Operations Research (OR) has been the cornerstone of supply chain optimisation and logistics management for decades. The classical OR paradigm, founded on the pioneering work of Dantzig [1] in linear programming and Hitchcock [2] in the transportation problem, assumes that all parameters, such as travel time, transportation cost, and demand, are crisp, deterministic, and precisely known. This assumption has enabled elegant mathematical formulations and computationally efficient solutions across a wide range of industries and application domains.

The transportation problem, in its classical formulation, seeks to minimise the total cost of shipping a commodity from a set of supply nodes to a set of demand nodes, subject to supply and demand constraints. It is a special case of linear programming and can be solved efficiently by the simplex method or the more specialised transportation simplex algorithm. Decades of development built upon this foundation, including multi-commodity extensions, vehicle routing problems, and dynamic routing algorithms, have made classical OR indispensable to modern logistics management.

However, a fundamental tension exists between the mathematical elegance of classical OR and the messy, information-poor reality of urban logistics in developing economies. In complex urban environments like Lagos, the core assumption of parameter crispness breaks down. Modern logistics systems operate in contexts characterised by unpredictable traffic congestion due to rapid urbanisation and population pressure, spontaneous road maintenance and infrastructure changes with minimal prior notice, weather-induced delays and hazardous driving conditions, supply and demand fluctuations, and incomplete or outdated route information.

Traditional approaches attempt to handle uncertainty through three main paradigms. Stochastic Programming: Dantzig [3] introduces probabilistic models where parameters follow known probability distributions. Robust Optimisation [4] addresses worst-case scenarios by optimising against the worst realisation within an uncertainty set. Fuzzy Set Theory [5] assigns membership degrees to variables, allowing gradual transitions between fully true and fully false states. While each has proven valuable in specific contexts, all three share a common limitation: they cannot adequately quantify the genuinely indeterminate: Aspects of urban logistics that are not merely vague or variable, but fundamentally unknown at the time of the decision. Neutrosophic Logic directly addresses this limitation by treating indeterminacy as an explicit, independent component. Introduced by Smarandache [6] as a generalisation of both classical and fuzzy logic, Neutrosophic Sets assign three independent membership components to every variable: T (Truth-membership), the degree to which a statement is true; I (Indeterminacy-membership), the degree of genuine unknownness; and F (Falsity-membership), the degree to which a statement is false. Unlike probability theory and fuzzy logic, these three components are independent and not constrained to sum to unity, providing a more nuanced and mathematically rigorous approach to decision-making under deep uncertainty.

Neutrosophic OR has been applied to supply chain management [7], inventory control [8], and graph-based routing problems [9], but its application to a specific named Nigerian urban logistics corridor, with route-level, time-of-day-differentiated parameterisation, remains unexplored prior to this study. This research fills that gap.

Lagos is the commercial capital of Nigeria and one of the fastest-growing megacities in the world. With a metropolitan population exceeding 15 million people and a daily commuter base that generates over 20 million vehicle-trips per day National Bureau of Statistics (NBS) [10], Lagos represents one of the most complex and challenging urban transport environments in sub-Saharan Africa. The city's road network, designed for a far smaller population, is chronically overburdened, and traffic congestion is endemic across the majority of arterial routes during morning and evening peak hours.

The economic consequences of this congestion are substantial. The Lagos State Ministry of Transportation [11] estimates that traffic congestion costs the Lagos economy approximately 4 billion naira daily in lost productivity, fuel wastage, and vehicular wear. For the logistics sector specifically, unoptimised routing contributes to increased fuel costs, driver overtime, delayed deliveries, spoilage of perishable goods, and lost business due to unreliable delivery windows. Alamutu and Hammed [12] document the systemic nature of these challenges, noting that Lagos State Traffic Management Authority (LASTMA) interventions provide only partial relief given the structural mismatch between road capacity and traffic demand.

Nwaigwe et al. [13] identify the principal causes of Lagos traffic congestion as: Inadequate road infrastructure relative to vehicle population growth; encroachment by pedestrians, hawkers, and informal traders on road space; frequent vehicular breakdowns on major arterials; insufficient public transport capacity leading to private vehicle over-dependence; and poor traffic signal management. These causes collectively produce a transport environment where travel times are not merely variable but genuinely indeterminate, where even experienced commuters cannot reliably predict whether a given route will take 20 or 60 minutes on a given morning.

The Yaba–University of Lagos (UNILAG) corridor exemplifies this broader challenge in microcosm. Yaba is a dense commercial and transit hub serving thousands of commuters daily, including students, staff, and delivery personnel travelling to the University of Lagos. Three distinct route alternatives exist, each with markedly different traffic profiles, infrastructure quality, and exposure to indeterminate delay events. Table 1 below summarises key statistics that motivate this research.

These figures underscore the severity of the problem and the practical need for routing decision-support tools that go beyond classical deterministic models. A routing model that assumes peak-hour travel on the Yaba-Herbert Macaulay Way route takes a fixed 28 minutes, when in reality it can range from 12 to 55 minutes, will systematically produce suboptimal recommendations.

Transportation companies and logistics distributors in Lagos face significant operational challenges that classical OR models are ill-equipped to address. These challenges manifest across three interconnected dimensions.

Financial Losses: Inefficient routing leads to increased fuel costs, driver overtime, and delayed deliveries, resulting in substantial losses. When vehicles encounter unforeseen delays, congestion, unexpected road reconstruction, or severe weather, classical models cannot adapt dynamically because they encode only the nominal most-likely travel time and provide no mechanism for managing downside risk.

Operational Rigidity: Classical routing algorithms assume fixed travel times and costs. They are unable to accommodate the dynamic, volatile nature of Lagos urban traffic, where conditions can shift materially within minutes. The same route that took 18 minutes at 11 am may take 55 minutes at 6 pm under peak-hour congestion plus a road incident.

Incomplete Models: Existing deterministic models reduce complex urban logistics to overly simplistic representations. They ignore or undervalue the "unknown unknowns", genuinely indeterminate factors such as undocumented flooding risk on canal-adjacent routes, informal market spillover on local roads, or unpredictable roadside construction with no formal commencement notification.

The fundamental research problem is: Classical optimisation models fail to provide robust, adaptive solutions for logistics in volatile environments because they treat indeterminacy as an outlier rather than an inherent, quantifiable characteristic of the problem space. This study's central hypothesis is that Neutrosophic Operations Research, by explicitly incorporating indeterminacy as a core variable through the Single-Valued Neutrosophic Sets (SVNS) framework, will produce superior and more risk-aware routing recommendations for Lagos logistics networks compared to classical and fuzzy baselines.

This research addresses the following specific questions:

I. What are the key indeterminate variables affecting logistics and travel times across the Yaba-UNILAG route alternatives?b II. How can SVNS be mathematically formulated to represent travel times, costs, and route reliability for this corridor?

III. To what extent does a Neutrosophic model outperform classical deterministic and fuzzy models in route optimisation for the Yaba–UNILAG corridor?

IV. How sensitive is the neutrosophic model's route recommendation to the decision-maker's risk-aversion parameter (λ)?

V. What practical guidelines can be developed for implementing Neutrosophic OR in real-world Lagos logistics operations?

Aim: To design, formulate, and evaluate a Neutrosophic Route Reliability framework for the Yaba–UNILAG corridor that explicitly accounts for indeterminate variables and provides superior, risk-aware routing recommendations compared to classical deterministic and fuzzy baselines.

This study is significant to academic researchers, logistics operators, and commuters on the Yaba-UNILAG corridor, offering a reproducible, mathematically grounded routing framework. The work is scoped to this single corridor and to a literature-grounded simulated dataset rather than continuously logged field data; these constraints, and the priorities they suggest for future extension, are discussed further.

This study adopts a quantitative modelling research design. The research involves: construction of a mathematical model; parameterisation using a literature-grounded simulated dataset; computational validation through worked examples and full scenario analysis; and comparative evaluation against established baselines. The design follows the applied mathematical modelling research framework: Problem identification, model formulation, parameterisation, solution, and validation.

No primary field data collection was conducted. A simulated dataset was constructed from documented, verifiable facts about the Yaba–UNILAG corridor, as disclosed fully in Section 3.3. This design choice is consistent with the scope of a B.Sc. final year project and is made explicit as a methodological constraint rather than concealed.

This research is deliberately scoped, on supervisory guidance, to a single well-defined logistics and commuter corridor: The Yaba–University of Lagos (UNILAG) corridor in Lagos State. This approach preserves the core motivation of the study while making data construction, model validation, and presentation tractable within the scope of a B.Sc. final year project.

Three real, verifiable route alternatives connect Yaba to UNILAG, each presenting a distinct profile of traffic volume, road quality, infrastructure exposure, and informational opacity.

UNILAG is treated throughout as a single destination node reached via three distinct street-level paths rather than three physically identical endpoints. Routes A and C both terminate at the Main Gate on University Road, Route A approaching via Herbert Macaulay Way, Route C via the Sabo–Abule-Oja residential streets, while Route B terminates separately at the Back Gate via Murtala Muhammed Way and Alagomeji. The comparison is therefore between three real, verifiable street-level paths to campus, not between three routes converging on one identical gate.

Trip-level, continuously logged travel-time data for these specific routes is not publicly available, and access to commercial fleet telematics was outside the scope of this project. A literature-grounded simulated dataset was therefore constructed. For each route and scenario, three travel-time bands, best-case, most-likely, and worst-case, and an associated historical frequency split were assigned, anchored to documented and verifiable facts about the corridor.

The anchoring facts include: documented peak-hour travel-time ranges on Yaba arterials from Nwaigwe et al. [13]; average free-flow speeds and congestion-factor multipliers from Alamutu and Hammed [12]; route-specific flooding risk from LASURA; and the researcher's own knowledge of the corridor as a UNILAG student. The model's mathematical structure and decision logic are fully validated by this exercise; specific parameter values would be refined, not restructured, by future primary data collection.

The opacity term ε is the most novel element of the indeterminacy formulation and requires explicit justification. It captures the proportion of each route's variability that comes from genuinely undocumented or unobservable factors, not statistical randomness, but true unknownness. ε is higher for routes where more delay risk comes from factors that are not routinely monitored, reported, or predictable from historical data. To assign ε in a principled and reproducible manner, a four-criterion scoring procedure was developed. Each route was scored on each criterion on a scale of 0 to 3, and the total score was normalised. Table 8 presents this procedure.

Conservative rounding was applied to reflect the fact that the Yaba–UNILAG corridor is better-documented overall than more peripheral Lagos routes. The normalisation range is consistent with opacity-equivalent terms in the range 0.10–0.30 reported in analogous formulations by Abdel-Basset et al. [8].

For each route r ∈ {A, B, C} and scenario s ∈ {Morning Peak, Midday, Evening Peak, Night}, travel time is represented as the SVNS element ⟨T(r,s), I(r,s), F(r,s)⟩. The derivation procedure follows five steps.

Step 1 (Raw inputs). For each route-scenario, specify t_best, t_likely, t_worst (travel times in minutes) and frequency estimates freq(best), freq(likely), freq(worst) with freq(best) + freq(likely) + freq(worst) ≈ 1. Step 2. Component derivation:

I. T = freq(likely): Confidence that typical conditions hold.

II. F = freq(worst): Probability of severe delay.

III. I = min(1.0, 0.15 × [(t_worst − t_best) / t_likely] + ε): Residual genuine unknownness.

Step 3. Deneutrosophicated expected travel time:

E[t] = Tₙ × t_likely + Iₙ × [(t_likely + t_worst)/2] + Fₙ × t_worst, (3)

where Tₙ = T/(T+I+F), Iₙ = I/(T+I+F), Fₙ = F/(T+I+F).

Step 4. Scalar measures:

Score: S = (2T + I − F) / 4 Accuracy: A = T – F. (4)

Step 5. Route selection via risk-adjusted cost:

Cost(λ) = E[t] × (1 + λ × (1 − A)). (5)

The optimal route r* for scenario s at risk-aversion level λ is argmin_r Cost(λ). The model is evaluated at λ = 0.3, 0.5, and 0.7, and for sensitivity analysis at λ ∈ {0.1, 0.2, 0.3, 0.372, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}.

I. Classical (deterministic): Selects the route with the lowest most-likely time t_likely, ignoring variability or failure risk entirely.

II. Fuzzy (triangular defuzzification): Selects the route with the lowest centroid = (t_best + t_likelya+ t_worst) / 3, incorporating spread symmetrically but without distinguishing indeterminacy from falsity.

III. Neutrosophic (proposed): Selects the route with the lowest Risk-Adjusted Cost(λ), evaluated across nine λ values for full sensitivity analysis.

Worked Example 1 (Route B, Morning Peak). This section demonstrates the full five-step procedure for Route B (Alagomeji) during Morning Peak.

Worked Example 2 (Route B vs Route C, Evening Peak). This example demonstrates the sensitivity crossover finding: the model recommendation changes depending on λ. Routes B and C have similar expected times but materially different accuracy values, making this the most instructive scenario.

Worked Example 3 (All three Routes, Night/Off-peak). This example provides a full three-route comparison for the Night/Off-Peak scenario. When conditions are stable, all three models converge on the same recommendation, validating that the neutrosophic framework does not impose spurious complexity when the situation does not warrant it.

Table 12 presents the full set of computed SVNS components, scalar measures, and expected travel times for all 12 Route-scenario combinations.

This section discusses their implications for OR theory and considers their practical significance for logistics operators and commuters on the Yaba–UNILAG corridor.

Best/most-likely/worst travel time bands (minutes) and empirical frequency splits used as model inputs, per route per scenario. See Section 3.3 for full disclosure and justification.

Table A1. Simulated dataset (full).

Scenario Route Best (min) Likely (min) Worst (min) Freq(best) Freq(likely) Freq(worst) Morning Peak Route A 12 28 55 0.10 0.45 0.45 (7–10 am) Morning Peak Route B 14 24 40 0.20 0.55 0.25 (7–10 am) Morning Peak Route C 15 22 38 0.25 0.45 0.30 (7–10 am) Midday (11 Route A 10 16 30 0.35 0.50 0.15 am–3 pm) Midday (11 Route B 12 18 28 0.35 0.50 0.15 am–3 pm) Midday (11 Route C 13 19 32 0.30 0.45 0.25 am–3 pm) Evening Peak Route A 13 30 60 0.08 0.42 0.50 (5–8 pm) Evening Peak Route B 15 26 45 0.18 0.52 0.30 (5–8 pm) Evening Peak Route C 16 24 42 0.22 0.43 0.35 (5–8 pm) Night/Off- Route A 8 11 20 0.55 0.35 0.10 peak (8 pm–6 am) Night/Off- Route B 10 14 22 0.50 0.40 0.10 peak (8 pm–6 am) Night/Off- Route C 12 16 26 0.40 0.40 0.20 peak (8 pm–6 am)

Computed travel time/cost under each model, per route per scenario. Classical and Fuzzy columns show travel time in minutes. Neutrosophic columns show Risk-Adjusted Cost. Asterisk (*) marks the recommended route per scenario per model.

Table B1. Full model comparison output.

Scenario Route Classical (min) Fuzzy (min) Neutro. (λ=0.3) Neutro. (λ=0.5) Neutro. (λ=0.7) Morning Peak Route A 28 31.7 54.0 62.2 70.5 Morning Peak Route B 24 26.0 36.2* 40.4* 44.6* Morning Peak Route C 22* 25.0* 36.3 41.2 46.1 Midday Route A 16* 18.7* 24.4* 27.0* 29.7* Midday Route B 18 19.3 25.2 28.0 30.7 Midday Route C 19 21.3 30.1 34.0 37.9 Evening Peak Route A 30 34.3 60.8 70.7 80.6 Table B1. Continued.

Scenario Route Classical (min) Fuzzy (min) Neutro. (λ=0.3) Neutro. (λ=0.5) Neutro. (λ=0.7) Evening Peak Route B 26 28.7 41.6 46.8* 52.1* Evening Peak Route C 24* 27.3* 41.3* 47.3 53.3 Night/Off-peak Route A 11* 13.0* 17.2* 19.2* 21.3* Night/Off-peak Route B 14 15.3 20.0 22.3 24.6 Night/Off-peak Route C 16 18.0 24.7 27.9 31.0

* Indicates the minimum-cost (recommended) route for that scenario and model.

The complete Python implementation used to compute all results is provided below. The code requires only Python 3.x standard libraries (json, numpy) and is fully self-contained.

Neutrosophic Route Reliability Model: Yaba -> UNILAG Corridor Author: Morounfolu Akinyele | Matric: 210806132 Supervisor: Dr. S. A. Adebisi | Department of Mathematics, UNILAG

import json, numpy as np ROUTES = {"A": "Herbert Macaulay Way", "B": "Alagomeji", "C": "Abule-Oja"} SCENARIOS= ["Morning Peak (7-10 am)", "Midday (11 am-3 pm)", "Evening Peak (5-8 pm)", "Night/Off-peak (8 pm-6 am)"] DATA = { "A": {"Morning Peak (7-10 am)":(12,28,55,0.10,0.45,0.45), "Midday (11 am-3 pm)":(10,16,30,0.35,0.50,0.15), "Evening Peak (5-8 pm)":(13,30,60,0.08,0.42,0.50), "Night/Off-peak (8 pm-6 am)":(8,11,20,0.55,0.35,0.10)}, "B": {"Morning Peak (7-10 am)":(14,24,40,0.20,0.55,0.25), "Midday (11 am-3 pm)":(12,18,28,0.35,0.50,0.15), "Evening Peak (5-8 pm)":(15,26,45,0.18,0.52,0.30), "Night/Off-peak (8 pm-6 am)":(10,14,22,0.50,0.40,0.10)}, "C": {"Morning Peak (7-10 am)":(15,22,38,0.25,0.45,0.30), "Midday (11 am-3 pm)":(13,19,32,0.30,0.45,0.25), "Evening Peak (5-8 pm)":(16,24,42,0.22,0.43,0.35), "Night/Off-peak (8 pm-6 am)":(12,16,26,0.40,0.40,0.20)},

OPACITY = {"A":0.12, "B":0.16, "C":0.22} LAMBDAS = [0.1,0.2,0.3,0.372,0.4,0.5,0.6,0.7,0.8,0.9] def derive_tif(b,l,w,fb,fl,fw,eps):

T=fl; F=fw spread=(w-b)/l if l else 0 I=min(1.0, 0.15*spread+eps) return round(T,4),round(I,4),round(F,4) def expected_time(b,l,w,T,I,F):

tot=T+I+F; Tn,In,Fn=T/tot,I/tot,F/tot return Tn*l+In*((l+w)/2)+Fn*w def score(T,I,F): return (2*T+I-F)/4 def accuracy(T,I,F): return T-F def adj_cost(et,A,lam): return et*(1+lam*(1-A)) results={} for r in ROUTES:

results[r]={} for s in SCENARIOS:

b,l,w,fb,fl,fw=DATA[r][s] T,I,F=derive_tif(b,l,w,fb,fl,fw,OPACITY[r]) et=expected_time(b,l,w,T,I,F) A=accuracy(T,I,F) costs={lam:round(adj_cost(et,A,lam),2) for lam in LAMBDAS} results[r][s]={"T": T, "I": I, "F": F, "expected_time_min":round(et,1), "score":round(score(T,I,F),4), "accuracy":round(A,4), "risk_adjusted_costs": costs} with open("results_full.json", "w") as f:

json.dump(results,f,indent=2) print("Results saved to results_full.json") print("\nRoute Recommendations:") for s in SCENARIOS:

print(f"\n{s}:") for lam in [0.3,0.5,0.7]:

costs={r:results[r][s]["risk_adjusted_costs"][lam] for r in ROUTES} best=min(costs,key=costs.get) print(f" lambda={lam}: Route {best} (cost={costs[best]:.2f} min)")

Table D1 provides a concise glossary of all mathematical symbols and notation used throughout this study, together with their definitions and the section of first use.

Table D1. Glossary of mathematical notation.

Symbol Name Definition First Used ⟨T, I, F⟩ Neutrosophic triple SVNS representation with three Section 1.9 independent membership components T Truth membership Degree to which the most-likely Section 1.9 outcome holds; T = freq(likely) I Indeterminacy Degree of genuine unknownness; I = Section 1.9 membership min(1, 0.15 × spread + ε) F Falsity membership Degree of severe delay or route failure; Section 1.9 F = freq(worst) ε Opacity term Route-specific informational opacity Section 3.4 parameter in [0,1] t_best Best-case time Minimum travel time in minutes under Section 3.3 best-case conditions t_likely Most-likely time Most frequently occurring travel time Section 3.3 in minutes t_worst Worst-case time Maximum travel time in minutes Section 3.3 under worst-case conditions freq(x) Historical frequency Empirical frequency of travel-time Section 3.5 band x; frequencies sum to 1.0 Tₙ, Iₙ, Fₙ Normalised degrees T/(T+I+F), I/(T+I+F), F/(T+I+F) Section 3.5 respectively E[t] Expected travel De-neutrosophicated weighted mean Section 3.5 time travel time in minutes S Score function S = (2T + I − F) / 4; overall route Section 3.5 quality scalar. Range: [−0.25, 0.75] A Accuracy function A = T − F; net certainty/reliability Section 3.5 margin scalar. Range: [−1, 1] λ Risk-aversion Decision-maker risk sensitivity; λ ∈ Section 3.5 parameter [0,1]. λ=0 risk-neutral; λ→1 risk-averse λ* Crossover point Value of λ at which model Section 3.8 recommendation switches between two routes Cost(λ) Risk-adjusted cost Cost(λ) = E[t] × (1 + λ × (1 − A)); Section 3.5 route comparison criterion r* Optimal route Route minimising Cost(λ) for a given Section 3.5 scenario and λ cᵢⱼ Unit transportation Classical OR: cost per unit shipped Section 2.1 cost from source i to destination j xᵢⱼ Flow variable Classical OR: quantity shipped from Section 2.1 source i to destination j μ_A(x) Fuzzy membership Degree to which element x belongs to Section 2.3 fuzzy set A; μ_A: X → [0,1]

The following terms are used throughout this study with specific technical meanings as defined below.

Table E1. Operational definitions of key terms.

Term Symbol Definition SVNS SVNS A representation of a quantity as a triple ⟨T, I, F⟩ where each component takes a single value in [0,1]. Unlike probability, T, I, and F are not required to sum to 1, allowing simultaneous representation of partial truth, genuine indeterminacy, and partial falsity.

Truth membership T The degree to which the most-likely outcome is expected to hold. Estimated from the historical frequency of the most-likely travel-time band: T = freq(likely).

Indeterminacy I The degree to which the outcome is genuinely unknown.

membership Computed as I = min(1, 0.15 × [(worst − best) / likely] + ε), incorporating both time-spread and route-specific informational opacity.

Falsity membership F The degree to which severe delay or route failure is expected.

Estimated from the historical frequency of the worst-case travel-time band: F = freq(worst).

Opacity term ε A route-specific parameter capturing the proportion of route variability attributable to genuinely undocumented or unobservable factors. Derived via a four-criterion scoring procedure in Section 3.4.

Deneutrosophication N/A The process of converting a neutrosophic ⟨T, I, F⟩ representation into a single actionable crisp number for decision-making, analogous to defuzzification in fuzzy set theory.

Score function S S = (2T + I − F) / 4. A scalar measure of overall route quality.

Higher S indicates a more favourable and reliable route.

Accuracy function A A = T − F. A scalar measure of net certainty: how much more likely a good outcome is than a bad one. Higher A indicates lower downside risk.

Risk-aversion parameter λ A tunable decision parameter in [0,1] representing how strongly a decision-maker penalises low-accuracy routes. λ = 0 is risk-neutral; λ → 1 is highly risk-averse.

Risk-adjusted cost Cost(λ) Cost(λ) = E[t] × (1 + λ × (1 − A)). Expected travel time scaled by a penalty that increases with risk aversion and decreases with route accuracy.

Crossover point λ* The value of λ at which the model recommendation switches between routes. Computed algebraically by equating the cost expressions of two competing routes.