Mathematical Problems in Engineering

Volume 2015, Article ID 925675, 12 pages

http://dx.doi.org/10.1155/2015/925675

## A Hybrid Approach to the Optimization of Multiechelon Systems

Institute of Management and Control Systems, Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland

Received 7 August 2014; Accepted 28 September 2014

Academic Editor: Kannan Govindan

Copyright © 2015 Paweł Sitek and Jarosław Wikarek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In freight transportation there are two main distribution strategies: direct shipping and multiechelon distribution. In the direct shipping, vehicles, starting from a depot, bring their freight directly to the destination, while in the multiechelon systems, freight is delivered from the depot to the customers through an intermediate points. Multiechelon systems are particularly useful for logistic issues in a competitive environment. The paper presents a concept and application of a hybrid approach to modeling and optimization of the Multi-Echelon Capacitated Vehicle Routing Problem. Two ways of mathematical programming (MP) and constraint logic programming (CLP) are integrated in one environment. The strengths of MP and CLP in which constraints are treated in a different way and different methods are implemented and combined to use the strengths of both. The proposed approach is particularly important for the discrete decision models with an objective function and many discrete decision variables added up in multiple constraints. An implementation of hybrid approach in the system using Eplex library is presented. The Two-Echelon Capacitated Vehicle Routing Problem (2E-CVRP) and its variants are shown as an illustrative example of the hybrid approach. The presented hybrid approach will be compared with classical mathematical programming on the same benchmark data sets.

#### 1. Introduction

In the modern freight transportation there are two main distribution strategies: direct shipping and multiechelon distribution. In the direct shipping, vehicles, starting from a depot, bring their freight directly to the destination, while in the multiechelon systems, freight is delivered from the depot to the customers through an intermediate point.

The majority of multiechelon systems presented in the literature usually explicitly consider the routing problem at the last level of the transportation system, while a simplified routing problem is considered at higher levels [1].

In recent years multiechelon systems have been introduced in different areas:(i)logistics enterprises and express delivery service companies under competitions;(ii)hypermarkets and supermarkets products distribution;(iii)multimodal freight transportation;(iv)supply chains;(v)delivery in logistic competition;(vi)E-commerce and home delivery services under competitions;(vii)city and public logistics.

The vast majority of models of optimization in freight transportation and logistics industry have been formulated as the mixed integer programming (MIP) or mixed integer linear programming (MILP) problems and solved using the operations research (OR) methods [2]. Their structures are similar and proceed from the principles and requirements of mathematical programming (MP) [2, 3].

Unfortunately, high complexity of decision-making models and their integer nature contribute to the poor efficiency of OR methods. Therefore a new approach to solving these problems was proposed. As the best structure for the implementation of this approach, a declarative environment was chosen [4, 5].

It seems that better results will be obtained by the use of the declarative constraint programming paradigms (CP/CLP) especially in modeling. The CP-based environments have advantage over traditional methods of mathematical modeling in that they work with a much broader variety of interrelated constraints and allow producing “natural” solutions for highly combinatorial problems.

The main contribution of this paper is hybrid approach (mixed CP with MP paradigms) to modeling and optimization of the Multi-Echelon Capacitated Vehicle Routing Problems or the similar vehicle routing problems. In addition, some extensions and modifications to the standard Two-Echelon Capacitated Vehicle Routing Problems (2E-CVRP) are presented.

The paper is organized as follows. In Section 2 the literature related to Multi-Echelon Vehicle Routing Problems has been reviewed. Next section is about our motivation and contribution. In Section 4 the concept of hybrid approach to modeling and solving and the solution hybrid framework have been presented. Then, the general description of Multi-Echelon Vehicle Routing Problems and mathematical model of 2E-CVRP has been discussed. Finally test instances for 2E-CVRP with extension variants and some computational results were discussed in Section 6.

#### 2. Literature Review

The Vehicle Routing Problem (VRP) is used to design an optimal route for a fleet of vehicles to serve a set of customers’ orders (known in advance), given a set of constraints. The VRP is used in supply chain management in the physical delivery of goods and services. The VRP is of the NP-hard type.

Nowadays, the VRP literature offers a wealth of heuristic and metaheuristic approaches, which are surveyed in the papers of [6, 7] because exact VRP methods have a size limit of 50–100 orders depending on the VRP variant and the time-response requirements.

There are several variants and classes of VRP like the capacitated VRP (CVRP), VRP with Time Windows (VRPTW), and Dynamic Vehicle Routing Problems (DVRP), sometimes referred to as Online Vehicle Routing Problems and so forth [6].

Different distribution strategies are used in freight transportation. The most developed strategy is based on the direct shipping: freight starts from a depot and arrives directly to customers. In many applications and real situations, this strategy is not the best one and the usage of a multiechelon and particular two-echelon distribution system can optimize several features as the number of the vehicles, the transportation costs, loading factor, and timing.

In the literature the multiechelon system and the two-echelon system in particular refer mainly to supply chain and inventory problems [1]. These problems do not use an explicit routing approach for the different levels, focusing more on the production and supply chain management issues. The first real application of a two-tier distribution network optimizing the global transportation costs is due to [8] and is related to the city logistics area. They developed a two-tier freight distribution system for congested urban areas, using small intermediate platforms, called satellites (intermediate points for the freight distribution). This system is developed for a specific situation and a generalization of such a system has not already been formulated. The complete mathematic model of The Two-Echelon Capacitated Vehicle Routing Problem (2E-CVRP) with the solution for sample test data in the classical approach has been proposed by [7], complemented with the method for boosting the computing efficiency (see Section 5).

The increasing role of supply chains and their urban parts evokes a need to focus greater attention on this issue in modeling and efficient optimization methods, in particular.

#### 3. Motivation and Contribution

Based on [2, 5–7, 9–11] and our previous work [3, 4, 12] we observed some advantages and disadvantages of both (CP/MP) paradigms.

An integrated approach of constraint programming/constraint logic programming (CP/CLP) and mixed integer programming/mixed integer linear programming (MIP/MILP) can help to solve optimization problems that are intractable with either of the two methods alone [13–15]. Although Operations Research (OR) and Constraint Programming (CP) have different roots, the links between the two environments have grown stronger in recent years.

Approaches known from the literature are based mostly on the division of the main problem into sub-problems and iteratively solving each of them in the proper CP/CLP or MP/MILP technique. This is usually a collection of many local optimization points of feasible solutions. Other approaches are based on a “blind” transformation for the CLP to the MILP model. In most cases, this results in an explosion of the number of constraints and variables, which has a negative impact on the effectiveness of optimization. In the proposed hybrid approach, a very important element is the transformation of the initial problem and its solution in the field of domains, which takes place in CP/CLP environment. Then the converted and “slimmed down” problem is solved in the MILP environment, thus creating a global approach to optimization [14, 16].

Both MIP/MILP and finite domain CP/CLP involve variables and constraints. However, the types of the variables and constraints that are used, and the way the constraints are solved, are different in the two approaches [13, 15].

MIP/MILP relies completely on linear equations and inequalities in integer variables; that is, there are only two types of constraints: linear arithmetic (linear equations or inequalities) and integer (stating that the variables have to take their values in the integer numbers). In finite domain CP/CLP, the constraint language is richer. In addition to linear equations and inequalities, there are various other constraints such disequalities, nonlinear and symbolic (*alldifferent*,* disjunctive*,* cumulative*, etc.) constraints. In both MIP/ MILP and CP/CLP, there is a group of constraints that can be solved with ease and a group of constraints that are difficult to solve. The easily solved constraints in MIP/MILP are linear equations and inequalities over rational numbers.

Integer constraints are difficult to solve using mathematical programming methods and often the real problems of MIP/MILP make them NP-hard.

In CP/CLP, domain constraints with integers and equations between two variables are easy to solve. The system of such constraints can be solved over integer variables in polynomial time. The inequalities between two variables, general linear constraints (more than two variables), and symbolic constraints are difficult to solve, which makes real problems in CP/CLP NP-hard. This type of constraints reduces the strength of constraint propagation. As a result, CP/CLP is incapable of finding even the first feasible solution.

Both environments use various layers of the problem (methods, the structure of the problem, data) in different ways. The approach based on mathematical programming (MIP/MILP) focuses mainly on the methods of optimization and, to a lesser degree, on the structure of the problem. However, the data is completely outside the model. The same model without any changes can be solved for multiple instances of data. In the approach based on constraint programming (CP/CLP), due to its declarative nature, the methods are already built-in. The data and structure of the problem are used for its modelling in a significantly greater extent.

To use so much different environments and a variety of functionalities such as modeling, optimization, and transformation, the declarative approach was adopted.

The motivation and contribution behind this work were to create a hybrid method for constrained decision problems modelling and optimization instead of using mathematical programming or constraint programming separately.

It follows from the above that what is difficult to solve in one environment can be easy to solve in the other.

Moreover, such a hybrid approach allows the use of all layers of the problem to solve it. In our approach, to modelling and optimisation, we proposed the environment, where:(i)knowledge related to the problem can be expressed as linear, logical, and symbolic constraints;(ii)the optimization models solved using the proposed approach can be formulated as a pure model of MIP/MILP or of CP/CLP, or it can also be a hybrid model;(iii)the problem is modelled in the constraint programming environment by CLP-based predicates, which is far more flexible than the mathematical programming environment/very important for decision-making problems under competitions;(iv)transforming the decision model to explore its structure has been introduced by CLP-based predicates;(v)constrained domains of decision variables, new constraints, and values for some variables are transferred from CP/CLP into MILP/MIP/IP by CLP-based predicates;(vi)optimization is performed by MP-based environment.As a result, a more effective hybrid solution environment for a certain class of decision and optimization problems (2E-CVRP or similar) was obtained.

#### 4. A Hybrid Solution Framework for Capacitated Vehicle Routing Problems (HSFCVRP)

Both environments have advantages and disadvantages. Environments based on the constraints such as CLPs are declarative and ensure a very simple modeling of decision problems, even those with poor structures if any. In the CLP a problem is described by a set of logical predicates. The constraints can be of different types (linear, nonlinear, logical, binary, etc.). The CLP does not require any search algorithms. This feature is characteristic of all declarative backgrounds, in which modeling of the problem is also a solution, just as it is in Prolog, SQL, and so on. The CLP seems perfect for modeling any decision problem.

Numerous MP models of decision-making have been developed and tested, particularly in the area of decision optimization. Constantly improved methods and mathematical programming algorithms, such as the simplex algorithm, branch and bound, and branch-and-cost, have become classics now [2].

The proposed method’s strength lies in high efficiency of optimization algorithms and a substantial number of tested models. Traditional methods when used alone to solve complex problems provide unsatisfactory results. This is related directly to different treatment of variables and constraints in those approaches (Section 3).

This schema of the hybrid solution framework for Capacitated Vehicle Routing Problems (HSFCVRP) and the concept of this framework with its predicates (P1–P7) are presented in Figure 1. The names and descriptions of the CLP predicates and the implementation environment are shown in Table 1.