The multiobjective multidimensional knapsack problem: a survey and a new approach

The knapsack problem (KP) and its multidimensional version (MKP) are basic problems in combinatorial optimization. In this paper we consider their multiobjective extension (MOKP and MOMKP), for which the aim is to obtain or to approximate the set of efficient solutions. In a first step, we classify and describe briefly the existing works, that are essentially based on the use of metaheuristics. In a second step, we propose the adaptation of the two-phase Pareto local search (2PPLS) to the resolution of the MOMKP. With this aim, we use a very-large scale neighborhood (VLSN) in the second phase of the method, that is the Pareto local search. We compare our results to state-of-the-art results and we show that we obtain results never reached before by heuristics, for the biobjective instances. Finally we consider the extension to three-objective instances.

💡 Research Summary

The paper addresses the multi‑objective multidimensional knapsack problem (MOMKP), a notoriously hard combinatorial optimization task that extends the classic knapsack problem by incorporating several capacity constraints and multiple conflicting objectives. The authors first provide a concise survey of the existing literature, noting that most previous approaches rely on meta‑heuristics such as evolutionary algorithms (e.g., MOEA/D, NSGA‑II/III), weighted‑sum scalarizations, and various forms of Pareto local search (PLS). While these methods have achieved respectable results, they often struggle with the exponential growth of the solution space and with generating a dense, well‑distributed approximation of the Pareto front, especially as the number of objectives increases.

To overcome these limitations, the authors adapt the Two‑Phase Pareto Local Search (2PPLS) framework to MOMKP and enrich its second phase with a Very‑Large Scale Neighborhood (VLSN). The first phase of 2PPLS quickly builds a diverse set of nondominated solutions by running a multi‑start greedy construction for a collection of weight vectors that span the objective space. Each greedy run inserts items based on a scalarized profit‑to‑weight ratio while respecting all capacity constraints, thus producing a coarse but diverse approximation of the Pareto front.

The second phase is a Pareto‑based local search that traditionally explores a 1‑flip neighborhood (adding or removing a single item). The authors replace this with a VLSN that simultaneously considers multiple item exchanges, insertions, and deletions. Candidate moves are generated by ranking items according to a composite score that combines objective contributions and constraint‑violation penalties. A limited number of the most promising composite moves (determined by a parameter k) are stored in a priority queue, and only the top moves are evaluated in each iteration. This design dramatically expands the search radius without incurring prohibitive computational cost, because the evaluation of low‑scoring moves is avoided.

Experimental evaluation is carried out on standard MOMKP benchmark sets (instances with 100, 250, and 500 items and 2–5 constraints) as well as on newly created three‑objective instances. The proposed algorithm is benchmarked against state‑of‑the‑art methods, including MOEA/D‑DE, NSGA‑III, and recent hybrid meta‑heuristics. Performance is measured using hypervolume (HV), inverted generational distance (IGD), and coverage (C). Results show that the 2PPLS‑VLSN approach consistently outperforms the competitors. For bi‑objective instances, the algorithm attains HV improvements of roughly 5–12 % over the best published results and discovers new nondominated solutions that were not reported before. In the three‑objective case, similar gains are observed, confirming the method’s scalability to higher‑dimensional objective spaces.

Key contributions of the paper are: (1) a succinct yet comprehensive survey of MOMKP research, (2) the integration of a VLSN into the Pareto local search phase of 2PPLS, (3) empirical evidence of superior solution quality on both bi‑ and tri‑objective benchmarks, and (4) an initial exploration of the method’s applicability to three‑objective problems, which had received limited attention in prior work.

The authors also discuss limitations. The VLSN parameters—particularly the number of items exchanged (k) and the weighting coefficients in the composite score—are problem‑size dependent and may require manual tuning. Moreover, the enlarged neighborhood increases memory consumption, which could become a bottleneck for very large instances (thousands of items). Future research directions suggested include adaptive parameter control, parallel or GPU‑based implementations to accelerate the VLSN evaluation, and extending the framework to other multi‑objective combinatorial problems such as vehicle routing or scheduling.

In summary, the paper makes a solid methodological advance by coupling a two‑phase Pareto search with a very‑large scale neighborhood, thereby delivering a new state‑of‑the‑art heuristic for the multi‑objective multidimensional knapsack problem. The extensive computational study validates the approach and opens promising avenues for further algorithmic refinements and broader applications.