The literature is rich for symmetric VRPs and poor for asymmetric VRPs, although the symmetric VRPs is considered as a special case of asymmetric VRPs. The exact methods of asymmetric VRPs have a weak performance on symmetric VRPs. Furthermore, the methods designed for symmetric VRP instances may not be adapted easily to solve asymmetric VRPs [70].
Surprisingly, ADVRP is not studied like other types of VRPs. There are a few papers that discuss this problem see [99, 113].
• The first paper was in 1984 by Laporte, Desrochers and Nobert [108]. It presents two
exact algorithms for DVRP. One of them is based on Gomory cutting planes and the other one is based on branch and bound. They deal with symmetric instances (Eu- clidean and non-Euclidean) where Euclidean means that the distance matrix satisfies
the triangle inequality. They conclude that solving non-Euclidean instances is easier than solving Euclidean, where both algorithms are able to find the optimal solution with up to 50 customers in Euclidean cases and 60 in non-Euclidean instances. In addition, the cutting plane algorithm performs better than branch and bound algo- rithm. Moreover, in both algorithms, solving instances become more difficult when the maximum distance allowed is decreased.
• Laporte, Nobert and Desrochers in 1985 present an integer linear programming al-
gorithm to solve VRP with distance and capacity constraints. They use relaxation constraints and subtour elimination constraints. They solve the model with up to 60 customers [112] using Euclidean and non-Euclidean instances.
• The third paper was in 1987 by Laporte et. al [113]. It is considered as the first
paper with Asymmetric DVRP in the operations research literature. They use similar techniques to Laporte et. al (see [110]) which is originally considered as an extension to the algorithm of Carpaneto and Toth for TSP [28].
An exact algorithm for solving ADVRP is developed in [113]. It uses the branch and bound method where the relaxation problem is the modified assignment problem. They extend the distance matrix based on the technique of Lenstra and Rinnooy (see [120]) by adding (m − 1) dummy depots where m represents the number of vehicles. The solution is feasible to ADVRP if two conditions are satisfied:
– the solution contains m hamiltonian circuits.
– the length for each of them is less than or equal to the maximum distance allowed.
In the case that the infeasible solution is obtained, the infeasible circuit is eliminated by adding a new constraint. This means the illegal subtour is eliminated by branching this infeasible subproblem into subproblems.
They find the first feasible solution by adapting Clarke and Wright’s algorithm (see [32]). If this is not able to provide a feasible solution then the upper bound (U B) is set to: U B = m × Dmax where Dmax represents the maximum distance allowed. If
the total length of a subtour is greater than Dmax, then it has to be eliminated. They
eliminate illegal tours by excluding arcs.
They use randomly generated instances, with two types of distance matrices, those satisfying and not satisfying the triangle inequality. This method is able to solve up to 100 customer problems for ADVRP. They conclude that solving tighter problems are more difficult.
• The forth paper published in 1992 by Li et. al, see [122], considers two objective func-
tions to DVRP: minimize total distance and minimize the number of vehicles used. They transform the DVRP into a multiple traveling salesman problem with time win- dows (mTSPTW), where the time window constraint [ai, bi] for any customer i means
that it is not allowed to serve customer i before ai or after bi. In other words, the
vehicle has to wait until time ai to start before dealing with customer i. For details on
time windows with VRP see Section 1.1.
In order to enable the transformation, the distance constraint is used as a time window constraint [0, Dmax] for all customers, and another copy of the depot is added to the
graph. The time window for the first depot is [0, 0] (departure depot), and the time window for the last depot is [0, Dmax] (arrival depot). It is solved using a column
generation approach. They present and analyze the worst case performance for DVRP with a heuristic and provide results with up to 100 customers. The comparison includes the length of the initial tour and the value of the lower bound.
• Conference paper by Almoustafa et.al in 2009 [7]. An old branch-and-bound method
(suggested by Laporte et al. in 1987) is revised and modified. This method is based on reformulating the distance–constrained vehicle routing problem into a traveling sales- man problem and use of the assignment problem (AP) as a lower bounding procedure. The Hungarian algorithm is used to find the solution to AP (an efficient implementa- tion of Hungarian method for AP). In [7] branching based on tolerances and costs are used in two algorithms.
is found, the performance of tolerance-based algorithms is better than the performance of cost-based algorithms. On the other hand, the opposite is held when the CPU time is considered. Both algorithms are able to find optimal solution up to 200 customers.
• Kara emphasized in 2011 that there are still a limited number of published papers on
DVRP in this area of literature [98, 99, 100]. Kara’s technical report [99] displayed the existing formulations and presented new formulations for DVRP:
– flow based formulation. – vertex based formulation.
All new formulations have O(n2) binary variables and O(n2) constraints and it can
be used by commercial solvers such as CPLEX. The flow based formulation performs better than vertex based formulation according to the computational times. On the other hand, the vertex based formulation provides better lower bounds than flow based formulation [98].
Kara recommends flow based formulation to solve small and moderate-sized cases, while vertex based formulation to be used to improve heuristic procedures for DVRP [98]. Finally the proposed formulations by Kara can be adapted by adding other constraints to DVRP.
1.3.2 Our Approach
Our target is to increase the size of instances that can be solved exactly by our approach to solve ADVRP. In addition, our target is to propose a simple and robust algorithm. In this thesis we propose three results.
Firstly, we present a general flow-based formulation to solve ADVRP. This formulation is more general than Kara formulation [98], since it does not require the distance matrix to satisfy the triangle inequality. It produces a solution faster than the adapted formulation. In addition, we are able to improve the quality of the objective function in case the optimal solution is not reached because of stopping conditions.
Secondly, we use tolerance based branching rules [6] and try to improve it in different ways. First of all by using CPLEX as a lower bounding procedure to solve AP and comparing that with the Hungarian algorithm. We find that the Hungarian algorithm produces a solution in shorter CPU time when compared with CPLEX for solving AP. In other words, there is a big gap, in terms of time, between using the Hungarian algorithm and CPLEX to solve AP.
Tolerance based branching rules method is fast but memory consuming, and could stop before optimality is proven. Therefore, we introduce randomness in choosing the node of the search tree in cases where we have more than one choice. If an optimal solution is not found and restart is required due to memory issues, we restart our procedure. In this way, we get a multistart branch and bound method.
Computational experiments show that we are able to exactly solve large test instances with up to 1000 customers. So, despite the simplicity, this proposed algorithm is capable of solving the largest instances ever solved in literature. As far as we know, those instances are much larger than instances considered for other VRP models and exact solution approaches from recent literature. For example CVRP is not always able to solve instances optimally with more than 200 customers [48].
In order to compare our approach we use a commercial IP solver (CPLEX) to get the optimal solution of the ADVRP. Using CPLEX solver to obtain the optimal solution to ADVRP faces some difficulties for two reasons. The first reason is related to the CPU time which is too long and the second reason is related to the larger instances which can’t be uploaded due to lack of memory.
Thirdly, we develop heuristic based on VNS to find a good feasible solution in case our exact multistart branch and bound method stops because of memory or stopping conditions. We use the route-first-cluster-second approach to transfer TSP solution to ADVRP solution. Unsatisfactory results are obtained. The reason for not getting expected good results could be the route-first-cluster-second approach.
1.4
Thesis Overview
The structure of this thesis is as follows:
• Chapter 1 we present classification of VRPs based on constraints (unconstrained VRP
and constrained VRP), then we explain in more detail definitions and formulations of the main constrained VRPs: Capacitated VRP; distance-constrained VRP; VRP with time windows; VRP with backhauls; and VRP with pickup and delivery. In addition, basic formulation types for VRPs are presented.
• Chapter 2 we provide basic information about the solution methods: Exact methods
that find the optimal solution such as: branch and bound, cutting plane, branch and cut, column generation, cut and solve; branch-and-cut-and-price, branch-and-price, and dynamic programming; Classical Heuristics that find approximate solution such as: constructive heuristics, two phase methods, improvement heuristics; Metaheuristics are classified into three groups, and are presented with basic information as follows:
1. Local search based metaheuristics: such as multi-start method, simulated an- nealing (SA), tabu search (TS), greedy randomized adaptive search procedure (GRASP), neural networks (NN), variable neighborhood search (VNS), and guided local search (GLS).
2. Population Based (natural inspired): genetic algorithm (GA), evolutionary al- gorithm (EA), scatter search (SS), Ant colony optimization (ACO), and Path Relinking (PR).
3. Hybrid metaheuristics.
In addition, we provide more information on VNS, since it is used in Chapter 5 to find feasible solutions to ADVRP. We propose different variants of VNS types and their algorithms.
• Chapter 3 we present three formulations for ADVRP: adapted bus school routing prob-
lem (ABSRP), Kara formulation, and our general formulation. We explain the differ- ence between Kara formulation and our general formulation then we compare between
adapted formulation and our general formulation with an illustrative example. Finally, we present computational results and the conclusion.
• Chapter 4 Multistart Branch and Bound for ADVRP (MSBB − ADVRP): a simple intro-
duction is presented in section 4.1, then mathematical programming formulations of ADVRP is in section 4.2. We discuss in section 4.3 single start branch and bound for ADVRP and most of the relevant basic concepts which are used later: upper bound, lower bound, branching rules, the algorithm, and an illustrative example. A descrip- tion of the multi start method used to solve ADVRP is given in section 4.4 with the algorithm and an example. An efficient implementation and data structure are given in section 4.5. Computational results are provided in section 4.6 with methods compared, numerical analysis and summary tables of results. Section 4.7 contains the conclusion and future research directions. For more information on the detailed tables of results see Appendix A.
• Chapter 5 Variable neighborhood search (VNS), we explain our VNS based heuristic
for solving ADVRP. The initialization algorithms are presented in section 5.1, and the main algorithm VNS-ADVRP is explained with more detail in section 5.2 and illustrated with an example. The last two sections present the obtained results and the analysis beyond them, conclusion and future research. For more information on the detailed tables of results see Appendix B.
• Chapter 6 contains a summary of thesis conclusions and possible future research where
we suggest some ideas for further research.
• Appendix A contains tables of results for Multistart Branch and Bound for ADVRP
in Chapter 4.
Solution Methods
Three types of algorithms are used to solve any VRP:
• Exact algorithms which look for an optimal solution. Such methods include branch
and bound, cutting plane, branch and cut, column generation, cut and solve, branch- and-cut-and-price, branch-and-price, and dynamic programming.
• Classical heuristics which search for a good feasible solution without guarantee of opti-
mality. Such methods include constructive heuristics, two phase methods, improvement heuristics.
• Metaheuristics or framework for building heuristics. They are classified in this the-
sis into three groups: local search based metaheuristics, population based (natural inspired), and hybrid metaheuristics.
For surveys of solution methods for VRPs we refer to [49, 106, 111, 114, 160].