A cross-efficiency model based on super-efficiency for ranking units through the TOPSIS approach and its extension to the interval case

One of the most important approaches for ranking decision-making units in data envelopment analysis is the cross-efficiency method. The main idea of cross-efficiency is to use data envelopment analysis in peer evaluation, instead of only a self evaluation. However, in the cross-efficiency method, optimal weights corresponding to evaluation of decision-making units may not be unique. In this research with modifying the cross-efficiency method we are going to overcome this problem. Then with regard to the changes and using the TOPSIS method we present a new super-efficient method to rank all decision-making units. Furthermore, we extend the proposed method to the case that data are intervals.     

The interval number of a planar graph: Three intervals suffice

Suppose each vertex of a graph G is assigned a subset of the real line consisting of at most t closed intervals. This assignment is called a t-interval representation of G when vertex v is adjacent to vertex w if and only if some interval for v intersects some interval for w. The interval number i(G) of a graph G is the smallest number t such that G has a t-interval representation. It is proved that i(G) ≤ 3 whenever G is planar and that this bound is the best possible. The related concepts of displayed interval number and depth-r interval number are discussed and their maximum values for certain classes of planar graphs are found.     

Efficiency ranking of decision making units in data envelopment analysis by using TOPSIS-DEA method

Data envelopment analysis methods classify the decision making units into two groups: efficient and inefficient ones. Therefore, the fully ranking all DMUs is demanded by most of the decision makers. However, data envelopment analysis and multiple criteria decision making units are developed independently and designed for different purposes. However, there are some applications in problem solving such as ranking, where these two methods are combined. Combination of multiple criteria decision making methods with data envelopment analysis is a new idea for elimination of disadvantages when applied independently. In this paper, first the new combined method is proposed named TOPSIS-DEA for ranking efficient units which not only includes the benefits of both data envelopment analysis and multiple criteria decision making methods, but also solves the issues that appear in former methods. Then properties and advantages of the suggested method are discussed and compared with super efficiency method, MAJ method, statistical-based model (CCA), statistical-based model (DR/DEA), cross-efficiency—aggressive, cross-efficiency—benevolent, Liang et al.’s model, through several illustrative examples. Finally, the proposed methods are validated.     

Measuring the performances of decision-making units using interval efficiencies

Efficiency is a relative measure because it can be measured within different ranges. The traditional data envelopment analysis (DEA) measures the efficiencies of decision-making units (DMUs) within the range of less than or equal to one. The corresponding efficiencies are referred to as the best relative efficiencies, which measure the best performances of DMUs and determine an efficiency frontier. If the efficiencies are measured within the range of greater than or equal to one, then the worst relative efficiencies can be used to measure the worst performances of DMUs and determine an inefficiency frontier. In this paper, the efficiencies of DMUs are measured within the range of an interval, whose upper bound is set to one and the lower bound is determined through introducing a virtual anti-ideal DMU, whose performance is definitely inferior to any DMUs. The efficiencies turn out to be all intervals and are thus referred to as interval efficiencies, which combine the best and the worst relative efficiencies in a reasonable manner to give an overall measurement and assessment of the performances of DMUs. The new DEA model with the upper and lower bounds on efficiencies is referred to as bounded DEA model, which can incorporate decision maker (DM) or assessor's preference information on input and output weights. A Hurwicz criterion approach is introduced and utilized to compare and rank the interval efficiencies of DMUs and a numerical example is examined using the proposed bounded DEA model to show its potential application and validity.     

The problem of ranking nonreusable interval objects specified by three points

Three methods for preference-based ranking of nonreusable objects are described in the case when the possible results of their use are represented as pessimistic, optimistic, and most likely estimates. The methods rely on the approximation of the binary probability preference relation by binary preference relations with respect to specially designed characteristics based on the above three estimates, namely, the median, dominant, and most likely values. The methods are verified using Monte Carlo simulation. It is shown that the median and dominant preference relations ensure a relatively high degree of approximation accuracy in most cases, while the binary preference relation with respect to the most likely value leads to a considerable reduction in the accuracy of approximation.     

Performance ranking of units considering ideal and anti-ideal DMU with common weights

A characteristic of traditional DEA CCR mode is that it allows DMUs to measure their maximum efficiency score with the most favorable weights. Thus, it would have some shortcomings, for example, the efficiencies of different DMUs obtained by different sets of weights may be unable to be compared and ranked on the same basis. Besides, there are always more than one DMU to be evaluated as efficient because of the flexibility in the selection of weights; it would cause the situation that all DMUs cannot be fully discriminated. With the research gaps, in this paper, we propose two models considering ideal and anti-ideal DMU to generate common weights for performance evaluation and ranking. Finally, two examples of Asian lead frame firms and flexible manufacturing systems are illustrated to examine the validity of the proposed methods.     

An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection

This paper presents a consensus model for group decision making with interval multiplicative and fuzzy preference relations based on two consensus criteria: (1) a consensus measure which indicates the agreement between experts’ preference relations and (2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by using the relative projections of individual preference relations on the collective one, which are obtained by extending the relative projection of vectors. First, the weights of experts are determined by the relative projections of individual preference relations on the initial collective one. Then using the weights of experts, all individual preference relations are aggregated into a collective one. The consensus and proximity measures are calculated by using the relative projections of experts’ preference relations respectively. The consensus measure is used to guide the consensus process until the collective solution is achieved. The proximity measure is used to guide the discussion phase of consensus reaching process. In such a way, an iterative algorithm is designed to guide the experts in the consensus reaching process. Finally the expected value preference relations are defined to transform the interval collective preference relation to a crisp one and the weights of alternatives are obtained from the expected value preference relations. Two numerical examples are given to illustrate the models and approaches.     

DEA and the discriminant analysis of ratios for ranking units

The purpose of this study is to develop a new method which provides for given inputs and outputs the best common weights for all the units that discriminate optimally between the efficient and inefficient units as pregiven by the Data Envelopment Analysis (DEA), in order to rank all the units on the same scale. This new method, Discriminant Data Envelopment Analysis of Ratios (DR/DEA), presents a further post-optimality analysis of DEA for organizational units when their multiple inputs and outputs are given. We construct the ratio between the composite output and the composite input, where their common weights are computed by a new non-linear optimization of goodness of separation between the two pregiven groups. A practical use of DR/DEA is that the common weights may be utilized for ranking the units on a unified scale. DR/DEA is a new use of a two-group discriminant criterion that has been presented here for ratios, rather than the traditional discriminant analysis which applies to a linear function. Moreover, non-parametric statistical tests are employed to verify the consistency between the classification from DEA (efficient and inefficient units) and the post-classification as generated by DR/DEA.     

Covering graphs by the minimum number of equivalence relations

An equivalence graph is a vertex disjoint union of complete graphs. For a graphG, let eq(G) be the minimum number of equivalence subgraphs ofG needed to cover all edges ofG. Similarly, let cc(G) be the minimum number of complete subgraphs ofG needed to cover all its edges. LetH be a graph onn vertices with maximal degree ≦d (and minimal degree ≧1), and letG= \(\bar H\) be its complement. We show that $$\log _2 n - \log _2 d \leqq eq(G) \leqq cc(G) \leqq 2e^2 (d + 1)^2 \log _e n.$$ The lower bound is proved by multilinear techniques (exterior algebra), and its assertion for the complement of ann-cycle settles a problem of Frankl. The upper bound is proved by probabilistic arguments, and it generalizes results of de Caen, Gregory and Pullman.     

The maximum interval number of graphs with given genus

The interval number of a graph G, denoted i(G), is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t compact real intervals. It is known that every planar graph has interval number at most 3 and that this result is best possible. We investigate the maximum value of the interval number for graphs with higher genus and show that the maximum value of the interval number of graphs with genus g is between ?√g? and 3 + ?√3g?. We also show that the maximum arboricity of graphs with genus g is either 1 + ?√3g? or 2 + ?√3g?.     

The method of medians in the problem of ranking interval objects specified by three points

A method for ranking interval objects is proposed and analyzed; the characteristics of those objects are represented by pessimistic, optimistic, and most probable estimates. The method is based on the approximation of the binary probability preference relation by the binary median preference relation. The method is verified using statistical modeling (the Monte Carlo method). The proposed approach can be used for ranking nonreusable and reusable objects.     

Some relations on the ordering of trees by minimal energies between subclasses of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n?2 (3≤d≤n?3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n?2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11.     

One-sided weighted integral approximation of characteristic functions of intervals by polynomials on a closed interval

We consider the problem of one-sided weighted integral approximation on the interval [?1, 1] to the characteristic functions of intervals (a, 1] ? (?1, 1] and (a, b) ? (?1, 1) by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate the difficulties arising in the case of an open interval.     

The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms.This procedure allows the construction of a tree for each given genus containing all semigroups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some light to a former conjecture saying that the number of semigroups of a given genus increases with the genus.We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.     

Explicit relations between pair correlation of zeros and primes in short intervals

In this paper we obtain a quantitative version of the well known theorem by Goldston and Montgomery about the equivalence between the asymptotic behaviors of the mean-square of primes in short intervals, and of the pair-correlation function of the zeros of the Riemann zeta function.     

Examination and prediction of fog and haze pollution using a Multi-variable Grey Model based on interval number sequences

In this study, a new Multivariable Grey Model (1,m) aimed at interval grey number sequences with known possibility functions is built using the kernel and degree of greyness under new definitions. Based on the new model, formulae are deduced to calculate and predict the upper and lower bounds of interval grey numbers. Since the grey system model and fog- and haze-prone weather have the same characteristics of uncertainty, this model was applied to simulate and predict the measurable indicators of fog and haze in Nanjing, China. We selected visibility data and particulate matter data with a diameter of 2.5 µm to build a new Multivariable Grey Model (1,2) with a new kernel and degree of greyness sequence. In addition, we established the traditional Multivariable Grey Model (1,2) with the original kernel and degree of greyness and the Auto-Regressive Integrated Moving Average Model (1,1,0). The results show that the new Multivariable Grey Model (1,2) has the best simulation and prediction effects among the three models, with average relative errors of simulation and prediction at 1.32% and 0.32%, respectively. To further verify the validity and feasibility of the proposed model, we added another real-world example to establish the three models mentioned above. The results prove that the proposed model has evidently superior performance to another two models.     

A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.