Algebraic solution of project scheduling problems with temporal constraints
N. Krivulin, S. Gubanov
TL;DR
This work addresses temporal project scheduling under start-start, start-finish, and finish-start constraints with box bounds, aiming for analytic closed-form solutions rather than iterative linear-programming methods. It develops a general constrained tropical optimization framework for objective functions in vector form, then specializes to a rank-one (unit-rank) matrix to obtain simplified, explicit solutions. The key contributions include a complete analytic solution for minimizing conjugate quadratic forms under linear/box constraints, the derivation of explicit expressions for the optimum value $\theta$ and the solution set $\bm{x}=\bm{G}\bm{u}$, and the application to project scheduling problems with makespan and maximum start-time deviation objectives. A concrete vaccination scheduling example demonstrates the method’s practical impact, producing a unique optimal schedule with start times $\bm{x}$ and finish times $\bm{y}$ achieving the minimum makespan, and illustrating how the approach yields compact, parameterized representations suitable for efficient software implementation. Overall, the paper shows that tropical optimization can yield direct analytic solutions with $O(n^{3})$ complexity, offering valuable insights and a scalable alternative to traditional LP-based scheduling in operations research.
Abstract
New solutions for problems in optimal scheduling of activities in a project under temporal constraints are developed in the framework of tropical algebra which deals with the theory and application of algebraic systems with idempotent operations. We start with a constrained tropical optimization problem that has an objective function represented as a vector form given by an arbitrary matrix, and that can be solved analytically in a closed but somewhat complicated form. We examine a special case of the problem when the objective function is given by a matrix of unit rank, and show that the solution can be sufficiently refined in this case, which results in an essentially simplified analytical form and reduced computational complexity of the solution. We exploit the obtained result to find complete solutions of project scheduling problems to minimize the project makespan and the maximum absolute deviation of start times of activities under temporal constraints. The constraints under consideration include ``start--start'', ``start--finish'' and ``finish--start'' precedence relations, release times, release deadlines and completion deadlines for activities. As an application, we consider optimal scheduling problems of a vaccination project in a medical centre.