# 5 – Solutions to the DEA model

The efficiency of the target unit in a set can be obtained by solving M3. The solution to this LP provides a measure of the relative efficiency of the target unit and the weights leading to that efficiency. These weights are the most favourable ones from the point of view of the target unit. To obtain the efficiencies of the entire set of units it is necessary to solve a linear program focusing on each unit in turn. Clearly as the objective function is varying from problem to problem the weights obtained for each target unit may be different. The efficiencies obtained for the full set of depots are shown in Table l.

Table 1

 Depot Efficiency 19 1.00 15 1.00 14 1.00 12 1.00 9 0.96 5 0.95 2 0.94 16 0.91 10 0.89 20 0.84 13 0.83 6 0.83 1 0.82 3 0.82 7 0.71 4 0.65 11 0.63 17 0.55 8 0.52 18 0.42

A concern with the DEA model is that if all units can adopt their most favourable weights, they may all appear efficient. With the depot data we can see that there is in fact a considerable difference in efficiencies. At one extreme depot 18 has an efficiency of only 0.42. This can broadly be interpreted as saying that depot 18 should have been able to support its activity levels with only 42% of its resources. In fact 12 of the depots have an efficiency below 0.9 indicating a fair degree of discrimination. This degree of discrimination is similar in scale to that reported by Thanassoulis, Dyson and Foster(4) and in other published studies.

In solving each linear program the solution technique will attempt to make the efficiency of the target unit as large as possible. This search procedure will terminate when either the efficiency of the target unit or the efficiency of one or more other units hits the upper limit of 1. Thus for an inefficient unit at least one other unit will be efficient with the target unit’s set of weights. These efficient units are known as the peer group for the inefficient unit. Table 2 shows the peers for depot 18.

Table 2 Peer Units for Depot 18

 Depot 18 Input or Output Depot 12 Depot 15 Depot 19 4.0 – stock 2.0 2.0 3.0 6.0 – wages 4.0 3.0 4.0 25.0 + issues 45.0 20.0 45.0 38.0 + receipts 40.0 50.0 67.0 20.0 + reqs 44.0 15.0 32.0

(Note – indicates an input and + an output).

It is sometimes useful to scale the data on the peer units so that a better comparison of the inefficient unit with the peer units can be made. In Table 3 the peers have been scaled on one of the inputs, so that each peer unit uses no more of an input than the inefficient unit. In this example inefficiency is clearly demonstrated as all the outputs of the peer units are greater than those of depot 18. (These comparisons assume constant returns to scale).

Table 3 Scaled Peer Units for Depot 18

 Depot 18 Input or Output Depot 12 Depot 15 Depot 19 4.0 scale 1.500 2.000 1.33 6.0 – stock 3.0 4.0 4.0 25.0 + issues 67.5 40.0 60.0 38.0 + receipts 60.0 100.0 89.3 20.0 + reqs 66.0 30.0 42.7

Finally for each inefficient unit the linear programming solution will provide a set of target inputs and outputs. The targets correspond to either a pro rata decrease in inputs or increase in outputs. For a unit at an extreme of the data set a pro rata decrease on inputs may be insufficient and will need to be coupled with an increase on one or more of the outputs or further decreases in certain inputs. This will be illustrated in the subsequent section. For the inefficient depot 18 the targets are shown in Table 4.

Table 4 Targets for Depot 18 (efficiency 0.42)

 Variable Actual Target – stock 4.0 1.7 – wages 6.0 2.5 + issues 25.0 25.3 + receipts 38.0 38.0 + reqs 20.0 20.0

The above targets simply consist of a reduction in inputs of 42% of their current levels plus a small increase in issues. It will often be the case that some inputs or outputs are uncontrollable and therefore targets associated with them are meaningless. Banker and Morey(5) address the problem where some inputs or outputs are exogenously fixed, and Thanassoulis and Dyson(6) show how a variety of different targets can be obtained for an inefficient unit.

The solution to the DEA model thus provides a relative efficiency measure for each unit in the set, a subset of peer units for each inefficient unit, and a set of targets for each inefficient unit.

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