Iterated variable neighborhood search for the capacitated clustering problem

Abstract

The NP-hard capacitated clustering problem (CCP) is a general model with a number of relevant applications. This paper proposes a highly effective iterated variable neighborhood search (IVNS) algorithm for solving the problem. IVNS combines an extended variable neighborhood descent method and a randomized shake procedure to explore effectively the search space. The computational results obtained on three sets of 133 benchmarks reveal that the proposed algorithm competes favorably with the state-of-the-art algorithms in the literature both in terms of solution quality and computational efficiency. In particular, IVNS discovers an improved best known result (new lower bounds) for 28 out of 83 most popular instances, while matching the current best known results for the remaining 55 instances. Several essential components of the proposed algorithm are investigated to understand their impacts on the performance of algorithm.

Heuristic search to the capacitated clustering problem

“…The starting point for the search is a feasible solution generated by means of two randomized construction methods similar to those used in [17,24]. The first method consists in two stages, where the first stage performs the following steps: (i) randomly select a node v from the set of unassigned nodes, and randomly choose a cluster g from the set of clusters whose lower capacity constraint is not satisfied; (ii) allocate v to cluster g. The two steps are repeated until all clusters satisfy the lower capacity constraint.…”

“…As previously mentioned, the neighborhood exploited by FLS is defined by a joint use of three basic move operators, which have previously been employed in [7,10,24,29,30]. These operators are briefly described as follows:…”

“…OneMove operator: Given a solution s = {C 1 , C 2 , ..., C p }, OneM ove transfers a node v from its original cluster i to another cluster j such that the capacity constraint is respected, i.e., |C i | − w v ≥ L i and |C j | + w v ≤ U j . To rapidly evaluate the gain value for each candidate move, our algorithm employs a fast incremental evaluation technique similar to that used in [6,24,34,37]. The main idea is to maintain an incremental matrix γ, where each element γ[v][g] represents the sum of the edge weights between v and other nodes located in cluster g of the current solution, i.e., γ[v][g] = u∈Cg c uv .…”

“…Aside from the diversification incurred during InfLS, FLS additionally incorporates a shake procedure similar to that used in [24]. The shake procedure is applied periodically, and consists in performing a random move from the combined feasible OneM ove and SwapM ove neighborhoods.…”

“…To evaluate the performance of FITS on the first two sets of instances (DB and RanReal), we provide comparisons with several state-of-the-art algorithms including GRASP [29], TS [29], GRASP+TS [29], IVNS [24], GVNS [7] and SGVNS [7]. For a fair comparison with these state-of-the-art algorithms, we use their corresponding source codes and run them on our computing platform under the same computing conditions as described in Section 4.2.…”