Improving the frontier in DEA models

ISSN 1064-5624, Doklady Mathematics, 2016, Vol. 94, No. 3, pp. 715–719. © Pleiades Publishing, Ltd., 2016. Published in Russian in Doklady Akademii Nauk, 2016, Vol. 471, No. 4, pp. 398–402. CONTROL THEORY Improving the Frontier in DEA Models1 V. E. Krivonozhkoa,b*, F. R. Førsundc,**, and A. V. Lycheva*** Presented by Academician of the RAS V.A. Gelovani April 25, 2016 Received July 1, 2016 Abstract—Some inadequate results may appear in the DEA models as in any other mathematical model. In the DEA scientific literature several methods were proposed to deal with these difficulties. In our previous paper, we introduced the notion of terminal units. It was also substantiated that only terminal units form necessary and sufficient sets of units for smoothing the frontier. Moreover, some relationships were established between terminal units and other sets of units that were proposed for improving the frontier. In this paper we develop a general algorithm for improving the frontier. The construction of algorithm is based on the notion of terminal units. Our theoretical results are verified by computational results using real-life data sets and also confirmed by graphical examples. DOI: 10.1134/S1064562416060181 INTRODUCTION AND PROBLEM STATEMENT The DEA (Data Envelopment Analysis) models like any other mathematical model may produce inadequate results when they apply to the real-life problems. In the DEA scientific literature several methods were proposed to deal with such difficulties. Already Farrell [1] introduced artificial observations in the primal space of inputs and outputs in order to secure convex isoquants. Another way to improve the frontier is to insert restrictions on the dual variables. An elegant and subtle approach was developed in the DEA models, which was based on incorporating domination cones in the dual formulations of DEA models. A number of outstanding papers developed applications of domination cones to the DEA models [2–5]. However, it may be difficult for a decision-maker to determine cones in the space that is a dual to the space of inputs and outputs where a production possibility set is constructed [6]. 1 The article was translated by the authors. aNational University of Science and Technology “MISiS”, Moscow, Russia bDepartment of Information Technologies and Computing Systems, Russian Academy of Sciences, Moscow, Russia cDepartment of Economics, University of Oslo, Oslo, Norway * e-mail: [email protected] ** e-mail: [email protected] *** e-mail: [email protected] Bougnol and Dulá [7] introduced their definition of anchor units for the case of variable returns to scale and multiple inputs and outputs. They also elaborated algorithm for finding anchor units. However, their algorithm may generate just usual efficient units that cannot be used for smoothing the frontier. The paper [8] developed further the super-efficiency method for discovering anchor units in the BCC model. At the same time, their method does not reveal all efficient units that may be the point of departure for improving the frontier in BCC models. Moreover, their method of frontier improvement may turn initially efficient units into inefficient ones. The notion of terminal units was defined in [9, 10]; it was substantiated that only terminal units give a necessary and sufficient set of units as a basis for smoothing the frontier in the DEA models. Moreover, some relationships between different sets of units that may cause inadequacies in the DEA models were established. In this paper we developed a general algorithm for smoothing the frontier in the DEA models. We take the notion of terminal units as a point of departure for construction of algorithm. Our theoretical results are verified by computational experiments using real-life data sets and also illustrated by graphical examples. Consider a set of n observations of actual production units (Xj, Yj), j = 1, 2, …, n, where the vector of outputs Yj = (y1j, y2j, … , yrj) ≥ 0, j = 1, 2, …, n, is produced from the vector of inputs Xj = (x1j, x2j, …, xmj) ≥ 0, j = 1, 2, …, n. The primal input-oriented BCC model [11] can be written in the form min θ 715 716 KRIVONOZHKO et al. subject to n ∑X λ j j + S − = θ X 0, j =1 n ∑Y λ j + j − S = Y 0, (1) j =1 n ∑λ j j =1 s k− ≥ 0, = 1, λ j ≥ 0 , j = 1,2,…, n, k = 1,2,…, m, s i+ ≥ 0, i = 1,2,…, r, where Xj = (x1j, x2j, …, xmj) and Yj = (y1j, y2j, …, yrj) represent the observed inputs and outputs of production units j = 1, 2, …, n, S − = (s1−, s 2−, … , s m− ) and S + = (s1+, s 2+, … , s r+ ) are vectors of slack variables. In this primal model the efficiency score θ of the production unit ( X 0,Y 0 ) is found; ( X 0,Y 0 ) is any unit from the set of production units (Xj, Yj), j = 1, 2, …, n. Notice that we do not use an infinitesimal constant ε (a non-Archimedean quantity) explicitly in the DEA models, since we suppose that each model is solved in two stages in order to separate efficient and weakly efficient units. Definition 1 [6]. Unit (X0, Y0) ∈ T is called efficient with respect to the input-oriented BCC model if any optimal solution of (1) satisfies: (a) θ* = 1, (b) all slacks s k– , s i+ , k = 1, 2, …, m, i = 1, 2, …, r are zero. If the first condition (a) in Definition 1 is satisfied, then unit (X0, Y0) is called input weakly efficient with respect to the BCC input-oriented model. Definition 2 [6]. Activity (X ', Y ') ∈ T is weakly Pareto efficient if and only if there is no (X, Y) ∈ T such that X < X ' and Y > Y '. The production possibility set T for the BCC model can be written in the form  T = ( X ,Y ) X ≥  n n ∑ X λ ,Y ≤ ∑Y λ , j j =1 j j j j =1 (2)  λ j = 1, λ j ≥ 0, j = 1,2, ..., n .  j =1 Definition 3 [9]. We call an efficient (vertex) unit terminal unit if an infinite edge is going out from this unit. According to [9] only vectors of the following forms d k = (d k ,0) ∈ E m + r , k = 1, … , m , g i = −(0, g i ) ∈ E m + r , i = 1, 2, …, r can be the direction vectors of infinite n ∑ edges of set T, where d k = (0, ...1, ...,0) ∈ E m (the unity is in kth position) and g i = (0, ...,1, ...,0) ∈ E r (the unity is in ith position). A set of such direction vectors for given terminal unit we call terminal directions associated with this unit. We denote the set of terminal units with respect to the production possibility set (2) by Tterm. The models for determination all terminal units of set T are given in the paper [9]. In the paper [9] it was proved that only terminal units give a necessary and sufficient set of units as a basis for creating artificial units in order to improve the frontier. MAIN RESULTS Under the elaboration of the algorithm for smoothing the frontier we stick to the following principles: (a) all efficient units have to stay efficient after the frontier transformation; (b) every inefficient unit will be projected on the efficient part of the frontier. First of all, all terminal units are determined. Models for discovering of such units are described in [9]. Then two-dimensional sections are constructed for every terminal unit. For our purposes we need three types of sections. Let us define a section of the frontier with a twodimensional plane [12] Sec( X o,Y o, d1, d 2 ) = { ( X ,Y ) ( X ,Y ) ∈ Pl( X o,Y o, d1, d 2 ) ∩ WEff PT } , where Pl( X o,Y o, d1, d 2 ) is a two-dimensional plane going through point ( X o,Y o ) and it spanned by vectors d1, d 2 ∈ E m + r , WEffpT is a set of weakly Pareto efficient units. In our exposition we will use the following three types of sections. 1. Input isoquant, section S1. In this case we take the following directions d1 = (e p,0) ∈ E m + r , d 2 = (e s ,0) ∈ E m + r , where e p and e s are m -identity vectors with a one in position p and s , respectively. 2. Output isoquant, section S2. In this case vectors for cutting the frontier are determined as follows d1 = (0, e p ) ∈ E m + r , d 2 = (0, e s ) ∈ E m + r , e p and e s are r -identity vectors with a one in position p and s , respectively. 3. Section S3 reflects the dependence between variables y p and x s . For construction of such dependence we took directions: d1 = (0, e p ) ∈ E m + r , where e p is r identity vector with a one in position p , d 2 = (e s ,0) ∈ E m + r , e s is m -identity vector with a one in position s . Figure 1 represents an input isoquant for some terminal unit Z k . If artificial unit C is inserted somewhere in the region limited by rays Z k B , Z k A and axis DOKLADY MATHEMATICS Vol. 94 No. 3 2016 IMPROVING THE FRONTIER Ox i , then unit Z k becomes just an efficient unit. Such operations can be accomplished for every terminal unit and for every type of sections going through this unit and that were described above. Observe that components of artificial unit C coincide with corresponding components of unit Z k except coordinates that correspond to variables x i and x j . In other words, unit C belongs to the section that is going through point Z k and is determined by variables x i and x j . Similar figures can be depicted for other sections. For our purpose, it is sufficient to consider only three types of sections described above. Now, we describe a general scheme of the Algorithm for smoothing the frontier in DEA models. ALGORITHM Part 1. Smoothing Terminal Units 1. Compute efficiency scores for all production units j = 1, 2, …, n . 2. Find terminal units, i.e., determine set of terminal units Tterm . 3. For every terminal unit j ∈ Tterm do item 4. 4. For every terminal direction do item 5. 5. For every two-dimensional section that contains this direction do: (a) Insert an artificial unit on the two-dimensional section outside the PPS in the current iteration. (b) Compute efficiency scores for all units. If the number of an efficient unit is less than the original number, then move the artificial unit closer to the frontier, go to the beginning of step (b). (c) Store the new artificial unit. 6. Include all artificial units in the set of production units of the PPS. Part 2. Correction of the First Part 1. Compute efficiency scores for all production units including artificial ones. 2. Find units that were efficient and become inefficient. 3. Find artificial units that caused the situations in the previous item. 4. While there exist artificial units that have to be corrected do: (a) Move all artificial units closer to the frontier. (b) Compute efficiency scores. 5. Delete inefficient artificial units. 6. Compute efficiency scores for all units including also artificial production units. DOKLADY MATHEMATICS Vol. 94 No. 3 2016 717 xj ~ T Zl D1 С1 Zk A O C B D2 xi Fig. 1. Terminal unit Z k turns into just an efficient unit. Part 3. Smoothing the Weakly Efficient Faces of the Frontier 1. While there exist units that are projected on the weakly efficient faces do: (a) Move projection on the weakly efficient faces along the radial direction outside the PPS, create artificial unit from such projection, and insert this artificial unit in the current iteration in the PPS. (b) Compute efficiency scores. (c) If the number of efficient units decreases, then decrease the distance of the new artificial unit from the frontier, go to (b). (d) Store the new artificial unit. 2. Include all artificial units in the production possibility set. Part 4. Correction of the Third Part 1. Compute efficiency scores for all production units including artificial ones. 2. Find original units that were efficient and become inefficient. 3. Find artificial units that were inserted in the model for correction in the previous part 3. 4. While there exist artificial units that should be corrected do: (a) Move all such artificial units simultaneously closer to the frontier along the radial direction. (b) Recompute efficiency scores. 5. Remove all inefficient artificial units in the model. 6. Finally, compute efficiency scores for all units in the model. Observe that during the run of the parts 1 and 3 of the Algorithm some artificial units are inserted in the model. For this reason some efficient units may turn into inefficient ones since the configuration of the production possibility set (set of vertices, set of faces 718 KRIVONOZHKO et al. mon point, unit 149, this means that curve 1 consists of weakly efficient point of the frontier except unit 149. After running the Algorithm, all inefficient units are projected onto efficient faces of the frontier, which are determined by observed and artificial units. However, the number of observed units is always finite, and this number is not sufficient for construction of the production possibility set in the whole. For this reason, artificial units are introduced in order to transform weakly efficient faces into efficient ones. However, all originally efficient faces will remain efficient ones. This is the essence of the proposed algorithm. x2 2.0 2 1 1.5 149 1.0 0.5 0 0.5 1.0 1.5 2.0 x1 Fig. 2. Input isoquant for unit 149. and their mutual disposition) may be changed. For this reason two additional stages (parts 2 and 4) are introduced in the Algorithm in order to correct such cases. This can be accomplished by moving artificial units closer to the corresponding faces. The following assertion can be proved. Theorem. After the run of the Algorithm the following results will be obtained: (1) all originally efficient units will remain efficient; (2) all terminal units are transformed into just usual efficient ones; (3) all inefficient units are projected onto the efficient faces of the frontier. In our computational experiments we used the software FrontierVision. This program allows us to visualize the multidimensional production possibility set by means of constructing two- and three-dimensional sections of the frontier. We took data from 174 Russia bank’s financial accounts for January 2009. We used the following variables as inputs: working assets, time liabilities, and demand liabilities. As output variables we took: equity capital, liquid assets, fixed assets. The Algorithm inserted 412 artificial units in the original set of units in order to smooth the frontier. Figure 2 represents two input isoquants for unit 149 that are intersections of the six-dimensional production possibility set with two-dimensional planes for unit 149. This unit in the figure is shown by white circle. Other small circles represent orthogonal projections of actual and artificial units onto the section. The curve 1 shows input isoquant for the original set T . The curve 2 is built for the transformed set Tɶ . Directions of the two-dimensional plane are determined by the following inputs: x1 is demand liabilities and x2 is time liabilities. Input isoquants have only one com- CONCLUSIONS In this paper, we proposed a general algorithm for improving the frontier in the DEA models. Under the construction of the Algorithm we adhered to the following principles: (a) all efficient faces of the production possibility set have to remain efficient, (b) all inefficient units are projected on the efficient faces. Therefore the Algorithm includes new artificial units in such a way that all initially efficient units remain efficient. The production possibility set is a polyhedral set. For this reason an adding of even one artificial unit may change the configuration of this set significantly. In order to avoid such situations correction parts are included in the algorithm, which position new artificial units closer to the hyper surface if configuration of the production possibility set may be destroyed. Certainly this may be done by several methods. In our opinion, we have chosen the most simple and reliable method: movement of artificial units take place perpendicular to the efficient faces (parts 1 and 2 of the algorithm) or in radial directions (parts 3 and 4). Strictly speaking, only a general scheme of the Algorithm was described in detail, since Algorithm can be realized in many different forms that depend on original program modules used for Algorithm constructions, sizes of the datasets that has to be analyzed, socio-economic areas where DEA models are used and so on. Computational experiments using real-life datasets confirmed that the Algorithm works reliably and improves the frontier significantly. The improved frontier may significantly increase the accuracy of the DEA models. ACKNOWLEDGMENTS The research was carried out with financial support from the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISiS” (agreement no. 02.A03.21.004). The reported study was also partially supported by RFBR, research project no. 1407-00472. DOKLADY MATHEMATICS Vol. 94 No. 3 2016 IMPROVING THE FRONTIER REFERENCES 1. M. J. Farrell, J. R. Stat. Soc. Ser. A 120 (2), 253–281 (1957). 2. A. Charnes, W. W. Cooper, Z. M. Huang, et al., J. Econometrics 46 (1–2), 73–91 (1990). 3. P. L. Brockett, A. Charnes, W. W. Cooper, et al., Eur. J. Operat. Res. 98 (2), 250–268 (1997). 4. G. Yu, Q. Wei, and P. Brockett, Ann. Operat. Res. 66 (1), 47–89 (1996). 5. Q. Wei, H. Yan, and L. Xiong, Eur. J. Operat. Res. 190 (3), 855–876 (2008). 6. W. W. Cooper, L. M. Seiford, and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, 2nd ed. (Springer Science and Business Media, New York, 2007). DOKLADY MATHEMATICS Vol. 94 No. 3 2016 719 7. M.-L. Bougnol and J. H. Dula, Eur. J. Operat. Res. 192 (2), 668–676 (2009). 8. E. Thanassoulis, M. Kortelainen, and R. Allen, Eur. J. Operat. Res. 218 (1), 175–185 (2012). 9. V. E. Krivonozhko, F. R. Forsund, and A. V. Lychev, J. Product. Anal. 42 (2), 151–164 (2015). 10. V. E. Krivonozhko, F. R. Forsund, and A. V. Lychev, Ann. Operat. Res. (2015). doi 10.1007/s10479-0151875-8 11. R. D. Banker, A. Charnes, and W. W. Cooper, Manage. Sci. 30 (9), 1078–1092 (1984). 12. V. E. Krivonozhko, O. B. Utkin, A. V. Volodin, et al., J. Operat. Res. Soc. 55 (10), 1049–1058 (2004).