Atanu Sengupta

On comparing interval numbers

Abstract This paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then, on the basis of this, gives two approaches to compare any two interval numbers. The first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches. This index has also been called an optimistic decision makers preference index. The other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision makers point of view. Choice of different decision makers with different degrees of pessimism has also been considered.

Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming

Abstract The modern trend in Operations Research methodology deserves modelling of all relevant vague or uncertain information involved in a real decision problem. Generally, vagueness is modelled by a fuzzy approach and uncertainty by a stochastic approach. In some cases, a decision maker may prefer using interval numbers as coefficients of an inexact relationship. As a coefficient an interval assumes an extent of tolerance or a region that the parameter can possibly take. However, its use in the optimization problems is not much attended as it merits. This paper defines an interval linear programming problem as an extension of the classical linear programming problem to an inexact environment. On the basis of a comparative study on ordering interval numbers, inequality constraints involving interval coefficients are reduced in their satisfactory crisp equivalent forms and a satisfactory solution of the problem is defined. A numerical example is also given.

Fuzzy Preference Ordering of Interval Numbers in Decision Problems

In conventional mathematical programming, coefficients of problems are usually determined by the experts as crisp values in terms of classical mathematical reasoning. But in reality, in an imprecise and uncertain environment, it will be utmost unrealistic to assume that the knowledge and representation of an expert can come in a precise way. The wider objective of the book is to study different real decision situations where problems are defined in inexact environment. Inexactness are mainly generated in two ways (1) due to imprecise perception and knowledge of the human expert followed by vague representation of knowledge as a DM; (2) due to huge-ness and complexity of relations and data structure in the definition of the problem situation. We use interval numbers to specify inexact or imprecise or uncertain data. Consequently, the study of a decision problem requires answering the following initial questions: How should we compare and define preference ordering between two intervals?, interpret and deal inequality relations involving interval coefficients?, interpret and make way towards the goal of the decision problem? The present research work consists of two closely related fields: approaches towards defining a generalized preference ordering scheme for interval attributes and approaches to deal with some issues having application potential in many areas of decision making.

Solving the Shortest Path Problem with Interval Arcs

This paper presents an algorithm for the shortest path problem when the connected arcs in a transportation network are represented as interval numbers. The methodology proposed in this paper considers fuzzy preference ordering of intervals (Sengupta and Pal (2000), European Journal of Operational Research 127, 28–43) from pessimistic and optimistic decision maker’s point of view.

On Comparing Interval Numbers: A Study on Existing Ideas

The problem of comparing imprecise or uncertain quantities has been attempted by many researchers. In literature, as much as 40 schemes for ranking fuzzy quantities have been presented during last three decades with a lot of debates and contradictory claims. Inconsistency of the ranking orders among different approaches and the debates thereon were due to the different criteria of selections. Given two imprecise quantities, it is assumed that a set of precisely defined decision objectives are needed in order to decide, which of the two precedes or ranks above the other, or, which of the notions – the notion of preference or the notion of greater than (or less than) – is to be given the priority, or, whether an optimally selected (greatest/lowest) set satisfies an intuitive notion of being the most preferred (Wang & Kerre (1996, 2001a, 2001b)). In fuzzy literatures, we find some remarkable research papers that categorize and compare ranking strategies implicitly or explicitly on the basis of some criteria, such as the distinguishability (Bortolan & Degani (1985)), the rationality (Nakamura (1986)), and the fuzzy or linguistic presentation (Tong & Bonissone (1980) and Delgado et al. (1988)).

Travelling Salesman Problem with Interval Cost Constraints

The Travelling Salesman Problem (TSP) is one of the classical discrete (combinatorial) optimization problems, encountered in Operations Research (Lawler et al (1985), Clifford & Siu (1995)). TSP is also one of such problems considered as puzzles by mathematicians. Suppose a salesman has to visit n cities (or nodes) cyclically and in one tour he has to visit each city just once, and finish up where he started. In what order should he visit them to minimize the cost or distance or time travelled? Much of the work on the TSP is not motivated by direct applications, but rather by the fact that the TSP provides an ideal platform for the study of general methods that can be applied to a wide range of discrete optimization problems (Grotschel & Holland (1991), Reinelt (1994), Applegate et al (1998), Cook (web document), Hoffman (web document)).

Fuzzy Preference Ordering of Intervals

In the formulation of realistic problems, set of intervals may appear as coefficients in the inequality (or equality) constraints of an optimization problem or they may appear in the selection of a preferred alternative from among a set of many in a decision making problem. Consequently, there might be a key question related to the comparison of any two interval-numbers.

Interval Transportation Problem with Multiple Penalty Factors

The problems of distribution of goods from manufacturer to customer are generally described under a common heading, Transportation Problem (TP). The TP, originally developed by Hitchcock (1941), can be used when a firm tries to decide where to locate a new facility. Good financial decisions concerning facility location also attempt to minimize total transportation and production costs for the entire system. Moreover, there are many problems not exactly being called the TP but can be modelled alike.

Acceptability Index and Interval Linear Programming

This chapter defines an interval linear programming problem as an extension of the classical linear programming problem to an inexact environment. Let’s refer here a very good example (Tong (1994)) of using interval numbers in an optimization problem:

Concluding Remarks and the Future Scope

The use of interval parameters is possibly the simplest but yet an intuitive way to introduce data uncertainty for large and complex decision problems. The way of argument as well as the advantages in handling data uncertainty in the form of interval numbers has attracted many researchers. Consequently, the problem of interval number ordering has become a matter of perennial interest because of its direct relevance to the practical modelling and optimization in the real world processes under imprecise and uncertain environment. However, one main dilemma in using interval data for decision problems was perhaps in making the choice of an appropriate interval order relation. Earlier it was believed and was often argued that crisp intervals, theoretically, could only be partially ordered and hence all were not comparable. Nevertheless, when interval numbers are started to be aptly used as uncertain parameters in practical applications, the question of choice among the interval alternatives is started brewing. This unattended question of ranking any two interval numbers has created a new enthusiasm among the researchers during only the last two decades.