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Management Science, V. 51, No. 4, pp 655-661. http://dx.doi.org/10.1287/mnsc.1040.0290 Peer review of research proposals and articles is an essential element in research and development processes worldwide. Here we consider a problem that, to the best of our knowledge, has not been addressed until now: how to assign subsets of proposals to reviewers where the reviewers supply their evaluations through ordinal ranking. The solution approach we propose for this assignment problem maximizes the number of proposal pairs that will be evaluated by one or more reviewers. This new approach should facilitate meaningful aggregatin of partial rankings of subsets of proposals by multiple reviewers into a consensus ranking. We offer two ways to implement the approach: an integer programming set-covering model and a heuristic procedure. The effectiveness and efficiency of the two models are tested through an extensive simulation experiment.

In a recent paper, Golany (1988) proposes an interesting extension of Data Envelopment Analysis to the situation where ordinal relations among weights corresponding to certain dimensions exist. However, the development of the extended model contains mathematical errors and the proposed equivalence is incorrect. We establish the correct model equivalence and prove that weak ordinal relations require a nonstandard DEA model. We also show that the case of strict ordinal relations can be handled with the standard DEA model. These results hold for the CCR, BCC, additive, and multiplicative models and for relations involving both input and output multipliers.

This paper presents a general model for aggregating votes from a preferential ballot. The thrust of the model is to accord each candidate a fair assessment in terms of his overall standing vis-a-vis first place, second place, …, kth place votes. The form of the model is a combined index Σj = 1k Wjvij where vij is the number of the jth place votes received by the ith candidate. The weights Wj are assumed to form a monotonically decreasing sequence with Wj − Wj + 1 ≥ d(j, ε). These constraints correspond to the assurance region (AR) side constraints in the DEA framework. The properties of the model are examined in terms of this discrimination intensity function d, and in the special case that d(j, ε) = ε, our model is shown to be equivalent to the consensus models of Borda and Kendall.

This paper investigates the problem of obtaining a compromise/consensus from a set of ordinal rankings of n objects supplied by m committee members. Earlier work by Cook and Seiford (Cook, Wade D., Lawrence M. Seiford. 1978. Priority ranking and consensus formation. Management Sci. 24 (16) 1721–1732.) dealt with the problem of consensus when attention was restricted to complete rankings only. That is, no ties were permitted. This paper examines the general problem which allows for tied preferences. A convex polyhedral representation is given of the feasible solution space, and a branch-and-bound procedure is developed for determining an optimal ranking. Computational results for various problem sizes are presented. Generalizations, and directions for further research are discussed.

This paper presents a solution procedure for a class of discontinuous nonlinear knapsack problems. These problems have a single linear constraint and a restriction that each variable must be either zero or take on a value within a specified interval. The objective function is separable and each term is concave within the interval. Problems of this type arise in capital budgeting and a particular application in the scheduling of pavement maintenance is given. The branch-and-bound algorithm developed to solve the problem considers an approximation-relaxation at each step. Computational experience with the algorithm and a brief overview of applications of the model are given.

One of the best known and most widely referenced models for representing ordinal preferences is that due to Kemeny and Snell (Kemeny, J. G., L. J. Snell. 1962. Preference ranking: an axiomatic approach. Mathematical Models in the Social Sciences. Glnn, New York, 9–23.). This model is designed to accommodate pairwise comparison data with an l1 norm used to measure voter disagreement. While this model possesses many of the necessary properties for a social choice function, solution procedures developed to date have been capable of handling only small problems due to the difficulty of modelling the transitivity requirements of an optimal consensus ranking. This paper shows how the consensus formation problem for strict linear orderings can be modelled as a generalized network. Since efficient computer codes already exist for handling this special structure, this approach will permit the solution of much larger problems than has been the case previously.

This paper investigates the Borda-Kendall method for the determination of a consensus ranking. It is shown that in the case of ties the method does not perform as claimed. A “minimum variance” method for determining the consensus ranking is proposed and its properties examined. It is shown to be equivalent to the Borda-Kendall method if ties are not allowed. An algorithm to determine the “minimum variance” consensus ranking in the case of ties is described. Results obtained from the solution of problems of various sizes are discussed.

This paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament. The criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament. Such a situation is referred to as a violation. The objective, therefore, is to determine the Minimum Violations Ranking (MVR). While there are situations where this ranking would be allowed to contain ties among subsets of objects, we will concern ourselves herein with linear ordering (no ties). A series of examples are given where this requirement would seem to be appropriate. In order to put the MVR problem into proper perspective we introduce the concept of a distance on the set of tournaments. A set of natural axioms is presented which any such distance measure should obey, and it is proven that in the presence of these axioms a unique such measure exists. It is then shown that the MVR problem is equivalent to the minimum distance problem, which can be represented in several forms—in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem. This opens up a wide scope of possible solution procedures for the MVR problem. An optimal algorithm is presented along with computational results. In addition, various heuristics are discussed including an improved heuristic referred to as the Iterated Kendall method.

This paper investigates the problem of combining ordinal preferences, expressed as priority vectors, to form a consensus. An axiomatic structure relating to the concept of distance between rankings is developed, uniqueness of the distance measure is proven and its form derived. Adopting the median ranking as a form of consensus, it is shown that this ranking can be determined, in the case of complete rankings, by solving a certain assignment problem. The results are compared and contrasted to earlier work due to Arrow, Kendall, Kemeny and Snell, and Bogart.

In conventional ordinal ranking models the voter/ranker supplies an ordered set of preferences on a collection of objects without specifying any form of intensity of preference. For example, an executive committee of ten members is required to assign five candidates to five positions. The nature of the positions is such that position one requires the highest qualified candidate (relative to a given attribute), position two the second qualified person and so on. Each one of the members in the committee is required, therefore, to supply a ranking of the five candidates and from each individual preference the committee must come up with an aggregate or consensus ranking which dictates the assignments of the candidates to the jobs. In many situations, however, it is desirable to permit the individual to express some measure of intensity of preference. For example, suppose that six committee members agreed that candidate number one is slightly better than candidate number two, but four of them evaluated candidate number two as being much better than candidate number one. The question now is whether this situation is identical to the case where the six members evaluate candidate one as much better than candidate two and four members evaluate candidate two as slightly better than candidate one. The ordinal ranking model will not distinguish between the above two cases while it is clear that when taking account of the intensity of preference, the committee should end up with two different consensus rankings. This paper develops a model for aggregating ordinal rankings in which the voter is allowed to express such intensity of preferences, and a method to derive the consensus ranking is proposed.