International Journal of Algebra and Statistics Volume 5: 2(2016), 135–146 DOI :10.20454/ijas.2016.1181
Published by Modern Science Publishers Available at: http://www.m-sciences.com
Certain Applications of Fuzzy Parameterized Fuzzy Soft Sets in Decision-Making Problems Muhammad Riaza , Masooma Raza Hashmia a Department
of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan (Received: 18 September 2016; Accepted: 17 October 2016)
Abstract. In this paper, we present certain applications of fuzzy parameterized fuzzy soft sets (FPFS-sets) in decision-making problems. We define AND and OR operations for FPFS-sets. We study aggregation operator, reduct and comparison tables for FPFS-set. We present our methods in the form of some algorithms with the help of examples.
1. Introduction In our daily life many fields deals with uncertain data which may not be modeled by the classical mathematics. There are some interesting mathematical tools including fuzzy set theory developed by Zadeh [40] and soft set theory developed by Molodtsov [24] to deal with uncertanities. Recently many researchers used fuzzy soft set (FS-set) theory for decision processes. Maji et al. [21, 22] introduced some basic operations on soft sets. He used soft set theory in decision-making problems. Peyghan and Varol [27, 39] defined some interesting results on soft topology and fuzzy soft topology respectively. Chen et al. [8] presented some applications on parameterized reduction of soft sets. Feng et al. [16–19] studied soft sets, fuzzy soft set and rough sets to establish some significant results. Akram et al. [1–4] introduced various concepts including Bipolar Fuzzy Soft Lie algebras, Fuzzy soft K-algebras, Fuzzy soft Lie algebras and Fuzzy soft graphs. Ali et al. [6] suggested some operations on soft sets which became very useful in the field of soft set theory. Aslihan et al. [37, 38] presented some operations of soft sets and soft intersection semigroups. Shabir and Naz [36] established soft topology on soft set. Cagman et al. [9–11] presented some applications of decision-making problems based on soft topology and fuzzy parameterized fuzzy soft set theory. Samanta and Das [13–15] introduced soft real numbers, soft real set and soft metric spaces. Sut [23] presented some applications of FS-sets to the decision-making problems. Riaz et al. [29–31] discussed various concepts including soft σ-algebra, measurable soft set, measurable soft mappings and soft metric spaces. Idris and Serkan [41] presented some interesting results on FPFS-topological space. Borah and Hazarika [7] discussed some properties of FS-topology and also established some applications in Chemistry. Roy and Maji [33] studied fuzzy soft set and established a theoretic approach to decisionmaking problems. Soft set theory and fuzzy soft set theory has studied by many explorers in the last decade (See [9, 10, 13–16, 21, 22, 24, 36]).
2010 Mathematics Subject Classification. 54A05, 11B05, 54D30, 54A40, 06D72, 03E72 Keywords. Fuzzy soft set, FPFS-set, FPFS-AND operation, FPFS-OR operation, FPFS decision-making. Email addresses:
[email protected] (Muhammad Riaz),
[email protected] (Masooma Raza Hashmi)
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2. Preliminaries e Definition 2.1. [9, 39] A fuzzy soft set (FS-set) is a mapping λ : R → P(X) such that λA (ζ) = φ if ζ < A, where X is the set of universe and A ⊆ R, R is the set of parameters or attributes. It is denoted as (λ, A) given by e (λ, A) = {(ζ, λA (ζ)) : ζ ∈ R, λA (ζ) ∈ P(X)}. The value λA (ζ) is a fuzzy set known as ζ-approximate element of FS-set (λ, A) ∀ ζ ∈ R. The degree of membership of elements is taken in the interval [0, 1]. ] Then the pair (X, R) represents the family of all FS-sets on X with parameters from R and known as FS-class. e Definition 2.2. [9, 41] A fuzzy parameterized fuzzy soft set (FPFS-set) is a mapping γ : R → P(X) such that γA (ζ) = φ if µA (ζ) = 0, where X is the initial universe and A ⊆ R, R is the set of parameters or attributes. It is denoted by FA as given below e FA = {(µA (ζ)/ζ, γA (ζ)) : ζ ∈ R, γA (ζ) ∈ P(X); µA (ζ), γA (ρ) ∈ [0, 1], ρ ∈ X}. The value γA (ζ) is a fuzzy set known as ζ-element of FPFS-set FA ∀ ζ ∈ R. Example 2.3. Let us consider the set of healthy snacks given by X = {ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 }, where ρ1 = hard boiled eggs, ρ2 = bean salad, ρ3 = almonds, ρ4 = air popped popcorn, ρ5 = cherry tomatoes, ρ6 = soy chips, ρ7 = nonfat yogurt, ρ8 = frozen mangoes and frozen grapes. The set of attributes is given by, R = {ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 } where, ζ1 = adequacy, ζ2 = sustain energy level , ζ3 = helps in weight lose, ζ4 = for skin freshness, ζ5 = tasty , ζ6 = prevent diseases. We consider the fuzzy subset A on R given by A = {0.5/ζ1 .0.8/ζ2 , 0.6/ζ3 , 0.3/ζ5 }. Then FPFS-set is written by FA = {(0.5/ζ1 , {0.1/ρ3 , 0.3/ρ5 , 0.6/ρ7 }), (0.8/ζ2 , {0.7/ρ1 , 0.5/ρ4 , 0.3/ρ8 }), (0.6/ζ3 , {0.4/ρ3 , 0.2/ρ5 , 1/ρ7 }), (0.3/ζ5 , {0.4/ρ1 , 1/ρ4 , 0.2/ρ6 })}. The tabular form the FPFS-set is given below X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8
0.5/ζ1 0 0 0.1 0 0.3 0 0.6 0
0.8/ζ2 0.7 0 0 0.5 0 0 0 0.3
0.6/ζ3 0 0 0.4 0 0.2 0 1 0
0/ζ4 0 0 0 0 0 0 0 0
0.3/ζ5 0.4 0 0 1 0 0.2 0 0
0/ζ6 0 0 0 0 0 0 0 0
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Definition 2.4. [9, 41] Let FA be a FPFS-set over X. If λA (ζ) = φ ∀ ζ ∈ R then FA is known as A-empty FPFS-set. It is represented as FφA . If A = φ, then A-empty FPFS-set is called empty FPFS-set denoted as Fφ . Definition 2.5. [9, 41] Let FA be a FPFS-set over X. If γA (ζ) = X and µA (ζ) = 1 ∀ ζ ∈ R then FA is known as A-universal FPFS-set. It is represented as FAe. If A = R, then A-universal FPFS-set is said to be universal or absolute FPFS-set written as FeR . Definition 2.6. [41] Let FA and FB be two FPFS-sets. Then FA is called FPFS-subset of FB denoted by FA e ⊆FB if (i) µA (ζ) ≤ µB (ζ) (ii) γA (ζ) ⊆ γB (ζ), ∀ ζ ∈ R. Definition 2.7. [9, 41] Let FA and FB be two FPFS-sets. The union of two FPFS-sets FA and FB is written as FA e ∪FB and defined by µAe∪B (ζ) = max{µA (ζ), µB (ζ)} and γAe∪B (ζ) = {γA (ζ) ∪ γB (ζ)}, ∀ ζ ∈ R. Definition 2.8. [9, 41] Let FA and FB be two FPFS-sets. The intersection of two FPFS-sets FA and FB is written as FA e ∩FB and defined by µAe∩B (ζ) = min{µA (ζ), µB (ζ)} and γAe∩B (ζ) = {γA (ζ) ∩ γB (ζ)}∀ ζ ∈ R.
3. Applications of FPFS-set to decision-making In this section, we present some operations on FPFS-set and certain applications of decision-making based on FPFS-set. Definition 3.1. If FA and GB are two FPFS-sets then ”FA AND GB ” is a FPFS-set denoted by FA ∧ GB and is defined by FA ∧ GB = HA×B , where A × B is the cross product of two fuzzy sets. HA×B can be calculated as, µA×B (ζαβ ) = µA (ζα ) ∧ µB (ζβ ) ∀ µA (ζα )/ζα ∈ A, µB (ζβ )/ζβ ∈ B and ζ
ζ
ζ
αβ γA×B (ραβ ) = γζAα (ρα ) ∧ γBβ (ρβ ) ∀ γζAα (ρα ) ∈ FA , γBβ (ρβ ) ∈ GB .
Example 3.2. Let FA and FB be two FPFS-sets given by FA = {(0.7/ζ1 , {0.8/ρ1 , 0.6/ρ2 ), (0.5/ζ2 , {0.3/ρ1 , 0.7/ρ2 )}, FB = {(0.5/ζ3 , {0.4/ρ1 , 0.7/ρ2 ), (0.2/ζ4 , {0.8/ρ1 , 0.9/ρ2 )}. The tabular form of FA is given below X ρ1 ρ2
0.7/ζ1 0.8 0.6
0.5/ζ2 0.3 0.7
X ρ1 ρ2
0.5/ζ3 0.4 0.7
0.2/ζ4 0.8 0.9
Similarly the tabular form FB is given below
We apply AND operation on FA and FB . Then we get 2 ∗ 2 = 4 parameters of the form ζαβ = ζα ∧ ζβ ∀α = 1, 2 and β = 3, 4. We require the FPFS-set for the fuzzy set of parameters
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C = {µ(ζ13 )/ζ13 , µ(ζ14 )/ζ14 , µ(ζ23 )/ζ23 , µ(ζ24 )/ζ24 }, where µ(ζαβ ) denotes the degree of membership of the parameters of C ∀ α = 1, 2 and β = 3, 4. µ(ζαβ ) = min[µ(ζα ), µ(ζβ )]. This implies that C = {0.5/ζ13 , 0.2/ζ14 , 0.5/ζ23 , 0.2/ζ24 }. By applying AND operation we obtain FPFS-set given by FC = {(0.5/ζ13 , {0.4/ρ1 , 0.6/ρ2 }), (0.2/ζ14 , {0.8/ρ1 , 0.6/ρ2 }), (0.5/ζ23 , {0.3/ρ1 , 0.7/ρ2 }), (0.2/ζ24 , {0.3/ρ1 , 0.7/ρ2 })}. Thus the tabular form of FPFS-set FC is given below X ρ1 ρ2
0.5/ζ13 0.4 0.6
0.2/ζ14 0.8 0.6
0.5/ζ23 0.3 0.7
0.2/ζ24 0.3 0.7
Definition 3.3. If FA and GB are two FPFS-sets then ”FA OR GB ” is a FPFS-set denoted by FA ∨ GB and is defined by FA ∨ GB = HA×B , where A × B is the cross product of two fuzzy sets. HA×B can be calculated as, µA×B (ζαβ ) = µA (ζα ) ∨ µB (ζβ ) ∀ µA (ζα )/ζα ∈ A, µB (ζβ )/ζβ ∈ B and ζ
ζ
ζ
αβ γA×B (ραβ ) = γζAα (ρα ) ∨ γBβ (ρβ ) ∀ γζAα (ρα ) ∈ FA , γBβ (ρβ ) ∈ GB
Example 3.4. Let FA and FB be two FPFS-sets given as, FA = {(0.7/ζ1 , {0.8/ρ1 , 0.6/ρ2 ), (0.5/ζ2 , {0.3/ρ1 , 0.7/ρ2 )}, FB = {(0.5/ζ3 , {0.4/ρ1 , 0.7/ρ2 ), (0.2/ζ4 , {0.8/ρ1 , 0.9/ρ2 )}. FA can be written in the tabular form as follows: X ρ1 ρ2
0.7/ζ1 0.8 0.6
0.5/ζ2 0.3 0.7
0.5/ζ3 0.4 0.7
0.2/ζ4 0.8 0.9
FB can be written in the tabular form as follows: X ρ1 ρ2
We apply here OR operation on FA and FB . Then we obtain 2 ∗ 2 = 4 parameters of the form ζαβ = ζα ∨ ζβ ∀α = 1, 2 and β = 3, 4. We require the FPFS-set for the fuzzy set of parameters C = {µ(ζ13 )/ζ13 , µ(ζ14 )/ζ14 , µ(ζ23 )/ζ23 , µ(ζ24 )/ζ24 } where, µ(ζαβ ) denotes the degree of membership of the parameters of C ∀ α = 1, 2 and β = 3, 4. µ(ζαβ ) = max[µ(ζα ), µ(ζβ )]. This implies that C = {0.7/ζ13 , 0.7/ζ14 , 0.5/ζ23 , 0.5/ζ24 }. After applying ’OR’ operation we get FPFS-set the FC given as, FC = {(0.7/ζ13 , {0.8/ρ1 , 0.7/ρ2 }), (0.7/ζ14 , {0.8/ρ1 , 0.9/ρ2 }), (0.5/ζ23 , {0.4/ρ1 , 0.7/ρ2 }), (0.5/ζ24 , {0.8/ρ1 , 0.9/ρ2 })}. Finally the tabular form of FPFS-set FC becomes X ρ1 ρ2
0.7/ζ13 0.8 0.7
0.7/ζ14 0.8 0.9
0.5/ζ23 0.4 0.7
0.5/ζ24 0.8 0.9
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Now we present an algorithm for decision-making to get admission in well-reputed university of the world by using FPFS-set.
Example 3.5. Assume that a student is interested to get admission in a university. Let X = {ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 } be the set of eight universities of the world, where ρ1 = Stanford University (U.S.), ρ2 = Harvard University (U.S.), ρ3 = Massachusetts Institute of Technology (MIT) (U.S.), ρ4 = University of Cambridge (UK), ρ5 = Oxford University (UK), ρ6 = Swiss Federal Institute of Technology (ETH) (Switzerland), ρ7 = University of Tokyo (Japan), ρ8 = University of Toronto (Canada). The facilities which may be provided by these universities is given by a set R = {ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 , ζ11 , ζ12 }. The parameters ζi (i = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) stands for ζ1 = highly qualified faculty, ζ2 = healthy environment, ζ3 = in a good ranking position, ζ4 = library , ζ5 = cafeteria, ζ6 = hostels, ζ7 = international standard course work, ζ8 = best security system, ζ9 = computer and internet facility, ζ10 = transport, ζ11 = sports, ζ12 = highly furnished computer labs. Consider a fuzzy subset on the basis of his requirements given by A = {0.7/ζ2 , 0.6/ζ3 , , 0.7/ζ5 , 0.8/ζ6 , 0.9/ζ7 , 0.5/ζ8 , 0.4/ζ10 } of R. We use the algorithm given by Cagman in [9] which is written by Step 1: Construct a FPFS-set FA . Step 2: Find the aggregate fuzzy set F∗A of FA . Step 3: Find the largest membership grade max µF∗A (ρ). Step 1:We construct the following FPFS-set FA over X, FA ={(0.7/ζ2 , {0.8/ρ1 , 0.6/ρ2 , 0.5/ρ3 , 0.2/ρ4 , 0.3/ρ5 , 0.1/ρ6 , 0.2/ρ7 , 0.1/ρ8 }), (0.6/ζ3 , {0.8/ρ1 , 0.6/ρ2 , 0.5/ρ3 , 0.3/ρ4 , 0.2/ρ5 , 0.4/ρ6 , 0.3/ρ7 , 0.2/ρ8 }), (0.7/ζ5 , {0.9/ρ1 , 0.7/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.2/ρ5 , 0.3/ρ6 , 0.4/ρ7 , 0.4/ρ8 }), (0.8/ζ6 , {0.7/ρ1 , 0.5/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.3/ρ5 , 0.1/ρ6 , 0.2/ρ7 , 0.1/ρ8 }), (0.9/ζ7 , {0.8/ρ1 , 0.8/ρ2 , 0.7/ρ3 , 0.4/ρ4 , 0.5/ρ5 , 0.6/ρ6 , 0.4/ρ7 , 0.6/ρ8 }), (0.5/ζ8 , {0.5/ρ1 , 0.4/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.6/ρ5 , 0.4/ρ6 , 0.3/ρ7 , 0.5/ρ8 }), (0.4/ζ10 , {0.9/ρ1 , 0.7/ρ2 , 0.6/ρ3 , 0.5/ρ4 , 0.4/ρ5 , 0.3/ρ6 , 0.5/ρ7 , 0.4/ρ8 })} In tabular form the FPFS-set FA can be written as:
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X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8
0.7/ζ2 0.8 0.6 0.5 0.2 0.3 0.1 0.2 0.1
0.6/ζ3 0.8 0.6 0.5 0.3 0.2 0.4 0.3 0.2
0.7/ζ5 0.9 0.7 0.5 0.4 0.2 0.3 0.4 0.4
0.8/ζ6 0.7 0.5 0.4 0.2 0.3 0.1 0.2 0.1
0.9/ζ7 0.8 0.8 0.7 0.4 0.5 0.6 0.4 0.6
0.5/ζ8 0.5 0.4 0.5 0.4 0.6 0.4 0.3 0.5
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0.4/ζ10 0.9 0.7 0.6 0.5 0.4 0.3 0.5 0.4
Step 2: Now we find the aggregate fuzzy set by using the formula: FA∗ = {µFA∗ (ρ)/ρ : ρ ∈ X} where
µFA∗ (ρ) = Σζ∈R µA (ζ) γA (ρ)/|R|.
Thus we have FA∗ = {0.2236/ρ1 , 0.2391/ρ2 , 0.1579/ρ3 , 0.1266/ρ4 , 0.135/ρ5 , 0.1216/ρ6 , 0.1225/ρ7 , 0.125/ρ8 }. Step 3: The largest membership grade is max µFR∗ (ρ) = 0.2391. This shows that the university ρ2 has the greatest membership degree. Thus the first priority for the admission goes to Harvard University (U.S.), second priority goes to Stanford University (U.S.) with degree of membership 0.2236 and third priority goes to Massachusetts Institute of Technology (MIT) (U.S.) with degree of membership 0.1579. Example 3.6. In this example, we present an algorithm which is useful for a person who wants to purchase some shares of a multinational company by using FPFS-set. We extended the algorithm for FPFS-set which was used for FS-set given in [11]. The algorithm has the following steps: Step 1: Construct a FPFS-set FA . Step 2: Find cardinal set cFA of FA . Step 3: Find aggregate F∗A of FA . Step 4: Find the best alternative from F∗A that has the largest membership degree described by max µF∗A (ρ). Consider the following set of companies: X = {ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 , ρ9 , ρ10 }, where ρ1 = Microsoft, ρ2 = IBM, ρ3 = Nestle, ρ4 = Procter and Gamble (P and G), ρ5 = Coca Cola, ρ6 = PepsiCo, ρ7 = Citi Group, ρ8 = Sony, ρ9 = HEWLETT PACKARD (HP), ρ10 = APPLE INC. We consider the following set of parameters for multinational companies: R = {ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 , ζ8 , ζ9 , ζ10 } where, ζ1 = Productive organization, ζ2 = World wide, ζ3 = Ownership and control, ζ4 = Oligopolistic Powers,
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ζ5 = Marketing superiority, ζ6 = High efficiency, ζ7 = Sophisticated Technology, ζ8 = Better Quality of Products, ζ9 = Professional Management. We choose a fuzzy subset of R on the basis of his requirements. The degrees of memberships for the selected parameters represent demands in percentages. A = {0.7/ζ2 , 0.6/ζ3 , , 0.7/ζ5 , 0.8/ζ6 , 0.9/ζ7 , 0.5/ζ8 } of R. Step 1: We construct the following FPFS-set FA over X. FA ={(0.7/ζ2 , {0.9/ρ1 , 0.6/ρ2 , 0.5/ρ3 , 0.2/ρ4 , 0.3/ρ5 , 0.1/ρ6 , 0.2/ρ7 , 0.1/ρ8 , 0.1/ρ9 , 0.1/ρ10 }), (0.6/ζ3 , {0.9/ρ1 , 0.6/ρ2 , 0.5/ρ3 , 0.3/ρ4 , 0.2/ρ5 , 0.4/ρ6 , 0.3/ρ7 , 0.2/ρ8 , 0.1/ρ9 , 0.1/ρ10 }), (0.7/ζ5 , {0.9/ρ1 , 0.7/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.2/ρ5 , 0.3/ρ6 , 0.4/ρ7 , 0.4/ρ8 , 0.1/ρ9 , 0.1/ρ10 }), (0.8/ζ6 , {0.9/ρ1 , 0.5/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.3/ρ5 , 0.1/ρ6 , 0.2/ρ7 , 0.1/ρ8 , 0.1/ρ9 , 0.1/ρ10 }), (0.9/ζ7 , {0.9/ρ1 , 0.8/ρ2 , 0.7/ρ3 , 0.4/ρ4 , 0.5/ρ5 , 0.6/ρ6 , 0.4/ρ7 , 0.6/ρ8 , 0.1/ρ9 , 0.1/ρ10 }), (0.5/ζ8 , {0.8/ρ1 , 0.4/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.6/ρ5 , 0.4/ρ6 , 0.3/ρ7 , 0.5/ρ8 , 0.1/ρ9 , 0.1/ρ10 })} Step 2: The cardinal is computed by the formula cFA = {µcFR (ζ) /ζ : ζ ∈ A} where µcFR (ζ) = Σρ∈X µA (ζ) γA (ρ)/|X| Then we have cFA = {0/ζ1 , 0.217/ζ2 , 0.216/ζ3 , 0/ζ4 , 0.28/ζ5 , 0.232/ζ6 , 0.459/ζ7 , 0.205/ζ8 , 0/ζ9 }. Step 3: We use here this formula to find the aggregate fuzzy set, |R| ∗ MF∗A = MFA ∗ MtcFA
(1)
where MFA , McFA and MF∗A are matrices representation of FA , cFA and F∗A respectively. Then we obtain the matrix of F∗A by using (1). 0 0.9 0.9 0 0.9 0.9 0.9 0.8 0 0 0.1586 0 0.6 0.6 0 0.7 0.5 0.8 0.4 0 0.217 0.1134 0 0.5 0.5 0 0.5 0.4 0.7 0.5 0 0.0970 0.216 0 0.2 0.3 0 0.4 0.2 0.4 0.4 0 0 0.0591 0 0.3 0.2 0 0.2 0.3 0.5 0.6 0 0.0651 0.28 = MF∗A = 1/9 0.0636 0 0.1 0.4 0 0.3 0.1 0.6 0.4 0 0.232 0 0.2 0.3 0 0.4 0.2 0.4 0.3 0 0.0568 0 0.1 0.2 0 0.4 0.1 0.6 0.5 0 0.459 0.0642 0.205 0 0.1 0.1 0 0.1 0.1 0.1 0.1 0 0 0.0178 0.0178 0 0.1 0.1 0 0.1 0.1 0.1 0.1 0 Thus we have F∗A = {0.1586/ρ1 , 0.1134/ρ2 , 0.0970/ρ3 , 0.0591/ρ4 , 0.0651/ρ5 , 0.0636/ρ6 , 0.0568/ρ7 , 0.0642/ρ8 , 0.0178/ρ9 , 0.0178/ρ10 }. Step 4: In the last we choose the greatest degree of membership i.e. max µFR∗ (ρ) = 0.1586. This shows that the applicant ρ1 has the greatest membership degree. So the 1st preference is Microsoft, the 2nd preference is IBM and the third preference is Nestle. Example 3.7. Assume that a person wants to purchase a smart phone from a market. We introduce an algorithm for FPFS-set to decision-making problem. Our algorithm is an extension of the algorithm used in [21] for soft set. The algorithm consists of the following steps: Step 1: Construct a FPFS-set FA . Step 2: Find the reduct of FA .
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Step 3: Construct the table for weighted choice values. Step 4: Find the best alternative from this set that has the largest membership degree by max µF∗A (ρ). Suppose that the collection of mobiles phones is X = {ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 } The set of parameters represents the qualities of mobile phones. R = {ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 }, where, ρ1 = iphone 6 ρ2 = Nokia lumia(930) ρ3 = Samsung Galaxy Note ρ4 = Qmobile Noir E8 ρ5 = Huawei Nexus 6p ρ6 = LG phones V10 ρ7 = Motorola V3i ρ8 = Sony Xperia Z5 Premium and ζ1 = Beautiful, ζ2 = Low cost, ζ3 = Expensive, ζ4 = Long battery timing, ζ5 = Metallic body, ζ6 = Less weight, ζ7 = Durable glass screen. He choose the following parameters ’beautiful’, ’long battery timing’, ’less weight’ and durable screen. Which constitute the fuzzy subset A = {0.8/ζ1 , 0.7/ζ4 , 0.8/ζ5 , 0.5ζ7 } Suppose that (FA ) be a fuzzy soft set given by, FA = {({0.8/ζ1 , {0.8/ρ1 , 0.7/ρ3 , 0.6/ρ4 , 0.5/ρ5 , 0.8/ρ8 }), (0.7/ζ4 , {0.8/ρ1 , 0.9/ρ2 , 0.4/ρ5 , 0.3/ρ6 , 0.1/ρ7 , 0.8/ρ8 }), (0.8/ζ5 , {0.8/ρ1 , 0.7/ρ3 , 0.6/ρ4 , 0.5/ρ5 , 0.8/ρ8 }), (0.5/ζ7 , {0.8/ρ1 , 0.3/ρ5 , 0.7/ρ7 , 0.8/ρ8 })). The tabular form of FPFS-set can be written as: X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8
0.8/ζ1 0.8 0 0.7 0.6 0.5 0 0 0.8
0.7/ζ4 0.8 0.9 0 0 0.4 0.3 0.1 0.8
0.8/ζ5 0.8 0 0.7 0.6 0.5 0 0 0.8
0.5/ζ7 0.8 0 0 0 0.3 0 0.7 0.8
Clearly, from the above table we can see that {0.8/ζ1 , 0.7/ζ4 , 0.5/ζ7 } and {0.7/ζ4 , 0.8/ζ5 , 0.5/ζ7 } are two reducts of A ={0.8/ζ1 , 0.7/ζ4 , 0.8/ζ5 , 0.5/ζ7 } we have to choose only one. Let Q = {0.8/ζ1 , 0.7/ζ4 , 0.5/ζ7 } be the reduct which we choose. We want to find out the choice values so, the reduct FPFS-set can be write in tabular form as:
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X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8
0.8/ζ1 0.8 0 0.7 0.6 0.5 0 0 0.8
0.7/ζ4 0.8 0.9 0 0 0.4 0.3 0.1 0.8
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0.5/ζ7 0.8 0 0 0 0.3 0 0.7 0.8
The degrees of membership value of the parameters of Q are also called their weights, so we have not need to defined separately weighted values for the parameters. By using this concept we take degrees of membership as weighted values here : for parameter ’beautiful’ π1 = 0.8 , for parameter ’long battery timing’ π2 = 0.7 , for parameter ’ durable glass screen’ π3 = 0.5. Now we represent the above table in the form of membership table with weighted choice values as: X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8
ζ1 , π1 = 0.8 0.8 0 0.7 0.6 0.5 0 0 0.8
ζ4 , π2 = 0.7 0.8 0.9 0 0 0.4 0.3 0.1 0.8
ζ7 ,π3 = 0.5 0.8 0 0 0 0.3 0 0.7 0.8
weighted choice value 1.6 0.63 0.56 0.48 0.83 0.21 0.42 1.6
This is clear from the given table that the person should decide ρ1 or ρ8 i.e iphone or sony. His second choice should be ρ5 i.e Qmobile and his third priority is ρ2 i.e Nokia. Example 3.8. Next we present an application for decision-making in which a man wants to buy a car. The algorithm used in this example is modification of algorithm in [33]. The modified algorithm consists of the following steps: Step 1: Construct the FPFS-sets FA and FB . Step 2: Apply ’AND’ operation to FA and FB then we get FPFS-set FC for desired set of parameters. Step 3: Construct the comparison table for FC by using some modifications. Step 4: Calculate the final score table by using the row-sum and column-sum. Step 5: Find the best alternative from this table that has maximum score. The set of cars having different colors and prizes given as: X = {ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 }, where ρ1 = Toyota, ρ2 = Honda, ρ3 = Suzuki, ρ4 = Audi, ρ5 = Wolks Wagon, ρ6 = Mercedes, ρ7 = FAW. The set of parameters is given as: R = {ζ1 , ζ2 , ζ3 , ζ4 , ζ5 , ζ6 , ζ7 } where, ζ1 = gun metallic, ζ2 = graphite gray, ζ3 = bronze mice, ζ4 = black, ζ5 = expensive, ζ6 = very expensive,
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ζ7 = cheap. Let A = {0.7/ζ1 , 0.6/ζ2 , 0.4/ζ3 , 0.8/ζ4 } and B = {0.6/ζ5 , 0.7/ζ6 , 0.5/ζ7 } be the fuzzy subsets of R where, fuzzy set A represents the colors of cars and the fuzzy set B represent the prizes of cars. We construct here the FPFS-set FA given as: FA = {(0.7/ζ1 , {0.6/ρ1 , 0.3/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 }), ({0.6/ζ2 , {0.7/ρ1 , 0.2/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.1/ρ5 , 0.9/ρ6 , 0.2/ρ7 }), ({0.4/ζ3 , {0.7/ρ1 , 0.2/ρ2 , 0.6/ρ3 , 0.3/ρ4 , 0.2/ρ5 , 0.8/ρ6 , 0.3/ρ7 }), ({0.8/ζ4 , {0.6/ρ1 , 0.1/ρ2 , 0.6/ρ3 , 0.1/ρ4 , 0.1/ρ5 , 0.9/ρ6 , 0.4/ρ7 })} Now we construct here the FPFS-set FB given as: FB = {({0.6/ζ5 , {0.6/ρ1 , 0.3/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 }), ({0.7/ζ6 , {0.7/ρ1 , 0.2/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.1/ρ5 , 0.9/ρ6 , 0.2/ρ7 }), ({0.5/ζ7 , {0.7/ρ1 , 0.2/ρ2 , 0.6/ρ3 , 0.3/ρ4 , 0.2/ρ5 , 0.8/ρ6 , 0.3/ρ7 })} We consider these two FPFS-sets FA and FB and we apply here ’AND’ operation then we get 4 ∗ 3 = 12 parameters of the form ζαβ = ζα ∧ ζβ ∀α = 1, 2, 3, 4 and β = 5, 6, 7. We require the FPFS-set for the fuzzy set of parameters C = {µ(ζ15 )/ζ15 , µ(ζ17 )/ζ17 , µ(ζ25 )/ζ25 , µ(ζ26 )/ζ26 , µ(ζ37 )/ζ37 , µ(ζ45 )/ζ45 } where, µ(ζαβ ) denotes the degree of membership of the parameters of C ∀ α = 1, 2, 3, 4 and β = 5, 6, 7. µ(ζαβ ) = min[µ(ζα ), µ(ζβ )]. This implies that C = {0.6/ζ15 , 0.5/ζ17 , 0.6/ζ25 , 0.6/ζ26 , 0.4/ζ37 , 0.6/ζ45 }. After applying ’AND’ operation we get FPFS-set the FC given as: FC = {(0.6/ζ15 , {0.6/ρ1 , 0.3/ρ2 , 0.5/ρ3 , 0.4/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 }), (0.5/ζ17 , {0.6/ρ1 , 0.2/ρ2 , 0.5/ρ3 , 0.3/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 }), (0.6/ζ25 , {0.6/ρ1 , 0.2/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 }), (0.6/ζ26 , {0.7/ρ1 , 0.2/ρ2 , 0.4/ρ3 , 0.2/ρ4 , 0.1/ρ5 , 0.9/ρ6 , 0.2/ρ7 }), (0.4/ζ37 , {0.7/ρ1 , 0.2/ρ2 , 0.6/ρ3 , 0.3/ρ4 , 0.2/ρ5 , 0.8/ρ6 , 0.3/ρ7 }), (0.6/ζ45 , {0.6/ρ1 , 0.1/ρ2 , 0.5/ρ3 , 0.1/ρ4 , 0.1/ρ5 , 0.8/ρ6 , 0.2/ρ7 })}. The tabular form of FPFS-set FC can be written as: X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7
0.6/ζ15 0.6 0.3 0.5 0.4 0.1 0.8 0.2
0.5/ζ17 0.6 0.2 0.5 0.3 0.1 0.8 0.2
0.6/ζ25 0.6 0.2 0.4 0.2 0.1 0.8 0.2
0.6/ζ26 0.7 0.2 0.4 0.2 0.1 0.9 0.2
0.4/ζ37 0.7 0.2 0.6 0.3 0.2 0.8 0.3
0.6/ζ45 0.6 0.1 0.5 0.1 0.1 0.8 0.2
We here find the comparison-table of above FPFS-set FC by modifying the algorithm which is given by Roy and Maji in [33]. Now for modification we multiply each column of the table by its corresponding degree of membership of the parameter. Which gives us: X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7
0.6/ζ15 0.36 0.18 0.30 0.24 0.06 0.48 0.12
0.5/ζ17 0.30 0.10 0.25 0.15 0.05 0.40 0.10
0.6/ζ25 0.36 0.12 0.24 0.12 0.06 0.48 0.12
0.6/ζ26 0.42 0.12 0.24 0.12 0.06 0.54 0.12
0.4/ζ37 0.28 0.08 0.24 0.12 0.08 0.32 0.12
0.6/ζ45 0.36 0.06 0.30 0.06 0.06 0.48 0.12
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The comparison table is given below : X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7
ρ1 6 0 0 0 0 6 0
ρ2 6 6 6 6 2 6 5
ρ3 6 0 6 0 0 6 0
ρ4 6 3 6 6 1 6 4
ρ5 6 6 6 6 6 6 6
ρ6 0 0 0 0 0 6 0
ρ7 6 4 6 5 0 6 6
Now we calculate the column-sum and row-sum after that we calculate the score for each ρα ∀ α = 1, 2, 3, 4, 5, 6, 7. X ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7
row-sum(rα ) 36 19 30 23 9 42 21
column-sum(cα ) 12 38 18 32 42 6 33
Score(sα ) 24 -19 12 -9 -33 36 -12
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