A Polar Representation for Complex Interval Numbers

The present work defines the basic elements for the introduction to the Study of Complex variables under the mathematical interval context with the goal of using it as a foundation for the understanding of pure mathematical problems, associating the mathematical interval to support the results. The present article contributes to the complex interval theory taking into consideration Euler’s Identity and redefining the polar representation of interval complex numbers. In engineering, the present article could be used as a subsidy for many applications where complex variable theory is applicable and requires accurate results

Total interval numbers of complete r-partite graphs

AbstractA multiple-interval representation of a graph G is a mapping f which assigns to each vertex of G a union of intervals on the real line so that two distinct vertices u and v are adjacent if and only if f(u)∩f(v)≠∅. We study the total interval number of G, defined asI(G)=min∑v∈V#f(v):fisamultiple-intervalrepresentationofG,where #f(v) is the minimum number of intervals whose union is f(v). We give bounds on the total interval numbers of complete r-partite graphs. Exact values are also determined for several cases

Fuzziness in analytic network process under interval numbers for criteria and alternatives

Because of the lack of data or knowledge or limited time, decision makers could not express their experiences exactly, perhaps they prefer interval numbers for such situations. Whenever uncertainty is involved in the decision making process, fuzzy and stochastic models would be arisen. Recently, fuzzy theory for Multiple Attribute Decision Making (MADM) under interval numbers has attracted a lot of researchers. This paper deals with a fuzzy MADM approach under interval numbers. We propose to apply the approach for Analytic Network Process (ANP) as a new class of decision making methods. The interval numbers are formed for criteria weights and values that have effect on alternatives’ values. The process of this method is clarified by an example

A new view to uncertainty in Electre III method by introducing interval numbers

The Electre III is a widely accepted multi attribute decision making model, which takes into account the uncertainty and vagueness. Uncertainty concept in Electre III is introduced by indifference, preference and veto thresholds, but sometimes determining their accurate values can be very hard. In this paper we represent the values of performance matrix as interval numbers and we define the links between interval numbers and concordance matrix .Without changing the concept of concordance, in our propose concept, Electre III is usable in decision making problems with interval numbers

Double sequences of interval numbers defined by Orlicz functions

We define and study λ2-convergence of double sequences of interval numbers dened by Orlicz function and λ2-statistical convergence of double sequences of interval numbers. We also establish some inclusion relations between them

On the hull and interval numbers of oriented graphs

In this work, for a given oriented graph $D$, we study its interval and hull numbers, denoted by ${in}(D)$ and ${hn}(D)$, respectively, in the geodetic, ${P_3}$ and ${P_3^*}$ convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph $D$, we prove that ${hn_g}(D)\leq m(D)-n(D)+2$ and that there is a strongly oriented graph such that ${hn_g}(D) = m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allows us to deduce polynomial-time algorithms to compute ${hn_{P_3}}(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ${in_g}(D)\leq k$ or ${hn_g}(D)\leq k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether ${hn_{P_3}}(D)\leq k$ or ${hn_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is bipartite; that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split. We also argue that the interval and hull numbers in the oriented $P_3$ and $P_3^*$ convexities can be computed in polynomial time for graphs of bounded tree-width by using Courcelle's theorem

Some class of generalized entire sequences of Modal Interval numbers

The history of modal intervals goes back to the very first publications on the topic of interval calculus. The modal interval analysis is used in Computer graphics and Computer Aided Design (CAD), namely the computation of narrow bounds on Bezier and B-Spline curves. Since modal intervals are used in different fields, we have constructed a new sequence space  of modal intervals. Also , we have given some new definitions and theorems about the sequence space  of modal interval numbers

Supplier evaluation in manufacturing environment using compromise ranking method with grey interval numbers

Evaluation of proper supplier for manufacturing organizations is one of the most challenging problems in real time manufacturing environment due to a wide variety of customer demands. It has become more and more complicated to meet the challenges of international competitiveness and as the decision makers need to assess a wide range of alternative suppliers based on a set of conflicting criteria. Thus, the main objective of supplier selection is to select highly potential supplier through which all the set goals regarding the purchasing and manufacturing activity can be achieved. Because of these reasons, supplier selection has got considerable attention by the academicians and researchers. This paper presents a combined multi-criteria decision making methodology for supplier evaluation for given industrial applications. The proposed methodology is based on a compromise ranking method combined with Grey Interval Numbers considering different cardinal and ordinal criteria and their relative importance. A ‘supplier selection index’ is also proposed to help evaluation and ranking the alternative suppliers. Two examples are illustrated to demonstrate the potentiality and applicability of the proposed method