Keywords

1 Introduction

Parallel computing is a kind of computation to make large computational tasks smaller, allocate them to multiple computing units, and perform the computations concurrently. Recently, parallel computing has been adopted to shorten computation time in meta-heuristic algorithms [1]. Some studies conducted fitness calculations of an existing meta-heuristic algorithm in a parallel manner while other studies suggested a coarse-grained parallel algorithm in which a group of population is evolved independently under a computing unit (e.g., a core) and search information is shared only at migration phase. Artina et al. [2] proposed a parallelization method of non-dominated sorting genetic algorithms-II (NSGA- II) [3] for multi-objective optimal design of water distribution systems. Abu-Lebdeh et al. [4] compared coarse-grained parallel genetic algorithms (PGAs) and cellular PGA in terms of their performances on a transportation system optimization problem. Jung et al. [5] was the first to propose a coarse-grained parallel harmony search (PHS), which employed a hierarchical migration topology. That is, the temporal global best solution identified at the migration phase is broadcasted/transferred to individual harmony searches.

There exists few more questions to answer on the best condition for PHS. For instance, how often migration phase should be exhibited, what kind of migration topology should be set up, and how many and what kind of solutions should be migrated.

In this study, we perform a sensitivity analysis on the migration frequency (MF) of PHS to investigate its impact on the final solution quality. The PHS with three different MFs are compared with that without migration (i.e., zero MF) in a water and wastewater system planning problem.

2 Parallel Harmony Search

Harmony search (HS) [6, 7] which was first introduced in 2001 is a novel meta-heuristic algorithm inspired by improvisation of musical instrumental players. In HS new solution can be generated by three operators such as harmony memory consideration, pitch adjustment, and random selection. Harmony memory (HM), which is a space for storing good solutions, keeps updated by replacing the worst solution with the new solution if it is better than the worst one.

In PHS with migration, HSs are evolved independently and concurrently until migration phase. Migration phase includes both identifying the overall best solution and broadcasting it to other HSs except the HS which it is from, then serial evolutions are carried out until next migration phase. The PHS’s efficiency can be affected by how often migration phase is exhibited (migration frequency, MF), what kind of migration topology should be set up (migration topology), and how many solutions should be migrated (migration size). In this study, PHS with zero MF and three MFs are examined and compared in terms of the solution quality to identify the most efficient migration through the optimization of a multi-scenario planning problem.

3 Application

3.1 Study Area

PHS with migration is examined through the planning problem of a developing area in southwest US which finds the most cost effective water and wastewater infrastructure design and expansion over staged planning periods. Total number of pipes is 333 per period and commercial pipe size are 0, 6, 8, 10, 12, 16, 20, 24, 30, 36, 48, 54, 60, 72 inches. The design flow, the design head, and the number of pump in operation under peak and average demand conditions should be determined for each 36 pump stations. In addition, plant capacity is determined to a potential location of each 6 satellite wastewater plants. The decisions are made at three phase: 2010–2020, 2020–2030, and 2030–2050. Total number of decision variables is 1449 \((=3*(333+4*36+6))\).

3.2 Results

Three performance metrics, the average, best, and worst solutions, of PHS with and without migration were compared. Three MFs are examined for the latter: migration every 25, 100, and 200 iterations until total 500 iterations, resulting in the migration phases of 20, 5, and 2 times, respectively. Note that the identical HS parameters were used for each HS in PHS. The performance metrics were obtained from 20 independent optimization runs of each PHS.

Figure 1 shows the average total cost changes of the four approaches over iterations. The solution by PHS with migration every 25 iterations is always worse than any other PHSs during total computation. PHS with migration every 100 iterations shows the best solutions until approximately 200 iterations, however, after that the solutions get worse than before. At 500 iterations, which is termination criteria, PHS with migration every 200 iterations has the best solution, 1.688 (109 USD), and PHS with migration every 25 iterations has the worst solution, 1.767 (109 USD). Final solution found by PHS with no migration was good rather than PHS with migration every 25, 100 iterations.

Fig. 1.
figure 1

Average variation of solution costs.

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In Table 1, the PHSs without and with migration every 200 iterations are compared with respect to three performance metrics (Table 1). The worst solution cost of the latter was 0.23 percents less than that of the former. More significant gap between the two algorithms was found in the best solution. That is, PHS with migration every 200 iterations obtained 1.79 percents less cost. It was found that PHS with the migration produce better solution than PHS without migration overall.

Table 1. Performance metrics (Unit: 10\(^9\) USD)
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4 Summary and Conclusions

This study conducted a sensitivity analysis on the migration frequency of parallel harmony search. In PHS with migration, individual HS is computed independently under processors in a workstation and every certain iterations migration phase occurs to identify the overall best solution and broadcast it to other HSs. The PHS is applied to multi-period and multi-scenarios planning of water and wastewater infrastructure in a developing area of southwest US.

First, PHSs without and with migration were compared in terms of three types of results: the mean, worst and best solution. It is found that overall PHS with migration has better solution than PHS without migration. Second, PHS with no migration and PHSs with migration every 25, 100, and 200 iterations are compared in terms of the average evolution of solution cost by iteration. The result shows that very frequent migration does not guarantee the best performance, in other words, it is not that the more frequently the migration phase occurs, the better the final solution is. In conclusion, migration frequency is a problem-specific parameter which should be calibrated for the problem of interest.

In the future studies, sensitivity analysis on the migration topology and migration size should be carried out to find the best efficiency of migration. Ensemble of various algorithms not only harmony search should be conducted for parallel computing.