Applied Soft Computing 47 (2016) 438–448 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc Relations between complex vague soft sets Ganeshsree Selvachandran a,∗ , P.K. Maji b , Israa E. Abed c , Abdul Razak Salleh c a Department of Actuarial Science and Applied Statistics, Faculty of Business & Information Science, UCSI University, Jalan Menara Gading, 56000 Cheras, Kuala Lumpur, Malaysia b Department of Mathematics, B.C. College, Asansol, West Bengal, India c School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia a r t i c l e i n f o Article history: Received 11 April 2016 Received in revised form 27 May 2016 Accepted 31 May 2016 Available online 21 June 2016 MSC: 08A99 08A72 Keywords: Complex vague soft set Relations between complex vague soft sets Complex fuzzy set Vague soft set Fuzzy set a b s t r a c t Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies. © 2016 Elsevier B.V. All rights reserved. 1. Introduction The complex number set allows us to solve many problems that traditionally cannot be solved by using the real number set, such as improper integrals that represent electrical resistance in the field of engineering. Thus applying soft, fuzzy and hybrid sets to complex num- bers is an essential step to incorporate the advantages of complex numbers into the notion of soft sets, fuzzy sets, and their generalizations. This work initiated by Ramot et al. [1], who introduced the concept of complex fuzzy sets which extend the notion of fuzzy sets, are made possible by adding a phase term that describes the periodicity of the elements with respect to time. The range of Ramot’s complex fuzzy sets is not limited to [0,1] but rather extends to the unit circle in the complex plane. The theory behind the development of complex fuzzy sets says that in many instances a second dimension must be added to the expression of the membership value of an element or object which is particularly important in situations wherein the elements of the set vary with time. Examples include meteorological time series such as sunspot cycles which describe the number of sunspots that appear on the surface of the sun as a function of time and economic time series such as the fluctuation of stock prices on a daily or hourly basis or the effects of certain financial factors on the economy of a country or a region. Among other notable research in this relatively unexplored area are the application of complex fuzzy sets to traditional fuzzy logic via the introduction of complex fuzzy logic by Ramot et al. [2] and Dick [3] who further expanded the theory of complex fuzzy logic by introducing several new operators pertaining to this theory. The study of uncertainties and nonlinear problems and by extension, the modelling of uncertainties and nonlinear problems has always been a major area in the study of mathematics. Over the years, many techniques and methods have been proposed as tools to be used to find the solutions of problems that are nonlinear or vague in nature, with every method introduced superior to its predecessors. The study of nonlinear problems is of particular interest to engineers, physicists, mathematicians and other scientists as most systems in the real world are inherently nonlinear in nature and often appear as chaotic, unpredictable and sometimes even counterintuitive. Applied mathematical techniques and artificial intelligence are the most commonly used methodologies used to handle perturbation and chaotic ∗ Corresponding author. E-mail addresses: ganeshsree86@yahoo.com (G. Selvachandran), pabitra maji@yahoo.com (P.K. Maji). http://dx.doi.org/10.1016/j.asoc.2016.05.055 1568-4946/© 2016 Elsevier B.V. All rights reserved.