60 Punto de fluidez máx 0 C
Capítulo 2. Sistema de mantenimiento de las Centrales de Fuel Su diagnóstico
Consider a CRS technology TCRS with m inputs and s outputs. The observed units are
denoted as pairs (Xj,Yj), =1,...,n, where j∈Rm and j∈Rs. Vectors
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Assumptions 4.1 and 4.2. Let X and Y be the input and output matrices consisting of the input and output vectors Xj and
Yj, respectively.
Following Podinovski and Førsund (2010), it can be assumed that all inputs and outputs can be divided into three disjoint sets: A, B and C. The analysis in this study is concerned with the elasticity of response of the factors in the set B with respect to marginal changes of the factors in the set A, provided the inputs and outputs in the set C do not change. The set A is not empty and may include both inputs and outputs. Two scenarios are considered for set B: the set B contains only outputs or only inputs. This section deals with the elasticity measures for the output scenario. The case of inputs is considered in Section 4.6.
A more general and apparently symmetrical case of the set B containing both inputs and outputs can also be considered. However, the resulting notion of elasticity of the factors in B with respect to A will generally not apply to all efficient units. This is because an efficient unit may not necessarily produce the maximum proportion of its mixed input-output bundle B for the given mixed input-output bundle A. This makes the exposition and interpretation more technical, and is not pursued in the current study.
Assume that the sets A and B are not empty, the set B contains only outputs, the set A may contain either inputs or outputs, or both inputs and outputs. The set C contains the remaining inputs and outputs not included in the sets A and B, and can be empty. Then any unit
(X0,Y0)∈TCRS can be represented as (X0,Y0)=(X0A,X 0 C,Y 0 A,Y 0 B,Y 0 C), (4.1)
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where the superscripts indicate the sub-vectors of X0 and Y0 corresponding to the sets A, B
and C. If the sets A and C do not contain inputs or outputs, the corresponding sub-vectors are omitted.
For any unit (X0,Y0) in the form (4.1), the response of the outputs in the set B to marginal changes of the inputs and/or outputs in the set A is defined only if such a change is feasible in the given technology. This leads to the following two definitions.
Definition 4.1. A proportional marginal increase of vectors X0A and Y0
A is feasible in
technology TCRS if there exists an α >1 such that, for any α ∈[1,α], there exists a β ≥0
(depending on α) for which
(αX0A,X 0 C,αY 0 A,βY 0 B,Y 0 C)∈T CRS. (4.2)
Definition 4.2. A proportional marginal reduction of vectors X0A and Y0
A is feasible in
technology TCRS if there exists a ˆα ∈[0,1) such that, for any α ∈[ ˆα,1], there exist a β ≥0
(depending on α) for which (4.2) holds.
Following Podinovski and Førsund (2010), in order to define the elasticity of response of the output vector Y0B to marginal changes of the vectors
X0 A and
Y0
A, first consider the output
response function β(α)=max{β (αX0A,X 0 C,αY 0 A,βY 0 B,Y 0 C)∈T CRS} (4.3)
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in some neighbourhood of α=1. If a proportional marginal increase or reduction of the vectors X0A and
Y0
A is not feasible in T
CRS (in the sense of Definitions 4.1 and 4.2), the
function β(α) is undefined in the right or left neighbourhoods of α =1, respectively.
Let XA, XC, YA, YB and YC be the sub-matrices of X and Y corresponding to the inputs and outputs included in the sets A, B and C. The output response function β(α) defined in (4.3) is the optimal value in the following linear program, where β is a variable and α is a fixed value: β(α)=max β (4.4) Subject to XAλ ≤ αX0 A XCλ ≤X0 C −YAλ ≤ −αY 0 A −YBλ+βY 0 B≤0 −YCλ ≤ −Y 0 C λ≥ 0, β sign free
It is common in the DEA literature to define elasticity measures only for efficient units. A unit (X0,Y0) is efficient if there exists no other unit (X′,Y′) in the technology such that, on the component-wise basis, X0≥ ′X , Y0≤ ′Y and (X0,Y0)≠(X′,Y′). The fact that a unit is technically efficient (that is its radial input or output efficiency is equal to 1), does not guarantee the efficiency of the unit. Testing for efficiency requires the utilization of a two- stage optimisation procedure or an equivalent method (Cooper et al. 2006; Thanassoulis
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2001). For practical purposes, a unit is efficient if it coincides with its efficient target, as reported by most DEA programs.
Because the concern is the elasticity of response of a specific subset B of outputs, the overall efficiency of the unit is not required, and we only need the unit (X0,Y0) to be efficient in the production of its output vector Y0B. This is stated below.
Assumption 4.1. (Selective radial efficiency with respect to the output set B). The function β(α) is finite at α =1, and β(1)=1.
Theorem 4.1. If the unit (X0,Y0)∈TCRS is efficient and the vector Y0B has at least one
strictly positive component then Assumption 4.1 is satisfied.
Since zero outputs are allowed in DEA models, the efficiency of the unit itself is not sufficient for the definition of elasticity. For example, if in an efficient unit (X0,Y0) output 1 is equal to 1 and output 2 is equal to zero, the elasticity of response of output 2 to output 1 is undefined.
Assumption 4.1 means that the unit (X0,Y0), which may be efficient or inefficient, produces the maximum proportion β=1 of the output vector Y0B possible in the technology for the fixed levels of its inputs and outputs included in the sets A and C. Any unit (X0,Y0) that satisfies Assumption 4.1 is located on the boundary of the technology TCRS but not
necessarily on its efficient part (efficient frontier). Even though Assumption 4.1 allows the definition of elasticity measures at some inefficient units, this is different from defining
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Although Assumption 4.1 is needed for the theoretical development of elasticity measures (and can be verified by solving program (4.4) above), in practice no extra effort is required for checking whether this assumption is true. The linear programs developed below in Theorem 4.2 for the calculation of elasticities are self-testing in this respect: according to Theorem 4.3, these become infeasible if Assumption 4.1 is not satisfied.
Following Podinovski and Førsund (2010), if Assumption 4.1 is satisfied and the required derivatives exist, the following definition can be given.
Definition 4.3. The right-hand (left-hand) elasticity of response of the output vector Y0B with respect to marginal proportional changes of the vectors X0A and
Y0
A is the right (left)
derivative of the function β(α) at α =1:
εA,B+ (X
0,Y0)=β′+(1), (4.5)
εA,B− (X
0,Y0)=β′−(1). (4.6)
The existence of the required one-sided derivatives in (4.5) and (4.6) is established by Theorem 4.2 below. As discussed in Podinovski and Førsund (2010), Definition 4.3 is consistent with conventional definitions of production economics. In particular, this includes the scale elasticity and partial elasticities as special cases.
The following result in Theorem 4.2 extends Proposition 1 of Podinovski and Førsund (2010) to the case of CRS technology. Its proof is given in Appendix A.
Theorem 4.2. Consider any unit (X0,Y0)∈TCRS that satisfies Assumption 4.1. (The unit
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(a) If a proportional marginal increase of vectors X0A and
Y0
A is feasible in technology T
CRS,
then the right-hand elasticity εA,B+ (X0,Y0) exists, is finite and can be calculated as follows:
εA,B + (X 0,Y0)=min ν AX 0 A−µAY 0 A (4.7.1) Subject to νAX 0 A+νCX 0 C −µAY 0 A−µCY 0 C =1 (4.7.2) νAXA+νCXC −µAYA−µBYB−µCYC ≥0 (4.7.3) µBY 0 B=1 (4.7.4) νA,νC,µA,µB,µC ≥0 (4.7.5)
(b) If a proportional marginal reduction of vectors X0A and Y0
A is feasible in technology
TCRS, then the left-hand elasticity εA,B− (X0,Y0) exists, is finite and can be calculated by
changing the minimisation to maximisation in program (4.7), that is
εA,B− (X 0,Y0)=max ν AX 0 A−µAY 0 A (4.8.1) Subject to νAX 0 A+νCX 0 C−µAY 0 A−µCY 0 C =1 (4.8.2) νAXA+νCXC −µAYA−µBYB−µCYC ≥0 (4.8.3) µBY 0 B=1 (4.8.4) νA,νC,µA,µB,µC ≥0 (4.8.5)
(c) If a proportional marginal increase (reduction) of vectors X0A and
Y0
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Comparing (4.7) and (4.8), it can be observed that εA,B+ (X0,Y0)≤εA,B− (X
0,Y0), provided both one-sided elasticities exist. In the case of equality, the output response function β(α) is differentiable at α=1 and we can define the elasticity εA,B(X0,Y0) as the derivative β′(1).
Formally, the use of Theorem 4.2 requires checking Assumption 4.1 first, which can be done by solving model (4.4), where α =1. However, in practical computations this is not necessary. The following result shows that a violation of Assumption 4.1 is equivalent to the infeasibility of linear programs (4.7) and (4.8). Note that programs (4.7) and (4.8) have the same feasible set, and the feasibility of one of them implies the feasibility of the other.
Theorem 4.3. Assumption 4.1 is true at (X0,Y0) if and only if both linear programs (4.7) and (4.8) are feasible.
Theorem 4.3 means that programs (4.7) and (4.8) can in practice be solved for all units, efficient and inefficient. If, for a particular unit (X0,Y0), a linear optimizer indicates an infeasible program (4.7) or (4.8), Assumption 4.1 does not hold and the notion of elasticity is not defined at this unit.
It is worth noting that Theorem 4.3 applies to the case of VRS as well and complements the development of Podinovski and Førsund (2010) where the case β(1)≠1 is not considered. In the case of VRS, the linear programs (4.7) and (4.8) are modified to incorporate the dual multiplier µ0 to the convexity constraint, as in Podinovski and Førsund (2010).