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 MultiEchelon 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
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 [
In recent years multiechelon systems have been introduced in different areas:
logistics enterprises and express delivery service companies under competitions;
hypermarkets and supermarkets products distribution;
multimodal freight transportation;
supply chains;
delivery in logistic competition;
Ecommerce and home delivery services under competitions;
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 [
Unfortunately, high complexity of decisionmaking 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 [
It seems that better results will be obtained by the use of the declarative constraint programming paradigms (CP/CLP) especially in modeling. The CPbased 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 MultiEchelon Capacitated Vehicle Routing Problems or the similar vehicle routing problems. In addition, some extensions and modifications to the standard TwoEchelon Capacitated Vehicle Routing Problems (2ECVRP) are presented.
The paper is organized as follows. In Section
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 NPhard type.
Nowadays, the VRP literature offers a wealth of heuristic and metaheuristic approaches, which are surveyed in the papers of [
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 [
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 twoechelon 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 twoechelon system in particular refer mainly to supply chain and inventory problems [
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.
Based on [
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 [
Approaches known from the literature are based mostly on the division of the main problem into subproblems 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 [
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 [
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 (
Integer constraints are difficult to solve using mathematical programming methods and often the real problems of MIP/MILP make them NPhard.
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 NPhard. 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 builtin. 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:
knowledge related to the problem can be expressed as linear, logical, and symbolic constraints;
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;
the problem is modelled in the constraint programming environment by CLPbased predicates, which is far more flexible than the mathematical programming environment/very important for decisionmaking problems under competitions;
transforming the decision model to explore its structure has been introduced by CLPbased predicates;
constrained domains of decision variables, new constraints, and values for some variables are transferred from CP/CLP into MILP/MIP/IP by CLPbased predicates;
optimization is performed by MPbased environment.
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 decisionmaking 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 branchandcost, have become classics now [
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
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
Description of CLP predicates.
Predicate  Description 

P1 
The implementation of the model in CLP, the term representation of the problem in the form of predicates. 


P2 
The transformation of the original problem aimed at extending the scope of constraint propagation. The transformation uses the structure of the problem. The most common effect is a change in the representation of the problem by reducing the number of decision variables and the introduction of additional constraints and variables, changing the nature of the variables, and so forth. 


P3 
Constraint propagation for the model: constraint propagation is one of the basic methods of CLP. As a result, the variable domains are narrowed, and in some cases, the values of variables are set, or even the solution can be found. 


P4 
Generation by the AG: 
Merging files generated by predicate AG into one file. It is a model file format in MP format.  


P5 
Finding the consistent area based on information from the CLP. 


P6 
The solution of the model from the P4 by MP solver. 


P7 
Solution transfer from EPLEX to CLP (predicate 
The scheme of the hybrid solution framework for Capacitated Vehicle Routing Problems (HSFCVRP).
From a variety of tools for the implementation of the CP/CLP,
The TwoEchelon Capacitated Vehicle Routing Problem (2ECVRP) is an extension of the classical Capacitated Vehicle Routing Problem (CVRP) where the delivery depotcustomers pass through intermediate depots (called satellites). As in CVRP, the goal is to deliver goods to customers with known demands, minimizing the total delivery cost in the respect of vehicle capacity constraints. 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 [
In 2ECVRP, the freight delivery from the depot to the customers is managed by shipping the freight through intermediate depots. Thus, the transportation network is decomposed into two levels (Figure
Example of 2ECVRP transportation network.
From a practical point of view, a 2ECVRP system operates as follows (Figure
freight arrives at an external/first/base zone, the depot, where it is consolidated into the 1stlevel vehicles, unless it is already carried into a fully loaded 1stlevel vehicles;
each 1stlevel vehicle travels to a subset of satellites that will be determined by the model and then it will return to the depot;
at a satellite, freight is transferred from 1stlevel vehicles to 2ndlevel vehicles.
The formal mathematical model (MILP) was taken from [
Summary indices, parameters, and decision variables.
Symbol  Description 

Indices  



Number of satellites 

Number of customers 

Deport 

Set of satellites 

Set of customers 


Parameters  



Number of the 1stlevel vehicles 

Number of the 2ndlevel vehicles 

Capacity of the vehicles for the 1st level 

Capacity of the vehicles for the 2nd level 

Demand required by customer 

Cost of the arc( 

Cost of loading/unloading operations of a unit of freight in satellite 


Decision variables  



An integer variable of the 1stlevel routing is equal to the number of 1stlevel vehicles using arc( 

A binary variable of the 2ndlevel routing is equal to 1 if a 2ndlevel vehicle makes a route start from satellite 

The freight flow arc( 

The freight arc( 

A binary variable that is equal to 1 if the freight to be delivered to customer 
One of the most important features that characterize the hybrid approach is the ease of modeling and transformation of the problem. The transformation is usually used to reduce the size of the problem and increase the efficiency of the search for a solution. In this case the transformation is based on the transition from arc to the route notation. During the transformation in the CLP the TSP, traveling salesman problem, is repeatedly solved and only the best routes in terms of costs are generated. In the process of transformation, the capacity vehicles constraints and those resulting from the set of orders are taken into account at both first and second level. For 2ECVRP variants, time and logic constraints are also included.
The obtained optimization model after the transformation
Summary indices, parameters, and decision variables for transformed model.
Symbol  Description 

Indices  



Number of satellites 

Number of customers 

Number of possible routes from depot to satellites (CLPdetermined) 

Number of possible routes from satellites to customers (CLPdetermined) 

Satellite index 

Depotsatellite route index 

Customer index 

Satellitecustomer route index 

Number of the 1stlevel vehicles 

Number of the 2ndlevel vehicles 


Input parameters  



Cost of loading/unloading operations of a unit of freight in satellite 

Demand required by customer 

Total demand for route 

Route 

Route 

If 

If satellite or receipient 

Capacity of the vehicles for the 1st level 


Decision variables  



If the tour takes place along the route 

If the tour takes place along the route 


Computed quantities  



Total demand for route 
Decision variables and constraints before
Before transformation  After transformation  Description 

Decision variables  




Transformation of decision variables level 1 from the arc model arc( 
 


Transformation of decision variables level 2 from the arc model arc( 
 
 


Constraints  


( 

Objective function after transformation, different decision variables, the same in terms of the essence and functionality. 
( 

Number of 1type resources (CLPdetermined) 
( 
—  Supply balance equation for 1level nodes is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated. 
( 

Number of 2type resources (CLPdetermined) 
( 
—  Vehicle balance equation for level 2 is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated. 
( 

Supply balance for satellites. 
( 

Number of tours for level 1 resulting from the capacity of vehicles. 
( 
—  Supply balance constraint for recipients is not required. In the route model, the supply volume is calculated for the route. 
( 
—  Supply volume constraint resulting from the vehicle capacity is unnecessary for level 2. The routes are generated only for the allowable capacities. 
( 
—  No return loads from satellite to depot ( 
( 
—  No return loads from the customer to satellite ( 
( 
—  No 
( 

No overlapping deliveries to customers. 
( 
—  This is ensured by the route model. 
( 

Integer and binary 
( 
—  Additional constraints are not necessary in the model with routes. 
For the final validation of the proposed hybrid approach, the benchmark data for 2ECVRP was selected. 2ECVRP, a well described and widely discussed problem, corresponded to the issues to which our approach was applied.
The instances for computational examples were built from the existing instances for CVRP [
Numerical experiments were conducted for the same data in three runs. The first run was a classical implementation of models (
P1 ≈ 1 000 000;
P2 ≈ 1 400 000;
P3 ≈ 22 700.
The results of numerical examples for 2ECVRP.
En13k4  HSFCVRP (P3)  MP + EdgeCuts (P2)  MP (P1)  


Fc 

Fc 

Fc  
En13k401  17,36  280  600^{*}  280  600^{*}  280 
En13k402  17,22  286  600^{*}  286  600^{*}  286 
En13k403  15,39  284  600^{*}  284  600^{*}  284 
En13k404  10,09  218  44  218  65  218 
En13k405  9,58  218  48  218  108  218 
En13k406  11,05  230  78  230  154  230 
En13k407  9,16  224  39  224  64  224 
En13k408  13,03  236  46  236  75  236 
En13k409  13,22  244  67  244  93  244 
En13k410  14,08  268  107  268  183  268 
En13k411  18,91  276  159  276  600^{*}  276 
En13k412  20,38  290  600^{*}  290  600^{*}  290 
En13k413  15,14  288  600^{*}  288  600^{*}  288 
En13k414  9,53  228  29  228  67  228 
En13k415  9,38  228  42  228  86  228 
En13k416  11,48  238  61  238  90  238 
En13k417  10,38  234  40  234  64  234 
En13k418  10,28  246  52  246  79  246 
En13k419  11,30  254  78  254  126  254 
En13k420  12,14  276  76  276  487  276 
En13k421  15,11  286  600^{*}  286  600^{*}  286 
En13k422  9,97  312  600^{*}  312  600^{*}  312 
En13k423  15,36  242  51  242  50  242 
En13k424  14,39  242  54  242  92  242 
En13k425  10,38  252  67  252  121  252 
En13k426  12,19  248  36  248  67  248 
En13k427  12,02  260  51  260  69  260 
En13k428  24,09  268  53  268  65  268 
En13k429  17,11  290  83  290  94  290 
En13k430  15,00  300  104  300  136  290 
En13k431  16,27  246  61  246  84  246 
En13k432  10,28  246  100  246  600^{*}  246 
En13k433  15,17  258  93  258  123  258 
En13k434  11,00  252  48  252  55  252 
En13k435  8,92  264  40  264  52  264 
En13k436  11,11  272  97  272  138  272 
En13k437  16,06  296  109  296  213  296 
En13k438  16,69  304  124  304  600^{*}  304 
En13k439  12,58  248  58  248  65  248 
En13k440  11,50  254  27  254  38  254 
En13k441  16,19  256  58  256  79  256 
En13k442  14,20  262  58  262  74  262 
En13k443  14,34  262  62  262  64  262 
En13k444  15,28  262  40  262  41  262 
En13k445  15,14  262  32  262  55  262 
En13k446  11,42  280  135  280  600^{*}  280 
En13k447  12,20  274  95  274  142  274 
En13k448  13,17  280  76  280  257  280 
En13k449  11,16  280  79  280  117  280 
En13k450  12,30  280  63  280  83  280 
En13k451  14,97  280  48  280  62  280 
En13k452  15,30  292  63  292  98  292 
En13k453  12,33  300  66  300  150  300 
En13k454  14,28  304  94  304  600^{*}  304 
En13k455  14,19  310  216  310  600^{*}  310 
En13k456  17,05  310  60  310  162  310 
En13k457  14,13  326  221  326  600^{*}  326 
En13k458  9,17  326  78  326  600^{*}  326 
En13k459  12,02  326  56  326  112  326 
En13k460  13,91  326  42  326  68  326 
En13k461  12,20  338  600^{*}  338  600^{*}  338 
En13k462  10,05  350  79  350  365  350 
En13k463  11,92  350  83  350  239  350 
En13k464  10,13  358  122  358  600^{*}  358 
En13k465  12,94  358  219  358  600^{*}  358 
En13k466  11,91  400  600^{*}  400  600^{*}  400 
Fc: the optimal value of the objective function.
The logical relationship between mutually exclusive variables was taken into account, which in realworld distribution systems means that the same vehicle cannot transport two types of selected goods or two points cannot be handled at the same time.
Those constraints result from technological, marketing, sales safety or competitive reasons. Only declarative application environments based on constraint satisfaction problem (CSP) make it possible to implement of this type of constraint. Table
The results of numerical examples for 2ECVRP with logical constraints.
En13k4  Fc 



exCustomer^{*} 

En13k401  284  15,36  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k407  240  7,16  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k411  290  16,91  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k420  280  13,14  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k426  270  10,72  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k432  270  10,88  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k433  276  14,124  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k440  284  11,23  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k446  308  11,12  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
En13k454  334  14,28  21  788  2,3; 2,4; 2,6; 2,7; 1,8; 1,9 
The final stage of the research was to optimize TwoEchelon Capacitated VRP with Time Windows (2ECVRPTW). This problem is the extension of 2ECVRP where time windows on the arrival or departure time at the satellites and/or at the customers are considered. The time windows can be hard or soft. This variant of the 2ECVRP is extremely important in a competitive environment.
In the first case the time windows cannot be violated, while in the second one if they are violated a penalty cost is paid. 2ECVRPTW has been implemented in a hybrid environment. This was followed by the optimization problem under the time constraints (time windows). There have been experiments with both windows hard and windows soft. The results are shown in Table
(a) The results of numerical examples for 2ECVRPTW (hard windows). (b) The results of numerical examples for 2ECVRPTW (soft windows, penalty = 30).
En13k4 



40  50  60  70  80  90  100  110  130  150  160  
En13k401  —  —  —  —  —  —  280  280  280  280  280 
En13k407  —  224  224  224  224  224  224  224  224  224  224 
En13k411  —  —  304  276  276  276  276  276  276  276  276 
En13k420  —  294  280  276  276  276  276  276  276  276  276 
En13k426  —  248  248  248  248  248  248  248  248  248  248 
En13k432  —  —  262  246  246  246  246  246  246  246  246 
En13k433  —  258  258  258  258  258  258  258  258  258  258 
En13k440  —  284  284  254  254  254  254  254  254  254  254 
En13k446  —  —  308  308  280  280  280  280  280  280  280 
En13k454  —  —  —  324  304  304  304  304  304  304  304 
En13k4 
 

40  50  60  70  80  90  100  110  130  150  160  
En13k401  358  354  346  346  310  310  280  280  280  280  280 
En13k407  270  224  224  224  224  224  224  224  224  224  224 
En13k411  306  306  304  276  276  276  276  276  276  276  276 
En13k420  366  294  280  276  276  276  276  276  276  276  276 
En13k426  292  248  248  248  248  248  248  248  248  248  248 
En13k432  322  278  262  246  246  246  246  246  246  246  246 
En13k433  336  258  258  258  258  258  258  258  258  258  258 
En13k440  344  284  284  254  254  254  254  254  254  254  254 
En13k446  344  310  308  308  280  280  280  280  280  280  280 
En13k454  342  334  334  324  304  304  304  304  304  304  304 
(a) Example of 2ECVRP transportation network for En13k420 instance. (b) Example of 2ECVRP transportation network for En13k420 instance with logic constraints. (c) Example of 2ECVRPTW transportation network for En13k420 instance.
The efficiency of the proposed approach is based on the reduction of the combinatorial problem and using the best properties of both environments. The hybrid approach (Table
In addition to solving larger problems faster, the proposed approach provides virtually unlimited modeling options with many types of constraints. Therefore, the proposed solution is recommended for decisionmaking problems under competitions and that has a structure similar to the presented models (Section
Further work will focus on running the optimization models with nonlinear and other logical constraints, multiobjective, uncertainty, and so on, in the hybrid optimization framework. The planned experiments will employ HSFCVRP for TwoEchelon Capacitated VRP with Satellites Synchronization, 2ECVRP with Pickup and Deliveries, and other VRP issues in Supply Chain Sustainability [
The authors declare that there is no conflict of interests regarding the publication of this paper.