A New Algorithm For The Triangulation Of Input–Output Tables In Economics

Developed by Leontief in the 1930s, input-output models have become an indispensable tool for economists and policy-makers. They provide a framework upon which researchers can systematically analyze the interrelations among the sectors of an economy. In an input-output model, a table is constructed where each entry represents the flow of goods between each pair of sectors. Special features of the structure of this matrix are revealed by a technique called triangulation. It is shown to be equivalent to the linear ordering problem (LOP), which is an $\mathcal{NP}$-hard combinatorial optimization problem. Due to its complexity, it is essential in practice to search for quick approximate (heuristic) algorithms for the linear ordering problem. In addition to the triangulation of input-output tables, the LOP has a wide range of applications in practice. In this chapter, we develop a new heuristic procedure to find high quality solutions for the LOP. The proposed algorithm is based on a Greedy Randomized Adaptive Search Procedure (GRASP), which is one of the most effective heuristics for solving combinatorial and global optimization problems to date. We propose an improved solution technique by using a new local search scheme and integrating a path-relinking procedure for intensification. We tested our implementation on the set of 49 real-world instances of input-output tables in LOLIB22. In addition, we tested a set of 30 large randomly-generated instances due to Mitchell.18 Most of the LOLIB instances were solved to optimality within 0.87 seconds on average. The average gap for the Mitchell instances was 0.0173% with an average running time of 21.98 seconds. The results prove the efficiency and high-quality of the algorithm.