In the case study, smart display units originated at electronics manufacturing centers in Oragadam and Sriperumbudur and were assigned to freight depots in Kanchipuram and Vellore. Deliveries could use either an automated electric truck or an automated electric van over one of two designated routes for each source–destination pairing. The model selected shipment quantities from the available combinations; it did not generate new road routes, determine sequences of stops or simulate driving behavior.

Its scope was therefore a mathematical allocation model for a specified network, not a field trial of automated vehicles. Two competing objectives were minimized simultaneously. The first combined direct transportation expenses, automation costs and energy-related costs.

The second sought to reduce total transportation time. The researchers represented the tradeoff by treating the cost and time objectives as parties in a cooperative bargaining calculation, with a deviation term favoring proximity to ideal outcomes and a satisfaction term favoring plans that perform well across both objectives. In the payoff calculation, the cost-only optimum was ₹147,480 with a transportation-time value of 13,076.1 hours, while the time-only optimum was ₹169,615.76 with 10,159.20 hours.

The distinct optima showed that minimizing one measure did not produce the same allocation as minimizing the other. Uncertain inputs were represented with triangular Fermatean fuzzy numbers, a framework designed to accommodate both membership and non-membership information under the constraint that the cubes of the two degrees do not exceed one. This permits assessments that would be excluded by intuitionistic or Pythagorean fuzzy frameworks in some cases.

The study modeled transportation costs, unit time and energy-consumption costs as fuzzy because market fluctuations, congestion, road conditions and weather can make such values difficult to specify precisely. Route distances, battery capacities, vehicle payloads, supply quantities and destination demands were treated as exact values. The case used two vehicle categories, each with defined capacity and battery limits, and a small set of designated routes rather than a comprehensive road network.

Because directly solving a bi-objective model with fuzzy coefficients would be computationally complex, the researchers first applied a score function to transform each triangular Fermatean fuzzy number into a deterministic equivalent. They then solved the cost and time objectives separately to identify their ideal and anti-ideal values. The hybrid Fermatean fuzzy game-theoretic method normalized each objective against its corresponding ideal and anti-ideal values, making deviations dimensionless despite the objectives having different units.

It then combined these normalized disagreements with Fermatean fuzzy satisfaction and dissatisfaction measures. The resulting single-objective model was implemented in LINGO with its global solver, converting the original bi-objective problem into a computationally tractable optimization task. The selected plan sent 2.0542 quintals from Oragadam to Kanchipuram by automated electric truck on route 1, 57.9457 quintals from Oragadam to Vellore by truck on route 2, 42.9457 quintals from Sriperumbudur to Kanchipuram by truck on route 1, and 7.0542 quintals from Sriperumbudur to Vellore by automated electric van on route 1.

The quantities satisfied the stated supply, demand, vehicle-capacity, battery-range and non-negativity constraints. For this allocation, the estimated transportation cost of ₹148,490 comprised ₹108,901.64 in vehicle movement and distribution expenses, ₹34,575.84 in automation costs and ₹5,012.52 in energy-consumption costs. A separate optimality test found no feasible allocation that could improve both cost and time simultaneously, confirming that the result was Pareto optimal for the two stated objectives.

In a comparison using the same deterministic model and normalization values, the Fermatean fuzzy programming method returned a cost of ₹169,607.90 and a time of 13,076.1 hours. A weighted sum method with equal weights of 0.5 for the normalized objectives returned ₹148,490 and 13,076.1 hours. The hybrid game-theoretic result matched the weighted sum method’s cost but reduced its time estimate by 133.16 hours, or about 1.02%.

Satisfaction levels for the two objectives were 99.32% and 72.60% under the hybrid method, compared with 99.32% and 71.29% under the weighted sum approach. Its degree of closeness was 0.1109, lower than the weighted sum method’s 0.1149 and the fuzzy programming method’s 0.1768; the paper treats a smaller degree of closeness as preferable. These comparisons concern the model’s calculated allocations and objective values, not measured operational performance in a live logistics system.

The computational tests indicated that the method could handle larger arrays of sources, destinations, vehicles and routes. Reported solution times rose from 0.01 seconds for a 2 × 2 × 2 × 2 instance with 16 decision variables and 12 structural constraints to 0.05 seconds for a 3 × 3 × 3 × 3 instance, 0.18 seconds for a 4 × 4 × 4 × 4 instance and 0.76 seconds for a 5 × 5 × 5 × 5 instance with 625 decision variables and 60 structural constraints. Sensitivity tests preserved feasibility when supply increased by 1% to 10% and when demand decreased by 1% to 10%.

With a 10% supply increase, the cost estimate fell by approximately 1.07% while time rose by approximately 0.69%. With a 10% demand reduction, time fell from 12,942.94 hours to 9,606.514 hours, while cost changed from ₹148,490 in the main results to ₹148,032.70 in the sensitivity table. These results describe the mathematical model’s response to parameter changes, not measured changes in an operating logistics network.

The study’s main limitation is that its sustainability and deployability claims extend beyond the variables included in the optimization. The model omits charging-station placement, communication-network requirements, data-transfer security, cybersecurity threats and unauthorized vehicle control. It also treats several uncertainty sources through fuzzy parameters rather than stochastic distributions, handles a limited product case and evaluates routes that are already predefined.

The authors note that lowering energy-related costs and time may indirectly reduce battery consumption and environmental impacts associated with electricity generation, but the case study did not measure emissions, fuel displacement, noise reduction or air-quality changes. They identify charging constraints, intrusion detection, multiple products, multilevel warehouse planning and stochastic uncertainty as possible extensions. Subject to those limitations, the case study shows how a Fermatean fuzzy game-theoretic method can convert uncertain cost and time inputs into one computationally tractable allocation for automated electric vehicle logistics.

Subject of Research: Fuzzy multi-objective optimization for automated electric vehicle transportation planning

Article Title: A sustainable Fermatean fuzzy bi-objective four-dimensional transportation model for automated electric vehicle logistics planning using a Fermatean fuzzy game-theoretic method

Article References: Kumar, E., & Dhanapal, A. (2026). A sustainable Fermatean fuzzy bi-objective four-dimensional transportation model for automated electric vehicle logistics planning using a Fermatean fuzzy game-theoretic method. Frontiers in Sustainability, 7, Article 1944956. https://doi.org/10.3389/frsus.2026.1944956

Image Credits: AI Generated

DOI: 10.3389/frsus.2026.1944956

Keywords: automated electric vehicles, logistics optimization, Fermatean fuzzy sets, game theory, four-dimensional transportation, supply chain management, Pareto optimization, uncertain inputs, smart display units, sustainable logistics, sustainable, Fermatean