Generalized DEA Model for Multiple Criteria Decision
8.2 Basic DEA Models
In the following discussion, we assume that there exist n DMUs to be evalu-ated. Each DMU consumes varying amounts of m different inputs to producep
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different outputs. Specifically, DMUj consumes amountsxj := (xij) of inputs (i= 1,· · ·, m)and produces amountsyj := (ykj)of outputs(k= 1,· · ·, p). For these constants, which generally take the form of observed data, we assumexij >0 for eachi= 1,· · ·, mandykj >0for eachk= 1,· · ·, p. Further, we assume that there are no duplicated units in the observed data. Thep×noutput matrix for the nDMUs is denoted byY, and them×ninput matrix for thenDMUs is denoted byX. xo := (x1o,· · ·, xmo)andyo := (y1o,· · ·, ypo)are amounts of inputs and outputs of DMUo, which is evaluated. In addition,εis a small positive number (“non-Archimedean”) and1= (1,· · ·,1)is a unit vector.1
For convenience, the following notations for vectors inIRp+mwill be used:
zo>zj ⇐⇒ zio > zij, i= 1,· · ·, p+m, zo=zj ⇐⇒ zio =zij, i= 1,· · ·, p+m,
zo≥zj ⇐⇒ zio =zij, i= 1,· · ·, p+m but zo 6=zj.
So far, several DEA models have been developed. Among them, the CCR model (Charnes et al., 1978; 1979), the BCC model (Banker et al., 1984), and the FDH model (Tulkens, 1993) are well known as basic DEA models. These models are based on the domination structure in the primal form, and moreover these are characterized by how to determine the production possibility set in the dual form;
the convex cone, the convex hull, and the free disposable hull for the observed data, respectively. These models are further discussed in the following subsections.
8.2.1 The CCR model
The CCR model, which was suggested by Charnes et al. (1978), is a fractional linear programming problem. It can be solved by being transformed into an equiv-alent linear programming problem. Therefore, the primal problem (CCR) with an input-oriented model can be formulated as the following:
maximize
1Archimedean property : Ifx∈IR,y∈IRandx >0, then there exists a positive integernsuch thatnx > y. Non-Archimedeanεis a small positive number not satisfying Archimedean property.
Generalized DEA Model for Multiple Criteria Decision Making 149 The dual problem (CCRD) to the problem (CCR) is given by
minimize
θ,λ,sx,sy
θ−ε(1Tsx+1Tsy) (CCRD) subject to Xλ−θxo+sx=0,
Y λ−yo−sy =0, λ=0, sx=0, sy =0,
θ∈IR, λ∈IRn, sx∈IRm,sy ∈IRp. The ‘efficiency’ in the CCR model is introduced as follows:
Definition 1. (CCR-efficiency) A DMUois CCR-efficient if and only if the optimal valuePp
k=1µ∗kykoto the problem (CCR) equals one. Otherwise, the DMUois said to be CCR-inefficient.
Definition 2. (CCRD-efficiency) A DMUois CCRD-efficient if and only if for the optimal solution(θ∗, λ∗, s∗x, s∗y)to the problem (CCRD), the following two con-ditions are satisfied:
(i) θ∗ is equal to one;
(ii) the slack variabless∗xands∗yare all zero.
Otherwise, the DMUois CCRD-inefficient.
Note that the above two definitions are equivalent due to the well known duality of linear programming.
Additionally, the production possibility set P1 in the dual form of the CCR model is the convex cone (or conical hull) generated by the observed data, which implies that the scale efficiency of a DMU is constant, that is to say, it involves constant returns to scale. Therefore,P1can be denoted by
P1= n
(y,x)|Yλ=y, Xλ5x, λ=0 o
.
and the definition of CCR-efficiency (or CCRD-efficiency) can be transformed into the following:
Definition 3. DMUois said to be Pareto efficient inP1if and only if there does not exist(y,x)∈P1such that(y,−x)≥(yo,−xo).
It is readily seen that the Pareto efficiency in P1 is equivalent to the CCR-efficiency. Figure 8.1 shows a geometric interpretation on the relation between the primal form of CCR model and the dual one.
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( a ) P r i m a l f o r m ( b ) D u a l f o r m
O u t p u t
I n p u t
O u t p u t
I n p u t
Figure 8.1. CCR efficient frontier and production possibility set generated by the CCR model from the observed data.
8.2.2 The BCC model
The BCC model of Banker et al. (1984) is formulated similarly to that for the CCR model. The dual problem for the BCC model is obtained by adding the convexity constraint1Tλ= 1to the dual problem (CCRD) and thus, the variableuoappears in the primal problem. The efficiency degree of a DMUowith respect to the BCC model can be measured by solving the problem.
maximize
µk, νi, uo
Xp k=1
µkyko−uo (BCC)
subject to
Xm i=1
νixio = 1, Xp
k=1
µkykj− Xm i=1
νixij−uo50, j = 1,· · ·, n, µk=ε, νi=ε, k= 1,· · ·, p; i= 1,· · ·, m.
The dual problem (BCCD) to the problem (BCC) is formulated as follows:
minimize
θ,λ,sx,sy
θ−ε(1Tsx+1Tsy) (BCCD) subject to Xλ−θxo+sx=0,
Y λ−yo−sy =0, 1Tλ= 1,
λ=0, sx=0, sy =0,
θ∈IR, λ∈IRn, sx∈IRm,sy ∈IRp.
Generalized DEA Model for Multiple Criteria Decision Making 151 The ‘efficiency’ in the BCC model is given by the following two definitions which are equivalent to each other due to the duality of linear programming.
Definition 4. (BCC-efficiency) A DMUois BCC-efficient if and only if the opti-mal value Pp
k=1µ∗kyko−u∗o
to the problem (BCC) equals one. Otherwise, the DMUois said to be BCC-inefficient.
Definition 5. (BCCD-efficiency) A DMUois BCCD-efficient if and only if for an optimal solution(θ∗, λ∗, s∗x, s∗y)to the problem (BCCD), the following two con-ditions are satisfied:
(i) θ∗ is equal to one;
(ii) the slack variabless∗xands∗yare all zero.
Otherwise, the DMUois said to be BCCD-inefficient.
The presence of the constraint1Tλ= 1in the dual problem (BCCD) yields that the production possibility setP2in the BCC model is the convex hull generated by the observed data. Therefore,P2can be obtained as
P2= n
(y,x)|Yλ=y, Xλ5x,1Tλ= 1,λ=0 o
.
and the definition of BCCD-efficiency can be transformed into the following:
Definition 6. DMUois said to be Pareto efficient inP2if and only if there does not exist(y,x)∈P2such that(y,−x)≥(yo,−xo).
It is readily seen that the Pareto efficiency in P2 is equivalent to the BCC-efficiency. Figure 8.2 shows a geometric interpretation on the relation between the primal form of BCC model and the dual one.
8.2.3 The FDH model
The FDH model by Tulkens (1993) is formulated as follows:
minimize
θ,λ,sx,sy
θ−ε(1Tsx+1Tsy) (FDHD) subject to Xλ−θxo+sx=0,
Y λ−yo−sy =0,
1Tλ= 1; λj ∈ {0, 1} for each j= 1,· · ·, n, λ=0, sx=0, sy =0,
θ∈IR, λ∈IRn, sx∈IRm,sy ∈IRp.
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( a ) P r i m a l f o r m ( b ) D u a l f o r m
O u t p u t
I n p u t
O u t p u t
I n p u t
Figure 8.2. BCC efficient frontier and production possibility set generated by the BCC model from the observed data.
Here, however, it is seen that the problem (FDHD) is a mixed integer program-ming problem, and hence the traditional linear optimization methods cannot apply to it. An optimal solution is obtained by means of a simple vector comparison procedure at the end.
For a DMUo, the optimal solutionθ∗ to the problem (FDHD) is equal to the valueR∗odefined by
R∗o = min
j∈D(o) max
i=1,···m
xij xio
, (8.1)
whereD(o) =
j|xj ≤xo and yj =yo, j = 1,· · ·, n .
R∗ois substituted forθ∗ as the efficiency degree for DMUoin the FDH model.
The ‘efficiency’ in the FDH model is given by the following.
Definition 7. (FDH-efficiency) A DMUois FDH-efficient if and only ifR∗o equals to one. IfR∗o<1, the DMUois said to be FDH-inefficient.
Definition 8. (FDHD-efficiency) A DMUois FDHD-efficient if and only if for an optimal solution(θ∗, λ∗, s∗x, s∗y)to the problem (FDHD), the following two con-ditions are satisfied:
(i) θ∗ is equal to one;
(ii) the slack variabless∗xands∗yare all zero.
Otherwise, the DMUois said to be FDHD-inefficient.
Generalized DEA Model for Multiple Criteria Decision Making 153
( a ) P r i m a l f o r m ( b ) D u a l f o r m
O u t p u t
I n p u t O u t p u t
I n p u t
Figure 8.3. FDH efficient frontier and production possibility set generated by the FDH model from the observed data.
It can be seen that the above two definitions are equivalent to each other, and the production possibility setP3, which is a free disposable hull, is given by
P3= n
(y,x)|Yλ=y, Xλ5x, 1Tλ= 1, λj ∈ {0,1}, j= 1,· · ·, n o
. (8.2) Besides, the definition of FDH-efficiency (or FDHD-efficiency) can be trans-formed into the following:
Definition 9. DMUois said to be Pareto efficient inP3if and only if there does not exist(y,x)∈P3such that(y,−x)≥(yo,−xo).
It is shown that the Pareto efficiency inP3 is equivalent to the FDH-effciency.
Figure 8.3 shows a geometric interpretation on the relation between the primal form of FDH model and the dual one.