A Non-Linear Optimization Model for the Multi-Depot Multi-Supplier Vehicle Routing Problem with Relaxed Time Windows

Abstract

In the realm of supply chain logistics, the Multi-Depot Multi-Supplier Vehicle Routing Problem (MDMSVRP) poses a significant challenge in optimizing the transportation process to minimize costs and enhance operational efficiency. This problem involves determining the most cost-effective routes for a fleet of vehicles to deliver goods from multiple suppliers to multiple depots, considering various constraints and non-linear relationships. The routing problem (RP) is a critical element of many logistics systems that involve the routing and scheduling of vehicles from a depot to a set of customer nodes. One of the most studied versions of the RP is the Vehicle Routing Problem with Time Windows (VRPTW), in which each customer must be visited at certain time intervals, called time windows. In this paper, it is considered that there are multiple depots (supply centers) and multiple suppliers, along with a fleet of vehicles. The goal is to efficiently plan routes for these vehicles to deliver goods from the suppliers to various customers while considering relaxed time windows. This research is intended to establish a new relaxation scheme that relaxes the time window constraints in order to lead to feasible and good solutions. In addition, this study develops a discrete optimization model as an alternative model for the time-dependent VRPTW involving multi-suppliers. This research also develops a metaheuristic algorithm with an initial solution that is determined through time window relaxation.

1. Introduction

In the context of supply chain logistics, the Multi-Depot Multi-Supplier Vehicle Routing Problem (MDMSVRP) poses a significant challenge in optimizing the transportation process to minimize costs and enhance operational efficiency. Traditional approaches to the MDMSVRP often encounter difficulties in meeting strict time window constraints imposed on delivery routes, leading to suboptimal solutions and operational inefficiencies. To address this challenge, this paper aims to develop a novel non-linear optimization model that integrates relaxed time window constraints into the routing problem framework.

The objective of the proposed optimization model is to efficiently plan routes for a fleet of vehicles operating from multiple depots and serving multiple suppliers and customers, while considering relaxed time windows for deliveries. The relaxation of the time window constraints allows for greater flexibility in route planning, enabling the optimization model to adapt to real-world logistical constraints and uncertainties.

Key considerations in the development of the non-linear optimization model include the formulation of objective functions that minimize the total transportation costs, taking into account factors such as vehicle capacities, supplier availability, and varying delivery demands. Additionally, constraints related to vehicle capacities, depot and supplier capacities, and customer demand satisfaction are incorporated to ensure the feasibility and practicality of the generated routing solutions.

Mathematically, the Vehicle Routing Problem (VRP) can be written in terms of a graph as follows. Suppose that $G=(V,A)$ is a digraph where $V=\{0, \dots , n\}$ is a set of nodes and $A=\left\{\right(i,j): i,j\}\in V,i\ne j\}$ is the set of edges. Node $0$ represents the origin, while the other nodes represent clients. A fleet consisting of m indistinguishable transportation means, each with a capacity of $Q$, is stationed at the origin. The volume of each vehicle is determined a priori and is a decision variable. Every customer $i$ has a request $q_i$ that is non-negative. The cost matrix $c_{ij}$ is defined at $A$. To simplify matters, all travel expenses, distances, and travel durations are presumed to be equal. The VRP involves crafting $m$ vehicle routes, each commencing and concluding at the depot, ensuring that each customer is serviced precisely once by a single vehicle, the total route requests do not exceed $Q$, and the total route length does not exceed the limit $L$ previously set. In the symmetric case, i.e., when $c_{ij}=c_{ji}$ for all $(i,j)\in A$, the solution search is usually conducted using the set of edges $E=\left\{\right(i,j):i, j\in V,I<j\}$.

Ref. [1] was the first to introduce the VRP to solve the routing problem in the delivery of petrol between a central depot along with several satellite service stations provided by the terminal. To read more about the variants of the problem with different attributes, the reader can refer to the comprehensive reviews of [2,3,4,5].

Due to the lingering theoretical issues and the perpetual increase in insights from logistical operations, research on the VRP is still ongoing. The version of the VRP mentioned most often is the Vehicle Routing Problem with Time Windows (VRPTW), introduced by [6], where each client is required to be visited within a certain time interval. Time restriction constraints arise as a result of product constraints (such as product usable dates) or production limits or may be required by the customer because of their inventory policies. Along with customer time constraints, the travel durations encompass the transit times between all customer locations or between the customers and the central depot. Vehicles must serve customers within a predetermined period of time at the minimum cost. Vehicles are allowed to arrive at customers prior to the commencing of the specified time frame but must wait if they arrive before the customer is ready to be served and are not permitted to arrive late or after the time window ends.

Multi-supplier engagement makes the TDVRTW even more complex. In order to pick up goods from different suppliers with different time windows, the vehicle is required to adhere to the scheduled stops at various suppliers, considering their differing opening hours. In addition, some suppliers may insist on being visited for a certain period of time. Meeting the delivery requirements with time windows is not easy as the delivery process is usually influenced by the flow of traffic.

This study develops a discrete optimization model as an alternative model for the time-dependent VRP with time windows and multi-supplier involvement. This research also develops a metaheuristic algorithm with an initial solution determined through time window relaxation.

2. Related Work

The VRP was first introduced by [1] to solve the routing problem of delivering petrol from a central depot to several satellite service stations. The VRP is a fundamental combinatorial optimization challenge in logistics and supply chain management, involving determining the most efficient routes for a fleet of vehicles to serve a set of customers while minimizing the costs and meeting various constraints [7]. The classical VRP aims to optimize the routing plans for a fleet of homogeneous vehicles to visit customers with known demands, starting and ending at a depot [8].

Various VRP variants address different operational scenarios and constraints. The Capacitated Vehicle Routing Problem (CVRP) involves designing minimum-cost routes for a fleet of vehicles with a limited capacity to serve customers [9]. The Open Vehicle Routing Problem (OVRP) relaxes the constraint of visiting each customer exactly once, allowing for more flexible routing solutions [10]. Additionally, the VRP with a Heterogeneous Fleet considers vehicles with different capacities and capabilities [11].

The significance of the VRP in logistics and supply chain management lies in its ability to optimize transportation operations, reduce costs, and improve the service quality. By efficiently planning vehicle routes, companies can enhance their delivery performance, minimize their fuel consumption, and reduce their carbon emissions [12]. Integrating VRP solutions with inventory management and facility location decisions can lead to more streamlined and cost-effective supply chain networks.

Advanced optimization techniques, such as metaheuristics and heuristic algorithms, have been developed to efficiently tackle large-scale VRPs. These approaches help to find near-optimal solutions for complex VRP instances, considering real-world constraints and operational requirements. Moreover, the continuous evolution of VRP research and the exploration of new variants contribute to enhancing the applicability and effectiveness of route optimization in logistics and supply chain operations.

In solving VRP variants, various algorithms and methods have been developed to address the complexity of these optimization challenges. These approaches can be broadly categorized into exact algorithms, heuristics, and metaheuristics, each offering unique advantages and trade-offs regarding solution quality and computational efficiency. Exact algorithms, such as branch-and-cut, aim to find the optimal solution by exhaustively exploring the solution space. While these methods guarantee optimality, they are often computationally intensive and may struggle with large-scale VRP instances [13]. On the other hand, heuristics provide quick and practical solutions by employing rules of thumb or intuitive strategies to guide the search process. Classical heuristic algorithms like the Palgunadi algorithm have been utilized to efficiently tackle VRP instances [14]. These methods sacrifice optimality for speed and are particularly useful for real-time decision-making in logistics and supply chain management.

Metaheuristics, including genetic algorithms [15,16], ant colony optimization [17,18], and simulated annealing [19], offer a middle ground between exact algorithms and heuristics. These techniques leverage probabilistic and iterative processes to explore the solution space and converge towards near-optimal solutions. Genetic algorithms have shown success in solving VRP variants with time windows, while simulated annealing has been applied to design vehicle routes for specific distribution systems [20]. Ant colony optimization has been utilized to address a class of VRPs, demonstrating the effectiveness of swarm intelligence in route optimization [21].

In the literature, different approaches have been explored to tackle VRP variants, including deterministic, stochastic, and fuzzy methods. Deterministic approaches rely on precise input data and deterministic algorithms to find solutions, offering a clear and deterministic path to optimization. Stochastic approaches introduce randomness or uncertainty into the optimization process, considering probabilistic factors such as demand variability or travel time fluctuations. Fuzzy approaches, on the other hand, deal with imprecise or vague information, allowing for flexible decision-making in the face of uncertainty [13]. The study conducted in [22] employed a multinomial logit model within the framework of a random utility model (RUM), utilizing empirical data derived from floating car data (FCD). The research primarily focused on optimizing routes by considering path and route criteria, rather than strictly adhering to time window constraints, which are typically emphasized in conventional VRP models.

Due to the lingering theoretical issues and the perpetual increase in insights from logistical operations, research on the VRP is still ongoing. One of the most frequently mentioned variants of the VRP is the Vehicle Routing Problem with Time Windows (VRPTW), introduced by [6], where each client must be visited within a certain time interval. Time restriction constraints arise from product constraints (such as product usable dates), production limits, or customer requirements due to their inventory policies. Along with customer time constraints, the travel durations encompass the transit times between all customer locations or between the customers and the central depot. Vehicles must serve customers within a predetermined period at the minimum cost. Vehicles are allowed to arrive at customers before the commencing of the specified time frame but must wait if they arrive before the customer is ready to be served and are not permitted to arrive late or after the time window ends.

The Relaxed Time Windows Vehicle Routing Problem (RTWVRP) is a variant of the classic Vehicle Routing Problem (VRP) that relaxes the strict time window constraints, allowing more flexibility in scheduling deliveries or pickups within a specific time frame. This relaxation of the time windows aims to improve the overall efficiency of the routing process by providing a buffer for drivers to accommodate unforeseen delays or optimize their routes based on real-time conditions.

Several studies have focused on different aspects of VRPs with time windows. The authors of [23] introduced a method using Lagrangian relaxation to solve soft-constrained VRPs, showcasing its applicability to various VRP types, including the Capacitated VRP with Time Windows (CVRPTW). The authors of [24] highlighted the importance of time window constraints in VRPs, emphasizing the need to consider delivery time limits in VRP optimization, especially in scenarios like the VRP with Time Windows (VRPTW).

Moreover, the research in [25] discussed VRPs with multiple fuzzy time windows based on a time-varying traffic flow, illustrating the complexity of incorporating dynamic factors into routing models. Additionally, the studies in [26,27] provide comprehensive overviews of the VRPTW, emphasizing its significance in the realm of heuristic and mathematical optimization research.

In addressing the RTWVRP specifically, the literature offers valuable insights into related problem variants. For instance, ref. [28] extends algorithms to handle soft time window constraints, allowing for violations at a cost, which could be adapted to the RTWVRP setting. Furthermore, ref. [29] proposes a unified model framework for the multi-attribute consistent periodic VRP, incorporating time windows, time dependence, and consistency, which could serve as a basis for the development of solutions for the RTWVRP.

Ref. [30] proves that the VRPTW is a problem characterized as NP-hard, presenting formidable computational challenges in determining the optimal solution, particularly for large-scale instances. Consequently, various heuristics and metaheuristics have been proposed, such as in [31,32,33]. The most effective metaheuristics utilize refinement procedures based on a local search for optimization and direct most of the computational effort to the serial exploration of neighborhoods. Thus, efficient displacement evaluation becomes critical to algorithmic performance and scalability.

In this context, employing intermediate measures with loose time constraints would not be practical as a direct means to ensure the availability of numerous initial solutions. Several relaxation schemes have been proposed in prior research, including techniques such as penalizing deviations from customer arrival times [34,35], path consistency [36], or refund penalties based on the travel time [37]. However, inducing a state of relaxation may also lead to nuanced assessments of displacement [34] and extended search spaces. This research aims to establish a new relaxation scheme that can lead to a feasible and good solution.

Furthermore, in most supply chain operations, vehicles often encounter difficulties in arriving at customers on time, which can be caused by traffic jams. Considering the general assumption in the VRP that the travel time is constant, using the basic model of the VRPTW to overcome this difficulty will result in a non-optimal or even non-feasible solution [38]. The VRPTW that considers the influence of traffic conditions in its modeling is called the Time-Dependent Vehicle Routing Problem with Time Windows (TDVRPTW), introduced in [39]. The authors formulated this problem as mixed integer programming (MIP) and proposed a metaheuristic genetic algorithm.

The optimization of coordination among multiple depots and suppliers in Vehicle Routing Problems (VRPs) requires further exploration and development. While the existing literature has addressed various aspects of VRPs with multiple depots and suppliers, there is still room for advancements in optimizing the coordination and efficiency of such complex supply chain networks.

For instance, ref. [40] introduces extensions like the Multi-Depot VRP with non-linear costs, considering subcontractors with individual costs and limited delivery radii. This emphasizes the importance of incorporating multiple depots and subcontractors into VRP solutions to enhance the operational efficiency and cost-effectiveness.

Moreover, the research in [41] focuses on solving VRPs involving heterogeneous fleets of vehicles, multi-depot subcontractors, and pickup/delivery time window constraints. This highlights the need to consider diverse factors such as the vehicle types, depot locations, and time constraints when optimizing multi-depot VRPs.

Additionally, ref. [42] proposes a solution framework for the coordination of centralized multi-tier multiple supplier networks, addressing challenges such as demand uncertainties. This underscores the significance of developing strategies to manage interactions between multiple suppliers and depots within a coordinated supply chain framework.

Furthermore, studies like [41] explore real-world VRPs with multi-depot subcontractors, heterogeneous vehicle fleets, and varying time windows and locations. Such research contributes to understanding the complexities involved in multi-depot VRPs and the need for tailored optimization approaches.

This study aims to develop a discrete optimization model as an alternative approach to addressing the Time-Dependent VRP with Time Windows and the involvement of multiple suppliers. Additionally, the research proposes a metaheuristic algorithm, with the initial solution derived through the relaxation of the time windows. The primary objective is to optimize the distribution of goods from various suppliers to multiple depots and ultimately to customers, while minimizing the overall transportation cost. This cost encompasses the transportation expenses, dependent on the distance and vehicle usage, as well as the penalty costs incurred from deviations from the specified time windows. All of this is achieved while adhering to a predefined set of constraints.

3. Problem Description

3.1. Problem Statement

In the MVRPTW, the problem consists of a single depot, represented as $o$, serving as both the origin and destination for all vehicle routes. There is a homogeneous vehicle fleet wherein all vehicles possess a capacity of $Q$ units, with the fleet consisting of $K$ available vehicles.

The customer set is denoted by $N=\{1, \dots , n\}$. A distance exists, $d_{ij}$, and a travel time, $t_{ij}$, associated with each pair $i,j\in N\cup \left\{o\right\}$. Each customer $i$ has a request $q_i$, revenue $g_i$, service time $s_i$, and time window $\left[a_i, b_i\right]$, where a_i denotes the minimum start time and $b_i$ represents the maximum start time when servicing customer $i$. This implies that if the vehicle reaches customer $i$ before $a_i$, it is obligated to wait. Without loss of generality, it is assumed that the vehicle initiates service to the customer at the earliest opportunity. The depot’s service time is defined as $s_0=0$, and all transportation routes must conform to the time window specified for the depot, $\left[a_0, b_0\right]$. Thus, no vehicle can depart from the depot before $a_0$ or access it after $b_0$. The time restriction represents the duration $W$ of workdays. It is assumed that $b_i+s_i+d_{io}\le b_o, \forall i\in N$.

Each vehicle can perform several routes during the working day. This means that it is able to perform one route, reload at the depot, and depart for the next route, until the end of the working day. Route $r$ is defined by the order of visits to a subset of customers $N_r\subseteq N$. It is feasible if the number of requests of all customers included in $N_r$ does not exceed the capacity of the vehicle and if the sequence of visits is structured to ensure the feasibility of visiting each customer within a specified time frame. In this model, it is also considered that the initiation of service for all customers along the route cannot be delayed beyond a certain threshold $t_{max}$, denoting the maximum time unit after the route is started. $R$ denotes the collection of viable routes. Each route additionally encompasses an associated setup time to factor in. Prior to departing the depot to execute route $r$, the vehicle takes $\beta \sum _{i\in N_r}{s}_i$ time units to load, with $\beta \in {\mathbb{R}}^{+}$.

Please acknowledge that visiting all customers might not be feasible due to the finite number of available vehicles. Nevertheless, the objective is consistently to serve as many customers as feasible.

3.2. Description of Model

The problem can be expressed in a fully directed graph $G=\left(V, A\right)$, with $V=N\cup \left\{o\right\}$ as a set of vertices and $A=\left\{\left(i, j\right) : i,j\in V\right\}$ as a set of arcs. In this description, there is a binary variable representing the subscriber to the route that determines the sequential pair of routes. The binary variables $x_{ij}^r$ and $y_i^r$, respectively, indicate that arc $\left(i,j\right)$ and customer $i$ belong to route $r$, while the binary variable $z_{rs}$ determines whether any vehicle traveling route $r$ is followed by route s within weekdays. The notation $r<s$ means that the same vehicle is assigned to perform route $s$ after performing route $r$. The variable $t_i^r$ represents the start time of service for customer $i$, if served by route $r$, and $t_o^r$ and $t_o^{\prime r}$ represent the start and end times of route $r$, respectively. Suppose that $M$ is a large enough number. The concise formulation for the MVRPTW is stated as follows.

Minimize

$$\sum _{r\in R}\sum _{\left(i, j\right)\in A}{d}_{ij}{x}_{ij}^r-\alpha \sum _{r\in R}\sum _{i\in N}{g}_i{y}_i^r$$

(1)

with constraints

$$\sum _{j\in V}{x}_{ij}^r=y_i^r \forall i\in N,\forall r\in R$$

(2)

$$\sum _{r\in R}{y}_i^r\le 1 \forall i\in N$$

(3)

$$\sum _{i\in V}{x}_{ih}^r-\sum _{j\in V}{x}_{hj}^r=0 \forall h\in N,\forall r\in R$$

(4)

$$\sum _{i\in V}{x}_{oi}^r=1 \forall r\in R$$

(5)

$$\sum _{i\in V}{x}_{io}^r=1 \forall r\in R$$

(6)

$$\sum _{j\in N}{x}_{ij}=1 i\in N,i\ne 0,i\ne j$$

(7)

$$\sum _{i\in N}{x}_{ij}=1 j\in N,j\ne 0,i\ne j$$

(8)

$$\sum _{i\in N}{q}_i{y}_i^r\le Q \forall r\in R$$

(9)

$$q{y}_i^r\le \sum _{i\in N}{q}_i^r{x}_{ij}^r \forall r\in R$$

(10)

$$x_{ij}^m\left(l_i^m+u_i^m+s_i+t_{ij}-l_j^m\right)=0 \forall m\in K_m,\left(i,j\right)\in A$$

(11)

$$a_i{y}_i^r\le t_i^r\le b_i{y}_i^r \forall i\in N,\forall r\in R$$

(12)

$$t_o^r\ge \beta \sum _{i\in N}{s}_i{y}_i^r \forall r\in R$$

(13)

$$t_i^r\le t_o^r+t_{\max } \forall i\in N,\forall r\in R$$

(14)

$$t_o^s+M\left(1-z_{rs}\right)\ge t_o^{\prime r}+\beta \sum _{i\in N}{s}_i{y}_i^s \forall r,s\in R, r<s$$

(15)

$$\sum _{r\in R}\sum _{s\in R|r<s}{z}_{rs}\ge \left|R\right|-K$$

(16)

$$x_{ij}^r\in \left\{0, 1\right\} \forall \left(i,j\right)\in A,\forall r\in R$$

(17)

$$y_i^r\in \left\{0, 1\right\} \forall i\in N,\forall r\in R$$

(18)

$$z_{rs}\in \left\{0, 1\right\} \forall r,s\in R,r<s$$

(19)

$$t_i^r\ge 0 \forall i\in N,\forall r\in R$$

(20)

The objective function aims to minimize the total transportation cost while maximizing the customer service by balancing the distance traveled and associated penalties. Equation (2) maintains consistency between the routes and customer visits, while Equation (3) ensures that each customer is visited at most once. Flow conservation is enforced by Equation (4), ensuring continuity in routes. Equations (5) and (6) guarantee that all routes start and end at the depot.

Equations (7) and (8) ensure that each customer is visited exactly once by any vehicle, preventing redundancy. Capacity constraints are enforced by Equation (9), which ensures that the vehicle loads do not exceed capacity. Equations (10) and (11) manage the customer demand and service timing, ensuring feasible assignments and adherence to schedules. Equation (12) ensures that the service times fall within the specified time windows, while Equation (13) aligns the service start times with depot preparation.

Equations (14) and (15) enforce maximum route time limits and the correct sequencing of routes within a working day. The fleet size is controlled by Equation (16), ensuring that the number of routes does not exceed the available vehicles. Finally, Equations (17)–(20) define the binary nature of the decision variables, sequence routes logically, and ensure non-negative service start times, maintaining the overall feasibility and coherence of the model.

4. Methods for Optimization Based on Active Constraints

This study examines a category of algorithms where the determination of the direction around the active constraint boundary depends on the correlation between an orthogonal $Z$ matrix and a normal constraint matrix. Consequently, if $\widehat{A}x=\widehat{b}$ represents the most recent set of active constraints of size $n-s$, $Z$ is an $n\times s$ matrix structured as follows:

$$\widehat{A}Z=0$$

(21)

The following steps outline the essential tasks to be carried out in each iteration, ensuring the generation of a suitable descent direction, denoted as p.

• Calculate the reduced gradient:

$$g_A=Z^{T}g$$

2.

Generate approximations to reduce the Hessian, particularly focusing on

$$G_A\doteqdot Z^{T}GZ$$

3.

Obtain approximations for systems of equations

$$Z^{T}GZ{p}_A={-Z}^{T}g$$

(22)

4.

by resolving the system

$$G_A{p}_A={-g}_A$$

5.

Establish the direction to achieve $p=Z{p}_A$.

6.

Utilize a line search to identify the nearest approximation to $a^{*}$ where

$$f\left(x+{\alpha }^{*}p\right)=\underset{\begin{array}{c}\alpha \\ \left\{x+\alpha p \mathrm{feasiabel}\right\}\end{array}}{\min }f\left(x+\alpha p\right)$$

Besides possessing a full column rank, such as (27) being the sole algebraic constraint on Z, there are various possible forms for $Z$. The form that $Z$ takes parallels the procedure itself, notably

$$Z=\left[\begin{array}{c}-W\\ I\\ 0\end{array}\right]=\left.\left[\begin{array}{c}{-b}^{-1}S\\ I\\ 0\end{array}\right]\right\}\begin{array}{c}m\\ s\\ n-m-s\end{array}$$

(23)

This is a fundamental explanation for the analysis in the subsequent section, emphasizing that it exclusively operates computationally using the $S$ and triangular $\left(LU\right)$ factorizations of $B$. The complete computation of the $Z$ matrix is not undertaken.

$Z$, characterized by its orthonormal column vectors $(Z^{T}Z=I)$, is considered for various reasons. The primary benefit of the $Z$ transformation is its ability to avoid introducing redundant conditions into the problem reduction (as outlined in steps A–D above, particularly Equation (22)). This approach has been implemented in software where the accumulation of $Z$ occurs in the form of a dense matrix. The LDV factorization of the matrix $\left[\begin{array}{cc}B& S\end{array}\right]$ facilitates its extension to a sparsely distributed linear form:

$$\left[\begin{array}{cc}B& S\end{array}\right]=\left[\begin{array}{cc}L& O\end{array}\right]DV$$

In this context, $L$ represents a triangular matrix and $D^{1/2}V$ is normal, $D$ is diagonal, and $V$ is orthogonal, with $L$ and $V$ being accumulated through products. However, if $S$ contains a significant number of columns, this factorization will consistently result in a much denser matrix compared to the $LU$ factorization of $B$. Therefore, the decision to proceed with $Z$ in Equation (23) is based on performance considerations. Simultaneously, it is important to recognize (due to $B^{-1}$’s unwelcome appearance) that B requires careful handling to optimize the performance.

This section provides a condensed description of the optimization algorithm. Suppose that we are in possession of the following items:

1.

$\left[\begin{array}{ccc}B& S& N\end{array}\right]x=b, l\le x\le u$ is fulfilled by a feasible vector $x$;

2.

The corresponding function value $f\left(x\right)$ and the gradient vector $g\left(x\right)={\left[\begin{array}{ccc}{g}_B& g_S& g_N\end{array}\right]}^T$;

3.

The number of superbasis variables, $s(0\le s\le n-m)$;

4.

Factorization, $LU$, on the base matrix $B m\times m$;

5.

The factorization, $RTR$, of the quasi-Newton approach to the $s\times s$ matrix is $Z^{T}GZ$ (it should be noted that $G$, $Z$, and $Z^{T}GZ$ are never truly counted);

6.

A vector $rr$ that meets $B^T\pi =g_B$;

7.

The reduced gradient vector $h=g_S-S^T\pi$;

8.

$TOLRG$ and $TOLDJ$ both have small positive convergence tolerances.

In order to solve the model, we employ the generalized reduced gradient method, which initially uses the Lagrange function and then proceeds according to the algorithm.

The algorithm then proceeds as follows.

Step 1.

(Convergence testing in a known subspace). If $‖h‖>\mathrm{TOLRG}$, proceed to step 3.

Step 2.

(“PRICE”, i.e., calculate the Lagrange multiplier, add one superbase).

a.

Determine $\lambda =g_N-N^T\pi$.

b.

Choose ${\lambda }_{q_1}<-\mathrm{TOLDJ} \left({\lambda }_{q_2}>+\mathrm{TOLDJ}\right)$, $\lambda$’s largest element that corresponds to the variables in its upper (lower) bound. If not, STOP; Kuhn–Tucker’s essential requirements for an optimal solution have been met.

c.

If this is not the case

i.

Select $q=q_1$ or $q_2$ based on $\left|{\lambda }_{q_1}\right|=\max \left(\left|{\lambda }_{q_1}\right|,\left|{\lambda }_{q_2}\right|\right)$;

ii.

Insert $a_q$ as the new column $S$;

iii.

Insert ${\lambda }_1$ as a new $h$ element;

iv.

Sum up a new relevant column to $R$.

d.

Multiply S by 1.

(Note: MINOS also has a DOUBLE PRICE alternative, which provides several non-basic variables to be a superbase).

Step 3.

(Determine the search direction, $p=Z{p}_s$).

a.

Complete $R^{T}R{p}_S=-h$.

b.

Complete $\mathrm{LU} p_B=-S{p}_S$.

c.

Make $p=\left[\begin{array}{c}{p}_B\\ p_S\\ 0\end{array}\right]$.

Step 4.

(Test Ratio, “CHUZR”).

a.

If ${\alpha }_{\max }\ge 0$, the highest $\alpha$ value of $x+\alpha p$ is feasible.

b.

If ${\alpha }_{\max }=0$, proceed to step 7.

Step 5.

(Line search).

a.

Determine $\alpha$, an ${\alpha }^{*}$ approximation in which

$$F\left(x+{\alpha }^{*}p\right)={\min }_{0<\theta \le {\alpha }_{\max }}f(x+\theta p)$$

b.

Convert $x$ to $x+\alpha p$ and $f$ and $g$ to their respective values in the new $x$.

Step 6.

(Calculate the reduced slope, $\overline{h}=Z^{T}g$).

a.

Complete $U^T{L}^T\pi =g_B$.

b.

Determine the new reduced slope, $\overline{h}={g_S-S}^T\pi$.

c.

Using $\alpha ,p_S$ and metric-variable recursion on $R^{T}R$, modify $R$ and switch in reduced gradient $\overline{h}-h$.

d.

Set $\overline{h}-h$.

e.

If $\alpha <{\alpha }_{\max }$, proceeds to step 1. As there are no new constraints found, they persist within this subspace.

Step 7.

(Exchange base if required; eliminate one superbase). Here, $\alpha <{\alpha }_{\max }$ has reached one of its limits, and, for some $p(0<p\le m+s)$, the variable associated with the $p$ column of $\left[\begin{array}{cc}B& S\end{array}\right]$ has also attained one of its limits.

a.

In the case of a base variable exceeding the limit $(0<p\le m)$,

i.

Substitute the $p$-th column with the $q$-th column of $\left[\begin{array}{c}B\\ X_B^{\mathrm{T}}\end{array}\right]$ and $\left[\begin{array}{c}S\\ X_S^T\end{array}\right]$;

Assume that $q$ is chosen so that $B$ non-singular is maintained (this involves a $\pi p$ vector that fulfills $U^T{L}^T{\pi }_p=e_p$);

ii.

Changes to $L$, $U$, $R$, and $\pi$, as well as changes to $B$, to reflect these changes;

iii.

Identify the latest gradient at the bottom $h=g_S-S^T\pi$;

iv.

Go to (c).

b.

Otherwise, the variable superbase reaches its maximum value $(m<p\le m+s)$. Determine $q=p-m$.

c.

After reaching the desired limit, create the $q$-th variable in nonbasis $S$, accordingly:

i.

Eliminate the $q$-th column from $\left[\begin{array}{c}S\\ X_S^T\end{array}\right]$ and $\left[\begin{array}{c}R\\ h^T\end{array}\right]$;

ii.

To the triangular matrix, add $R$.

d.

Step 1 should be repeated after subtracting by one.

5. Problem Description for MVRPTW

A local retail chain, Informi Retailers, operates multiple depots and collaborates with several suppliers to ensure the timely delivery of goods to its stores across the province. The company faces the challenge of optimizing its delivery routes to minimize costs while maintaining its service quality. Each depot has a fleet of vehicles with specific capacity constraints, and each supplier has a set of delivery requirements with relaxed time windows. The goal is to develop a model that minimizes the total operational costs while ensuring that all deliveries are made within the allowed time frames.

5.1. Real Problem Example

Informi Retailers manages a distribution network consisting of five depots located in major cities within North Sumatra Province and collaborates with ten key suppliers. The depots are responsible for delivering goods to 200 retail stores spread across the country. Each supplier provides goods that need to be delivered to specific stores, with delivery time windows that are flexible by $\pm$2 h. The depots’ fleets consist of vehicles with varying capacities and fuel efficiencies. The objective is to minimize the combined transportation and inventory holding costs while ensuring that all deliveries are completed within the relaxed time windows.

5.2. Methodology

We develop a non-linear optimization model to solve the MDMS-VRPRTW, as formulated in Equations (1)–(20). The model incorporates variables representing the assignment of vehicles to routes, the sequence of deliveries, and the timing of each delivery. Then, we solve the model using the algorithm as mentioned in the Method section.

To provide a clear understanding of the specific parameters and constraints considered in this study, the data relevant to the problem are outlined below.

Problem Details:

• Number of vehicles—4;
• Number of customers—8;
• Number of routes—4;
• Multiple suppliers and depots;
• Flexible but constrained time windows for delivery.

6. Results and Discussion

The non-linear optimization model is applied to the Informi Retailers problem instance. Computational experiments demonstrate that the model effectively reduces the total operational costs compared to the company’s existing routing strategy. The relaxed time windows provide significant flexibility, allowing for the better utilization of the vehicle fleet and reducing the need for expedited deliveries. The sensitivity analysis highlights the impact of various parameters, such as the vehicle capacity and fuel efficiency, on the overall solution quality.

The number of iterations needed to obtain the optimal value is 238 with the objective value $5.16\times {10}^2$.

Table 1. Results of binary variable $x_{ij}^r$.

Route	Customer	Customer
		0	1	2	3	4	5	6	7	8
1	0		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00000		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	0.00000		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000	0.00000
	4	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000	0.00000	0.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000	0.00000
	7	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000
2	0		0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00000		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	1.00000	0.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		1.00000	0.00000	1.00000	0.00000	0.00000
	4	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000	1.00000	0.00000
	6	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000
	7	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000
3	0		0.00000	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000	0.00000
	1	0.00000		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000	1.00000
	4	0.00000	0.00000	0.00000	0.00000		0.00000	0.00000	1.00000	0.00000
	5	0.00000	0.00000	0.00000	1.00000	0.00000		0.00000	0.00000	0.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000
	7	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000
	8	0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
4	0		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00000		1.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		0.00000	0.00000	0.00000	1.00000	0.00000
	4	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000	0.00000	1.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000
	7	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000

and

Table 2. Results of binary variable $y_i^r$.

Vehicle	Customer	Customer
		0	1	2	3	4	5	6	7	8
1	0		0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	0.00000		0.00000	0.00000	1.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000
	4	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000	0.00000
	7	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000
2	0		0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	1.00000	0.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		0.00000	1.00000	1.00000	0.00000	0.00000
	4	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	0.00000	1.00000	0.00000		0.00000	0.00000	0.00000
	6	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000		0.00000	0.00000
	7	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		1.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000	0.00000
3	0		0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		0.00000	0.00000	0.00000	0.00000	1.00000
	4	0.00000	0.00000	0.00000	0.00000		0.00000	0.00000	1.00000	0.00000
	5	0.00000	0.00000	0.00000	1.00000	0.00000		0.00000	0.00000	0.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000
	7	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000
	8	0.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000
4	0		1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00000		1.00000	0.00000	1.00000	0.00000	0.00000	0.00000	0.00000
	2	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	3	0.00000	0.00000	0.00000		0.00000	0.00000	0.00000	1.00000	0.00000
	4	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000	0.00000	0.00000
	5	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000	0.00000	1.00000
	6	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000		0.00000	0.00000
	7	1.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000		0.00000
	8	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	1.00000	0.00000

present the results from the model for the binary variables $x_{ij}^r$ and $y_i^r$, respectively.

Table 3. Start time from each node variable $t_i^r$.

Route	Customer
	1	2	3	4	5	6	7	8
1	39.74026	69.48052	40.00000	20.64935	20.00000	10.25974	10.00000	19.87013
2	40.00000	40.00000	60.00000	20.00000	10.00000	30.00000	10.00000	20.00000
3	0.00000	0.00000	30.00000	0.00000	0.00000	59.87013	110.00000	100.12987
4	60.25974	10.51948	0.00000	99.35065	110.00000	19.87013	0.00000	0.00000

presents the variable $t_i^r$ to denote the start time of service for customer $i$, if served by route $r$.

7. Conclusions

This paper addresses a company operating a fleet of vehicles tasked with delivering multiple products from various suppliers to a set of customers, without strict delivery time constraints. The objective is to optimize the vehicle routing to minimize the overall transportation costs, which include the travel distance, vehicle utilization, and deviations from the expected delivery times, while ensuring that the customer demand is met and the relaxed time windows are adhered to.

To tackle this problem, we formulated it as a combinatorial optimization problem. We proposed a hybrid approach that begins with the development of a generalized reduced gradient method, designed to find “near” integer feasible solutions. Following this, a feasible neighborhood search method was implemented, focusing on minimizing the deterioration in the objective function. This approach allows for the deeper exploration of the solution space and reduces the risk of becoming trapped in local optima, making it particularly effective in complex or highly constrained scenarios.

The VRPRTW demonstrates superior flexibility and efficiency in vehicle utilization and route planning. By reducing the pressure to meet strict deadlines, it allows for more cost-effective routing solutions. The feasible neighborhood search approach further enhances the performance, providing a balanced strategy that controls the risks while potentially leading to better overall solutions.