Chance constrained programming approaches to ... - Semantic Scholar
European Journal of Operational Research 155 (2004) 487–501 www.elsevier.com/locate/dsw
Stochastics and Statistics
Chance constrained programming approaches to congestion in stochastic data envelopment analysis William W. Cooper a
a,*
, H. Deng a, Zhimin Huang b, Susan X. Li
b
The Red McCombs School of Business, University of Texas at Austin, Austin, TX 78712-1174, USA b School of Business, Adelphi University, Garden City, NY 11530, USA Received 8 August 2001; accepted 25 September 2002
Abstract The models described in this paper for treating congestion in DEA are extended by according them chance constrained programming formulations. The usual route used in chance constrained programming is followed here by replacing these stochastic models with their ‘‘deterministic equivalents.’’ This leads to a class of non-linear problems. However, it is shown to be possible to avoid some of the need for dealing with these non-linear problems by identifying conditions under which they can be replaced by ordinary (deterministic) DEA models. Examples which illustrate possible uses of these approaches are also supplied in an Appendix A. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Inefficiency; Congestion; DEA (data envelopment analysis); Chance constrained programming
1. Introduction This paper forms one part of a series of continuing research efforts. A previous paper treated the topic of stochastic characterizations of efficiency and inefficiency in DEA using chance constrained programming formulations and constructs. See Cooper et al. (2002a). Here a similar chance constrained approach to efficiency is undertaken but attention is centered on ‘‘congestion’’ as one (particularly severe) form of inefficiency.
*
Corresponding author. Tel.: +1-512-471-1822; fax: +1-512471-0587. E-mail address: [email protected] (W.W. Cooper).
A focus on this one type of inefficiency may be justified as follows. Congestion has been an underresearched topic in the economic theory of production even though it can be of importance when its use is associated with a need for augmenting inputs to serve important objectives besides output maximization. As noted in Cooper et al. (2001b), for instance, congestion is used in China to deal with the need for providing employment for a large labor force, with some 16,000,000–18,000,000 new entrants each year. In addition, recent exchanges in the literature that deal with congestion have been accompanied by new developments that have opened additional topics for research. See the exchange between Cherchye et al. (2001) and Cooper et al. (2001a). See also the exchange between F€are and Grosskopf (2000) and Cooper et al. (2000).
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00901-3
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As noted above, congestion is a particularly severe form of inefficiency in that input increases are associated with output decreases in the manner formalized in the following definition. Congestion: Congestion is present in the performance of DMUo , the DMU being evaluated, when input increases give rise to decreases in the outputs that are maximally attainable or, conversely, when input decreases are accompanied by increases in the output amounts that are maximally attainable. We distinguish between congestion and ‘‘technical inefficiency’’––sometimes referred to as ‘‘waste’’–– which is defined as follows. Inefficiency (Technical): Inefficiency is present in the performance of DMUo if the evidence shows that it is possible to improve some of its inputs or outputs without worsening any of its other inputs or outputs. Note that nothing is said about an input reduction being accompanied by any output augmentation. To see what is involved we turn to Fig. 1 in Appendix A where the coordinates in parentheses record input amounts (on the left) and output amounts (on the right) for the one input used and the one output produced by each of five decision making units (DMUs) A–E. Now refer to point C and observe that its output level at y ¼ 2 is the same as for B. However, B attains this output with x ¼ 2 units of this input whereas C utilizes x ¼ 3 units. Hence, without worsening or improving the output attained at C it is possible to reduce input from x ¼ 3 to 2 by moving from C to B. This means that the evidence shows C to be inefficient because it is found to be possible to decrease its input without worsening output. The congestion in D, it may be noted, is a more severe form of inefficiency because the maximal output y ¼ 2 is not attainable without reducing the input for D from x ¼ 5 to 3. Remark. It is to be noted that we are not using prices, unit costs or other such relative weights.
Hence we are dealing only with ‘‘technical inefficiency’’ and not with other types of inefficiency such as ‘‘allocative inefficiency,’’ etc., which require a use of relative weights to arrive at their evaluations. Finally, we introduce the following definition of (technical) efficiency. Efficiency: The performance of DMUo is to be characterized as technically efficient if and only if the evidence shows that it is not possible to improve some of its inputs or outputs without worsening some of its other inputs or outputs. Points A and B in Fig. 1, as well as all of the points on the boundary shown by the solid line between them have this property. In the literature of DEA this is referred to as the ‘‘efficiency frontier.’’ No point not on a boundary can have this efficiency property and no point on the boundaries between B and C or between C and D have this property. Hence only the points on the efficiency frontier represented by the solid line between A and B are efficient. With these definitions in hand we begin to provide the stochastic (¼ probabilistic) characterizations of congestion that are the special focus of this paper. This is done as follows. First we introduce the models we use to identify congestion in deterministic form. Next we undertake our chance constrained formulations with accompanying definitions and developments. We then reduce these formulations to ‘‘deterministic equivalents.’’ Finally, we introduce simplifying assumptions that allow us to identify conditions under which appearances of congestion with associated probabilities can be determined in a straightforward manner. A summary and conclusion is followed by Appendix A which provides illustrative examples. 2. A two-model approach We start with the deterministic models we use to evaluate congestion. Assume that there are n
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
DMUs to be evaluated which we associate with points, xj ¼ ðx1j ; . . . ; xmj ÞT and yj ¼ ðy1j ; . . . ; ysj ÞT that represent (m 1) and (s 1) input and output vectors, respectively, for each DMUj , j ¼ 1; . . . ; n, and where the superscript T indicates the transpose of these vectors. Now, to maintain contact with Cooper et al. (2002a), we begin with the following version of a BCC model: ! s m X X þ max / þ e sr þ si ; r¼1
s:t:
/yr0
n X
i¼1
yrj kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
xij kj þ s i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð1Þ
Thus (1) is used to determine technical efficiency and inefficiency. To determine whether congestion is present we proceed from an optimal solution of (1) as follows. If the two conditions specified in the definition of DEA efficiency are satisfied then neither technical inefficiency nor congestion is evidenced by the performance recorded for DMUo . If either condition fails to be satisfied then DMUo is inefficient. Then, to see whether congestion is present we use the solution of (1) to erect a second problem as follows. For an optimal solution (/ , k , sþ , s ) of (1), we can reexpress its constraints in the following form: n X / yr0 þ sþ ¼ yrj kj ; r ¼ 1; . . . ; s; ð2Þ r
j¼1 n X
489
j¼1
kj ¼ 1;
j¼1
0 6 kj ;
sþ r ; si ; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m: Here xij and yrj represent input and output amount recorded for j ¼ 1; . . . ; n DMUs and j ¼ o designates one of the DMUj as the DMUo to be evaluated relative to all of the data (including the data on DMUo ) and e > 0 is a ‘‘non-Archimeadean element’’ defined to be smaller than any positive real number. This means that e is not a real number. The standard procedure is to avoid any need for explicitly assigning a value to e by using the following two-stage process. Stage one: maximize / while ignoring the slacks, sþ r , si , in the objective. Stage two: replace / with / ¼ max / in (1) and maximize the sum of the slacks. Then determine whether DMUo is efficient or inefficient in accordance with the following definition. Definition (DEA Efficiency). DMUo is efficient if and only if the following two conditions are both satisfied: (i) / ¼ 1, (ii) sþ r ¼ si ¼ 0, 8 r, i. Here and hereafter is used to designate an optimum value.
xi0 s i ¼
n X
xij kj ;
i ¼ 1; . . . ; m:
ð3Þ
j¼1
Then we can use the values on the left in (2) and (3) to define new outputs and inputs, y^r0 , x^i0 , as in the following:
y^r0 ¼ / yr0 þ sþ r P yr0 ;
x^i0 ¼ xi0 s i 6 xi0 ;
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m:
ð4Þ ð5Þ
The expressions (4) and (5) are referred to as the ‘‘CCR projection formulas’’ because they project the observed yr0 and xi0 into y^r0 and x^i0 on the efficient frontier. As proved in Charnes et al. (1981), none of the coordinates for this projected point can be improved without worsening some (one or more) of its other coordinates. Thus Dyr0 ¼ y^r0 yr0 P 0 represents the amount of inefficiency in the rth output for DMUo , and Dxi0 ¼ xi0 x^i0 P 0 represents the amount of inefficiency in the ith input for DMUo . Hence, in accordance with the above definition, inefficiency is present if and only if Dyr0 6¼ 0 or Dxi0 6¼ 0 for some r or i. Note, therefore, that the amounts as well as the sources of all of these inefficiencies are identified for each DMUj ¼ DMUo by reference to these Dyr0 , Dxi0 values. Remark. As can be seen, this definition adapts the verbal definition of efficiency given in the preceding section by relating it to the above DEA model.
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Hence, to reflect this specialization, we refer to this as ‘‘DEA Efficiency.’’ As described in Cooper et al. (2001b), these y^r0 and x^r0 values together with the yrj and xij as defined in (1) are used to erect the following new problem: m X max d i ; i¼1
s:t:
y^r0 ¼ / yr0 þ sþ r ¼
n X
yrj kj ;
j¼1
r ¼ 1; . . . ; s;
x^i0 ¼ xi0 s i ¼
n X
xij kj d i ;
j¼1
i ¼ 1; . . . ; m; n X 1¼ kj ;
ð6Þ
j¼1
s i P di ;
i ¼ 1; . . . ; m;
with all variables constrained to be non-negative. See also Cooper et al. (1996b) where this formulation was first introduced. Finally, to identify the congesting inputs and to estimate their amounts, we utilize the i ¼ 1; . . . ; m input constraints shown in the last set of constraints at the bottom of (6) to obtain
sc ¼ s i i di ;
i ¼ 1; . . . ; m;
ð7Þ
where di is obtained from an optimal solution to (6). sc is then the ‘‘congesting amount’’ in the i ‘‘total slack’’ associated with s in input i ¼ i 1; . . . ; m, as obtained from (1), and d is the i (maximum) amount of this total slack that can be assigned to ‘‘purely technical’’ (non-congesting) inefficiency, as obtained from (6). Remark. Reference to Fig. 1 may help again to clarify what is happening. Model (1) identifies B as the efficient point for use in evaluating E. (This is confirmed by noting that (4) and (5) yield x^i0 as the coordinates of B.) Use of (4) and (5) provide the expression on the left in (6)––viz., y^r0 and x^i0 ––as well as the values of s for the bottom-most coni straints. The di in (6) are also slacks. These slacks are to be ‘‘backed out’’ as far as the equations for the output constraints allow in (6). Hence di must be consistent with y^ ¼ 2 and condition 2 P d i .
Here 2 ¼ s ¼ 4 2 represents the slack associated with the maximization for the e term in the objective of (1). Hence the best that can be done in ‘‘backing out’’ the slack from (1) moves the solu tion from B to C in (6) giving d ¼ 3 2 ¼ 1 as the maximal value. To obtain sc we therefore simply subtract this value of d from s ¼ 2 to obtain sc ¼ 2 1 ¼ 1 as the amount of this input that is i congesting. The value of d i ¼ 1, on the other hand, represents the amount of the slack associated with the ‘‘technical inefficiency’’ that is exhibited by moving from B to C. We can now further clarify the last sentence in the definition of congestion given in Section 1. Suppose E is translated horizontally to a new position to the left of G in Fig. 1. This identifies it with a point under the boundary line between B and C. From this position it is possible to raise output to the maximum level allowed by this boundary without reducing input. No such output increase can be achieved from E in its present position without decreasing all of the congesting amount of input, sc ¼ 1. Thus, the objective in (2) is to back i out all of the slack that is consistent with y^ ¼ 2, the maximum output, as determined from (1). The remainder is the congesting amount in the ‘‘total slack,’’ s , as also determined from (1). 3. A one-model approach Cooper et al. (2002b) introduced a ‘‘one model’’ alternative to the two-model approach incorporated in (1) and (6) above to treat congestion. To see how this is done we first note that for an op timal solution (/ , k , sþ , s ) of (1), we can use c si ¼ si di , as defined in (7), to rewrite (6) as m X sc min i ; i¼1
s:t:
/ yr0 þ sþ r ¼
n X
yrj kj ;
r ¼ 1; . . . ; s;
j¼1
xi0 sc i ¼ 1¼
n X
n X
xij kj ;
i ¼ 1; . . . ; m;
j¼1
kj ;
j¼1
0 6 kj ;
sc i ; j ¼ 1; . . . ; n; i ¼ 1; . . . ; m:
ð8Þ
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
This model can be regarded as part of a twostage procedure analogous to the one we described for the second stage in dealing with the nonArchimedean element e > 0 in (1). Here, however, (8) can also be regarded as being derived from the following modification of (1): max
/þe
s X
sþ r
r¼1
s:t:
/yr0
n X
m X
! sc ; i
i¼1
yrj kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
xij kj þ sc i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð9Þ
j¼1 n X
kj ¼ 1;
j¼1
0 6 kj ;
þ sc i ; sr ; j ¼ 1; . . . ; n; i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s:
following theorem represents the promised improvement. Theorem 2. Congestion is present if and only if for an optimal solution (/ , k , sþ , sc ) of (9), there is at least one sc > 0 (1 6 i 6 m). i Proof. (Necessary condition) It is obvious by the congestion definition. (Sufficient condition) We need to show that for sc > 0 (1 6 i0 6 m), we must have / > 1 or there i0 is at least one sþ r > 0 (1 6 r 6 s). Suppose to the contrary that / ¼ 1 and sþ r ¼ 0 _ _þ þ for all r ¼ 1; . . . ; s. Let / ¼ / ¼ 1, s r ¼ sr ¼ 0, _ _c r ¼ 1; . . . ; s, s i ¼ 0, i ¼ 1; . . . ; m, k j ¼ 0 for j 6¼ 0 _ _ _ _þ _c and k 0 ¼ 1. Then (/ , k , s , s ) is a feasible solution of (9) and ! s m X X _ _þ _c / þe sr si ¼ / r¼1
>/ þe As can be seen, the objective of (1) is here modified c by replacing þs i with si in the objective of (9). c See also the replacement of s in the i with si constraints. Now we can reason as follows. From our above analysis, let us suppose that (/ , k , sþ , sc ) is an þ optimal solution of (9). Then / and sr are part of an optimal solution of (1), and (k , sc ) is an optimal solution of (8). The congesting amount of input i ¼ 1; . . . ; m, is then represented by sc in (9) i in accordance with the following theorem as taken from Cooper et al. (2002b).
491
i¼1
m X
!
sc i
i¼1
¼/ þe
s X r¼1
sþ r
m X
! sc i
:
i¼1
This contradicts the assumption that (/ , k , sþ , sc ) is an optimal solution of (9). Therefore, we have the following combined theorem on inefficiency and congestion.
Theorem 3. Suppose (/ , k , sþ , sc ) is an optimal solution of (9). Then:
(i) / > 1 and there is at least one sc >0 i (1 6 i 6 m), (ii) there exists at least one sþ r > 0 (1 6 r 6 s) and c at least one si > 0 (1 6 i 6 m).
(a) If / > 1, then DMUo is inefficient. (b) If there exists at least one sþ r > 0 (1 6 r 6 s), then DMUo is inefficient. (c) If there exists at least one sc > 0 (1 6 i 6 m), i then DMUo is inefficient and congestion is present. (d) If / ¼ 1, sþ ¼ 0 and sc ¼ 0, then DMUo is on a frontier.
Next we improve the theorem by showing that if we have at least one sc > 0 (1 6 i 6 m), then it i guarantees / > 1 or there exists at least one sþ r > 0 (1 6 r 6 s), i.e., congestion is present. The
In conclusion we note that (d) in Theorem 3 guarantees only that DMUo is on a frontier. It does not guarantee that it will be on an efficient portion of the frontier.
Theorem 1. Congestion is present if and only if in an optimal solution (/ , k , sþ , sc ) of (9), at least one of the following two conditions is satisfied:
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4. Stochastic models We follow the notation in Cooper et al. (1996a) T T and let ~xj ¼ ð~x1j ; . . . ; ~xmj Þ and y~j ¼ ð~ y1j ; . . . ; y~sj Þ represent (m 1) and (s 1) random input and T output vectors, and xj ¼ ðx1j ; . . . ; xmj Þ and T yj ¼ ðy1j ; . . . ; ysj Þ stand for the corresponding vectors of expected values of input and output for each DMUj , j ¼ 1; . . . ; n. That is, we utilize these expected values in place of the observed values in (1). See Olesen and Petersen (1995) for an alternate approach which uses the means of a series of observations to obtain ‘‘confidence interval’’ estimates. Let us consider all input and output components to be jointly normally distributed in the following chance constrained version of a stochastic DEA model: max s:t:
/; ( P
n X
) y~rj kj P /~ yr0
P 1 a;
j¼1
r ¼ 1; . . . ; s; ( P
n X
) ~xij kj 6 ~xi0
P 1 a;
ð10Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1
kj P 0;
j ¼ 1; . . . ; n:
Here, P means ‘‘Probability’’ and a is a predetermined number between 0 and 1. We now use this model to define stochastic efficiency as follows. Definition (Stochastic Efficiency). DMUo is stochastic efficient if and only if the following two conditions are both satisfied: (i) / ¼ 1; (ii) slack values are all zeros for all optimal solutions, where (ii) refers to all alternate optima because the second stage optimization associated with e > 0 in (1) is not used in (10).
The ~xij ¼ ~xi0 , y~rj ¼ y~r0 values for DMUo appear on the left as well as on the right inside the braces of (10). Hence, we can always get a solution with / ¼ 1, k0 ¼ 1 and 8kj ¼ 0 (j 6¼ 0) with all slacks zero. However, this solution need not be maximal. It follows that a maximum with / > 1 in (10) for any sample of j ¼ 1; . . . ; n observations means that the DMUo being evaluated is not efficient because, to the specified level of probability defined by a, the evidence will then show that all outputs of DMUo can be increased to / y~r0 > y~r0 , r ¼ 1; . . . ; s, without violating the output constraints by using a convex combination of other DMUs which will also satisfy n X
~xij kj 6 ~xi0 ;
i ¼ 1; . . . ; m:
ð11Þ
j¼1
To the indicated degree of probability no input is worsened by this increase in all of DMUo Õs outputs. Thus, to the same level of probability, these output increases will not require an augmentation in any of the input amounts used by DMUo . Indeed, any positive slacks in the ith input constraint permit a decrease in the corresponding ~xi0 . Now let us discuss positive slacks in the rth output chance constraint. Suppose 1r > 0 is an ‘‘external slack’’––i.e., a slack which can be inserted in the inequality outside the braces to achieve equality for the rth output chance constraint, viz., ) ( n X P ð12Þ y~rj kj /~ yr0 P 0 ¼ ð1 aÞ þ 1r : j¼1
There must then exist a number sþ r > 0 such that ( ) n X P y~rj kj /~ yr0 P sþ ¼ 1 a: ð13Þ r j¼1
This value of sþ r permits a still further increase in y~r0 for any sample of observations without worsening any other input or output. It is easy to see that 1r ¼ 0 if and only if sþ r ¼ 0. In a manner similar to our treatment of the output constraints, suppose ni > 0, used as the external slack for ith input chance constraint, satisfies
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
( P
n X
) ~xij kj ~xi0 6 0
¼ ð1 aÞ þ ni :
ð14Þ
max
/þe
j¼1
Then there must exist a positive number s i > 0 such that ( ) n X ~xij kj þ si 6 ~xi0 ¼ 1 a: P ð15Þ j¼1
( s:t:
P
j¼1
r ¼ 1; . . . ; s; ( P
n X
) ~xij kj þ
s i
6 ~xi0
¼ 1 a;
ð16Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m: This leads to the following Definition. DMUo is stochastic efficient to the indicated levels of probability if and only if the following two conditions are both satisfied: (i) / ¼ 1, (ii) sþ r ¼ si ¼ 0, 8 i, r. The stochastic model in (16) is evidently a generalization of the BCC model in (1). In a similar manner, it is natural to generalize the ‘‘onemodel’’ approach to congestion represented in (9) to the following stochastic version:
sþ r
r¼1 n X
y~rj kj
m X
! sc i
i¼1
/~ yr0 P sþ r
; ) ¼ 1 a;
j¼1
( P
To the indicated level of probability this value of s xi0 for any set of sample i permits a decrease in ~ observations without worsening any other input or output. Again it is easy to show that ni ¼ 0 if and only if s i ¼ 0. Therefore, if we introduce the nonArchimedean infinitesimal, e > 0, stochastic efficiencies and inefficiencies can be characterized by the following stochastic version of (1): ! s m X X þ max / þ e sr þ si ; r¼1 i¼1 ( ) n X þ y~rj kj /yr0 P sr ¼ 1 a; s:t: P
s X
493
r ¼ 1; . . . ; s; ) n X c ~xij kj þ si 6 ~xi0 ¼ 1 a;
ð17Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1 c kj P 0; sþ r P 0; si P 0; j ¼ 1; . . . ; n; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
Therefore, we have the following results as obvious generalizations of Theorems 2 and 3. Theorem 4. Congestion is present for DMUo to the prescribed level of probability in stochastic model (16) if and only if for an optimal solution (/ , k , sþ , sc ) of (17), there exists at least one sc >0 i0 (1 6 i0 6 m). Theorem 5. At an optimum of (17) we have the following: (a) If / > 1, then DMUo is stochastic inefficient. (b) If there exists at least one sþ r0 > 0 (1 6 r0 6 s), then DMUo is stochastic inefficient. (c) If there exists at least one sc > 0 (1 6 i0 6 m), i0 then DMUo is stochastic inefficient and congestion is present. (d) If / ¼ 1, sþ ¼ 0 and sc ¼ 0, then DMUo is on a segment of the stochastic frontier, i.e., DMUo 2 E [ E0 [ F , where E and E0 represent the collection of observed values for DMUs represented by extreme and nonextreme points on the efficiency frontier and F represents the points which are on a portion of the frontier that is not efficient. See Charnes et al. (1991, 1986). To make this all more concrete we follow Cooper et al. (1996a), and assume that inputs and outputs are random variables with a multivariate normal distribution and known parameters. We
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also restrict attention to the class of ‘‘zero-order decision rules’’ (see Charnes et al. (1958) for a discussion of these and other decision rules). Remark. Choices of multivariate normal distributions and zero-order rules are less restrictive than might at first appear to be the case. Transformations are available for bringing other types of distributions into approximately normal form––as was done in Charnes et al. (1958), for instance, when it was found necessary to treat highly skewed distributions with log-normal approximations. We can also interpret our zero-order decision rules as a series of one-period-at-a-time applications, with appropriate models, to allow for changing realizations and probabilities, and regard these as approximations to the more complex solution procedures involved in developing higher order ‘‘conditional’’ decision rules to deal with the full scale treatment of dynamics.
where the yrj and xij (including yr0 and xi0 ) are the means of these variables. (As noted earlier, these means are assumed to be known.) Similarly, the deterministic equivalent of (17) can be represented by ! s m X X þ c max / þ e sr si ; r¼1
s:t:
n X
/yr0
r ¼ 1; . . . ; s; n X
1 I xij kj þ sc i U ðaÞri ðkÞ ¼ xi0 ;
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1 c sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
sþ r ,
s i
We now show how to obtain the / , from deterministic equivalents of the stochastic models represented in (16) and (17). With normal distributions and zero order decision rules we can obtain a deterministic equivalent for (16) which can be represented by
max
/þe
þ
m X
r¼1
s:t:
/yr0
n X
Here U is the standard normal distribution function and U1 , its inverse, is the so-called ‘‘fractile function,’’ Finally, 2
ðr0r ð/; kÞÞ ¼
XX i6¼0
! s i
ð19Þ
j¼1
5. Deterministic equivalents
sþ r
1 0 yrj kj þ sþ r U ðaÞrr ð/; kÞ ¼ 0;
j¼1
kj P 0;
s X
i¼1
ki kj Covð~ yri; y~rj Þ
j6¼0
þ 2ðk0 /Þ
;
X
ki Covð~ yri ; y~r0 Þ
i6¼0
i¼1 2
þ ðk0 /Þ Varð~ yr0 Þ
1 0 yrj kj þ sþ r U ðaÞrr ð/; kÞ ¼ 0;
j¼1
and
r ¼ 1; . . . ; s; n X
xij kj þ
s i
1
U
ðaÞrIi ðkÞ
¼ xi0 ;
j¼1
i ¼ 1; . . . ; m; n X
ð18Þ
ðrIi ðkÞÞ2 ¼
XX j6¼0
kj kk Covð~xij;~xik Þ
k6¼0
þ 2ðk0 1Þ
X
kj Covð~xij ; ~xi0 Þ
j6¼0
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
2
þ ðk0 1Þ Varð~xi0 Þ; where we have separated out the terms for DMUo because they appear on both sides of the expres-
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
max
/þe
s X r¼1
s:t:
0 /yr0
n X
sþ r
m X
! s i
;
i¼1
yrj0 kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ s i ¼ xi0 ;
i ¼ 1; . . . ; m;
j¼1 n X
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; and model (19) can be written
!
sions in (16) ff. Thus, / , sþ and the sc can r , si i be determined from (18) and (19) where the data (means and variances) are all assumed to be known. Because of the functional forms of r0r ð/; kÞ and rIi ðkÞ it is obvious that models (18) and (19) are non-linear programming problems. Let x0r and xIr be non-negative variables. Replace r0r ð/; kÞ by x0r and rIi ðkÞ by xIr in both (18) and (19), and add two quadratic equality constraints, ðx0r Þ2 ¼ ðr0r ð/; kÞÞ2 and ðxIr Þ2 ¼ ðrIi ðkÞÞ2 , to both (18) and (19), then (18) and (19) are both transformed to easily solvable quadratic programming problems. From (18) and (19) we can see that if the predetermined value of a is equal to 0.5, then U1 ðaÞ ¼ 0 and stochastic efficiencies, inefficiencies and congestion can be characterized by the BCC model (1) and congestion model (9) using the mean values of inputs and outputs. Thus a ¼ 0:5 may be regarded as indicating indifference between stochastic and deterministic approaches and classifications of stochastic efficiencies, inefficiencies and congestion are the same as in the deterministic situation with the input and output means. To simplify matters in a different manner let us assume that only DMUo has random variations in its inputs and outputs, i.e., rIi0 6¼ 0, r0r0 6¼ 0, rIij ¼ 0, and r0rj ¼ 0 (j 6¼ 0) for all i and r. In this case, model (18) can be written
max
/þe
s X
sþ r
r¼1 0 /yr0
s:t:
n X
m X
495
sc i
;
i¼1
yrj0 kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ sc i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð21Þ
j¼1 n X
kj ¼ 1;
j¼1
kj P 0;
c sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; where 0 ¼ yr0 r0r0 U1 ðaÞ; yr0
yrj0
¼ yrj ;
j 6¼ o; r ¼ 1; . . . ; s;
x0i0 ¼ xi0 þ rIi0 U1 ðaÞ; x0ij
¼ xij ;
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m;
j 6¼ o; i ¼ 1; . . . ; m:
ð22Þ
ð23Þ
With these assumptions model (20) is the deterministic equivalent of stochastic model (16) and model (21) is the deterministic equivalent of (17). Reasons for us to consider random variations only in DMUo are as follows: First, treating more than one DMU in this manner leads to deterministic equivalents with the more complicated relations that have been discussed in detail in Cooper et al. (2002a). The simpler approach used here allows us to arrive at analytical results and characterizations in the straightforward manner that we discuss in the next section. Second it opens possible new routes for effecting ‘‘sensitivity analyses’’ which we remark upon as follows.
ð20Þ Remark. We are referring to the ‘‘sensitivity analyses’’ that are to be found in Charnes and Neralic (1990), Charnes et al. (1992), Charnes et al. (1996) and Seiford and Zhu (1998). In the terminology of the survey article by Cooper et al. (2001c), these sensitivity analyses are directed to analyzing allowable limits of data variations for only one DMU at a time and hence contrast with other approaches to sensitivity analysis in DEA that
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allow all data for all DMUs to be varied simultaneously until at least one DMU changes its status from efficient to inefficient, or vice versa. These sensitivity analyses are entirely deterministic. Our chance constrained approach can be implemented by representations that are similar in form to those used in sensitivity analysis but the conceptual meanings are different. A chance constrained programming problem can be solved by a deterministic equivalent, as we have just shown, but the issue originally addressed in the chance constrained formulation is different and this introduces elements, such as the risk associated with a, that are nowhere present in these sensitivity analyses. This can be illustrated by Fig. 1 in the Appendix A where, as previously noted, point C is technically inefficient but does not display congestion. This point is evidently very sensitive to changes in its output (but not its input) data. In the terminology of Charnes et al. (1996), it has a ‘‘zero radius of stability.’’ If its output is raised in any positive amount, C becomes efficient. Alternatively, if C is lowered it becomes an example of congestion. All these arguments are from a sensitivity analysis point of view. If we consider changes in point C to involve only random variations, these characterizations will change. They will depend not on the change in output level, but will depend rather on the specified probability level a. When a ¼ 0:5, random variations in the coordinates of point C do not have any impact on its efficiency, inefficiency or congestion characterizations. Hence it is satisfactory to employ the deterministic model (1) since, with this choice of a, the user is indifferent to the possible presence of inefficiency (or congestion) stochastically. This is different from the sensitivity analysis results. When a is taken between 0 and 0.5, point C will be efficient in the stochastic sense irrespective of the random variations (see Theorem 7(a), below). This is again different from the result of the sensitivity analysis discussed above. When a is assigned a value between 0.5 and 1, point C will be inefficient with congestion in the stochastic sense present–– no matter what the direction of random variations (see Theorem 8(b), below). Thus, in all cases the choice of a plays the critical role.
6. Results We are now in position to draw forth some further theorems as follows. Theorem 6. For a ¼ 0:5. Congestion is present for DMUo in input–output mean model (1) if and only if congestion is present for DMUo in stochastic model (16). Proof. As already noted this follows from the fact that U1 ð0:5Þ ¼ 0 for the standard normal distribution. See the discussion following (19). Theorem 7. For 0 < a < 0:5. (a) Suppose congestion is not present for DMUo in input–output mean model (1), then congestion is also not present for DMUo in stochastic model (16). (b) Suppose congestion is present for DMUo and DMUo 2 F in input–output mean model (1), then congestion is not present for DMUo and DMUo 2 E in stochastic model (16). (c) Suppose congestion is present for DMUo and DMUo 2 N in input–output mean model (1), then congestion is also present for DMUo and DMUo 2 N in stochastic model (16) and 0 sc ¼ sc þ rIi0 U1 ðaÞ is the congesting amount i i of the ith input of DMUo in stochastic model (16) for i ¼ 1; . . . ; m, if c rIi0 < min bi ; si =ðU1 ðaÞÞ and
1 r0r0 < bþ r =ðU ðaÞÞ;
where sc is the congesting amount of the ith i inputPof DMUo P in input–output mean model (1) m and sr¼1 bþ þ r i¼1 bi is the optimal value of s m X X bþ b max r þ i ; r¼1
s:t:
n X j¼1 n X j¼1 n X
i¼1
yrj kj bþ r P yr0 ;
r ¼ 1; . . . ; s;
xij kj þ b i 6 xi0 ;
i ¼ 1; . . . ; m;
kj ¼ j¼1 bþ r P 0;
1;
b i P 0; kj P 0; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n: ð24Þ
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
Proof (a) Consider (20) as the deterministic equivalent of (16). For 0 < a < 0:5, we have U1 ðaÞ < 0. Hence, referring to (22) and (23) we have y00 P y0 and x00 6 x0 . Therefore improvement for DMUo from (x0 ; y0 ) to (x00 ; y00 ) implies that congestion should also not be present for DMUo in stochastic model (16) when it is not present in mean model (1). (b) From Cooper et al. (2002a) we know that DMUo 2 E in stochastic model (16), consequently there is no congestion at DMUo in stochastic model (16). (c) From Cooper et al. (2002a) we know that DMUo 2 N in stochastic model (16). Therefore there are still improvements that can be made for output y00 of DMUo in deterministic equivalent (20) of stochastic model (16). Notice 0 that xi0 x0i0 ¼ rIi0 U1 ðaÞ < sc Let sc ¼ i . i 0 1 I c sc þ r U ðaÞ, then s is the congesting i i i0 amount of the ith input of DMUo in stochastic 0 model (16). Since sc > 0, congestion is present at i DMUo in stochastic model (16). Theorem 8. For 1 > a > 0:5. (a) Suppose congestion is present for DMUo in input–output mean model (1), then congestion is also present for DMUo in stochastic model 0 (16). Furthermore, sc ¼ sc þ rIi0 U1 ðaÞ is i i the congesting amount of the ith input of DMUo in stochastic model (16), where the sc i (i ¼ 1; . . . ; m) are the input congesting values as determined from input–output model (1). (b) Suppose DMUo 2 F and DMUo is an extreme point in input–output mean model (1), then DMUo 2 N and congestion is present for DMUo in stochastic model (16). Furthermore, 0 1 I sc ¼ s i i þ ri0 U ðaÞ is the congesting amount of the ith input of DMUo in stochastic model (16), where s i (i ¼ 1; . . . ; m) are the optimal input slack values obtained from input–output model (1) Proof (a) This is similar to the proof of Theorem 7(a). (b) From Cooper et al. (2002a) we know that DMUo 2 N in stochastic model (16). Since
497
DMUo is the only DMU with random variations in inputs and outputs and it is an extreme point on F in input–output model (1), any reductions in its outputs will move it from the 0 frontier of model (1). Notice that yr0 ¼ yr0 r0r0 U1 ðaÞ < yr0 and x0i0 ¼ xi0 þ rIi0 U1 ðaÞ > xi0 . 0 1 I While sc ¼ s i i þ ri0 U ðaÞ is the congesting amount of the ith input of DMUo in stochastic model (16). Theorem 9. For 0 < a < 0:5. (a) Suppose congestion is present for DMUo in stochastic model (16), then congestion is also present at DMUo in input–output mean model 0 (1). Furthermore, sc ¼ sc rIi0 U1 ðaÞ is i i the congesting amount of the ith input of DMUo 0 in input–output mean model (1), where the sc i (i ¼ 1; . . . ; m) are the optimal input congesting amounts in stochastic model (16). (b) Suppose DMUo 2 F and DMUo is an extreme point in stochastic model (16), then DMUo 2 N and congestion is present for DMUo in input–output mean model (1). Fur 0 thermore, sc ¼ sc rIi0 U1 ðaÞ is the i i congesting amount of the ith input of DMUo in input–output model (1), where 0 sc (i ¼ 1; . . . ; m) are the optimal input i slack values in stochastic model (16). Proof. This is similar to the proof of Theorem 8 when one replaces (xj ; yj ) by (x0j ; yj0 ), and notices that ðx0j ; yj0 Þ ¼ ðxj ; yj Þ for j 6¼ o. Theorem 10. For 1 > a > 0:5. (a) Suppose congestion is not present for DMUo in stochastic model (16), then congestion is also not present for DMUo in input–output mean model (1). (b) Suppose congestion is present for DMUo and DMUo 2 F in stochastic model (16), then congestion is not present for DMUo and DMUo 2 E in input–output mean model (1). (c) Suppose congestion is present for DMUo and DMUo 2 N in stochastic model (16), then congestion is also present for DMUo and DMUo 2 N in input–output mean model (1) 0 and sic ¼ sc rIi0 U1 ðaÞ is the congesting i
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amount of the ith input of DMUo in input–output mean model (1) for i ¼ 1; . . . ; m, if
c0 rIi0 < min b =U1 ðaÞ i ; si and
1 r0r0 < bþ i =U ðaÞ; 0
where sc is the congesting amount of the ith i input of DMU Pmo instochastic model (16), and Ps þ b þ is the optimal value of r¼1 r i¼1 bi max
s X
bþ r þ
r¼1
s:t:
n X
m X
b i ;
i¼1 0 yrj0 kj bþ r P yr0 ;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ b i 6 xi0 ;
i ¼ 1; . . . ; m;
j¼1 n X
kj ¼ 1;
j¼1
bþ r P 0;
b i P 0; kj P 0; r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m; j ¼ 1; . . . ; n:
ð25Þ
Proof. It is similar to the proof of Theorem 7 when one replaces (xj ; yj ) by (x0j ; yj0 ). This concludes the analytical developments. Examples are supplied in the Appendix A to help clarify meanings and possible uses of these results.
means. In other cases recourse must be had to the stochastic models we have introduced. Even in the latter class of cases, however, the task of identification and estimation may be accomplished by reference to deterministic equivalent models like those in (18) and (19). Although one must then deal with a non-linear programming problem, the task can be reduced to solving a quadratic programming problem. See the discussion following (19). The developments in this paper are part of a larger family of chance constrained programming approaches which seek to extend DEA to other contexts – see Land et al. (1992, 1993, 1994) – or to provide alternatives to presently available approaches in sensitivity analyses and other areas. For further discussions of chance constrained programming and other alternatives, see Sengupta (1995, pp. 142 ff). More research is also needed, of course, and some of this research might be directed to relaxing some of the assumptions used in this paper such as the need for assuming normal distributions with known means and variances. See Allen et al. (1972). Finally, we note that the development of our stochastic models involved switches from using the observed values of xij , yrj and xi0 , yr0 to a use of the means of the distributions that generated them. This opens a path for research pointed toward more forward looking uses of DEA in which probabilities of the occurrences of inefficiencies (and congestion) may be used to anticipate and perhaps forestall their occurrences.
Acknowledgements 7. Summary and conclusion In the early sections of this paper we covered the topic of congestion as treated in the DEA literature. We then incorporated these deterministic formulations in corresponding chance constrained programming models. We next showed how the task of identifying congestion may be accomplished with deterministic models rather than their chance constrained (stochastic) counterparts under suitable assumptions. In many cases this may be accomplished very simply from a knowledge of the
W.W. Cooper wishes to express his appreciation to the IC2 Institute of the University of Texas and the RGK Foundation in Austin, Texas, for support of his research. Z.M. Huang would like to acknowledge a PresidentÕs Faculty Development Grant of Adelphi University for support of his research, and S.X. Li would like to acknowledge the Research Sabbatical leave Grant of Adelphi University for support of her research. Thanks are also due to an anonymous referee for improvements in an earlier version of this paper.
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
499
Appendix A In order to illustrate some of our theoretical results, let us reconsider Fig. 1 and start by providing the following numerical illustrations to what was said in the remarks with which Section 5 concluded. Using only points C and E we consider only two cases with a ¼ 0:20 and 0.80. We also assume that rI ¼ r0 ¼ 0:5. Fig. 2. Random variations in C.
Case 1. a ¼ 0:20. From a cumulative normal distribution table, we have U1 ðaÞ ¼ 0:84 and therefore rI U1 ðaÞ ¼ r0 U1 ðaÞ ¼ 0:42 for use in the following example. Assume that only point C has random variations in its input and output. This situation is depicted in Fig. 2. Based on (22) and (23), the adjusted input and output for point C with a ¼ 0:2 is C 0 : x0 ¼ 2:58 and y 0 ¼ 2:42:
ðA:1Þ
0
The relation of C to all other points is depicted in Fig. 3. It is obvious that C 0 is efficient and hence no congestion is present at C 0 . Consequently, there is no congestion at C in stochastic model (16). This is consistent with Theorem 7(a). Now assume only point E has random variations in its input and output. Based on (22) and (23), the adjusted input and output for point E0 is E0 : x0 ¼ 3:58 and y 0 ¼ 1:42:
Fig. 3. Deterministic equivalent solution for C ¼ C 0 when a ¼ 0:2.
ðA:2Þ
Its relationships with all other points are depicted in Fig. 4. Using (21), it is found that E0 is not efficient and input congestion for E0 is equal to 0.58. Fig. 4. Deterministic equivalent solution for E ¼ E0 when a ¼ 0:2.
This is consistent with Theorem 7(c) as we show in the following. Utilizing (24) on point E, with coordinates as given in Fig. 1, we have the following linear pro gram to determine bþ and b : max s:t: Fig. 1. Congestion and technical inefficiency. Source: P.L. Brockett, W.W. Cooper, Hong Chul Shin and Yuying Wang. ‘‘Inefficiency and Congestion in Chinese Production before and after the 1978 Reforms,’’ Socio-Economic Planning Sciences, 32 (1998).
bþ þ b ; 0:5kA þ 2kB þ 2kC þ 1kD þ 1kE bþ P 1; 1kA þ 2kB þ 3kC þ 5kD þ 4kE þ b 6 4; kA þ kB þ kC þ kD þ kE ¼ 1; ðA:3Þ kA ; kB ; kC ; kD ; kE ; bþ ; b P 0:
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This gives bþ ¼ 1 and b ¼ 2, so that, as prescribed in Theorem 7(c),
minðb ; sc Þ=ðU1 ðaÞÞ ¼ 1:19;
bþ =ðU1 ðaÞÞ ¼ 1:19;
where sc ¼ 1 from Fig. 1 on point E. Thus, all conditions for Theorem 7(c) are satisfied and we have s
c0
¼s
c
I
1
þ r U ðaÞ ¼ 1 0:42 ¼ 0:58:
Fig. 6. Deterministic equivalent solution for E ¼ E0 when a ¼ 0:8.
This is the amount of input congestion for E in stochastic model (16).
on (22) and (23), the adjusted input and output values for point E are as depicted in Fig. 6.
Case 2. a ¼ 0:80. In this case, we again have U1 ðaÞ ¼ 0:84 and I 1 r U ðaÞ ¼ r0 U1 ðaÞ ¼ 0:42. Now we return to point C and assume that only this point has random variations in its input and output. Utilizing (22) and (23), the adjusted input and output values for point C are
E0 : x0 ¼ 4:42 and y 0 ¼ 0:58:
C 0 : x0 ¼ 3:42 and y 0 ¼ 1:58
ðA:4Þ
as shown in Fig. 5. It is easy to see from Fig. 5 (using (21)) that C 0 is not efficient and input congestion for C 0 is equal to 1.42. Consequently, point C is stochastically inefficient. This is consistent with Theorem 8(b). In order to see this, we notice that the optimal slack value for input at C is s ¼ 1. Then based on Theorem 8(b), 0
sc ¼ s þ rI U1 ðaÞ ¼ 1 þ 0:42 ¼ 1:42 is the input congesting amount for C in stochastic model (16). Now assume that only the point E in Fig. 1 has random variations in its input and output. Based
Fig. 5. Deterministic equivalent solution for C ¼ C 0 when a ¼ 0:8.
ðA:5Þ
It is easy to see from Fig. 6 (using (21)) that E0 is not efficient and input congestion for E0 is equal to 1.42. This is consistent with Theorem 8(a). Notice from Fig. 1 that the input congesting amount for E is sc ¼ 1. Therefore, utilizing Theorem 8(a), we have 0
sc ¼ sc þ rI U1 ðaÞ ¼ 1 þ 0:42 ¼ 1:42:
References Allen, F., Broswell, M., Rao, P.V., 1972. Distribution free approximations for chance constraints. Operations Research 22, 610–621. Charnes, A., Cooper, W.W., Rhodes, E., 1981. Evaluating program and managerial efficiency: An application of data envelopment analysis to program follow through. Management Science 27, 668–697. Charnes, A., Cooper, W.W., Symonds, G.H., 1958. Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Management Science 4, 235–263. Charnes, A., Cooper, W.W., Thrall, R.M., 1991. A structure for classifying and characterizing efficiencies and inefficiencies in data envelopment analysis. Journal of Productivity Analysis 2, 197–237. Charnes, A., Cooper, W.W., Thrall, R.M., 1986. Identifying and classifying efficiencies and inefficiencies in data envelopment analysis. Operations Research Letters 5, 105–116. Charnes, A., Haag, S., Jaska, P., Semple, J., 1992. Sensitivity of efficiency classifications in the additive model of data envelopment analysis. International Journal of Systems Science 23, 789–798. Charnes, A., Neralic, L., 1990. Sensitivity analysis of the additive model in data envelopment analysis. European Journal of Operational Research 48, 332–341.
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501 Charnes, A., Rousseau, J., Semple, J., 1996. Sensitivity and stability of efficiency classifications in data envelopment analysis. Journal of Productivity Analysis 7, 5–18. Cherchye, C., Kuosmanen, T., Post, T., 2001. Alternative treatments of congestion in DEA: A rejoinder to Cooper, Gu and Li. European Journal of Operational Research 132, 75–80. Cooper, W.W., Gu, B., Li, S., 2001a. Note: Alternative treatments of congestion––A response to the Cherchye, Kuosmanen and Post critique. European Journal of Operational Research 132, 85–87. Cooper, W.W., Deng, H., Gu, B., Li, S., Thrall, R.M., 2001b. Using DEA to improve the management of congestion in Chinese industries (1981–1997). Socio-Economic Planning Sciences 35, 1–16. Cooper, W.W., Li, S., Seiford, L.M., Tone, K., Thrall, R.M., Zhu, J., 2001c. Sensitivity and stability analysis in DEA: Some recent developments. Journal of Productivity Analysis 15, 217–246. Cooper, W.W., Deng, H., Huang, Z.M., Li, S.X., 2002a. Change constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis. Journal of the Operational Research Society 53, 1347–1356. Cooper, W.W., Deng, H., Huang, Z.M., Li, S.X., 2002b. A onemodel approach to congestion in data envelopment analysis. Socio-Economic Planning Sciences 36, 231–238. Cooper, W.W., Huang, Z., Li, S., 1996a. Satisficing DEA models under chance constraints. The Annals of Operations Research 66, 279–295.
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Cooper, W.W., Thompson, R.G., Thrall, R.M., 1996b. Introduction: Extensions and new developments in DEA. Annals of Operations Research 66, 3–46. Cooper, W.W., Seiford, L.M., Zhu, J., 2000. Slacks and congestion: Response to a comment by R. F€are and S. Grosskopf. Socio-Economic Planning Sciences 35, 1–11. F€are, R., Grosskopf, S., 2000. Slacks and congestion: A comment. Socio-Economic Planning Sciences 34, 27–33. Land, K.C., Lovell, C.A.K., Thore, S., 1993. Chance-constrained data envelopment analysis. Managerial and Decision Economics 14, 541–554. Land, K.C., Lovell, C.A.K., Thore, S., 1994. Productivity and efficiency under capitalism and state socialism: An empirical inquiry using chance-constrained data envelopment analysis. Technological Forecasting and Social Change. Land, K.C., Lovell, C.A.K., Thore, S., 1992. Productivity and efficiency under capitalism and state socialism: The chanceconstrained programming approach. In: Pestieau, P. (Ed.), Public Finance in a World of Transition, Proceedings of the 47th Congress of the International Institute of Public Finance. pp. 109–121. Olesen, O.B., Petersen, N.C., 1995. Chance constrained efficiency evaluation. Management Science 41, 442–457. Seiford, L.M., Zhu, J., 1998. Stability regions for maintaining efficiency in data envelopment analysis. European Journal of Operational Research 108 (1), 127–139. Sengupta, J.K., 1995. Dynamics of Data Envelopment Analysis. Kluwer Academic Publishers, Norwell, MA, pp. 142 ff.
Stochastics and Statistics
Chance constrained programming approaches to congestion in stochastic data envelopment analysis William W. Cooper a
a,*
, H. Deng a, Zhimin Huang b, Susan X. Li
b
The Red McCombs School of Business, University of Texas at Austin, Austin, TX 78712-1174, USA b School of Business, Adelphi University, Garden City, NY 11530, USA Received 8 August 2001; accepted 25 September 2002
Abstract The models described in this paper for treating congestion in DEA are extended by according them chance constrained programming formulations. The usual route used in chance constrained programming is followed here by replacing these stochastic models with their ‘‘deterministic equivalents.’’ This leads to a class of non-linear problems. However, it is shown to be possible to avoid some of the need for dealing with these non-linear problems by identifying conditions under which they can be replaced by ordinary (deterministic) DEA models. Examples which illustrate possible uses of these approaches are also supplied in an Appendix A. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Inefficiency; Congestion; DEA (data envelopment analysis); Chance constrained programming
1. Introduction This paper forms one part of a series of continuing research efforts. A previous paper treated the topic of stochastic characterizations of efficiency and inefficiency in DEA using chance constrained programming formulations and constructs. See Cooper et al. (2002a). Here a similar chance constrained approach to efficiency is undertaken but attention is centered on ‘‘congestion’’ as one (particularly severe) form of inefficiency.
*
Corresponding author. Tel.: +1-512-471-1822; fax: +1-512471-0587. E-mail address: [email protected] (W.W. Cooper).
A focus on this one type of inefficiency may be justified as follows. Congestion has been an underresearched topic in the economic theory of production even though it can be of importance when its use is associated with a need for augmenting inputs to serve important objectives besides output maximization. As noted in Cooper et al. (2001b), for instance, congestion is used in China to deal with the need for providing employment for a large labor force, with some 16,000,000–18,000,000 new entrants each year. In addition, recent exchanges in the literature that deal with congestion have been accompanied by new developments that have opened additional topics for research. See the exchange between Cherchye et al. (2001) and Cooper et al. (2001a). See also the exchange between F€are and Grosskopf (2000) and Cooper et al. (2000).
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00901-3
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As noted above, congestion is a particularly severe form of inefficiency in that input increases are associated with output decreases in the manner formalized in the following definition. Congestion: Congestion is present in the performance of DMUo , the DMU being evaluated, when input increases give rise to decreases in the outputs that are maximally attainable or, conversely, when input decreases are accompanied by increases in the output amounts that are maximally attainable. We distinguish between congestion and ‘‘technical inefficiency’’––sometimes referred to as ‘‘waste’’–– which is defined as follows. Inefficiency (Technical): Inefficiency is present in the performance of DMUo if the evidence shows that it is possible to improve some of its inputs or outputs without worsening any of its other inputs or outputs. Note that nothing is said about an input reduction being accompanied by any output augmentation. To see what is involved we turn to Fig. 1 in Appendix A where the coordinates in parentheses record input amounts (on the left) and output amounts (on the right) for the one input used and the one output produced by each of five decision making units (DMUs) A–E. Now refer to point C and observe that its output level at y ¼ 2 is the same as for B. However, B attains this output with x ¼ 2 units of this input whereas C utilizes x ¼ 3 units. Hence, without worsening or improving the output attained at C it is possible to reduce input from x ¼ 3 to 2 by moving from C to B. This means that the evidence shows C to be inefficient because it is found to be possible to decrease its input without worsening output. The congestion in D, it may be noted, is a more severe form of inefficiency because the maximal output y ¼ 2 is not attainable without reducing the input for D from x ¼ 5 to 3. Remark. It is to be noted that we are not using prices, unit costs or other such relative weights.
Hence we are dealing only with ‘‘technical inefficiency’’ and not with other types of inefficiency such as ‘‘allocative inefficiency,’’ etc., which require a use of relative weights to arrive at their evaluations. Finally, we introduce the following definition of (technical) efficiency. Efficiency: The performance of DMUo is to be characterized as technically efficient if and only if the evidence shows that it is not possible to improve some of its inputs or outputs without worsening some of its other inputs or outputs. Points A and B in Fig. 1, as well as all of the points on the boundary shown by the solid line between them have this property. In the literature of DEA this is referred to as the ‘‘efficiency frontier.’’ No point not on a boundary can have this efficiency property and no point on the boundaries between B and C or between C and D have this property. Hence only the points on the efficiency frontier represented by the solid line between A and B are efficient. With these definitions in hand we begin to provide the stochastic (¼ probabilistic) characterizations of congestion that are the special focus of this paper. This is done as follows. First we introduce the models we use to identify congestion in deterministic form. Next we undertake our chance constrained formulations with accompanying definitions and developments. We then reduce these formulations to ‘‘deterministic equivalents.’’ Finally, we introduce simplifying assumptions that allow us to identify conditions under which appearances of congestion with associated probabilities can be determined in a straightforward manner. A summary and conclusion is followed by Appendix A which provides illustrative examples. 2. A two-model approach We start with the deterministic models we use to evaluate congestion. Assume that there are n
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
DMUs to be evaluated which we associate with points, xj ¼ ðx1j ; . . . ; xmj ÞT and yj ¼ ðy1j ; . . . ; ysj ÞT that represent (m 1) and (s 1) input and output vectors, respectively, for each DMUj , j ¼ 1; . . . ; n, and where the superscript T indicates the transpose of these vectors. Now, to maintain contact with Cooper et al. (2002a), we begin with the following version of a BCC model: ! s m X X þ max / þ e sr þ si ; r¼1
s:t:
/yr0
n X
i¼1
yrj kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
xij kj þ s i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð1Þ
Thus (1) is used to determine technical efficiency and inefficiency. To determine whether congestion is present we proceed from an optimal solution of (1) as follows. If the two conditions specified in the definition of DEA efficiency are satisfied then neither technical inefficiency nor congestion is evidenced by the performance recorded for DMUo . If either condition fails to be satisfied then DMUo is inefficient. Then, to see whether congestion is present we use the solution of (1) to erect a second problem as follows. For an optimal solution (/ , k , sþ , s ) of (1), we can reexpress its constraints in the following form: n X / yr0 þ sþ ¼ yrj kj ; r ¼ 1; . . . ; s; ð2Þ r
j¼1 n X
489
j¼1
kj ¼ 1;
j¼1
0 6 kj ;
sþ r ; si ; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m: Here xij and yrj represent input and output amount recorded for j ¼ 1; . . . ; n DMUs and j ¼ o designates one of the DMUj as the DMUo to be evaluated relative to all of the data (including the data on DMUo ) and e > 0 is a ‘‘non-Archimeadean element’’ defined to be smaller than any positive real number. This means that e is not a real number. The standard procedure is to avoid any need for explicitly assigning a value to e by using the following two-stage process. Stage one: maximize / while ignoring the slacks, sþ r , si , in the objective. Stage two: replace / with / ¼ max / in (1) and maximize the sum of the slacks. Then determine whether DMUo is efficient or inefficient in accordance with the following definition. Definition (DEA Efficiency). DMUo is efficient if and only if the following two conditions are both satisfied: (i) / ¼ 1, (ii) sþ r ¼ si ¼ 0, 8 r, i. Here and hereafter is used to designate an optimum value.
xi0 s i ¼
n X
xij kj ;
i ¼ 1; . . . ; m:
ð3Þ
j¼1
Then we can use the values on the left in (2) and (3) to define new outputs and inputs, y^r0 , x^i0 , as in the following:
y^r0 ¼ / yr0 þ sþ r P yr0 ;
x^i0 ¼ xi0 s i 6 xi0 ;
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m:
ð4Þ ð5Þ
The expressions (4) and (5) are referred to as the ‘‘CCR projection formulas’’ because they project the observed yr0 and xi0 into y^r0 and x^i0 on the efficient frontier. As proved in Charnes et al. (1981), none of the coordinates for this projected point can be improved without worsening some (one or more) of its other coordinates. Thus Dyr0 ¼ y^r0 yr0 P 0 represents the amount of inefficiency in the rth output for DMUo , and Dxi0 ¼ xi0 x^i0 P 0 represents the amount of inefficiency in the ith input for DMUo . Hence, in accordance with the above definition, inefficiency is present if and only if Dyr0 6¼ 0 or Dxi0 6¼ 0 for some r or i. Note, therefore, that the amounts as well as the sources of all of these inefficiencies are identified for each DMUj ¼ DMUo by reference to these Dyr0 , Dxi0 values. Remark. As can be seen, this definition adapts the verbal definition of efficiency given in the preceding section by relating it to the above DEA model.
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Hence, to reflect this specialization, we refer to this as ‘‘DEA Efficiency.’’ As described in Cooper et al. (2001b), these y^r0 and x^r0 values together with the yrj and xij as defined in (1) are used to erect the following new problem: m X max d i ; i¼1
s:t:
y^r0 ¼ / yr0 þ sþ r ¼
n X
yrj kj ;
j¼1
r ¼ 1; . . . ; s;
x^i0 ¼ xi0 s i ¼
n X
xij kj d i ;
j¼1
i ¼ 1; . . . ; m; n X 1¼ kj ;
ð6Þ
j¼1
s i P di ;
i ¼ 1; . . . ; m;
with all variables constrained to be non-negative. See also Cooper et al. (1996b) where this formulation was first introduced. Finally, to identify the congesting inputs and to estimate their amounts, we utilize the i ¼ 1; . . . ; m input constraints shown in the last set of constraints at the bottom of (6) to obtain
sc ¼ s i i di ;
i ¼ 1; . . . ; m;
ð7Þ
where di is obtained from an optimal solution to (6). sc is then the ‘‘congesting amount’’ in the i ‘‘total slack’’ associated with s in input i ¼ i 1; . . . ; m, as obtained from (1), and d is the i (maximum) amount of this total slack that can be assigned to ‘‘purely technical’’ (non-congesting) inefficiency, as obtained from (6). Remark. Reference to Fig. 1 may help again to clarify what is happening. Model (1) identifies B as the efficient point for use in evaluating E. (This is confirmed by noting that (4) and (5) yield x^i0 as the coordinates of B.) Use of (4) and (5) provide the expression on the left in (6)––viz., y^r0 and x^i0 ––as well as the values of s for the bottom-most coni straints. The di in (6) are also slacks. These slacks are to be ‘‘backed out’’ as far as the equations for the output constraints allow in (6). Hence di must be consistent with y^ ¼ 2 and condition 2 P d i .
Here 2 ¼ s ¼ 4 2 represents the slack associated with the maximization for the e term in the objective of (1). Hence the best that can be done in ‘‘backing out’’ the slack from (1) moves the solu tion from B to C in (6) giving d ¼ 3 2 ¼ 1 as the maximal value. To obtain sc we therefore simply subtract this value of d from s ¼ 2 to obtain sc ¼ 2 1 ¼ 1 as the amount of this input that is i congesting. The value of d i ¼ 1, on the other hand, represents the amount of the slack associated with the ‘‘technical inefficiency’’ that is exhibited by moving from B to C. We can now further clarify the last sentence in the definition of congestion given in Section 1. Suppose E is translated horizontally to a new position to the left of G in Fig. 1. This identifies it with a point under the boundary line between B and C. From this position it is possible to raise output to the maximum level allowed by this boundary without reducing input. No such output increase can be achieved from E in its present position without decreasing all of the congesting amount of input, sc ¼ 1. Thus, the objective in (2) is to back i out all of the slack that is consistent with y^ ¼ 2, the maximum output, as determined from (1). The remainder is the congesting amount in the ‘‘total slack,’’ s , as also determined from (1). 3. A one-model approach Cooper et al. (2002b) introduced a ‘‘one model’’ alternative to the two-model approach incorporated in (1) and (6) above to treat congestion. To see how this is done we first note that for an op timal solution (/ , k , sþ , s ) of (1), we can use c si ¼ si di , as defined in (7), to rewrite (6) as m X sc min i ; i¼1
s:t:
/ yr0 þ sþ r ¼
n X
yrj kj ;
r ¼ 1; . . . ; s;
j¼1
xi0 sc i ¼ 1¼
n X
n X
xij kj ;
i ¼ 1; . . . ; m;
j¼1
kj ;
j¼1
0 6 kj ;
sc i ; j ¼ 1; . . . ; n; i ¼ 1; . . . ; m:
ð8Þ
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
This model can be regarded as part of a twostage procedure analogous to the one we described for the second stage in dealing with the nonArchimedean element e > 0 in (1). Here, however, (8) can also be regarded as being derived from the following modification of (1): max
/þe
s X
sþ r
r¼1
s:t:
/yr0
n X
m X
! sc ; i
i¼1
yrj kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
xij kj þ sc i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð9Þ
j¼1 n X
kj ¼ 1;
j¼1
0 6 kj ;
þ sc i ; sr ; j ¼ 1; . . . ; n; i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s:
following theorem represents the promised improvement. Theorem 2. Congestion is present if and only if for an optimal solution (/ , k , sþ , sc ) of (9), there is at least one sc > 0 (1 6 i 6 m). i Proof. (Necessary condition) It is obvious by the congestion definition. (Sufficient condition) We need to show that for sc > 0 (1 6 i0 6 m), we must have / > 1 or there i0 is at least one sþ r > 0 (1 6 r 6 s). Suppose to the contrary that / ¼ 1 and sþ r ¼ 0 _ _þ þ for all r ¼ 1; . . . ; s. Let / ¼ / ¼ 1, s r ¼ sr ¼ 0, _ _c r ¼ 1; . . . ; s, s i ¼ 0, i ¼ 1; . . . ; m, k j ¼ 0 for j 6¼ 0 _ _ _ _þ _c and k 0 ¼ 1. Then (/ , k , s , s ) is a feasible solution of (9) and ! s m X X _ _þ _c / þe sr si ¼ / r¼1
>/ þe As can be seen, the objective of (1) is here modified c by replacing þs i with si in the objective of (9). c See also the replacement of s in the i with si constraints. Now we can reason as follows. From our above analysis, let us suppose that (/ , k , sþ , sc ) is an þ optimal solution of (9). Then / and sr are part of an optimal solution of (1), and (k , sc ) is an optimal solution of (8). The congesting amount of input i ¼ 1; . . . ; m, is then represented by sc in (9) i in accordance with the following theorem as taken from Cooper et al. (2002b).
491
i¼1
m X
!
sc i
i¼1
¼/ þe
s X r¼1
sþ r
m X
! sc i
:
i¼1
This contradicts the assumption that (/ , k , sþ , sc ) is an optimal solution of (9). Therefore, we have the following combined theorem on inefficiency and congestion.
Theorem 3. Suppose (/ , k , sþ , sc ) is an optimal solution of (9). Then:
(i) / > 1 and there is at least one sc >0 i (1 6 i 6 m), (ii) there exists at least one sþ r > 0 (1 6 r 6 s) and c at least one si > 0 (1 6 i 6 m).
(a) If / > 1, then DMUo is inefficient. (b) If there exists at least one sþ r > 0 (1 6 r 6 s), then DMUo is inefficient. (c) If there exists at least one sc > 0 (1 6 i 6 m), i then DMUo is inefficient and congestion is present. (d) If / ¼ 1, sþ ¼ 0 and sc ¼ 0, then DMUo is on a frontier.
Next we improve the theorem by showing that if we have at least one sc > 0 (1 6 i 6 m), then it i guarantees / > 1 or there exists at least one sþ r > 0 (1 6 r 6 s), i.e., congestion is present. The
In conclusion we note that (d) in Theorem 3 guarantees only that DMUo is on a frontier. It does not guarantee that it will be on an efficient portion of the frontier.
Theorem 1. Congestion is present if and only if in an optimal solution (/ , k , sþ , sc ) of (9), at least one of the following two conditions is satisfied:
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4. Stochastic models We follow the notation in Cooper et al. (1996a) T T and let ~xj ¼ ð~x1j ; . . . ; ~xmj Þ and y~j ¼ ð~ y1j ; . . . ; y~sj Þ represent (m 1) and (s 1) random input and T output vectors, and xj ¼ ðx1j ; . . . ; xmj Þ and T yj ¼ ðy1j ; . . . ; ysj Þ stand for the corresponding vectors of expected values of input and output for each DMUj , j ¼ 1; . . . ; n. That is, we utilize these expected values in place of the observed values in (1). See Olesen and Petersen (1995) for an alternate approach which uses the means of a series of observations to obtain ‘‘confidence interval’’ estimates. Let us consider all input and output components to be jointly normally distributed in the following chance constrained version of a stochastic DEA model: max s:t:
/; ( P
n X
) y~rj kj P /~ yr0
P 1 a;
j¼1
r ¼ 1; . . . ; s; ( P
n X
) ~xij kj 6 ~xi0
P 1 a;
ð10Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1
kj P 0;
j ¼ 1; . . . ; n:
Here, P means ‘‘Probability’’ and a is a predetermined number between 0 and 1. We now use this model to define stochastic efficiency as follows. Definition (Stochastic Efficiency). DMUo is stochastic efficient if and only if the following two conditions are both satisfied: (i) / ¼ 1; (ii) slack values are all zeros for all optimal solutions, where (ii) refers to all alternate optima because the second stage optimization associated with e > 0 in (1) is not used in (10).
The ~xij ¼ ~xi0 , y~rj ¼ y~r0 values for DMUo appear on the left as well as on the right inside the braces of (10). Hence, we can always get a solution with / ¼ 1, k0 ¼ 1 and 8kj ¼ 0 (j 6¼ 0) with all slacks zero. However, this solution need not be maximal. It follows that a maximum with / > 1 in (10) for any sample of j ¼ 1; . . . ; n observations means that the DMUo being evaluated is not efficient because, to the specified level of probability defined by a, the evidence will then show that all outputs of DMUo can be increased to / y~r0 > y~r0 , r ¼ 1; . . . ; s, without violating the output constraints by using a convex combination of other DMUs which will also satisfy n X
~xij kj 6 ~xi0 ;
i ¼ 1; . . . ; m:
ð11Þ
j¼1
To the indicated degree of probability no input is worsened by this increase in all of DMUo Õs outputs. Thus, to the same level of probability, these output increases will not require an augmentation in any of the input amounts used by DMUo . Indeed, any positive slacks in the ith input constraint permit a decrease in the corresponding ~xi0 . Now let us discuss positive slacks in the rth output chance constraint. Suppose 1r > 0 is an ‘‘external slack’’––i.e., a slack which can be inserted in the inequality outside the braces to achieve equality for the rth output chance constraint, viz., ) ( n X P ð12Þ y~rj kj /~ yr0 P 0 ¼ ð1 aÞ þ 1r : j¼1
There must then exist a number sþ r > 0 such that ( ) n X P y~rj kj /~ yr0 P sþ ¼ 1 a: ð13Þ r j¼1
This value of sþ r permits a still further increase in y~r0 for any sample of observations without worsening any other input or output. It is easy to see that 1r ¼ 0 if and only if sþ r ¼ 0. In a manner similar to our treatment of the output constraints, suppose ni > 0, used as the external slack for ith input chance constraint, satisfies
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
( P
n X
) ~xij kj ~xi0 6 0
¼ ð1 aÞ þ ni :
ð14Þ
max
/þe
j¼1
Then there must exist a positive number s i > 0 such that ( ) n X ~xij kj þ si 6 ~xi0 ¼ 1 a: P ð15Þ j¼1
( s:t:
P
j¼1
r ¼ 1; . . . ; s; ( P
n X
) ~xij kj þ
s i
6 ~xi0
¼ 1 a;
ð16Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m: This leads to the following Definition. DMUo is stochastic efficient to the indicated levels of probability if and only if the following two conditions are both satisfied: (i) / ¼ 1, (ii) sþ r ¼ si ¼ 0, 8 i, r. The stochastic model in (16) is evidently a generalization of the BCC model in (1). In a similar manner, it is natural to generalize the ‘‘onemodel’’ approach to congestion represented in (9) to the following stochastic version:
sþ r
r¼1 n X
y~rj kj
m X
! sc i
i¼1
/~ yr0 P sþ r
; ) ¼ 1 a;
j¼1
( P
To the indicated level of probability this value of s xi0 for any set of sample i permits a decrease in ~ observations without worsening any other input or output. Again it is easy to show that ni ¼ 0 if and only if s i ¼ 0. Therefore, if we introduce the nonArchimedean infinitesimal, e > 0, stochastic efficiencies and inefficiencies can be characterized by the following stochastic version of (1): ! s m X X þ max / þ e sr þ si ; r¼1 i¼1 ( ) n X þ y~rj kj /yr0 P sr ¼ 1 a; s:t: P
s X
493
r ¼ 1; . . . ; s; ) n X c ~xij kj þ si 6 ~xi0 ¼ 1 a;
ð17Þ
j¼1
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1 c kj P 0; sþ r P 0; si P 0; j ¼ 1; . . . ; n; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
Therefore, we have the following results as obvious generalizations of Theorems 2 and 3. Theorem 4. Congestion is present for DMUo to the prescribed level of probability in stochastic model (16) if and only if for an optimal solution (/ , k , sþ , sc ) of (17), there exists at least one sc >0 i0 (1 6 i0 6 m). Theorem 5. At an optimum of (17) we have the following: (a) If / > 1, then DMUo is stochastic inefficient. (b) If there exists at least one sþ r0 > 0 (1 6 r0 6 s), then DMUo is stochastic inefficient. (c) If there exists at least one sc > 0 (1 6 i0 6 m), i0 then DMUo is stochastic inefficient and congestion is present. (d) If / ¼ 1, sþ ¼ 0 and sc ¼ 0, then DMUo is on a segment of the stochastic frontier, i.e., DMUo 2 E [ E0 [ F , where E and E0 represent the collection of observed values for DMUs represented by extreme and nonextreme points on the efficiency frontier and F represents the points which are on a portion of the frontier that is not efficient. See Charnes et al. (1991, 1986). To make this all more concrete we follow Cooper et al. (1996a), and assume that inputs and outputs are random variables with a multivariate normal distribution and known parameters. We
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also restrict attention to the class of ‘‘zero-order decision rules’’ (see Charnes et al. (1958) for a discussion of these and other decision rules). Remark. Choices of multivariate normal distributions and zero-order rules are less restrictive than might at first appear to be the case. Transformations are available for bringing other types of distributions into approximately normal form––as was done in Charnes et al. (1958), for instance, when it was found necessary to treat highly skewed distributions with log-normal approximations. We can also interpret our zero-order decision rules as a series of one-period-at-a-time applications, with appropriate models, to allow for changing realizations and probabilities, and regard these as approximations to the more complex solution procedures involved in developing higher order ‘‘conditional’’ decision rules to deal with the full scale treatment of dynamics.
where the yrj and xij (including yr0 and xi0 ) are the means of these variables. (As noted earlier, these means are assumed to be known.) Similarly, the deterministic equivalent of (17) can be represented by ! s m X X þ c max / þ e sr si ; r¼1
s:t:
n X
/yr0
r ¼ 1; . . . ; s; n X
1 I xij kj þ sc i U ðaÞri ðkÞ ¼ xi0 ;
i ¼ 1; . . . ; m; n X
kj ¼ 1;
j¼1 c sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
sþ r ,
s i
We now show how to obtain the / , from deterministic equivalents of the stochastic models represented in (16) and (17). With normal distributions and zero order decision rules we can obtain a deterministic equivalent for (16) which can be represented by
max
/þe
þ
m X
r¼1
s:t:
/yr0
n X
Here U is the standard normal distribution function and U1 , its inverse, is the so-called ‘‘fractile function,’’ Finally, 2
ðr0r ð/; kÞÞ ¼
XX i6¼0
! s i
ð19Þ
j¼1
5. Deterministic equivalents
sþ r
1 0 yrj kj þ sþ r U ðaÞrr ð/; kÞ ¼ 0;
j¼1
kj P 0;
s X
i¼1
ki kj Covð~ yri; y~rj Þ
j6¼0
þ 2ðk0 /Þ
;
X
ki Covð~ yri ; y~r0 Þ
i6¼0
i¼1 2
þ ðk0 /Þ Varð~ yr0 Þ
1 0 yrj kj þ sþ r U ðaÞrr ð/; kÞ ¼ 0;
j¼1
and
r ¼ 1; . . . ; s; n X
xij kj þ
s i
1
U
ðaÞrIi ðkÞ
¼ xi0 ;
j¼1
i ¼ 1; . . . ; m; n X
ð18Þ
ðrIi ðkÞÞ2 ¼
XX j6¼0
kj kk Covð~xij;~xik Þ
k6¼0
þ 2ðk0 1Þ
X
kj Covð~xij ; ~xi0 Þ
j6¼0
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m;
2
þ ðk0 1Þ Varð~xi0 Þ; where we have separated out the terms for DMUo because they appear on both sides of the expres-
W.W. Cooper et al. / European Journal of Operational Research 155 (2004) 487–501
max
/þe
s X r¼1
s:t:
0 /yr0
n X
sþ r
m X
! s i
;
i¼1
yrj0 kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ s i ¼ xi0 ;
i ¼ 1; . . . ; m;
j¼1 n X
kj ¼ 1;
j¼1
kj P 0;
sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; and model (19) can be written
!
sions in (16) ff. Thus, / , sþ and the sc can r , si i be determined from (18) and (19) where the data (means and variances) are all assumed to be known. Because of the functional forms of r0r ð/; kÞ and rIi ðkÞ it is obvious that models (18) and (19) are non-linear programming problems. Let x0r and xIr be non-negative variables. Replace r0r ð/; kÞ by x0r and rIi ðkÞ by xIr in both (18) and (19), and add two quadratic equality constraints, ðx0r Þ2 ¼ ðr0r ð/; kÞÞ2 and ðxIr Þ2 ¼ ðrIi ðkÞÞ2 , to both (18) and (19), then (18) and (19) are both transformed to easily solvable quadratic programming problems. From (18) and (19) we can see that if the predetermined value of a is equal to 0.5, then U1 ðaÞ ¼ 0 and stochastic efficiencies, inefficiencies and congestion can be characterized by the BCC model (1) and congestion model (9) using the mean values of inputs and outputs. Thus a ¼ 0:5 may be regarded as indicating indifference between stochastic and deterministic approaches and classifications of stochastic efficiencies, inefficiencies and congestion are the same as in the deterministic situation with the input and output means. To simplify matters in a different manner let us assume that only DMUo has random variations in its inputs and outputs, i.e., rIi0 6¼ 0, r0r0 6¼ 0, rIij ¼ 0, and r0rj ¼ 0 (j 6¼ 0) for all i and r. In this case, model (18) can be written
max
/þe
s X
sþ r
r¼1 0 /yr0
s:t:
n X
m X
495
sc i
;
i¼1
yrj0 kj þ sþ r ¼ 0;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ sc i ¼ xi0 ;
i ¼ 1; . . . ; m;
ð21Þ
j¼1 n X
kj ¼ 1;
j¼1
kj P 0;
c sþ r P 0; si P 0; j ¼ 1; . . . ; n;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; where 0 ¼ yr0 r0r0 U1 ðaÞ; yr0
yrj0
¼ yrj ;
j 6¼ o; r ¼ 1; . . . ; s;
x0i0 ¼ xi0 þ rIi0 U1 ðaÞ; x0ij
¼ xij ;
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m;
j 6¼ o; i ¼ 1; . . . ; m:
ð22Þ
ð23Þ
With these assumptions model (20) is the deterministic equivalent of stochastic model (16) and model (21) is the deterministic equivalent of (17). Reasons for us to consider random variations only in DMUo are as follows: First, treating more than one DMU in this manner leads to deterministic equivalents with the more complicated relations that have been discussed in detail in Cooper et al. (2002a). The simpler approach used here allows us to arrive at analytical results and characterizations in the straightforward manner that we discuss in the next section. Second it opens possible new routes for effecting ‘‘sensitivity analyses’’ which we remark upon as follows.
ð20Þ Remark. We are referring to the ‘‘sensitivity analyses’’ that are to be found in Charnes and Neralic (1990), Charnes et al. (1992), Charnes et al. (1996) and Seiford and Zhu (1998). In the terminology of the survey article by Cooper et al. (2001c), these sensitivity analyses are directed to analyzing allowable limits of data variations for only one DMU at a time and hence contrast with other approaches to sensitivity analysis in DEA that
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allow all data for all DMUs to be varied simultaneously until at least one DMU changes its status from efficient to inefficient, or vice versa. These sensitivity analyses are entirely deterministic. Our chance constrained approach can be implemented by representations that are similar in form to those used in sensitivity analysis but the conceptual meanings are different. A chance constrained programming problem can be solved by a deterministic equivalent, as we have just shown, but the issue originally addressed in the chance constrained formulation is different and this introduces elements, such as the risk associated with a, that are nowhere present in these sensitivity analyses. This can be illustrated by Fig. 1 in the Appendix A where, as previously noted, point C is technically inefficient but does not display congestion. This point is evidently very sensitive to changes in its output (but not its input) data. In the terminology of Charnes et al. (1996), it has a ‘‘zero radius of stability.’’ If its output is raised in any positive amount, C becomes efficient. Alternatively, if C is lowered it becomes an example of congestion. All these arguments are from a sensitivity analysis point of view. If we consider changes in point C to involve only random variations, these characterizations will change. They will depend not on the change in output level, but will depend rather on the specified probability level a. When a ¼ 0:5, random variations in the coordinates of point C do not have any impact on its efficiency, inefficiency or congestion characterizations. Hence it is satisfactory to employ the deterministic model (1) since, with this choice of a, the user is indifferent to the possible presence of inefficiency (or congestion) stochastically. This is different from the sensitivity analysis results. When a is taken between 0 and 0.5, point C will be efficient in the stochastic sense irrespective of the random variations (see Theorem 7(a), below). This is again different from the result of the sensitivity analysis discussed above. When a is assigned a value between 0.5 and 1, point C will be inefficient with congestion in the stochastic sense present–– no matter what the direction of random variations (see Theorem 8(b), below). Thus, in all cases the choice of a plays the critical role.
6. Results We are now in position to draw forth some further theorems as follows. Theorem 6. For a ¼ 0:5. Congestion is present for DMUo in input–output mean model (1) if and only if congestion is present for DMUo in stochastic model (16). Proof. As already noted this follows from the fact that U1 ð0:5Þ ¼ 0 for the standard normal distribution. See the discussion following (19). Theorem 7. For 0 < a < 0:5. (a) Suppose congestion is not present for DMUo in input–output mean model (1), then congestion is also not present for DMUo in stochastic model (16). (b) Suppose congestion is present for DMUo and DMUo 2 F in input–output mean model (1), then congestion is not present for DMUo and DMUo 2 E in stochastic model (16). (c) Suppose congestion is present for DMUo and DMUo 2 N in input–output mean model (1), then congestion is also present for DMUo and DMUo 2 N in stochastic model (16) and 0 sc ¼ sc þ rIi0 U1 ðaÞ is the congesting amount i i of the ith input of DMUo in stochastic model (16) for i ¼ 1; . . . ; m, if c rIi0 < min bi ; si =ðU1 ðaÞÞ and
1 r0r0 < bþ r =ðU ðaÞÞ;
where sc is the congesting amount of the ith i inputPof DMUo P in input–output mean model (1) m and sr¼1 bþ þ r i¼1 bi is the optimal value of s m X X bþ b max r þ i ; r¼1
s:t:
n X j¼1 n X j¼1 n X
i¼1
yrj kj bþ r P yr0 ;
r ¼ 1; . . . ; s;
xij kj þ b i 6 xi0 ;
i ¼ 1; . . . ; m;
kj ¼ j¼1 bþ r P 0;
1;
b i P 0; kj P 0; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n: ð24Þ
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Proof (a) Consider (20) as the deterministic equivalent of (16). For 0 < a < 0:5, we have U1 ðaÞ < 0. Hence, referring to (22) and (23) we have y00 P y0 and x00 6 x0 . Therefore improvement for DMUo from (x0 ; y0 ) to (x00 ; y00 ) implies that congestion should also not be present for DMUo in stochastic model (16) when it is not present in mean model (1). (b) From Cooper et al. (2002a) we know that DMUo 2 E in stochastic model (16), consequently there is no congestion at DMUo in stochastic model (16). (c) From Cooper et al. (2002a) we know that DMUo 2 N in stochastic model (16). Therefore there are still improvements that can be made for output y00 of DMUo in deterministic equivalent (20) of stochastic model (16). Notice 0 that xi0 x0i0 ¼ rIi0 U1 ðaÞ < sc Let sc ¼ i . i 0 1 I c sc þ r U ðaÞ, then s is the congesting i i i0 amount of the ith input of DMUo in stochastic 0 model (16). Since sc > 0, congestion is present at i DMUo in stochastic model (16). Theorem 8. For 1 > a > 0:5. (a) Suppose congestion is present for DMUo in input–output mean model (1), then congestion is also present for DMUo in stochastic model 0 (16). Furthermore, sc ¼ sc þ rIi0 U1 ðaÞ is i i the congesting amount of the ith input of DMUo in stochastic model (16), where the sc i (i ¼ 1; . . . ; m) are the input congesting values as determined from input–output model (1). (b) Suppose DMUo 2 F and DMUo is an extreme point in input–output mean model (1), then DMUo 2 N and congestion is present for DMUo in stochastic model (16). Furthermore, 0 1 I sc ¼ s i i þ ri0 U ðaÞ is the congesting amount of the ith input of DMUo in stochastic model (16), where s i (i ¼ 1; . . . ; m) are the optimal input slack values obtained from input–output model (1) Proof (a) This is similar to the proof of Theorem 7(a). (b) From Cooper et al. (2002a) we know that DMUo 2 N in stochastic model (16). Since
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DMUo is the only DMU with random variations in inputs and outputs and it is an extreme point on F in input–output model (1), any reductions in its outputs will move it from the 0 frontier of model (1). Notice that yr0 ¼ yr0 r0r0 U1 ðaÞ < yr0 and x0i0 ¼ xi0 þ rIi0 U1 ðaÞ > xi0 . 0 1 I While sc ¼ s i i þ ri0 U ðaÞ is the congesting amount of the ith input of DMUo in stochastic model (16). Theorem 9. For 0 < a < 0:5. (a) Suppose congestion is present for DMUo in stochastic model (16), then congestion is also present at DMUo in input–output mean model 0 (1). Furthermore, sc ¼ sc rIi0 U1 ðaÞ is i i the congesting amount of the ith input of DMUo 0 in input–output mean model (1), where the sc i (i ¼ 1; . . . ; m) are the optimal input congesting amounts in stochastic model (16). (b) Suppose DMUo 2 F and DMUo is an extreme point in stochastic model (16), then DMUo 2 N and congestion is present for DMUo in input–output mean model (1). Fur 0 thermore, sc ¼ sc rIi0 U1 ðaÞ is the i i congesting amount of the ith input of DMUo in input–output model (1), where 0 sc (i ¼ 1; . . . ; m) are the optimal input i slack values in stochastic model (16). Proof. This is similar to the proof of Theorem 8 when one replaces (xj ; yj ) by (x0j ; yj0 ), and notices that ðx0j ; yj0 Þ ¼ ðxj ; yj Þ for j 6¼ o. Theorem 10. For 1 > a > 0:5. (a) Suppose congestion is not present for DMUo in stochastic model (16), then congestion is also not present for DMUo in input–output mean model (1). (b) Suppose congestion is present for DMUo and DMUo 2 F in stochastic model (16), then congestion is not present for DMUo and DMUo 2 E in input–output mean model (1). (c) Suppose congestion is present for DMUo and DMUo 2 N in stochastic model (16), then congestion is also present for DMUo and DMUo 2 N in input–output mean model (1) 0 and sic ¼ sc rIi0 U1 ðaÞ is the congesting i
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amount of the ith input of DMUo in input–output mean model (1) for i ¼ 1; . . . ; m, if
c0 rIi0 < min b =U1 ðaÞ i ; si and
1 r0r0 < bþ i =U ðaÞ; 0
where sc is the congesting amount of the ith i input of DMU Pmo instochastic model (16), and Ps þ b þ is the optimal value of r¼1 r i¼1 bi max
s X
bþ r þ
r¼1
s:t:
n X
m X
b i ;
i¼1 0 yrj0 kj bþ r P yr0 ;
r ¼ 1; . . . ; s;
j¼1 n X
0 x0ij kj þ b i 6 xi0 ;
i ¼ 1; . . . ; m;
j¼1 n X
kj ¼ 1;
j¼1
bþ r P 0;
b i P 0; kj P 0; r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m; j ¼ 1; . . . ; n:
ð25Þ
Proof. It is similar to the proof of Theorem 7 when one replaces (xj ; yj ) by (x0j ; yj0 ). This concludes the analytical developments. Examples are supplied in the Appendix A to help clarify meanings and possible uses of these results.
means. In other cases recourse must be had to the stochastic models we have introduced. Even in the latter class of cases, however, the task of identification and estimation may be accomplished by reference to deterministic equivalent models like those in (18) and (19). Although one must then deal with a non-linear programming problem, the task can be reduced to solving a quadratic programming problem. See the discussion following (19). The developments in this paper are part of a larger family of chance constrained programming approaches which seek to extend DEA to other contexts – see Land et al. (1992, 1993, 1994) – or to provide alternatives to presently available approaches in sensitivity analyses and other areas. For further discussions of chance constrained programming and other alternatives, see Sengupta (1995, pp. 142 ff). More research is also needed, of course, and some of this research might be directed to relaxing some of the assumptions used in this paper such as the need for assuming normal distributions with known means and variances. See Allen et al. (1972). Finally, we note that the development of our stochastic models involved switches from using the observed values of xij , yrj and xi0 , yr0 to a use of the means of the distributions that generated them. This opens a path for research pointed toward more forward looking uses of DEA in which probabilities of the occurrences of inefficiencies (and congestion) may be used to anticipate and perhaps forestall their occurrences.
Acknowledgements 7. Summary and conclusion In the early sections of this paper we covered the topic of congestion as treated in the DEA literature. We then incorporated these deterministic formulations in corresponding chance constrained programming models. We next showed how the task of identifying congestion may be accomplished with deterministic models rather than their chance constrained (stochastic) counterparts under suitable assumptions. In many cases this may be accomplished very simply from a knowledge of the
W.W. Cooper wishes to express his appreciation to the IC2 Institute of the University of Texas and the RGK Foundation in Austin, Texas, for support of his research. Z.M. Huang would like to acknowledge a PresidentÕs Faculty Development Grant of Adelphi University for support of his research, and S.X. Li would like to acknowledge the Research Sabbatical leave Grant of Adelphi University for support of her research. Thanks are also due to an anonymous referee for improvements in an earlier version of this paper.
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Appendix A In order to illustrate some of our theoretical results, let us reconsider Fig. 1 and start by providing the following numerical illustrations to what was said in the remarks with which Section 5 concluded. Using only points C and E we consider only two cases with a ¼ 0:20 and 0.80. We also assume that rI ¼ r0 ¼ 0:5. Fig. 2. Random variations in C.
Case 1. a ¼ 0:20. From a cumulative normal distribution table, we have U1 ðaÞ ¼ 0:84 and therefore rI U1 ðaÞ ¼ r0 U1 ðaÞ ¼ 0:42 for use in the following example. Assume that only point C has random variations in its input and output. This situation is depicted in Fig. 2. Based on (22) and (23), the adjusted input and output for point C with a ¼ 0:2 is C 0 : x0 ¼ 2:58 and y 0 ¼ 2:42:
ðA:1Þ
0
The relation of C to all other points is depicted in Fig. 3. It is obvious that C 0 is efficient and hence no congestion is present at C 0 . Consequently, there is no congestion at C in stochastic model (16). This is consistent with Theorem 7(a). Now assume only point E has random variations in its input and output. Based on (22) and (23), the adjusted input and output for point E0 is E0 : x0 ¼ 3:58 and y 0 ¼ 1:42:
Fig. 3. Deterministic equivalent solution for C ¼ C 0 when a ¼ 0:2.
ðA:2Þ
Its relationships with all other points are depicted in Fig. 4. Using (21), it is found that E0 is not efficient and input congestion for E0 is equal to 0.58. Fig. 4. Deterministic equivalent solution for E ¼ E0 when a ¼ 0:2.
This is consistent with Theorem 7(c) as we show in the following. Utilizing (24) on point E, with coordinates as given in Fig. 1, we have the following linear pro gram to determine bþ and b : max s:t: Fig. 1. Congestion and technical inefficiency. Source: P.L. Brockett, W.W. Cooper, Hong Chul Shin and Yuying Wang. ‘‘Inefficiency and Congestion in Chinese Production before and after the 1978 Reforms,’’ Socio-Economic Planning Sciences, 32 (1998).
bþ þ b ; 0:5kA þ 2kB þ 2kC þ 1kD þ 1kE bþ P 1; 1kA þ 2kB þ 3kC þ 5kD þ 4kE þ b 6 4; kA þ kB þ kC þ kD þ kE ¼ 1; ðA:3Þ kA ; kB ; kC ; kD ; kE ; bþ ; b P 0:
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This gives bþ ¼ 1 and b ¼ 2, so that, as prescribed in Theorem 7(c),
minðb ; sc Þ=ðU1 ðaÞÞ ¼ 1:19;
bþ =ðU1 ðaÞÞ ¼ 1:19;
where sc ¼ 1 from Fig. 1 on point E. Thus, all conditions for Theorem 7(c) are satisfied and we have s
c0
¼s
c
I
1
þ r U ðaÞ ¼ 1 0:42 ¼ 0:58:
Fig. 6. Deterministic equivalent solution for E ¼ E0 when a ¼ 0:8.
This is the amount of input congestion for E in stochastic model (16).
on (22) and (23), the adjusted input and output values for point E are as depicted in Fig. 6.
Case 2. a ¼ 0:80. In this case, we again have U1 ðaÞ ¼ 0:84 and I 1 r U ðaÞ ¼ r0 U1 ðaÞ ¼ 0:42. Now we return to point C and assume that only this point has random variations in its input and output. Utilizing (22) and (23), the adjusted input and output values for point C are
E0 : x0 ¼ 4:42 and y 0 ¼ 0:58:
C 0 : x0 ¼ 3:42 and y 0 ¼ 1:58
ðA:4Þ
as shown in Fig. 5. It is easy to see from Fig. 5 (using (21)) that C 0 is not efficient and input congestion for C 0 is equal to 1.42. Consequently, point C is stochastically inefficient. This is consistent with Theorem 8(b). In order to see this, we notice that the optimal slack value for input at C is s ¼ 1. Then based on Theorem 8(b), 0
sc ¼ s þ rI U1 ðaÞ ¼ 1 þ 0:42 ¼ 1:42 is the input congesting amount for C in stochastic model (16). Now assume that only the point E in Fig. 1 has random variations in its input and output. Based
Fig. 5. Deterministic equivalent solution for C ¼ C 0 when a ¼ 0:8.
ðA:5Þ
It is easy to see from Fig. 6 (using (21)) that E0 is not efficient and input congestion for E0 is equal to 1.42. This is consistent with Theorem 8(a). Notice from Fig. 1 that the input congesting amount for E is sc ¼ 1. Therefore, utilizing Theorem 8(a), we have 0
sc ¼ sc þ rI U1 ðaÞ ¼ 1 þ 0:42 ¼ 1:42:
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