Numerous set-theoretic frameworks have been developed to handle uncertainty. In this paper, we focus on Rough Set theory. A rough set approximates a target subset by computing lower and upper bounds from equivalence classes, thereby capturing both certain and uncertain membership [16]. Related approaches include Fuzzy Sets [28], Intuitionistic Fuzzy Sets [3], Neutrosophic Sets [21, 22], Plithogenic Sets [23, 24], Graded Rough Sets [7, 27], Multigranular Rough Sets [18], and Soft Sets [15]. These models have been extensively studied in various ap-plications, including decision making and artificial intelligence.

Extensions of Rough Sets (such as the HyperRough Set and the n-SuperHyperRough Set) have also been proposed. The HyperRough Set extends classical rough set theory by incorporating multiple attribute domains. An n-SuperHyperRough Set further generalizes the concept by using iterative power sets of attribute values to produce more nuanced approximations under uncertainty [11].

In this paper, we introduce the HyperRough TOPSIS method and the SuperHyperRough TOPSIS method, and we examine their underlying mathematical structures. TOPSIS is a well-established decision-making technique, and our proposed methods constitute generalized extensions of the classical Rough TOPSIS approach. We hope that these contributions will help advance research in decision-making theory.

The definitions of the Rough Set, HyperRough Set, and SuperHyperRough Set are presented below.

Definition 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 (Rough Set Approximation). [17] Let X be a nonempty universe of discourse, and let R ⊆ X ×X be an equivalence relation (also called an indiscernibility relation) on X. The relation R partitions X into disjoint equivalence classes, denoted by [x]R for each x ∈ X, where [x]R = {y ∈ X | (x, y) ∈ R}.

For any subset U ⊆ X, the lower approximation U and the upper approximation U are defined by:

(1) Lower Approximation:

U = {x ∈ X | [x]R ⊆ U}.

This set contains all elements whose entire equivalence class is contained within U; these elements definitely belong to U.

(2) Upper Approximation:

U = {x ∈ X | [x]R ∩ U ̸= ∅}.

This set contains all elements whose equivalence class has a nonempty intersection with U; these elements possibly belong to U.

Thus, the pair (U,U) forms the rough set representation of U, satisfying U ⊆ U ⊆ U.

The HyperRough Set extends rough set theory by incorporating multiple attributes. Its formal definition is given below [11, 9].

Definition 3 (HyperRough Set). [11] Let X be a nonempty finite universe, and let T1, T2, . . . , Tn be n distinct attributes with corresponding domains J1, J2, . . . , Jn. Define the Cartesian product J = J1 × J2 × · · · × Jn.

Let R ⊆ X × X be an equivalence relation on X, with [x]R denoting the equivalence class of x. A HyperRough Set over X is a pair (F,J), where:

• F : J → P(X) is a mapping that assigns to each attribute value combination a = (a1, a2, . . . , an) ∈ J a subset F(a) ⊆ X.

• For each a ∈ J, the rough set approximations of F(a) are defined as F(a) = {x ∈ X | [x]R ⊆ F(a)}, F(a) = {x ∈ X | [x]R ∩ F(a) ̸= ∅}.

Here, F(a) comprises all elements whose equivalence classes are completely contained within F(a), while F(a) contains elements whose equivalence classes intersect F(a). Additionally, the following properties hold for all a ∈ J:

• F(a) ⊆ F(a).

• If F(a) = ∅, then F(a) = F(a) = ∅.

• If F(a) = X, then F(a) = F(a) = X.

Example 1 (HyperRough Set in Patient Diagnosis). (cf.[2, 1]) Let X = {P1, P2, P3, P4, P5, P6, P7, P8} be eight patients in a hospital, and consider two attributes:

T1 = “Symptom Group”, J1 = {Respiratory, Gastrointestinal}, T2 = “Test Result”, J2 = {Positive, Negative}.

Define J = J1 × J2 = { (s, t) | s ∈ J1, t ∈ J2} , and let the equivalence relation R group patients by their ward:

[P1]R = {P1, P2}, [P3]R = {P3, P4}, [P5]R = {P5, P6, P7}, [P8]R = {P8}.

Define F : J → P(X) by F(Respiratory, Positive) = {P1, P3, P5}, F(Respiratory, Negative) = {P2, P4}, F(Gastrointestinal, Positive) = {P6, P7}, F(Gastrointestinal, Negative) = {P8}.

Then for each a ∈ J we compute F(a) = { x ∈ X | [x]R ⊆ F(a)}, F(a) = { x ∈ X | [x]R ∩ F(a) ̸= ∅}.

Concretely:

F(Resp+) = ∅, F(Resp+) = {P1, P2, P3, P4, P5, P6, P7}, F(Resp–) = ∅, F(Resp–) = {P1, P2, P3, P4}, F(Gastro+) = ∅, F(Gastro+) = {P5, P6, P7}, F(Gastro–) = ∅, F(Gastro–) = {P8}.

One checks:

F(a) ⊆ F(a), F(a) = F(a) = ∅ if F(a) = ∅, F(a) = F(a) = X if F(a) = X.

An n-SuperHyperRough Set generalizes rough sets by using power sets of attribute values to produce nuanced approximations under uncertainty[10, 8, 11]. The definition of n-SuperHyperRough Sets is described as follows.

Definition 4 (n-SuperHyperRough Set). [11] Let X be a nonempty finite universe, and let T1, T2, . . . , Tn be n distinct attributes with respective domains J1, J2, . . . , Jn. For each attribute Ti, let P(Ji) denote its power set. Define the set of all possible attribute value combinations as J = P(J1) × P(J2) × · · · × P(Jn).

Let R ⊆ X × X be an equivalence relation on X. An n-SuperHyperRough Set over X is a pair (F,J), where:

• F : J → P(X) is a mapping that assigns to each attribute value combination A = (A1, A2, . . . , An) ∈ J (with Ai ⊆ Ji for all i) a subset F(A) ⊆ X.

• For each A ∈ J, the lower and upper approximations are defined as F(A) = {x ∈ X | [x]R ⊆ F(A)}, F(A) = {x ∈ X | [x]R ∩ F(A) ̸= ∅}.

Thus, F(A) consists of all elements whose equivalence classes are entirely contained in F(A), and F(A) includes those elements whose equivalence classes intersect F(A). The following properties hold for all A ∈ J:

• F(A) ⊆ F(A).

• If F(A) = ∅, then F(A) = F(A) = ∅.

• If F(A) = X, then F(A) = F(A) = X.

• For any A, B ∈ J, F(A ∩ B) ⊆ F(A) ∩ F(B), F(A ∪ B) ⊇ F(A) ∪ F(B).

Example 2 (n-SuperHyperRough Set in Smartphone Selection). (cf.[14, 4]) Let X = {S1, S2, S3, S4, S5, S6} be six smartphone models. Two attributes:

T1 = “Brand”, J1 = {A, B}, T2 = “Screen Size”, J2 = {Small, Medium, Large}.

We form J = P(J1) × P(J2), so each element A = (A1, A2) has A1 ⊆ J1, A2 ⊆ J2. Let the equivalence relation R group models by release-year cohort:

[S1]R = {S1, S3}, [S2]R = {S2, S6}, [S4]R = {S4, S5}.

Define F : J → P(X) on four representative combinations:

F({A}, {M,L}) = {S1, S2}, F({B}, {S, M}) = {S3, S4, S5}, F({A, B}, {L}) = {S2, S4, S6}, F(∅, ∅) = ∅.

Then for each A ∈ J, F(A) = {x ∈ X | [x]R ⊆ F(A)}, F(A) = {x ∈ X | [x]R ∩ F(A) ̸= ∅}.

For example:

F({A}, {M,L}) = ∅, F({A}, {M,L}) = {S1, S2, S3, S6}, F({B}, {S, M}) = {S4, S5}, F({B}, {S, M}) = {S1, S3, S4, S5}, F({A, B}, {L}) = {S2, S4}, F({A, B}, {L}) = {S2, S4, S6}, F(∅, ∅) = F(∅, ∅) = ∅.

One verifies the general properties:

F(A) ⊆ F(A), F(A ∩ B) ⊆ F(A) ∩ F(B), F(A ∪ B) ⊇ F(A) ∪ F(B).

Research on decision-making and multi-criteria decision-making (MCDM) has gained significant attention in recent years[12, 13, 6]. Among various approaches, this paper focuses on the TOPSIS method. The Rough TOPSIS Method is a multi-criteria decision-making technique that employs interval-based approximations derived from rough set theory to effectively represent and manage uncertainty [26, 5, 20, 19]. The following section presents the formal definition of the method, along with a concrete illustrative example.

Definition 5 (Rough TOPSIS Method). [26] Let A = {A1, . . . , Am} be a set of alternatives and C = {c1, . . . , cn} a set of criteria. Each performance rating is given as a rough number x ˜ij = xij, xij , i = 1, . . . , m, j = 1, . . . , n.

The Rough TOPSIS method proceeds as follows:

(1) Construct decision matrix.

X ˜ = (˜ x ) .

ij m×n

(2) Normalize. For benefit criteria, h x i ij xij r ˜ij = q , q .

Pm Pm xij xij

i=1 i=1

(For cost criteria one takes reciprocals before normalization.) (3) Determine weights. Let w ˜ = ( ˜ w1, . . . , w ˜n) be a weight vector of rough numbers w ˜j = [wj, wj]. (4) Weighted normalized matrix. Compute v ˜ij = ˜ wj × r ˜ij, via interval multiplication.

(5) Ideal solutions.

v ˜j = max vij, max vij , v ˜j = min vij, min vij .

i i i i

(6) Distance measure. For two rough numbers a ˜ = [a, a], ˜ b = [b, b], define ˜ 2 2 d(˜ a, b) = (a − b) + (a − b) .

(7) Distance to ideals.

D = t d(˜ v , v ˜ ) , D = t d(˜ v , v ˜ ) .

(8) Closeness coefficient.

CCi = , and rank alternatives in descending order of CCi.

Example 3. Consider three alternatives A1, A2, A3 evaluated on two benefit criteria c1, c2. The normalized rough decision matrix and crisp weight vector w = (0.4, 0.6) are [0.50, 0.70] [0.60, 0.80] r ˜ = [0.60, 0.80] [0.50, 0.70] , w = (0.4, 0.6).

ij

[0.40, 0.60] [0.70, 0.90] Compute the weighted normalized matrix [0.20, 0.28] [0.36, 0.48] v ˜ = w · r ˜ = [0.24, 0.32] [0.30, 0.42] .

ij j ij

[0.16, 0.24] [0.42, 0.54] The RPIS and RNIS are v ˜ = [0.24, 0.32], [0.42, 0.54] , v ˜ = [0.16, 0.24], [0.30, 0.42] .

Calculating distances and closeness coefficients yields CC1 = 0.50, CC2 = 0.40, CC3 = 0.60, so the final ranking is A3 ≻ A1 ≻ A2.

The definition of the HyperRough TOPSIS method is presented below.

Definition 6 (HyperRough TOPSIS Method). Let • A = {A1, . . . , Am} be the set of m alternatives, • C = {c1, . . . , cn} be the set of n criteria, • R1, . . . , Rk be k equivalence relations (levels) on the universe U, • For each (i, j) the performance rating under cj for Ai is the hyperrough number

1 1 2 2 k k

x ˆij = [xij, xij], [xij, xij], . . . , [xij, xij] .

Then HyperRough TOPSIS consists of:

(1) Hyper-Decision Matrix:

ˆ j=1,...,n X = (ˆ xij)i=1,...,m.

(2) Level-wise Normalization: For each criterion cj and level ℓ ∈ {1, . . . , k} define

uX ℓ t ℓ 2 Lj = (xpj) ,

then for benefit-type cj set xij xij rij = , rij = , Lj Lj and assemble

1 1 k k

r ˆij = [rij, rij], . . . , [rij, rij] .

(For cost-type criteria, replace xij by 1/xij before normalizing.) (3) Hyperrough Weights: Let each weight be

1 1 k k ℓ ℓ

w ˆj = [wj , wj ], . . . , [wj , wj ] , wj + wj /2 = 1 ∀ℓ.

(4) Weighted Normalized Matrix: For each (i, j) multiply intervals level-wise:

ℓ ℓ ℓ ℓ ℓ ℓ

vij = wj rij, vij = wj rij, so

1 1 k k

v ˆij = [vij, vij], . . . , [vij, vij] .

(5) Positive and Negative Hyper-Ideal: For each criterion cj and each level ℓ set

+,ℓ ℓ +,ℓ ℓ

vj = max vij, vj = max vij,

i=1,...,m i=1,...,m −,ℓ ℓ −,ℓ ℓ

vj = min vij, vj = min vij.

i=1,...,m i=1,...,m

Collect into v ˆj and v ˆj as k-tuples.

ℓ ℓ ℓ

ℓ k ˆ k (6) Hyperrough Distance: For any two hyperrough numbers a ˆ = ([a , a ]) and b = ([b , b ]) , define

ℓ=1 ℓ=1

uX X

ℓ 2 ℓ ℓ 2

ˆ t ℓ d(ˆ a, b) = a − b + a − b .

ℓ=1 ℓ=1

(7) Separation Measures: For each alternative Ai set

D = t d v ˆ , v ˆ , D = t d v ˆ , v ˆ .

(8) Closeness Coefficient and Ranking:

CCi = , Rank the alternatives in descending order of CCi.

Example 4. HyperRough TOPSIS for m = 3, n = 2, k = 2.

Data:

c1 c2 A1 : ([2, 4], [3, 5]) ([5, 7], [4, 6]) A2 : ([3, 6], [2, 4]) ([6, 8], [5, 7]) A3 : ([1, 3], [4, 6]) ([4, 6], [3, 5]) Weights:

w ˆ1 = ([0.3, 0.5], [0.2, 0.4]), w ˆ2 = ([0.4, 0.6], [0.3, 0.5]).

Step by step:

√ √ √ √ √ √

1 2 2 2 2 2 2 2 1 2 2 2

(1) Compute L1 = 4 + 6 + 3 = 61, L1 = 5 + 4 + 6 = 77, L2 = 7 + 8 + 6 = 149,

2 2 2 2

L2 = 6 + 7 + 5 = 110.

(2) Normalize each entry at both levels, e.g. r11 = 2/ 61 ≈ 0.256, r11 = 4/ 61 ≈ 0.512, etc., building all r ˆij.

(3) Multiply by weights to get v ˆij, e.g. v11 = 0.3 × 0.256 = 0.0768, v11 = 0.5 × 0.512 = 0.256, etc.

(4) Determine for each (j, ℓ) the positive and negative ideals, e.g. v1 = max{0.0768, . . . }, etc.

(5) Compute distances d(ˆ vij, v ˆj ) by plugging into the two-sum formula.

± − + −

(6) Sum to obtain Di and then CCi = Di /(Di + Di ).

(7) Rank: suppose numeric evaluation yields CC1 = 0.42, CC2 = 0.68, CC3 = 0.35, then A2 ≻ A1 ≻ A3. Theorem 1 (Boundedness of Closeness Coefficient). For each alternative Ai in HyperRough TOPSIS, the closeness coefficient CCi =

satisfies 0 ≤ CCi ≤ 1.

Proof : By construction, the separation measures Di and Di are nonnegative:

+ t + 2 − t − 2 Di = d(ˆ vij, v ˆj ) ≥ 0, Di = d(ˆ vij, v ˆj ) ≥ 0.

Hence the sum Di + Di is positive unless both are zero (in which case CCi is undefined but by convention set to 1/2). Otherwise, − − + Di 0 ≤ Di ≤ Di + Di =⇒ 0 ≤ ≤ 1.

Theorem 2 (Reduction to Rough TOPSIS). When the number of levels k = 1, the HyperRough TOPSIS method coincides exactly with the ordinary Rough TOPSIS method.

Proof : With k = 1, each hyperrough number x ˆij = [xij, xij] is a single interval. Then:

qP

1 1 2

• Level-wise normalization uses Lj = xpj , same as in Rough TOPSIS.

• Weighted normalization, ideal solutions, distance

d([a, a], [b, b]) = (a − b) + (a − b) coincide.

± − + −

• Separation measures Di and CCi = Di /(Di + Di ) are identical to those in Rough TOPSIS.

Thus every step of HyperRough TOPSIS reduces to the corresponding step of Rough TOPSIS. □

Theorem 3 (Scale Invariance). Let c > 0 be a constant. If for a fixed criterion cj and level ℓ we replace every performance rating

ℓ ℓ ′ ℓ ℓ

x ˆij = (. . . , [xij, xij], . . . ) 7−→ x ˆij = (. . . , [c xij, c xij], . . . ), then all closeness coefficients CCi remain unchanged.

Proof : Under this scaling, the normalization denominator

ℓ ℓ 2 ′ ℓ ℓ 2 ℓ

Lj = (xpj) 7−→ Lj = (c xpj) = c Lj.

Thus the normalized intervals

ℓ ℓ ℓ ′ ℓ ℓ ℓ ℓ

rij = xij/Lj 7−→ rij = (c xij)/(c Lj) = rij, and similarly for rij. Hence the weighted normalized values, ideals, distances d(·, ·), and separation measures Di are unchanged, so ′ Di CCi = = CCi.

Theorem 4 (Permutation Invariance). The closeness coefficients {CCi} are invariant under any permutation of the set of criteria C or of the levels {1, . . . , k}.

Proof : Permuting criteria cj simply reorders the summation index in

uX ± t ± 2 Di = d(ˆ vij, v ˆj ) ,

which does not affect the value. Permuting levels ℓ within each hyperrough number

1 1 k k

v ˆij = [vij, vij], . . . , [vij, vij] corresponds to relabeling the terms in the distance formula

uX

ℓ ℓ ℓ

ˆ t ℓ 2 2 d(ˆ a, b) = (a − b ) + (a − b ) ,

which is symmetric in the summands. Therefore both Di and CCi remain unchanged. □

The definition of the SuperHyperRough TOPSIS method is presented below.

Definition 7 (SuperHyperRough TOPSIS Method). Let • A = {A1, . . . , Am} be a set of m alternatives, • C = {c1, . . . , cn} be a set of n criteria, • R1, . . . , Rk be k equivalence relations on the universe U (levels of indiscernibility), • Q = {q1, . . . , qp} be p contexts (e.g. scenarios or expert opinions), • For each triple (i, j, α), let

(α) 1,(α) 1,(α) k,(α) k,(α)

x ˜ij = [xij , xij ], . . . , [xij , xij ] be the k-level hyperrough evaluation of alternative Ai under criterion cj in context qα.

Define the superhyperrough number for (i, j) as the mapping x ˆij : Q −→ I × · · · × I , x ˆij(qα) = ˜ xij , | {z }

where I denotes the set of closed real intervals. Then the SuperHyperRough TOPSIS Method consists of:

(1) Aggregate contexts: Treat each x ˆij as a single object in the superhyperrough set SHR(U; R1, . . . , Rk, Q) = x ˆ : Q → (I) .

(2) Normalization & weighting: For each context α, normalize the intervals in x ˜ij and multiply by

(α) k

rough-interval weights w ˜j ∈ (I) , exactly as in HyperRough TOPSIS.

(3) Super-ideal solutions: For each criterion cj and each context qα compute positive/negative ideals

± (α)

v ˜j by taking component-wise maxima/minima across i = 1, . . . , m.

(4) Superhyperrough distance: For any two superhyperrough numbers a, ˆ ˆ b ∈ SHR define u p p uX X X X

ℓ,(α) 2 ℓ,(α) ℓ,(α) 2

ˆ t ℓ,(α) D(ˆ a, b) = a − b + a − b .

α=1 ℓ=1 α=1 ℓ=1 (5) Separation measures: For each Ai define

D = t D v ˆ , v ˆ , D = t D v ˆ , v ˆ .

(6) Closeness coefficient & ranking:

CCi = , rank alternatives by descending CCi.

Example 5 (SuperHyperRough TOPSIS in Supply Chain Partner Selection). Consider a manufacturing firm that must choose one of three potential logistics partners A = {Partner1, Partner2, Partner3} based on two criteria:

C = {Cost (c1), Delivery Reliability (c2)}.

The firm evaluates each partner under two indiscernibility levels (vehicle type and region):

R1 : equivalence by vehicle fleet (small, large), R2 : equivalence by delivery region (local, national). Furthermore, it obtains expert opinions under two contexts:

(Market Demand) (Fuel Price)

Q = {Context , Context }.

Each evaluation x ˜ij is a pair of intervals—one per level—e.g. for Partner1 on Cost under Market Demand:

(1) [1200, 1500] : cost interval for small vehicles (R1), x ˜11 = [1200, 1500], [1100, 1400] , where [1100, 1400] : cost interval for large vehicles (R2).

A complete table of raw hyperrough evaluations is:

(1) (2)

x ˜ij (Q = Market Demand) x ˜ij (Q = Fuel Price) Ai\cj c1 c2 c1 c2 Partner1 ([1200, 1500], [1100, 1400]) ([0.85, 0.95], [0.80, 0.90]) ([1300, 1550], [1150, 1420]) ([0.80, 0.90], [0.75, 0.85]) Partner2 ([1000, 1300], [900, 1200]) ([0.80, 0.88], [0.78, 0.86]) ([1050, 1350], [920, 1250]) ([0.82, 0.92], [0.77, 0.87]) Partner3 ([1100, 1400], [1000, 1300]) ([0.88, 0.96], [0.83, 0.93]) ([1150, 1450], [1020, 1320]) ([0.85, 0.95], [0.81, 0.91])

Step 1. Normalization & Weighting. For each context α and each pair of intervals, normalize by the Euclidean norm over all partners and levels, then multiply by the decision maker’s rough-interval weights. Assume crisp weights w1 = 0.6 for cost and w2 = 0.4 for reliability, converted to identical intervals at both levels. Step 2. Super-ideal Solutions. Compute the positive and negative super-ideal for each (cj, qα) by taking component-wise maxima and minima of the weighted normalized intervals across i = 1, 2, 3.

Step 3. Superhyperrough Distance. For each partner Ai, calculate u 2 2 u 2 2 uX X uX X

+ (α) +, (α) 2 − (α) −, (α) 2

D = t d(˜ v , v ˜ ) , D = t d(˜ v , v ˜ ) ,

j=1 α=1 j=1 α=1

where d(·, ·) is the interval-distance at each level.

Step 4. Closeness Coefficient & Ranking. Compute

CCi = , i = 1, 2, 3,

and rank partners in descending order of CCi.

In this example, numerical calculation yields CC1 ≈ 0.42, CC2 ≈ 0.58, CC3 ≈ 0.75, so the firm would select Partner3 as the optimal logistics provider under varying market demand and fuel price conditions.

Example 6 (SuperHyperRough TOPSIS in Cloud Provider Selection). A technology firm must choose one of three cloud providers A = {CloudA, CloudB, CloudC} based on two criteria:

C = {Monthly Cost (c1), Average Latency (c2)}.

They consider two levels of indiscernibility:

R1 : by data-center region (US vs EU), R2 : by service tier (Standard vs Premium), and solicit two expert opinions (contexts):

Q = {TechExpert, FinanceExpert}.

The raw 2-level hyperrough evaluations x ˜ij are

(TechExpert) (FinanceExpert)

x ˜ij x ˜ij Ai\cj c1 c2 c1 c2 CloudA ([1000, 1200], [1500, 1700]) ([100, 150], [90, 140]) ([950, 1150], [1450, 1650]) ([105, 155], [95, 145]) CloudB ([1100, 1300], [1400, 1600]) ([120, 170], [110, 160]) ([1150, 1350], [1500, 1700]) ([125, 175], [115, 165]) CloudC ([900, 1100], [1300, 1500]) ([130, 180], [100, 150]) ([1000, 1200], [1350, 1550]) ([135, 185], [105, 155])

Step 1. Normalization & Weighting. For each context α ∈ Q and each level ℓ ∈ {1, 2} compute uX ℓ,(α) t ℓ,(α) 2 Lj = xij ,

ℓ,(α)

then normalize each interval by dividing its endpoints by Lj . Use crisp weights w1 = 0.7 (cost) and w2 = 0.3 (latency), converted to identical rough-intervals at both levels. Multiply level-wise to obtain weighted normalized hyperrough numbers v ˜ij .

Step 2. Super-ideal Solutions. For each criterion cj and context α, set

+, ℓ,(α) ℓ,(α) +, ℓ,(α) ℓ,(α)

vj = max vij , vj = max vij ,

−, ℓ,(α) ℓ,(α) −, ℓ,(α) ℓ,(α) ±,(α)

and similarly vj = mini vij , vj = mini vij , yielding positive/negative super-ideals v ˜j .

Step 3. Superhyperrough Distance. Define for any two superhyperrough objects a, ˆ b : Q → (I) the distance v uX X

ℓ,(α) 2 ℓ,(α) ℓ,(α) 2

ˆ t ℓ,(α) D(ˆ a, b) = a − b + a − b .

α∈Q ℓ=1

Then for each provider Ai compute u 2 u 2 D = t D v ˜ , v ˜ , D = t D v ˜ , v ˜ .

Step 4. Closeness Coefficient & Ranking. Calculate

CCi = ,

and rank CloudA, CloudB, CloudC by descending CCi. Numerical evaluation yields CCA ≈ 0.42, CCB ≈ 0.55, CCC ≈ 0.68, so the firm selects CloudC as the optimal provider under varying technical and financial scenarios.

Theorem 5 (SuperHyperRough Structure). The set of all decision objects {x ˆij : Q → (I) } forms an n-SuperHyperRough Set over U with respect to R1, . . . , Rk and contexts Q.

Proof : By construction, each x ˆij is a mapping from the finite set of contexts Q into the k-fold product of interval-valued rough approximations. Such mappings are exactly the elements of the n-SuperHyperRough Set SHR(U; R1, . . . , Rk, Q) = P(J1) × · · · × P(Jn) −→ P(U), where each context yields a hyperrough pair of approximations. Closure under intersection and union (context-wise) follows from the properties of rough approximations at each level and context, verifying the superhyperrough set axioms. □

Theorem 6 (Generalization of HyperRough TOPSIS). If the number of contexts p = 1, then the SuperHy-perRough TOPSIS Method reduces exactly to the HyperRough TOPSIS Method.

Proof : When p = 1, the superhyperrough number x ˆij is constant on Q, i.e. x ˆij(q1) = x ˜ij . All steps—normalization, ideal computation, distance, separation and closeness coefficient—coincide with those of HyperRough TOPSIS applied to the single context. Hence the procedures are identical. □

Theorem 7 (Boundedness of Closeness Coefficient). Under the SuperHyperRough TOPSIS Method, for each alternative Ai the closeness coefficient CCi =

satisfies 0 ≤ CCi ≤ 1.

Proof : By construction, the separation measures

D = t D v ˆ , v ˆ , D = t D v ˆ , v ˆ

− + + − −

are nonnegative. Hence Di ≥ 0 and Di ≥ 0, so Di + Di ≥ Di ≥ 0. It follows that − Di 0 ≤ Di =⇒ 0 ≤ ≤ 1.

If Di + Di = 0, by convention one sets CCi = 0.5, which also lies in [0, 1]. □

Theorem 8 (Reduction to HyperRough TOPSIS). If there is only one context (p = 1), then the SuperHy-perRough TOPSIS Method coincides exactly with the HyperRough TOPSIS Method.

Proof : When p = 1, each superhyperrough number x ˆij is constant on Q = {q1}, i.e.

x ˆij(q1) = ˜ xij , and all steps—normalization, weighting, ideal computation, superhyperrough distance, separation measures, and closeness coefficient—reduce to those defined for HyperRough TOPSIS applied to the single context. Therefore the two procedures are identical. □

Theorem 9 (Scale Invariance). Let c > 0 be a constant. If for a fixed criterion cj, level ℓ, and context qα we replace every evaluation

ℓ,(α) ′ ℓ,(α)

x ˜ij = [x, x] 7−→ x ˜ij = [c x, c x], then all closeness coefficients CCi remain unchanged.

Proof : Under this scaling, the normalization denominator for that level-context

ℓ,(α) ℓ,(α) 2 ′ ℓ,(α) ℓ,(α) 2 ℓ,(α)

Lj = (xij ) 7−→ Lj = (c xij ) = c Lj .

Thus each normalized interval endpoint

ℓ,(α) ℓ,(α)

xij c xij

ℓ,(α) ℓ,(α)

rij = 7−→ = rij ,

ℓ,(α) ℓ,(α)

Lj c Lj

ℓ,(α)

and similarly for rij . All subsequent weighted values, ideals, distances D, separation measures, and hence CCi remain unchanged. □

Theorem 10 (Permutation Invariance). The closeness coefficients {CCi} are invariant under any permutation of the criteria C or of the contexts Q.

Proof : Permuting the criteria c1, . . . , cn simply reorders the summation index in

uX

± ± 2

D = t D v ˆ , v ˆ ,

i ij j

which does not change its value. Permuting the contexts q1, . . . , qp corresponds to relabeling the terms in the superhyperrough distance uX X ˆ t 2 D(ˆ a, b) = (· · · ) ,

α=1 ℓ=1

which is symmetric in the summands. Therefore both separation measures and all CCi remain unchanged. □