	0
RC102	1846.74	301,424	0	2.04	301,424	0
RC201	197.04	201,575	0	127.63	201,575	0
RC202	353.42	101,758	0	1.30	101,758	0
RC203	1765.47	101,334	0	3568.57	101,334	0
RC204	221.58	101,124	0	238.19	101,124	0
RC205	786.61	101,671	0	0.81	101,671	0
RC206	29.23	101,608	0	9.38	101,608	0
Mean	279.78	174,754.26	0.00	200.95	174,754.26	0.00

, we present the results, respectively.

For the set of 10 customer instances, Table 4 shows that Gurobi reached the optimal solution in all instances, requiring an average computing time of $8.7$ s. Additionally, all instances were optimally solved using the TS algorithm, which required less computing time than the Gurobi solver, with an average of $1.0$ s. For the set of 20 customer instances, Table 5 shows that Gurobi reached the optimal solution in only 34 out of a total of 56 instances, requiring an average computing time of $279.7$ s. The TS algorithm also reached the optimal solution in the same instances as the Gurobi solver, with an average computing time of $200.9$ s. Therefore, the TS algorithm is capable of reaching the optimal solution for instances with 10 and 20 customers and is faster than Gurobi, on average.

Finally, in

Table 6. Results for the set of 20 customer instances in which the Gurobi solver did not reached the optimal solutions.

	Gurobi				Tabu Search
		60 min			1 min			10 min			60 min
Instance	OF	GAP	TT	NV	OF	TT	NV	OF	TT	NV	OF	TT	NV
C102	200,775	0.0002	775	2	200,775	775	2	200,775	775	2	200,775	775	2
C103	200,750	0.0005	750	2	200,750	750	2	200,750	750	2	200,750	750	2
C104	200,758	0.0003	758	2	200,769	769	2	200,753	753	2	200,753	753	2
C109	200,757	0.0002	757	2	200,757	757	2	200,757	757	2	200,757	757	2
R102	501,621	0.1994	1621	5	501,637	1637	5	501,628	1628	5	501,621	1621	5
R103	401,555	0.2495	1555	4	401,561	1561	4	401,555	1555	4	401,555	1555	4
R104	301,410	0.3324	1410	3	301,410	1410	3	301,409	1409	3	301,404	1404	3
R107	301,427	0.3322	1427	3	301,462	1462	3	301,427	1427	3	301,427	1427	3
R108	301,370	0.3323	1370	3	301,403	1403	3	301,373	1373	3	301,370	1370	3
R109	301,653	0.0003	1653	3	301,676	1676	3	301,676	1676	3	301,676	1676	3
R110	301,462	0.3322	1462	3	301,486	1486	3	301,486	1486	3	301,486	1486	3
R111	301,612	0.3327	1612	3	301,541	1541	3	301,541	1541	3	301,541	1541	3
R112	301,319	0.3325	1319	3	301,399	1399	3	301,319	1319	3	301,319	1319	3
R211	201,300	0.4976	1300	2	101,301	1301	1	101,271	1271	1	101,271	1271	1
RC103	301,386	0.3325	1386	3	301,386	1386	3	301,386	1386	3	301,386	1386	3
RC104	301,399	0.3328	1399	3	301,399	1399	3	301,399	1399	3	301,399	1399	3
RC105	401,628	0.2490	1628	4	401,629	1629	4	401,629	1629	4	401,629	1629	4
RC106	301,455	0.0000	1455	3	301,455	1455	3	301,455	1455	3	301,455	1455	3
RC107	301,357	0.3322	1357	3	301,357	1357	3	301,357	1357	3	301,357	1357	3
RC108	301,263	0.3327	1263	3	301,263	1263	3	301,263	1263	3	301,263	1263	3
RC207	201,390	0.4981	1390	2	101,641	1641	1	101,637	1637	1	101,637	1637	1
RC208	101,052	0.0002	1052	1	101,075	1075	1	101,070	1070	1	101,070	1070	1
Mean	283,122.67	0.23	1304.49	2.82	274,051.45	1324.18	2.73	274,041.64	1314.36	2.73	274,040.95	1313.68	2.73

, we present the results for the set of 20 customer instances that did not reach optimal solutions within one hour of computing time using the Gurobi solver. These results are compared with the solutions of the TS algorithm using 1, 10, and 60 min of computation time. For each non-optimal instance, we present the result for the objective function ($OF$), and additionally, we report the total travel time ($TT$) and the number of vehicles ($NV$). The results show that, with only 1 min of computing time, the TS algorithm outperforms the results obtained by the Gurobi solver with 60 min of computing time. Moreover, there are no significant improvements, considering longer computational times of 10 and 60 min. Regarding NV, the most important dimension in the objective function, the best results are achieved at 1 min of computation time. For 10 and 60 min, there are only slight improvements in total travel time. These results highlight the efficiency of the TS algorithm in generating high-quality solutions in low computation times.

6.4. Analysis of Results Obtained by Proposed KPIs

This subsection aims to analyze the operational impact of service consistency in routing plans, considering the proposed ConVRPTW model and the standard VRPTW model. Therefore, this evaluation is conducted by comparing the results in scenarios with and without service consistency using the proposed key performance indicators (KPIs).

Table 7. KPI results for 10 customer instances.

	ConVRPTW						VRPTW
Instance	$\mathit{NV}$	$\overline{\mathit{WT}}$	$\overline{\mathit{CU}}$	$\overline{\mathit{CVRL}}$	$\overline{\mathit{CVVC}}$	$\overline{\mathit{CVCV}}$	$\mathit{NV}$	$\overline{\mathit{WT}}$	$\overline{\mathit{CU}}$	$\overline{\mathit{CVRL}}$	$\overline{\mathit{CVVC}}$	$\overline{\mathit{VCCV}}$
C101	2	1460.50	0.59	0.14	0.59	0.36	2	1463.00	0.59	0.16	0.66	0.53
C102	1	2124.00	0.86	-	-	-	1	1936.00	0.86	-	-	-
C103	2	1063.00	0.49	0.65	0.75	0.86	2	1314.50	0.49	0.65	0.82	0.82
C104	2	1255.00	0.48	0.42	0.65	0.54	2	720.00	0.48	0.67	0.83	0.85
C105	2	1727.00	0.56	0.33	0.64	0.60	2	1355.00	0.56	0.31	0.57	0.53
C106	2	965.00	0.46	0.70	0.83	0.88	2	857.00	0.46	0.84	0.85	0.85
C107	2	1562.50	0.48	0.49	0.75	0.85	2	876.00	0.48	0.65	0.80	0.81
C108	2	1624.00	0.47	0.25	0.37	0.36	2	990.00	0.47	0.51	0.76	0.75
C109	2	985.00	0.50	0.31	0.70	0.71	2	797.50	0.50	0.49	0.75	0.77
C201	1	10,633.00	0.32	-	-	-	1	10,633.00	0.32	-	-	-
C202	1	11,681.00	0.23	-	-	-	1	11,681.00	0.23	-	-	-
C203	1	11,998.00	0.23	-	-	-	1	11,998.00	0.23	-	-	-
C204	1	8809.00	0.26	-	-	-	1	8809.00	0.26	-	-	-
C205	1	10,969.00	0.30	-	-	-	1	10,969.00	0.30	-	-	-
C206	1	10,594.00	0.34	-	-	-	1	10,594.00	0.34	-	-	-
C207	1	8800.00	0.31	-	-	-	1	8800.00	0.31	-	-	-
C208	1	9643.00	0.34	-	-	-	1	9643.00	0.34	-	-	-
R101	4	222.75	0.18	0.93	1.02	0.93	4	223.75	0.18	0.92	1.01	0.90
R102	3	302.33	0.28	0.54	0.70	0.50	3	292.33	0.28	0.51	0.61	0.54
R103	2	334.00	0.36	0.23	0.38	0.28	2	309.50	0.36	0.47	0.62	0.42
R104	2	236.00	0.47	0.09	0.21	0.11	2	201.50	0.47	0.30	0.42	0.33
R105	3	251.00	0.31	0.40	0.68	0.43	3	246.00	0.31	0.28	0.39	0.27
R106	2	244.50	0.39	0.24	0.34	0.28	2	247.50	0.39	0.37	0.43	0.36
R107	2	212.00	0.47	0.12	0.18	0.16	2	237.00	0.47	0.16	0.20	0.22
R108	2	71.00	0.42	0.49	0.55	0.52	2	76.50	0.42	0.72	0.70	0.70
R109	2	160.50	0.45	0.10	0.26	0.16	2	178.50	0.45	0.28	0.44	0.40
R110	2	115.00	0.40	0.44	0.66	0.55	2	113.00	0.40	0.61	0.71	0.68
R111	2	255.50	0.44	0.15	0.21	0.17	2	263.00	0.44	0.17	0.32	0.17
R112	2	94.00	0.41	0.43	0.67	0.54	2	62.00	0.41	0.59	0.73	0.67
R201	1	2364.00	0.17	-	-	-	1	2364.00	0.17	-	-	-
R202	1	2466.00	0.19	-	-	-	1	2466.00	0.19	-	-	-
R203	1	2168.00	0.16	-	-	-	1	2168.00	0.16	-	-	-
R204	1	2026.00	0.16	-	-	-	1	2026.00	0.16	-	-	-
R205	1	2120.00	0.19	-	-	-	1	2120.00	0.19	-	-	-
R206	1	2151.00	0.19	-	-	-	1	2150.00	0.19	-	-	-
R207	1	2128.00	0.14	-	-	-	1	2128.00	0.14	-	-	-
R208	1	2056.00	0.17	-	-	-	1	2056.00	0.17	-	-	-
R209	1	1998.00	0.17	-	-	-	1	1998.00	0.17	-	-	-
R210	1	2412.00	0.19	-	-	-	1	2412.00	0.19	-	-	-
R211	1	1538.00	0.18	-	-	-	1	1538.00	0.18	-	-	-
RC101	3	333.33	0.61	0.10	0.31	0.36	3	187.33	0.61	0.60	0.64	0.67
RC102	3	252.67	0.54	0.24	0.39	0.53	3	205.67	0.54	0.60	0.64	0.71
RC103	2	327.00	0.75	0.11	0.13	0.17	2	277.50	0.75	0.11	0.23	0.34
RC104	3	91.33	0.58	0.49	0.67	0.69	3	141.67	0.58	0.59	0.64	0.72
RC105	3	292.67	0.52	0.23	0.43	0.53	3	195.67	0.52	0.59	0.64	0.69
RC106	3	245.33	0.59	0.23	0.37	0.40	3	184.00	0.59	0.59	0.62	0.64
RC107	3	130.33	0.47	0.35	0.57	0.67	3	84.00	0.47	0.88	0.96	1.01
RC108	3	123.00	0.51	0.59	0.69	0.72	3	135.67	0.51	0.60	0.68	0.76
RC201	1	2354.00	0.24	-	-	-	1	2354.00	0.24	-	-	-
RC202	1	2298.00	0.34	-	-	-	1	2298.00	0.34	-	-	-
RC203	1	2581.00	0.30	-	-	-	1	2581.00	0.30	-	-	-
RC204	1	2118.00	0.30	-	-	-	1	2118.00	0.30	-	-	-
RC205	1	2285.00	0.30	-	-	-	1	2285.00	0.30	-	-	-
RC206	1	1983.00	0.30	-	-	-	1	1983.00	0.30	-	-	-
RC207	1	2197.00	0.27	-	-	-	1	2209.00	0.27	-	-	-
RC208	1	1679.00	0.28	-	-	-	1	1684.00	0.28	-	-	-
Mean	1.70	2519.81	0.37	0.35	0.53	0.49	1.70	2468.50	0.37	0.51	0.63	0.61

and

Table 8. KPI results for 20 customer instances.

	ConVRPTW						VRPTW
Instance	$\mathit{NV}$	$\overline{\mathit{WT}}$	$\overline{\mathit{CU}}$	$\overline{\mathit{CVRL}}$	$\overline{\mathit{CVVC}}$	$\overline{\mathit{CVCV}}$	$\mathit{NV}$	$\overline{\mathit{WT}}$	$\overline{\mathit{CU}}$	$\overline{\mathit{CVRL}}$	$\overline{\mathit{CVVC}}$	$\overline{\mathit{VCCV}}$
C101	3	716.33	0.43	0.57	0.68	0.65	3	716.33	0.43	0.57	0.68	0.65
C102	2	819.00	0.63	0.29	0.29	0.15	2	1031.50	0.63	0.19	0.11	0.22
C103	2	989.50	0.59	0.27	0.12	0.16	2	893.00	0.59	0.27	0.12	0.16
C104	2	962.50	0.70	0.31	0.20	0.04	2	905.50	0.70	0.27	0.13	0.09
C105	2	743.00	0.62	0.27	0.14	0.09	2	743.00	0.62	0.27	0.14	0.09
C106	2	806.00	0.66	0.21	0.11	0.19	2	806.00	0.66	0.21	0.11	0.19
C107	3	401.00	0.41	0.60	0.66	0.67	3	401.00	0.41	0.60	0.66	0.67
C108	2	487.50	0.58	0.27	0.12	0.17	2	533.00	0.58	0.27	0.12	0.17
C109	2	191.00	0.70	0.29	0.23	0.08	2	236.00	0.70	0.29	0.23	0.08
C201	1	8022.00	0.31	-	-	-	1	8022.00	0.31	-	-	-
C202	1	8896.00	0.35	-	-	-	1	8896.00	0.35	-	-	-
C203	1	6828.00	0.41	-	-	-	1	6825.00	0.41	-	-	-
C204	1	8834.00	0.36	-	-	-	1	8834.00	0.36	-	-	-
C205	1	6819.00	0.37	-	-	-	1	6814.00	0.37	-	-	-
C206	1	6029.00	0.40	-	-	-	1	6029.00	0.40	-	-	-
C207	1	5846.00	0.40	-	-	-	1	5846.00	0.40	-	-	-
C208	1	6232.00	0.37	-	-	-	1	6232.00	0.37	-	-	-
R101	6	253.50	0.15	0.46	0.60	0.54	6	253.50	0.15	0.46	0.60	0.54
R102	5	264.20	0.14	0.39	0.63	0.53	5	271.80	0.14	0.39	0.63	0.53
R103	4	305.75	0.23	0.42	0.51	0.45	4	306.75	0.23	0.42	0.51	0.45
R104	3	159.33	0.30	0.19	0.25	0.32	3	109.33	0.30	0.13	0.24	0.25
R105	4	256.00	0.21	0.30	0.28	0.27	4	256.00	0.21	0.30	0.28	0.27
R106	4	243.25	0.23	0.26	0.31	0.26	4	243.25	0.23	0.26	0.31	0.26
R107	3	230.00	0.31	0.16	0.28	0.28	3	217.33	0.31	0.16	0.28	0.28
R108	3	160.00	0.29	0.27	0.36	0.30	3	143.67	0.29	0.16	0.25	0.31
R109	3	119.00	0.32	0.15	0.36	0.17	3	119.00	0.32	0.15	0.36	0.17
R110	3	112.33	0.32	0.16	0.34	0.26	3	115.67	0.32	0.16	0.34	0.26
R111	3	119.67	0.34	0.12	0.22	0.20	3	119.67	0.34	0.11	0.23	0.20
R112	3	147.00	0.35	0.10	0.15	0.16	3	145.67	0.35	0.12	0.20	0.25
R201	2	2385.50	0.10	0.35	0.37	0.22	2	2385.50	0.10	0.35	0.37	0.22
R202	1	1445.00	0.19	-	-	-	1	1459.00	0.19	-	-	-
R203	1	2108.00	0.17	-	-	-	1	2108.00	0.17	-	-	-
R204	1	1524.00	0.18	-	-	-	1	1524.00	0.18	-	-	-
R205	1	1689.00	0.16	-	-	-	1	1689.00	0.16	-	-	-
R206	1	1569.00	0.19	-	-	-	1	1579.00	0.19	-	-	-
R207	1	1942.00	0.18	-	-	-	1	1942.00	0.18	-	-	-
R208	1	1468.00	0.19	-	-	-	1	1468.00	0.19	-	-	-
R209	1	1311.00	0.20	-	-	-	1	1311.00	0.20	-	-	-
R210	1	1713.00	0.17	-	-	-	1	1713.00	0.17	-	-	-
R211	2	1886.00	0.10	0.10	0.14	0.15	1	1284.00	0.21	-	-	-
RC101	4	188.00	0.31	0.51	0.76	0.75	4	188.00	0.31	0.51	0.76	0.75
RC102	3	217.00	0.53	0.08	0.29	0.32	3	209.67	0.53	0.08	0.29	0.32
RC103	3	160.00	0.55	0.07	0.35	0.40	3	165.00	0.55	0.07	0.35	0.40
RC104	3	105.33	0.53	0.05	0.47	0.44	3	101.33	0.53	0.05	0.47	0.44
RC105	4	148.25	0.44	0.41	0.62	0.63	4	152.00	0.44	0.41	0.62	0.63
RC106	3	134.00	0.56	0.09	0.37	0.39	3	141.33	0.56	0.09	0.37	0.39
RC107	3	126.00	0.51	0.03	0.37	0.44	3	126.00	0.51	0.03	0.37	0.44
RC108	3	97.00	0.51	0.17	0.43	0.46	3	107.33	0.51	0.17	0.43	0.46
RC201	2	2279.50	0.16	0.35	0.28	0.21	2	2279.50	0.16	0.35	0.28	0.21
RC202	1	1434.00	0.30	-	-	-	1	1424.00	0.30	-	-	-
RC203	1	2131.00	0.29	-	-	-	1	2131.00	0.29	-	-	-
RC204	1	2325.00	0.31	-	-	-	1	2325.00	0.31	-	-	-
RC205	1	1560.00	0.27	-	-	-	1	1560.00	0.27	-	-	-
RC206	1	1198.00	0.31	-	-	-	1	1210.00	0.31	-	-	-
RC207	2	1701.50	0.16	0.40	0.72	0.73	1	1032.00	0.32	-	-	-
RC208	1	1624.00	0.28	-	-	-	1	1624.00	0.28	-	-	-
Mean	2.16	1793.95	0.34	0.26	0.36	0.34	2.13	1773.28	0.35	0.25	0.35	0.33

report the results regarding the number of vehicles $\left(NV\right)$, the average daily waiting time per vehicle $\left(\overline{WT}\right)$, the average daily capacity used per vehicle $\left(\overline{CU}\right)$, the average coefficient of variation in route length $\left(\overline{CVRL}\right)$, the average coefficient of variation in vehicle capacity $\left(\overline{CVVC}\right)$, and the average coefficient of variation in customers per vehicle $\left(\overline{CVCV}\right)$. The ConVRPTW and VRPTW models were solved using the Gurobi solver, with a time limit of one hour.

It is important to note that the coefficients of variation measure the differences in workload between the vehicles. For this reason, it is not possible to analyze these coefficients in solutions that only use a single vehicle. Likewise, a solution using one vehicle cannot be compared to one that uses two or more vehicles.

6.4.1. KPIs Results for the Set of 10 Customer Instances

Analyzing the KPIs for the ConVRPTW and VRPTW models from Table 7, regarding the $\overline{WT}$, it is observed that $42.86\%$ of the instances are equal, $35.71\%$ of the time the ConVRPTW outperforms the VRPTW, and $21.43\%$ of the time the result is inverse. Additionally, on average, the ConVRPTW generates a $\overline{WT}$ $54.31$ units higher than the VRPTW. However, considering that the total travel time in all solutions is substantially higher, it is concluded that the ConVRPTW does not present significantly higher $\overline{WT}$ than the VRPTW. However, there are instances in which the ConVRPTW generates considerably higher $\overline{WT}$, reaching extreme cases where it exceeds the other model by more than 500 $\overline{WT}$ units. In conclusion, although the $\overline{WT}$ tends to be equal between the two models in most cases, the VRPTW presents more instances where it outperforms the ConVRPTW in terms of $\overline{WT}$. However, on average, this model is outperformed by the VRPTW by a small margin. Therefore, regarding waiting time, both models are similar on average, with some specific exceptions.

Regarding the average daily capacity used per vehicle $\left(\overline{CU}\right)$, it is observed that the results are consistent across all instances, indicating that, on average, the ConVRPTW and VRPTW models perform equally in distributing demand among vehicles. This equality is explained by the fact that, although the solutions for each model may assign customers to vehicles differently, utilized capacity tends to balance out when the average is calculated across all routes and days in the planning horizon.

The indicators $\overline{CVRL}$, $\overline{CVVC}$, and $\overline{CVCV}$ measure the workload imbalance in terms of route length, utilized vehicle capacity, and customers per vehicle, respectively. In most cases, the VRPTW model shows significantly higher values than those obtained by the ConVRPTW model, indicating that the VRPTW tends to generate more unbalanced routes in terms of workload. In contrast, the ConVRPTW model achieves more balanced solutions due to the driver consistency constraint. When evaluating the differences in route balance, it is observed that, on average, ConVRPTW solutions are $15.93\%$ superior to $\overline{CVRL}$, $10.59\%$ superior to $\overline{CVVC}$, and $11.55\%$ superior to $\overline{CVCV}$. These results highlight ConVRPTW’s ability to generate more balanced routes, contributing to a better workload distribution and strengthening system performance.

In conclusion, for the set of 10 customer instances, it can be identified that each of the coefficients of variation of the ConVRPTW model is significantly higher than those of the VRPTW, demonstrating the advantage that service consistency has in generating balanced solutions in terms of workload.

6.4.2. KPIs Results for the Set of 20 Customer Instances

Analyzing the KPIs for the ConVRPTW and VRPTW models from Table 8, regarding the $\overline{WT}$, it is observed that in $57.14\%$ of the instances, the $\overline{WT}$ are equals, while in $19.64\%$ of the cases, the ConVRPTW model presents higher values, and in the remaining $23.21\%$, the VRPTW outperforms the ConVRPTW. Additionally, when considering the average $\overline{WT}$ value across all instances, the ConVRPTW has slightly higher values. Therefore, both models exhibit comparable performance in this metric, with a marginal advantage for the VRPTW.

Regarding the average daily capacity used per vehicle $\left(\overline{CU}\right)$, it is observed that the results are not precisely equal between the ConVRPTW and VRPTW models. This is due to instances R211 and RC207, where the ConVRPTW model fails to reach the optimal solution and generates an additional route, compared to the VRPTW model, affecting the overall average. By excluding these instances from the evaluation, the results show that both models distribute demand similarly, with equivalent $\left(\overline{CU}\right)$ across all other instances.

Regarding the metrics $\overline{CVRL}$, $\overline{CVVC}$, and $\overline{CVCV}$, the results show minimal differences between both models. On average, ConVRPTW generates solutions with values slightly higher than VRPTW, but the difference never exceeds $1\%$. This suggests that ConVRPTW performs very similarly to VRPTW in terms of load balance, generating routes that are as balanced as those of the traditional model.

Finally, the analysis indicates that, although ConVRPTW does not clearly outperform VRPTW in any of the evaluated metrics, it achieves comparable results regarding $\overline{WT}$ and the coefficients of variation. This demonstrates that ConVRPTW can generate solutions that are as effective as a traditional VRPTW model based on daily optimization.

7. Conclusions

This work proposed a tabu search (TS) metaheuristic for solving the ConVRPTW problem, obtaining highly competitive results compared to the Gurobi algorithm. Experiments conducted with 10 and 20 customers demonstrated that TS can achieve high-quality solutions in all instances analyzed, resulting in a GAP of $0\%$. This validates its effectiveness in terms of solution quality. Furthermore, runtime analysis shows that the TS metaheuristic offers a significant advantage in computational efficiency. This highlights its potential for tackling more complex problems where exact methods require significant computational time.

Regarding the performance of the ConVRPTW model compared with the traditional VRPTW model, the results presented show an advantage in incorporating service consistency into a routing plan, as evaluated by the KPIs, thereby improving vehicle workload balance. Furthermore, this model offers additional advantages inherent to service consistency, directly impacting customer satisfaction and driver productivity. Incorporating consistency into route design enhances the overall performance of a routing plan in terms of load balancing, thereby reducing excessively long or poorly distributed routes. Furthermore, although service consistency may result in slightly longer waiting times, the impact is small compared to the benefits of route balancing and customer service perception.

The approach proposed in this research offers an alternative to assessing the actual performance of a delivery system for last-mile logistics, moving away from the traditional approach that relies solely on costs. It constitutes a comprehensive approach that integrates service quality and operational performance, striking a balance between operating costs, system performance, and customer experience.

From a practical perspective, the results highlight clear trade-offs for decision-makers in last-mile logistics. While incorporating service consistency into routing may increase total travel time by around 2–4%, it simultaneously reduces workload imbalances between drivers by up to 30%, as reflected in the coefficients of variation in route length, vehicle capacity, and customers served. This trade-off demonstrates that a marginal increase in operational costs can yield significant improvements in task balance, supporting fairer and more predictable working conditions for drivers. In practice, this translates into reduced fatigue, higher driver satisfaction, and lower turnover, ultimately decreasing hidden costs associated with workforce instability.

Furthermore, the KPI-based analysis shows that delivery systems designed under the ConVRPTW framework generate more homogeneous workloads across days and vehicles. Compared with traditional VRPTW solutions, ConVRPTW routes exhibit on average 15.9% lower variability in distance traveled, 10.6% lower variability in vehicle load utilization, and 11.6% lower variability in customers served per route. For practitioners, this means that the proposed approach ensures more equitable use of fleet capacity, facilitates operational planning, and reduces risks associated with overburdened vehicles or drivers. These quantified improvements provide strong evidence that the ConVRPTW model not only maintains efficiency but also adds value through fairer resource allocation.

Overall, the study offers managers actionable insights into how consistency-based planning can strike a balance between efficiency, cost, and equity. The results suggest that companies willing to accept small increases in travel distance can secure substantial benefits in workload fairness and long-term sustainability. Future research could extend this analysis to different operational scenarios (e.g., varying fleet sizes, customer densities, or service time constraints) to provide practitioners with a richer basis for evaluating the applicability of the ConVRPTW model in diverse real-world settings. Finally, other future work involves developing a study incorporating disruptive events into the ConVRPTW model to strengthen the delivery system against unexpected events. The ability of a balanced system to continue operating in the face of events such as accidents, breakdowns, or vehicle theft is established as a key differentiating factor. In this context, route balancing plays a fundamental role, as it facilitates the efficient redistribution of customers assigned to an affected vehicle, ensuring a timely response to demand. This approach reinforces service continuity and helps maintain high customer satisfaction levels. Furthermore, the driver’s prior knowledge of assigned routes enables a swift response to disruptions, optimizing delivery coordination and minimizing negative impacts on the system. Considering this characteristic in the ConVRPTW model would generate a routing plan to increase customer satisfaction and service continuity for the delivery system.