# A Memetic Algorithm for Periodic Capacitated Arc Routing Problem

TL;DR: A new Memetic Algorithm (MA) is proposed that adopts a new solution representation scheme and a novel crossover operator, and a Route-Merging procedure is devised and embedded in the algorithm to tackle the insensitive objective of PCARP.

Abstract: This paper investigates the Periodic Capacitated Arc Routing Problem (PCARP), which is often encountered in the waste collection application. PCARP is an extension of the well-known Capacitated Arc Routing Problem (CARP) from a single period to a multi-period horizon. PCARP is a hierarchical optimization problem which has a primary objective (minimizing the number of vehicles ) and a secondary objective (minimizing the total cost ). An important factor that makes PCARP challenging is that its primary objective is little affected by existing operators and thus difficult to improve. We propose a new Memetic Algorithm (MA) for solving PCARP. The MA adopts a new solution representation scheme and a novel crossover operator. Most importantly, a Route-Merging (RM) procedure is devised and embedded in the algorithm to tackle the insensitive objective . The MA with RM (MARM) has been compared with existing meta-heuristic approaches on two PCARP benchmark sets and a real-world data set. The experimental results show that MARM obtained better solutions than the compared algorithms in much less time, and even updated the best known solutions of all the benchmark instances. Further study reveals that the RM procedure plays a key role in the superior performance of MARM.

## Summary (4 min read)

### Introduction

- This is derived from the fact that in the waste collection application, the untreated waste after one service will be retained until the next service reaches.
- Besides the constraints, PCARP is different from CARP in the objectives as well.

### II. NOTATIONS AND PROBLEM DEFINITION

- In PCARP, a connected graph G(V,E) and a p-period horizon are given, where V and E are the vertex and edge sets.
- Constraints (9) and (10) indicate that each task is served no more than once in each period.
- The reason lies in that CARP only considers minimizing tc while PCARP considers minimizing mnv prior to minimizing tc.
- Compared with tc, mnv is much less sensitive to the search operators, including the crossover and local search operators.
- During the local search, the existing operators generally define a small neighborhood around the current solution by moving only one or two tasks.

### IV. MEMETIC ALGORITHM WITH ROUTE-MERGING

- It can be viewed as a class of population-based meta-heuristic approaches that incorporates local search procedures with the traditional genetic algorithms, and has been successfully applied to many realworld problems (e.g., [16], [28], [38]) with better solutions achieved and the ability of exploring the solution spaces more efficiently than traditional genetic algorithms.
- In the field of PCARP, the only two meta-heuristic approaches LMA and SS can both be viewed as adopting the framework of MA by combining global search operators with local search process.

### A. Framework of the Algorithm

- At first, the population pop is set empty.
- To keep the diversity of the population, identical solutions, also called clones, are not allowed in the population throughout the search process.
- Once an initial solution has been generated, it is compared with all the solutions in pop.
- The search process stops after Gmax generations.
- Next, the authors will describe the details of MARM, including the solution representation and evaluation, solution initialization, crossover operator and local search process.

### B. Solution Representation and Evaluation

- Different representation schemes will build different fitness landscapes in the solution space, and thus lead to different difficulties to search for the global optimum.
- For an edge task, the two corresponding IDs have the same serving costs, deadheading costs, demand vectors, service frequencies and allowed period combination sets, which are exactly those of the edge task itself.
- It should be noted that under the explicit task encoding scheme, the capacity constraints may be violated and infeasible solutions may appear during the search.
- Thus, it is no longer appropriate to use stochastic ranking procedure since it can only maintain a set of relatively good solutions, but cannot tell which solution is the best.
- 2) For infeasible solutions, the one with less constraint violation is better;.

### C. Solution Initialization

- In the initialization phase, each initial solution is generated by the following three steps: Step 1) Randomly choose a period combination for each task from its allowed period combination set; Step 2).
- For each period, apply the path scanning heuristic [18] to all the task that should be served in that period to generate a corresponding single-period sub-solution; Step 3) Combine all the single-period sub-solutions to form a complete PCARP solution.
- The path scanning heuristic was proposed by Golden et al. [18] in 1983 for the Vehicle Routing Problem (VRP), which is the node routing counterpart of CARP, and was extended by Lacomme et al. in 2002 to solve CARP [26].
- Therefore, employing the path scanning heuristic in the solution initialization can generate good PCARP initial solutions and accelerate the convergence of MARM.
- Therefore, the feasibility of the final solution can be guaranteed, and the final solution should be no worse than the best initial solution.

### D. Crossover Operator

- Since a new solution representation scheme is employed in MARM, a corresponding crossover operator should be designed.
- To this end, the authors extend the Route-Based Crossover (RBX) operator [35] from the case of CARP to PCARP.
- For each period of Sx, remove the tasks whose period combinations have been changed and should no longer be served in that period; Step 7).
- Note that the insertion of a task may increase the tc and tvd.
- If more than one such position exists, one of them is chosen arbitrarily; Step 8) Return Sx. 2.

### E. Local Search

- Sx undergoes the local search process with a predefined probability Pls.
- The neighborhood N (S) is generated by the single-insertion, double-insertion and swap operators.
- The above three operators are described as follows: Single-insertion: move a task service from its original position to another; Double-insertion: move two adjacent task services from their original positions to another; Swap: exchange the positions of two task services.
- In all the three operators, both directions of the involved tasks are considered.
- Note that the period combinations of the involved tasks may change due to the movements.

### V. EXPERIMENTAL STUDIES

- Two sets of experiments have been carried out to evaluate the performance of MARM.
- The first experiment is carried out on two relatively simple test sets, i.e., the pgdb and pval test sets [10] generated from the corresponding gdb and val CARP benchmark sets.
- MARM is applied to them and the results are compared with that obtained by LMA (provided in [26]) and SS (provided in [10]).
- Besides, to observe the influence of the RM procedure on the performance of MARM, the authors remove it from the framework of MARM and compare the resultant algorithm with MARM on the pgdb and pval test sets.
- The real-world data set was first generated by Brandão and Eglese [7] and consists of 10 large CARP instances defined on a road network in Lancashire, U.K.

### A. Experimental Setup

- The pgdb and pval test sets used in the first experiment were generated by Chu et al. in [10] by extending two well-known CARP benchmark sets, i.e., the gdb and val sets from singleperiod case to multi-period case.
- If the service frequency of (u, v) is 2 or 3, then consecutive services over the horizon (e.g., (Monday, Tuesday) and (Monday, Tuesday, Wednesday)) are forbidden.
- Unfortunately, this set is unavailable online, nor could the authors implement it due to the ambiguous description provided in the original literature.
- The real-world data set used in the second experiment, called the pG set, is extended from the G set generated by Brandão and Eglese in [7], which consists of 10 large CARP instances based on a road network in Lancashire, U.K.

### B. Experimental Results on the Benchmark Sets

- In the first experiment, MARM is compared with LMA and SS on the pgdb and pval test sets.
- From the column headed “Best”, it can be observed that the best performance of MARM was better than LMA, SS, and even the best known results.
- Table V presents the average runtime (in seconds) of each compared algorithm on the two test sets.
- In summary, the authors can conclude that embedding RM can enhance the ability of the algorithm to find better solutions, especially those with smaller mnv’s.

### D. Results on the Real-World Data Set

- For the pG real-world data set, LMA, SS, and MARM were implemented for 30 independent runs, and their best and average performance are shown in Tables VIII and IX.
- For each compared algorithm, Table VII gives the mnv and tc of the best solution obtained in the 30 independent runs, while Table IX presents the mean and standard deviation of the mnv and tc of the 30 final solutions obtained by the 30 runs (in the same form as in Tables III and IV).
- Besides, the properties of each instance, including the number of vertices, edges and services over the horizon and LBV are provided.
- From Table VIII, it is seen that MARM was able to achieve solutions with smaller mnv than that of LMA and SS on all the 6 pG instances.
- This verifies the superiority of MARM in optimizing the primary objective mnv.

### E. Summary and Further Discussion

- The experimental studies showed that MARM outperformed the other two compared algorithms significantly, especially in terms of the primary objective mnv.
- Besides, it was found that the average tc obtained by MARM was also smaller than those of the compared algorithms.
- Therefore, smaller tc’s can be obtained through the more efficient local search process.
- This observation indirectly implies that MARM is not an ideal approach to CARP.
- Thus, the authors suggest practitioners employ MARM only in the case of PCARP.

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