Fuzzy dynamic programming sharpens decisions under layered uncertainty

A new mathematical framework for optimizing decisions under uncertainty combines q-rung picture fuzzy numbers with fuzzy dynamic programming, producing feasible, multi-stage solutions while retaining membership, hesitation and non-membership information. The study, described as research rather than a clinical or field experiment, reports stronger comparative scores than previously published approaches in two numerical optimization problems and keeps the selected policy unchanged in sensitivity tests across several values of the model’s uncertainty parameter. The results are computational rather than evidence that the method improves outcomes in an operating business, power system or supply chain.

The central idea is to make uncertainty an explicit part of both a linear program and its sequence of decisions. Conventional linear programming assigns numerical coefficients and bounds to an objective, constraints and decision variables. That format can be awkward when available information is incomplete, ambiguous or contradictory. Q-rung picture fuzzy numbers, abbreviated q-RPFNs, add a third layer to the representation: alongside positive membership and non-membership, they carry a neutral or hesitation degree. A q value controls the admissible combinations of these components under the condition that the sum of their q-powered values does not exceed one. Larger q therefore represents a different mathematical treatment of uncertainty rather than a directly measured improvement in real-world accuracy.

The authors embed this representation in a staged optimization method. Each decision variable becomes an activity assigned to a stage, while the state records the resources still available after preceding allocations. At every stage, the procedure tests a continuous range of possible values, combines the current contribution with the best attainable value for later stages, and retains the alternative with the highest score. The backward formulation uses a Bellman-style recurrence: the value at a stage combines a fuzzy objective coefficient and current activity level with the recursively calculated value of the remaining resources after subtracting the activity’s resource requirements. In this way, uncertainty can be carried through states, transitions and rewards rather than reduced to a single approximate number at the outset.

The mathematical foundation includes a score function, S(α) = (μ^q + 1 − ν^q)/2, which converts a q-RPFN into a comparable value using membership μ and non-membership ν. The hesitation component remains part of the underlying fuzzy calculation, although it does not appear directly in this ranking formula. The paper also derives the permitted bounds for each activity from resource limits and coefficients, using q-rung picture fuzzy division, and evaluates the objective through the corresponding fuzzy multiplication. A stated advantage of this construction is that the model returns q-RPFN-valued decisions and objective values, preserving a richer description of the solution than a single crisp optimum.

Two worked applications illustrate the method. The first models the choice of a regional commercial facility across four candidate locations and four attributes: skilled labor, transport and logistics efficiency, safety and regulatory risk, and proximity to raw-material suppliers. The resulting q-RPFN decision values are [0.954, 0.562, 1], [0.7, 0.5, 0.9], [0.8, 0.6, 0.85] and [0.92, 0.5, 0.88], with a reported optimal objective of [0.11, 0.45, 0.93]. The second application considers four prospective suppliers over three evaluation periods, covering operational stability, logistics and delivery reliability, and quality and safety compliance. It reports a fuzzy objective of [0.000697303277, 0.3698652013, 0.250143097]. These are constructed numerical examples based on supplied matrices, not reported deployments with measured business results.

A sensitivity analysis tests how the selected policies respond as q varies from 1 to 5. In the first example, a three-stage production allocation case retains the policy (2, 1, 1) at every tested q, while its reported score rises from 0.6842 at q = 1 to 0.7415 at q = 5. The corresponding feasible upper bound changes from 2.15 to 2.40 before further tests at q = 4 and 5. In a four-stage capital-budgeting case, the policy changes once: it is (1, 0, 2, 1) at q = 1 but becomes (1, 1, 2, 1) at q = 2 and remains there through q = 5. Its score rises from 0.6120 to 0.6750 over that range, showing that invariance of decisions is not universal across the two examples.

The runtime measurements are similarly narrow but consistent within the reported test. The first application records 0.042, 0.041, 0.043, 0.042 and 0.044 seconds for q = 1 through 5. The second records 0.088, 0.087, 0.089, 0.088 and 0.090 seconds. The paper’s sensitivity section treats each scalar power calculation as constant-time, yielding an overall form of O(nMX) for n stages, M discretized states and X candidate decisions. A separate complexity discussion, however, characterizes exponentiation as O(q), combines fuzzy arithmetic and defuzzification as O(mnq), and gives the dynamic component as O(q·ΣS_tA_t), or O(qTn²) under its stated structured assumption. The paper therefore presents two complexity accounts, and the claim that runtime is completely independent of q should be read alongside the later analysis that assigns a linear q factor.

Against two earlier methods, the proposed approach receives higher score values in the paper’s comparison. It reports a score of 0.0736 for Problem 1 and 0.46872 for Problem 2, both above the comparison values shown in its table. Those gains are associated with the broader domain permitted by q-rung picture fuzzy arithmetic and the recursive treatment of sequential choices. However, the comparison is confined to the paper’s scoring framework and examples; a higher fuzzy score is not automatically a higher observed profit, lower cost, faster delivery or better clinical result. The paper also acknowledges that adding fuzzy structure and staged search can increase computational burden, that standardized software for q-RPFNs remains limited, and that the mathematical representation may be difficult for non-specialists to interpret.

For science and engineering audiences, the work’s significance lies in joining two established but separate toolkits: a richer language for contradictory information and an algorithm for choices whose consequences unfold over several stages. The approach could be examined in applications involving resource allocation, scheduling, supplier selection, investment planning, energy management or other sequential decisions in which coefficients and constraints are uncertain. The authors identify multi-objective optimization, type-2 fuzzy sets, machine-learning acceleration and testing across additional domains as future directions. Until such extensions are tested with independent data and real deployments, the strongest supported conclusion is narrower but still notable: within the reported mathematical examples, q-RPFN arithmetic and fuzzy dynamic programming generate feasible ranked solutions, preserve richer uncertainty information, and can retain the same policy across moderate changes in q.

Subject of Research: Fuzzy dynamic programming for multi-stage linear optimization under layered uncertainty

Article Title: Enhanced optimization using q-rung picture fuzzy numbers and dynamic programming

Article References: Prakash, Y. (2026). Enhanced optimization using q-rung picture fuzzy numbers and dynamic programming. Discover Artificial Intelligence, 6(1), Article 1572. https://doi.org/10.1007/s44163-026-02137-8

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02137-8

Keywords: q-rung picture fuzzy sets, fuzzy dynamic programming, uncertain optimization, linear programming, multi-stage decisions, fuzzy arithmetic, decision support, sensitivity analysis, computational mathematics, Enhanced, optimization, q-rung