Adaptive Tabu search and variable neighborhood descent for packing unequal circles into a square

Abstract

This paper proposes a hybrid metaheuristic for Packing Unequal Circles into a Square (PUCS), denoted by Adaptive Tabu search and Variable Neighborhood Descent (ATS-VND). The metaheuristic is an adaptive combination of Tabu search (TS) and Variable Neighborhood Descent (VND). It is also an extension to TS-VND (Tabu search and Variable Neighborhood descent). Supplementary neighborhoods (SNs) used in VND procedures are often complex and time-consuming. To reduce the employments of SNs in TS-VND, the metaheuristic proposes an adaptive tradeoff mechanism. It divides the unitary VND procedure and TS procedure into different levels respectively, and schedules them dynamically according to their efficiencies. Since VND procedures that use complex SNs are only scheduled when other more efficient options have failed, their employments are reduced considerably. The metaheuristic is applied to the problem of Packing Unequal Circles into a Square (PUCS), and enhanced by an Iterated Local search (ILS) framework to form the final IATS-VND algorithm. Experimental results show that IATS-VND is effective to the problem. For a total of 68 instances chosen from two well established benchmark sets, it improves the world records on 60 instances and matches the other 8 within a reasonable time. Further experimental results also show that the new adaptive mechanism reduces the employments of supplementary neighborhoods considerably, and hence can improve the efficiency of TS-VND. Although we only use the hybrid metaheuristic for PUCS, the basic idea is general and may be applicable to other similar problems.