Production Possibility Set (PPS)

Figure 2.1 illustrates how PPS is defined and how DEA works in principle with a simple one input and one output example.

Figure 2.1. Illustration of Production Possibility Set

In Figure 2.1, the observed units A, B, C, D and E are plotted on the graph. The input-output correspondences lying on linear segments AB, BC and CD are enveloping the data and they are feasible. Since operating at A and operating at B are both possible, in principle, it is reasonable to deduce through interpolation that to operate at input-output correspondences between those points is also possible (Thanassoulis, 2001). Another assumption that can be made in order to define the PPS is that it is always possible to use more input and produce less output than observed (free disposability principle), in other words, it is possible to operate inefficiently. Therefore, the PPS consists of units on the piece-wise linear boundary FABCDG and all units to the right and below of this boundary.

In the given PPS, the piece-wise linear boundary ABCD is the efficient frontier since units on this boundary are relatively the best performing units. Units on segments AF and DG are

feasible but not efficient in Pareto sense since units A and D dominate them, respectively. For the units on AF, it is possible to produce more of output with the same input as the unit A has achieved. Similarly, for the units on DG, it is possible to produce the same output with less input as at unit G. Unit E lies below the efficient frontier ABCD, and thus operating inefficiently relative to the other observed units. It is outperformed by all other observed units, as well as the hypothetical unit E1. The unit E1, which is an interpolation between units A and B, is hypothetically producing the same amount of output with the less input than unit E. In input terms, the unit E1 can be a target for unit E. In principle, the efficiency of unit E is calculated by the ratio HE1/HE.

To generalize the basic assumptions underlying the PPS in DEA, consider a set of n Decision Making Units (DMUs), J =

{

}

produce s outputs. The observed units are denoted as pairs(Xj_,Yj_),

j∈J, where vectors

Xj∈R+ m _and

Yj∈R+

s_{. The Production Possibility Set, denoted by T, is defined as the set of}

input and output vectors (X,Y) such that it is possible to produce _Y≥0 from _X ≥0.

Conceptually, the Production Possibility Set in DEA is defined as the minimum technology that satisfies the following production axioms (Banker et al., 1984; Podinovski, 2004a):

Axiom 2.1. Feasibility of observed data. (Xj_,Yj_{)∈T , for any} j∈J.

Axiom 2.2. Convexity. The set T is convex.

Axiom 2.3. Free disposability. (X,Y)∈T , _Y ≥ ′Y ≥0 and _X ≤ ′X implies (X′,Y′)∈T .

DEA models are built under different Returns to Scale (RTS) assumptions. Depending on the RTS assumption, the production technology, and thus the production axioms of the PPS

CCR (Charnes Cooper Rhodes) model. The CCR approach assumes a constant returns-to- scale technology (CRS) and so, proportionality between inputs and outputs. This means that scaled inputs and outputs of DMUs with same proportion are members of the technology. For example, in a two inputs and two outputs case, if we raise both inputs by 10% and expect the outputs to rise by 10%, then CRS assumption is appropriate approach to incorporate for this case. Under CRS assumption, it is assumed that the operating scale of a unit does not have an effect on its efficiency. The CCR approach is modified by Banker et al. (1984) and named as BCC (Banker Charnes Cooper) model, which is assuming variable returns-to-scale (VRS). BCC approach ignores the proportionality assumption.

Figure 2.2. Illustration of Production Possibility Set under Constant Returns-to-Scale

The production possibility set for CRS technology is illustrated in Figure 2.2 with the same units as in Figure 2.1. As seen in Figure 2.2, for our one input and one output example, under CRS technology, the efficient boundary has a linear form starting form the origin different than the frontier in the VRS technology as given by Figure 2.1. The PPS is defined as the set of units on or below the ray OBJ. Unit B is the only unit operating efficiently

relative to others. For unit E, the efficient target in terms of inputs is the hypothetical unit E2 and the efficiency of E is KE2/KE.

Production Axioms 2.1, 2.2 and 2.3, stated above define the VRS technology. For the PPS under CRS assumption, proportionality axiom (Axiom 2.4) additional to above 3 axioms is considered, which is referred as “Ray Unboundness” in Banker et al. (1984).

Axiom 2.4.Proportionality. (X,Y)∈T and α ≥0 implies (αX,αY)∈T.

One of the main advantages of DEA is that it allows the user to deal with multiple inputs and outputs. More inputs and outputs mean more dimensions for the technology. In the presence of multiple inputs and outputs the PPS has an unbounded polyhedral shape under VRS technology and a conical shape under CRS technology.