Propagation of uncertainty by the possibility theory in Choquet integral ...

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 55, NO. 3, JUNE 2006

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Propagation of Uncertainty by the Possibility Theory in Choquet Integral-Based Decision Making: Application to an E-Commerce Website Choice Support Afef Denguir-Rekik, Gilles Mauris, and Jacky Montmain

Abstract—This paper is concerned with some aspects of uncertainty evaluations in multicriteria decision making (MCDM) in the framework of e-commerce website recommendation. The emphasis is on the interest of handling uncertainty with possibility distributions in the MCDM process where evaluations coming from the users present variability. Thus, the authors consider the propagation of possibility distributions through the multicriteria aggregation made by a Choquet integral that takes into account the interactions between the decision-making criteria. To support the recommendation process, location and uncertainty indicators of possibility distributions are defined, as well as their marginal contributions to the aggregated result. The proposed approach is applied here to the problem of the choice of an e-commerce website for purchase purposes, but it can also be used for dealing with uncertainty in other complex problems. Index Terms—Choquet integral aggregation, decision-making support system, e-commerce website choice, possibility theory, uncertainty propagation.

πi πy Π(A) N (A) P (A) F P ∗ (A) P∗ (A) E ∗ (π)

N OMENCLATURE = {s1 , . . . , sl , . . . , sq }. Set of q e-commerce websites to be evaluated. = [ml1 , ml2 , . . . , mln ]. Vector of satisfaction degrees ml of sl with respect to the n considered elementary criteria. Overall evaluation of ml . yl CI(ml ) 2-Additive Choquet integral of the elements of ml : y l = CI(ml ). Coefficient of the importance of the criterion i (also νi called Shapley coefficient). Coefficient of interaction between the criteria i and j. Iij H(ml ) Simplex domain corresponding to the ranking of the elements of ml . Linear coefficient of the importance of criterion i in ∆µli the simplex domain defined by ml . S

E∗ (π)

MD(π) ∆(π) C(πi ) πyk πik C(πik ) C∆i

Manuscript received July 4, 2005; revised March 3, 2006. A. Denguir-Rekik and J. Montmain are with the Commissariat à l’Energie Atomique Site EERIE, URC Ecole des Mines d’Alès, Nîmes 30035, France (e-mail: [email protected]; [email protected]). G. Mauris is with the Laboratoire d’Informatique, Systèmes, Traitement de l’Information et de la Connaissance, Université de Savoie, Annecy BP 806 74016, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2006.873803

Possibility distribution associated to the evaluation mi of criterion i. Possibility distribution associated to the overall evaluation y. = supx∈A π(x). Possibility measure of the set A for a possibility distribution π. = 1 − supx∈A π(x). Necessity measure of the set A fora possibility distribution π. = x∈A p(x)dx. Probability measure of the set A for a probability x distribution p. F (x) = −∞ p(x)dx. Cumulative probability function associated to a probability distribution p. Upper probability measure of the set A defined by the possibility measure Π(A). Lower probability measure of the set A defined by the necessity measure N (A). +∞ = −∞ xdF∗ (x)dx. Mean value of the upper probability distribution defined by a possibility distribution +∞ π. = −∞ xdF ∗ (x)dx. Mean value of the lower probability distribution defined by a possibility distribution π. = (E ∗ (π) + E∗ (π))/2. Location indicator of a possibility distribution π. = E ∗ (π) − E∗ (π). Uncertainty indicator of a possibility distribution π. Contribution of the possibility distribution πi to the overall possibility distribution πy . Decomposition part of the overall possibility distribution πy in the simplex domain k. Decomposition part of the elementary possibility distribution πi in the simplex domain k. Contribution of the possibility distribution πik to the possibility distribution πyk . Contribution of the elementary uncertainty indicator ∆(πi ) to the overall uncertainty indicator ∆(πy ). I. I NTRODUCTION

I

N RECENT years, there has been a growing interest in the need for designing intelligent systems to address complex problems in engineering, business, and social applications. To

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deal with the considered applications, there is often a need to fuse information coming from a variety of sources (e.g., physical sensors, measurement systems, and experts’ opinions) and having different formats (e.g., numerical, statistical, and linguistic) [1]. Moreover, due to their nature, the sources are different when it comes to the reliability and uncertainty of the information produced. Therefore, one of the most challenging issues in complex systems is to handle real-world uncertainties effectively. In this line, this paper is a contribution to the estimation and propagation of uncertainty in complex decision problems. Indeed, the final aim of any measurement process rests on the acquisition of evidence that enables understanding and formulating decisions about some problems [2]. More specifically, this paper addresses the problem of high-quality decisions in web-shopping activities and makes propositions to improve the quality of information available to consumers. Indeed, in this emerging domain as well as in many other areas such as medicine, geology, and robotics, human experts are essential, but they are not able to solve all the numerous related problems. It is therefore desirable to develop computer-based systems that incorporate available knowledge to provide high-level advice to people trying to solve these problems. In this sense, the proposed methodology can be applied to other decisionmaking situations within the context of justification of decisions with regard to the problem of uncertainty and the risks involve. In fact, at the basic level, the assessment and presentation of the effects of uncertainty for complex systems can be viewed in a generic way as the study of functions of the form y = f (m) where the function f represents the model, m = [m1 , m2 , . . . , mn ] is a vector of basic information inputs, and y is the high-level information output. In practice, f can be quite complex (i.e., a nonlinear function, a computer program involving complex processing, e.g., fuzzy, neuronal, and evolutionary). Here, we study the case where f is a Choquet integral. Indeed, in many problems, the best known and most extensively used function f is a linear function, e.g., a weighted mean, but it does not allow considering interactions between variables that are often present in applications. Therefore, in this paper, to improve the decision-making support, we first propose to consider the 2-additive Choquet integral that allows considering mutual interactions for pairs of variables. Next, the goal of an uncertainty analysis is to determine the uncertainty of y that results from the uncertainty in the elements of m and further how the uncertainty of m affects the uncertainty of y through f (sensitivity analysis). To carry out this analysis, the uncertainty in the elements mi must be characterized. Here, the elements mi are assumed to be characterized by a possibility distribution πi . This uncertainty representation will not be deeply discussed here with the other ones (e.g., probability, fuzzy random variables, and intervals [3]–[8]), but some reasons will be referred to in Section IV through our application domain. Here, in the particular application context, the elements of m are criteria evaluations (within the range [0, 20]) provided by the users of e-commerce websites, and y is the

overall evaluation also within the range [0, 20]. Providing the decision maker with a global uncertainty evaluation (e.g., a propagated possibility distribution) guarantees that no information is lost in the information fusion processing. However, it is not necessarily easily understandable. As a decision-making support system is aimed at representing and improving the way people use their reasoning and data processing abilities, significant and relevant indicators for aiding the decision maker must be added. The proposed indicators are based upon the descriptive measures on the propagated distribution such as location and uncertainty, which are reinterpreted in the framework of decision making. Indeed, the analysis is viewed under a linear decomposition of the indicators for y into components derived from the indicators of m; the size of these components provides the decision maker with indications of the importance and variability of the elements of m. This paper is organized as follows. Section II recalls the propagation mechanism of possibility distributions into a 2-additive Choquet integral. Section III addresses the issue of how to support the decision makers in this multicriteria decision-making (MCDM) possibility representation of uncertainty. In this view, we propose to describe possibility distributions in terms of interpretable indicators. Only location and uncertainty indicators are considered in this paper. The interpretation in terms of marginal contributions of each elementary indicator on the overall indicators constitutes the basis of the proposed recommendation functionalities. Section IV illustrates our purpose with e-recommendation applications. II. 2-A DDITIVE C HOQUET I NTEGRAL A GGREGATION OF P OSSIBILITY D ISTRIBUTIONS A. Choquet Integral Aggregation Often, the decision maker problem is to make a tradeoff between the different criteria evaluations involved in the considered problem. This leads to consider compromise operators (the aggregated evaluation is between the minimum and the maximum of the elementary evaluations) and thus to disqualify those that model severe or tolerant behavior (such as t-norms and t-conorms). Thus, according to these considerations, the operators of the Choquet integral CI family [9] are particularly well adapted because they include a lot of generalized mean operators (i.e., those included between the min and the max operators). Moreover, they can be written under the form of a conventional weighed mean modified by effects coming from the interactions between elementary evaluations. According to the application context, we will consider here a particular case of Choquet fuzzy integrals known as the 2-additive Choquet integral that only considers interactions by pairs. In the following, the definition of the 2-additive Choquet integral and its principal properties will be briefly recalled. The expression of the 2-additive Choquet fuzzy integral can be written in the form

CI(ml ) =

n i=1

νi mli −

1 l Iij mi − mlj . 2 i>j

(1)

DENGUIR-REKIK et al.: POSSIBILITY THEORY IN CHOQUET INTEGRAL-BASED DECISION MAKING

Fig. 1.

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Example of uncertain elementary evaluations.

This equation involves two types of parameters [10]: 1) weight of each elementary performance expression in relation to all the other contributions to the overall evaluation n by the so-called Shapley coefficients νi that satisfy i=1 νi = 1, which is a natural condition for decision makers; 2) interaction coefficients Iij of any pair of performance criteria (i, j) that range within [−1, 1], which are listed as follows: a) positive Iij implies that the criteria are complementary (positive synergy); b) negative Iij implies that the criteria are redundant (negative synergy); c) null Iij implies that no interaction exists and the criteria are independent; thus, νi s act as weights in acommon weighed mean [(1) becomes CI(ml ) = n l i=1 νi mi ]. An important point is that the CI has a linear form on the simplex domain H(ml ) corresponding to the ranking defined by the elements of ml . The CI can thus be written as [11] CI(ml ) =

n

∆µli · mli

(2)

i=1

with ∆µli = νi +

1 1 Iij − Iij . 2 j>i 2 j C∆(P dtv) > C∆(DT ime) > C∆(P rice). Thus, customers are aware of the possible problems with the payment security. For the website manager of CDiscount.com, one recommendation is to improve services in terms of payment security. The uncertainty of delivery time is higher but has less influence on the overall uncertainty.

DENGUIR-REKIK et al.: POSSIBILITY THEORY IN CHOQUET INTEGRAL-BASED DECISION MAKING

TABLE VIII INDICATORS OF Cdiscount.com

TABLE IX INDICATORS OF Amazon.com

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we have proposed to associate uncertainty indicators to each distribution to give the user an idea about the variability of the evaluations of other people. Furthermore, these indicators allow evaluating the impact of the uncertainty associated with each criterion on the decision-making process and explaining how the uncertainty inherent to a criterion can contribute to the final result. For the sake of illustration, we have considered a process aimed at supporting customers in the choice of a suitable e-retailer for their purchase, but the methodology can also be used for dealing with uncertainty in other complex systems. For the time being, we have only considered monomodal possibility distributions. Work concerning the case of bimodal possibility distributions is currently in progress, which will allow considering controversial aspects of customers’ evaluations. A PPENDIX A

TABLE X INDICATORS OF Priceminister.com

We have for Amazon.com C∆(P ytsec) > C∆(P dtv) > C∆(P rice) > C∆(DT ime). Note that once the criteria payment security and the product variety have the same evaluations, they have different uncertainty contributions due to their varying importance in aggregation. To have a global ranking of these different websites, one issue is to arrange them according to their location values MD. Thus, we have Amazon.com Priceminister.com CDiscount.com (where is a ranking operator). Moreover, if we look at their uncertainty indicator, in this particular case, we find the same ranking. Thus, the recommendation to the customers is to choose the Amazon.com website for their purchase, even though the delivery time is bad for this site. Thus, if the customers’ requirements regarding delivery time are higher than the population average, they are advised to select Priceminister. In fact, all these tables allow customers to adjust their choices according to the additional pieces of information they contain. V. C ONCLUSION This paper is concerned with some aspects of uncertainty in an MCDM process. More specifically, we have emphasized the interest of considering possibility distributions instead of precise quantitative evaluations. A method for propagating these possibility distributions using generalized weighed mean aggregation operators such as the Choquet integral has been discussed. It allows taking interactions into account. In addition,

Proposition 1: Let us consider a piecewise linear possibility distribution π. This distribution can thus be written as π = k=1,...,p π k , where π k are the linear adjacent possibility (not necessarily normalized); then, ∆(π) = p distributions k ∆(π ). k=1 Proof: Let us consider a generic distribution π plotted in Fig. 4. Let us first define E ∗1 and E∗1 for π 1 that is a nonnormalized possibility distribution. In this case, we take for dF1∗ the following degenerated probability distribution that is the weighed sum of a uniform probability distribution and a Dirac one, i.e., dF1∗ (x) = π(a1)U[a,a1] (x) + (1 − π(a1))δ(x − a1). Thus, E∗1 = π(a1)(a1 + a)/2 + (1 − π(a1))a1. Obviously, E ∗1 = a1. With the same reasoning, we have for 3 π the expressions E∗3 = c3 and E ∗3 = π(c3)(c3 + d)/2 + (1 − π(c3))c3. For the distribution π 2 , we have dF2∗ (x) = (1 − π(a1)) × U[a1,b] (x) + π(a1)δ(x − a1) and dF∗2 (x) = (1 − π(c3)) × U[c,c3] (x) + π(c3)δ(x − c3). Therefore, E∗2 = (1 − π(a1))(a1 + b)/2 + π(a1)a1 and E ∗2 = (1 − π(c3))(c + c3)/2 + π(c3)c3. For the whole distribution π, applying the definitions of E ∗ and E∗ leads to E∗ = π(a1)(a + a1)/2 + (1 − π(a1))(a1 + b)/2 and E ∗ = (1 − π(c3))(c + c3)/2 + π(c3)(c3 + d)/2. Thus, E∗ = E∗1 − (1 − π(a1))a1 + E∗2 − π(a1)a1, which gives E∗ = E∗1 − a1 + E∗2 = E∗1 − E ∗1 + E∗2 . In other respect, E ∗ = E ∗2 − π(c3)c3 + E ∗3 − (1 − π(c3))c3, which gives E ∗ = E ∗2 + E ∗3 − c3 = E ∗2 + E ∗3 − E∗3 . In conclusion, we have E ∗ − E∗ = E ∗1 − E∗1 + E ∗2 − E∗2 + ∗3 E − E∗3 , and finally, ∆(π) = ∆(π 1 ) + ∆(π 2 ) + ∆(π 3 ). This result can be easily generalized for distributions having any number of linear pieces. A PPENDIX B Proposition 2: Let us consider the propagated possibility distribution πy and its associated uncertainty indicator ∆(πy ); then, ∆(πy ) = pk=1 ni=1 ∆µki ∆(πik ). p Proof:k From Proposition 1, we have ∆(πy ) = k=1 ∆(πy ).

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Fig. 4. Example of possibility distribution decomposition.

Then, due to the linearity in each simplex domain, the uncertainty indicator of the partial propagated distribution πyk is then given as ∆(πyk ) = ni=1 ∆µki ∆(πik ). Therefore, ∆(πy ) = pk=1 ni=1 ∆µki ∆(πik ) holds. R EFERENCES [1] L. Valet, G. Mauris, and P. Bolon, “A statistical overview of recent literature in information fusion,” IEEE Aerosp. Electron. Syst. Mag., vol. 16, no. 3, pp. 7–14, Mar. 2001. [2] D. Hall, Mathematical Techniques in Multi-Sensor Data Fusion. Norwood, MA: Artech House, 1992. [3] S. Rabinovich, Measurement Errors and Uncertainties. New York: Springer-Verlag, 1995, 296 pages. [4] Guide for the Expression of Uncertainty in Measurement, 1993, ISO 1993, 99 pages. [5] G. Mauris, V. Lasserre, and L. Foulloy, “A fuzzy approach for the expression of uncertainty in measurement,” Measurement, vol. 29, no. 3, pp. 165–177, 2001. [6] S. Ferson and L. Ginzburg, “Different methods are needed to propagate ignorance and variability,” Reliab. Eng. Syst. Saf., vol. 54, no. 2/3, pp. 133–144, Nov./Dec. 1996. [7] M. Urbanski and J. Wasowski, “Fuzzy approach to the theory of measurement inexactness,” Measurement, vol. 34, no. 1, pp. 67–74, 2003. [8] A. Ferrero and S. Salicone, “The random-fuzzy variables: A new approach to the expression of uncertainty,” IEEE Trans. Instrum. Meas., vol. 53, no. 5, pp. 1370–1377, Oct. 2004. [9] M. Grabisch, “The application of fuzzy integrals in multi-criteria decision making,” Eur. J. Oper. Res., vol. 89, no. 3, pp. 445–456, 1996. [10] ——, “k-ordered discrete fuzzy measures and their representation,” Fuzzy Sets Syst., vol. 92, no. 2, pp. 167–189, Dec. 1997. [11] A. Akharraz, J. Montmain, and G. Mauris, “A project decision support system based on an elucidative fusion system,” in Proc. 5th Int. Conf. Inf. Fusion, Annapolis, MD, 2002, pp. 593–599. [12] M. Grabisch, H. T. Nguyen, and E. A. Walker, Fundamentals of Uncertainty Calculi With Applications to Fuzzy Inference. Dordrecht, The Netherlands: Kluwer, 1994. [13] T. Kroupa, “On application of Choquet integral in possibilistic information theory,” Neural Netw. World, vol. 13, no. 5, pp. 541–548, 2003. [14] R. Yang, Z. Wang, P. Heng, and K. Leung, “Fuzzy numbers and fuzzification of the Choquet integral,” Fuzzy Sets Syst., vol. 153, no. 1, pp. 95–113, Jul. 2005. [15] L. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst., vol. 1, no. 1, pp. 3–28, Jan. 1978. [16] A. Akharraz, J. Montmain, and G. Mauris, “Elucidation and risk expressions of a movie recommendation based on a multi-criteria aggregation with a Choquet integral,” in Proc. Int. Conf. IPMU, Perugia, Italy, 2004, pp. 617–624. [17] A. Akharraz, J. Montmain, A. Denguir, and G. Mauris, “Information system and decisional risk control for a cybernetic modeling of project management,” in Proc. 5th Int. Conf. Comput. Sci. MCO, Metz, France, 2004, pp. 407–414. [18] H. Prade and D. Dubois, “The mean value of a fuzzy number,” Fuzzy Sets Syst., vol. 24, no. 3, pp. 279–300, Dec. 1987.

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Afef Denguir-Rekik was born in Tunis, Tunisia, in 1978. She received the Master’s degree in 2002 from the Faculté des Sciences de Tunis, Tunis, and is currently working toward the Ph.D. degree at the University of Savoie, Annecy, France, and at the Génie Informatique et Ingénierie de Production, Ecole des Mines d’Alès, Nimes, France. Her research interests include fuzzy logic and possibility theory for multicriteria decision making.

Gilles Mauris received the Master’s degree from the Ecole Normale Supérieure de Cachan, Cachan, France, in 1985 and the Ph.D. degree from the University of Savoie, Annecy, France, in 1992. He is currently an Associate Professor of electrical engineering at the Laboratoire d’Informatique, Systèmes, Traitement de l’Information et de la Connaissance, University of Savoie. His research interests include fuzzy logic and possibility theory for information processing in the instrumentation and measurement area.

Jacky Montmain was born in Lyon, France, in 1964. He received the Master’s degree from the Ecole Nationale Supérieure d’Ingénieurs Electriciens de Grenoble, Grenoble, France, in 1987 and the Ph.D. degree from the National Polytechnic Institute, France, in 1992, both in control theory. He is currently a Professor at the URC Ecole des Mines d’Alès, Nîmes, France. His research interests include the application of artificial intelligence techniques to model-based diagnosis and supervision and multicriteria and fuzzy approaches to decision making.