Application of lattice ordered multi-fuzzy soft set in forecasting process

1 Introduction

Garrett Birkhoff [2] started the general development of lattice theory in the mid 1930’s. Zadeh [14] proferred fuzzy set theory as an extension of classical set theory in 1965. In addition, this concept of fuzziness has been widely used to characterize the problems in many fields such as medicine, engineering, economics and so on. In 1999, Molodtsov [6, 5] originated the notion of soft set theory as one of the newly-emerging mathematical tool to handle with uncertainty and ambiguity associated with the real world data based structures. Moreover, its application has boomed in various fields such as decision making, forecasting, demand analysis, information sciences and texture classification. Maji et al. [4, 9] established fuzzy soft set theory which was considered as a generalization of soft set theory. In 2011, Cagman et al.[1] formulated application of fuzzy soft set theory in decision making method.

In 2011, Sabu sebastian and Ramakrishnan [10] propounded the new innovative idea of the multi-fuzzy set which is a extension of fuzzy set theory. Multi-fuzzy sets using ordinary fuzzy sets as building blocks, its membership function is an ordered sequence of ordinary fuzzy membership functions. With the establishment of multi-fuzzy set, its application has been applied commonly in various situations such as image processing, pattern recognition, decision making and approximation of vague data.

By combining the soft set and multi-fuzzy set model, Yong Yang [13] et al. initiated the new innovative idea of multi-fuzzy soft set and studied its basic operations. They also focused the application of this notion in decision making. Forecasting [3] is a process of estimating a future event by casting forward past data. The past data are systematically combined in a predetermined way to obtain the estimate of the future. In [11], Vimala formulated a new technique of fuzzy lattice ordered group in 2014. In 2015, Muhammad Irfan Ali et al. [7] originated the notion of lattice ordered soft set and discussed its algebraic properties. Vimala et al. [8, 12] conferred fuzzy soft cardinality in lattice ordered fuzzy soft group and its application in decision making problem and also constructed lattice ordered interval-valued hesitant fuzzy soft sets and its real life application in 2018.

The rest of this paper is organized in the following manner: In Section 2, some necessary definitions are given. In Section 3, the definition of lattice ordered multi-fuzzy soft set and some of its algebraic operations are discussed. In Section 4, an application of lattice ordered multi-fuzzy soft set has been described. Finally, we conclude this paper inSection 5.

2 Preliminaries

Definition 2.1. [14] Let X be a non-empty set, then a fuzzy set μ over X is a function from X into [0, 1].

Definition 2.2. [6] A pair (F,A) is called a soft set over U, where F is a mapping given by F:A →P(U).

In other words, a soft set over U is a mapping from parameters to P(U), and it is not a set, but a parameterized family of subsets of U. For e ∈ A, F(e) may be considered as the set of e-approximate elements of the soft set (F, A).

Definition 2.3. [4] Let P ˜ ( U ) be the set of all fuzzy subsets of U. A pair ( F ˜ , A ) is called a fuzzy soft set over U, where F ˜ is a mapping given by F ˜ : A → P ˜ ( U ) .

Definition 2.4. [10] Let k be a positive integer. A multi-fuzzy set A ˜ in U is a set of ordered sequences

A ˜ = {u/(μ₁ (u) , μ₂ (u) , . . . . . . , μ _i  (u) , . . . . . μ _k  (u)) : u ∈ U}, where μ i ∈ P ˜ ( U ) , i = 1, 2,...., k.

The function μ A ˜ = ( μ 1 , μ 2 , . . . . . , μ k ) is called the multi-membership function of multi-fuzzy set A ˜ , k is called the dimension of A ˜ . The set of all multi-fuzzy sets of dimension k in U is denoted by M ^k FS (U).

Definition 2.5. [13] A pair ( F ˜ , A ) is called a multi-fuzzy soft set of dimension k over U, where F ˜ is a mapping given by F ˜ : A → M K FS ( U ) .

A multi-fuzzy soft set is a mapping from parameters to M ^k FS (U). It is a parameterized family of multi-fuzzy subsets of U. For e ∈ A, F ˜ ( e ) may be considered as the set of e-approximation elements of the multi-fuzzy soft set ( F ˜ , A ) .

Definition 2.6. [13] Let A, B ∈ E. Let ( F ˜ , A ) and ( G ˜ , B ) be two multi-fuzzy soft sets of dimension k over U. ( F ˜ , A ) is said to be a multi-fuzzy soft subset of ( G ˜ , B ) if

A ⊆ B, and

In this case, we write ( F ˜ , A ) ⊆ ( G ˜ , B ) .

Definition 2.7. [13] Two multi-fuzzy soft sets ( F ˜ , A ) and ( G ˜ , B ) of dimension k over U are said to be multi-fuzzy soft equal, if ( F ˜ , A ) is a multi-fuzzy soft subset of ( G ˜ , B ) and ( G ˜ , B ) is a multi-fuzzy soft subset of ( F ˜ , A ) .

In this case, we write ( F ˜ , A ) = ( G ˜ , B ) .

Definition 2.8. [13] Union of two multi-fuzzy soft sets ( F ˜ , A ) and ( G ˜ , B ) of dimension k over U is the multi-fuzzy soft set ( H ˜ , C ) where C = A ∪ B and for all e ∈ C, H ˜ ( e ) = { F ˜ ( e ) if e ∈ A - B G ˜ ( e ) if e ∈ B - A F ˜ ( e ) ⊔ G ˜ ( e ) if e ∈ A ∩ B We write ( F ˜ , A ) ∪ ˜ ( G ˜ , B ) = ( H ˜ , C ) .

Definition 2.9. [13] Intersection of two multi-fuzzy soft sets ( F ˜ , A ) and ( G ˜ , B ) of dimension k with A ∩ B ≠ φ over U is the multi-fuzzy soft set ( H ˜ , C ) , Where C = A ∩ B, for all e ∈ C and H ˜ ( e ) = F ˜ ( e ) ⊓ G ˜ ( e )

We write ( F ˜ , A ) ∩ ˜ ( G ˜ , B ) = ( H ˜ , C ) .

Definition 2.10. [13] If ( F ˜ , A ) and ( G ˜ , B ) are two multi-fuzzy soft sets of dimension k over U, the ( F ˜ , A ) AND ( G ˜ , B ) is defined by ( F ˜ , A ) ∧ ( G ˜ , B ) = ( H ˜ , A × B ) , where H ˜ ( α , β ) = F ˜ ( α ) ⊓ G ˜ ( β ) for all (α, β) ∈ A × B.

Definition 2.11. [13] If ( F ˜ , A ) and ( G ˜ , B ) are two multi-fuzzy soft sets of dimension k over U, the ( F ˜ , A ) OR ( G ˜ , B ) is defined by ( F ˜ , A ) ∨ ( G ˜ , B ) = ( O ˜ , A × B ) , where O ˜ ( α , β ) = F ˜ ( α ) ⊔ G ˜ ( β ) for all (α, β) ∈ A × B.

3 On lattice ordered multi-fuzzy soft set

In this section, the concept of lattice approach on multi-fuzzy soft set theory is introduced in an ordered sequence manner.

Throughout this paper, X ˜ refers to the initial universe, P( X ˜ ) is the power set of X ˜ , E ˜ is a set of parameters and A ˜ ⊆ E ˜ . M n FS ( X ˜ ) is a set of all multi-fuzzy sets of dimension n in X ˜ .

Definition 3.1. Let X ˜ be an initial universe set and A ˜ be a set of parameters. A multi-fuzzy soft set ( F , A ˜ ) of dimension n over X ˜ is called a lattice ordered multi-fuzzy soft set of dimension n over X ˜ if for the mapping F : A ˜ → M n FS ( X ˜ ) , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] , ∀ ( p i ) n , ( p j ) n ∈ A ˜ , where (p _i )  ⁿ , (p _j )  ⁿ denotes n dimension of p _i , p _j and is written as L - M n FSS .

Example 3.2. Let X ˜ = {x₁, x₂} be the set of male and female unemployees and A ˜ = {p₁, p₂, p₃} be the set of parameters, where p₁ stands for ’Taluk’ which consist of Ambarnath, Nankhari, Piriyapatna, p₂ stands for ’District’ which consist of Thane, Shimla, Mysore, p₃ stands for ’State’ which consist of Maharastra, Himachal Pradesh, Karnataka.

We define a L - M 3 FSS as follows,

- 8.5 pt ( F , A ˜ ) = { F [ ( p 1 ) 3 ] = { x 1 ( 0.1 , 0.4 , 0.3 ) , x 2 ( 0.5 , 0.1 , 0.6 ) } ,        F [ ( p 2 ) 3 ] = { x 1 ( 0.4 , 0.6 , 0.7 ) , x 2 ( 0.6 , 0.3 , 0.8 ) } ,

       F [ ( p 3 ) 3 ] = { x 1 ( 0.8 , 1.0 , 0.9 ) , x 2 ( 0.9 , 0.5 , 1.0 ) } }

Then ( F , A ˜ ) forms an L - M n FSS over X ˜ .

Definition 3.3. Let F [ ( p i ) n ] = {x/(μ _{p i1}  (x), μ p i 2 ( x ) , . . . . . . . , μ p in ( x ) ) : x ∈ X ˜ } and F [ ( p j ) n ] = { x / ( μ p j 1 ( x ) , μ p j 2 ( x ) , . . . . . , μ p jn ( x ) ) : x ∈ X ˜ } in ( F , A ˜ ) . Then the following operations holds:

( p i ) n ≤ ( p j ) n ⇒ F [ ( p i ) n ] ⊑ F [ ( p j ) n ] iff μ ( p i ) n ( x ) ≤ μ ( p j ) n ( x ) , ∀ ( p i ) n , ( p j ) n ∈ A ˜ .

Proposition 3.4. Let ( F , A ˜ ) be an L - M n FSS over X ˜ . Then the following are equivalent,

(p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] , ∀ ( p i ) n , ( p j ) n ∈ A ˜ .

Proposition 3.5. Let ( F , A ˜ ) be an L - M n FSS over X ˜ . Then for all ( p i ) n , ( p j ) n ∈ A ˜ , F [ ( p i ) n ] ∨ F [ ( p j ) n ] and F [ ( p i ) n ] ∧ F [ ( p j ) n ] are the least upper bound and greatest lower bound of F [ ( p i ) n ] and F [ ( p j ) n ] respectively.

Definition 3.6. Let ( F , A ˜ ) be an L - M n FSS over X ˜ . Then ( F , A ˜ ) is called a multi-fuzzy soft chain if F [ ( p i ) n ] ⊑ F [ ( p j ) n ] or F [ ( p j ) n ] ⊑ F [ ( p i ) n ] for all F [ ( p i ) n ] , F [ ( p j ) n ] ∈ ( F , A ˜ ) .

Proposition 3.7. Every multi-fuzzy soft chain is an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) be a multi-fuzzy soft chain.

since any two elements are comparable,

i.e., F [ ( p i ) n ] ⊑ F [ ( p j ) n ] or F [ ( p j ) n ] ⊑ F [ ( p i ) n ] and for all ( p i ) n , ( p j ) n ∈ A ˜ .

Then F [ ( p i ) n ] ∨ F [ ( p j ) n ] and F [ ( p i ) n ] ∧ F [ ( p j ) n ] exist in ( F , A ˜ ) .

Hence ( F , A ˜ ) is an L - M n FSS over X ˜ .□

Proposition 3.8. Let ( F , A ˜ ) be an L - M n FSS over X ˜ . Then the following operations holds:

Idempotent: F [ ( p i ) n ] ∨ F [ ( p i ) n ] = F [ ( p i ) n ] and F [ ( p i ) n ] ∧ F ( p i ) n ] = F [ ( p i ) n ] , for all ( p i ) n ∈ A ˜ .

Theorem 3.9. Union of two L - M n FSS s over X ˜ is also an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then for all ( p i ) n , ( p j ) n ∈ A ˜ , ( p i ) n ≤ ( p j ) n implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and for all ( p i ) n , ( p j ) n ∈ B ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that G [ ( p i ) n ] ⊑ G [ ( p j ) n ] .

Let ( F , A ˜ ) ∪ ( G , B ˜ ) = ( M , C ˜ ) , where C ˜ = A ˜ ∪ B ˜ , for all ( p m ) n ∈ C ˜ and M [ ( p m ) n ] = F [ ( p m ) n ] ∨ G [ ( p m ) n ] .

Case 1:

If ( p i ) n , ( p j ) n ∈ A ˜ - B ˜ then F [ ( p i ) n ] ⊑ F [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ

Case 2:

If ( p i ) n , ( p j ) n ∈ B ˜ - A ˜ then G [ ( p i ) n ] ⊑ G [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ

Case 3:

If ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ then F [ ( p i ) n ] ∨ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∨ G [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ .

From this we get, for all ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ∨ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∨ G [ ( p j ) n ] .

Then (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that M [ ( p i ) n ] ⊑ M [ ( p j ) n ] ∀ ( p i ) n , ( p j ) n ∈ C ˜ .

Hence ( M , C ˜ ) is an L - M n FSS over X ˜ □ .

Theorem 3.10. Intersection of two L - M n FSS s over X ˜ is also an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then for all ( p i ) n , ( p j ) n ∈ A ˜ , ( p i ) n ≤ ( p j ) n implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and for all ( p i ) n , ( p j ) n ∈ B ˜ , ( p i ) n ≤ ( p j ) n implies that G [ ( p i ) n ] ⊑ G [ ( p j ) n ] .

Let ( F , A ˜ ) ∩ ( G , B ˜ ) = ( N , C ˜ ) , where C ˜ = A ˜ ∩ B ˜ , for all ( p n ) n ∈ C ˜ and N [ ( p n ) n ] = F [ ( p n ) n ] ∧ G [ ( p n ) n ] .

From this we get, for all ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ , ( p i ) n ≤ ( p j ) n implies that F [ ( p i ) n ] ∧ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∧ G [ ( p j ) n ] .

Then (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that N [ ( p i ) n ] ⊑ N [ ( p j ) n ] ∀ ( p i ) n , ( p j ) n ∈ C ˜ .

Hence ( N , C ˜ ) is an L - M n FSS over X ˜ □ .

Theorem 3.11. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ . Then ( F , A ˜ ) ∨ ( G , B ˜ ) is an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then for all ( p i ) n , ( p j ) n ∈ A ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and for all ( p i ) n , ( p j ) n ∈ B ˜ , G [ ( p i ) n ] ⊑ G [ ( p j ) n ] .

Let ( F , A ˜ ) ∨ ( G , B ˜ ) = ( O , C ˜ ) , where C ˜ = A ˜ × B ˜ , for all ( ( p i ) n , ( p k ) n ) ∈ A ˜ × B ˜ and O ( ( p i ) n , ( p k ) n ) = F [ ( p i ) n ] ∨ G [ ( p k ) n ] .

Now, (p _i )  ⁿ  ≤ (p _j )  ⁿ and (p _k )  ⁿ  ≤ (p _l )  ⁿ  ⇒ ((p _i )  ⁿ , (p _k )  ⁿ ) ≤ ((p _j )  ⁿ , (p _l )  ⁿ ) and F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and G [ ( p k ) n ] ⊑ G [ ( p l ) n ] ⇒ F [ ( p i ) n ] ∨ G [ ( p k ) n ] ⊑ F [ ( p j ) n ] ∨ G [ ( p l ) n ] .

Then ((p _i )  ⁿ , (p _k )  ⁿ ) ≤ ((p _j )  ⁿ , ( p l ) n ) ⇒ O ( ( p i ) n , ( p k ) n ) ⊑ O ( ( p j ) n , (p _l )  ⁿ ) for all ((p _i )  ⁿ , (p _k )  ⁿ ), ( ( p j ) n , ( p l ) n ) ∈ A ˜ × B ˜ .

Hence ( F , A ˜ ) ∨ ( G , B ˜ ) is an L - M n FSS over X ˜ □ .

Theorem 3.12. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ . Then ( F , A ˜ ) ∧ ( G , B ˜ ) is an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then for all ( p i ) n , ( p j ) n ∈ A ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and for all ( p i ) n , ( p j ) n ∈ B ˜ , ( p i ) n ≤ ( p j ) n implies that G [ ( p i ) n ] ⊑ G [ ( p j ) n ] .

Let ( F , A ˜ ) ∧ ( G , B ˜ ) = ( P , C ˜ ) , where C ˜ = A ˜ × B ˜ , for all ( ( p i ) n , ( p k ) n ) ∈ A ˜ × B ˜ and P ( ( p i ) n , ( p k ) n ) = F [ ( p i ) n ] ∧ G [ ( p k ) n ] .

Now, (p _i )  ⁿ  ≤ (p _j )  ⁿ and (p _k )  ⁿ  ≤ (p _l )  ⁿ  ⇒ ((p _i )  ⁿ , (p _k )  ⁿ ) ≤ ((p _j )  ⁿ , (p _l )  ⁿ ) and F [ ( p i ) n ] ⊑ F [ ( p j ) n and G [ ( p k ) n ] ⊑ G [ ( p l ) n ] ⇒ F [ ( p i ) n ] ∧ G [ ( p k ) n ] ⊑ F [ ( p j ) n ] ∧ G [ ( p l ) n ] .

Then ( ( p i ) n , ( p k ) n ) ≤ ( ( p j ) n , ( p l ) n ) ⇒ P ( ( p i ) n , ( p k ) n ) ⊑ P ( ( p j ) n , ( p l ) n ) for all ( ( p i ) n , ( p k ) n ) , ( ( p j ) n , ( p l ) n ) ∈ A ˜ × B ˜ .

Hence ( F , A ˜ ) ∧ ( G , B ˜ ) is an L - M n FSS over X ˜ □ .

Theorem 3.13. Extended intersection of two L - M n FSS s over X ˜ is also an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then for all ( p i ) n , ( p j ) n ∈ A ˜ , ( p i ) n ≤ ( p j ) n implies that F ⊑ F [ ( p j ) n ] and for all ( p i ) n , ( p j ) n ∈ B ˜ , ( p i ) n ≤ ( p j ) n implies that G [ ( p i ) n ] ⊑ G [ ( p j ) n ] .

Let ( F , A ˜ ) ∩ e ( G , B ˜ ) = ( R , C ˜ ) , where C ˜ = A ˜ ∪ B ˜ , for all ( p r ) n ∈ C ˜ and R [ ( p r ) n ] = F [ ( p r ) n ] ∧ G [ ( p r ) n ] .

Case 1:

If ( p i ) n , ( p j ) n ∈ A ˜ - B ˜ then F [ ( p i ) n ] ⊑ F [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ

Case 2:

If ( p i ) n , ( p j ) n ∈ B ˜ - A ˜ then G [ ( p i ) n ] ⊑ G [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ

Case 3:

If ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ then F [ ( p i ) n ] ∧ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∧ G [ ( p j ) n ] for (p _i )  ⁿ  ≤ (p _j )  ⁿ

From this we get, for all ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ∧ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∧ G [ ( p j ) n ] .

Then (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that R [ ( p i ) n ] ⊑ R [ ( p j ) n ] ∀ ( p i ) n , ( p j ) n ∈ C ˜ .

Hence ( R , C ˜ ) is an L - M n FSS over X ˜ □ .

Theorem 3.14. Restricted union of two L - M n FSS s over X ˜ is an L - M n FSS over X ˜ .

Proof. Let ( F , A ˜ ) and ( G , B ˜ ) be two L - M n FSS s over X ˜ .

Then (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ⊑ F [ ( p j ) n ] and (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that G [ ( p i ) n ] ⊑ G [ ( p j ) n ] for all ( p i ) n , ( p j ) n ∈ A ˜ .

Let ( F , A ˜ ) ∩ ɛ ( G , B ˜ ) = ( S , C ˜ ) , where C ˜ = A ˜ ∩ B ˜ , for all ( p s ) n ∈ C ˜ and S [ ( p s ) n ] = F [ ( p s ) n ] ∧ G [ ( p s ) n ] .

From this we get, for all ( p i ) n , ( p j ) n ∈ A ˜ ∩ B ˜ , (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that F [ ( p i ) n ] ∨ G [ ( p i ) n ] ⊑ F [ ( p j ) n ] ∨ G [ ( p j ) n ] .

Then (p _i )  ⁿ  ≤ (p _j )  ⁿ implies that S [ ( p i ) n ] ⊑ S [ ( p j ) n ] ∀ ( p i ) n , ( p j ) n ∈ C ˜ .

Hence ( S , C ˜ ) is an L - M n FSS over X ˜ □ .

4 Application of L - M n FSS in forecasting process

Consider the alternatives x₁, x₂ to represent two different products in a company and p₁, p₂, p₃ be the set of parameters, where p₁ stands for ’Day’ which consist of day1, day2, day3, day 4, day5, day6, day7, day8, p₂ stands for ’Week’ which consist of week1, week2, week3, week4, week5, week6, week7, week8 and p₃ stands for ’Month’ which consist of month1, month2, month3, month4, month5, month6, month7, month8 respectively.

The following L - M 8 FSS ,

- 10 pt ( F , A ˜ ) = { F [ ( p 1 ) 8 ] = { x 1 ( 0.4 , 0.3 , 0.1 , 0.5 , 0.4 , 0.6 , 0.2 , 0.6 ) , x 2 ( 0.1 , 0.2 , 0.3 , 0.2 , 0.4 , 0.5 , 0.7 , 0.1 ) } , F [ ( p 2 ) 8 ] = { x 1 ( 0.5 , 0.3 , 0.8 , 0.6 , 0.4 , 0.7 , 0.4 , 0.8 ) , x 2 ( 0.6 , 0.3 , 0.4 , 0.5 , 0.8 , 0.7 , 0.7 , 0.5 ) } , F [ ( p 3 ) 8 ] = { x 1 ( 0.5 , 0.4 , 0.9 , 0.7 , 0.5 , 0.8 , 0.4 , 0.9 ) , x 2 ( 0.8 , 0.7 , 0.5 , 0.9 , 0.8 , 1.0 , 0.7 , 1.0 ) } } represents the profit of two products for last year.

Now the company decided to find the product in which it has to investing more for this year.

For this, we should detect the better forecast for day, week and month respectively.

In calculating 3-day/week/month moving average forecast, the average of the first three day/week/month is taken to represent the forecast value for the fourth day/week/month. Leaving the first data value, we take the average of the next three data to represent the forecast value for the fifth day/week/month. In this procedure, the moving average forecast values are calculated. In a similar way, we determine the 4-day/week/month and 5-day/week/month moving average forecast.

After calculating moving average forecast, we compute forecast error. For this, we compute the difference between the observed value and the forecast value.

Mean squared error is the average of the sum of squared forecast errors and is often used as measure of the accuracy of forecasts.

In this present work, concept of lattice ordered multi-fuzzy soft set has been initiated and someoperations of lattice ordered multi-fuzzy soft set are also examined. Further we have constructed an application of this concept by using forecasting approach. In this approach, one can get better solution in real life situations. To extend this work, one can investigate the theory of modular and distributive lattice ordered multi-fuzzy soft sets.

Conflicts of interest

The authors declare no conflicts of interest.