A hybrid method for decision making with dependence & feedback under incomplete information

decision


INTRODUCTION
Multi-criteria decision making (MCDM) refers to making preference decisions over the available alternatives that are characterized by multiple, usually conflicting attributes [1].It occurs in a variety of actual situations, such as economic analysis, strategic planning, forecasting, medical diagnosis, supply chain management and many other areas.Due to the increasing complexity of the socio-economic environment, this has made it even more difficult for decision making, which is mainly shown in two aspects.
On the one hand, due to the complexity and the situation of uncertainty in decision, the information about attribute weights provided by decision makers is usually incompletely known.Some of recent research on the topic incorporates generalized interval-valued fuzzy numbers [2], triangular fuzzy number [3], intuitionistic fuzzy set [4,5], 2-tuple linguistic [6] and others.Although these literatures are very good for solving the incomplete weight information under uncertain environment, they are associated with an inherent limitation, which is inadequacy of the parameterization tool associated with these theories.Yet, soft set which was initiated by [7], a new mathematical tool can deal with uncertainties, which is free from the above limitations.
In recent years, the research on soft set theory has achieved great progress in theoretical aspect.At the same time, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making.Maji and Roy [8] introduced the definition of reduct-soft-set and described the application of soft set theory as a problem in decision-making.Mushrif et al. [9] proposed a new classification algorithm of the natural textures, which was based on the notions of soft set theory.Zou and Xiao [10] presented data analysis approaches of soft set under incomplete information.Roy and Maji [11] proposed a novel method of object recognition from an imprecise multi-observer data and a decision making application of fuzzy soft set.Although the algorithm was proved incorrectly by Kong et al. [12], fuzzy soft sets and multi-observer concepts are valuable to successive researchers.Cagman and Enginoglu [13] defined products of soft sets and uni-int decision function.By using these new definitions, they constructed a uni-int decision making method which selected a set of optimum elements from the alternatives.Feng et al. [14] presented an adjustable approach to fuzzy soft set based decision making and enhanced it with illustrations.Although fuzzy soft set has been progressive in decision making, few literatures concentrated on the decision making with incomplete weight information.Therefore, we constructed a decision model based on fuzzy soft set to determine unknown weight in this paper.
On the other hand, MCDM is also involved in determining the optimal alternative among multiple, conflicting and interactive criteria [15].For most previous literature, they assumed that attributes were mutually independent.However, in real-life situation there exist dependence and feedback effects simultaneously among criteria, while making decisions.The analytic network process (ANP) which was proposed in [16,17], overcomes the problem of dependence and feedback effect among criteria or alternatives.Though ANP has been widely used in various applications, there are still two main problems which were highlighted by [18].To deal with this problem, we use FCM to express dependence and feedback effect among criteria in order to overcome the preferential dependent and shortcomings of ANP.
Based on above analysis, it is hard for decision makers to make a good decision using a simple weighted method because of attribute weights with interaction effects and incomplete information.So it will be an interesting and important research topic as few literatures considered them simultaneously.To fill this gap, we presented a hybrid multi-criteria decision making approach based on fuzzy soft set and FCM.The proposed method not only enriches fuzzy soft set theory, but also expands the field of decision making.
As for the remainder of this paper, it is organized as follows: Section 2 will review some basic concepts related to fuzzy soft set and fuzzy cognitive maps.In Section 3, we will provide a new hybrid model to determine the attribute weights considering the incomplete information, the independent and feedback effect among criteria.A fuzzy soft set will also be introduced for multiple attribute decision making problems.Next, a case study is developed to demonstrate on how to apply the proposed approach in Section 4. Finally, conclusion and remarks will be in Section 5.

THEORETICAL BACKGROUND
In this section, we will briefly review the basic theoretical background on fuzzy soft set and fuzzy cognitive maps.The values of the connection weights can be organized in a matrix, The reasoning process of FCM can be expressed as , [19]: ( ( ) Among them, ( 1)    If the reasoning process achieves one of the following three states, one has reached a steady state, and ends the iteration: output concept value has stabilized at a fixed value; changes of the values have shown signs of cyclical; chaotic state has appeared, that is, the concept value is uncertain and random.

A HYBRID AND RATIONAL METHOD FOR MULTI-CRITERIA DECISION MAKING BASED ON FUZZY SOFT SET AND FCM
This section presents a hybrid and rational approach to tackle multiple criteria decision making problems with incomplete weight information in the context of fuzzy soft sets.Suppose that there exist an alternative set U={h 1 ,h 2 ,…,h n }, consisting of n non-inferior alternatives, and an attribute set E={e 1 ,e 2 ,…,e m }.Each alternative is assessed on the m attributes.The decision problem is to select a most preferred alternative from set U based on the overall assessments of all alternatives on the m attributes.Due to the complexity and the situation of uncertainty in decision, the information about attribute weights provided by the decision makers is usually incompletely known and the dependent and feedback effect among criteria cannot be ignored.
In the following, we proposed a hybrid and rational method that offers a possibility for handing these issues simultaneously.The framework of decision making can be shown in Figure 2.

A rational model for determining initial criteria weight
In this section, we will discuss all kinds of optimization models considering decision makers' rationality to determine the weights of attributes.
Case 1: The information about attribute weights is incompletely known 1) The decision maker is completely optimistic Due to the complexity and uncertainty of decision situation, the information about attribute weights provided by decision makers is usually incompletely known.Under [20] and [21] inspiration, we establish the following optimization model to minimize ( ) )( ( ( ( Subject to: Form 1: A weak ranking: {w j1 ≥ w j2 }, j 1 ≠j 2 ; Form 2: A strict ranking: Form 3: A ranking with multiples: Based on the above optimization model, we can see that the derived weight vector is overestimated because every alternative is often given its most ideal weighted condition. 2) If the decision maker is completely pessimistic, we used optimization model which minimized the deviation from NIS for all alternative.We establish the following optimization model to minimize ( ) )( ( ( ( Subject to:  ( , ,..., )

A method for determining criteria global weight
In this section, we will determine the criteria global weight.The procedure is shown in the following steps: Step 1: Depict the fuzzy cognitive maps to indicate the influence among criteria by the experts and determine the connection matrix W on the basis of FCM.
Step 2: Learn the connection matrix W using PSO (particle swarm optimization) algorithm and calculate Eq. ( 4) to obtain the steady-state matrix W*.
Step 3: Derive the global weight vector.In order to derive the global weights, we should first normalized the local weight vector ( ) initial w and the steady-state matrix (W*) as follows [18] 1 where λ is the largest element of w initial and γ is the largest row sum of W*. we can the global vector by using the following weighting equation [20]:

Evaluation and selection of alternatives
The evaluation and selection of alternative is conducted in this section.Here we applied fuzzy soft set, which is a novel mathematical tool for dealing uncertainty, to evaluate alternative.The detailed procedures are as follows: Step 1: Construct the resultant weighted fuzzy soft set ( ,( ))  In this section, we illustrate this hybrid evaluation and selection process by using a case study (adapted from [23]) as an example.A high-tech company which manufactures electronic products intends to evaluate and select a supplier of USB connectors from four suppliers ( 1 2 3 4 , , , h h h h ) conforming to the basic conditions of choice.In order to select the most suitable candidate, the decision makers take four attributes into account.They are the strong ability of delivery management 1 e , the high integrated service capability 2 e , the high quality management capability According to the market forecast, the attribute value of each alternative is expressed as fuzzy soft set and the decision making data are shown in Table 1.The structure of this performance evaluation problem is shown in Figure .3.The best supplier will be according to the above decision-making information.The proposed method is applied to this problem and the computational procedure is summarized as Figure 3.
Step 1: Identify the initial of attributes cording to the incomplete weights information.Step 4: According to Eq.( 11),( 12),( 13), we can obtain the final global weights and calculate the normalized global weights: , , , which is shown in Table 2.  , and thus the most desirable supplier is h 4 Case 2: The information about the attribute weights is completely unknown Utilize Eq.( 10) and get the attribute initial weights., and thus the most desirable supplier is h 4 .In this paper, we have investigated a hybrid approach to tackle multiple criteria decision making problems with incomplete weight information in the context of fuzzy soft sets.To determine the criteria weights, the proposed approach consists of two stages.Firstly, in order to derive the initial weights, the novel decision making models determining the unknown weight vector have been developed.Secondly, we used FCM to deal with dependent and feedback effect among criteria.After completing these two stages, we can derive the global criteria weights.Next we will apply fuzzy soft sets in evaluating and selecting the most desirable alternative.Finally, a case study is presented to examine the practicality of the proposed model.The proposed hybrid method has a clear logic and has less loss of information than other literatures.

Figure 1 .
Figure 1.A simple fuzzy cognitive map

A
is the value of concept C j at simulation step k; w ij is the weight of the interconnection between concept C j and C i ; f is a threshold function, which is used to ensure the node concept value in the interval [0,1].The threshold function f can be bivalent (f (x)=0 or 1), trivalent (f (x)= -1, 0 or 1), tangent hyperbolic (f (x)=tanh(x)) or the unipolar sigmoid function (f (x)=1/(1+e -cx )), where c 0 determines the steepness of the continuous function f.The sigmoid function is typically used when the concept interval is [0,1].Hyperbolic function is used when concepts can be negative and their values belong to the interval [-1,1].Thus the selection of threshold function depends on the description of concepts.

1 2 1 (
, ,..., ) , 0, 1, 2,..., , 1 is a set of constraints concerning the unknown weight information.For the sake of simplicity, they can only take the following forms, for i≠j: model, the derived weight vector is too low estimated because every alternative is often given smallest weighted So we establish the optimization model:

Step 2 :
According to the Eq.(3), we can calculate the comprehensive

Figure 3 .
Figure 3.An fuzzy cognitive map