Decomposition-Based Memetic Algorithm for Multiobjective Capacitated Arc Routing Problem
Summary (4 min read)
Introduction
- Given a graph with some edges and arcs required to be served (called tasks) and a number of vehicles with limited Manuscript received October 6, 2009; revised February 7, 2010 and April 14, 2010.
- Therefore, heuristics and meta-heuristics are promising approaches in such a situation in order to obtain acceptable solutions in time.
- Contributions are now needed to fill this gap in literature.
- Essentially, the MO-CARP lies in the reign of multiobjective optimization.
A. Multiobjective CARP
- CARP is defined on a graph G(V, E, A), where V , E, and A stand for the set of vertices, edges, and arcs (directed edges), respectively.
- For an edge task (vi, vj), service in either direction is acceptable.
- All the IDs are unique positive integers.
- There are two numbers associated with each edge task, the one out of the parenthesis denotes the task ID of the direction traversed by the route, while the other one denotes its inversion.
- Constraints (6), which are also called the capacity constraints, indicate that the total demand served by each vehicle does not exceed its capacity.
B. Evolutionary Multiobjective Optimization Revisited
- →Rn consists of n objective functions that are conflicting with each other, also known as F.
- Unlike in single-objective optimization problems, the fitness of a solution needs to be assigned according to multiple criteria in a multiobjective optimization problem.
- On the other hand, when applying an existing MOEA, it is difficult to search effectively in the solution space of MO-CARP, although the EMO issues are deeply considered.
- Therefore, it is reasonable to increase the synergy between the two directions so that both of their drawbacks can be overcome.
- The authors consider incorporating an existing SO-CARP approach and various strategies proposed for EMO issues.
III. EMO Issues in MO-CARP
- MO-CARP is a combinatorial problem that tries to find a set of Pareto optimal feasible solutions in a discrete and finite solution space subject to a number of constraints.
- The hybridization of EA with local search has been reported to be quite efficient for solving combinatorial problems including SO-CARP (see [18], [22]).
- This hybridized approach is also called MA.
- That is, how to identify a solution in the neighborhood to replace the current solution.
- The first three issues are commonly considered in EMO, while the last one must be addressed when local search is employed.
A. Fitness Assignment in MO-CARP
- The existing strategies for fitness assignment in EMO can be categorized into three types: 1) the criterion-based (see [27]); 2) domination-based (see [24]); and 3) decompositionbased (see [29]) methods.
- Previous studies on numerical test functions showed that the criterion-based methods will overlook the intermediate regions of the Pareto front, while the domination-based methods will not.
- On the other hand, the decomposition-based methods are based on the assumption that each Pareto optimal solution can be seen as the optimal solution to a scalar optimization subproblem.
- This is not true in the case of MOCARP.
- Furthermore, due to the discreteness of the Pareto front in MO-CARP, one Pareto optimal solution may be the optimal solution of multiple decomposed subproblems.
B. Diversity Preservation in MO-CARP
- The niching technique, cell-based methods and crowding distance method are three typical existing strategies for diversity preservation.
- Therefore, they are expected to be able to maintain diversity in MO-CARP as well.
- Among them, the performance of the niching technique and cell-based methods are parameter-dependent, i.e., they largely depend on the parameters such as the sharing parameter in the niching technique and the cell size in the cell-based methods.
- This is based on the assumption that different subproblems can reach different optimal solutions.
- As a result, the diversity can no longer be maintained in this way.
C. Elitism in MO-CARP
- The elitism mechanism can be implemented by either storing the nondominated solutions in an archive or combining the parents and offsprings for selection.
- The archive strategy can be further divided into two types: the solutions stored in the archive do or do not influence the search process.
- There is no big difference when they are adopted in the test functions and MO-CARP.
- Therefore, they can be applied to MO-CARP in exactly the same way as to the test functions.
D. Evaluating Solutions During Local Search in MO-CARP
- Identifying the best neighboring solution is essentially equivalent to assigning fitness to each solution and then selecting the one with the best fitness.
- The criterion-based and domination-based strategies divide the solutions into different fronts, each of which consists of solutions with the same fitness.
- In addition to the decomposition-based methods, aggregating objective functions into a single one has been a commonly used idea in the literature (see [32], [33]).
- MOEA/D is the only algorithm that can address the last issue in MO-CARP because its distinctive decomposition-based framework provides a natural way to employ local search.
- Therefore, the authors propose a multiobjective MA, named D-MAENS, to address all four issues properly.
IV. D-MAENS
- D-MAENS adopts a decomposition-based framework which is analogous with that of MOEA/D.
- It decomposes the original MO-CARP into a number of scalar subproblems using the weighted sum approach with a set of uniformly distributed weight vectors.
- A population of individuals whose size equals the number of the subproblems is maintained.
- When solving each subproblem, the evolutionary operators and local search are applied to the individuals corresponding to the neighboring subproblems of the current one.
- Then, the MAENS ingredients will be briefly introduced in Section IV-B.
A. Decomposition-Based Framework
- In the decomposition framework, the original MO-CARP is first decomposed into a number of SO-CARPs by the weighted sum approach.
- Then, N subpopulations are constructed, each for a subproblem.
- Thus, the information of the subproblems whose weight vectors are close to that of the current subproblem should be helpful for solving the current subproblem.
- Sort the solutions in the set Z = X ∪ Y by the fast nondominated sorting procedure and crowding distance approach of NSGA-II [24].
- Finally, in contrast to the solution replacement of MOEA/D, in which a solution of a subproblem can only be replaced by a solution generated for the same subproblem, D-MAENS combines all the solutions together and compares them regardless of which subproblem they belong to.
B. MAENS Component
- The previous section describes a general framework that solves a MO-CARP by decomposing it into a number of single-objective subproblems.
- As a MA, MAENS is characterized by five issues: 1) the solution representation; 2) the evolutionary operator; 3) the local search operator; 4) the evaluation schemes in the evolutionary phase; and 5) the evaluation schemes in the local search.
- In the above definition, f1(S) and f2(S) can be directly set to the two objective functions (i.e., ctot(S) and cmax(S)) of the MO-CARP.
- In case some infeasible solutions are generated, both (7) and total violation to the capacity constraints (denoted as tvl(S)) will be considered.
- As for local search, (7) is used as the objective function, while infeasible solutions are handled in a way more efficient than stochastic ranking.
C. Comparisons Between D-MAENS and LMOGA
- Comparing D-MAENS with the only existing approach to MO-CARP, LMOGA [23], it can be seen that they have the same selection operator, which is the combination of the fast nondominated sorting procedure and the crowding distance approach.
- They are totally different in the ways of generating offspring.
- Second, although LMOGA also transforms the MOCARP into a SO-CARP by weighted sum when carrying out local search, the weight vector is determined based on the location of the objective vector of the offspring in the objective space.
- Moreover, D-MAENS and LMOGA employ different solution representations and evaluation schemes.
- When a solution is evaluated, it is first split into a set of feasible routes so that the additional cutting cost induced is minimized.
V. Experimental Studies
- In order to evaluate the efficacy of incorporating the strength of SO-CARP approaches and EMO strategies, the authors compared D-MAENS with LMOGA and NSGA-II on three benchmark test instances.
- Here, LMOGA was chosen since it was the only published algorithm proposed for MO-CARP and represented the way of extending an approach of SO-CARP to solve MOCARP.
- NSGA-II served as a representative of the traditional MOEAs without local search and represented the way of directly using an existing MOEA for MO-CARP.
A. Experimental Setup
- The gdb set was generated by DeArmon in [39] and consists of 23 instances, most of which are small-size instances.
- It contains 34 instances based on ten different graphs.
- The val instances have larger problem sizes than the gdb instances.
- In their experiments, the solution representation and crossover operator of D-MAENS were directly employed in NSGA-II, and the mutation operator was a random implementation of the single insertion operator.
- The authors set the parameters in such a way that the three algorithms shared the same key parameters, such as population size, crossover rate, and mutation rate.
B. Performance Measures
- The performance of a MOEA is usually evaluated from two aspects.
- The following three metrics are used.
- Furthermore, the Pareto optimal solutions themselves may even be distributed non-uniformly.
- Therefore, in their experiments, they are defined as the leftmost and rightmost solutions among the nondominated solutions obtained by all the 30 runs of the compared algorithms.
- The larger the IH , the closer the corresponding nondominated set is to the Pareto front.
C. Experimental Results
- Tables II-IV present the average value of ID, and IH over the 30 independent runs of the compared algorithms on the three test sets, respectively.
- For each instance and each metric, the result that is significantly better than others is in boldface (with smallest ID and , while with greatest IH ).
- It is shown from the tables that D-MAENS performed significantly better than the others on 71 out of the total 81 instances, including 18 out of the 23 gdb instances, 31 out of the 34 val instances, and 22 out of the 24 egl instances.
- This shows that for these two instances, D-MAENS and LMOGA both consistently found all the best nondominated solutions in all 30 runs.
- Besides, D-MAENS covered the objective space more completely than LMOGA.
VI. Conclusion
- The authors investigated a MO-CARP that considers minimizing the total cost and the makespan as two objectives.
- Integrating a competitive algorithm for SO-CARP into this framework, a novel algorithm called D-MAENS has been developed.
- This verified the efficacy of combining conventional EMO techniques with domain-specific search algorithms.
- In most realworld applications such as winter gritting, many other factors need to be considered, e.g., the time window constraints, the intermediate facilities and the time-dependent service costs.
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Citations
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Cites background from "Decomposition-Based Memetic Algorit..."
...The MOEA/D variant in [1] and most Pareto dominance algorithms may not be able to solve such MOPs in an efficient way....
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...Some MOEA/D variants have been proposed for dealing with various MOPs [7], [8]....
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...MOEA/D-DE is an efficient and effective implementation of MOEA/D for continuous MOPs proposed in [1]....
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...In such a way, good population diversity can be achieved, which is essential for solving some MOPs....
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514 citations
Cites background from "Decomposition-Based Memetic Algorit..."
...…such as the knapsack problem (Li and Landa-Silva, 2011; Ke et al., 2010; Ishibuchi et al., 2010), the traveling salesman problem (Li and Landa-Silva, 2011; Zhang et al., 2010b), the flow-shop scheduling problem (Chang et al., 2008) and the capacitated arc routing problem (Mei et al., 2011)....
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..., 2008), and the capacitated arc routing problem (Mei et al., 2011)....
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References
37,111 citations
"Decomposition-Based Memetic Algorit..." refers background or methods in this paper
...(LMOGA) [14] and NSGA-II [15] on the gdb test set in terms of several performance metrics....
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...proposed a hybrid algorithm by combining the SO-CARP approach proposed by them earlier [8] and the nondominated sorting genetic algorithm II (NSGA-II) [15], which is one of the most commonly used EA framework for MOP....
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...Combine the parent and offspring populations together and sort them by the fast nondominated sorting procedure and crowding distance approach of NSGA-II [15]....
[...]
22,704 citations
"Decomposition-Based Memetic Algorit..." refers methods in this paper
...1) stand for the shortest path from the former vertex to the latter vertex, which can be obtained by Dijkstra’s algorithm [16]....
[...]
7,512 citations
6,657 citations
"Decomposition-Based Memetic Algorit..." refers background in this paper
...That is, the diversity can be naturally preserved by the “diversity” among sub-problems [21]....
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5,062 citations
"Decomposition-Based Memetic Algorit..." refers background in this paper
...[20] to indicate the volume covered by the nondominated solutions....
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Frequently Asked Questions (11)
Q2. What are the future works mentioned in the paper "Decomposition-based memetic algorithm for multiobjective capacitated arc routing problem" ?
Therefore, their future work will focus on incorporating these factors into the CARP model.
Q3. What are the three typical strategies for diversity preservation in MO-CARP?
The niching technique, cell-based methods and crowding distance method are three typical existing strategies for diversity preservation.
Q4. What are the main factors that need to be considered in the MO-CARP?
in most realworld applications such as winter gritting, many other factors need to be considered, e.g., the time window constraints, the intermediate facilities and the time-dependent service costs.
Q5. What is the way to address the last issue in MO-CARP?
MOEA/D is the only algorithm that can address the last issue in MO-CARP because its distinctive decomposition-based framework provides a natural way to employ local search.
Q6. What are the main reasons why MOEAs fail on SO-CARP?
their preliminary studies showed that the problem natures of SO-CARP such as the discrete search space, the lack of a natural definition of neighborhood and various constraints made successful algorithms for numerical optimization problems failed on SO-CARP.
Q7. What are the three types of strategies for fitness assignment in EMO?
The existing strategies for fitness assignment in EMO can be categorized into three types: 1) the criterion-based (see [27]); 2) domination-based (see [24]); and 3) decompositionbased (see [29]) methods.
Q8. What are the parameters that affect the performance of the niching technique and cell-based?
Among them, the performance of the niching technique and cell-based methods are parameter-dependent, i.e., they largely depend on the parameters such as the sharing parameter in the niching technique and the cell size in the cell-based methods.
Q9. What is the cost of serving a task?
Note that the serving cost is only induced by serving a task, the authors have cserv(vi, vj) > 0 ⇐⇒ d(vi, vj) > 0 and cserv(vi, vj) = 0 ⇐⇒ d(vi, vj) = 0. m vehicles with an identical capacity Q are based at the depot vs ∈ V to serve the tasks.
Q10. Why is it difficult to apply a MOEA developed for one problem to another?
due to different structures of combinatorial optimization problems, it is often difficult to directly apply a MOEA developed for one problem to another.
Q11. Why did MAENS spend more time than other approaches?
This is due to the high computational cost of the Merge-Split operator employed in the local search phase of MAENS, which also made MAENS much more timeconsuming than other approaches for SO-CARP [22].