To model diverse real-world phenomena, a range of uncertainty-handling concepts has been actively studied, including Fuzzy Sets, Rough Sets, Intuitionistic Fuzzy Sets, Paraconsistent Sets, Neutrosophic Sets, Hyperneutrosophic Sets, Plithogenic Sets, and others. Among these extensions of fuzzy sets, Hyperfuzzy Sets are of particular significance. A hyperfuzzy set extends the notion of fuzzy sets to a hierarchical structure, enabling a more refined and flexible representation of uncertainty. In addition, the notion of a HyperRough Set generalizes the classical setting to multi-attribute data by assigning, to each attribute profile, a subset of the universe and then taking rough approximations with respect to a fixed indiscernibility relation. However, research on real-life applications of Hyperfuzzy Sets and HyperRough Sets remains limited. This paper explores application examples drawn from real-world scenarios by examining system integration using the Hyperfuzzy Set and HyperRough Set frameworks. Note that Health-Care system integration is the process of connecting distinct subsystems or components into a unified, functional, and efficient whole.
A fuzzy set on a universe U is specified by a membership function µA : U → [0, 1], which replaces crisp, binary inclusion with graded degrees of belonging and thus captures uncertainty at finer resolution
Rough set theory provides a framework for handling uncertainty by approximating a target set using its lower and upper approximations defined with respect to an indiscernibility (equivalence) relation on the universe
A HyperRough Set generalizes the classical rough set model to multi-attribute contexts: for each attribute profile, one first assigns a subset of the universe and then constructs the corresponding lower and upper approximations under a fixed indiscernibility relation
The foregoing makes clear that advancing the theory and practice of Fuzzy, Hyperfuzzy, and SuperHyperFuzzy Sets, as well as Rough and HyperRough Sets, is of substantial importance. However, especially for Hyper-fuzzy/SuperHyperFuzzy and HyperRough models, comprehensive application studies remain limited. To help close this gap, the present paper develops illustrative, real-world use cases focused on system integration within the Hyperfuzzy and HyperRough frameworks. For clarity, by health-care system integration we mean the process of connecting distinct subsystems or components into a unified, functional, and efficient whole.
This section provides an introduction to the foundational concepts and definitions required for the discussions in this paper. It should be noted that only finite sets are considered throughout this paper.
A fuzzy set assigns to each element a membership degree in [0, 1], thereby capturing uncertainty through more granular membership levels rather than a strict binary classification
Definition 2.1 (Universal Set). A universal set, denoted by U, is the set that contains all elements under consideration in a particular context. Every set discussed is assumed to be a subset of U.
Definition 2.2 (Powerset). (see
In particular, both the empty set ∅ and S itself lie in P(S).
Definition 2.3 (Fuzzy Set).
Definition 2.4 (Hyperfuzzy Set).
Example 2.5 (Hyperfuzzy Set in Health Care: Sepsis-risk assessment under multiple scorers). Let X = {p1, p2, p3} be three inpatients evaluated for early sepsis. Three independent scorers are used: SOFA (s1), qSOFA (s2), and an ML model (s3). A hyperfuzzy set µ ˜ : X → P ˜([0, 1]) assigns to each patient the set of admissible membership degrees (risk levels) reported by the scorers:
µ ˜(p1) = {0.72, 0.69, 0.80}, µ ˜(p2) = {0.31, 0.40}, µ ˜(p3) = {0.55, 0.61, 0.58}.
Two useful summaries are min ˜ µ(p1) = 0.69, max ˜ µ(p1) = 0.80 (similarly for p2, p3), which bound per-patient risk while retaining the full hyperfuzzy information as sets of degrees.
2.2| (m, n)-superhyperfuzzy set
An (m, n)-superhyperfuzzy set maps each m-level subset to a family of n-level membership sets, modeling hierarchical recursive uncertainty and complexity
Definition 2.6 (n-th powerset).
When excluding the empty set, write Pn ∗(X) = Pn(X) \ {∅}.
Example 2.7 (A concrete instance of the n-th powerset). Let X = {a, b}. Then the first powerset is P1(X) = P(X) = ∅, {a}, {b}, {a, b} , so |P1(X)| = 2|X| = 4.
The second powerset collects subsets of P1(X):
P2(X) = P P1(X) , which has 24 = 16 elements. Typical elements of P2(X) (i.e., subsets of P1(X)) are A1 = {{a}}, A2 = {{b}, {a, b}}, A3 = P1(X) (the full set), A4 = ∅.
Thus Ai ∈ P2(X) for i = 1, 2, 3, 4.
The third powerset is again a powerset, now of P2(X):
P3(X) = P P2(X) , so |P3(X)| = 216. An element of P3(X) is a set of elements of P2(X); for instance, B = { A1, A2 } ∈ P3(X).
If the empty set is excluded at each stage, the nonempty variants are P1 ∗(X) = {{a}, {b}, {a, b}} and P2 ∗(X) = P2(X) \ {∅}, etc. In particular, A4 = ∅ ∈ P / 2 ∗(X) but A1, A2, A3 ∈ P2 ∗(X). This illustrates the progression:
P P P
X −−→ P1(X) −−→ P2(X) −−→ P3(X), where each level consists of sets built from the objects of the previous level.
Definition 2.8 (m, n-SuperHyperFuzzy Set). (cf.
In particular, Pm ∗ (X) is the family of all nonempty elements of the m-th iterated powerset of X, and Pn ∗([0, 1]) is defined analogously. Then an (m, n)-SuperHyperFuzzy Set on X is a function µ ˜m,n : P∗ (X) −→ P ˜∗ [0, 1] , A 7→ µ ˜m,n(A), where P ˜∗([0, 1]) denotes the collection of all nonempty subsets of Pn([0, 1]). Thus each A ∈ P∗ (X) is assigned a nonempty family of membership-degree sets µ ˜m,n(A) ⊆ Pn([0, 1]), capturing hierarchical uncertainty across both the m- and n-levels.
Example 2.9 (SuperHyperFuzzy Set in Health Care: Discharge readiness for patient groups). Let X be the current census on a ward and consider nonempty patient groups A ∈ P1 ∗(X). Define an (m, n) = (1, 1)- SuperHyperFuzzy map µ ˜1,1 : P∗(X) −→ P ˜∗([0, 1]), where µ ˜1,1(A) is a family of membership-degree sets contributed by distinct policies. For instance, for A = {p1, p2} (“step-down candidates”):
µ ˜1,1(A) = [0.60, 0.80] (clinical guideline), {0.55, 0.70} (bed-management heuristic) .
Thus one group-level input produces a set of admissible degree-sets, letting clinicians reconcile multiple governance sources while keeping their structure explicit.
Rough set theory offers a principled method for treating uncertainty by approximating a target set through its lower and upper envelopes with respect to an indiscernibility (equivalence) relation on the universe
Definition 2.10 (Rough Approximation).
Then lowR(Y ) ⊆ Y ⊆ upR(Y ). It is convenient to write the boundary region BR(Y ) := upR(Y ) \ lowR(Y ).
Example 2.11 (Rough Set in Health Care: Isolation decision under indiscernibility). Let X = {x1, . . . , x6} be patients in an ED. Define an indiscernibility relation R by shared (age band, symptom cluster):
C1 = {x1, x2}, C2 = {x3, x4}, C3 = {x5, x6} (R-equivalence classes).
Let Y = {“requires respiratory isolation”} be the target, with Y = {x1, x3, x6}. Then the R-lower and R-upper approximations are lowR(Y ) = {Ci : Ci ⊆ Y } = ∅, upR(Y ) = {Ci : Ci ∩ Y ̸= ∅} = C1 ∪ C2 ∪ C3 = X, so the boundary BR(Y ) = X \ ∅ = X (decision is uncertain in every class). If, instead, Z = {x1, x2, x3, x4}, then lowR(Z) = C1 ∪ C2 = {x1, x2, x3, x4}, upR(Z) = C1 ∪ C2 = {x1, x2, x3, x4}, BR(Z) = ∅, exhibiting a case with no ambiguity.
The notion of a HyperRough Set generalizes the classical setting to multi-attribute data by assigning, to each attribute profile, a subset of the universe and then taking rough approximations relative to a fixed indiscernibility relation
Definition 2.12 (HyperRough Set).
Fix an equivalence relation R ⊆ X × X (indiscernibility on X). A HyperRough Set on X is a pair (F, J) where F : J −→ P(X) assigns to each attribute vector a = (a1, . . . , an) ∈ J a subset F(a) ⊆ X. For each a ∈ J, the rough approximations of F(a) are lowR F(a) = { x ∈ X : [x]R ⊆ F(a) }, upR F(a) = { x ∈ X : [x]R ∩ F(a) = ̸ ∅ }.
These satisfy, for all a ∈ J, lowR F(a) ⊆ upR F(a) , lowR(∅) = upR(∅) = ∅, lowR(X) = upR(X) = X, and are monotone in the argument: if F(a) ⊆ F(b) then lowR(F(a)) ⊆ lowR(F(b)) and upR(F(a)) ⊆ upR(F(b)).
Example 2.13 (HyperRough Set in Health Care: High-risk triage across attribute profiles). Let X = {x1, . . . , x5} be admitted patients. Attributes are T1 = Fever ∈ {yes, no} and T2 = Comorbidity ∈ {yes, no}, so J = T1 ×T2 = {(y, y), (y, n), (n, y), (n, n)}. Define F : J → P(X), where F(a) collects patients deemed high risk under profile a: F(y, y) = {x1, x3}, F(y, n) = {x2}, F(n, y) = {x4}, F(n, n) = ∅.
Let R be indiscernibility by ward: C1 = {x1, x2}, C2 = {x3, x4, x5}. For a = (y, y) we have F(a) = {x1, x3}, hence lowR F(a) = ∅ (no Ci contained in F(a)), upR F(a) = C1 ∪ C2 = {x1, x2, x3, x4, x5}.
If we refine F to F ′(y, y) = C2 = {x3, x4, x5}, then lowR(F ′(y, y)) = upR(F ′(y, y)) = C2, illustrating how HyperRough sets encode profile-wise targets F(a) while rough approximations are taken uniformly with respect to the fixed R.
System Integration connects diverse subsystems or components into a unified system, ensuring seamless function-ality, interoperability, efficiency, and coordinated operations
Example 3.1 (Smart Building Energy Management). Consider integrating three subsystems in a smart building: U = {HVAC, Lighting, Security}.
Define a Hyperfuzzy set of energy-efficiency demands Henergy = ( u, µ ˜energy(u) ) | u ∈ U , where µ ˜energy : U → P e([0, 1]) is a fuzzy-set-valued membership:
µ ˜energy(HVAC) = [0.7, 0.9], µ ˜energy(Lighting) = [0.5, 0.8], µ ˜energy(Security) = [0.3, 0.5].
System integration proceeds by computing the hyperfuzzy intersection of energy demands with occupancy patterns, yielding adaptive control rules.
Example 3.2 (Industrial IoT Fault Detection). In an Industrial IoT network, let U = {Sensor1, Sensor2, Actuator}.
Define the Hyperfuzzy fault-likelihood set Hfault = u, µ ˜fault(u) | u ∈ U , with µ ˜fault(u) = [ µ(u), µ(u) ] ⊆ [0, 1], e.g. µ ˜fault(Sensor1) = [0.2, 0.4], µ ˜fault(Sensor2) = [0.6, 0.8], µ ˜fault(Actuator) = [0.1, 0.3]. Integration uses hyperfuzzy aggregation to trigger maintenance alerts when combined likelihood exceeds a threshold interval.
Example 3.3 (Healthcare Information System). Integrate Electronic Health Records, Monitoring Devices, and Billing:
U = {EHR, Monitor, Billing}.
Define a Hyperfuzzy trust-level set Htrust = u, µ ˜trust(u) | u ∈ U , where µ ˜trust(u) = [TL(u), TU (u)] ⊆ [0, 1], for instance µ ˜trust(EHR) = [0.8, 0.95], µ ˜trust(Monitor) = [0.6, 0.85], µ ˜trust(Billing) = [0.4, 0.7]. System integra-tion applies hyperfuzzy composition to ensure only data with sufficiently high trust intervals are shared across modules.
Example 3.4 (Smart Agriculture Irrigation Management). Integrate three subsystems in a precision-farming setup:
U = {SoilSensor, WeatherStation, IrrigationController}.
Define the Hyperfuzzy moisture-need set Hmoisture = u, µ ˜moisture(u) | u ∈ U , with µ ˜moisture(SoilSensor) = [0.2, 0.5], µ ˜moisture(WeatherStation) = [0.3, 0.6], µ ˜moisture(IrrigationController) = [0.4, 0.8].
Hyperfuzzy integration computes the intersection of soil and weather demands to drive adaptive irrigation schedules.
Example 3.5 (Supply Chain Risk Integration). Consider three elements of a supply chain:
U = {SupplierReliability, InventoryLevel, TransportAvailability}.
Define the Hyperfuzzy risk-assessment set Hrisk = u, µ ˜risk(u) | u ∈ U , where µ ˜risk(SupplierReliability) = [0.1, 0.3], µ ˜risk(InventoryLevel) = [0.4, 0.7], µ ˜risk(TransportAvailability) = [0.2, 0.5].
Hyperfuzzy aggregation yields a combined risk interval; thresholds trigger contingency integration protocols.
Example 3.6 (Autonomous Vehicle Sensor–Control Coordination). Integrate perception and control modules: U = {PerceptionModule, LocalizationModule, ControlModule}.
Define the Hyperfuzzy confidence-level set Hconfidence = u, µ ˜conf(u) | u ∈ U , with example intervals µ ˜conf(PerceptionModule) = [0.6, 0.9], µ ˜conf(LocalizationModule) = [0.5, 0.8], µ ˜conf(ControlModule) = [0.7, 0.95].
System integration uses hyperfuzzy fusion to adjust control commands based on the weakest-link confidence interval.
Example 3.7 (Smart Grid Orchestration via (m, n) = (2, 1)–SuperHyperfuzzy Set). Universe and levels. Let the device universe be X = {PVA, BATA, LOADA, PVB, BATB, LOADB}.
We model orchestratable bundles of device–groups during a heatwave as elements of P2 ∗(X) (nonempty families of nonempty subsets of X). Consider A = A1, A2 ∈ P2 ∗(X), A1 = {PVA, BATA}, A2 = {PVB, LOADB}, representing two coupled sub-bundles to be co-dispatched.
Membership as a family of degree sets. Choose (m, n) = (2, 1). Define a SuperHyperfuzzy membership µ ˜2,1(A) ∈ P ˜1 ∗([0, 1]) (a nonempty family of subsets of [0, 1]) encoding dispatchability from two estimators (EST1, EST2): n o µ ˜2,1(A) = [0.68, 0.82], [0.74, 0.88] .
SHF–AND (envelope aggregator). For a family S ⊆ P1([0, 1]), define ANDmin(S) := min inf S, min sup S .
S∈S S∈S
Then the integrated readiness interval is µ ˜ready(A) = ANDmin µ ˜2,1(A) = min(0.68, 0.74), min(0.82, 0.88) = [0.68, 0.82].
Action policy. Let thresholds τauto = 0.75 and τreview = 0.65.
Auto–orchestrate if L ≥ τauto, Orchestrate w/ operator review if L < τauto ≤ U and U ≥ τreview, Defer if U < τreview, where [L, U] = [0.68, 0.82]. Since L = 0.68 < 0.75 but U = 0.82 ≥ 0.75, Proceed with orchestration behind operator review.
Interpretation. Both estimators support feasibility up to 0.82, but the lower bound 0.68 (from EST1) prevents fully automatic co-dispatch.
Example 3.8 (Warehouse Multi-Robot Task Allocation via (m, n) = (1, 2)–SuperHyperfuzzy Set). Universe and levels. Let X = {r1, r2, r3, r4, r5} be robots. Consider a team A = {r1, r2, r4} ∈ P1 ∗(X).
We choose (m, n) = (1, 2) so that membership collects scenario-wise families of degree-sets.
Membership as a family of families of degree sets. Define µ ˜1,2(A) ∈ P ˜∗([0, 1]) with two scenarios: µ ˜1,2(A) = Σ1 = {[0.70, 0.85], [0.65, 0.80]}, Σ2 = {[0.55, 0.72]} .
Here Σ1 = normal peak hours (two estimators), Σ2 = degraded comms.
Two-stage SHF–AND. First compute an envelope per scenario Σ:
E(Σ) := ANDmin(Σ) = min inf S, min sup S .
S∈Σ S∈Σ
Thus E(Σ1) = [0.65, 0.80], E(Σ2) = [0.55, 0.72]. Adopt a robust (worst-case) aggregator across scenarios: RobAND µ ˜1,2(A) := min inf E(Σ), min sup E(Σ) = [0.55, 0.72].
Action policy. With τauto = 0.70 and τsafe = 0.60, Auto–allocate if L ≥ τauto, Supervised allocate if τsafe ≤ L < τauto and U ≥ τauto, Defer/replan if L < τsafe.
For [L, U] = [0.55, 0.72], L < 0.60, so Defer allocation and replan under degraded communications.
Interpretation. The level-2 structure explicitly captures adverse scenarios, driving a conservative integration decision.
Here, we present example application cases of system integration using HyperRough Sets.
Example 4.1 (Smart Building Integration: HVAC–Lighting–Access). Let the universe X = {z1, . . . , z6} denote six building zones. Define the indiscernibility relation R by same floor, with equivalence classes [z1]R = {z1, z2}, [z3]R = {z3, z4}, [z5]R = {z5, z6}.
Let attributes be Occ ∈ {Low, High} (occupancy) and Per ∈ {Day, Night} (period), so J = {Low, High} × {Day, Night}. For the profile a = (High, Night), suppose the integration target is the set of zones requiring cooling + lighting boost:
F(a) = {z1, z2, z4, z5} ⊆ X.
Then, using the rough approximations w.r.t. R, lowR F(a) = {x ∈ X : [x]R ⊆ F(a)} = {z1, z2}, upR F(a) = {x ∈ X : [x]R ∩ F(a) = ̸ ∅} = {z1, z2, z3, z4, z5, z6} = X, BR F(a) = upR(F(a)) \ lowR(F(a)) = {z3, z4, z5, z6}.
Integration policy.
• Apply immediate HVAC/lighting actions to lowR(F(a)) = {z1, z2} (definite need).
• For boundary zones BR(F(a)), enable sensors and short-interval rechecks; escalate only if demand persists. • Zones outside upR(F(a)) receive no change (none in this instance).
This realizes a floor-consistent control strategy: decisive actions occur only when an entire floor-class warrants them, while ambiguous floors are handled conservatively.
Example 4.2 (Hospital IT Integration: EHR–Monitors–Lab Systems). Let X = {r1, . . . , r8} denote record fragments across three systems (EHR, bedside Monitor, Lab). Define R by same patient ID, yielding [r1]R = {r1, r2}, [r3]R = {r3, r4}, [r5]R = {r5, r6}, [r7]R = {r7, r8}.
Let attributes be patient Consent ∈ {No, Yes} and clinical Criticality ∈ {Low, High}; thus J = {No, Yes} × {Low, High}. For a = (Yes, High), take the target set of records eligible for cross-module sharing with alerts: F(a) = {r1, r2, r3, r6, r7}.
Then lowR F(a) = {x : [x]R ⊆ F(a)} = {r1, r2} (only the full class {r1, r2} ⊆ F(a)), upR F(a) = {x : [x]R ∩ F(a) = ̸ ∅} = {r1, . . . , r8} = X, BR F(a) = X \ {r1, r2} = {r3, r4, r5, r6, r7, r8}.
Integration policy.
• Auto-sync and notify across EHR, Monitor, Lab for lowR(F(a)) (patient-level completeness guaranteed). • For the boundary BR(F(a)), gate sharing behind consent/role checks and clinician override; queue reconcil-iation jobs to complete missing counterparts.
• Deny propagation outside upR(F(a)) (none here).
Thus, only patients with complete consented high-criticality bundles are integrated automatically; partial bundles are handled safely.
Example 4.3 (Supply-Chain Integration: WMS–TMS–ERP Coordination). Let X = {s1, . . . , s9} represent shipments. Define R by same route group, with [s1]R = {s1, s2, s3}, [s4]R = {s4, s5, s6}, [s7]R = {s7, s8, s9}.
Attributes: Weather ∈ {Mild, Severe}, Urgency ∈ {Low, High}; hence J = {Mild, Severe} × {Low, High}. For a = (Severe, High), let the target be shipments to reroute or expedite:
F(a) = {s1, s2, s3, s6, s7}.
Then lowR F(a) = {x : [x]R ⊆ F(a)} = {s1, s2, s3} (entire route group {s1, s2, s3} ⊆ F(a)), upR F(a) = {x : [x]R ∩ F(a) = ̸ ∅} = {s1, . . . , s9} = X, BR F(a) = X \ {s1, s2, s3} = {s4, s5, s6, s7, s8, s9}.
Integration policy.
• WMS–TMS–ERP immediately co-execute rerouting and capacity reservation for lowR(F(a)) (route-consistent action).
• For BR(F(a)), request carrier/weather confirmation and inventory checks; commit actions only if the remaining route members flip into F(a).
• No actions for items outside upR(F(a)) (none here).
This enforces route-coherent interventions while avoiding fragmented decisions on partially affected routes.
A healthcare system is an integrated network that delivers services, coordinates data, finances care, and optimizes population health outcomes effectively
Example 5.1 (HealthCare-System Integration Using Hyperfuzzy Sets — Sepsis Early-Warning Pipeline). Subsystems and universe. Let U = {EHR, BedsideMonitor, Lab}.
Define a hyperfuzzy set of trust for a real-time sepsis early-warning pipeline:
Htrust = {(u, µ ˜trust(u)) | u ∈ U}, µ ˜trust(u) ⊆ [0, 1] (interval).
For a particular ICU context, set the interval memberships (higher = more trustworthy):
µ ˜trust(EHR) = [0.85, 0.95], µ ˜trust(BedsideMonitor) = [0.70, 0.90], µ ˜trust(Lab) = [0.60, 0.80].
Integration operator (hyperfuzzy AND). Using the Gödel t-norm (min) lifted to intervals, [li, ui] i=1 := min li, min ui .
G i i
Hence the pipeline readiness interval is µ ˜ready = [0.85, 0.95], [0.70, 0.90], [0.60, 0.80] = min(0.85, 0.70, 0.60), min(0.95, 0.90, 0.80) = [0.60, 0.80].
Action policy (explicit thresholds). Let τauto = 0.75, τreview = 0.65.
Decision rule:
Auto-integrate if L ≥ τauto, Integrate with human review if L < τauto ≤ U and U ≥ τreview, Defer/block if U < τreview.
With [L, U] = [0.60, 0.80], we have L = 0.60 < 0.75 but U = 0.80 ≥ 0.75, so the second clause applies:
Proceed with integration behind clinician review (alert gated).
All three subsystems jointly support the pipeline (U = 0.80), but the weakest link (Lab lower bound 0.60) prevents fully automatic action; the hyperfuzzy interval captures this uncertainty explicitly.
Example 5.2 (HealthCare-System Integration Using Hyperfuzzy Sets — Remote Diabetes Titration). Subsys-tems and universe. Let U = {EHRMedicationList, PharmacyDispense, WearableAdherence}.
Define a hyperfuzzy data-reliability set Hreli = {(u, µ ˜reli(u)) | u ∈ U}, µ ˜reli(u) = [l(u), u(u)] ⊆ [0, 1].
For a telehealth visit:
µ ˜reli(EHRMedicationList) = [0.90, 0.98], µ ˜reli(PharmacyDispense) = [0.92, 0.99], µ ˜reli(WearableAdherence) = [0.85, 0.95].
Integration operator (product t-norm on intervals). Using the product t-norm TΠ(x1, . . . , xn) = i xi lifted to intervals, h i ^ n Y Y [li, ui] i=1 := li, ui .
Thus the titration-readiness interval equals ^ n o µ ˜titrate = [0.90, 0.98], [0.92, 0.99], [0.85, 0.95]
= 0.90 · 0.92 · 0.85, 0.98 · 0.99 · 0.95 .
Explicitly, L = 0.90 × 0.92 × 0.85 = 0.702, U = 0.98 × 0.99 × 0.95 = 0.92169, so µ ˜titrate = [0.702, 0.92169].
Action policy (dose-change gating). Let the auto-titration threshold be τauto = 0.70 and the safety floor be τsafe = 0.60.
Auto-titrate if L ≥ τauto, Issue recommendation + confirm if L < τauto ≤ U and U ≥ τsafe, Do not titrate if U < τsafe.
Since L = 0.702 ≥ 0.70, we obtain Auto-titrate is enabled (high-confidence integration across systems).
High reliability across EHR, pharmacy, and adherence yields a tight, high readiness interval; the product t-norm enforces that all subsystems must be strong to pass the automatic threshold.
Example 5.3 (Perioperative Scheduling Integration via (m, n) = (2, 1)). Universe and levels. Let X = {OR1, AnesthesiaTeam, SterileProc, PACU}.
For a specific case, coordinatable bundles of unit-groups are in P2 ∗(X). Consider A = {OR1, SterileProc}, {AnesthesiaTeam, PACU} .
Membership as readiness families. Choose (m, n) = (2, 1). Let µ ˜2,1(A) collect checklists and feeds (room turnover, tray status, staff availability):
µ ˜2,1(A) = [0.80, 0.95], [0.70, 0.90], [0.60, 0.85] .
Integration and policy. Envelope:
µ ˜ready(A) = AND min µ ˜2,1(A) = [min(0.80, 0.70, 0.60), min(0.95, 0.90, 0.85)] = [0.60, 0.85].
With thresholds τbook = 0.75, τhold = 0.65:
Auto–book if L ≥ 0.75, Book w/ charge–nurse confirm if L < 0.75 ≤ U and U ≥ 0.65, Hold if U < 0.65.
Here L = 0.60 < 0.65 but U = 0.85 ≥ 0.75 ⇒ Book behind charge–nurse confirmation; expedite tray turnaround.
The weakest readiness feed (sterilization) pulls the lower bound down, preventing an automatic commit.
Example 5.4 (Multimodal Cancer Diagnosis Integration via (m, n) = (1, 2)). Universe and levels. Let diagnostic subsystems be X = {Radiology, Pathology, Genomics}.
For a tumor board candidate, consider A = {Radiology, Pathology} ∈ P1 ∗(X). Choose (m, n) = (1, 2) so membership aggregates scenario families.
Membership (scenario families). Define µ ˜1,2(A) with two scenarios:
µ ˜1,2(A) = Σ1 = {[0.80, 0.92], [0.75, 0.90]}, Σ2 = {[0.58, 0.72]} , where Σ1 = complete imaging + preliminary histology; Σ2 = pending stains / discordant read.
Two-stage envelope and decision. Per-scenario envelopes: E(Σ1) = [0.75, 0.90], E(Σ2) = [0.58, 0.72]. Robust aggregator:
µ ˜integrate(A) = RobAND µ ˜1,2(A) = [min(0.75, 0.58), min(0.90, 0.72)] = [0.58, 0.72].
Policy with τauto = 0.70, τreview = 0.60:
Queue for tumor board with radiology–pathology consensus review (not auto-escalated).
While one scenario is strong, the level-2 family captures a plausible weaker path; robust integration mandates expert review.
This short paper explored application examples drawn from real-world scenarios by examining system integration using the Hyperfuzzy Set and HyperRough Set frameworks. In the future, we plan to pursue research on application cases of system integration within the settings of Neutrosophic Sets
No external funding or financial support was provided for this study.
The authors wish to thank all colleagues and mentors whose feedback and encouragement enriched this work. We are grateful to the community of researchers whose foundational contributions informed our developments. Special appreciation goes to the institutions that offered resources and technical infrastructure throughout this project.
This manuscript presents purely conceptual work without empirical data. Scholars interested in these ideas are invited to undertake experimental or case-study research to substantiate and extend the proposed frameworks.
This paper involves no human or animal subjects and thus did not require ethics committee review or approval.
We use generative AI and AI-assisted tools for tasks such as English grammar checking, and We do not employ them in any way that violates ethical standards.
The authors declare that there are no competing interests concerning the content or publication of this article.
The theoretical models and propositions herein have not yet been subjected to practical validation. Readers should independently verify all citations and be aware that inadvertent inaccuracies may remain. The opinions expressed are those of the authors and do not necessarily represent the views of affiliated organizations.