Abstract
The contributions of the paper are threefold: (i) compare with mathematical rigour the data envelopment analysis (DEA) model of Charnes, Cooper, and Rhodes and the Farrell model exhibiting constant returns to scale, (ii) reinterpret the contribution of Farrell and Fieldhouse that extended the analysis to variables returns to scale and establish the connection with the approach in Banker, Charnes, and Cooper, and (iii) provide graphical visualization of properties of the frontier function. Both papers by Farrell emphasized the importance of graphical visualization of non-parametric frontier functions, but, to our knowledge, this is seldom followed up in the literature. We use a graphical package (EffiVision) with a numerical representation of the frontier functions, representing the contemporary development of visualization. By making suitable cuts through the DEA frontier in multidimensional space, various graphical representations of features of economic interest can be done. Development of ray average cost function and scale elasticity plots are novel illustrations.
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References
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Acknowledgements
A first draft was written while Professor Vladimir Krivonozhko visited the Frisch Centre for four weeks in January and February 2007 on a grant from the Frisch Centre. Helpful comments from two anonymous referees are gratefully acknowledged.
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Appendix: Proof of Theorem 1
Appendix: Proof of Theorem 1
Suppose that we obtain an optimal solution of problem (1). Let J B be the index set of optimal basic variables, and let λ j *,j∈J B, be the optimal variables. Optimization theory (Dantzig, 1998) ensures that in this optimal solution vector not more than m of the λ j * will be strictly positive and the rest will be zero.
Let us consider the hyperplane H going through vectors P j ,j∈J B, which form the optimal basis. Such basis always exists and contains precisely m vectors. Farrell introduced artificial points in problem (1) to ensure the existence of feasible bases. Observe that hyperplane H does not contain the origin, since otherwise vectors P j , j∈J B, would be linearly dependent. Then, point P k can be represented in the form
Introducing λ j ′=λ j */η 1, the expression after the second equality sign is obtained, and introducing the vector P k ′ yields the expression after the third equality sign.
From (1) and (A.1), it follows that
So, we associate the value function (1) with the distance along line OP k ′ from point P k to point P k ′ that belongs to the convex combination of vectors P j , j∈J B.
It can be shown that
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a)
if η 1<1, then point P k and the origin are on the same side of hyperplane H,
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b)
if η 1>1, then hyperplane H separates the origin and the point P k ,
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c)
if η 1=1, then points P k ′ and P k coincide.
Hence, the value function of problem (1) reaches at least the value η 1. The case η 1<1 is not possible, because the feasible solution λ k =1, λ j =0, j≠k, increases the value function I 1=1, contradicting the optimality of solution λ j *, j∈J B, I 1 *=η 1 of problem (1).
Observe that the finite optimal solution of problem (1) exists as there is a finite set of feasible bases of problem (1).
Now, turning to problem (3), it follows from (A.1) and (A.2) that variables λ j =λ j ′ if j∈J B, λ j =0 if j∉J B, and θ=1/η 1 form a feasible solution of problem (3). Hence, the following relations hold:
Conversely, consider optimization problem (3). From the first part of the proof it follows that problem (3) has a feasible solution. So, according to convex analysis (Nikaido, 1968), there exists a unique point P k ′ on the segment OP k , which belongs to some facet of the polyhedron (3), this point is represented in the form
where P j , j∈J 1, are vertices of the feasible set (3) which determines this facet. Because point P k ′ is unique on the segment OP k , problem (3) has an optimal solution λ j =λ j ′ if j∈J 1, λ j =0 otherwise, I 2=θ. (Note that, though the point P k ′ is unique, solution vectors in the variables λ j *, j∈J B, may not be unique, they may constitute a whole set of optimal solutions.)
The case θ=1 takes place when points P k and P k ′ coincide. From (A.1) and (A.4), it follows that variables λ j =(1/θ)λ j ′ if j∈J 1, λ j =0 if j∉J 1, form a feasible solution of problem (1). Hence, for the optimal value function of problem (1) the following relations hold:
Thus, two opposite inequalities (A.3) and (A.5) establish that I 2=1/I 1. Relations (4) follow immediately from (A.1) and (A.4). This completes the proof. □
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Førsund, F., Kittelsen, S. & Krivonozhko, V. Farrell revisited–Visualizing properties of DEA production frontiers. J Oper Res Soc 60, 1535–1545 (2009). https://doi.org/10.1057/jors.2008.185
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DOI: https://doi.org/10.1057/jors.2008.185