International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013 © 2013 TFSA 488 Two Interpretations of Multidimensional RDM Interval Arithmetic - Multiplication and Division Andrzej Piegat and Marek Landowski Abstract 1 The paper presents two possible interpretations and realization ways of interval multiplication and division: the possibilistic, unconditional interpreta- tion that is of great meaning for fuzzy arithmetic and fuzzy systems, and the probabilistic, conditional in- terpretation that requires either knowledge of prob- ability density distributions or assumptions concern- ing these distributions. The possibilistic interpreta- tion has a great significance not only for fuzzy arith- metic but also for other sciences that use it such as Computing with Words, Grey Systems, etc. These two interpretations are explained in frame of a new, multidimensional RDM interval-arithmetic. The pos- sibility of realization of interval-arithmetic opera- tions in two ways is an argument for reconciliation of two competing scientific groups that propagate two approaches to uncertainty modeling: the probabilistic and possibilistic one. For many years Professor Zadeh has been claiming in his publications that both approaches are not contradictory but rather com- plementary. Keywords: Computing with words, fuzzy arithmetic, granular computing, interval arithmetic, interval equa- tions, uncertainty theory. 1. Introduction For many scientists and engineers interval arithmetic seems to be a less important area of science. However, its importance grows with the development of the uncer- tainty theory [1]. Scientists realize more and more that taking into account the parameter and variable uncer- tainty in mathematical models is necessary. Data uncer- tainty is met everywhere. Let us consider as an example Corresponding Author: Andrzej Piegat is with the Faculty of Com- puter Science and Information Technology, West Pomeranian Uni- versity of Technology, Zolnierska 49, 71-210 Szczecin, Poland. E-mail: apiegat@wi.zut.edu.pl Marek Landowski is with the Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500 Szczecin, Poland. E-mail: m.landowski@am.szczecin.pl Manuscript received 22 July 2013; revised 20 Nov. 2013; accepted 18 Dec. 2013. the charging process of the car battery [16], where L 0 [Ah] means the initial battery charge, L(t) [Ah] means the charge after time t [h] of charging, and i [A] means the charging current. The charging process can be de- scribed by (1). it L t L   0 ) ( (1) If we want to determine the charging time T [h] of a battery that is discharged, then (2) should be used.   i L L T T / 0   (2) In formula (2) L T means the required end charge of the battery. However, in this problem we mostly do not know precisely but only approximately the initial state L 0 [ 0 0 , L L ] as e.g. L 0 [4,7] Ah. Similarly, the charg- ing current is not constant and varies during the charging process. It can be approximated as i[ i i , ], e.g. i[3,4] A. Therefore, to calculate the required charging time T [h] we should not use (2) but rather the interval formula (3) ] 4 , 3 [ ] 7 , 4 [ 90 ] , [ ] , [ ] , [ 0 0     i i L L L T T T (3) The mathematical battery-model is simple. However, in practical problems much more complicated models occur, e.g. ship movement model, airplane, rocket, space-ship movement models, economical, medical, mechanical, environmental models, etc. In most of these models occur uncertainties and taking them into account is necessary if we want to get realistic results. This situa- tion explains the increasing interest in uncertainty theory [1], Grey Systems [8], Granular Computing [11], Com- puting with Words [21, 22], fuzzy systems [7, 12] in science areas that allow for processing uncertain data and modeling uncertain systems. Interval arithmetic is the basic arithmetic of approximate data, because inter- val [ x x, ] is the simplest and a very convenient way of elicitation and modeling uncertainty. This arithmetic is also used as a basis by other sciences dealing with un- certainty. And so, all calculation results delivered by fuzzy arithmetic [3, 6] have to be consistent with results delivered by interval arithmetic in respect of their sup- ports (it results from the -cut method). The same refers to probabilistic arithmetic [4, 5, 19], Grey Systems, etc. The so called “precise” models as e.g. battery model (1) are at present called “academic” or “laboratory” models, because they are not rather realistic. They make illusion