# Soft Expert Sets

Authors: Shawkat Alkhazaleh; Abdul Razak Salleh
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Many researchers have studied this theory, and they created some models to solve problems in decision making and medical diagnosis, but most of these models deal only with one expert. This causes a problem with the user, especially with those who use questionnaires in their work and studies. In our model, the user can know the opinion of all experts in one model. So, in this paper, we introduce the concept of a soft expert set, which will more effective and useful. We also define its basic operations, namely, complement, union intersection AND, and OR. Finally, we show an application of this concept in decision-making problem.


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## Body

## 1. Introduction

Most of the problems in engineering, medical science, economics, environments, and so forth, have various uncertainties. Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. After Molodtsov’s work, some operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3, 4]. Alkhazaleh et al. [5] introduced the concept of soft multisets as a generalization of soft set. They also defined in [6, 7] the concepts of possibility fuzzy soft set and fuzzy parameterized interval-valued fuzzy soft set and gave their applications in decision making and medical diagnosis. Many researchers have studied this theory, and they created some models to solve problems in decision making and medical diagnosis, but most of these models deal only with one expert, and if we want to take the opinion of more than one expert, we need to do some operations such as union, intersection, and so forth. This causes a problem with the user, especially with those who use questionnaires in their work and studies. In our model the user can know the opinion of all experts in one model without any operations. Even after any operation on our model the user can know the opinion of all experts. So in this paper we introduce the concept of a soft expert set, which will be more effective and useful. We also define its basic operations, namely, complement, union intersection AND and OR and study their properties. Finally, we give an application of this concept in a decision-making problem.


## 2. Preliminaries

In this section, we recall some basic notions in soft set theory. Molodtsov [1] defined soft set in the following way. Let U be a universe and E be a set of parameters.  Let P(U) denote the power set of U and A⊆E.Definition 2.1 (see [1]).
A pair(F,A) is called a soft set over U, where F is a mapping F:A→P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε∈A,F(ε) may be considered as the set of ε-approximate elements of the soft set (F,A).The following definitions are due to Maji et al. [3].Definition 2.2.
For two soft sets(F,A) and (G,B) over  U, (F,A) is called a soft subset of (G,B) if(i)
A⊂B,(ii)
for allε∈A,F(ε),andG(ε) are identical approximations.This relationship is denoted by  (F,A)⊂̃(G,B). In this case, (G,B) is called a soft superset of (F,A).Definition 2.3.
Two  soft  sets(F,A)  and  (G,B)  over  a common  universe U  are said  to  be soft  equal  if  (F,A)  is  a soft  subset  of  (G,B)  and  (G,B) is a soft  subset  of  (F,A).Definition 2.4.
LetE={e1,e2,…,en} be a set of parameters.  The NOT set of E denoted by ΊE is defined by ΊE={Ίe1,Ίe2,…,Ίen} where Ίei= not ei,for alli.Definition 2.5.
Thecomplement  of  a  soft  set (F,A)  is  denoted  by  (F,A)c and  is  defined  by  (F,A)c=(Fc,ΊA)  where  Fc:ΊA→P(U)  is  a mapping  given  by  Fc(α)=U-F(Ία),for allα∈ΊA.Definition 2.6.
A  soft  set(F,A)  over  U  is  said  to  be  a  NULL  soft  set denoted  by  Φ,iffor allε∈A,F(ε)=∅, (null-set).Definition 2.7.
A soft set(F,A) over U is said to be an absolute soft set, denoted by Ã,iffor allε∈A,F(ε)=U.Definition 2.8.
If(F,A) and (G,B) are two soft sets then (F,A) AND (G,B) denoted  by  (F,A)∧(G,B), is defined by
(2.1)(F,A)∧(G,B)=(H,A×B),
where  H(α,β)=F(α)∩G(β),for all(α,β)∈A×B.Definition 2.9.
If(F,A) and (G,B) are two soft sets, then (F,A) OR (G,B) denoted by (F,A)∨(G,B), is defined by
(2.2)(F,A)∨(G,B)=(O,A×B),
where  O(α,β)=F(α)∪G(β),for all(α,β)∈A×B.Definition 2.10.
Theunion of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C) where  C=A∪B, and for allε∈C,
(2.3)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,F(ε)∪G(ε)ifε∈A∩B.The following definition is due to Ali et al. [8] since they discovered that Maji et al.’s definition of intersection in [3] is not correct.Definition 2.11.
Theextended intersection of two soft sets (F,A) and (G,B) over a common universe U is the soft set  (H,C) where  C=A∪B, and for allε∈C,
(2.4)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,F(ε)∩G(ε)ifε∈A∩B.


## 3. Soft Expert Set

In this section, we introduce the concept of a soft expert set, and give definitions of its basic operations, namely, complement, union, intersection, AND, and OR. We give examples for these concepts. Basic properties of the operations are also given.LetU  be a universe, E a set of parameters, and X a set of experts (agents). Let O be a set of opinions, Z=E×X×O and A⊆Z.Definition 3.1.
A pair(F,A) is called a soft expert set over U, where F is a mapping given by
(3.1)F:A⟶P(U),
where P(U) denotes the power set of U.Note 3.2.
For simplicity we assume in this paper, two-valued opinions only in setO, that is, O={0=disagree,1=agree}, but multivalued opinions may be assumed as well.Example 3.3.
Suppose that a company produced new types of its products and wishes to take the opinion of some experts about concerning these products. LetU={u1,u2,u3,u4} be a set of products, E={e1,e2,e3} a set of decision parameters where ei(i=1,2,3) denotes the decision “easy to use,” “quality,” and “cheap,” respectively, and let X={p,q,r} be a set of experts.
Suppose that the company has distributed a questionnaire to three experts to make decisions on the company's products, and we get the following:(3.2)F(e1,p,1)={u1,u2,u4},F(e1,q,1)={u1,u4},F(e1,r,1)={u3,u4},F(e2,p,1)={u4},F(e2,q,1)={u1,u3},F(e2,r,1)={u1,u2,u4},F(e3,p,1)={u3,u4},F(e3,q,1)={u1,u2},F(e3,r,1)={u4},F(e1,p,0)={u3},F(e1,q,0)={u2,u3},F(e1,r,0)={u1,u2},F(e2,p,0)={u1,u2,u3},F(e2,q,0)={u2,u4},F(e2,r,0)={u3},F(e3,p,0)={u1,u2},F(e3,q,0)={u3,u4},F(e3,r,0)={u1,u2,u3}.
Then we can view the soft expert set(F,Z) as consisting of the following collection of approximations:
(3.3)(F,Z)={((e1,p,1),{u1,u2,u4}),((e1,q,1),{u1,u4}),((e1,r,1),{u3,u4}),((e2,p,1),{u4}),((e2,q,1),{u1,u3}),((e2,r,1),{u1,u2,u4}),((e3,p,1),{u3,u4}),((e3,q,1),{u1,u2}),((e3,r,1),{u4}),((e1,p,0),{u3}),((e1,q,0),{u2,u3}),((e1,r,0),{u1,u2}),((e2,p,0),{u1,u2,u3}),((e2,q,0),{u2,u4}),((e2,r,0),{u3}),((e3,p,0),{u1,u2}),((e3,q,0),{u3,u4}),((e3,r,0),{u1,u2,u3})}.
Notice that in this example the first expert, p, “agrees” that the “easy to use” products are u1,u2, and u4. The second expert, q, “agrees” that the “easy to use” products are u1 and u4, and the third expert, r, “agrees” that the “easy to use” products are u3 and u4. Notice also that all of them “agree” that product u4 is “easy to use.”Definition 3.4.
For two soft expert sets(F,A) and (G,B) over U,(F,A) is called a soft expert subset of (G,B) if(i)
A⊆B,(ii)
for allε∈B,G(ε)⊆F(ε).This relationship is denoted by (F,A)⊆̃(G,B). In this case (G,B) is called a soft expert superset of (F,A).Definition 3.5.
Two soft expert sets(F,A) and (G,B) over U are said to be equal if  (F,A)  is a soft expert subset of  (G,B)  and (G,B) is a soft expert subset of  (F,A).Example 3.6.
Consider Example3.3. Suppose that the company took the opinion of the experts once again after the products have been in the market for a month.
Suppose(3.4)A={(e1,p,1),(e2,p,0),(e3,p,1),(e1,q,1),(e2,q,1),(e3,q,0),(e1,r,0),(e2,r,1),(e3,r,1)},B={(e1,p,1),(e2,p,0),(e1,q,1),(e2,q,1),(e1,r,0),(e2,r,1)}.
Clearly B⊂A. Let (F,A) and (G,B) be defined as follows:
(3.5)(F,A)={((e1,p,1),{u1,u2,u4}),((e1,q,1),{u1,u4}),((e2,q,1),{u1,u3}),((e2,r,1),{u1,u2,u4}),((e3,p,1),{u3,u4}),((e3,r,1),{u4}),((e1,r,0),{u1,u2}),((e2,p,0),{u1,u2,u3}),((e3,q,0),{u3,u4})},(G,B)={((e1,p,1),{u1,u4}),((e1,q,1),{u4}),((e2,q,1),{u1,u3}),((e2,r,1),{u1}),((e1,r,0),{u2}),((e2,p,0),{u1,u3})}.
Therefore  (G,B)⊆̃(F,A).Definition 3.7.
LetE be a set of parameters and X a set of experts. The NOTset of  Z=E×X×O  denoted  by ΊZ, is defined  by ΊZ={(Ίei,xj,ok),∀i,j,k} where Ίei is not ei.Definition 3.8.
Thecomplement  of  a soft expert set (F,A)  is  denoted  by  (F,A)c and  is  defined  by  (F,A)c=(Fc,ΊA)  where  Fc:ΊA→P(U)  is  a mapping  given  by  Fc(α)=U-F(Ία),for allα∈ΊA.Example 3.9.
Consider Example3.3. Then
(3.6)(F,A)c={((Ίe1,p,1),{u3}),((Ίe1,q,1),{u2,u3}),((Ίe1,r,1),{u1,u2}),((Ίe2,p,1),{u1,u2,u3}),((Ίe2,q,1),{u2,u4}),((Ίe2,r,1),{u3}),((Ίe3,p,1),{u1,u2}),((Ίe3,q,1),{u3,u4}),((Ίe3,r,1),{u1,u2,u3}),((Ίe1,p,0),{u1,u2,u3}),((Ίe1,q,0),{u1,u4}),((Ίe1,r,0),{u3,u4}),((Ίe2,p,0),{u4}),((Ίe2,q,0),{u1,u3}),((Ίe2,r,0),{u1,u2,u4}),((Ίe3,p,0),{u3,u4}),((Ίe3,q,0),{u1,u2}),((Ίe3,r,0),{u4})}.Definition 3.10.
Anagree-soft expert set(F,A)1 over  U  is  a soft expert subset of (F,A) defined as follows:
(3.7)(F,A)1={F1(α):α∈E×X×{1}}.Example 3.11.
Consider Example3.3. Then the agree-soft expert set(F,A)1 over  U  is
(3.8)(F,A)1={((e1,p,1),{u1,u2,u4}),((e1,q,1),{u1,u4}),((e1,r,1),{u3,u4}),((e2,p,1),{u4}),((e2,q,1),{u1,u3}),((e2,r,1),{u1,u2,u4}),((e3,p,1),{u3,u4}),((e3,q,1),{u1,u2}),((e3,r,1),{u4})}.Definition 3.12.
Adisagree-soft expert set(F,A)0 over  U  is a soft expert subset of (F,A) defined as follows:
(3.9)(F,A)0={F0(α):α∈E×X×{0}}.Example 3.13.
Consider Example3.3. Then the disagree-soft expert set(F,A)0 over  U  is
(3.10)(F,A)0={((e1,p,0),{u3}),((e1,q,0),{u2,u3}),((e1,r,0),{u1,u2}),((e2,p,0),{u1,u2,u3}),((e2,q,0),{u2,u4}),((e2,r,0),{u3}),((e3,p,0),{u1,u2}),((e3,q,0),{u3,u4}),((e3,r,0),{u1,u2,u3})}.Proposition 3.14.
If(F,A) is a soft expert set over  U,  then(i)
((F,A)c)c=(F,A),(ii)
(F,A)1c=(F,A)0,(iii)
(F,A)0c=(F,A)1.Proof.
The proof is straightforward.Definition 3.15.
Theunion of two soft expert sets (F,A) and (G,B) over U denoted by (F,A)∪̃(G,B), is the soft expert set (H,C) where C=A∪B, and for allε∈C,
(3.11)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,F(ε)∪G(ε)ifε∈A∩B.Example 3.16.
Consider Example3.3. Let
(3.12)A={(e1,p,1),(e2,p,0),(e3,p,1),(e1,q,1),(e2,q,1),(e3,q,0),(e1,r,0),(e2,r,1),(e3,r,1)},B={(e1,p,1),(e2,p,0),(e3,p,0),(e1,q,1),(e2,q,1),(e3,q,1),(e1,r,0),(e2,r,1)}.
Suppose (F,A) and (G,B)  are two soft expert sets over U such that
(3.13)(F,A)={((e1,p,1),{u1,u2,u4}),((e1,q,1),{u1,u4}),((e2,q,1),{u1,u3}),((e2,r,1),{u1,u2,u4}),((e3,p,1),{u3,u4}),((e3,r,1),{u4}),((e1,r,0),{u1,u2}),((e2,p,0),{u1,u2,u3}),((e3,q,0),{u3,u4})},(G,B)={((e1,p,1),{u1,u3,u4}),((e1,q,1),{u3}),((e2,q,1),{u2,u3}),((e2,r,1),{u2,u3}),((e1,r,0),{u2}),((e3,p,0),{u1,u2,u4}),((e3,q,0),{u2,u3}),((e2,p,0),{u2,u3})}.
Therefore
(3.14)(F,A)∪̃(G,B)=(H,C)={((e1,p,1),U),((e1,q,1),{u1,u3,u4}),((e2,q,1),{u1,u2,u3}),((e2,r,1),U),((e3,p,1),{u3,u4}),((e3,r,1),{u4}),((e1,r,0),{u1,u2}),((e2,p,0),{u1,u2,u3}),((e3,q,0),{u2,u3,u4}),((e3,p,0),{u1,u2,u4})}.Proposition 3.17.
If(F,A), (G,B), and (H,C) are three soft expert sets over U, then(i)
(F,A)∪̃(G,B)=(G,B)∪̃(F,A),(ii)
(F,A)∪̃((G,B)∪̃(H,C))=((F,A)∪̃(G,B))∪̃(H,C).Proof.
The proof is straightforward.Definition 3.18.
Theintersection of two soft expert sets  (F,A) and (G,B) over U denoted  by (F,A)∩̃(G,B) is the soft expert set  (H,C) where  C=A∪B,for allε∈C, and
(3.15)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,F(ε)∩G(ε)ifε∈A∩B.Example 3.19.
Consider Example3.16. Then
(3.16)(F,A)∩̃(G,B)=(H,C)={((e1,p,1),{u1,u4}),((e1,q,1),∅),((e2,q,1),{u3}),((e2,r,1),{u2}),((e3,p,1),{u3,u4}),((e3,r,1),{u4}),((e1,r,0),{u2}),((e2,p,0),{u2,u3}),((e3,q,0),{u3}),((e3,p,0),{u1,u2,u4})}.Proposition 3.20.
If(F,A), (G,B), and (H,C) are three soft expert sets over U,  then(i)
(F,A)∩̃(G,B)=(G,B)∩̃(F,A),(ii)
(F,A)∩̃((G,B)∩̃(H,C))=((F,A)∩̃(G,B))∩̃(H,C).Proof.
The proof is straightforward.Proposition 3.21.
If(F,A), (G,B), and (H,C) are three soft expert sets over U, then(i)
(F,A)∪̃((G,B)∩̃(H,C))=((F,A)∪̃(G,B))∩̃((F,A)∪̃(H,C)),(ii)
(F,A)∩̃((G,B)∪̃(H,C))=((F,A)∩̃(G,B))∪̃((F,A)∩̃(H,C)).Proof.
The proof is straightforward.Definition 3.22.
If(F,A) and (G,B)  are two soft expert sets over U then (F,A) AND (G,B) denoted by (F,A)∧(G,B), is defined by
(3.17)(F,A)∧(G,B)=(H,A×B),
where H(α,β)=F(α)∩̃G(β),for all(α,β)∈A×B.Example 3.23.
Consider Example3.3. Let
(3.18)A={(e1,p,1),(e2,p,0),(e1,r,0),(e2,r,1)},B={(e1,p,1),(e1,r,0),(e2,r,1)}.
Suppose (F,A) and (G,B)  are two soft expert sets over U such that
(3.19)(F,A)={((e1,p,1),{u1,u2,u4}),((e1,r,0),{u1,u2}),((e3,q,0),{u3,u4}),((e2,r,1),{u1,u2,u4})}(G,B)={((e1,p,1),{u1,u3,u4}),((e1,r,0),{u2}),((e2,r,1),{u2,u3})}.
Therefore
(3.20)(F,A)∧(G,B)=(H,A×B)={(((e1,p,1),(e1,p,1)),{u1,u4}),(((e1,p,1),(e1,r,0)),{u2}),(((e1,p,1),(e2,r,1)),{u2}),(((e1,r,0),(e1,p,1)),{u1}),(((e1,r,0),(e1,r,0)),{u2}),(((e1,r,0),(e2,r,1)),{u2}),(((e3,q,0),(e1,p,1)),{u3,u4}),(((e3,q,0),(e1,r,0)),∅),(((e3,q,0),(e2,r,1)),{u3}),(((e2,r,1),(e1,p,1)),{u1,u4}),(((e2,r,1),(e1,r,0)),{u2}),(((e2,r,1),(e2,r,1)),{u2})}.Definition 3.24.
If(F,A) and (G,B) are two soft expert sets then (F,A) OR (G,B) denoted by (F,A)∨(G,B), is defined by
(3.21)(F,A)∨(G,B)=(O,A×B),
where O(α,β)=F(α)∪G(β),for all(α,β)∈A×B.Example 3.25.
Consider Example3.23. Then
(3.22)(F,A)∨(G,B)=(H,A×B)={(((e1,p,1),(e1,p,1)),U),(((e1,p,1),(e1,r,0)),{u1,u2,u4}),(((e1,p,1),(e2,r,1)),U),(((e1,r,0),(e1,p,1)),U),(((e1,r,0),(e1,r,0)),{u1,u2}),(((e1,r,0),(e2,r,1)),{u1,u2,u3}),(((e3,q,0),(e1,p,1)),{u1,u3,u4}),(((e3,q,0),(e1,r,0)),{u2,u3,u4}),(((e3,q,0),(e2,r,1)),{u2,u3,u4}),(((e2,r,1),(e1,p,1)),U),(((e2,r,1),(e1,r,0)),{u1,u2,u4}),(((e2,r,1),(e2,r,1)),{u1,u2,u4})}.Proposition 3.26.
If(F,A)  and (G,B) are two soft expert sets over U, then(i)
((F,A)∧(G,B))c=(F,A)c∨(G,B)c,(ii)
((F,A)∨(G,B))c=(F,A)c∧(G,B)c.Proof.
See Maji et al. [3].Proposition 3.27.
If(F,A), (G,B), and (H,C) are three soft expert sets over U, then(i)
(F,A)∧((G,B)∧(H,C))=((F,A)∧(G,B))∧(H,C),(ii)
(F,A)∨((G,B)∨(H,C))=((F,A)∨(G,B))∨(H,C),(iii)
(F,A)∨((G,B)∧(H,C))=((F,A)∨(G,B))∧((F,A)∨(H,C)),(iv)
(F,A)∧((G,B)∨(H,C))=((F,A)∧(G,B))∨((F,A)∧(H,C)).Proof.
Straightforward from Definitions3.22 and 3.24.


## 4. An Application of Soft Expert Set

Maji et al. [4] applied the theory of soft sets to solve a decision-making problem using rough mathematics. In this section, we present an application of soft expert set theory in a decision-making problem. The problem we consider is as below.Assume that a company wants to fill a position. There are eight candidates who form the universeU={u1,u2,u3,u4,u5,u6,u7,u8}. The hiring committee considers a set of parameters, E={e1,e2,e3,e4,e5} where the parameters ei(i=1,2,3,4,5) stand for “experience,” “computer knowledge,” “young age,” “good speaking,” and “friendly,” respectively.  Let X={p,q,r} be a set of experts (committee members). Suppose(4.1)(F,Z)={((e1,p,1),{u1,u2,u4,u7,u8}),((e1,q,1),{u1,u4,u5,u8}),((e1,r,1),{u1,u3,u4,u6,u7,u8}),((e2,p,1),{u3,u5,u8}),((e2,q,1),{u1,u3,u4,u5,u6,u8}),((e2,r,1),{u1,u2,u4,u7,u8}),((e3,p,1),{u3,u4,u5,u7}),((e3,q,1),{u1,u2,u5,u8}),((e3,r,1),{u1,u7,u8}),((e4,p,1),{u1,u7,u8}),((e4,q,1),{u1,u4,u5,u8}),((e4,r,1),{u1,u6,u7,u8}),((e5,p,1),{u1,u2,u3,u5,u8}),((e5,q,1),{u1,u4,u5,u8}),((e5,r,1),{u1,u3,u5,u7,u8}),((e1,p,0),{u3,u5,u6}),((e1,q,0),{u2,u3,u6,u7}),((e1,r,0),{u2,u5}),((e2,p,0),{u1,u2,u4,u6,u7}),((e2,q,0),{u2,u7}),((e2,r,0),{u3,u5,u6}),((e3,p,0),{u1,u2,u6,u8}),((e3,q,0),{u3,u4,u6,u7}),((e3,r,0),{u2,u3,u4,u5,u6}),((e4,p,0),{u2,u3,u4,u5,u6}),((e4,q,0),{u2,u3,u6,u7}),((e4,r,0),{u2,u3,u4,u5}),((e5,p,0),{u4,u6,u7}),((e5,q,0),{u2,u3,u6,u7}),((e5,r,0),{u2,u4,u6})}.In Tables1 and 2 we present the agree-soft expert set and disagree-soft expert set, respectively, such that if ui∈F1(ε) then uij=1 otherwise uij=0, and if ui∈F0(ε) then uij=1 otherwise uij=0 where uij are the entries in Tables 1 and 2.Table 1
Agree-soft expert set.
Uu1u2u3u4u5u6u7u8(e1,p)11010011(e2,p)00101001(e3,p)00111010(e4,p)10000011(e5,p)11101001(e1,q)10011001(e2,q)10111101(e3,q)00111010(e4,q)10011001(e5,q)10011001(e1,r)10110111(e2,r)11010011(e3,r)10000011(e4,r)10000111(e5,r)10101011cj=∑iuijc1=12c2=3c3=7c4=9c5=9c6=3c7=9c8=13Table 2
Disagree-soft expert set.
Uu1u2u3u4u5u6u7u8(e1,p)00100100(e2,p)11010110(e3,p)11000101(e4,p)01111100(e5,p)00010110(e1,q)01100110(e2,q)01000010(e3,q)11000101(e4,q)01100110(e5,q)01100110(e1,r)01001000(e2,r)00101100(e3,r)01111100(e4,r)01111000(e5,r)01010100kj=∑iuijk1=3k2=12k3=8k4=6k5=6k6=12k7=6k8=2The followingalgorithm may be followed by the company to fill the position.Algorithm 4.1.
(1)
input the soft expert set(F,Z),(2)
find an agree-soft expert set and a disagree-soft expert set,(3)
findcj=∑iuij for agree-soft expert set,(4)
findkj=∑iuij for disagree-soft expert set,(5)
findsj=cj-kj,(6)
findm, for which sm=maxsj.Thensm is the optimal choice object.  If m has more than one value, then any one of them could be chosen by the company using its option.Now we use this algorithm to find the best choices for the company to fill the position. From Tables1 and 2 we have Table 3.Table 3cj=∑iuijkj=∑iuijsj=cj-kjc1=12k1=3s1=9c2=3k2=12s2=-9c3=7k3=8s3=-1c4=9k4=6s4=3c5=9k5=6s5=3c6=3k6=12s6=-9c7=9k7=6s7=3c8=13k8=2s8=11Thenmaxsj=s8, so the committee will choose candidate 8 for the job.



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*Source: 757868-2012-01-04.xml*