Data envelopment analysis with missing data

Abstract

A first systematic attempt to use data containing missing values in data envelopment analysis (DEA) is presented. It is formally shown that allowing missing values into the data set can only improve estimation of the best-practice frontier. Technically, DEA can automatically exclude the missing data from the analysis if blank data entries are coded by appropriate numerical values.

Appendix. Proofs of theorems

Proof of Theorem 1

• Organised in three parts:

  • 1) T _UB⊆T _IDEAL: From the perspective of identity (1), the only difference between sets T _UB and T _IDEAL concerns the constraint for output j. For T _IDEAL this constraint reads as

    

    Since output j is missing for DMU k in T _UB, the constraint for output j reads as

    

    Recall that the true but unknown value Y _kj must be non-negative in T _IDEAL. Thus, for any given weights λ, the value of admissible y _j in T _UB must always be less than or equal to the corresponding value of y _j in T _UB. Since the two sets are otherwise identical, we have proved that T _UB⊆T _IDEAL.

  • 2) T _FIRMk⊆T _UB: Set T _FIRMk is obtained by setting λ _k =0 in (1). Thus, output constraints of T _FIRMk are of form

    

    Comparing constraints (A2) and (A3), we observe that the constraints for output j are in effect identical in both T _UB and T _FIRMk (the first one assigns the output of DMU k equal to zero, the latter one the weight of DMU k). However, the constraints for outputs s≠j are different: set T _FIRMk imposes constraint of type (A3) for all outputs, while the output constraints of T _UB are of form

    

    For any given weights λ, the constraints of type (A3) imply a maximum value of output s that is less than or equal to the maximum value allowed by constrain (A4). Since the two sets are identical except for the output constraints, the two sets are nested as T _FIRMk⊆T _UB.

  • 3) T _Yj ⊆T _UB: Set T _Yj is obtained by setting Y _nj =0 for all DMUs n. Thus, the constraint for output j in T _Yj reads as

    

    Comparing constraints (A2) and (A4), we see that for any given weights λ, the admissible values of output j in set T _UB are greater than or equal to the zero value implied by T _Yj . Since the two sets are otherwise identical, the two sets are nested as T _Yj ⊆T _UB. □

Proof of Theorem 2

• Observe that technology T _Yj is obtained by imposing constraint u _j =0 in (2). To prove the equivalence, we show that using reference technology T _UB the DEA problems of Table 2 always yield optimal solutions where output weights u ^* satisfy u _j ^*=0.

  Consider the input-oriented problem (the left column of Table 2). The objective function reads as

  

  Since DMU k must have produced a strictly positive amount of at least one output, say output i. Note that the value of sum (A6) will always increase if we increase weight u _i and simultaneously decrease weight u _j . Thus, the optimal solution of the input efficiency problem will always satisfy u _j ^*=0. The same argument directly applies to the output efficiency problem. □

Proof of Theorem 3

• In this case the evaluated input–output vector of (ie that of DMU l) is identical for both φ _UB ^l and φ _FIRMk ^l. The only difference concerns the production possibilities frontiers. Thus, the result follows directly from the fact that T _FIRMk⊆T _UB⊆T _IDEAL (see Theorem 1). □