Preference score of units in the presence of ordinal data G.R. Jahanshahloo a , M. Soleimani-damaneh a , A. Mostafaee b, * a Department of Mathematics, Teacher Training University, Tehran, Iran b Department of Mathematics, North-Tehran Branch, Islamic Azad University, Tehran, Iran Accepted 2 January 2007 Communicated by Prof. L. Marek-Crnjac Abstract This study deals with the ordinal data in the performance analysis framework and provides a weight-restricted DEA model to obtain the preference score of each unit under assessment. The obtained scores are used to rank DMUs. Fur- thermore, to decrease the complexity of the provided model, the number of the constraints is decreased by some linear transformations. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Data envelopment analysis (DEA) is a very useful mathematical programming-based technique for evaluating the efficiency of a set of peer decision making units (DMUs) when the data are known exactly [6]. In many real world sit- uations, it happens that the aforementioned assumption is violated. One of these situations happens when the input– output data are preference information. In this case the input–output data are ordinal, and it is better to rank the units in an ordinal way, as well, because providing a quantity measure when the data are qualitative in nature does not seem in order. Many authors have studied embedding ordinal data into the DEA framework (see, e.g. [5,7,9]). In this paper we present a new approach to obtain the preference score of units with ordinal data. Our approach is a new one in that we do not consider y rj as having numerical values. In previous works (see, e.g. [7,9]), some mathematical operations are performed on the ordinal data, e.g., y rA < y rB is transformed to y rB  y rA P v r or y rB P v r y rA ðv r > 1Þ. In [9], for instance, when DMU k is under evaluation and strong ordinal relations are imposed, Zhu sets y rk ¼ 1 and considers y rj ¼ L jk r ðj ¼ k þ 1; ... ; nÞ, where y rk and y rj are exact data. But it seems incorrect to assign exact values to qualitative data. This is while our assumption in the present paper is that y rA < y rB only implies that y rB is more important to the DMU’s management than y rA . We demonstrate this in the form of scale rates. In the customary studies an efficiency score is assigned to each DMU, while it seems better to rank DMUs based on some preference scores. In this study we propose a new idea for treating ordinal data in DEA models, which assigns a preference score to each DMU and ranks them based upon these preference scores. In fact, these preference scores only give the preference position, and not anything more. The preference score of each unit is obtained via solving a 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.142 * Corresponding author. Tel.: +98 21 6636 9355. E-mail address: mostafaee_m@yahoo.com (A. Mostafaee). Chaos, Solitons and Fractals 39 (2009) 214–221 www.elsevier.com/locate/chaos