Péter Majlender

A possibilistic approach to selecting portfolios with highest utility score

The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz, has been one of the most important research fields in modern finance. In this paper we will assume that: (i) each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of the portfolios; and (ii) the rates of return on securities are modelled by possibility distributions rather than probability distributions. We will present an algorithm of complexity o(n3) for finding an exact optimal solution (in the sense of utility scores) to the n-asset portfolio selection problem under possibility distributions.

On weighted possibilistic mean and variance of fuzzy numbers

Dubois and Prade defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. Carlsson and Fuller defined an interval-valued mean value of fuzzy numbers, viewing them as possibility distributions. In this paper, we shall introduce the notation of weighted interval-valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval-valued probabilistic mean. We shall also introduce the notations of crisp weighted possibilistic mean value, variance and covariance of fuzzy numbers, which are consistent with the extension principle. Furthermore, we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory.

A fuzzy approach to R&D project portfolio selection

A major advance in the development of project selection tools came with the application of options reasoning in the field of Research and Development (R&D). The options approach to project evaluation seeks to correct the deficiencies of traditional methods of valuation through the recognition that managerial flexibility can bring significant value to projects. Our main concern is how to deal with non-statistical imprecision we encounter when judging or estimating future cash flows. In this paper, we develop a methodology for valuing options on R&D projects, when future cash flows are estimated by trapezoidal fuzzy numbers. In particular, we present a fuzzy mixed integer programming model for the R&D optimal portfolio selection problem, and discuss how our methodology can be used to build decision support tools for optimal R&D project selection in a corporate environment.

On interactive fuzzy numbers

In this paper we will introduce a measure of interactivity between marginal distributions of a joint possibility distribution C as the expected value of the interactivity relation between the γ-level sets of its marginal distributions.

OWA operators with maximal Rényi entropy

This paper lays bare a parametric class of ordered weighted averaging (OWA) operators with maximal Renyi entropy OWA weights for any level of orness. In particular, we shall formulate an extended approach to decision making with OWA operators by introducing considerations of maximizing information content in the aggregation according to a general parametric class of dispersion measures.

On possibilistic correlation

In 2004, Fuller and Majlender introduced the notion of covariance between fuzzy numbers by their joint possibility distribution to measure the degree to which they interact. Based on this approach, in this paper we will present the concept of possibilistic correlation representing an average degree of interaction between marginal distributions of a joint possibility distribution as compared to their respective dispersions. Moreover, we will formulate the classical Cauchy-Schwarz inequality in this possibilistic environment and show that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart. In particular, applying the idea of transforming level sets of possibility distributions into uniform probability distributions, we will point out a fundamental relationship between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation.

Strategic investment planning by using dynamic decision trees

We represent strategic planning problems by dynamic decision trees, in which the nodes are projects that can be deferred or postponed for a certain period of time. Using the theory of real options an enhanced, option pricing method is provided, where the relationship of investments is taken into consideration as well, and. the optimal path of the decision tree, i.e. the path with the biggest real option value in the end of the planning period will be identified. The theory developed implicitly contains not only the deferral or cancellation flexibility of projects but also the possibility of considering vague informations, which needs to be taken into account when (long-time range) financial decisions are made.

Fuzzy black and scholes real options pricing

Giga-investments are very large projects with long life cycles, which make them both risk investments and endeavours built on imprecise and uncertain assumptions. Still, the classical approach to judging the profitability of the investments is the calculation of the net present values (NPV) of estimated cash flows. This involves the use of fairly precise numbers, which somehow are estimated for life cycles of 15-25 years in an uncertain context. Real option valuation (ROV) has been proposed as an alternative and complement to the NPV calculations. Flexibility introduced with the ROV methods can be used for strategic purposes and better investment decisions can be made, when there are some degrees of flexibility available for the process of deciding on investment alternatives. Still, the problem with unrealistic expectations on precision has not been solved. It appears clear that a high level of precision in cash flow estimates will be misleading for the decision makers, but the methods used have required the precision used. We introduce a new method, the fuzzy ROV (or FROV), which is built on the use of fuzzy numbers and possibility distributions, and which has the capability to tackle imprecision and uncertainty in a direct and explicit way.

Comments on the paper: “On the properties of equidifferent OWA operator” by Xinwang Liu

Abstract In this communication, we will point out an interesting correspondence between a recent paper “On the properties of equidifferent OWA operator” [International Journal of Approximate Reasoning, in press, doi:10.1016/j.ijar.2005.11.003 ] by Liu, and an earlier result “On obtaining minimal variability OWA operator weights” [Fuzzy Sets and Systems 136 (2003) 203–215] by Fuller and Majlender.

Fuzziness in Supply Chain Management

In this chapter, we shall present some theoretic and practical aspects of employing fuzzy logic and possibility theory in Supply Chain Management (SCM). We will present a wide point of view to our topics by introducing basic concepts of supply chain management and fuzzy logic without requiring any background knowledge from the reader in these areas. First, we will generally present the topic of supply chain management along with some of its classical methods that can be used to work with problems in that area. Then, we will introduce fuzzy logic to supply chain management by incorporating possibility distributions in the mathematical models. Doing the mathematical formulations we will also present methods for dealing with the bullwhip effect using possibility distributions.