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
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; E-commerce 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 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 [
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
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 [
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 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 [
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 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 [
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 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: 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 CLP-based predicates, which is far more flexible than the mathematical programming environment/very important for decision-making problems under competitions; transforming the decision model to explore its structure has been introduced by CLP-based predicates; 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; 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.
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 [
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 |
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P1 |
The implementation of the model in CLP, the term representation of the problem in the form of predicates. |
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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. |
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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. |
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P4 |
Generation by the AG: |
Merging files generated by predicate AG into one file. It is a model file format in MP format. | |
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P5 |
Finding the consistent area based on information from the CLP. |
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P6 |
The solution of the model from the P4 by MP solver. |
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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 Two-Echelon Capacitated Vehicle Routing Problem (2E-CVRP) is an extension of the classical Capacitated Vehicle Routing Problem (CVRP) where the delivery depot-customers 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 2E-CVRP, 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 2E-CVRP transportation network.
From a practical point of view, a 2E-CVRP system operates as follows (Figure freight arrives at an external/first/base zone, the depot, where it is consolidated into the 1st-level vehicles, unless it is already carried into a fully loaded 1st-level vehicles; each 1st-level 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 1st-level vehicles to 2nd-level vehicles.
The formal mathematical model (MILP) was taken from [
Summary indices, parameters, and decision variables.
Symbol | Description |
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Indices | |
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Number of satellites |
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Number of customers |
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Deport |
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Set of satellites |
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Set of customers |
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Parameters | |
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Number of the 1st-level vehicles |
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Number of the 2nd-level vehicles |
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Capacity of the vehicles for the 1st level |
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Capacity of the vehicles for the 2nd level |
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Demand required by customer |
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Cost of the arc( |
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Cost of loading/unloading operations of a unit of freight in satellite |
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Decision variables | |
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An integer variable of the 1st-level routing is equal to the number of 1st-level vehicles using arc( |
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A binary variable of the 2nd-level routing is equal to 1 if a 2nd-level vehicle makes a route start from satellite |
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The freight flow arc( |
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The freight arc( |
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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 2E-CVRP 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 |
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Indices | |
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Number of satellites |
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Number of customers |
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Number of possible routes from depot to satellites (CLP-determined) |
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Number of possible routes from satellites to customers (CLP-determined) |
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Satellite index |
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Depot-satellite route index |
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Customer index |
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Satellite-customer route index |
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Number of the 1st-level vehicles |
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Number of the 2nd-level vehicles |
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Input parameters | |
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Cost of loading/unloading operations of a unit of freight in satellite |
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Demand required by customer |
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Total demand for route |
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Route |
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Route |
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If |
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If satellite or receipient |
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Capacity of the vehicles for the 1st level |
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Decision variables | |
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If the tour takes place along the route |
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If the tour takes place along the route |
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Computed quantities | |
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Total demand for route |
Decision variables and constraints before
Before transformation | After transformation | Description |
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Decision variables | ||
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Transformation of decision variables level 1 from the arc model arc( |
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Transformation of decision variables level 2 from the arc model arc( |
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Constraints | ||
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( |
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Objective function after transformation, different decision variables, the same in terms of the essence and functionality. |
( |
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Number of 1-type resources (CLP-determined) |
( |
— | Supply balance equation for 1-level nodes is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated. |
( |
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Number of 2-type resources (CLP-determined) |
( |
— | Vehicle balance equation for level 2 is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated. |
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Supply balance for satellites. |
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Number of tours for level 1 resulting from the capacity of vehicles. |
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— | Supply balance constraint for recipients is not required. In the route model, the supply volume is calculated for the route. |
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— | Supply volume constraint resulting from the vehicle capacity is unnecessary for level 2. The routes are generated only for the allowable capacities. |
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— | No return loads from satellite to depot ( |
( |
— | No return loads from the customer to satellite ( |
( |
— | No |
( |
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No overlapping deliveries to customers. |
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— | This is ensured by the route model. |
( |
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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 2E-CVRP was selected. 2E-CVRP, 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 2E-CVRP.
E-n13-k4 | HSFCVRP (P3) | MP + Edge-Cuts (P2) | MP (P1) | |||
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Fc |
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Fc |
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Fc | |
E-n13-k4-01 | 17,36 | 280 | 600* | 280 | 600* | 280 |
E-n13-k4-02 | 17,22 | 286 | 600* | 286 | 600* | 286 |
E-n13-k4-03 | 15,39 | 284 | 600* | 284 | 600* | 284 |
E-n13-k4-04 | 10,09 | 218 | 44 | 218 | 65 | 218 |
E-n13-k4-05 | 9,58 | 218 | 48 | 218 | 108 | 218 |
E-n13-k4-06 | 11,05 | 230 | 78 | 230 | 154 | 230 |
E-n13-k4-07 | 9,16 | 224 | 39 | 224 | 64 | 224 |
E-n13-k4-08 | 13,03 | 236 | 46 | 236 | 75 | 236 |
E-n13-k4-09 | 13,22 | 244 | 67 | 244 | 93 | 244 |
E-n13-k4-10 | 14,08 | 268 | 107 | 268 | 183 | 268 |
E-n13-k4-11 | 18,91 | 276 | 159 | 276 | 600* | 276 |
E-n13-k4-12 | 20,38 | 290 | 600* | 290 | 600* | 290 |
E-n13-k4-13 | 15,14 | 288 | 600* | 288 | 600* | 288 |
E-n13-k4-14 | 9,53 | 228 | 29 | 228 | 67 | 228 |
E-n13-k4-15 | 9,38 | 228 | 42 | 228 | 86 | 228 |
E-n13-k4-16 | 11,48 | 238 | 61 | 238 | 90 | 238 |
E-n13-k4-17 | 10,38 | 234 | 40 | 234 | 64 | 234 |
E-n13-k4-18 | 10,28 | 246 | 52 | 246 | 79 | 246 |
E-n13-k4-19 | 11,30 | 254 | 78 | 254 | 126 | 254 |
E-n13-k4-20 | 12,14 | 276 | 76 | 276 | 487 | 276 |
E-n13-k4-21 | 15,11 | 286 | 600* | 286 | 600* | 286 |
E-n13-k4-22 | 9,97 | 312 | 600* | 312 | 600* | 312 |
E-n13-k4-23 | 15,36 | 242 | 51 | 242 | 50 | 242 |
E-n13-k4-24 | 14,39 | 242 | 54 | 242 | 92 | 242 |
E-n13-k4-25 | 10,38 | 252 | 67 | 252 | 121 | 252 |
E-n13-k4-26 | 12,19 | 248 | 36 | 248 | 67 | 248 |
E-n13-k4-27 | 12,02 | 260 | 51 | 260 | 69 | 260 |
E-n13-k4-28 | 24,09 | 268 | 53 | 268 | 65 | 268 |
E-n13-k4-29 | 17,11 | 290 | 83 | 290 | 94 | 290 |
E-n13-k4-30 | 15,00 | 300 | 104 | 300 | 136 | 290 |
E-n13-k4-31 | 16,27 | 246 | 61 | 246 | 84 | 246 |
E-n13-k4-32 | 10,28 | 246 | 100 | 246 | 600* | 246 |
E-n13-k4-33 | 15,17 | 258 | 93 | 258 | 123 | 258 |
E-n13-k4-34 | 11,00 | 252 | 48 | 252 | 55 | 252 |
E-n13-k4-35 | 8,92 | 264 | 40 | 264 | 52 | 264 |
E-n13-k4-36 | 11,11 | 272 | 97 | 272 | 138 | 272 |
E-n13-k4-37 | 16,06 | 296 | 109 | 296 | 213 | 296 |
E-n13-k4-38 | 16,69 | 304 | 124 | 304 | 600* | 304 |
E-n13-k4-39 | 12,58 | 248 | 58 | 248 | 65 | 248 |
E-n13-k4-40 | 11,50 | 254 | 27 | 254 | 38 | 254 |
E-n13-k4-41 | 16,19 | 256 | 58 | 256 | 79 | 256 |
E-n13-k4-42 | 14,20 | 262 | 58 | 262 | 74 | 262 |
E-n13-k4-43 | 14,34 | 262 | 62 | 262 | 64 | 262 |
E-n13-k4-44 | 15,28 | 262 | 40 | 262 | 41 | 262 |
E-n13-k4-45 | 15,14 | 262 | 32 | 262 | 55 | 262 |
E-n13-k4-46 | 11,42 | 280 | 135 | 280 | 600* | 280 |
E-n13-k4-47 | 12,20 | 274 | 95 | 274 | 142 | 274 |
E-n13-k4-48 | 13,17 | 280 | 76 | 280 | 257 | 280 |
E-n13-k4-49 | 11,16 | 280 | 79 | 280 | 117 | 280 |
E-n13-k4-50 | 12,30 | 280 | 63 | 280 | 83 | 280 |
E-n13-k4-51 | 14,97 | 280 | 48 | 280 | 62 | 280 |
E-n13-k4-52 | 15,30 | 292 | 63 | 292 | 98 | 292 |
E-n13-k4-53 | 12,33 | 300 | 66 | 300 | 150 | 300 |
E-n13-k4-54 | 14,28 | 304 | 94 | 304 | 600* | 304 |
E-n13-k4-55 | 14,19 | 310 | 216 | 310 | 600* | 310 |
E-n13-k4-56 | 17,05 | 310 | 60 | 310 | 162 | 310 |
E-n13-k4-57 | 14,13 | 326 | 221 | 326 | 600* | 326 |
E-n13-k4-58 | 9,17 | 326 | 78 | 326 | 600* | 326 |
E-n13-k4-59 | 12,02 | 326 | 56 | 326 | 112 | 326 |
E-n13-k4-60 | 13,91 | 326 | 42 | 326 | 68 | 326 |
E-n13-k4-61 | 12,20 | 338 | 600* | 338 | 600* | 338 |
E-n13-k4-62 | 10,05 | 350 | 79 | 350 | 365 | 350 |
E-n13-k4-63 | 11,92 | 350 | 83 | 350 | 239 | 350 |
E-n13-k4-64 | 10,13 | 358 | 122 | 358 | 600* | 358 |
E-n13-k4-65 | 12,94 | 358 | 219 | 358 | 600* | 358 |
E-n13-k4-66 | 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 real-world 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 2E-CVRP with logical constraints.
E-n13-k4 | Fc |
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exCustomer* |
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E-n13-k4-01 | 284 | 15,36 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-07 | 240 | 7,16 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-11 | 290 | 16,91 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-20 | 280 | 13,14 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-26 | 270 | 10,72 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-32 | 270 | 10,88 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-33 | 276 | 14,124 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-40 | 284 | 11,23 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-46 | 308 | 11,12 | 21 | 788 | 2,3; 2,4; 2,6; 2,7; 1,8; 1,9 |
E-n13-k4-54 | 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 Two-Echelon Capacitated VRP with Time Windows (2E-CVRP-TW). This problem is the extension of 2E-CVRP 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 2E-CVRP 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. 2E-CVRP-TW 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 2E-CVRP-TW (hard windows). (b) The results of numerical examples for 2E-CVRP-TW (soft windows, penalty = 30).
E-n13-k4 |
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40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 130 | 150 | 160 | |
E-n13-k4-01 | — | — | — | — | — | — | 280 | 280 | 280 | 280 | 280 |
E-n13-k4-07 | — | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 |
E-n13-k4-11 | — | — | 304 | 276 | 276 | 276 | 276 | 276 | 276 | 276 | 276 |
E-n13-k4-20 | — | 294 | 280 | 276 | 276 | 276 | 276 | 276 | 276 | 276 | 276 |
E-n13-k4-26 | — | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 |
E-n13-k4-32 | — | — | 262 | 246 | 246 | 246 | 246 | 246 | 246 | 246 | 246 |
E-n13-k4-33 | — | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 |
E-n13-k4-40 | — | 284 | 284 | 254 | 254 | 254 | 254 | 254 | 254 | 254 | 254 |
E-n13-k4-46 | — | — | 308 | 308 | 280 | 280 | 280 | 280 | 280 | 280 | 280 |
E-n13-k4-54 | — | — | — | 324 | 304 | 304 | 304 | 304 | 304 | 304 | 304 |
E-n13-k4 |
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40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 130 | 150 | 160 | |
E-n13-k4-01 | 358 | 354 | 346 | 346 | 310 | 310 | 280 | 280 | 280 | 280 | 280 |
E-n13-k4-07 | 270 | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 | 224 |
E-n13-k4-11 | 306 | 306 | 304 | 276 | 276 | 276 | 276 | 276 | 276 | 276 | 276 |
E-n13-k4-20 | 366 | 294 | 280 | 276 | 276 | 276 | 276 | 276 | 276 | 276 | 276 |
E-n13-k4-26 | 292 | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 | 248 |
E-n13-k4-32 | 322 | 278 | 262 | 246 | 246 | 246 | 246 | 246 | 246 | 246 | 246 |
E-n13-k4-33 | 336 | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 | 258 |
E-n13-k4-40 | 344 | 284 | 284 | 254 | 254 | 254 | 254 | 254 | 254 | 254 | 254 |
E-n13-k4-46 | 344 | 310 | 308 | 308 | 280 | 280 | 280 | 280 | 280 | 280 | 280 |
E-n13-k4-54 | 342 | 334 | 334 | 324 | 304 | 304 | 304 | 304 | 304 | 304 | 304 |
(a) Example of 2E-CVRP transportation network for E-n13-k4-20 instance. (b) Example of 2E-CVRP transportation network for E-n13-k4-20 instance with logic constraints. (c) Example of 2E-CVRP-TW transportation network for E-n13-k4-20 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 decision-making 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 Two-Echelon Capacitated VRP with Satellites Synchronization, 2E-CVRP 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.