FLPs are fundamental in operations research and supply chain management. Among the most studied deterministic formulations are the Uncapacitated Facility Location Problem (UFLP) and the Capacitated Facility Location Problem (CFLP). The UFLP aims to minimize the sum of fixed facility opening costs and transportation costs, assuming no capacity limitations on facilities [9], [10]. Early contributions by Efroymson and Ray [11], Khumawala [12], and Cornuejols and Thizy [13] formalized these models and investigated solution properties. On the other hand, CFLP introduces capacity constraints, reflecting practical service limitations. Notable early studies by Kuehn and Hamburger [14] and Akinc and Khumawala [15] developed models and solution techniques addressing these constraints [16–20]. Furthermore, extensions to multi-level and two-stage facility location models have been explored to represent better hierarchical and multi-product supply chain systems [21].

In real-world applications, uncertainties in demand, costs, and potential disruptions significantly affect facility location decisions [22]. Two primary research streams address these challenges. Stochastic models treat uncertain parameters as random variables, optimizing for expected cost or utility. Mirchandani and Odoni [23] and Louveaux and Thisse [24] examined the effects of such uncertainty on facility siting and customer assignments [25–28]. Robust and reliable models focus on resilience against failures or disruptions. Cheng et al. [29] developed robust optimization models for UFLP and CFLP considering disruptions, while Snyder and Ülker [30] introduced scenario-based models emphasizing supply chain reliability [31], [32].

Fuzzy set theory has been widely applied to model imprecise and ambiguous data in FLPs. Many studies utilize fuzzy numbers to represent uncertain demands, costs, or distances. Researchers proposed fuzzy facility location models solved via α-cut and credibility-based approaches [33], [34]. Given the multi-criteria nature of facility location, fuzzy Multi-Criteria Decision Making (MCDM) techniques such as Analytic Hierarchy Process (AHP), TOPSIS, and ELECTRE have been combined with fuzzy logic to evaluate alternatives under vague criteria [35–37]. Significant contributions include Kahraman et al. [38], Kaboli et al. [39], and Chu [40]. Recent research integrates fuzzy programming with simulation and heuristic optimization to tackle non-linearities and complex spatial distributions [41–43].

Exact methods like Integer Linear Programming (ILP), branch-and-bound, and Lagrangian relaxation are standard for small and medium-sized FLPs. However, large-scale or complex FLPs require heuristics and metaheuristics such as GAs, tabu search, simulated annealing, and particle swarm optimization [44–46]. Hybrid and metaheuristic approaches combining mathematical programming with metaheuristics have improved performance [47]. Specifically, integrating fuzzy programming with evolutionary algorithms—such as hybrid genetic-local search methods—has demonstrated effectiveness for multi-objective and uncertain FLPs [48], [49].

Despite progress, several research gaps persist. There is a shortage of comprehensive studies combining fuzzy programming with advanced evolutionary/metaheuristic algorithms capable of solving large-scale FLPs under uncertainty. Furthermore, many existing approaches lack extensive computational validation on realistic, large datasets, limiting practical adoption. Additionally, more investigation is needed into multi-objective fuzzy FLPs incorporating real-world constraints such as reliability, disruptions, and multi-level supply chains.

The FLP under uncertainty is modeled here by incorporating fuzzy parameters that reflect the inherent vagueness in costs and demands. The goal remains to minimize the total cost, including the construction of facilities and the cost of serving customers, while respecting capacity constraints and assignment rules.

To capture uncertainty, the cost and demand parameters are expressed as triangular fuzzy numbers defined by a triplet (a, b, c), where:

I. a is the lower bound (minimum possible value).

II. b is the most plausible value (mode).

III. c is the upper bound (maximum possible value).

This representation allows flexible modeling of uncertainty without requiring precise probability distributions. To handle these fuzzy numbers computationally, the α-cut technique is applied. The α-cut of a fuzzy number at level α ∈ [0,1] extracts an interval:

α α α (1)

A = [aL, aU], where

aL = a + α(b − a), (2) aU = c − α(c − b). (3)

This converts the fuzzy problem into a family of interval-valued subproblems parameterized by α, enabling tractable optimization.

The problem seeks to minimize the total fuzzy cost, which is the sum of facility opening costs and service costs for all customers, considering the fuzzy nature of these parameters:

∼ ∼ ∼

minZ = ∑ f j yj + ∑ ∑ cij xij.

j∈J i∈I j∈J

Here, I and J denote the sets of customers and potential facility locations, respectively.

The model is subject to the following logical and capacity constraints:

I. Assignment constraint: each customer must be assigned to exactly one facility.

∑ xij = 1, for all i ∈ I.

II. Facility activation constraint: customers can only be assigned to open facilities.

xij ≤ yj, for all i ∈ I, j ∈ J. (6) III. Capacity constraint: the total fuzzy demand allocated to a facility cannot exceed its capacity.

∑ di xij ≤ sjyj, for all j ∈ J. (7)

3.5|α-Cut Transformation

The fuzzy model is transformed into crisp subproblems to enable solutions via classical optimization methods by applying the α-cut method to all fuzzy parameters. At a given α-level, the fuzzy numbers are replaced by their α-cut intervals, transforming the objective and constraints into interval forms.

For example, the fuzzy objective becomes a bi-objective problem minimizing the lower and upper bounds of the cost intervals:

α α α

minZL = ∑ fjL yj + ∑ ∑ cijL xij,

j∈J i∈I j∈J α α α

minZU = ∑ fjU yj + ∑ ∑ cijU xij.

j∈J i∈I j∈J Similarly, capacity constraints incorporate fuzzy demands' lower and upper bounds, ensuring feasibility across uncertainty levels.

Solving these subproblems for various values of α can generate solutions reflecting different confidence levels, from conservative (high α) to optimistic (low α) scenarios.

FLPs under fuzzy data conditions require optimization methods to handle combinatorial complexity and data uncertainty. Exact methods often become computationally intractable for large-scale instances, while pure heuristic methods may not fully capture fuzziness. GAs, known for their global search capability, combined with fuzzy programming via α-cut transformations, present a promising hybrid approach for efficient and effective solutions.

A candidate solution S is encoded as a binary chromosome consisting of:

I. Facility opening genes: y = (y1, y2, . . . , y∣J∣), where yj ∈ {0,1} indicates whether facility j is open.

II. Customer allocation genes: x = {xij}, where xij ∈ {0,1} denotes the assignment of customer i to facility j.

The chromosome satisfies the following

∑ xij = 1, for all i ∈ I, (10)

xij ≤ yj, for all i ∈ I, j ∈ J. (11)

Ensuring feasible allocation.

∼ ∼ ∼

Using the α-cut representation at level α, fuzzy parameters f j, cij, di are transformed to intervals:

∼ ∼ ∼

α α α α α α α α α (12) f = [f , f ], c = [c , c ], d = [d , d ].

j jL jU ij ijL ijU i iL iU

The total cost interval for solution S is:

α α α

Z (S) = [ZL (S), ZU(S)], (13)

where

α α α

ZL (S) = ∑ fjL yj + ∑ ∑ cijL xij, (14)

j∈J i∈I j∈J

α α α

ZU(S) = ∑ fjU yj + ∑ ∑ cijU xij.

(15)

j∈J i∈I j∈J

The scalar fitness function is defined as:

Fitness (S) = w × ZL (S) + (1 − w) × ZU(S), w ∈ [0,1], (16)

where w controls the trade-off between optimistic and pessimistic scenarios.

I. Selection: tournament selection with size k, favoring solutions with lower fitness values.

II. Crossover: two-point crossover exchanging contiguous subsequences in y and x, preserving feasibility. III. Mutation: bit-flip mutation applied independently to yj and xij with adaptive probabilities pc, pm.

To maintain feasibility after genetic operations:

I. Repair procedures:

− If a customer is assigned to multiple or no facilities, reassign to the nearest open facility.

− If facility capacity constraints are violated:

∑ > . (17)

Reallocate customers from overloaded facilities to underutilized ones.

IV. Penalty function: for infeasible solutions, the fitness is penalized by:

Penalty(S) = λ × ∑ max (0, ∑ diU xij − sjyj), (18)

j∈J i∈I

where λ is a penalty coefficient.

The penalized fitness is

Fitness ( ) = Fitness( ) + Penalty( ). (19)

Parameters pc, pm, and w are dynamically adjusted based on:

I. Population diversity D.

II. Convergence rate R.

III. Fuzziness level α.

For example:

pm = pm0 × (1 − D), (20) w = w0 + (1 − α)(w1 − w0), (21)

where pm0, w0, and w1 are initial tuning constants.

This hybrid approach combines global exploration via GAs and rigorous fuzzy uncertainty modeling via α-cut methods, balancing solution quality and computational efficiency.

To evaluate the effectiveness of the proposed hybrid algorithm, a comprehensive set of datasets was employed, encompassing both synthetic instances and real-world-inspired scenarios relevant to logistics and supply chain contexts.

I. Synthetic datasets were generated to systematically vary problem size, demand variability, and fuzziness levels. These datasets included facility counts ranging from 10 to 50, and customer counts from 50 to 200, allowing scalability assessment.

II. Real-world-inspired datasets were adapted from publicly available logistics benchmarks, incorporating practical constraints such as geographic location, transportation costs, and variable demand patterns. The fuzzy parameters were modeled based on historical fluctuations in operational costs and customer demand uncertainty

The algorithm was tested across multipleα-cut levels to reflect different confidence intervals in the fuzzy parameters:

α ∈ {0.1,0.5,0.9}, (22)

Lower α represents more optimistic scenarios, and higher α corresponds to conservative decision-making. Key algorithm parameters were set as follows after preliminary tuning:

I. Population size: 100 II. Maximum generations: 500 III. Crossover probability pc: dynamically varied between 0.6 and 0.9 IV. Mutation probability pm: dynamically adjusted from 0.01 to 0.05 V. Penalty coefficient λ: 1000 All experiments were conducted on a workstation equipped with an Intel Core i7 processor (3.4 GHz) and 16 GB RAM, implementing the algorithm in Python with optimization libraries.

Performance assessment focused on the following criteria:

I. Solution quality: measured by the total cost interval at each α-level, comparing lower and upper bounds. II. Computational time: recorded in seconds to evaluate efficiency and scalability.

III. Robustness under uncertainty: assessed by analyzing solution stability across varying α levels and demand fluctuations.

IV. Feasibility rate: percentage of feasible solutions generated throughout the evolutionary process.

The hybrid algorithm was benchmarked against:

I. A classical deterministic facility location model was solved via Mixed Integer Linear Programming (MILP) using CPLEX.

II. A fuzzy exact method employing α-cut based branch and bound, as described in the literature.

III. A GA variant ignoring fuzziness (treating fuzzy parameters as their modal values).

Comparisons focused on solution quality, computational efficiency, and robustness to uncertainty.

Results revealed clear trends:

I. The suggested hybrid continuously delivered solutions with lower cost ranges than traditional and non-fuzzy GA, mainly at the more conservative (higher) α values.

II. Computational time scaled approximately linearly with problem size, demonstrating acceptable efficiency for practical logistics applications.

III. Solutions exhibited increasing robustness as α increased, with allocations favoring facilities with higher capacity buffers to accommodate demand variability.

IV. Facility opening decisions adapted to the uncertainty level, with more facilities opened under high α scenarios to mitigate risk.

V. The penalty and repair mechanisms ensured a high feasibility rate (>95%) throughout the search, contributing to solution reliability.

The experiments demonstrate that using fuzzy logic and genetics improves the soundness of decisions when there is uncertainty, with affordable computational expenses.