Multiple Criteria Decision Making (MCDM) models are used to solve a number of decision making problems universally. Most of these methods require the use of integers as input data. However, there are problems which have indeterminate values or data intervals which need to be analysed. In order to solve problems with interval data, many methods have been reported. Through this study an attempt has been made to compare and analyse the popular decision making tools for interval data problems. Namely, ITOPSIS (Technique for Order Preference by Similarity to Ideal Solution), DITOPSIS, cross entropy, and interval VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) have been compared and a novel algorithm has been proposed. The new algorithm makes use of basic TOPSIS technique to overcome the limitations of known methods. To compare the effectiveness of the various methods, an example problem has been used where selection of best material family for the capacitor application has to be made. It was observed that the proposed algorithm is able to overcome the known limitations of the previous techniques. Thus, it can be easily and efficiently applied to various decision making problems with interval data.
Engineers and managers over the world are daily faced with problems that require the selection of the best alternative from among the feasible options. Such complications are called decision making problems and encompass a wide variety of applications from design, optimisation, allotment, and screening to name a few. Often these problems present alternatives where numerous conflicting constraints are to be considered while making a decision. The attributes associated are such that maximisation of one would lead to minimisation of others. Such problems have no absolute solution and require an optimisation of all the traits to present the best possible solution. This category of problems is known as Multiple Criteria Decision Making (MCDM) problems and various MCDM techniques are used to solve such dilemmas [
MODM techniques require the knowledge of functional relationship that exists between various attributes associated with the alternatives. This relationship is used to formulate figure of merits (FOM) that are used to quantify the desirability or performance of a given alternative [
Both, MODM and MADM techniques are popular decision making tools and are freely used by the scientific and industrial community for various decision making problems [
To the best of our knowledge, this is the first attempt to utilize indeterminate data for ranking purposes using a traditional approach.
Several techniques have been proposed to deal with problems having nondeterministic or interval data. Most of these techniques require the basic knowledge of TOPSIS and VIKOR which are two popular MADM selection and ranking tools. The basic underlying mechanism of the two techniques is as explained below.
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a MADM technique which was first proposed by Hwang and Yoon [
TOPSIS uses vector normalisation for scaling of data. TOPSIS is a very versatile and popular MADM tool among the scientific community. Since inception, many new methods have been proposed to modify the classical TOPSIS approach to suit specific problems. These include fuzzy TOPSIS [
Decision matrix with
Attributes  

Alternatives 




 




 




 









Construction of normalised decision matrix:
Construction of weighted normalised decision matrix:
Determination of the positive ideal and negative ideal solutions: the positive ideal solution
Calculate the distances
Determine relative closeness of alternatives to the ideal solution:
VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is a compromise ranking and selection technique. Like TOPSIS, VIKOR is also a MADM technique and implies similar principles of application. However, the two differ slightly in that VIKOR uses linear normalisation technique [
Calculation of normalised decision matrix (optional):
Determination of ideal and negative ideal solutions: the ideal solution
Calculation of utility measure and regret measure:
Determination of VIKOR index:
The alternative with the least value of VIKOR index
Many of the real life problems are multidimensional in nature. Most of those dimensions have a degree of uncertainty associated with them. This uncertainty causes the variation in data and leads to indeterminate outcome. When the degree of uncertainty is low we can assume a crisp value to represent the average outcome. However, when the degree of uncertainty is high we can no longer assume a crisp value but report the data in a ranged format (Table
Decision matrix with interval data.
Attributes  

Alternatives 




 





















The direct techniques on the other hand do not attempt to change the nature of data and thus are able to preserve information. Several methods have been reported for decision making using direct interval data. Some of the popular methods are being explained as follows.
ITOPSIS was proposed by Jahanshahloo et al. [
Calculation of normalised decision matrix:
Construction of weighted normalised decision matrix:
Determination of the ideal solutions:
Calculation of distances (separation) of alternatives from ideal solutions:
Determination of ranking index
DI TOPSIS or direct interval TOPSIS was proposed by Sevastjanov and Tikhonenko [
Calculation of normalised decision matrix:
Construction of weighted normalised decision matrix:
Determination of ideal solutions:
Calculation of separation from ideal solutions
The separation
Calculation of relative closeness to the ideal solution
Cross entropy is another method used to solve decision making problems with interval data sets [
Calculation of the normalised decision matrix:
Calculate the weighted normalised decision matrix:
Determination of the positive (
Calculation of cross entropy:
Calculation of relative closeness to the idea solution:
VIKOR was extended for decision models with interval data by Sayadi et al. [
Determination of the positive and negative ideal solutions:
Computation of
Compute the ranking interval
These intervals can now be compared using interval comparison methods as per the requirements of the decision maker. Under the comparison of two interval numbers,
If these interval numbers have no intersection, the minimum interval number is the one that has lower values. In other words, if
If two interval numbers are the same, both have the same priority.
In situations that
In a condition where
The purpose of this study was to enable the use of ranged or indeterminate data to be applied to point based decision making techniques without any appreciable loss of form or information. It is a common trend with the conventional interval data techniques to employ only the upper and lower limits of the spread for evaluation purposes. This is done because the statistical nature of the data is relatively incapable to alter the ranking by large. Thus, it not considered while the implementation of the decision technique. Exceptions to this rule are observed for fuzzy decision making.
Considering the aforementioned logic and after observing the data handling associated with direct methods, it can be concluded that most of the limitations can be overcome by slight modification to the approach itself. An interval
Proposed algorithm decision matrix.
Attributes  

Alternatives 








 








 








 













Figurative description of data conversion technique to be employed for proposed algorithm.
In order to test the effectiveness of the new approach, we incorporated the additional step to the conventional TOPSIS approach. TOPSIS has been chosen because it has been shown to be in good agreement with MODM results and experimental demonstrations through our previous works [
To compare the performance of different methods and the proposed algorithm we make use of an example problem. The given decision matrix (Table
Decision matrix for the problem.
Materials  Breakdown voltage (MV/mm)  Dielectric constant  

Lower  Upper  Lower  Upper  
Polymerbased nanodielectric  4  7  220  370 
Ferroelectric glass ceramic  3  7  170  370 
Metalglass nanocomposite  4  6  220  320 
Ferroelectric ceramic  3  6  170  320 
A twodimensional representation of TOPSIS, ITOPSIS, and DITOPSIS has been given in Figures
Graphical representation of TOPSIS with two attributes.
Graphical representation of ITOPSIS with two attributes.
Graphical representation of DITOPSIS with two attributes.
In ITOPSIS the data has intervals and as such cannot be represented as points on the plane. To this effect we use rectangles where the dimensions of the rectangle represent the range of the respective property (Figure
For DITOPSIS the ideal solutions are calculated as ranged data. The positive ideal solution is derived as the difference between the maxima of the upper and the lower limits of the associated attributes (Figure
The results of the ranking using various methods are given in Table
Rankings and index for applied techniques.
Materials  ITOPSIS  DITOPSIS  Cross entropy  Interval VIKOR  Proposed method  

Index  Rank  Index  Rank  Index  Rank  Index  Rank  Index  Rank  
Polymerbased nanodielectric  0.57884  1  1  1  1  1 

1  1.00000  1 
Metalglass nanocomposite  0.50000  2  0.5  2  0.76604  2 

2  0.64345  2 
Ferroelectric glass ceramic  0.50000  2  0.5  2  0.234306  3 

3  0.35655  3 
Ferroelectric ceramic  0.42116  4  0  4  0  4 

4 

4 
The proposed algorithm on the other hand suffers from none of the abovementioned drawbacks. It can differentiate between alternatives having overlapping data sets as well as data sets which have coinciding midpoints. Integers can be used as data points and they return a crisp value as the final ranking index. This enables the decision maker to easily rank and compare different alternatives with little effort. For the special case of data sets in which one completely overlaps the other and also has coincidental geometric centres, the proposed algorithm favours the data set with the lower spread. This is acceptable as a wider spread is generally associated with a higher value of uncertainty. Thus, for majority of the problems where the certainty of the outcome is important, selection of alternative with lesser data spread will be beneficial. For such problems, the proposed algorithm would be an excellent way to make decisions where the input data is nondeterministic or has interval values.
Through this study an attempt has been made to compare various decision making methods for problems having interval data. These methods are ITOPSIS, DITOPSIS, cross entropy, and interval VIKOR. Finally, a new algorithm has been proposed to solve interval data problems using basic TOPSIS approach. To compare the effectiveness of the techniques an example problem has been used. It was observed that ITOPSIS and DITOPSIS techniques have failed to distinguish between alternatives having coincident geometric data centre points. Cross entropy and interval VIKOR can easily distinguish the data points but have their own set of limitations. Cross entropy can only accept positive values and thus cannot be used with integer data. Interval VIKOR gives the result in the form of intervals, which then have to be compared individually making it computationally taxing for a large number of alternatives. The proposed algorithm does not have the limitations of the previous methods and thus can be easily and efficiently applied to problems with interval data.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Rahul Vaish acknowledges support from the Indian National Science Academy (INSA), New Delhi, through a grant by the Department of Science and Technology (DST), New Delhi, under INSPIRE faculty award2011 (ENG01).