Information Technology Journal

Year: 2014 | Volume: 13 | Issue: 5 | Page No.: 853-858
DOI: 10.3923/itj.2014.853.858

Application of Multiple Attribute Decision-making Approaches with Interval Numbers in Fields of Investment Decision

Sha Fu, Jian Chen, Hangjun Zhou, Dan Song and Yezhi Xiao

Abstract: With respect to the problem of multiple attribute decision-making with incomplete information on attribute weights to which the attribute values are given in terms of interval numbers, a decision method based on projection technique and prospect theory is proposed. The value function is built based on the distance formula of interval numbers, conbining with the idea of projection technique the multi-goal programming model is established to get a more accurate decision weights by using different approch on gains and lossed. Furthermore, the optimal solution based on the value of all programs sorts is figured out in the method with the comprehensive prospect values of each program calculated with the value function and decision-weighting function. Eventually, the feasibility and effectiveness of the proposed method is reasonably verified through solving a typical multi-attribute decision-making problem about the property buyers decision.

Keywords: multi-attribute decision-making, interval number, projection, Prospect theory and alternative sorts

INTRODUCTION

As an important part of modern decision-making science, MADM (Multiple Attribute Decision Making) is mainly used to solve multiple attributes in considering the case to choose the best alternatives or alternative sorts decision problems. This theory and method is widely used in engineering, technical, economic, military, management and many other fields. The complexity of the objective world and random of the decision maker, often lead to fuzzy decision data. In practice, decision-makers are hard to give a certain value for the evaluation of some criterions, in most cased they only give a range, so that the guideline value are usually in the form of interval number. Wan (2009) did a detailed study of Interval Multi-Attribute Decision-Making attitude index, for the multiple attribute decision making problems of both decision-makers preference information and attribute values were interval numbers. Sun and Zhang (2011) proposed interval fuzzy VIKOR method, VIKOR method uses linear normalization matrix during process of decision making. Interval numbers ranking is determined by the index of decision-maker’s attitude, the compromise solution is obtained under the condition of acceptable advantage and acceptable stability in decision making.

Currently, many scholars are give positive attention on the Multi-attribute Decision-Making problems, but this targeted decision analysis method are also rare. Zhang and Fan (2012) proposed a decision analysis method based on prospect theory for the risk-based hybrid multi-attribute decision making problems that the decision-makers give desired information. Fan et al. (2012) proposed a decision analysis method based on cumulative prospect theory for the hybrid multiple attribute decision making problems with policy makers expect. Wang et al. (2009) established the multi-objective selection model of travelers under uncertainty conditions based on prospect theory. Junhua and Qi (2011) proposed a multiple criteria decision making method based on prospect theory, for the uncertainty decision making problems when the guidelines preference value were interval numbers. Yang and Wanhua (2009) proposed a decision making method based on linear programming and fuzzy vector projection and established a linear programming model based on the weighted attribute value maximizing deviations. Wei and Yi (2007) proposed an interval evaluation method based on the projection method and given the calculation steps to solve the multiple attribute decision making problems of index value were interval numbers. Wu et al. (2008) established a nonlinear quadratic programming model to obtain attribute weights based on the projection method, to sort the programs in accordance with the projection in the interval ideal point. In a word, the multiple attribute decision making problems with decision makers desired information were better solved, As opposed to a poorer solution to multiple attribute decision making problems taking into account the makers’ risk attitudes and which attribute value were interval numbers, the weight information were incomplete.

Based on a comprehensive study of public achievements, this study proposed a decision method based on projection technique and prospect theory and the feasibility and effectiveness of the proposed method is verified through solving a typical multi-attribute decision-making problem.

MATERIALS AND METHODS

Prospect theory: "Prospect Theory" was put forward in 1979 by the Nobel laureate in economics Kahneman etc. The theory made a seminal contributions on cross scientific research of judgment and decision making, it mainly used in behavioral economics, management, financial investment and other fields. Prospect theory use value function v and decision weighting function π replace the expected utility and subjective probability in traditional expected utility theory to fully embodying decision-maker’s irrationality, so the decision were more consistent with the inherent habit of thinking in humanity itself (Li et al., 2012).

Kahneman and Tversky developed a new form of the value function v, it better reflected the preferences characteristics of decision-makers, as they were risk seeking when facing gain and facing averse when facing lose, thus its application is widely. The value function given by Tversky and others were power function, specifically stated as:

In the equation, Δx is the face value of the gains and losses, revenue is positive and loss is negative; α and β individually represent the unevenness degree of gains and losses on the regional value power function. 0<α<1, 0<β<1, if α and β are bigger, it shows the decision makers are more inclined to risky; θ is loss aversion coefficient, if θ>1, the decision makers are more sensitive for the loss (Ma et al., 2011).

Tversky and Kahneman (1992) and others believe that decision weights were some kind of subjective judgment given by decision makers according to the event's results appear probability p and its function π is expressed as follows:

Prospect value is jointly determined by the value function and decision weight function, that is:

Interval number and its calculation: Assume = [a^L, a^U] = {x|a^L≤x≤a^U}, a^L, a^U∈R; in which a^L represents the lower limit, a^U represents the upper limit and put closed interval = [a^L, a^U] on real number axis called interval number. In addition, if a^L = a^U, then, degenerates to a particular interval number, that is the real number (Su et al., 2010).

Assume = [a^L, a^U] and = [b^L, b^U] were positive interval numbers, k is a positive real number, the number of interval arithmetic rules are:

Definition 1: Assume interval number = [a^L, a^U] and = [b^L, b^U], if the norm ||-|| = |b^L-a^L|+|b^U-a^U|, call d (, ) = ||-|| is the distance of interval numbers and . Obviously, the greater the distance of and , d (, ) will be greater.

Definition 2: Assume interval number = [a^L, a^U] and = [b^L, b^U], in terms of interval number = [a^L, a^U] as a reference point, the prospect value function of interval number = [b^L, b^U] is:

Definition 3: Assume two interval numbers = [a^L, a^U] and = [b^L, b^U] and l_a = a^U-a^L, l_b = b^U-b^L, call:

is the possible degrees of (Ren and Gao, 2010).

Basic principle of projection
Definition 4: Assume two vectors were a = [a₁, a₂,..., a_m] and b = [b₁, b₂,...,b_m], call:

is the projection of vector a on vector b.

In Eq. 6, the left part of multiplication is cosine of the angle between the two vectors, the right part is the size of the comparison vector mode; the greater the value of Prjb(a), indicating that it will be closer between vectors a and b, on the contrary show more separation between them.

According to the analysis in literature about interval number and interval vector projection (Xu and Da, 2003), it can get the general interval vector projection equation is:

where, a_j = [a₁, a₂,…, a_m], a_j = [a_j^L, a_j^U]; b_j = [b₁, b₂, ..., b_m], b_j = [b_j^L, b_j^U]; 1≤j≤m.

SORTING METHOD BASED ON PROSPECT THEORY AND PROJECTION

In case of each attribute weights incomplete on information and its attribute values were interval numbers, introduce prospect theory into interval Multiple Attribute Decision Making, proposed the interval number Multi-attribute Decision-Making method based on this theory and the specific steps are as follows.

The most common attribute types are cost type and benefit type, the benefit type attribute value is the bigger the better and the cost type attribute value is the smaller the better. Where, I₁, I₂ respectively refers to the subscript set of benefit type attribute and cost type attribute (Yu et al., 2011), then establish the normalized interval numbers decision matrix method as follows.

For the benefit type attributes are:

For the cost type attributes are:

The positive ideal solution:

The negative ideal solution:

Wherein, i∈N, j∈N

The distance set of programs A_i to the positive ideal solution is:

The distance set of programs A_i to the negative ideal solution is:

Clearly, the more program A_I (i∈N) close to interval ideal point the better, (That is, d (A_i, ) is smaller, program A_i (i∈N) will be better). As the interval distance of program A_i (i∈N) to interval ideal point are interval numbers, we can compare the size of the interval numbers sort and can keep a large extent about the uncertainty of decision-making information, also in line with its environmental features.

Use negative ideal program as a reference point, the program is gains, thereby obtaining a positive prospect value:

wherein, e_j is the lower limit of the weight w_j, f_j is the upper limit of the weight w_j.

If the importance degree of these two objectives were consistent, the model I can be transformed into the following single-objective programming model, denote as model II:

Model II can get the optimal solutions by simulation calculation of Lingo and Visual C++ mixed programming, that is obtain the attribute weight vector w* = (w₁*, w₂*,…, w_n*)^T met the objective function max H (w).

wherein, π⁺ (w_j) and π¯ (w_j) were non-linear weighting function of gains and losses, r and δ were the bending process of decisions weighting function of gains and losses.

RESULTS AND DISCUSSION

Let’s take the investment decision problem (house purchase) as an example: There are 6 category residential area (object), expressed in A = {A₁, A₂, A₃, A₄, A₅, A₆}, respectively, the index set which buyers mainly considerate C = { C₁, C₂, C₃, C₄ }, respectively mean the weighted total distance C₁ (km) conform from the address to the school, the workplace, supermarkets, banks, hospitals and other locations, the average price of Property C₂ (unit area Price: ten thousand Yuan m^-2), Standard Index C₃ (the level) conform from integrated environment and lighting, housing area C₄ (m²) (Yang and Fang, 2012).

The direct observation number quantitative indicators of buyers about residential areas as shown in Table 1, to determine the objects can available for purchase.

In the above indicators, the weighted total distance C₁ and average price of Property C₂ are cost type index, housing standard index C₃ and housing size C₄ are benefit type index. As the known attribute weight information is: w = (w₁, w₂, w₃, w₄), each attribute weights range in incomplete information roughly are: 0.35≤w₁≤0.4, 0.15≤w₂≤0.25, 0.1≤w₃≤0.25, 0.2≤w₄≤0.3 (Fu et al., 2012) and:

The positive prospect value is:

By Lingo and Visual C++ mixed programming method to solve the model and get the weight vector of each attribute as follows:

Finally, according to the comprehensive future value of each program, obtain the sorted results were: A₃A₄A₅A₆A₂A₁. The results of this study were completely consistent with the literature (Yang and Fang, 2012), it proved this method is feasible and effective and The step and process of calculation and analysis shows that this method is more reasonable compared to other method in similar documents.

CONCLUSION

According to the prospect theory, take the decision maker’s positive and negative ideal point to attributes as reference point, develop the calculation method for gains and losses of interval numbers attribute value and sort the programs by calculating the comprehensive future value of each program. Introduced in the projection ideal by the determination of attribute weights and established a multi-objective programming model to obtain the corresponding weight. The decision making method proposed in this study has clear concept, is easy to understand and has good maneuverability and practicality. It offers a new way to solve multiple attribute decision making problems with interval numbers and has a strong practical significance and practical value.

ACKNOWLEDGMENTS

This study was supported by the Science and Technology Plan Project of Science and Technology Department of Hunan Province (Grant No. 2013FJ3079) and also supported by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12C0559). Supported by the construct program of the key discipline in hunan province.

REFERENCES