To solve the larger cases of MT-VRPB, a two-level VσS algorithm called 'two-level VNS' is developed. Therefore, the two-level VNS is designed to obtain a maximum balance of diversification and intensification during the research process.
Introduction
- Introduction and Motivation
- The Multiple Trip Vehicle Routing Problem with Backhauls
- Aims and Objectives of the Thesis
- Outline of the Thesis
This chapter focuses on introducing a new variant of VRP that is being studied in this thesis, i.e., the Multi-Travel Vehicle Routing Problem with Backhauls (MT-VRPB). Adaptation of two-level VNS and Mat-heuristic with VRPB and MT-.
The Vehicle Routing Problem: Models and Solution Methods
The Vehicle Routing Problem and its Variants
- The Evolution of the Vehicle Routing Problem
- Definition of the Vehicle Routing Problem
- VRP Variants
- The Periodic VRP
- The Multiple Trip VRP
- The Multiple-Depot VRP
- The Mix Fleet VRP
- The VRP with Time Windows
- The Split Delivery VRP
- The classical VRP with Backhauls
- The VRP with Mixed Deliveries and Pickups
- The VRP with Simultaneous Deliveries and Pickups
- Future of the VRP
The Periodic Vehicle Routing Problem (PVRP) is a generalization of the classic VRP that deals with the planning period aspect of the problem. There are a few variants of VRPB that have been modeled and studied in the literature.
Combinatorial Optimisation: Problems and Algorithms
- Combinatorial Optimisation Problems
- The term Algorithm
Nevertheless, it is notable that research is constantly trying to develop models that are closer to the real applications of the vehicle routing. Therefore, we recognize that the word solution has different meanings in different situations, where the best solution to a problem can be either an optimal solution or a feasible solution.
Solution Techniques for the Vehicle Routing Problems: An Overview
- Exact Methods
- Heuristic Methods
In real-life problems, it is better to be able to find an approximate solution to the real problem than to find the optimal solution to an approximate model problem. However, the methods for dealing with VRP and its variants are not limited to those mentioned.
Examples of Exact Methods
- The Branch-and-Bound Method
Backtracking (known as depth-first search) guides the search tree by branching down one side of the search tree and quickly finding a possible solution. The process terminates when all nodes in the search tree are pruned or resolved, with the upper and lower bounds of the unpruned subregions equal to the global minimum of the function.
Examples of Heuristic Methods
- Construction-based Heuristics
- The Sweep Algorithm
- Other Construction-based Heuristic for VRPs
- Intra- and Inter-Route Improvement Heuristics
- Transfer Heuristics
- Swap Heuristics
- Other Improvement Heuristics for VRPs
Two of the more commonly used transfer heuristics are briefly introduced in the following sections. In the VRP literature, there are several other improvements/local search heuristics that have been used to improve the solution of VRP and its various extensions.
Examples of Metaheuristic Methods
- Variable Neighbourhood Search
- Large Neighbourhood Search
- Other Metaheuristic Methods for VRPs
The metaheuristic method that was used in this study belongs to the category of local search methods. The iterative procedure used for VNS first performs an oscillation, then a local search method is applied and the comparison of the two solutions would yield a local optimum obtained.
Hybrid Methods
Ant Colony Optimization (ACO), also known as the Ant Systems (AS), introduced by Moyson and Manderick (1988), Colorni et al. For mathematics, see Jourdan et al. 2009) for the taxonomy of hybridizing exact and metaheuristic methods and a recent study of mathematics by Ball (2011).
Summary
The above MT-VRPB formulation can be adapted to allow for the following four variants. a) The MT-VRP: this can be achieved by simply setting the number of backhaul customers equal to zero using equation (4.14). In the literature, these refinement routines are implemented in two solution acceptance criterion strategies, that is, the first-improvement and the best-improvement.
Literature Review of the VRPB and the MT-VRP
An Overview of the VRPB
Another use of VRPB can be found in the food distribution industry, where food is distributed to stores (considered line customers) from distribution centers. In addition, VRPB applications can be found in many other real-world scenarios where the return of goods to a distribution center is involved, i.e. reverse logistics (Cuervo et al. 2014).
Solution Methods for the VRPB
- Exact Methods
- Heuristic Methods
- Metaheuristic Methods
- Studies in VRPB-related areas
Ghaziri and Osman (2006) proposed a self-organizing feature map (SOFM) methodology for VRP with backhaul which is based on the concept of Neural Networks. Zachariadis and Kiranoudis (2012) developed a local search heuristic for classical VRPB that explores rich solution neighborhoods (i.e., neighborhoods consisting of variable length customer sequences) and uses Fibonacci stored local search moves Heaps (Fibonacci heaps are essentially special types of priority queue structures that allow a program with capabilities such as fast insertion, deletion, and rollback).
An Overview of the MT-VRP
Another closely related formulation was subsequently presented in Azi et al. 2010) who developed a branch-and-price model based on a group packing formulation for MT-VRP with an additional aspect of time windows. Recently Mingozzi et al. 2013) developed two similar set-splitting formulations for MT-VRP.
Solution methods for the MT-VRP
- Exact Methods
- Heuristic Methods
- Metaheuristic Methods
- Studies in MT-VRP related areas
- Studies in which VRPB and MT-VRP are addressed in a combined way
The proposed heuristic is tested on the MT-VRP dataset proposed by Taillard et al. Olivera and Viera (2007) developed an adaptive memory programming (AMP) approach based on the AMP principle of Rochat and Taillard (1995) to solve the MT-VRP.
Summary
162 Table 6.10: Detailed results of comparison of CSMH versus two-level VNS for data set-1. 164 Table 6.11: Detailed results of comparison of CSMH versus two-level VNS for dataset-1.
The Multiple Trip Vehicle Routing Problem with Backhauls: Formulation and
The Multiple Trip Vehicle Routing Problem with Backhauls
- Description of the MT-VRPB
- Graph theoretical definition of the MT-VRPB
MT-VRPB is supposed to form a set of minimum cost timetables in which each customer (LH/BH) is visited exactly once along the routes (starting and ending at the same depot) included in the schedules. In the next section, we briefly review the exact options of the methods for MT-VRPB.
Exact methods options for the MT-VRPB
A route cost (duration) is equal to the sum of the travel costs (travel times) of the nodes crossed. The methods can be computationally expensive if the number of positions is not taken into account (Christofides, 1981b).
Mathematical Formulation of the MT-VRPB
- Formulation of the basic case
- Model complexity
- Model variants and restricted problems
Constraints (4.2) and (4.3) ensure that each customer is served exactly once (each customer has an incoming arc and each customer has an outgoing arc). However, the formulation can be extended to allow for the condition where the number of vehicles to be used is exactly the given number K.
Significance of the MT-VRPB
75 Restrictions (4.16) sets restrictions on each vehicle that must be used at most once (equal sign can be used if each vehicle must be used) and thus blocks the use of multiple trips of vehicles. d) Finally, the objective function in the above formulation can be changed from reducing the total distance traveled to reducing the number of vehicles. Or one can set the objective function shown in equation (4.17) as a primary objective and minimizing the total distance traveled as a secondary objective.
Utility of IBM ILOG CPLEX optimisation studio
In the following sections, we briefly look at the utility of CPLEX software, the details of generating new data sets, and our CPLEX solution approach for the MT-VRPB. These solvers can efficiently address and solve a variety of problems; that is, mixed integer programming (MIP), quadratic constrained programming (QCP), linear programming (LP), and quadratic programming (QP) problems.
Validation of the MT-VRPB formulation
The optimal solution contains three paths with a total distance of 30, where path and are served by vehicle 2 and path is served by vehicle 1. To ensure that the CPLEX solution validates our mathematical formulation, we generated all 6 possible feasible solutions of the test instance by manual enumeration and found the same solution produced by CPLEX.
Generation of a new data set for the MT-VRPB
82 Name: institution identification name; v: number of vehicles - starts with an integer between one and the maximum number of vehicles; Tnb: total number of vehicles in each case; : maximum driving time of type one for each vehicle; : maximum driving time of type two for each vehicle.
CPLEX Results and Analysis
- Relevance of the results
93 Table 4.10: Comparison of free fleet VRPB and MT-VRPB solutions according to. 94 Table 4.11: Comparison of free fleet VRPB and MT-VRPB solutions according to.
Summary
These neighborhood schemes are used in the swing and local search phases of the two-level VNS algorithm. 178 Table 7.2: Comparison of the best VRPB algorithms with two-level VNS (dataset-2) Algorithm runs # Best solution.
A Two-Level Variable Neighbourhood Search Algorithm for the Multiple-Trip
Two-Level VNS Algorithm: An Overview
- An overview of the algorithm
For the outer level we define as a subset of neighborhoods (oscillating in the outer level) and as a subset of local search refinement routines; and at the inner level as a complete set of neighborhoods (oscillation at the inner level) and as a complete set of local search refinement routines. Note that it represents a subset of the neighborhoods and a subset of the local search refinement routines used in the outer layer.
Initial solution (Phase I)
- Solving the Assignment Problem
In Figure 5.5, an illustrative example of the problem case eil21_50 is shown, which demonstrates the visual characteristics of the open-ended routes. Depot Linehaul Customers Backhaul Customers Figure 5.5: LH and BH open-ended routes (Problem instance eil22_50 of dataset-2).
Neighbourhoods used in the Two-Level VNS Algorithm (Phase II)
As shown in Figure 5.14, some (consecutive) line track nodes from route r1 are swapped with a line track node from route r2 to achieve a reduction in travel cost. Routes to 1-Insertion Depot Linehaul Client Backhaul Client Figure 5.10: An illustrative example of the 1-Insertion (inter-route) refinement routine.
Multi-layer local search optimiser framework (local search stage)
In the best improvement strategy, the best of all possible improvements is accepted at the end of the search cycle. Note that in the original implementation of the multilevel heuristic (Salhi and Sari, 1997), the best optimization strategy is used instead.
Solving the Bin Packing Problem (Phase III)
Bisection Method (Repair Mechanism): The Bisection method works in such a way that the capacity of the bin(s) is iteratively increased by a certain percentage until routes are packed into bin(s). For example, as you can see in Figure 5.19, the capacity of the bin(s) increases by 5% with each iteration; and suppose that the required capacity of bins is reached with a level increase of 25%; It then attempts to optimize bucket capacity using a decreasing/increasing percentage mechanism, for example starting with a decreasing percentage of -2.5%, which is then iteratively reduced/increased by half (i.e. and so on) until the overtime of the bin(s) have been reached. optimized.
Computational Experience
- Introduction and Computer Details
- Results and analysis
- Search diversification and intensification analysis
It can be observed (see table 5.2 and table 5.3) that quality solutions are found when the container capacity is relatively large and the number of containers is smaller. Although the number of cases where other neighborhoods lead to better quality solutions is slightly lower than these two, but their importance cannot be ignored as they also play a key role in diversifying the search and thus lead to better solutions.
Summary
The detailed results of two-level VNS against the most popular solutions are given in Table 7.4 and Table 7.5 for cluster-2 and cluster-3, respectively. The performance of the CSMH algorithm is compared with the two-level VNS algorithm, as well as with some of the best published algorithms described previously.
Solving the MT-VRPB using a Collaborative Sequential Mat-heuristic approach
The Mat-heuristic Approaches
- Matheuristics for VRPs: Brief Literature Review
In the second stage, the set partitioning formulation is solved on the set of routes obtained in the first stage. In the second stage, a set partitioning model is solved to select the best routes from the set of all routes.
The Collaborative Sequential Approach for the MT-VRPB
If the solution selected from the data structure is feasible with respect to the specified planning period T, then the MIPstart variables have no problem working with the MT-VRPB formulation. The C++ programming language code we used to add MIPstart to our model is provided in Appendix A .
Computational Experience
- Data Set
- The CSMH execution times
- The CSMH algorithm performance
- Comparison of the CSMH vs CPLEX results
- Comparison of the CSMH and the Two-Level VNS results
The CSMH algorithm found a large number of optimal solutions (38 out of 84) and a large number of existing solutions (12 out of 84). For the set of instances, the CSMH algorithm found a large number (more than half of the instances) of optimal solutions (46 out of 84) and a large number of new, best existing (feasible) solutions (18 out of 38 non-optimal ). .
Summary
The results produced by the two-level VNS and CSMH algorithms are compared to the best published solutions of benchmark cases of these problems from the literature. The detailed results of the CSMH algorithm are given in Table 7.8 and Table 7.9 for group-2 and group-3, respectively.
Adaptation of the Two-Level VNS and Mat-heuristic to the VRPB and the MT-
The case of the VRPB
- VRPB Formulation
- The Two-Level VNS Algorithm for the VRPB
- Details of the VRPB Computations and the Data sets
- Two-Level VNS VRPB Results and Analysis
- Solving the VRPB with Mat-heuristic (CSMH algorithm)
- CSMH VRPB Results and Analysis
For dataset-2, Two-Level VNS outperformed two of the algorithms in terms of the number of best-known solutions found and matched with ILS (2014). For dataset-2, the CSMH algorithm outperformed two of the algorithms in terms of the number of best-known solutions found; however, it did not outperform Two-Level VNS and ILS (2014).
The case of the MT-VRP
- Formulation of the Basic Case
- The Two-Level VNS methodology for the MT-VRP
- Details of MT-VRP Computations and the Data sets
- Two-Level VNS MT-VRP Results and Analysis
- Solving the MT-VRP with the Mat-heuristic (CSMH algorithm)
- CSMH MT-VRP Results and Analysis
Summary
Conclusions
Research Summary
Future Research
