Data Envelopment Analysis as a Kaizen Tool: SBM Variations Revisited

Slacks-based measure (SBM) (Tone (2001), Pastor et al. (1999)) has been widely utilized as a representative non-radial DEA model. In Tone (2010), I developed four variants of the SBM model where main concerns are to search the nearest point on the efficient frontiers of the production possibility set. However, in the worst case, a massive enumeration of facets of polyhedron associated with the production possibility set is required. In this paper, I will present a new scheme, called SBM-Max, for this purpose which requires a limited number of additional linear program solutions for each inefficient DMU. Although the point thus obtained is not always the nearest point, it is acceptable for practical purposes and from the point of computational loads. Inefficient DMUs can be improved to the efficient status with less inputreductions and less output-enlargement. Thus, this model proposes a Kaizen (improvement) tool by DEA.


Introduction
There are two types of models in DEA (Data Envelopment Analysis); radial and non-radial. Radial models are represented by the CCR (Charnes-Cooper-Rhodes) model. Basically they deal with proportional changes of inputs or outputs. As such, the CCR score reflects the proportional maximum input (output) reduction (expansion) rate which is common to all inputs (outputs). However, in real world businesses, not all inputs (outputs) behave in the proportional way. For example, if we employ labor, materials and capital as inputs, some of them are substitutional and do not change proportionally. Another shortcoming of the radial models is the neglect of slacks in reporting the efficiency score. In many cases, we find a lot of remaining non-radial slacks. So, if these slacks have an important role in evaluating managerial efficiency, the radial approaches may mislead the decision when we utilize the efficiency score as the only index for evaluating performance of DMUs.
In contrast, the non-radial SBM models put aside the assumption of proportionate changes in inputs and outputs, and deal with slacks directly. This may discard varying proportions of original inputs and outputs. The SBM models are designed to meet the following two conditions.
(1) Units invariant: The measure should be invariant with respect to the units of data (2) Monotone: The measure should be monotone decreasing in each slack in input and output. The original SBM model evaluates efficiency of DMUs referring to the furthest frontier point within a range. This results in the worst score for the objective DMU and the projection may go to a remote point on the efficient frontier which may be inappropriate as the reference. In Tone (2010), I developed four variants of the SBM model where main concerns are to search the nearest point on the efficient frontiers of the production possibility set. Mathematically, finding S belongs to a NP-hard problem, because it is a maximization problem of a convex function over a convex region. However, the projected point S indicates that we can attain an efficient status with less input reductions and less output expansions than the ordinary SBM (Min) models. We can say that the projection by the SBM-Max model represents a practical "Kaizen" (improvement) by DEA.
Referring these variations, several authors published new models. Among them, I introduce two important papers. Fukuyama et al. (2014) developed a least distance efficiency measure with the strong/weak monotonicity of the ratio form measure under several norms including 1-norm, 2-norm and ∞-norm. This model utilizes mixed-integer linear programming (MILP) to identify efficiency frontiers and hence a computational difficulty arises for large-scale problems.
Hadi-Vencheh et al. (2015) developed a new SBM model to find the nearest point on the efficient frontiers. They utilize the multiplier form model to find all supporting hyperplanes. It also utilizes software which uses fractional coefficients (high precision arithmetic) to avoid loss data. Hence, computational time increases for large-scale problems.
In order to apply DEA models to actual real world problems, we need to try many instances including selection of DMUs and input/output factors before attaining the final scheme of evaluation. For this purpose, allowable computation time and easy accessible software are desirable.
The motivation and purpose of this paper is to obtain nearly closest points on the efficient frontiers within foreseeable computation loads using only popular linear programming codes.
The rest of this paper is organized as follows. Section 2 introduces the ordinary SBM-Min model briefly. Section 3 presents the new SBM-Max model. Observations on this new model are described in Section 4. Two numerical examples are exhibited in Section 5. Section 6 concludes this paper. Although we present the model in non-oriented mode, we can treat input-and output-oriented model as well. As to returns-to-scale characteristics, we present the constant returns-to-scale (CRS) case. However we can deal with the variable returns-to-scale (VRS) model as well.

The SBM Min model
The SBM model was introduced by Tone (2001) (see also Pastor et al. (1999)). It has three variations, i.e. input-, output-and non-oriented. The non-oriented model indicates both input-and output-oriented.
Let the set of DMUs be { }

Production Possibility Set
The production possibility set is defined using the non-negative combination of the DMUs in the set J as: 1 1 ( , ) , , .
is called the intensity vector. The inequalities in (2) can be transformed into equalities by introducing slacks as follows: are respectively called input and output slacks.

Non-oriented SBM
Non-oriented or both-oriented SBM efficiency min

The SBM Max Model
In this section, we introduce the new non-oriented SBM-Max model. Step We denote these efficient DMUs as ( ) ( ) ( ) x y x y x y  , wher Neff is the number of efficient DMUs.

Step 3. Local reference set
For an inefficient DMU ( )
Step x y is efficient with respect to the efficient DMU set R eff . However, it does not always satisfy Pareto-Koopmans efficiency condition.

Observations
In this section, we discuss several characteristics of the algorithm in Section 3.

Distance and choice of the set h R
The set h R plays a central role in choosing referent DMUs for inefficient DMUs. Because our main concern is the projection to the nearest point on the efficient frontiers, we evaluate the distance between the DMU ( ) x y and efficient DMUs by (12), and choose the shortest distance DMU as the first candidate DMU. Then, we expand the referent set in the ascending order of distances. Thus, we can expect a close efficient point on the frontiers with high probability. If tie occurs in distances, we can choose any one at random.

Computational amount
Computations needed for this algorithm for an inefficient DMU are as follows.
Let t 1 and t 2 be the CPU time for solving a LP problem, respectively, with the (m + s) rows and n columns, and (m + s) rows and Neff columns. Since LP solution time is proportional to the number of columns. We can estimate roughly t 1 = ( n / Neff ) t 2 .
(20) Thus, the computational amount is polynomial order and we do not need other software, e.g., MILP and fractional arithmetic.

Consistency with the super-efficiency SBM measure
The SBM-Max model aims at getting to the nearest point on the efficient frontiers. This concept is in line with the super-efficiency SBM model (Tone (2002)) which solves the following program for an efficient DMU ( ) We can solve the super-efficiency SBM model by applying LP code just once, because this problem belongs to a convex programming, i.e., minimization of a convex function over a convex region. However, SBM-Max problem cannot be solved in this manner, because it is a maximization of a convex function over a convex region.

Addition of weights to input-and output-slacks
We can assign weights (wand w + ) to input and output slacks in the objective function of the above SBM models corresponding to the relative importance of items as follows: We can define the input-(output-) oriented Weighted-SBM models by neglecting the denominator (numerator) of the objective function in (22).

Numerical examples
In this section, we show two numerical examples, the first one is illustrative and the other deals with real data. All computations are executed using a PC with Intel Core i7-3770 CPU at 3.40 GHz 16 GB (RAM) and Microsoft Excel VBA (Visual Basic for Applications). A LP soft (revised simplex method) is coded by the author. We checked the results of the first example using LINGO (LINDO Systems Inc.) and had the same figures.

An illustrative example
We deal with the same data as one in Tone (2010). Table 1 displays the data with two inputs (Doctor and Nurse) and two outputs (Outpatient and Inpatient).

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BMSA Volume 16 Table 3 compares results with SBM-Max, SBM-Pseudo and SBM-Min scores. Inefficient DMUs increased their efficiency from SBM-Min to SBM-Max.    Table 5 reports data and projections along with deviations (%) in the case of SBM-Max.   Table 6 exhibits statistics of the dataset.

SBM scores
The SBM-Min model found that 66 hospitals among 707 are efficient (Neff = 66). Table 7 compares three scores. We found large differences between Max and Min models.   Table 8 shows the average of percentage deviations,│Data -Projection│* 100 / Data. It is observed large differences exist in SBM-Min, while small differences in SBM-Max.

Computational time
The computational time increases as the number of efficient DMUs (Neff) increases, because number of facets increases accordingly and we need to solve additional Neff LPs. In this example we have:

Conclusions
In this paper, we have developed the SBM-Max model which attempts to find nearly closest reference point on the efficient frontiers so that slacks are minimized, while the scores are maximized. Sacrificing the rigorous solutions, the proposed model utilizes a standard LP code and finds approximate solutions in allowable (polynomial) times. Inefficient DMUs can be improved to the efficient status with less input-reductions and less output-enlargement. Thus, this model proposes a Kaizen