Duality and efficiency computations in the cost efficiency model with price uncertainty

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Abstract

The calculation of cost efficiency requires complete and accurate information on the input prices at each decision making unit (DMU). In practice, however, exact knowledge of the relevant prices is difficult to come by, and prices may be subject to variation in the short term. To estimate the cost efficiency while taking price uncertainty into account, cone-ratio DEA models incorporating the available price information as weight restrictions can be applied. However, the literature lacks a clear explanation regarding the exact relationships between these two models. In this paper, through a duality study, we establish both the theoretical properties of these relationships and the characteristics of their efficiency solutions between cone-ratio DEA models and CE models, assuming there are imprecise price data. Based on the duality study, we also develop a new approach and design a lexicographic order algorithm to estimate the lower bounds of the cost efficiency measure. Our computational experiments indicate that the proposed models are robust and that the proposed algorithm is computationally simple.

Introduction

The concept of cost efficiency dates back to Farrell [12], who originated many of the ideas underlying data envelopment analysis (DEA). According to Farrell’s concept of CE, the calculation of cost efficiency requires input and output quantity data, as well as an exact knowledge of the input prices at each decision making unit (DMU).However, the data requirements to compute cost efficiency usually cannot be satisfied because exact knowledge of the prices is often difficult or impossible to obtain; thus, the actual applications are severely limited. In addition, prices can be (and often are) subject to variation over very short periods of time. The research on the estimation of cost efficiency with incomplete price information was initiated by Thompson et al. [22] and Schaffnit et al. [20]. To address situations in which prices may fluctuate within prescribed bounds, Thompson et al. [22] introduced constraints with lower and upper bounds, within which relative prices are expected to vary. In addition, the assurance region (AR) approach is used to obtain scores despite the lack of precise information regarding the input prices. It is known that, as increasingly severe constraints are placed on the weights, the derived measure of efficiency moves from one of relative technical efficiency to one of relative cost efficiency. In the absence of precise prices, Schaffnit et al. [20] also presented the DEA/AR models with input multiplier constraints, which contain information regarding the relevant prices, to estimate cost efficiency. It can be empirically demonstrated that, in the limit when the exact prices are known, the resulting objective function is identical to the measure of cost efficiency.

Further extensions have been performed. Kuosmanen and Post [13], [14] extended Farrell’s classical theoretical framework to address the situation in which price information is only available in incomplete form. Assuming incomplete price data in the form of a convex polyhedral cone, [13], [14] proposed new DEA models to derive both upper and lower bounds for Farrell’s cost efficiency, which can provide a better approximation of the true CE measure. To account for incomplete price information, Camanho and Dyson [7] applied standard weight restriction techniques in the form of input cone assurance regions. They derived the estimation of upper and lower bounds for the cost efficiency measure in both the most and least favorable price scenarios. These bounds can provide robust estimates of cost efficiency in situations of price uncertainty. MostafaeeA [17] considered the situation in which the input data, the output data and the input prices are imprecise and take the form of ranges. A pair of two-level mathematical programming problems can be formulated to obtain the upper and lower bounds of the CE, and these problems are based on the dual multiplier formulation of the CE model, which can be transformed into equivalent linear CE models. They prove that, when the input prices are represented by a convex set, the upper and lower bounds of the CE can be obtained at extreme points of the convex set. Fang and Li [11] both present some counterexamples to show that the theorems of MostafaeeA [17] are not correct in general and provide an alternative proof to clarify the reason.

However, some important work still remains undone for the cost efficiency with incomplete price data. When precise prices exist and are known, the CE model can be applied to measure the cost efficiency. In the absence of precise prices, the measure of the CE can be obtained via the incorporation of the available price information as weight constraints in the standard DEA model, e.g., by means of the cone-ratio (CR) DEA model ([22], [20], [7], [2]). Weight constraints may be derived from the price and cost information. However, the literature lacks a clear explanation of the exact relationship between these two approaches: the CR DEA models and the CE models, assuming there is imprecise price data. Inspired by Park [18], who first clarified the relationships between the multiplier and the envelopment IDEA models [10], [15], [23], [21], [4], and revealed how to solve them, in this paper, we establish theoretical properties of the relationships and characteristics of these models’ efficiency solutions between these two approaches: the CR DEA models and the CE models, assuming there is imprecise price data. We demonstrate that, whereas incorporating the imprecise price data directly into the CE model (the primal model) yields a lower bound for the efficiency, the incorporation of the same imprecise price data as weight restrictions into the CR DEA model (the dual model) yields an upper bound for the efficiency. Specifically, as the input prices get more accurate (i.e., as the multiplier cones get smaller), the upper and lower CE measures converge to the measure of Farrell’s CE model.

When the prices are imprecise, the cost efficiency measure calculated from the imprecise price data should be variable. To demonstrate that the variable efficiency always lies with an upper bound and a lower bound, the papers alluded to in the second paragraph of this section contain several proposed approaches to obtain both lower and upper bounds on cost efficiency. However, they were based on the dual model (the CR DEA model), and they did not concern the primal model (the CE model), which is currently under consideration. In this paper, we develop two distinct approaches that, together with the duality theory, make it possible to figure out the problem’s solutions. While the articles cited in the second paragraph of the introduction develop several different algorithms to obtain the lower bound, some of these studies have drawbacks or limitations. For instance, to obtain the lower bound, Kuosmanen and Post [13], [14] define theset WV, which may be non-convex, and their approach complicates the process of operationalizing the model they presented for computing the lower bound of CE. Camanho and Dyson [7] mentioned that the assessment of the lower bound in the least favorable light can lead to very low Pessimistic CE estimates, but they did not give a clear managerial interpretation in applications with multiple output dimensions. To overcome these shortcomings, in the current paper, we develop a new method to achieve a lower bound on the cost efficiency measure. In summation, the main contributions of this paper are as follows: (a) the development of a duality theory, which clarifies the relationship between the CE models and the cone-ratio models; (b) the development of a computational method to compute the lower bound for the CE measure.

The paper is organized as follows: Section 2 describes the CE model and the CR DEA model, which is needed in the next section. Section 3 develops the duality results, which clarify the relationships and characteristics of the efficiency solutions between the primal models (the CE model) and the dual models (the CR DEA model). A new approach and a lexicographic order algorithm are developed in Section 4, and the experimental results are also reported in Section 4. Section 5 concludes the paper.

Section snippets

Theoretical underpinnings

Consider a set of n DMUs, each using m inputs to produce s outputs. For each DMUj (j=1,,n), we denote, respectively, the input and output vectors as (Xj,Yj), j=1,,n, where Xj=(x1j,,xmj) and Yj=(y1j,,ysj). We also employ X to denote the m×n matrix of inputs and Y to denote the s×n matrix of outputs. We assume that X>0 and that Y>0.

Following Farrell [12], the cost efficiency measure for a DMU, when the input prices are known exactly, can be obtained by solving the following equation:θo=mini=

Duality results

We are motivated to develop the following propositions and proofs so that we can establish the theoretical relationship between the CR DEA model and the CE model with imprecise data.

In this section, we first study the duality for perfect efficiency in Definition 2 or the UCE-P Model (8). To accomplish this goal, let us formulate the following model:

The UMDEA-P model:a0=maxr=1sμryros.t.r=1sμryrji=1mvixij0j=1,,ni=1mvixio=1i=1,,mforsomevivk[wioLwkoU,wioUwkoL]i,k=1,,mk>ihere the term

Methods

According to Farrell’s concept of CE, the calculation of cost efficiency requires input and output quantity data as well as exact knowledge of the input prices at each decision making unit (DMU).In situations where there is incomplete price information, the classical CE models cannot be used. However, our duality results enable us to calculate the cost efficiency in this situation. We can use Model (6) to obtain the upper bound for the CE measure and Model (12) to yield the lower bound for the

Conclusions

Cost efficiency (CE), as a DEA model, evaluates the ability of a DMU to produce the current outputs at minimal cost, given the input price paid at each DMU. The calculation of cost efficiency requires that the prices are fixed and known at each DMU. However, an exact knowledge of the prices is often difficult to obtain in actual applications, and prices may be (and often are) subject to variation in the short term.

The assessment regarding price uncertainty has been effectively addressed by

Acknowledgements

This research was partly supported by the Fundamental Research Funds for the Central Universities, the Asian Research Center of Nankai University, the project sponsored by SRF for ROCS, SEM and National Natural Science Foundation of China, Research No. 61065009.

References (22)

  • C Schaffnit et al.

    Best practice analysis of bank branches: an application of DEA in a large Canadian bank

    European Journal of Operational Research

    (1997)
  • Cited by (14)

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    • Inverse DEA based on cost and revenue efficiency

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      Färe, Grosskopf, & Lovell, 2013) developed a linear programming based model to estimate the cost efficiency of DMUs. Some theoretical extensions are proposed in the literature in presence of price uncertainty (see for instance (Kuosmanen & Post, 2003), (Kuosmanen & Post, 2001), (Fang & Li, 2013), (Fang & Hecheng, 2013), (Mostafaee & Saljooghi, 2010)). Cost efficiency analysis is used in many real world application like banks ((Camanho & Dyson, 2005), (Weill, 2004), (Paradi & Zhu, 2013)), insurance ((Tone & Sahoo, 2005)), power plants ((Hiebert, 2002)), agriculture ((Rungsuriyawiboon & Hockmann, 2015)) etc.

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      In this paper, we focus on how to use imprecise data (i.e., a mixture of ordinal and interval data) to measure the performance of the DMUs from both optimistic and pessimistic perspectives. Numerous articles have been published on imprecise DEA (IDEA) (Amirteimoori & Kordrostami, 2005; Azizi, 2013b; Azizi & Ganjeh Ajirlu, 2011; Cook & Zhu, 2006; Cooper, Park, & Yu, 2001a; Despotis & Smirlis, 2002; Emrouznejad, Rostamy-Malkhalifeh, Hatami-Marbini, & Tavana, 2012; Fang & Hecheng, 2013; Kao, 2006; Kao & Liu, 2000; Kim, Park, & Park, 1999; Lee, Park, & Kim, 2002; Park, 2007; Smirlis, Maragos, & Despotis, 2006; Wang, Greatbanks, & Yang, 2005; Wang, Luo, & Liang, 2009; Zhu, 2003). Most of them, however, are radial models.

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