Interval Linear Programming and Optimization Techniques
Summary
Interval linear programming extends classical linear optimisation by allowing coefficients in the objective function and constraints to be specified as intervals rather than fixed values. This framework captures uncertainty arising from measurement errors, parameter estimation and fluctuating environments, transforming a single deterministic problem into a family of realisations. Solution methods range from deterministic equivalents—where intervals are conservatively approximated—to robust and fuzzy‐based techniques that seek solutions remaining optimal or near‐optimal under all admissible coefficient variations. Extensions include bilevel and multiobjective formulations, enabling hierarchical decision models and simultaneous optimisation of multiple criteria under uncertainty. Computational challenges stem from the explosion of feasible regions and the need to balance tractability with conservatism. Applications span supply chain design, energy system planning, financial portfolio selection, resource allocation and socio‐economic modelling, where reliable decision support under uncertain data is essential.
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Recent approaches to interval bilevel linear programming have introduced preference‐based indices to convert hierarchical uncertainty into deterministic bilevel formulations, enabling decision makers to specify δ‐optimal solutions under varying preference levels. In multiobjective applications, interval multiobjective linear programming has been employed to assess comparative well‐being metrics, generating possibly and necessarily efficient solution sets for real‐world socio‐economic data. Further work has focused on robust optimality analysis, defining necessary optimality degrees for non‐degenerate basic feasible solutions under fuzzy or interval coefficient perturbations, thus quantifying solution stability against coefficient fluctuations.
Interval Linear Programming and Optimization Techniques publication trend
The graph below shows the total number of articles in interval linear programming and optimization techniques across all publications each year (not limited to Nature Index journals).
Technical terms
Interval coefficient: A numerical parameter expressed as a lower and upper bound, representing uncertainty in model data.
Bilevel programming: A hierarchical optimisation structure with a leader (upper level) and follower (lower level), each solving an optimisation problem.
Multiobjective linear programming: An optimisation problem with two or more linear objectives, seeking trade‐off solutions rather than a single optimum.
Preference‐based index: A scalar measure reflecting how well an interval objective meets a decision maker’s desired target, used to rank interval solutions.
Necessary optimality degree: The extent to which a basic feasible solution remains optimal under all permissible perturbations of interval or fuzzy coefficients.
Possibly and necessarily efficient solution: In interval multiobjective problems, a possibly efficient solution is Pareto optimal for at least one realisation of intervals, while a necessarily efficient solution is Pareto optimal for all realisations.
References
- A novel approach based on preference-based index for interval bilevel linear programming problem. Journal of Inequalities and Applications (2017).
- Analysis of the Well-Being Levels of Students in Spain and Finland through Interval Multiobjective Linear Programming. Mathematics (2021).
- Robust optimality analysis of non-degenerate basic feasible solutions in linear programming problems with fuzzy objective coefficients. Fuzzy Optimization and Decision Making (2022).
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