Vol.:(0123456789) 1 3 Mathematical Sciences https://doi.org/10.1007/s40096-020-00349-2 ORIGINAL RESEARCH Cost eﬃciency measurement with price uncertainty: a data envelopment analysis F. Hosseinzadeh Lotﬁ 1 · A. Amirteimoori 3 · Z. Moghaddas 2 · M. Vaez‑Ghasemi 3 Received: 15 May 2019 / Accepted: 13 August 2020 © Islamic Azad University 2020 Abstract Data envelopment analysis (DEA) technique is commonly utilized for eﬃciency assessment in a variety of ﬁelds for both theoretical and applicational purposes. In classic cost eﬃciency measurement models, the input and output data and input prices should be known for each decision-making unit (DMU). However, in real-life markets the input prices are not pre- cisely deﬁned for DMUs. In this paper, we shed light on the fact that ﬁxed prices assumption cannot reﬂect the reality of situations, because market will force lower prices if greater amounts of a product are purchased. It can be said that discounts are automatically considered in these circumstances. To this end, an innovative idea is considered to modify the classic cost eﬃciency DEA model in order to investigate the situations of real-life markets. Then, by an empirical example, a comparison between the proposed approach and the classic cost eﬃciency model is provided. Keywords Data envelopment analysis · Cost eﬃciency · Mixed integer programming · Marginal value · Piecewise linear functions Mathematics Subject Classiﬁcation 90C · 65K05 Introduction Cost eﬃciency measures the ﬁrm’s success in choosing an optimal set of inputs by minimizing total input costs. It reﬂects the diﬀerential between the current cost of a DMU and the possible minimal cost. The concept of CE can be traced back to Farrell [1], and Fare et al. [2]. They have developed this concept through a linear programming (LP) model. They considered mathematical programming tech- niques and data development analysis technique, to deter- mine the cost eﬃciency score, relative to the observed best practices. It is signiﬁcant to note that the DEA analysis determines the minimum cost necessary for each DMU, which leads to the production of the observed output. They noted that the input and output data and input prices for each of the DMU should be deﬁned in this LP model. Farrell [1] put forward the cost eﬃciency evaluation, which reﬂects the cost reductions, where adjusting the prices is not possible. In the previous studies, CE measures the closeness of a unit’s cost to the cost of the best practice unit’s that would be produced with the same bundle of outputs. Some limitations in the Farrell CE measurement are dis- cussed in the literature. In the presence of diﬀerent prices between the DMUs, Tone [3] stated that such diﬀerences do not reﬂect the diﬀerential between the current cost of the DMU and the minimal cost estimated from other DMUs, and thus, he pointed out that the use of Farrell CE measure may not yield the appropriate results. In this regard, Tone [3] for overcoming this limitation, relaxed the ﬁxed prices assumption and proposed the assessment of the DMUs in the cost space. Moreover, Cooper et al. [4] noted that due to data requirements, CE can limit the value of prices, which may vary in the short term, so existence of the accurate prices is a daunting task. As CE measure is one of the crucial factors for managers to better making decisions, researchers pay much attention to this issue and try to make modiﬁcations for its theoretical * A. Amirteimoori ateimoori@iaurasht.ac.ir 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran 3 Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran