A fuzzy set assigns to each element in a universe a membership degree in the interval [01 –] , providing a mathematical framework for representing imprecision and vagueness [1–4].

Several advanced frameworks have been developed to extend or complement the classical fuzzy set theory. These include:

• Intuitionistic Fuzzy Sets [5–7], which assign both a membership and a non-membership degree to each element;

• Vague Sets [8–10], which refine the interpretation of uncertainty through intervals;

• Bipolar Fuzzy Sets [11–13], allowing positive and negative degrees of membership;

• Neutrosophic Sets [14–16], which incorporate degrees of truth, indeterminacy, and falsity;

• Neutrosophic Offsets [17, 18], which model deviation or offset from standard membership structures;

• Picture Fuzzy Sets [19–21], which include positive, neutral, and negative membership components;

• Hesitant Fuzzy Sets [22–24], which allow a set of possible membership values for each element;

• Quadri-Partitioned Neutrosophic Sets [25–27], which divide uncertainty into four interpretive components;

• Hyperneutrosophic Sets [28], which introduce a higher-order hierarchical structure to neutrosophic models;

• Plithogenic Sets [29–33], which generalize fuzzy sets by incorporating attribute contradiction degrees.

In this paper, we focus on the Hyperfuzzy Set [34–37] and its recursive generalization, the SuperHyperfuzzy Set [28, 38], both of which aim to capture more complex forms of uncertainty by introducing set-valued and hierarchical membership degrees.

Definition 0.1 (Base Set). A base set is the foundational set from which complex structures such as powersets and hyperstructures are derived. It is formally defined as:

= { | is an element within a specified domain}.

All elements in constructs like P ( ) or P ( ) originate from the elements of .

Definition 0.2 (Powerset). The powerset of a set , denoted P ( ), is the collection of all possible subsets of , including both the empty set and itself. Formally, it is expressed as:

P ( ) = { | ⊆ }.

Definition 0.3 ( -th Powerset). (cf. [39–42])

The -th powerset of a set , denoted P ( ), is defined iteratively, starting with the standard powerset. The recursive construction is given by:

P1( ) = P ( ), P +1( ) = P (P ( )), for ≥ 1.

Similarly, the -th non-empty powerset, denoted P ( ), is defined recursively as:

∗ ∗ ∗ ∗ ∗

P ( ) = P ( ), ( ) = P (P ( )).

1 +1

Here, P ( ) represents the powerset of with the empty set removed.

Example 0.4 (Real-World Example of -th Powerset). Let be the set of basic ingredients for a simple recipe:

= {tomato, cheese, basil}.

Then the first powerset P1( ) = ( ) is the collection of all possible ingredient combinations (including the empty set), for instance {tomato, basil} or {cheese}, each of which you can think of as a “recipe.”

The second powerset P2( ) = P P1( ) is the set of all possible “menus,” i.e. collections of recipes. For example, {tomato, basil}, {cheese} ∈ 2( ) represents a menu that offers two recipes.

Finally, the third powerset P3( ) = P P2( ) is the set of all possible “meal plans,” i.e. collections of menus—useful, for instance, when planning multi-day catering events.

In general, the -th powerset P ( ) models -level hierarchical groupings in everyday planning tasks.

Definition 0.5 (Fuzzy set). [1, 43] A fuzzy set in a non-empty universe is a mapping : → [0, 1]. A fuzzy relation on is a fuzzy subset in × . If is a fuzzy set in and is a fuzzy relation on , then is called a fuzzy relation on if (, ) ≤ min{ ( ), ( )} for all , ∈ .

Definition 0.6 (Hyperfuzzy Set). [34–37, 44] Let be a non-empty universe. A hyperfuzzy set ˜ on is defined by a mapping ˜ : −→ ˜( [0, 1]), where ˜( [0, 1]) denotes the collection of all non-empty subsets of the interval [0, 1].

For each element ∈ , ˜( ) ⊆ [0, 1] represents the set of possible membership degrees of in the set ˜. This formulation allows for representing uncertainty or variability in the degree of membership, extending the classical fuzzy set (which assigns a single real number in [0, 1]) to a set-valued interpretation.

Example 0.7 (Freshness Assessment via a Hyperfuzzy Set). Let be the set of ingredients in a simple salad:

= {tomato, cheese, basil}.

Three quality-control experts each give a fuzzy score in [0, 1] for the “freshness” of each ingredient. We model their variability as a hyperfuzzy set ˜ on , where ˜( ) = { scores provided by the three experts for } ⊆ [0, 1].

Concretely, ˜(tomato) = {0.7, 0.8, 0.9}, ˜(cheese) = {0.6, 0.85, 0.9}, ˜(basil) = {0.5, 0.75, 0.95}.

Here each set ˜( ) captures the range of expert judgments on how “fresh” the ingredient is, allowing us to represent uncertainty and disagreement among evaluators.

Definition 0.8 (, -SuperHyperFuzzy Set). [28, 45, 46] Let be a nonempty set and let , ∈ N0. Define the nonempty -th powerset of a set by

∗ ∗ ∗

P ( ) = , P ( ) = P P ( ) \ {∅}, ≥ 1.

0 −1

In particular, P ( ) is the family of all nonempty elements of the -th iterated powerset of , and P ( [0, 1]) is defined analogously. Then an (, )-SuperHyperFuzzy Set on is a function ∗ ˜ ∗ ˜, : P ( ) −→ P [0, 1] , ↦→ ˜,( ), ˜ ∗ ∗ where P ( [0, 1]) denotes the collection of all nonempty subsets of P ( [0, 1]). Thus each ∈ P ( ) is assigned a nonempty family of membership-degree sets ˜,( ) ⊆ P ( [0, 1]), capturing hierarchical uncertainty across both the - and -levels.

Example 0.9 (Project Team Performance via a (2, 2)–SuperHyperFuzzy Set). Let = {Yutaka, Masafumi, Shintaro} , and take = 2, = 2. Then P ( ) = P P ( ) \ {∅}, so one representative element is = {Yutaka, Masafumi}, {Masafumi, Shintaro} ∈ P ( ).

We assign to a nonempty family of second-level membership sets by ˜2,2( ) = {{0.7, 0.75}, {0.8}}, | {z }

{{0.6}, {0.85, 0.90}} ⊆ P ( [0, 1]).

| {z }

Here:

• Each inner set, for example {0.7, 0.75}, consists of possible membership scores from one evaluation method (e.g.

quarterly metrics).

• The outer set { 1, 2} captures two distinct evaluation scenarios (e.g. peer review vs. manager review).

Thus this (2, 2)–SuperHyperFuzzy Set models hierarchical uncertainty both in the team structure (level = 2) and in the membership-assessment process (level = 2).

Example 0.10 (Supply Chain Reliability Assessment via a (2, 1)–SuperHyperFuzzy Set). Let = {A, B, C} be a set of suppliers, and take = 2, = 1. Then P ( ) = { }, { }, { }, {, }, {, }, {, }, {, , } , and P ( ) = P P ( ) \ {∅}, so one representative element is = {, }, {, } ∈ P ( ), which models two overlapping supply-chain partnerships. We assign to a set of possible reliability scores by ˜2,1( ) = { 0.75, 0.85, 0.92} ⊆ P ( [0, 1]) = all nonempty subsets of [0, 1] .

Here each value represents the aggregate reliability (on-time delivery, defect-rate compliance, audit score, etc.) under a different evaluation scenario. Thus this (2, 1)–SuperHyperFuzzy Set captures hierarchical uncertainty in the supplier network structure (level = 2) while allowing multiple possible reliability judgments (level = 1).

Linear programming is an optimization technique that aims to maximize or minimize a linear objective function subject to a set of linear equality and inequality constraints(cf. [47–52]). Fuzzy linear programming generalizes this framework by incorporating fuzzy numbers into the objective coefficients and constraints, allowing for uncertainty in both parameters and feasible regions [53–57]. The definition of the Fuzzy Linear Programming is provided below.

Definition 0.11 (Fuzzy Linear Program). (cf. [53]) Let = ( 1, . . . , ) ∈ R be the decision vector. A fuzzy linear program (FLP) is specified by fuzzy numbers ˜ ( = 1, . . . , ), ˜ ( = 1, . . . , , = 1, . . . , ), ˜ ( = 1, . . . , ), and written as

˜ ( ) = ˜ −→ max, (1)

subject to ˜ ⪯ ˜ , = 1, . . . , . (2)

Here:

• “⊕” denotes the extended addition of fuzzy numbers via Zadeh’s extension principle:

⊕ ( ) = sup min{ ( ), ( )}.

• “⪯” is a chosen fuzzy inequality relation between fuzzy sets (e.g. the pessimistic “<” of Zimmermann).

• Each fuzzy number ˜ (resp. ˜ , ˜ ) is a convex normalized fuzzy set on R with membership ( ).

Example 0.12 (Fuzzy Production Planning). (cf. [58, 59]) A factory produces two products 1, 2 ≥ 0. Profit per unit is only approximately known:

˜ = [5, 6] , ˜ = [3, 4] ,

1 tr 2 tr

where [, ]tr denotes the triangular fuzzy number with support [, ] and peak at the midpoint. There is a single resource constraint: the available capacity is vague, ˜ = [ 90, 110 ] ,

tr and each unit of 1 consumes 2–3 units, of 2 consumes 1–2 units:

˜ = [2, 3] , ˜ = [1, 2] .

11 tr 12 tr

Thus the FLP is ˜ ( ) = ˜ ⊕ ˜ −→ max, ˜ ⊕ ˜ ⪯ , ˜ , ≥ 0.

1 1 2 2 11 1 12 2 1 2

Concretely, under the usual min–+ extension, the objective’s bounds are ( ) = 5 1 + 3 2, ( ) = 6 1 + 4 2, and the resource constraint becomes the fuzzy interval [ 2 1 + 1 2 , 3 1 + 2 2 ] ⪯ [ 90, 110 ].

One may solve by introducing a crisp satisfaction level ∈ [0, 1] and forming the corresponding parametric LP.

The definition of the HyperFuzzy Linear Programming is provided below.

Definition 0.13 (Hyperfuzzy Number). A hyperfuzzy number on R is a hyperfuzzy set ˜ : R −→ P ( [ ˜ 0, 1]), such that each -cut ˜ = { ∈ R | sup ˜( ) ≥ } is nonempty and compact. Hyperfuzzy numbers generalize classical fuzzy numbers by assigning to each real a nonempty compact subset ˜( ) ⊆ [0, 1] of possible membership degrees.

Definition 0.14 (Hyperaddition and Hyperscalar-Multiplication). Let , ˜ ˜ be hyperfuzzy numbers, and ≥ 0 a scalar. Define: Ø ( ˜ ⊕ ˜) ( ) = ˜ ( ) ∩ ˜ ( ) ,

( ⊗ ˜) ( ) = ˜ ( ),

interpreting the unions over all decompositions of . These extend the usual Zadeh extension principle to the hyperfuzzy setting.

Definition 0.15 (HyperLinear Program). Let = ( 1, . . . , ) ∈ R be the decision vector. A HyperLinear Program (HLP) is specified by hyperfuzzy numbers ˜ ( = 1, . . . , ), ˜ ( = 1, . . . , , = 1, . . . , ), ˜ ( = 1, . . . , ), and written as

˜ ( ) = ⊗ ˜ −→ , (3)

ℎ ≥0

subject to ⊗ ˜ ⪯ ˜ , = 1, . . . , , (4)

where “⪯ℎ” is a chosen hyperfuzzy inequality (e.g. comparing -cuts).

Example 0.16 (HyperLinear Production Planning). A factory produces two products 1, 2 ≥ 0. The profit coefficients are uncertain and modeled as hyperfuzzy numbers:

  {0.7, 0.8}, = 5,   {0.6}, = 3,     ˜ ( ) = {0.9}, = 6, ˜ ( ) = {0.7, 0.75}, = 4,  ∅, otherwise,  ∅, otherwise.

Resource consumption per unit is also hyperfuzzy:

  {0.8}, = 2,   {0.65}, = 1,     ˜ ( ) = {0.9}, = 3, ˜ ( ) = {0.85}, = 2,  ∅, otherwise,  ∅, otherwise,

and the available capacity is hyperfuzzy:

  {1.0}, = 90,   ˜( ) = {0.5}, = 110,  ∅, otherwise.

Thus the HyperLinear Program is

˜ ( ) = ⊗ ˜ ⊕ ⊗ ˜ −→ max, ⊗ ˜ ⊕ ⊗ ˜ ⪯ , ˜ , ≥ 0.

1 ℎ 1 ℎ 2 ℎ 2 1 ℎ 11 ℎ 2 ℎ 12 ℎ 1 2

Objective evaluation at = (10, 5). Possible profit values and membership degrees:

= 5 · 10 + 3 · 5 = 65 : = min{sup ˜ (5), sup ˜ (3)} = min{0.8, 0.6} = 0.6, = 5 · 10 + 4 · 5 = 70 : = min{0.8, 0.75} = 0.75, = 6 · 10 + 3 · 5 = 75 : = min{0.9, 0.6} = 0.6, = 6 · 10 + 4 · 5 = 80 : = min{0.9, 0.75} = 0.75.

Hence ˜ (10, 5) assigns {65 ↦→ 0.6, 70 ↦→ 0.75, 75 ↦→ 0.6, 80 ↦→ 0.75}.

Constraint evaluation at = (10, 5). Possible consumptions and memberships:

= 2 · 10 + 1 · 5 = 25 : = min{sup ˜ (2), sup ˜ (1)} = min{0.8, 0.65} = 0.65, = 2 · 10 + 2 · 5 = 30 : = min{0.8, 0.85} = 0.8, = 3 · 10 + 1 · 5 = 35 : = min{0.9, 0.65} = 0.65, = 3 · 10 + 2 · 5 = 40 : = min{0.9, 0.85} = 0.85.

Comparing to capacity:

sup ˜(90) = 1.0, sup ˜(110) = 0.5.

All consumptions ≤ 90 yield satisfaction min{, 1.0} = , so = (10, 5) is feasible with minimum satisfaction 0.65.

Thus this example demonstrates a HyperLinear Program’s evaluation under multiple profit and resource scenarios, with membership degrees quantified by the hyperfuzzy arithmetic ⊗ℎ, ⊕ℎ.

Theorem 0.17 (Generalization of Linear Programming). Every classical linear program is a special case of an HLP in which all hyperfuzzy numbers ˜ , ˜ , ˜ have singleton-valued membership sets. Conversely, if in an HLP all ˜ , ˜ , ˜ are hyperfuzzy numbers with point-singleton membership at each real argument, then the HLP reduces exactly to the corresponding classical LP.

Proof: (1) LP =⇒ HLP. Given a classical LP with crisp coefficients , , , define hyperfuzzy numbers by ( ( ( {1}, = , {1}, = , {1}, = , ˜ ( ) = ˜ ( ) = ˜ ( ) = ∅, otherwise, ∅, otherwise, ∅, otherwise.

Then for any , ⊗ ˜ has support exactly at the single scalar , and similarly each constraint reproduces ≤ . Thus the HLP (3)–(4) coincides with the original LP.

(2) HLP with singletons =⇒ LP. Conversely, if each hyperfuzzy number ˜ satisfies ˜ ( ) ⊆ {1} or ∅, then it encodes a unique crisp coefficient . The hyperaddition ⊕ℎ and hyperscalar-multiplication ⊗ℎ on singleton-valued hyperfuzzy sets reduce to ordinary addition and multiplication. Hence the HLP reduces to the classical LP with coefficients , , .

(3) Hyperfuzzy Structure. By Definitions 0.15 and of hyperfuzzy numbers, each coefficient mapping takes values in P ( [ ˜ 0, 1]), endowing the objective and constraint functions with a hyperfuzzy set structure. Therefore HLPs both generalize LPs and inherit the algebraic richness of hyperfuzzy set theory. □ 2.2|(, )-SuperHyperfuzzy Programming

The definition of the (, )-SuperHyperfuzzy Programming is provided below.

Definition 0.18 ((, )-SuperHyperfuzzy Number). Let , ≥ 0. Define recursively the nonempty -th powerset of R by

∗ ∗ ∗

P (R) = R, P (R) = P P (R) \ {∅}, ≥ 1.

0 −1

An (, )-superhyperfuzzy number is a mapping ˜ ∗ ˜ ∗ : P (R) −→ P ( [0, 1]), such that for each ∈ P (R), ( ) ⊆ P ( [0, 1]) is nonempty and compact, and for each ∈ (0, 1] the -cut ˜ ∗ ( ) ˜ = ∈ P (R) | sup ( ) ≥ ∗ ( ) ˜ ∗ is a nonempty compact subset of P (R). Here sup denotes the -fold supremum in P ( [0, 1]).

Definition 0.19 ((, )-SuperHyperaddition and (, )-SuperHyperscalar-Multiplication). Let , ˜ ˜ be (, )- superhyperfuzzy numbers and ≥ 0. For ∈ P (R), define

( ˜ ⊕ ˜) ( ) = ˜ ( ) ⊗ ˜ ( ) , + = { + | ∈ , ∈ },

, ∈ P (R)

and Ø ( ⊗ ˜) ( ) = ˜ ( ), = { | ∈ }.

∈ P (R)

Here the (, )-superhyperproduct ⊗, is given recursively by

⊗,1 = { · | ∈ , ∈ }, ⊗, = ⊗,−1 , ≥ 2.

∈ , ∈

Definition 0.20 ((, )-SuperHyperFuzzy Linear Program). Let = ( 1, . . . , ) ∈ R . An (, )-superhyperlinear program is specified by (, )-superhyperfuzzy numbers ˜ , ˜ , ˜ for = 1, . . . , , = 1, . . . , , and takes the form

˜ ( ) = ⊗ ˜ −→ , (5)

, ≥0 , ; =1

s.t. ⊗ ˜ ⪯ ˜ , = 1, . . . , , (6)

, ; =1

where ⪯, is a chosen ordering on P ( [0, 1]) (for instance, comparing all level– cuts).

Example 0.21 (A (1, 2)-SuperHyperFuzzy Linear Production Planning). Consider a factory producing two products ˜ ∗ ˜ ∗ 1, 2 ≥ 0. We model profit and resource data as (1, 2)-superhyperfuzzy numbers : P (R) = P (R) \ {∅} → P ( [0, 1]). For example:

˜ ({5}) = {{0.7, 0.75}, {0.8}}, {{0.9}} , ˜ ({6}) = {{0.85}} , and similarly for ˜ . Resource consumption ˜ and capacity ˜ are defined analogously.

Then the (1, 2)-SHLP is ˜ ( ) = ⊗ ˜ ⊕ ⊗ ˜ −→ max, ⊗ ˜ ⊕ ⊗ ˜ ⪯ , ˜ , ≥ 0.

1 1,2 1 1,2 2 1,2 2 1 1,2 11 1,2 2 1,2 12 1,2 1 2

Evaluating at = (10, 5) produces a nested family of possible profit values and satisfaction levels, reflecting one level of set-valued uncertainty in the coefficients ( = 1) and two levels in the membership degrees ( = 2).

Example 0.22 (A (0, 3)-SuperHyperFuzzy Linear Production Planning). Consider a factory producing two products 1, 2 ≥ 0. We model both profit and resource coefficients as (0, 3)-superhyperfuzzy numbers : P (R) = R → P ( [0, 1]):

 {{0.6, 0.65}, {0.7}}, {{0.75}, {0.8, 0.85}} , = 5,  | {z } | {z }   ˜ 1 2 1( ) =  {{{0.9}}}, = 6, ∅, otherwise,  {{0.5}, {0.55, 0.6}}, {{0.65}, {0.7, 0.75}} , = 3,  | {z } | {z }   ˜ 1 2 2( ) =  {{{0.8}}}, = 4, ∅, otherwise.

˜ ˜ ∗ Resource consumption per unit is similarly given by (0, 3)-superhyperfuzzy numbers : R → P ( [0, 1]), for instance

  {{0.7}}, {{0.8}, {0.85}} , = 2,   {{0.6}, {0.65}} , = 1,     ˜ ( ) = {{{0.9}}}, = 3, ˜ ( ) = {{{0.75}, {0.8}}}, = 2,  ∅, otherwise,  ∅, otherwise.

˜ ˜ ∗ Available capacity : R → P ( [0, 1]) is defined by

  {{{1.0}}}, = 100,   ˜( ) = {{{0.5}}}, = 120,  ∅, otherwise.

Then the (0, 3)-SHLP takes the form

˜ ( ) = ⊗ ˜ ⊕ ⊗ ˜ −→ , ⊗ ˜ ⊕ ⊗ ˜ ⪯ , ˜ , ≥ 0.

1 0,3 1 0,3 2 0,3 2 ≥0 1 0,3 11 0,3 2 0,3 12 0,3 1 2

Objective evaluation at = (10, 5). Compute

10 ⊗ ˜ = {{6, 6.5}, {7}}, {{7.5}, {8, 8.5}} , 5 ⊗ ˜ = {{2.5}, {2.75, 3}}, {{4}} .

0,3 1 0,3 2

| {z } | {z }

10 1 10 2

Their (0, 3)-superhyper-sum yields four nested sets—for example {{6, 6.5}, {7}} ∪ {{2.5}, {2.75, 3}}—capturing all combined profit possibilities.

Constraint evaluation at = (10, 5). Similarly,

10 ⊗ ˜ = {{7}}, {{8}, {8.5}} , 5 ⊗ ˜ = {{3}, {3.25}} .

0,3 11 0,3 12

Their (0, 3)-superhyper-sum is {{7, 3}, {7, 3.25}, {8, 3}, {8, 3.25}, {8.5, 3}, {8.5, 3.25}}. Since sup 100 = 1.0 and sup 120 = 0.5, all consumption sets satisfy the capacity constraint at full membership.

This example shows how a (0, 3)-SHLP captures three nested levels of uncertainty in both objective and constraints, producing a hierarchy of potential outcomes and feasibility degrees.

Example 0.23 (A (2, 2)-SuperHyperFuzzy Linear Production Planning). Consider a factory that produces two products 1, 2 ≥ 0. We model both profit coefficients and resource-consumption coefficients as (2, 2)–superhyperfuzzy numbers:

 {{0.6, 0.65}, {0.7}}, {{0.8}} , = 10,  | {z } | {z }  1 2 ˜ ( ) = {{0.9, 0.95}}, {{0.85}} , = 12,  | {z } | {z }  3 4  ∅, otherwise,

 {{0.5, 0.55}, {0.6}}, {{0.65}, {0.7, 0.75}} , = 8,  | {z } | {z }  1 2 ˜ ( ) = {{0.8}}, {{0.85, 0.9}} , = 6,  | {z } | {z }  3 4  ∅, otherwise.

Resource consumption per unit is similarly given by      {{0.7}, {0.75}}, {{0.8}} , = 4,  {{0.6}, {0.65, 0.7}} , = 3,  | {z } | {z }  | {z }  1 2  1 ˜ ( ) = {{0.85}}, {{0.9, 0.95}} , = 5, ˜ ( ) = {{0.8}, {0.85}} , = 4,  | {z } | {z }  | {z }  3 4  2  ∅, otherwise,  ∅, otherwise.

The available capacity is  {{1.0}}, {{0.5}} , = 100,  | {z } | {z }  1 2 ˜( ) = {{0.9}, {0.95}} , = 120,  | {z }

 ∅, otherwise.

Then the (2, 2)-SuperHyperLinear Program reads

˜ ( ) = ⊗ ˜ ⊕ ⊗ ˜ −→ ,

1 2,2 1 2,2 2 2,2 2 ≥0

s.t. ⊗ ˜ ⊕ ⊗ ˜ ⪯ , ˜ , ≥ 0.

1 2,2 11 2,2 2 2,2 12 2,2 1 2

Objective evaluation at ( 1, 2) = (2, 3).

20 ↦→ { 1, 2}, 24 ↦→ { 1, 2}, 2 ⊗ ˜ : 3 ⊗ ˜ :

2,2 1 2,2 2

24 ↦→ { 3, 4}, 18 ↦→ { 3, 4}.

Thus there are four profit scenarios:

  44 : { ∪ , ∪ , ∪ , ∪ },  1 1 1 2 2 1 2 2  38 : { ∪ , ∪ , ∪ , ∪ },

1 3 1 4 2 3 2 4

˜ (2, 3) =  48 : { ∪ , ∪ , ∪ , ∪ },  3 1 3 2 4 1 4 2  42 : { ∪ , ∪ , ∪ , ∪ }.

 3 3 3 4 4 3 4 4 Constraint evaluation at ( 1, 2) = (2, 3).

8 ↦→ { 1, 2}, 9 ↦→ { 1}, 2 ⊗ ˜ : 3 ⊗ ˜ :

2,2 11 2,2 12

10 ↦→ { 3, 4}, 12 ↦→ { 2}.

Their superhyper-sum yields four consumption scenarios:

17 : { 1 ∪ 1, 2 ∪ 1}, 18 : { 1 ∪ 2, 2 ∪ 2}, 19 : { 3 ∪ 1, 4 ∪ 1}, 22 : { 3 ∪ 2, 4 ∪ 2}.

Comparing to capacity ˜ (at 100 and 120), all consumptions ≤ 100 satisfy the constraint with full membership, so (2, 3) is feasible at degree 1.

This example demonstrates how a (2, 2)-SHLP captures two nested levels of set-valued uncertainty in both the objective and the constraints, producing a rich hierarchy of possible outcomes and satisfaction degrees.

Example 0.24 (A (1, 3)-SuperHyperFuzzy Linear Production Planning). Consider a factory producing two products 1, 2 ≥ 0. We model both profit coefficients and resource-consumption coefficients as (1, 3)–superhyperfuzzy numbers ˜ ∗ ˜ ∗ : P (R) → P ( [0, 1])

and ˜ ∗ ˜ ∗ : P (R) → P ( [0, 1]) . For example:

 {{0.7, 0.75}, {0.8}}, {{0.85}, {0.9, 0.95}} , = 10,  | {z } | {z }   ˜ 1 2 1( ) =  {{{0.9}}}, = 12, ∅, otherwise,

 {{0.5}, {0.55, 0.6}}, {{0.65}, {0.7, 0.75}} , = 8,  | {z } | {z }   ˜ 1 2 2( ) =  {{{0.8}}}, = 6, ∅, otherwise.

Similarly, resource consumption per unit is

 {{0.7}, {0.75}}, {{0.8}, {0.85}} , = 4,  | {z } | {z }   ˜ 1 2 11( ) =  {{{0.9}}}, = 5, ∅, otherwise,

  {{0.6}, {0.65, 0.7}} , = 3,  | {z } 12( ) =  {{{0.95}}}, = 4, ∅, otherwise.

The available capacity is   {{{1.0}}}, = 100,   ˜( ) = {{{0.5}}}, = 120,  ∅, otherwise.

Then the (1, 3)–SuperHyperLinear Program takes the form ˜ ( ) = ⊗ ˜ ⊕ ⊗ ˜ −→ ,

1 1,3 1 1,3 2 1,3 2 ≥0

subject to ⊗ ˜ ⊕ ⊗ ˜ ⪯ , ˜ , ≥ 0.

1 1,3 11 1,3 2 1,3 12 1,3 1 2

Objective evaluation at ( 1, 2) = (2, 3).

20 ↦→ { 1, 2}, 24 ↦→ { 1, 2}, 2 ⊗ ˜ : 3 ⊗ ˜ :

1,3 1 1,3 2

24 ↦→ {{{0.9}}}, 18 ↦→ {{{0.8}}}.

Hence the objective yields four profit-scenarios:

  44 : { ∪ , ∪ , ∪ , ∪ },  1 1 1 2 2 1 2 2  38 : { ∪ {{0.8}}, ∪ {{0.8}}}, ˜ (2, 3) =  48 : {{{0.9}}} ∪ , {{{0.9}}} ∪ ,  1 2  42 : {{{0.9}}} ∪ {{0.8}}.

Constraint evaluation at ( 1, 2) = (2, 3).

8 ↦→ { 1, 2}, 9 ↦→ { 1}, 2 ⊗ ˜ : 3 ⊗ ˜ :

1,3 11 1,3 12

10 ↦→ {{{0.9}}}, 12 ↦→ {{{0.95}}}.

Their superhyper-sum produces four consumption scenarii:

17 : { 1 ∪ 1, 2 ∪ 1}, 20 : { 1 ∪ {{0.95}}, 2 ∪ {{0.95}}}, 19 : {{{0.9}}} ∪ 1, 22 : {{{0.9}}} ∪ {{0.95}}.

Since all consumptions ≤ 100, each scenario satisfies the constraint at membership level drawn from ˜(100) = {{1.0}}. Thus (2, 3) is feasible with full satisfaction.

This example illustrates how a (1, 3)-SuperHyperLinear Program captures one level of set-valued uncertainty in coefficients and three nested levels in membership degrees, yielding a rich scenario analysis for both objective and constraints.

Theorem 0.25 (Reduction of (, )-SuperHyperFuzzy Linear Programs). Let , ≥ 0. An (, )-SuperHyperFuzzy Linear Program, as given in Definition 0.20, enjoys the following reductions:

(i) Recovery of HyperFuzzy LP. If we set = 0 and = 1, then P (R) = R,

P ( [0, 1]) = { ⊆ [0, 1] | ≠ ∅}, and the superhyper-operations ⊕0,1, ⊗0,1, ⪯0,1 reduce exactly to the hyperaddition, hyperscalar-multiplication, and fuzzy ordering of a HyperFuzzy Linear Program (Definition 0.15). Hence every HyperFuzzy LP is a special case of the (, )-SHLP.

˜ ∗ ˜ ∗ (ii) Reduction to level- = 1 in the codomain. Suppose each coefficient map : P (R) → P ( [0, 1]) actually ˜ ∗ ˜ ∗ takes values in the subfamily P ( [0, 1]) ⊆ P ( [0, 1]). Then all membership-degree operations ⊕,, ⊗,, ⪯, restrict to the level-1 versions ⊕,1, ⊗,1, ⪯,1, and the program reduces to an (, 1)-SuperHyperLinear Program.

(iii) Recovery of HyperFuzzy LP from real-valued domain. If, in addition, each coefficient map’s domain is P (R) = R, then the coefficient domain collapses to real numbers and the program further reduces to a HyperFuzzy Linear Program, with no remaining set-valued or hierarchical structure.

Proof: We verify each statement in turn.

(i) Recovery of HyperFuzzy LP. By choosing = 0 and = 1, the domain P (R) becomes R, and the codomain P ( [0, 1]) is exactly all nonempty subsets of [0, 1]. Under these parameters, the superhyper-addition ⊕0,1, superhyper-scalar multiplication ⊗0,1, and ordering ⪯0,1 coincide with the operations defined for a HyperFuzzy Linear Program (Definition 0.15). Thus every HyperFuzzy LP embeds in the (, )-SHLP framework.

˜ ˜ ∗ ∗ ˜ (ii) Reduction to level-1 codomain. If each factors through P ( [0, 1]), then for all ∈ P (R) we have ( ) ⊆ P ( [0, 1]). By definition of the superhyper-operations, this condition forces ⊕,, ⊗,, and ⪯, to act as ⊕,1, ⊗,1, and ⪯,1 respectively. Hence the original (, )-SHLP is equivalent to an (, 1)-SHLP.

(iii) Recovery of HyperFuzzy LP from real domain. Finally, if the domain also collapses via P (R) = R, then all coefficient ˜ ˜ ∗ maps reduce to : R → P ( [0, 1]), and the program becomes precisely a HyperFuzzy Linear Program with operations ⊕0,1, ⊗0,1, and ⪯0,1.

This completes the proof. □

Theorem 0.26 (Classical Linear Program as a Special Case). Let , ≥ 0. Suppose in an (, )-SuperHyperFuzzy Linear Program (Definition 0.20) each coefficient mapping ˜ ∗ ˜ ∗ : P (R) −→ P ( [0, 1]) satisfies ˜ ( ) = {{ }} for some fixed ∈ R (and similarly for ˜ , ˜ ).

Then the program (5)–(6) reduces exactly to the classical linear program ∑︁ ∑︁ max , s.t. ≤ , = 1, . . . , ,

=1 =1

where , are the crisp values arising from ˜ and ˜ .

Proof: Since each ˜ ( ) is the singleton { }, the superhyper-scalar multiplication ⊗ satisfies

⊗ ˜ = { · } ⊂ R, and the superhyper-addition ⊕, of singletons reduces to ordinary addition. Likewise, the constraint operations collapse to ≤ . Hence the (, )-SuperHyperLinear Program coincides with the classical LP. □

Theorem 0.27 ( –Cut Decomposition). For each ∈ (0, 1], define the –cuts of the coefficient mappings by ∗ ( ) ˜ , = ∈ P (R) | sup ( ) ≥ , ∗ ( ) ˜ ( ) ˜ , = ∈ P (R) | sup ( ) ≥ , , = ∈ R | sup ( ) ≥ .

Then the family of classical programs ∑︁ ∑︁ max max , s.t. max ≤ min , = 1, . . . , ,

≥0 ∈ , ∈ , ∈ , =1 =1

parameterized by , exactly describes the –level behavior of the (, )-SuperHyperLinear Program. In particular:

• The fuzzy feasible region of the (, )-SHLP is ∈ (0,1] { | feasible at level }.

• The fuzzy objective ˜ ( ) is recovered by collecting, for each , the optimal values of these crisp programs.

Proof: By definition of the –cut, , collects precisely those domain values whose membership supremum is at least . For any , the value of ⊗ ˜ at level is the set { · | ∈ }. Superhyper-addition of these sets then yields the interval min ∈ , max ∈ . Requiring this interval to lie below min , enforces the crisp inequality max ∈ ≤ min ∈ . Collecting over all recovers the full fuzzy program. □

Theorem 0.28 (Monotonicity of –Level Feasible Sets). Let 1, 2 ∈ (0, 1] with 1 < 2. Then

, ⊆ , , , ⊆ , , , ⊆ , ,

2 1 2 1 2 1

and consequently the 2–level feasible region is contained in the 1–level feasible region.

( ) ˜ ( ) ˜ ˜ Proof: By construction, if sup ( ) ≥ 2 > 1 then certainly sup ( ) ≥ 1. The same argument applies to and ˜ . Hence each higher– cut is nested inside the lower– cut, and the corresponding feasible regions satisfy the stated containment. □