Tropical optimization technique in bi-objective project scheduling under temporal constraints

Algebraic Solution of a Problem of Optimal Project Scheduling in Project Management

“…Several versions of problem ( 16) with either a reduced set of constraints or less general objective functions are examined in [16,15,17,18] using somewhat different techniques. Below we give a solution to the problem set in the general form at (16), which combines and summarizes results for the reduced versions indicated above.…”

“…We conclude with a discussion of the computational complexity, which is involved in the solution given by formulas (17), (18) and (19). First, we note that the evaluation of the Kleene matrix B * takes O(n 3 ) operations if the Floyd-Warshall algorithm applied to the graph of the matrix B is implemented (or O(n 4 ) operations when using direct matrix multiplications).…”

“…Consider the calculation of the minimum θ according to (17). We examine the first term on the right-hand side, which has the form of the sum 0≤i+j≤n−2 (h − B i p)(q − B j g).…”

“…An efficient technique to handle temporal scheduling problems is proposed and developed in [13,14,16,15,17], based on the formulation and solution of the scheduling problems as optimization problems in the tropical algebra setting (tropical optimization problems). This technique offers direct solutions in an analytical vector form ready for further analysis with formal methods and numerical computation with moderate complexity, and thus can complement and supplement existing algorithmic approaches.…”

Minimizing maximum lateness in two-stage projects by tropical optimization

“…In the framework of tropical algebra, many temporal project scheduling problems can be represented as optimization problems and then analytically solved using methods and techniques of tropical optimization in explicit form. Examples of the solution includes results on both single-criterion problems [21,23,24,25] and multicriteria problems [26].…”