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Data envelopment analysis of integer-valued inputs and outputs
Computers & Operations Research 33 (2006) 3004 – 3014 www.elsevier.com/locate/cor
Data envelopment analysis of integer-valued inputs and outputs Sebastián Lozano∗ , Gabriel Villa Department of Industrial Management, University of Seville, Spain Available online 30 March 2005
Abstract This paper addresses DEA scenarios whose inputs and outputs are naturally restricted to take integer values. Conventional DEA models would project the DMU onto targets that generally do not respect such type of integrality constraints. Although integer-valued inputs and outputs can be considered as a special case of ordinal inputs and outputs, the use of that type of models has many drawbacks. In this paper a MILP DEA model that guarantees the required integrality of the computed targets is proposed.
Scope and purpose Data envelopment analysis (DEA) is a well-known OR technique for evaluating the relative efficiency of a set of similar decision making units (DMU). The number of applications of DEA is large, covering fields as diverse as finance, health, education, manufacturing, transportation, etc. Conventional DEA models are based on Linear Programming and consider continuous inputs and outputs. However, there are many occasions in which some inputs and/or outputs can only take integer values. Normally, it is assumed that the real-valued results of DEA would be rounded somehow ex post. In this paper, a different approach is proposed, consisting in introducing new DEA concepts and models that explicitly take into account integer inputs and outputs, thus guaranteeing ex ante the feasibility of the results. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Data envelopment analysis; Integer values; Mixed-integer linear program
∗ Corresponding author. Escuela Superior de Ingenieros, Camino de los Descubrimientos, 41092 Sevilla, Spain. Fax:
+34 95448 7329. E-mail address: [email protected] (S. Lozano). 0305-0548/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.02.031
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1. Introduction Data envelopment analysis (DEA) is a well-known family of mathematical programming tools for assessing the relative efficiency of a set of comparable processing units (a.k.a. decision making units, DMU). For an extensive description of this technique the reader is referred to existing textbooks on the subject (e.g., [1,2]). One of the strong points of DEA is its non-parametric character, which means that only the observed inputs consumption values and outputs production amounts are needed in order to properly assess the relative efficiencies of the DMU. The way to do this is extrapolating, from the observed sample of inputs and outputs, a set of possible operating points, assuming some technology. The most common technologies are constant return to scale (CRS) and variable return to scale (VRS), both of which consider linear combinations of the inputs and outputs of the existing DMU. Let xij and ykj the amounts of input i and output k respectively, corresponding to DMU j for i = 1, 2, . . . , m and k = 1, 2, . . . , p and j = 1, 2, . . . , n. Let x¯j and y¯j the corresponding vector columns for DMU j and X = (x¯1 , x¯2 , . . . , x¯n ) and Y = (y¯1 , y¯2 , . . . , y¯n ) the complete input and output matrices. The production possibility sets corresponding to CRS and VRS technologies are TCRS = (x, ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j 0 ∀j x¯ X¯ y¯ Y ¯ , ⎧ ⎫ ⎨ ⎬ ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j 0 ∀j j = 1 x¯ X ¯ y¯ Y ¯ . TVRS = (x, ⎩ ⎭ j
In each of these scenarios, the efficient frontier (technical efficiency in the VRS case) is a subset of the corresponding production possibility set formed by all non-dominated operating points, i.e., eff TCRS = (x, ¯ y) ¯ ∈ TCRS : ∀(x¯ , y¯ ) ∈ TCRS [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TCRS , eff = (x, ¯ y) ¯ ∈ TVRS : ∀(x¯ , y¯ ) ∈ TVRS [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TVRS . TVRS It is relatively common in DEA for some of the inputs (e.g., number of employees, number of desktop computers, number of vehicles, etc.) and/or outputs (e.g., number of cases solved by a police unit, number of questions rightly answered by a student, etc.) be integer. Using the technologies defined above would allow such input or output dimensions to take non-integer values and that leads to the fact that the efficient operating point onto which a DMU is projected (with whatever orientation is used) is not feasible because its corresponding inputs or outputs are not integer. Not only there are occasions in which, because the amounts handled are small and/or the unit cost or importance of an input or output is big, heuristically rounding the continuous DEA projections (so that an integer-valued operating point is determined which can be used in practice) may not be reasonable. More importantly, devising a workable heuristic rounding routine is not at all trivial. Thus, in general, rounding up the fractional values of the output targets and/or rounding down the input targets obtained using a conventional DEA model may lead to an operating point that is out of the production possibility set. Fig. 1 shows a simple case with a single, constant input and two integer outputs in which rounding up the two fractional output values of the output-oriented target of DMU D leads to an infeasible projection D + . Analogously, Fig. 2 shows a simple case with two integer inputs and a single, constant output in which rounding down the two fractional values of the input-oriented target of DMU D leads to an infeasible projection D − .
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Fig. 1. Effects of rounding for single, constant input and two integer output case.
Fig. 2. Effects of rounding for two integer inputs and a single, constant output case.
From the above comments, it follows that it is safer to round down fractional output values and round up fractional input values. However, doing so leads to an operating point that, in general, may be weakly dominated by other integer-valued operating point in the production possibility set being therefore clearly inefficient. Thus, Fig. 1 shows how rounding down the output target of DMU D leads to a projection D − which is dominated by existing DMU B. Similarly, Fig. 2 shows how rounding up the input targets of DMU D leads to a projection D + which is dominated by existing DMU B .
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Instead of rounding the continuous DEA projection, one possibility would be to consider the integer inputs and outputs as ordinal, discrete (i.e., categorical) variables. Unfortunately, none of the existing ordinal-variables DEA approaches (namely [3–6]) can effectively handle the general case of multiple integer-valued inputs and outputs. Another possibility would be the free disposal hull (FDH) approach [8] which can handle integer inputs and outputs because the target must coincide with one of the existing DMU. Such alternative will be considered in the next sections where it is shown to be outperformed by the proposed approach. As a matter of fact, the approach proposed in this paper is simple and intuitive. In Section 2, an appropriate integer-constrained production possibility set is introduced, its corresponding efficient subset is defined and an input-oriented, envelop model that projects onto it is defined. Although we have assumed CRS, the definitions and the model can be trivially adapted to VRS or any other convex technology. The structure of the model is similar to a conventional DEA model (e.g., CCR-I [7]) except that its integrality constraints make it an MILP model. In Section 3 the proposed approach is illustrated on a problem from the literature and it is compared with the CCR-I solution and also with the FDH approach [8]. Finally, some conclusions are drawn.
2. Integer-valued DEA model Let I ={1, 2, . . . , m} and O ={1, 2, . . . , p} the sets of input and output dimensions respectively and let I ⊆ I and O ⊆ O the subsets of the corresponding dimensions that must be integer-valued. Naturally, all observed xij and ykj must be integer for all i ∈ I and k ∈ O . Let us assume that CRS apply and define an integer-valued CRS production possibility set TCRS
=
⎧ ⎨
(x, ˆ y) ˆ : ∃(1 , 2 , . . . , n )j 0 ∀j
⎩
xˆi
j
j xij
yˆk
j
j ykj
I
⎫ ⎬
xˆi integer ∀i ∈ . yˆk integer ∀k ∈ O ⎭
Definition. A DMU J is CRS integer-efficient if no other integer-valued operating point dominates it, [xˆ x¯ ] ∩ [yˆ y¯ ] ⇔ (x, i.e., if ∀(x, ˆ y) ˆ ∈ TCRS ˆ y) ˆ = (x¯J , y¯J ). J J Definition. The CRS integer-efficiency frontier is the set of non-dominated, integer-valued operating points, i.e.,
eff TCRS = (x, ˆ y) ˆ ∈ TCRS : ∀ (x, ¯ y) ¯ ∈ TCRS [x¯ x] ˆ ∩ [y¯ y] ˆ ⇔ (x, ¯ y) ¯ = (x, ˆ y) ˆ ⊂ TCRS . Proposition 1. If an existing DMU J is CRS efficient, then it is CRS integer-efficient, eff i.e., (x¯J , y¯J ) ∈ TCRS ⇒ (x¯J , y¯J ) ∈ (TCRS )eff .
Note that, although, according to this proposition, existing, CRS efficient DMU are also integerefficient, the converse is not true, i.e., there may be integer-efficient DMU which, when the integrality constraints are relaxed may be dominated by other operating points and therefore they are not CRS efficient.
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In order to assess the relative performance of the existing DMU in this CRS integer-valued scenario, and assuming an input orientation, the following MILP DEA model is proposed. MILP-DEA:
− + Min J − ε si + sk s.t.
i
j xij = xi
k
∀i,
j
xi = J xiJ − si− ∀i, j ykj = yk ∀k, j
yk = ykJ + sk+ ∀k, j 0 ∀j si− , xi 0 ∀i sk+ , yk 0 ∀k, J free xi integer ∀i ∈ I yk integer ∀k ∈ O . This model has n + 2(m + p) + 1 variables of which one is free and |I | + |O | are integer. The number of constraints is 2(m + p). It is the typical input-oriented two-phase model seeking the maximal radial reduction of all inputs and a subsequent exhaustion of all possible slacks along any input or output dimension. What makes this model different is the integrality constraints imposed on the integer-valued input and output dimensions. The optimal solution gives, for each DMU J : (a) Its CRS integer-efficiency score ∗J , − )∗ , (s + )∗ , (s + )∗ , . . . , (s + )∗ , (b) Additional slacks (s1− )∗ , (s2− )∗ , . . . , (sm p 1 2 ∗ , y ∗ , y ∗ , . . . , y ∗ ). (c) An integer-valued target (x1∗ , x2∗ , . . . , xm p 1 2 The proof of the following result is included in the appendix. Proposition 2. An existing DMU J is CRS integer-efficient if and only if ∗J = 1, (si− )∗ = 0 ∀i and (sk+ )∗ = 0 ∀k. Proposition 3. The CRS integer-efficiency score computed by the proposed model is never lower than ∗ the CRS efficiency score, i.e., ∗J (CRS J ) for all DMU J. This is a direct consequence of the fact that the integer-valued CRS production possibility set is included in the CRS one, i.e., TCRS ⊂ TCRS . On the other hand, with respect to the FDH [8] approach, the corresponding production possibility set and efficient subset are respectively ⎧ ⎨
⎫ ⎬
(x, ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j ∈ {0, 1}∀j j = 1 x¯ X ¯ y¯ Y ¯ , ⎩ ⎭ j eff TFDH = (x, ¯ y) ¯ ∈ TFDH : ∀(x¯ , y¯ ) ∈ TFDH [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TFDH . TFDH =
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eff correspond to existing DMU, the Since all the operating points in the FDH efficient subset TFDH projections obtained by this approach:
(a) will have integer values in the input and output dimensions I and O respectively, and (b) will be feasible in the proposed MILP-DEA model. Proposition 4. The FDH efficiency score is never lower than the CRS integer-efficiency score computed by the proposed model, i.e., ∗J (FDH )∗ for all DMU J . J
3. Illustration In order to illustrate the proposed approach an instance from the literature [9] will be used. The problem consists in evaluating the relative performance of 47 different cellular manufacturing configurations. Each alternative configuration is considered a DMU with two inputs (number of workers and number of machines) and three outputs (related to average WIP, average flow time and average worker utilization). The output measures were obtained by simulation and in the case of WIP and flow time must be transformed so that higher outputs mean better performance. Specifically, the actual output values used for those two dimensions are the inverse (multiplied by 105 and 103 , respectively) of the values computed by the simulation. Note that the two inputs are integer-valued. In [9] the problem is solved using a conventional CCR-I approach [7]. A heuristic rounding procedure is not appropriate since the unit cost of adding a worker or a machine is important. Table 1 shows the CRS efficiency score and the targets obtained by the CCR-I model. Note that, except for the five existing efficient DMU (namely, 9, 10, 14, 22 and 47) the rest have fractional values for their inputs. The targets obtained by the proposed model, which are shown in Table 2, have integer input values for all targets. The targets obtained by the FDH approach are shown in Table 3 and they also have integer input values for all DMU. Note that:
∗ ∗ FDH ∗ (a) CRS J J for all DMU J with an average difference between the CRS and MILP scores J of 0.019 and an average difference between the FDH and MILP scores of 0.128. (b) Existing CRS efficient DMU (i.e., 9, 10, 14, 22 and 47) are also CRS integer-efficient with two additional CRS integer-efficient DMU (namely 13 and 15) that are not CRS efficient. (c) In some cases (e.g., DMU 1 and 2) the solution obtained by the proposed approach coincides with the result of rounding the fractional CCR-I input targets to the nearest integer. However, that is not always the case. Thus, for example, for DMU 15, instead of rounding down the second input, it is rounded up (i.e., from 23.04 to 24) and for the first input the target consists not just in rounding it up because that is not inside the integer-efficient possibility set so that another unit of that input must be added (i.e., from 12.60 to 14). (d) Of the three approaches only MILP and FDH provide integer input targets with the MILP targets always involving less inputs than the FDH targets (2.89 workers and 3.15 machines less on average).
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Table 1 CRS efficiency scores and targets DMU
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Score
0.939 0.744 0.831 0.943 0.711 0.862 0.938 0.850 1.000 1.000 0.750 0.883 0.976 1.000 0.960 0.948 0.816 0.793 0.853 0.909 0.925 1.000 0.955 0.707 0.610 0.697 0.753 0.864 0.860 0.875 0.871 0.854 0.584 0.614 0.676 0.725 0.805 0.854 0.744 0.801 0.559 0.530 0.582 0.621 0.611 0.729 1.000
Targets # Workers
# Machines
WIP−1
Flow time−1
Utilization
9.88 6.29 9.02 12.16 5.90 10.36 12.97 14.03 6 9 6.83 7.99 10.61 13 12.60 22.75 5.71 7.52 8.04 9.68 11.65 14 13.74 14.40 4.88 6.89 7.39 11.10 12.03 12.62 13.43 22.20 5.26 6.28 6.88 7.43 11.93 13.66 11.14 21.64 5.59 5.83 6.38 7.52 8.13 12.40 29
17.84 14.88 16.62 18.86 15.64 18.97 20.64 18.70 24 24 18.01 21.20 23.43 24 23.04 22.75 20.40 19.83 21.34 22.73 23.12 25 23.89 17.67 15.87 18.12 19.57 22.48 22.37 22.75 22.64 22.20 15.77 16.58 18.24 19.57 21.74 23.05 20.10 21.64 16.22 15.37 16.89 18.01 17.72 21.15 24
451.06 97.95 381.53 505.56 6.60 448.03 551.92 508.65 1.93 3.53 12.43 8.01 250.63 546.45 549.45 639.79 1.97 14.32 8.70 161.03 408.16 662.25 634.08 484.73 1.95 16.21 9.09 366.30 497.51 578.03 602.41 624.22 1.97 10.66 8.19 14.59 526.31 613.49 510.21 608.45 3.93 11.46 8.68 107.41 204.08 562.30 693.48
151.79 37.43 129.22 169.63 8.39 151.45 185.19 170.65 2.72 10.75 11.21 11.07 89.91 185.19 185.61 214.59 3.95 12.55 11.34 60.84 140.40 222.22 212.77 162.61 4.31 12.47 10.76 126.70 168.81 194.46 202.13 209.37 5.31 10.07 9.94 12.53 177.62 205.85 171.65 204.08 6.50 9.84 9.55 41.73 72.67 188.68 232.56
37.47 39.04 37.67 35.54 43.08 42.11 39.40 32.20 63.55 66.23 49.49 58.41 59.42 54.72 50.74 30.42 54.56 54.48 58.76 59.37 55.21 50.36 47.55 28.66 42.83 49.71 53.88 54.33 51.25 47.57 44.44 29.68 42.92 45.59 50.22 53.76 47.32 45.28 42.06 28.93 44.27 42.23 46.47 47.47 44.62 41.75 24.39
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Table 2 CRS integer-efficiency scores and targets DMU
Score
Targets
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
0.947 0.750 0.850 0.950 0.727 0.909 0.955 0.864 1.000 1.000 0.792 0.917 1.000 1.000 1.000 0.958 0.857 0.800 0.880 0.920 0.960 1.000 0.960 0.720 0.625 0.731 0.769 0.885 0.885 0.885 0.885 0.885 0.593 0.630 0.704 0.741 0.815 0.889 0.778 0.815 0.586 0.552 0.621 0.655 0.621 0.765 1.000
# Workers 10 6 10 12 6 11 13 14 6 9 7 9 11 13 14 23 6 8 9 10 12 14 14 15 5 8 8 11 12 13 14 23 5 7 8 8 12 14 12 22 5 7 8 9 8 13 29
# Machines 18 15 17 19 16 20 21 19 24 24 19 22 24 24 24 23 21 20 22 23 24 25 24 18 16 19 20 23 23 23 23 23 16 17 19 20 22 24 21 22 17 16 18 19 18 22 24
WIP−1 459.83 98.81 396.86 508.35 40.84 499.38 560.89 515.78 1.93 3.53 104.21 110.15 312.81 546.45 596.66 646.81 60.98 26.77 92.48 195.17 475.96 662.25 637.41 495.07 17.24 128.88 63.11 408.43 504.77 590.34 613.43 646.81 14.01 58.53 103.04 70.38 533.28 637.84 556.36 618.69 52.48 80.29 148.42 231.04 224.50 585.31 693.48
Flow time−1 154.64 36.91 133.99 170.57 18.82 168.07 188.20 173.04 2.72 10.75 38.90 44.11 109.42 185.19 200.73 216.95 22.22 16.49 38.49 71.83 161.33 222.22 213.88 166.08 9.16 48.89 28.27 138.91 170.59 198.40 205.83 216.95 8.18 25.54 40.67 30.63 179.76 214.03 186.69 207.52 19.29 32.04 54.83 81.77 78.38 196.40 232.56
Utilization 37.47 39.04 37.67 36.18 43.08 42.11 40.39 33.10 63.55 66.23 49.49 58.41 59.42 54.72 50.74 30.75 54.56 54.48 58.76 59.37 55.21 50.36 47.55 28.68 42.83 49.71 53.88 54.33 51.25 47.57 44.61 30.75 42.92 45.59 50.22 53.76 47.32 47.48 42.06 29.41 44.27 42.23 46.47 47.47 44.62 43.27 24.39
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Table 3 CRS efficiency scores and targets DMU
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Score
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.800 1.000 1.000 1.000 1.000 0.929 0.962 0.962 0.923 0.926 0.963 0.923 0.889 0.889 0.926 0.889 0.889 0.900 0.897 0.857 0.828 0.828 0.862 1.000
Target DMU
1 2 3 4 5 6 7 7 9 10 11 12 13 14 15 16 17 18 19 13 21 22 22 4 25 26 13 21 14 22 22 16 17 26 13 13 14 22 15 16 10 26 13 13 13 22 47
Targets # Workers
# Machines
WIP−1
Flow time−1
Utilization
19 18 19 20 15 17 18 18 6 9 10 11 12 13 14 24 7 10 11 12 13 14 14 20 8 11 12 13 13 14 14 24 7 11 12 12 13 14 14 24 9 11 12 12 12 14 29
19 20 20 20 22 22 22 22 24 24 24 24 24 24 24 24 25 25 25 24 25 25 25 20 26 26 24 25 24 25 25 24 25 26 24 24 24 25 24 24 24 26 24 24 24 25 24
451.06 97.94 381.53 505.56 6.61 448.03 550.66 550.66 1.93 3.53 3.55 8.01 250.63 546.45 549.45 639.80 1.93 3.45 8.70 250.63 408.16 662.25 662.25 505.56 1.95 3.72 250.63 408.16 546.45 662.25 662.25 639.80 1.93 3.72 250.63 250.63 546.45 662.25 549.45 639.80 3.53 3.72 250.63 250.63 250.63 662.25 693.48
149.25 33.22 126.58 169.49 6.14 149.25 185.19 185.19 2.72 10.75 11.21 6.22 86.96 185.19 181.82 212.77 2.91 12.55 5.68 86.96 136.99 222.22 222.22 169.49 2.76 12.47 86.96 136.99 185.19 222.22 222.22 212.77 2.91 12.47 86.96 86.96 185.19 222.22 181.82 212.77 10.75 12.47 86.96 86.96 86.96 222.22 232.56
37.47 39.04 37.67 35.54 43.08 42.11 39.4 39.4 63.55 66.23 49.49 58.41 59.42 54.72 50.74 29.44 54.56 54.48 58.76 59.42 55.21 50.36 50.36 35.54 42.83 49.71 59.42 55.21 54.72 50.36 50.36 29.44 54.56 49.71 59.42 59.42 54.72 50.36 50.74 29.44 66.23 49.71 59.42 59.42 59.42 50.36 24.39
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About computing times, the CCR-I and FDH approaches require a negligible amount and the proposed MILP DEA approach takes 28.51 s when using the XA commercial LP code on a AMD-K6, 128 Mbytes RAM, 500 MHz desktop computer. Larger problems (with one hundred DMU, five inputs and five outputs) have also been solved with the proposed approach and it never took longer than a few minutes. 4. Summary and conclusions In this paper a simple and efficient MILP DEA model is proposed for handling the case, relatively common, in which some of the inputs and/or outputs are integer. Except for the existing efficient DMU, the targets computed by conventional DEA models generally have fractional values in those dimensions, which may not be valid. FDH or ordinal DEA approaches cannot be considered a workable solution approach in general. In this paper a MILP model has been proposed which guarantees integer-efficiency for all targets. Appropriate definitions and propositions for the scenario addressed have been introduced and an instance from the literature has been used to illustrate the usefulness of the approach. Although only the constant return to scale, input orientation scenario is presented, the approach can be appropriately extended to all other DEA variants (VRS, output orientation, non-oriented models, additive metric, nondiscretionary variables, etc.). Appendix Proposition 2. An existing DMU J is CRS integer-efficient if and only if ∗J = 1,
∗
− ∗ = 0 ∀i and sk+ = 0 ∀k. si Proof. We must prove both the necessity and the sufficiency. About the necessity, note that if ∗J < 1
∗
∗ , would not or ∃i si− > 0 or ∃k sk+ > 0 then the integer-valued target obtained would belong to TCRS coincide with DMU J but would dominate it and therefore DMU J would not be, by definition, CRS integer-efficient. About the sufficiency, assume that the does not hold and one will again arrive at a con ∗
proposition ∗ tradiction. Thus, assume that ∗J = 1, si− = 0 ∀i and sk+ = 0 ∀k but that DMU J is not CRS integer-efficient. That means, by definition, that there must be an integer-valued point (x, ˆ y) ˆ ∈
operating TCRS (x, ˆ y) ˆ = (x¯J , y¯J ) which dominates it, i.e., xˆ x¯J and yˆ y¯J . Since x, ˆ y ˆ ∈ T there must CRS be a vector (1 , 2 , . . . , n )j 0 ∀j such that xˆi j j xij ∀i and yˆk j j ykj ∀k. Also, there is a non-negative vector J = 1 j = 0 ∀j = J such that x¯iJ = j j xij ∀i and y¯kJ = j j ykj ∀k. Since xˆ x¯J and yˆ y¯J then there must also be a vector (ˆ 1 , ˆ 2 , . . . , ˆ n )ˆ j 0 ∀j such that xˆi = j ˆ j xij ∀i and yˆk = j ˆ j ykj ∀k. The solution to the proposed model using that vector leads to xˆi = ˆ J xiJ − sˆi− xiJ
yˆk = ykJ + sˆk+ ykJ
∀i, ∀k.
Since x, ˆ yˆ ∈ TCRS it must be integer-valued in the I and O dimensions and, since x, ˆ yˆ = (x¯J , y¯J ) at least one of the above inequalities must be strict, i.e., ˆ < 1 and/or ∃i sˆi− > 0 and/or ∃k sˆk+ > 0. What is
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sure is that ˆ J − ε
i
sˆi− +
k
sˆk+ < ∗J − ε
i
− ∗
sˆi
+
k
+ ∗
sˆk
= 1.
This means that we have found a feasible solution for the proposed model which has a better objective function value than the supposedly optimal solution, thus arriving at a contradiction and proving the sufficiency clause of the proposition. References [1] Cooper WW, Seiford LM, Tone K. Data envelopment analysis. Kluwer, 2000. [2] Thanassoulis E. Introduction to the theory and application of data envelopment analysis—a foundation text with integrated software. Kluwer, 2001. [3] Banker RJ, Morey RC. The use of categorical variables in data envelopment analysis. Management Science 1986;32(12): 1613–27. [4] Kamakura WA. A Note on the use of categorical variables in data envelopment analysis. Management Science 1988;34(10):1273–6. [5] Rousseau JJ, Semple JH. Categorical outputs in data envelopment analysis. Management Science 1993;39(3):384–6. [6] Cook WD, Kress M, Seiford LM. Data envelopment analysis in the presence of both quantitative and qualitative factors. Journal of the Operational Research Society 1996;47:945–53. [7] Charnes A, Cooper WW, Rhodes E. Measuring efficiency of decision making units. European Journal of Operational Research 1978;2:429–44. [8] Tulkens H. On FDH efficiency analysis: some methodological issues and applications to retail banking, courts, and urban transit. Journal of Productivity Analysis 1993;4:183–210. [9] Shafer SM, Bradford JW. Efficiency measurement of alternative machine component grouping solutions via data envelopment analysis. IEEE Transactions on Engineering Management 1995;42:159–65.
Data envelopment analysis of integer-valued inputs and outputs Sebastián Lozano∗ , Gabriel Villa Department of Industrial Management, University of Seville, Spain Available online 30 March 2005
Abstract This paper addresses DEA scenarios whose inputs and outputs are naturally restricted to take integer values. Conventional DEA models would project the DMU onto targets that generally do not respect such type of integrality constraints. Although integer-valued inputs and outputs can be considered as a special case of ordinal inputs and outputs, the use of that type of models has many drawbacks. In this paper a MILP DEA model that guarantees the required integrality of the computed targets is proposed.
Scope and purpose Data envelopment analysis (DEA) is a well-known OR technique for evaluating the relative efficiency of a set of similar decision making units (DMU). The number of applications of DEA is large, covering fields as diverse as finance, health, education, manufacturing, transportation, etc. Conventional DEA models are based on Linear Programming and consider continuous inputs and outputs. However, there are many occasions in which some inputs and/or outputs can only take integer values. Normally, it is assumed that the real-valued results of DEA would be rounded somehow ex post. In this paper, a different approach is proposed, consisting in introducing new DEA concepts and models that explicitly take into account integer inputs and outputs, thus guaranteeing ex ante the feasibility of the results. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Data envelopment analysis; Integer values; Mixed-integer linear program
∗ Corresponding author. Escuela Superior de Ingenieros, Camino de los Descubrimientos, 41092 Sevilla, Spain. Fax:
+34 95448 7329. E-mail address: [email protected] (S. Lozano). 0305-0548/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.02.031
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1. Introduction Data envelopment analysis (DEA) is a well-known family of mathematical programming tools for assessing the relative efficiency of a set of comparable processing units (a.k.a. decision making units, DMU). For an extensive description of this technique the reader is referred to existing textbooks on the subject (e.g., [1,2]). One of the strong points of DEA is its non-parametric character, which means that only the observed inputs consumption values and outputs production amounts are needed in order to properly assess the relative efficiencies of the DMU. The way to do this is extrapolating, from the observed sample of inputs and outputs, a set of possible operating points, assuming some technology. The most common technologies are constant return to scale (CRS) and variable return to scale (VRS), both of which consider linear combinations of the inputs and outputs of the existing DMU. Let xij and ykj the amounts of input i and output k respectively, corresponding to DMU j for i = 1, 2, . . . , m and k = 1, 2, . . . , p and j = 1, 2, . . . , n. Let x¯j and y¯j the corresponding vector columns for DMU j and X = (x¯1 , x¯2 , . . . , x¯n ) and Y = (y¯1 , y¯2 , . . . , y¯n ) the complete input and output matrices. The production possibility sets corresponding to CRS and VRS technologies are TCRS = (x, ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j 0 ∀j x¯ X¯ y¯ Y ¯ , ⎧ ⎫ ⎨ ⎬ ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j 0 ∀j j = 1 x¯ X ¯ y¯ Y ¯ . TVRS = (x, ⎩ ⎭ j
In each of these scenarios, the efficient frontier (technical efficiency in the VRS case) is a subset of the corresponding production possibility set formed by all non-dominated operating points, i.e., eff TCRS = (x, ¯ y) ¯ ∈ TCRS : ∀(x¯ , y¯ ) ∈ TCRS [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TCRS , eff = (x, ¯ y) ¯ ∈ TVRS : ∀(x¯ , y¯ ) ∈ TVRS [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TVRS . TVRS It is relatively common in DEA for some of the inputs (e.g., number of employees, number of desktop computers, number of vehicles, etc.) and/or outputs (e.g., number of cases solved by a police unit, number of questions rightly answered by a student, etc.) be integer. Using the technologies defined above would allow such input or output dimensions to take non-integer values and that leads to the fact that the efficient operating point onto which a DMU is projected (with whatever orientation is used) is not feasible because its corresponding inputs or outputs are not integer. Not only there are occasions in which, because the amounts handled are small and/or the unit cost or importance of an input or output is big, heuristically rounding the continuous DEA projections (so that an integer-valued operating point is determined which can be used in practice) may not be reasonable. More importantly, devising a workable heuristic rounding routine is not at all trivial. Thus, in general, rounding up the fractional values of the output targets and/or rounding down the input targets obtained using a conventional DEA model may lead to an operating point that is out of the production possibility set. Fig. 1 shows a simple case with a single, constant input and two integer outputs in which rounding up the two fractional output values of the output-oriented target of DMU D leads to an infeasible projection D + . Analogously, Fig. 2 shows a simple case with two integer inputs and a single, constant output in which rounding down the two fractional values of the input-oriented target of DMU D leads to an infeasible projection D − .
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Fig. 1. Effects of rounding for single, constant input and two integer output case.
Fig. 2. Effects of rounding for two integer inputs and a single, constant output case.
From the above comments, it follows that it is safer to round down fractional output values and round up fractional input values. However, doing so leads to an operating point that, in general, may be weakly dominated by other integer-valued operating point in the production possibility set being therefore clearly inefficient. Thus, Fig. 1 shows how rounding down the output target of DMU D leads to a projection D − which is dominated by existing DMU B. Similarly, Fig. 2 shows how rounding up the input targets of DMU D leads to a projection D + which is dominated by existing DMU B .
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Instead of rounding the continuous DEA projection, one possibility would be to consider the integer inputs and outputs as ordinal, discrete (i.e., categorical) variables. Unfortunately, none of the existing ordinal-variables DEA approaches (namely [3–6]) can effectively handle the general case of multiple integer-valued inputs and outputs. Another possibility would be the free disposal hull (FDH) approach [8] which can handle integer inputs and outputs because the target must coincide with one of the existing DMU. Such alternative will be considered in the next sections where it is shown to be outperformed by the proposed approach. As a matter of fact, the approach proposed in this paper is simple and intuitive. In Section 2, an appropriate integer-constrained production possibility set is introduced, its corresponding efficient subset is defined and an input-oriented, envelop model that projects onto it is defined. Although we have assumed CRS, the definitions and the model can be trivially adapted to VRS or any other convex technology. The structure of the model is similar to a conventional DEA model (e.g., CCR-I [7]) except that its integrality constraints make it an MILP model. In Section 3 the proposed approach is illustrated on a problem from the literature and it is compared with the CCR-I solution and also with the FDH approach [8]. Finally, some conclusions are drawn.
2. Integer-valued DEA model Let I ={1, 2, . . . , m} and O ={1, 2, . . . , p} the sets of input and output dimensions respectively and let I ⊆ I and O ⊆ O the subsets of the corresponding dimensions that must be integer-valued. Naturally, all observed xij and ykj must be integer for all i ∈ I and k ∈ O . Let us assume that CRS apply and define an integer-valued CRS production possibility set TCRS
=
⎧ ⎨
(x, ˆ y) ˆ : ∃(1 , 2 , . . . , n )j 0 ∀j
⎩
xˆi
j
j xij
yˆk
j
j ykj
I
⎫ ⎬
xˆi integer ∀i ∈ . yˆk integer ∀k ∈ O ⎭
Definition. A DMU J is CRS integer-efficient if no other integer-valued operating point dominates it, [xˆ x¯ ] ∩ [yˆ y¯ ] ⇔ (x, i.e., if ∀(x, ˆ y) ˆ ∈ TCRS ˆ y) ˆ = (x¯J , y¯J ). J J Definition. The CRS integer-efficiency frontier is the set of non-dominated, integer-valued operating points, i.e.,
eff TCRS = (x, ˆ y) ˆ ∈ TCRS : ∀ (x, ¯ y) ¯ ∈ TCRS [x¯ x] ˆ ∩ [y¯ y] ˆ ⇔ (x, ¯ y) ¯ = (x, ˆ y) ˆ ⊂ TCRS . Proposition 1. If an existing DMU J is CRS efficient, then it is CRS integer-efficient, eff i.e., (x¯J , y¯J ) ∈ TCRS ⇒ (x¯J , y¯J ) ∈ (TCRS )eff .
Note that, although, according to this proposition, existing, CRS efficient DMU are also integerefficient, the converse is not true, i.e., there may be integer-efficient DMU which, when the integrality constraints are relaxed may be dominated by other operating points and therefore they are not CRS efficient.
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In order to assess the relative performance of the existing DMU in this CRS integer-valued scenario, and assuming an input orientation, the following MILP DEA model is proposed. MILP-DEA:
− + Min J − ε si + sk s.t.
i
j xij = xi
k
∀i,
j
xi = J xiJ − si− ∀i, j ykj = yk ∀k, j
yk = ykJ + sk+ ∀k, j 0 ∀j si− , xi 0 ∀i sk+ , yk 0 ∀k, J free xi integer ∀i ∈ I yk integer ∀k ∈ O . This model has n + 2(m + p) + 1 variables of which one is free and |I | + |O | are integer. The number of constraints is 2(m + p). It is the typical input-oriented two-phase model seeking the maximal radial reduction of all inputs and a subsequent exhaustion of all possible slacks along any input or output dimension. What makes this model different is the integrality constraints imposed on the integer-valued input and output dimensions. The optimal solution gives, for each DMU J : (a) Its CRS integer-efficiency score ∗J , − )∗ , (s + )∗ , (s + )∗ , . . . , (s + )∗ , (b) Additional slacks (s1− )∗ , (s2− )∗ , . . . , (sm p 1 2 ∗ , y ∗ , y ∗ , . . . , y ∗ ). (c) An integer-valued target (x1∗ , x2∗ , . . . , xm p 1 2 The proof of the following result is included in the appendix. Proposition 2. An existing DMU J is CRS integer-efficient if and only if ∗J = 1, (si− )∗ = 0 ∀i and (sk+ )∗ = 0 ∀k. Proposition 3. The CRS integer-efficiency score computed by the proposed model is never lower than ∗ the CRS efficiency score, i.e., ∗J (CRS J ) for all DMU J. This is a direct consequence of the fact that the integer-valued CRS production possibility set is included in the CRS one, i.e., TCRS ⊂ TCRS . On the other hand, with respect to the FDH [8] approach, the corresponding production possibility set and efficient subset are respectively ⎧ ⎨
⎫ ⎬
(x, ¯ y) ¯ : ∃(1 , 2 , . . . , n ) j ∈ {0, 1}∀j j = 1 x¯ X ¯ y¯ Y ¯ , ⎩ ⎭ j eff TFDH = (x, ¯ y) ¯ ∈ TFDH : ∀(x¯ , y¯ ) ∈ TFDH [x¯ x] ¯ ∩ [y¯ y] ¯ ⇔ (x¯ , y¯ ) = (x, ¯ y) ¯ ⊂ TFDH . TFDH =
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eff correspond to existing DMU, the Since all the operating points in the FDH efficient subset TFDH projections obtained by this approach:
(a) will have integer values in the input and output dimensions I and O respectively, and (b) will be feasible in the proposed MILP-DEA model. Proposition 4. The FDH efficiency score is never lower than the CRS integer-efficiency score computed by the proposed model, i.e., ∗J (FDH )∗ for all DMU J . J
3. Illustration In order to illustrate the proposed approach an instance from the literature [9] will be used. The problem consists in evaluating the relative performance of 47 different cellular manufacturing configurations. Each alternative configuration is considered a DMU with two inputs (number of workers and number of machines) and three outputs (related to average WIP, average flow time and average worker utilization). The output measures were obtained by simulation and in the case of WIP and flow time must be transformed so that higher outputs mean better performance. Specifically, the actual output values used for those two dimensions are the inverse (multiplied by 105 and 103 , respectively) of the values computed by the simulation. Note that the two inputs are integer-valued. In [9] the problem is solved using a conventional CCR-I approach [7]. A heuristic rounding procedure is not appropriate since the unit cost of adding a worker or a machine is important. Table 1 shows the CRS efficiency score and the targets obtained by the CCR-I model. Note that, except for the five existing efficient DMU (namely, 9, 10, 14, 22 and 47) the rest have fractional values for their inputs. The targets obtained by the proposed model, which are shown in Table 2, have integer input values for all targets. The targets obtained by the FDH approach are shown in Table 3 and they also have integer input values for all DMU. Note that:
∗ ∗ FDH ∗ (a) CRS J J for all DMU J with an average difference between the CRS and MILP scores J of 0.019 and an average difference between the FDH and MILP scores of 0.128. (b) Existing CRS efficient DMU (i.e., 9, 10, 14, 22 and 47) are also CRS integer-efficient with two additional CRS integer-efficient DMU (namely 13 and 15) that are not CRS efficient. (c) In some cases (e.g., DMU 1 and 2) the solution obtained by the proposed approach coincides with the result of rounding the fractional CCR-I input targets to the nearest integer. However, that is not always the case. Thus, for example, for DMU 15, instead of rounding down the second input, it is rounded up (i.e., from 23.04 to 24) and for the first input the target consists not just in rounding it up because that is not inside the integer-efficient possibility set so that another unit of that input must be added (i.e., from 12.60 to 14). (d) Of the three approaches only MILP and FDH provide integer input targets with the MILP targets always involving less inputs than the FDH targets (2.89 workers and 3.15 machines less on average).
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Table 1 CRS efficiency scores and targets DMU
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Score
0.939 0.744 0.831 0.943 0.711 0.862 0.938 0.850 1.000 1.000 0.750 0.883 0.976 1.000 0.960 0.948 0.816 0.793 0.853 0.909 0.925 1.000 0.955 0.707 0.610 0.697 0.753 0.864 0.860 0.875 0.871 0.854 0.584 0.614 0.676 0.725 0.805 0.854 0.744 0.801 0.559 0.530 0.582 0.621 0.611 0.729 1.000
Targets # Workers
# Machines
WIP−1
Flow time−1
Utilization
9.88 6.29 9.02 12.16 5.90 10.36 12.97 14.03 6 9 6.83 7.99 10.61 13 12.60 22.75 5.71 7.52 8.04 9.68 11.65 14 13.74 14.40 4.88 6.89 7.39 11.10 12.03 12.62 13.43 22.20 5.26 6.28 6.88 7.43 11.93 13.66 11.14 21.64 5.59 5.83 6.38 7.52 8.13 12.40 29
17.84 14.88 16.62 18.86 15.64 18.97 20.64 18.70 24 24 18.01 21.20 23.43 24 23.04 22.75 20.40 19.83 21.34 22.73 23.12 25 23.89 17.67 15.87 18.12 19.57 22.48 22.37 22.75 22.64 22.20 15.77 16.58 18.24 19.57 21.74 23.05 20.10 21.64 16.22 15.37 16.89 18.01 17.72 21.15 24
451.06 97.95 381.53 505.56 6.60 448.03 551.92 508.65 1.93 3.53 12.43 8.01 250.63 546.45 549.45 639.79 1.97 14.32 8.70 161.03 408.16 662.25 634.08 484.73 1.95 16.21 9.09 366.30 497.51 578.03 602.41 624.22 1.97 10.66 8.19 14.59 526.31 613.49 510.21 608.45 3.93 11.46 8.68 107.41 204.08 562.30 693.48
151.79 37.43 129.22 169.63 8.39 151.45 185.19 170.65 2.72 10.75 11.21 11.07 89.91 185.19 185.61 214.59 3.95 12.55 11.34 60.84 140.40 222.22 212.77 162.61 4.31 12.47 10.76 126.70 168.81 194.46 202.13 209.37 5.31 10.07 9.94 12.53 177.62 205.85 171.65 204.08 6.50 9.84 9.55 41.73 72.67 188.68 232.56
37.47 39.04 37.67 35.54 43.08 42.11 39.40 32.20 63.55 66.23 49.49 58.41 59.42 54.72 50.74 30.42 54.56 54.48 58.76 59.37 55.21 50.36 47.55 28.66 42.83 49.71 53.88 54.33 51.25 47.57 44.44 29.68 42.92 45.59 50.22 53.76 47.32 45.28 42.06 28.93 44.27 42.23 46.47 47.47 44.62 41.75 24.39
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Table 2 CRS integer-efficiency scores and targets DMU
Score
Targets
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
0.947 0.750 0.850 0.950 0.727 0.909 0.955 0.864 1.000 1.000 0.792 0.917 1.000 1.000 1.000 0.958 0.857 0.800 0.880 0.920 0.960 1.000 0.960 0.720 0.625 0.731 0.769 0.885 0.885 0.885 0.885 0.885 0.593 0.630 0.704 0.741 0.815 0.889 0.778 0.815 0.586 0.552 0.621 0.655 0.621 0.765 1.000
# Workers 10 6 10 12 6 11 13 14 6 9 7 9 11 13 14 23 6 8 9 10 12 14 14 15 5 8 8 11 12 13 14 23 5 7 8 8 12 14 12 22 5 7 8 9 8 13 29
# Machines 18 15 17 19 16 20 21 19 24 24 19 22 24 24 24 23 21 20 22 23 24 25 24 18 16 19 20 23 23 23 23 23 16 17 19 20 22 24 21 22 17 16 18 19 18 22 24
WIP−1 459.83 98.81 396.86 508.35 40.84 499.38 560.89 515.78 1.93 3.53 104.21 110.15 312.81 546.45 596.66 646.81 60.98 26.77 92.48 195.17 475.96 662.25 637.41 495.07 17.24 128.88 63.11 408.43 504.77 590.34 613.43 646.81 14.01 58.53 103.04 70.38 533.28 637.84 556.36 618.69 52.48 80.29 148.42 231.04 224.50 585.31 693.48
Flow time−1 154.64 36.91 133.99 170.57 18.82 168.07 188.20 173.04 2.72 10.75 38.90 44.11 109.42 185.19 200.73 216.95 22.22 16.49 38.49 71.83 161.33 222.22 213.88 166.08 9.16 48.89 28.27 138.91 170.59 198.40 205.83 216.95 8.18 25.54 40.67 30.63 179.76 214.03 186.69 207.52 19.29 32.04 54.83 81.77 78.38 196.40 232.56
Utilization 37.47 39.04 37.67 36.18 43.08 42.11 40.39 33.10 63.55 66.23 49.49 58.41 59.42 54.72 50.74 30.75 54.56 54.48 58.76 59.37 55.21 50.36 47.55 28.68 42.83 49.71 53.88 54.33 51.25 47.57 44.61 30.75 42.92 45.59 50.22 53.76 47.32 47.48 42.06 29.41 44.27 42.23 46.47 47.47 44.62 43.27 24.39
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Table 3 CRS efficiency scores and targets DMU
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Score
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.800 1.000 1.000 1.000 1.000 0.929 0.962 0.962 0.923 0.926 0.963 0.923 0.889 0.889 0.926 0.889 0.889 0.900 0.897 0.857 0.828 0.828 0.862 1.000
Target DMU
1 2 3 4 5 6 7 7 9 10 11 12 13 14 15 16 17 18 19 13 21 22 22 4 25 26 13 21 14 22 22 16 17 26 13 13 14 22 15 16 10 26 13 13 13 22 47
Targets # Workers
# Machines
WIP−1
Flow time−1
Utilization
19 18 19 20 15 17 18 18 6 9 10 11 12 13 14 24 7 10 11 12 13 14 14 20 8 11 12 13 13 14 14 24 7 11 12 12 13 14 14 24 9 11 12 12 12 14 29
19 20 20 20 22 22 22 22 24 24 24 24 24 24 24 24 25 25 25 24 25 25 25 20 26 26 24 25 24 25 25 24 25 26 24 24 24 25 24 24 24 26 24 24 24 25 24
451.06 97.94 381.53 505.56 6.61 448.03 550.66 550.66 1.93 3.53 3.55 8.01 250.63 546.45 549.45 639.80 1.93 3.45 8.70 250.63 408.16 662.25 662.25 505.56 1.95 3.72 250.63 408.16 546.45 662.25 662.25 639.80 1.93 3.72 250.63 250.63 546.45 662.25 549.45 639.80 3.53 3.72 250.63 250.63 250.63 662.25 693.48
149.25 33.22 126.58 169.49 6.14 149.25 185.19 185.19 2.72 10.75 11.21 6.22 86.96 185.19 181.82 212.77 2.91 12.55 5.68 86.96 136.99 222.22 222.22 169.49 2.76 12.47 86.96 136.99 185.19 222.22 222.22 212.77 2.91 12.47 86.96 86.96 185.19 222.22 181.82 212.77 10.75 12.47 86.96 86.96 86.96 222.22 232.56
37.47 39.04 37.67 35.54 43.08 42.11 39.4 39.4 63.55 66.23 49.49 58.41 59.42 54.72 50.74 29.44 54.56 54.48 58.76 59.42 55.21 50.36 50.36 35.54 42.83 49.71 59.42 55.21 54.72 50.36 50.36 29.44 54.56 49.71 59.42 59.42 54.72 50.36 50.74 29.44 66.23 49.71 59.42 59.42 59.42 50.36 24.39
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About computing times, the CCR-I and FDH approaches require a negligible amount and the proposed MILP DEA approach takes 28.51 s when using the XA commercial LP code on a AMD-K6, 128 Mbytes RAM, 500 MHz desktop computer. Larger problems (with one hundred DMU, five inputs and five outputs) have also been solved with the proposed approach and it never took longer than a few minutes. 4. Summary and conclusions In this paper a simple and efficient MILP DEA model is proposed for handling the case, relatively common, in which some of the inputs and/or outputs are integer. Except for the existing efficient DMU, the targets computed by conventional DEA models generally have fractional values in those dimensions, which may not be valid. FDH or ordinal DEA approaches cannot be considered a workable solution approach in general. In this paper a MILP model has been proposed which guarantees integer-efficiency for all targets. Appropriate definitions and propositions for the scenario addressed have been introduced and an instance from the literature has been used to illustrate the usefulness of the approach. Although only the constant return to scale, input orientation scenario is presented, the approach can be appropriately extended to all other DEA variants (VRS, output orientation, non-oriented models, additive metric, nondiscretionary variables, etc.). Appendix Proposition 2. An existing DMU J is CRS integer-efficient if and only if ∗J = 1,
∗
− ∗ = 0 ∀i and sk+ = 0 ∀k. si Proof. We must prove both the necessity and the sufficiency. About the necessity, note that if ∗J < 1
∗
∗ , would not or ∃i si− > 0 or ∃k sk+ > 0 then the integer-valued target obtained would belong to TCRS coincide with DMU J but would dominate it and therefore DMU J would not be, by definition, CRS integer-efficient. About the sufficiency, assume that the does not hold and one will again arrive at a con ∗
proposition ∗ tradiction. Thus, assume that ∗J = 1, si− = 0 ∀i and sk+ = 0 ∀k but that DMU J is not CRS integer-efficient. That means, by definition, that there must be an integer-valued point (x, ˆ y) ˆ ∈
operating TCRS (x, ˆ y) ˆ = (x¯J , y¯J ) which dominates it, i.e., xˆ x¯J and yˆ y¯J . Since x, ˆ y ˆ ∈ T there must CRS be a vector (1 , 2 , . . . , n )j 0 ∀j such that xˆi j j xij ∀i and yˆk j j ykj ∀k. Also, there is a non-negative vector J = 1 j = 0 ∀j = J such that x¯iJ = j j xij ∀i and y¯kJ = j j ykj ∀k. Since xˆ x¯J and yˆ y¯J then there must also be a vector (ˆ 1 , ˆ 2 , . . . , ˆ n )ˆ j 0 ∀j such that xˆi = j ˆ j xij ∀i and yˆk = j ˆ j ykj ∀k. The solution to the proposed model using that vector leads to xˆi = ˆ J xiJ − sˆi− xiJ
yˆk = ykJ + sˆk+ ykJ
∀i, ∀k.
Since x, ˆ yˆ ∈ TCRS it must be integer-valued in the I and O dimensions and, since x, ˆ yˆ = (x¯J , y¯J ) at least one of the above inequalities must be strict, i.e., ˆ < 1 and/or ∃i sˆi− > 0 and/or ∃k sˆk+ > 0. What is
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sure is that ˆ J − ε
i
sˆi− +
k
sˆk+ < ∗J − ε
i
− ∗
sˆi
+
k
+ ∗
sˆk
= 1.
This means that we have found a feasible solution for the proposed model which has a better objective function value than the supposedly optimal solution, thus arriving at a contradiction and proving the sufficiency clause of the proposition. References [1] Cooper WW, Seiford LM, Tone K. Data envelopment analysis. Kluwer, 2000. [2] Thanassoulis E. Introduction to the theory and application of data envelopment analysis—a foundation text with integrated software. Kluwer, 2001. [3] Banker RJ, Morey RC. The use of categorical variables in data envelopment analysis. Management Science 1986;32(12): 1613–27. [4] Kamakura WA. A Note on the use of categorical variables in data envelopment analysis. Management Science 1988;34(10):1273–6. [5] Rousseau JJ, Semple JH. Categorical outputs in data envelopment analysis. Management Science 1993;39(3):384–6. [6] Cook WD, Kress M, Seiford LM. Data envelopment analysis in the presence of both quantitative and qualitative factors. Journal of the Operational Research Society 1996;47:945–53. [7] Charnes A, Cooper WW, Rhodes E. Measuring efficiency of decision making units. European Journal of Operational Research 1978;2:429–44. [8] Tulkens H. On FDH efficiency analysis: some methodological issues and applications to retail banking, courts, and urban transit. Journal of Productivity Analysis 1993;4:183–210. [9] Shafer SM, Bradford JW. Efficiency measurement of alternative machine component grouping solutions via data envelopment analysis. IEEE Transactions on Engineering Management 1995;42:159–65.