Pawlewski , J. Molina , V.

## Hybrid metaheuristics in combinatorial optimization

Julian , R. Silveira , R. Unland , S. Giroux Eds. Lecture Notes in Computer Science , , 6 — Integrating operations research in constraint programming. Annals of Operations Research , , 37 — Although OR and Constraint Programming CP have different roots, the links between the two environments have grown stronger in recent years. However, the types of the variables and constraints that are used, and the way the constraints are solved, are different in the two approaches Achterberg et al. In addition to linear equations and inequalities, there are various other constraints: disequalities, non-linear, symbolic all different, disjunctive, cumulative etc.

The system of such constraints can be solved over integer variables in polynomial time. This type of constraints reduces the strength of constraint propagation. Both approaches use various layers of the problem methods, the structure of the problem, data in different ways. However, the data is completely outside the model. The same model without any changes can be solved for multiple instances of data.

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The data and structure of the problem are used for its modelling in a significantly greater extent. The motivation and contribution behind this work were to create a novel integrated method for constrained decision problems modelling and optimization instead of using MP or constraint programming separately.

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It follows from the above that what is difficult to solve in one environment can be easy to solve in the other. Moreover, such an integrated approach allows the use of all layers of the problem to solve it. The integrated method is not inferior to its component elements applied separately. This is due to the fact that the number of decision variables and the search area are reduced. The extent of the reduction directly affects the effectiveness of the method. Unlike approaches presented in the literature Hooker, Hooker, J.

Hybrid modeling. Van Hentenryck Eds. New York, NY : Springer. In addition, the integration and transformation are performed at the implementation level using the proposed framework Section 5. In our approach to these problems we proposed the modelling, solving and optimization framework, where: knowledge related to the problem can be expressed as linear, logical and symbolic constraints;. As a result, a more effective 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 modelling of decision problems, even those with poor structures if any.

The problem is described by a set of logical predicates. The constraints can be of different types linear, non-linear, logical, binary, etc. The CLP does not require any search algorithms. This feature is characteristic of all declarative backgrounds, in which modelling of the problem is also a solution, just as it is in Prolog, SQL, etc. The CLP seems perfect for modelling 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 MP algorithms, such as the simplex algorithm, branch and bound, branch-and-cost, etc. 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. The names and descriptions of the phases and the implementation environment are shown in Table 1.

Figure 1. Detailed scheme of the ISF. ECLiPSe is an open-source software system for the cost-effective development and deployment of constraint programming applications.

The 2E-CVRP was chosen to assess and evaluate the proposed approach and its implementation deliberately, as it is known in the literature and its models are published as MILP problems Perboli et al. 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.

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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 2 : the first level connecting the depot d to intermediate depots s and the second one connecting the intermediate depots s to the customers c. The objective is to minimize the total transportation cost of the vehicles involved in both levels. Constraints on the maximum capacity of the vehicles and the intermediate depots are considered, while the timing of the deliveries is ignored.

Figure 2. From a practical point of view, a 2E-CVRP system operates as follows Figure 2 : freight arrives at an external zone, the depot, where it is consolidated into the 1st-level vehicles, unless it is already carried into a fully-loaded 1st-level vehicles;. The objective function minimizes the sum of the routing and handling operations costs. The objective function is composed of three parts. The first is the cost of transport to the first level, next to the second level. The loading at the depot, as well as unloading at the client is ignored.

## Constraint programming and operations research

This results from the fact that they are fixed and do not affect the optimization process. Constraint 5 force each 2nd-level route to begin and end to one satellite and the balance of vehicles entering and leaving each customer. The number of the routes in each level must not exceed the number of vehicles for that level, as imposed by constraints 2 and 4. The flows balance on each network node is equal to the demand of this node, except for the depot, where the exit flow is equal to the total demand of the customers, and for the satellites at the second-level, where the flow is equal to the demand unknown assigned to the satellites which provide constraints 6 and 8.

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Moreover, constraints 6 and 8 forbid the presence of sub-tours not containing the depot or a satellite, respectively. In fact, each node receives an amount of flow equal to its demand, preventing the presence of sub-tours. Consider, for example, that a sub-tour is present between the nodes i , j and k at the 1st level. It is easy to check that, in such a case, does not exist any value for the variables Q 1 ij , Q 1 jk and Q 1 ki satisfying the constraints 6 and 8.

## Michael A. Trick

The capacity constraints are formulated in constraints 7 and 9 , for the first level and the second level, respectively. Constraints 10 and 11 do not allow residual flows in the routes, making the returning flow of each route to the depot first level and to each satellite second level equal to 0. Constraint 16 assign each customer to one and only one satellite, while constraints 14 and 15 indicate that there is only one 2nd-level route passing through each customer and connect the two levels.

Constraint 17 allows starting a second-level route from a satellite k only if a 1-level route has served it. Constraints from 17 to 20 result from the character of the MILP-formulated problem. Additional constraints were introduced by Perboli et al. They strengthen the continuous relaxation of the flow model. In particular, authors in Perboli et al. The edge cuts explicitly introduce the well-known sub-tours elimination constraints derived from the Traveling Sales Problem. They can be expressed as constraint The inequalities explicitly forbid the presence in the solution of sub-tours not containing the depot, already forbidden by the constraint 8.

The number of potential valid inequalities is exponential, so that each customer reduces the flow of an amount equal to its demand d i — constraints 22 and A possibility to transform the model in the CLP environment phase PH2 is an important aspect of that approach. The our transformation of this model in this integrated approach focused on the resizing of Y k,i,j decision variable by introducing additional imaginary volume of freight shipped from the satellite and re-delivered to it.

Such transformation resulted in two facts.