Optimization engineering techniques for the exact solution of NP-hard combinatorial optimization problems

In optimization engineering, the engineer might have different approaches to solve a problem and he/she asks himself/herself, " Which is the best exact approach for finding the optimal solution? " The best results are often obtained by using hybrid algorithms, i.e., by combining different approaches because the most effective algorithm to be used for finding the optimal solution of a given problem strongly depends on the specific instance to be solved. In this paper, the design of effective exact enumeration algorithms for finding the optimal solution of a given NP-hard combinatorial optimization problem is considered using combining the commonly used approaches, i.e., dynamic programming, branch-and-bound, branch-and-cut. To illustrate the above-mentioned approach, the 0 ¯ 1 knapsack problem and asymmetric traveling salesman problem are considered. The mathematical models of these problems are given by an integer linear program. I think this is a good paper to see the importance of combining different approaches to improve the solution methodology to an optimization problem. The proposed method for the 0-1 knapsack problem (KP) is to combine dynamic programming with tight upper bounds, obtaining a new algorithm. Tight upper bounds are derived by imposing cardinality constraints. These additional constraints on the maximum and minimum cardinality of an optimal solution are generated from extended covers and are relaxed with the original capacity constraint leading to a new 0 ¯ 1 knapsack problem. The new problem tends to be much easier to solve than the original one, since the LP upper bounds for this problem are generally tight. An optimal solution to the transformed problem yields an upper bound for the original problem, but generally it also produces a feasible solution to the original problem, thus solving the original KP to optimality. The enumeration part of this algorithm is based on the dynamic programming recursion, but the initial problem is chosen as a collection of items which fit together well with respect to some heuristic algorithms. Moreover, when the number of states in the dynamic programming gets too high, the lower bound is improved by pairing states with items not in the algorithm. This usually results in a tightening of the lower bound and thus in additional fathoming of states. The average performances of the hybrid, the branch-and-bound and dynamic programming approaches have been experimentally evaluated by using five classes of randomly generated data instances. The hybrid approach is clearly superior to all the branch-and-bound and …