Solving the picker routing problem in multi-block high-level storage systems using metaheuristics

Abstract

This study aims to minimize the travel time in multi-block high-level storage systems considering height level constraints for picking devices to leave aisles. Considering these operating environments, the formulation of minimum travel times between each pair of storage positions is proposed and the picker routing problem (PRP) is solved by means of Genetic Algorithms (GA) and Ant Colony Optimization (ACO). A parameter tuning is performed for both metaheuristics, and the performance of the GA and ACO is compared with the optimal solution for small-sized problems demonstrating the reliability of the algorithms solving the PRP. Then, the performance of the GA and ACO is tested under several warehouse configurations and pick-list sizes obtaining that both metaheuristics provide high-quality solutions within short computing times. It is concluded that the GA outperforms the ACO in both efficiency and computing time, so it is recommended to implement the GA to solve the PRP in joint order picking problems.

Appendices

Appendix A. Pseudo-code for GA, ACO, and LS

Appendix B. Parameter tuning for GA and ACO

For the parameter tuning of the proposed GA, the parameter values found in several studies of the PRP shown in Table

Table 7 GA parameters for the PRP proposed in the literature

7 were taken as a basis. From these values, ${C}_{r}$= 70%, 80%, 90%; ${M}_{r}$= 5%, 10%, 20%; $P=$ 20, 40; $G=$ 20, 40 were considered. Accordingly, more importance is placed on the crossover and mutation operator because they represent the most relevant operators of a GA, and the values of P and G are relatively small to provide solutions in short computing times. The elitism rate and the immigration rate depend directly on the crossover rate and are calculated as ${{E}_{r}=I}_{r}=(1-{C}_{r})/2$. These values provide ${C}_{r}\times {M}_{r}\times P\times G$ = 36 scenarios (3 × 3 × 2 × 2), and in each scenario, 10 replicas are generated for a total of 360 experimental runs. The warehouse configuration consists of 3 blocks, 10 aisles, 15 storage locations per aisle and five height levels for a total of 4,500 storage locations. The picking device utilizes a horizontal speed of 2.0 m/s and a vertical speed of 0.4 m/s, and 20 picking locations must be visited on each tour.

Table

Table 8 Results for GA parameters

8 shows that the combinations providing the best results in travel time are C36:(Cr = 0.9, Mr = 0.2, P = 40, G = 40), C24:(Cr = 0.8, Mr = 0.2, P = 40, G = 40) and C12:(Cr = 0.7, Mr = 0.2, P = 40, G = 40) in which the combination of mutation rate, population and generations is the same; while the combinations offering worst travel time performance are C13:(Cr = 0.8, Mr = 0.05, P = 20, G = 20), C1:(Cr = 0.7, Mr = 0.05, P = 20, G = 20) and C5:(Cr = 0.7, Mr = 0.1, P = 20, G = 20) sharing identical values of population size and generations. The experiments show the greatest difference in travel time depends on the ${M}_{r}$ parameter, generating savings of 1.73% when comparing ${M}_{r}$ = 0.2 with ${M}_{r}$ = 0.05, while the parameter ${C}_{r}$ does not offer significant changes in the average travel time but the computing time is increased by 12.6% when ${C}_{r}$ increases from 0.7 to 0.9.

Regarding the population size, considering P = 20 chromosomes generates an average picking time of 450.2 s and considering P = 40 generates a travel time of 446.5 s, representing savings of 0.83% and an increase of 36% in computing time. Likewise, by increasing the number of iterations from 20 to 40, the travel time is reduced by 0.55%, and the computing time is increased by 37%. Therefore, the combination C24 is selected because it offers an efficiency similar to the efficiency of the combination C36 and yields lower minimum values than those of the combination C36. Accordingly, the parameter values for the proposed GA are ${C}_{r}$=0.8, ${M}_{r}$=0.2, P = 40, G = 40, and consequently ${E}_{r}$=0.1 and ${I}_{r}$=0.1, ensuring better performance in travel time at a reasonable computing cost. Then, using the selected parameter values, it is identified that the average number of iterations required to obtain the global best is G = 15, so as a stopping criterion the GA finish when 15 consecutive iterations fail to improve the global best.

In order to obtain the parameters ensuring the best performance for the proposed ACO, Table

Table 9 ACO parameters for the PRP in the literature

9 present the parameter values proposed by several authors in the literature and used as the basis for parameter tuning. To reduce the computing time of the ACO, which is usually one of the main pitfalls of this metaheuristic, other parameter values are considered, like K = 10 and a number of iterations equal to 10. For the constant Q, the value Q = 1 is taken by default (Wang et al. 2012). Therefore $\left(\alpha ,\beta \right)\times \rho \times K\times iterations$ = 36 scenarios (2 × 3 × 3 × 2) are generated, performing 10 replicas in each scenario for a total of 360 experimental runs. The same warehouse configuration, picking device, and the number of picking locations proposed for the GA parameter tuning were used.

According to Table

Table 10 Results for ACO parameters

10, the parameter combinations showing the best results in travel time are C36:(α = 1, β = 5, $\rho$=0.9, m = 30, iter = 30), C35:(α = 1, β = 5, $\rho$=0.9, m = 30, iter = 10), C30:(α = 1, β = 5, $\rho$=0.5, m = 30, iter = 30) and C24:(α = 1, β = 5, $\rho$=0.15, m = 30, iter = 30); while the parameter combinations offering the worst travel time performance are C8:(α = 1, β = 2, $\rho$=0.5, m = 10, iter = 30), C13:(α = 1, β = 2, $\rho$=0.9, m = 10, iter = 10). The experiments show the greatest difference in picking time is obtained changing the parameters α and β, since (α = 1, β = 5) generates savings of 9% when compared to (α = 1, β = 2).

Regarding the pheromone evaporation (ρ), no relevant differences are observed for the average travel time and computing time. As for the number of ants (K), the average travel time with 10, 20 and 30 ants is 498.6, 489.3 and 481.3 respectively, then 1.9% reductions in travel time are obtained by increasing K from 10 to 20, and 1.6% reductions by increasing K from 20 to 30. The computing time increases significantly as K increases, increasing by 90% when K increases from 10 to 20, and increasing by 51% when K increases from 20 to 30 ants. On the other hand, when varying the number of iterations from 10 to 30, only a 0.1% reduction is obtained on average, which is insignificant compared to the increase in computing cost of 207%. These results show that increasing the number of iterations does not contribute to improving the efficiency of the algorithm and acceptable improvements are obtained as the number of ants increases, demanding a trade-off in which an increase in the solution quality implies a high computing cost. Therefore, the selected parameter combination is C2, which offers an efficiency similar to the efficiency of the parameter combination C1 and generate computing time savings of 64%. Consequently, the parameter values for the proposed ACO are α = 1, β = 5, $\rho$=0.9, K = 30, iter = 10 guaranteeing high performance and short computing time for the algorithm.

For the initialization of the ACO, the values of ${\tau }_{0}$ shown in Table

Table 11 Initial pheromone value ${\tau }_{0}$ for the ACO

11 were tested for the same warehouse configuration and number of picking locations used for the tuning of the primary parameters of the ACO. The ACO was executed 10 times for each ${\tau }_{0}$ to identify the value that provides the shortest average travel time. It is observed that the worst ACO performance is obtained when ${\tau }_{0}=0$, generating premature convergence in the first iteration in all the instances evaluated, while the best performance occurs when ${\tau }_{0}={\eta }_{ij}$ because the edges with less travel time will have a greater pheromone trail at the beginning of the algorithm, and the ants will tend to select the shorter edges in the construction of the picking route. This value for ${\tau }_{0}$ is variable in each routing problem depending on the warehouse configuration and the velocities of the picking device. Likewise, when considering ${\tau }_{0}={\eta }_{ij}$ the ACO requires between 1 and 5 iterations, which allows establishing a stopping criterion with which the algorithm finishes after five consecutive iterations without improving the best global. By including this stopping criterion, the quality of the solution is preserved and the computing time is reduced on average in 22.6%.

Appendix C. Performance of proposed algorithms versus optimal solution

Table

Table 12 GA and ACO performance compared with the optimal solution

12 shows the experimental results of the proposed GA and ACO compared to the optimal solution for a number of picks ranging from 3 to 9. If the number of picks equals n, then n(n-1)! picking routes are created and tested, determining the route providing the minimal distance.

Appendix D. Extended results of travel time and computing time

Table

Table 13 Overall travel time results

13 shows the experimental results for the performance of the GA, ACO, LS and SS regarding travel time, and Table

Table 14 Expected computing time (minutes) for the GA

14 presents the expected computing time of the GA for joint order picking problems.