Both algorithms reached the optimal (when available) or best-known solutions of the instances with n ⩽ 30.
Section 4 describes in detail the methodology developed for the solution of the proposed instance.
This fact implies the existence of equivalent optimal solutions to the instance that, on the one hand, delay the finding of a primal bound and, on the other hand, force the Optimizer to explore many more nodes in the tree search.
Since the problem is NP-hard, optimal solution for the instances of realistic size cannot be obtained within a reasonable amount of computational time using exact solution approaches.
It means if we generate initial solution of the large instances with PSO at 90% of needed time, we can improve the computation time of the proposed algorithm by almost 40% which is really significant and helpful at large size instances.
Moreover, summarizing the above discussion we can also perform each call to recursion (z ⁎, k ⁎, c ⁎, Γ, N ) in O (Γ | N | c ⁎ ) time: Indeed, the main effort in the recursion for k ⁎ ≥ Γ is the execution of Solve_RKP (c ⁎, Γ, N 1 ) requiring O (Γ | N | / 2 c ⁎ ) time and the solution of an instance of (KP) with item set N 2 which requires only O (| N | / 2 c ⁎ ) time.
The Min-Cost Bipartite Matching problem (MCM) is, given a matching instance (X, Y, w), to find the minimum cost of a matching between X and Y with respect to w. Denote by MCM X, Y, w) the solution of the MCM problem for the instance (X, Y, w), and call a matching whose cost equals to the solution optimal.
Numerous problems in physics, optimization and economics can be reduced to finding a solution of the equilibrium problem (for instance, see [28]).
In comparison to many existing heuristic approaches, GPA is able to find the optimal solution for most of the instances reported.
It has been shown that metaheuristics are able to tackle instances of problems that are believed to be hard in general, by exploring usually large solution search spaces of the instances.
