Overall profit Malmquist productivity index under data uncertainty
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Akbarian, Dariush Article Overall profit Malmquist productivity index under data uncertainty Financial Innovation Provided in Cooperation with: Springer Nature Suggested Citation: Akbarian, Dariush (2020) : Overall profit Malmquist productivity index under data uncertainty, Financial Innovation, ISSN 2199-4730, Springer, Heidelberg, Vol. 6, Iss. 1, pp. 1-20, https://doi.org/10.1186/s40854-020-0170-0 This Version is available at: https://hdl.handle.net/10419/237193 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Financia l Innovation Akbarian Financial Innovation (2020) 6:6 https://doi.org/10.1186/s40854-020-0170-0 RESEARCH Open Access Overall profit Malmquist productivity index under data uncertainty Dariush Akbarian Correspondence: [email protected] Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran Abstract The calculation of the overall profit Malmquist productivity index (MPI) requires precise and accurate information on the input, output, input-output prices of each decision making unit (DMU). However, in many situations, some inputs and/or outputs and input-output prices are imprecise. As such, we consider the overall profit MPI problem when the input, output, and input-output prices are imprecise and vary over intervals, showing that method (MCM 54: 2827–2838, 2011) has some shortfalls. To remedy these shortfalls, we propose another method for measuring the overall profit MPI when the inputs, outputs, and price vectors vary over intervals. That is, to calculate the overall profit efficiency intervals, cone-ratio data envelopment analysis models can be applied to the incorporated information as weight restrictions. Further, we provide a new approach to calculating the upper bound of the overall profit efficiency of each DMU. A numerical example is provided for illustrating the proposed method. Keywords: Data envelopment analysis, Imprecise data, Profit Malmquist productivity index Introduction The Malmquist productivity index (MPI) is one of the most popular approaches to measuring productivity changes over time and was introduced by Malmquist (1953). Under data envelopment analysis (DEA), productivity is defined as the ratio between efficiency and is measured by MPI for the same decision making unit (DMU) in two different periods. Caves et al. (1982a;1982b) proposed an MPI as the ratio of two input distance functions to calculate the relative performance of a DMU in different periods. Färe et al. (1994) extended the approach of Caves et al. (1982a) and constructed an MPI directly using input and output data as the geometric mean of the MPIs calculated in the two base periods. Using Farrell’s (1957) methodology for the measurement of efficiency and that of Caves et al. (1982a) on the measurement of productivity, Färe et al. (1994) constructed an MPI directly from input and output data using DEA. However, this conventional profit MPI requires the input-output quantity and exact input-output prices to be available. However, in many situations, some inputs and/or outputs and input-output prices have imprecise data. Therefore, conventional profit MPI models are not suitable or applicable to measuring overall profit MPI. Asmild et al. (2007) presented a framework in which DEA was used to measure the overall efficiencies of different behavioral objectives. Furthermore, they showed how this framework could be applied to assess the effectiveness of more general behavioral goals. © The Author(s). 2020 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Akbarian Financial Innovation (2020) 6:6 Page 2 of 20 These objectives are revenue maximization, cost minimization, and profit maximization. Asmild et al. (2007) clarified the relationships between various cone-ratio DEA (CRDEA) models and those used to measure overall efficiency. Aghayi et al. () evaluated the MPI of DMUs with desirable and undesirable interval outputs. To deal with data uncertainty, a fuzzy approach was proposed by Wanke et al. (2016), who also calculated the efficiency of banks. Mashayekhi and Omrani (2016) used a fuzzy approach for sorting genetic algorithms with uncertain data. Salehpour and Aghayi (2015) calculated revenue efficiency under price uncertainty to find a solution to minimizing the worst-case performance with uncertain data. Hatami-Marbini et al. (2018) developed an overarching evaluation process for estimating the RTS of DMUs under imprecise DEA (IDEA), where the input and output data lie within bounded intervals. For more on IDEA, see Shabani et al. (2019), Ebrahimi (2018), Fazelabdolabadi (2019), Toloo et al. (2018), Shokouhi et al. (2014), Hatami-Marbini et al. (2017), Ureña et al. (2019), Zhang et al. (2019), Kao et al. (2014) and the references therein. The MPI computation using DEA with uncertain data has not been studied widely in the literature. For instance, Emrouznejad et al. (2011) studied the overall profit MPI using DEA with fuzzy and interval data. They extended the model (39) of Asmild et al. (2007) and proposed two methods for measuring the overall profit MPI when the input, output, and price vectors are fuzzy or vary over intervals (see Emrouznejad et al. (2011), models (3a) and (3b)). To the best of our knowledge, to compute the overall profit MPI of each DMU, the profit efficiency at time period t[(t+1)must be computed using the technology and input-output prices at time period t+1](t)(see (Tohidi et al. 2010; Tohidi et al. 2014). However, in Emrouznejad et al.’s (2011) method, the input-output prices in periods tand t+1 are used simultaneously, meaning the results are not reasonable (see subsection 2.2. for details). As such, this paper overcomes the shortfall of Emrouznejad et al.’s model (3b) (Emrouznejad et al. 2011). Park (2001) introduced an approach to deal with IDEA involving variable input and output. He considered the multiplier and envelopment IDEA models (Cooper et al. 1999;Leeetal.2002) and clarified the relationships between them. These models yield an upper and a lower bound on efficiency, respectively. Mostafaee and Saljooghi (2010) extended the classical cost efficiency models to include data uncertainty. However, Fang and Li (2012) showed that Mostafaee and Saljooghi’s (2010) approach had some drawbacks. Then, Fang and Li (2013) extended Park’s approach and presented an alternative IDEA method to calculate an upper and a lower bound of cost efficiency measurement in the presence of imprecise price inputs. Based on the studies of Park (2001) and Emrouznejad et al. (2011), this paper introduces alternative methods for measuring the overall profit MPI when the input, output, and input-output prices are uncertain and also measures the lower and upper bounds of overall profit MPIs. We show that the upper bound of the overall profit efficiency is obtained by incorporating uncertain data as interval data directly into the overall profit efficiency models. In addition, the lower bound of the overall profit efficiency is achieved by incorporating the same uncertain data as weight restrictions into a CR-DEA model. The main contributions of this paper are as follows: (a) we calculate the overall profit MPI assuming that input, output, and input-output prices are imprecise; (b) we characterized these imprecise data with interval methods; (c) we propose models to measure the overall profit efficiency in adjacent periods, which has a reasonable interpretation (see model (4)); (d) we establish new models to compute the
Akbarian Financial Innovation (2020) 6:6 Page 3 of 20 upper bounds for the profit efficiency measures; (e) we extend the model using CR-DEA; and (f) we demonstrate the practical aspects of our model using a numerical example. The remainder of this paper is organized as follows. “Overall profit efficiency and MPIs” section presents an overview of overall profit efficiency and Malmquist indices. “Main results” section proposes models to calculate the lower and upper bounds of the profit efficiency of each DMU within the period and for adjusted periods. In “Computational aspects” section, we develop new methods to calculate the upper bounds of the overall profit efficiency of each DMU. A numerical example is also provided in “Computational aspects” section. Finally, “Conclusions” section concludes the paper. Overall profit efficiency and MPIs MPIs measure the productivity change of a DMU between two different time periods. Färe et al. (1994,1992) developed an input based non-parametric Malmquist index using DEA. This DEA-based Malmquist productivity can be extended to measure the productivity changes of DMUs over time. Here, we discuss the overall profit efficiency and overall profit MPIs. Overall profit efficiency Consider a set of nDMUs associated with minputs and soutputs. Particularly, DMUj (j∈J=1, ..., n)consumes amount xij of input iand produces amount yrj of output r.LetXj=(x1j, ..., xmj),whereXj≥0&Xj= 0andYj=(y1j, ..., yrj)and where Yj≥0&Yj= 0. In addition, cand rare the input and output price vectors, respectively, for DMUj(j∈J=1, ..., n),wherec≥0andr≥0, c= 0, and r= 0. Asmild (2007) presented the following model for measuring the overall profit efficiency of DMUo=(xo,yo),(o=1, ..., n): max rT oy rT oyo−cT ox cT oxo s.t.−j∈Jλjyj+y≤0 j∈Jλjxj−x≤0 j∈Jλj=1, λj≥0 (1) where x,y,andλj,j∈Jare variables and the objective function of this linear program is to maximize the difference between the revenue and cost ratios for a given price vector pT o=cT o,rT ofor the DMUounder assessment. Superscript Tstands for a transposed vector. The following definition and theorem refer to (Toloo et al. 2008). Definition 1 DMUois overall profit efficient if in model (1), rT oy∗ rT oyo−cT ox∗ cT oxo=0. Theorem 1 For every optimal solution (x∗,y∗,λ∗)of (1), we have rT oy∗ rT oyo−cT ox∗ cT oxo≥0. Overall profit MPIs Emrouznejad et al. (2011) used the following model to measure the overall profit efficiency in the adjacent period:
Akbarian Financial Innovation (2020) 6:6 Page 4 of 20 Dp oxq o,yq o|p,q=t,t+1, p= q=max ϕ−θ s.t.ϕrp jTyp 0≤rq jTYqλ,∀j, θcp jTxp 0≤cq jTXqλ,∀j, λ≥0, (2) where Xpand Ypare the input and output matrices of the observed data for period p, respectively. To the best of our knowledge, to compute the overall profit MPI of DMUo, the profit efficiency of DMUp o=xp o,yp omust be computed using the technology and input-output prices at period q,(p,q=t,t+1, p= q) (see Tohidi et al. (2010;2014). However, in (2), the input-output prices in period pand qare used simultaneously. To overcome this shortfall, this paper introduces variable returns to scale overall profit efficiency in the within and adjacent periods as (3)and(4), respectively: Bp oxp o,yp o|p=t,t+1=max rp oTy rp oTyp o −cp oTx (cp o)Txp o s.t.−j∈Jλjyp j+y≤0 j∈Jλjxp j−x≤0 j∈Jλj=1, λ≥0 (3) Dq oxp o,yp o|p,q=t,t+1, p= q=max rq oTy rq oTyp o −cq oTx cq oTxp o s.t.−j∈Jλjyq j+y≤0 j∈Jλjxq j−x≤0 j∈Jλj=1, λ≥0 (4) where xp jand yp jare the input and output of DMUjin period p,respectively. Models (3)and(4) have clear interpretations. Model (3) calculates the profit efficiency of DMUp ousing the technology and input-output prices in period pand model (4)calculates the profit efficiency of DMUp ousing the technology and input-output prices in period q,(p,q=t,t+1, p= q). The following definitions and theorem refer to (Emrouznejad et al. 2011). Theorem 2 For every optimal solution (x∗,y∗,λ∗)of (3), we have rp oTy∗ rp oTyp o −cp oTx∗ cp oTxp o ≥0. Proof Model (3) has a feasible solution λo=1, λj=0, j= o. Hence, the optimal objective, denoted by rp oTy∗ rp oTyp o −cp oTx∗ cp oTxp o , is greater than or equal to 0, i.e., rp oTy∗ rp oTyp o −cp oTx∗ cp oTxp o ≥ 0. Remarks 1 The objective function values for model (4) can be less than or equal to zero. Definition 2 DMUois overall efficient if Dt+1 oxt+1 o,yt+1 o=0and Dt oxt o,yt o=0. Definition 3 The efficiency scores of models (3)and(4) are respectively computed as follows:
Akbarian Financial Innovation (2020) 6:6 Page 5 of 20 (i) If rp oTy∗ rp oTyp o −cp oTx∗ cp oTxp o ≥0 ,then ρ=1 1+rp oTy∗ rp oTyp o −cp oTx∗ cp oTxp o . (ii) If rq oTy∗ rq oTyp o −cq oTx∗ cq oTxp o ≤0 ,then ρ=1+rq oTy∗ rq oTyp o −cq oTx∗ cq oTxp o . Obviously, if ρ=1DMUois efficient and if ρ<1, DMUois inefficient. Definition 4 The overall profit MPI of DMUois defined as follows: Mo= ρt oxt+1 o,yt+1 o ρt oxt o,yt o× ρt+1 oxt+1 o,yt+1 o ρt+1 oxt o,yt o. Therefore, the following three conditions hold: (i) Mo>1 , increase productivity and observe progress; (ii) Mo<1 , decrease productivity and observe regress; and (iii) Mo=1 , no change in productivity at time t+1 compared to t. Main results Here, we consider the overall profit efficiency and overall profit MPI of DMUo,o=1, ..., n when input, output, and input-output prices are uncertain and also define an interval for the overall profit MPI of DMUo. We reiterate that there are nDMUs under consideration. Assume that xpL ij ,xpU ij and ypL kj ,ypU kj are the intervals of input iand output kof DMUj, (j∈J)in period p, respectively. Additionally, cpL io ,cpU io ,andrpL ko ,rpU ko are the intervals of the input-output prices of input iand output kof DMUo,o=1, ..., nin period p, respectively. Models (3)and(4) can be extended to the overall profit efficiency models (5) and (6) with data uncertainty, respectively: Within −period time Bp oxp o,yp o|p=t,t+1=max rp oTy rp oTyp o −cp oTx cp oTxp o s.t.−j∈Jλjyp j+y≤0 j∈Jλjxp j−x≤0 j∈Jλj=1 cp io ∈cpL io ,cpU io ,i=1, ..., m rp ko ∈rpL ko ,rpU ko ,k=1, ..., s xp ij ∈xpL ij ,xpU ij ,i=1, ..., m yp kj ∈ypL kj ,ypU kj ,k=1, ..., s λj≥0, j∈J. (5)
Akbarian Financial Innovation (2020) 6:6 Page 6 of 20 Adjacent-period time Dq oxp o,yp o|p,q=t,t+1, p= q=max rq oTy rq oTyp o −cq oTx cq oTxp o s.t.−j∈Jλjyq j+y≤0 j∈Jλjxq j−x≤0 j∈Jλj=1 cq io ∈cqL io ,cqU io ,i=1, ..., m rq ko ∈rqL ko ,rqU ko ,k=1, ..., s xq ij ∈xqL ij ,xqU ij ,i=1, ..., m yq kj ∈yqL kj ,yqU kj ,k=1, ..., s xp io ∈xpL io ,xpU io ,i=1, ..., m yp ko ∈ypL ko ,ypU ko ,k=1, ..., s. λj≥0, j∈J (6) It can be observed that models (5)and(6) are nonlinear programming programs because of data uncertainty. Of particular importance is how to solve the newly constructed profit efficiency models with data uncertainty in (5)and(6). To illustrate these issues, we introduce the following definitions, which are similar to those of Park (2001). Definition 5 (Potential profit efficiency in the within-period time) The DMUoto be evaluated is potentially profit efficient in the within-period if and only if there exists at least one set of prices cp o∈cpL o,cpU oand rp o∈rpL o,rpU oand at least one set of input-output data satisfying xp ij ∈xpL ij ,xpU ij and yp kj ∈ypL kj ,ypU kj (j∈J),sothatB p∗ o=01in model (5). Definition 6 (Perfect profit efficiency in the within-period time) The DMUoto be evaluated is perfectly profit efficient in the within-period if and only if, for all cp o∈cpL o,cpU oand rp o∈rpL o,rpU oand all input-output data, xp ij ∈xpL ij ,xpU ij and yp kj ∈ypL kj ,ypU kj (j∈J), Bp∗ o=0are satisfied in model (5). Definition 7 (Potential profit efficiency in the adjacent period) The DMUoto be evaluated is potentially profit efficient in the adjacent period if and only if there exists at leastonesetofpricesc q o∈cqL o,cqU oand rq o∈rqL o,rqU oand at least one set of inputoutput data satisfying xq ij ∈xqL ij ,xqU ij ,y q kj ∈yqL kj ,yqU kj (j∈J),x p io ∈xpL io ,xpU io and yp ko ∈ypL ko ,ypU ko (p,q=t,t+1, p= q),sothatD q∗ o=0in model (6). Definition 8 (Perfect profit efficiency in the adjacent period) The DMUoto be evaluated is perfectly profit efficient in the adjacent-period time if and only if, for all cq o∈cqL o,cqU o and rq o∈rqL o,rqU oand all input-output data, xq ij ∈xqL ij ,xqU ij ,y q kj ∈yqL kj ,yqU kj (j∈J), xp io ∈xpL io ,xpU io and yp ko ∈ypL ko ,ypU ko (p,q=t,t+1, p= q)are satisfied so that Dq∗ o=0 in model (6). 1Superscript * indicates optimality.
Akbarian Financial Innovation (2020) 6:6 Page 7 of 20 In definitions 5 and 7, the profit efficiency of DMUois measured for some data, while definitions 6 and 8 refer to the profit efficiency of DMUofor all data. Therefore, perfect profit efficiency is measured in a more rigid manner than potential profit efficiency. In the spirit of Park (2001), we can represent these definitions using the following mathematical formulations, where term UPEW-S (UPEW-P) refers to the uncertain profit efficiency of DMUp o=xp o,yp owith the technology and prices at time p, the within period, for some (for perfect (all)). Additionally, term UPEA-S (UPEA-P) refers to uncertain profit efficiency of DMUp o=xp o,yp owith technology and prices at time q,theadjacentperiod, for some (for perfect (all)) (p,q=t,t+1, p= q): The UPEW-S model: Bp oxp o,yp o|p=t,t+1=max rp oTy rp oTyp o −cp oTx cp oTxp o s.t.−j∈Jλjyp j+y≤0 j∈Jλjxp j−x≤0 j∈Jλj=1 for some{cp io ∈cpL io ,cpU io ,i=1, ..., m} for some{rp ko ∈rpL ko ,rpU ko ,k=1, ..., s} for some{xp ij ∈xpL ij ,xpU ij ,i=1, ..., m} for some{yp kj ∈ypL kj ,ypU kj ,k=1, ..., s} λj≥0. (7) The UPEW-P model: Model (7)withBp oin place of Bp oand “for all” in place of “for some.” (8) The UPEA-S model: Dq oxp o,yp o|p,q=t,t+1, p= q=max rq oTy rq oTyp o −cq oTx cq oTxp o s.t.−j∈Jλjyq j+y≤0 j∈Jλjxq j−x≤0 j∈Jλj=1 for somecq io ∈[cqL io ,cqU io ], i=1, ..., m for somerq ko ∈[rqL ko ,rqU ko ], k=1, ..., s for somexq ij ∈[xqL ij ,xqU ij ], i=1, ..., m for someyq kj ∈[yqL kj ,yqU kj ], k=1, ..., s for somexp io ∈[xpL io ,xpU io ], i=1, ..., m for someyp ko ∈[ypL ko ,ypU ko ], k=1, ..., s λj≥0. (9) The UPEA-P model: Model (9)withDq oin place of Dq oand “for all” in place of “for some.” (10)
Akbarian Financial Innovation (2020) 6:6 Page 8 of 20 Clearly, Bp o≥Bp o(Dq o≥Dq o)becausethefeasibleregionofmodel(8)(10)isalways contained within the feasible region of model (7)(9). Using models (7)and(8), we can obtain the interval of profit efficiency of DMUoin the within-period as Bp o,Bp o. Additionally, by models (9)and(10), the interval of profit efficiency of DMUoin the adjacent period can be obtained as Dq o,Dq o.Bp oDq ois the lower bound of the interval overall profit efficiency of DMUofrom the pessimistic viewpoint in the within period (adjacent period) and Bp oDq ois the upper bound of the interval overall profit efficiency of DMUofrom the optimistic viewpoint in the within period (adjacent period). Remarks 2 It is clear that model (5) is equivalent to the UPEW-S model (7)andmodel (6) is equivalent to the UPEA-S model (9). Becauseofthenotionofmaximization “for some” and “for all” inthepermissibledata, the UPEW-S (7), UPEW-P (8), UPEA-S (9), and UPEA-P models (10) are equivalent to the two-level mathematical programs (11), (12), (13), and (14), respectively: Bp o=max max rp oTy rp oTyp o −cp oTx cp oTxp o cp io ∈cpL io ,cpU io rp ko ∈rpL ko ,rpU ko xp ij ∈xpL ij ,xpU ij yp kj ∈ypL kj ,ypU kj s.t.−j∈Jλjyp j+y≤0 j∈Jλjxp j−x≤0 j∈Jλj=1 λj≥0. (11) Model (11)butwith“Bp o=min” in place of “Bp o=max”. (12) Dp o=max max rq oTy rq oTyp o −cq oTx cq oTxp o cq io ∈cqL io ,cqU io rq ko ∈rqL ko ,rqU ko xq ij ∈xqL ij ,xqU ij yq kj ∈yqL kj ,yqU kj xp io ∈xpL io ,xpU io yp ko ∈ypL ko ,ypU ko s.t.−j∈Jλjyq j+y≤0 j∈Jλjxq j−x≤0 j∈Jλj=1 λj≥0. (13) Model (13)butwith“Dp o=min” in place of “Dp o=max”. (14)
Akbarian Financial Innovation (2020) 6:6 Page 15 of 20 at time periods t and t+1(in the within period). Model (8) calculates the same profit efficiency of DMUofrom the pessimistic viewpoint. A similar argument can be put forward for model UPEA-P (10), two-level model (14), and CR-DEA model (26) (in the adjacent period). Definition 9 The lower and upper bounds of the overall profit MPIs are obtained as follows: M=ρt+1 t ρt t ×ρt+1 t+1 ρt t+1 , M=ρt+1 t ρt t ×ρt+1 t+1 ρt t+1 ,whereρp pρp p,p=t,t+1represents the optimistic (pessimistic) efficiency in the within period and is computed by model (7)(8) and definition 3. Additionally, ρp qρp q,p,q=t,t+1, p= q represents the optimistic (pessimistic) efficiency in the adjacent-period time, and is computed by model (9)(10) and definition 3. Theorem 3 Any M ≤M≤M can be considered as the overall profit MPI for DMUo. Proof See (Emrouznejad et al. 2011). Emrouznejad et al. (2011) divided the overall MPI of any DMUointo six classes, as follows: •No change in productivity class. This class includes all the DMUs with constant productivity, that is, Eo={DMUj:Mj=Mj=1}. •Fully increasing productivity class. This class includes all the DMUs with increasing productivity and observed progress under the pessimistic viewpoint, that is, E++ ={DMUj:1<Mj≤Mj}. •Fully decreasing productivity class. This class includes all the DMUs with decreasing productivity and observed regress under the optimistic viewpoint, that is, E−− ={DMUj:Mj≤Mj<1}. •Partially increasing productivity class. This class includes all the DMUs with increasing productivity under the optimistic viewpoint and no change in productivity under the pessimistic viewpoint, that is, E+={DMUj:Mj=1, Mj>1}. •Partially decreasing productivity class. This class includes all the DMUs with decreasing productivity under the pessimistic viewpoint and no change in productivity under the optimistic viewpoint, that is, E−={DMUj:Mj<1, Mj=1}. •Partially increasing–decreasing productivity class. This class includes all the DMUs with increasing productivity under the optimistic viewpoint and decreasing productivity under the pessimistic viewpoint, that is, E={DMUj:Mj<1<Mj}. Computational aspects As mentioned in the previous section, we can use CR-DEA models (25)and(26) to obtain the lower bounds of the overall profit efficiency of DMUofrom the pessimistic viewpoint in the within Bp oand adjacent periods (Dq o), respectively. Regarding the upper bounds, models (27)and(28) are nonlinear two-level programs and cannot be converted to linear one-level programs. Therefore, we propose new methods to achieve the upper bounds as follows. We define θ(v,μ) =
Akbarian Financial Innovation (2020) 6:6 Page 16 of 20 min μo|μo≥m i=1vp ixp ij −s k=1μp kyp kj,vp i,μp k≥0. It can be shown that θ(v,μ) is piecewise linear, piecewise continuous, and a convex function. Moreover, the feasible spaces of (27)and(28) are bounded. As such, models (27)and(28) have bounded optimal solutions that occur on boundary of the feasible spaces. Therefore, we have the following propositions: Proposition 1 The optimal objective value of (27)isequalto: max s k=1 γ(j)∗ kyUp kj −yLp kj + s k=1 μ(j)p∗ kyLp kj − m i=1 v(j)p∗ ixLp ij − m i=1 ω(j)∗ ixUp ij −xLp ij ,j=1, ..., n, (29) where γ(j)∗,v(j)p∗,μ(j)p∗,ω(j)∗(j=1, ..., n)are the optimal solutions of the following linear model: max s k=1 γkyUp kj −yLp kj + s k=1 μp kyLp kj − m i=1 vp ixLp ij − m i=1 ωixUp ij −xLp ij ,j=1, ..., n s.t. s k=1 γkyUp ko −yLp ko + s k=1 μp kyLp ko =1 m i=1 ωixUp io −xLp io + m i=1 vp ixLp io =1 cpL io cpU ko vp k≤vp i≤cpU io cpL ko vp ki,k=1..., m,k>i rpL lo rpU κo μp κ≤μp l≤rpU lo rpL κo μp κl,κ=1, ..., s,κ>l 0≤γk≤μp k 0≤ωi≤vp i μp k≥0, k=1, ..., s vp i≥0, i=1, ..., m. (30) Proposition 2 The optimal objective value of (28)isequalto: max s k=1 γ(j)∗ kyUq kj −yLq kj + s k=1 μ(j)q∗ kyLq kj − m i=1 v(j)q∗ ixLq ij − m i=1 ω(j)∗ ixUq ij −xLq ij ,j=1, ..., n, (31) where γ(j)∗,v(j)q∗,μ(j)q∗,ω(j)∗(j=1, ..., n)are the optimal solutions of the following linear model: max s k=1 γkyUq kj −yLq kj + s k=1 μq kyLq kj − m i=1 vq ixLq ij − m i=1 ωixUq ij −xLq ij ,j=1, ...n
Akbarian Financial Innovation (2020) 6:6 Page 17 of 20 Table 1 Input and output data for the five DMUs in Example 1 at times tand t+1. Extracted from Emrouznejad et al. (2011) DMUjx1jx2jy1jy2j tt+1tt+1tt+1tt+1 1 (12, 15) (10, 14) (0.21, 0.48) (0.32, 0.5) (138, 144) (130, 140) (21, 22) (20, 23) 2 (10, 17) (11, 15) (0.1, 0.7) (0.21, 0.4) (143, 159) (137, 150) (28, 35) (24, 30) 3 (4, 5) (3, 7) (0.16, 0.35) (0.22, 0.42) (157, 198) (146, 160) (21, 29) (20, 30) 4 (19, 22) (14, 23) (0.12, 0.19) (0.31, 0.39) (158, 181) (159, 170) (21, 25) (25, 32) 5 (14, 15) (17, 18) (0.06, 0.09) (0.1, 0.17) (157, 180) (160, 189) (28, 40) (18, 35) s.t. s k=1 γkyUp ko −yLp ko + s k=1 μq kyLp ko =1 m i=1 ωixUp io −xLp io + m i=1 vq ixLp io =1 cqL io cqU ko vq k≤vq i≤cqU io cqL ko vq ki,k=1..., m,k>i rqL lo rqU κo μq κ≤μq l≤rqU lo rqL κo μq κl,κ=1, ..., s,κ>l 0≤γk≤μq k 0≤ωi≤vq i μq k≥0, k=1, ..., s vq i≥0, i=1, ..., m μ0free. (32) In model (30)(32) we maximize the linear objective functions individually and then calculate the highest using (29)(31). Example. We consider five DMUs with two inputs and two outputs, as per Table 1. Table 2shows the interval price vectors at time tand t+1. Using models (25)and (30), the interval of profit efficiency for DMUoin the within period is Bp o,Bp oand using models (26)and(32) the interval of profit efficiency for DMUoin the adjacent period time is Dq o,Dq o. The overall profit MPIs are shown in Table 3.Asper Table 3,DMU4is classified in the fully increasing productivity class, which is the Table 2 Numerical example 1 DMUj(r1j,r1j)(r2j,r2j)(c1j,c1j)(c2j,c2j) tt+1tt+1tt+1tt+1 1 [10, 12] [11, 15] [30, 35] [23, 30] [100, 110] [110, 115] [50, 55] [40, 50] 2 [9, 10] [8, 9] [27, 28] [24, 29] [110, 115] [114, 117] [40, 44] [42, 45] 3 [8, 9] [5, 9] [25, 27] [25, 26] [105, 110] [102, 109] [42, 45] [42, 50] 4 [9, 11] [7, 12] [29, 31] [23, 26] [107, 115] [114, 115] [50, 57] [49, 52] 5 [10, 11] [10, 14] [28, 31] [24, 30] [111, 117] [110, 114] [47, 62] [42, 52] The price vector data for the five DMUs in Example 1 at times tand t+1
Akbarian Financial Innovation (2020) 6:6 Page 18 of 20 Table 3 Numerical example 1 DMUj(Bt j(xt j,yt j),Bt j(xt j,yt j)) (Bt+1 j(xt+1 j,yt+1 j),Bt+1 j(xt+1 j,yt+1 j)) (Dt j(xt+1 j,yt+1 j),Dt j(xt+1 j,yt+1 j)) (Dt+1 j(xt j,yt j),Dt+1 j(xt j,yt j)) MjMj 1 [0.733,1.404] [0.493, 1.547] [0.239,1.548] [0.036,1.279] 0.459 1.887 2 [0.918,1.435] [0.143,1.673] [0.190,1.412] [0.453,0.765] 0.1434 1.450 3 [0.687, 1.001] [0.998,1.09] [0.0381,1.565] [0.503,1.674] 0.246 2.975 4 [0.322,1.376] [0.383, 1.402] [0.323,1.316] [0.100,1.807] 1.665 1.809 5 [0.209,1.173] [0.001,1.143] [0.361,1.322] [0.296,1.354] 0.217 1.614 The interval of profit efficiency and overall profit MPI of the five DMUs in Example 1
Akbarian Financial Innovation (2020) 6:6 Page 19 of 20 observed progress under the pessimistic viewpoint. Other DMUs are classified in the partially increasing–decreasing productivity class and they thus have increasing productivity under the optimistic viewpoint and decreasing productivity under the pessimistic viewpoint. DMU3has the highest productivity progress of 2.975 under the optimistic viewpoint and DMU5the highest productivity decrease of 0.217. According to the optimistic viewpoint, all DMUs can be ranked by their productivity progress in the order DMU3DMU1DMU4DMU5DMU2. However, according to the pessimistic viewpoint, the productivity regress is in the order DMU4DMU1DMU3DMU5 DMU2. Obviously, the productivity increase ranking may differ from the productivity decrease one. Conclusions Conventional DEA can be used to compute the productivity changes of a DMU over time under the profit MPI model, provided that the input, output, input costs, and output prices are known and exact for each DMU. However, in many situations, some inputs and/or outputs and input-output prices are imprecise. However, the conventional profit MPI model is not suitable to deal with inexact prices. Emrouznejad et al. (2011) studied the overall profit MPI using DEA with imprecise data and proposed two novel methods for measuring overall profit MPI. In this paper, we showed their method has some shortfalls. To overcome these shortfalls, we reformulated the conventional profit MPI model as an IDEA model by incorporating the available information into profit efficiency models and the same information into CR-DEA models in the form of a cone-ratio weight restriction. Additionally, the lower bounds of profit efficiency were easily calculated by solving a linear one-level program. Regarding the upper bounds, we proposes a new approach of solving nlinear programming problems for each bound. This is the penalty we pay to calculate the upper bound of the overall profit efficiency when the data are inexact. We also presented a numerical example to demonstrate the applicability of the proposed framework. Abbreviations CR-DEA: Cone-ratio DEA; DEA: Data envelopment analysis; DMU: Decision making unit; IDEA: Imprecise DEA; MPI: Malmquist productivity index; UPEA-P: Uncertain profit efficiency of DMUoin the adjacent period for perfect (all); UPEA-S: Uncertain profit efficiency of DMUoin the adjacent period for some; UPEW-P: Uncertain profit efficiency of DMUoin the within period for perfect; UPEW-S: Uncertain profit efficiency of DMUoin the within-period time for some Acknowledgments The author would like to thank the four anonymous reviewers and the editor for their insightful comments and suggestions. Authors’ contributions The author read and approved the final manuscript. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Availability of data and materials Data used in this paper were extracted from Emrouznejad et al., “An overall profit Malmquist productivity index with fuzzy and interval data,” Mathematical and Computer Modeling 2011; 54: 2827–2838. Competing interests The author declares that he has no competing interests. Received: 25 April 2019 Accepted: 6 January 2020
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