Study on Returns to Scale Consistency Between the Weak BCC Inefficient DMUs and Their Projection in DEA

Abstract

This paper discusses estimation of returns to scale (RTS) of Weak BCC inefficient DMUs in DEA. RTS generally has an ambiguous meaning if DMU is not on Weak BCC efficient frontier. Researchers adopt projection method for this problem. Theoretically, a Weak BCC inefficient DMU and its projection should exhibit the same RTS nature. Banker et~al. (Eur J Oper Res 88:583–585, 1996b)’s projection, however, may give inconsistent estimation of RTS in some cases. For accurate RTS estimation of Weak BCC inefficient DMUs, this paper establishes Weak BCC projection and strong BCC projection. It is proved that, for a Weak BCC inefficient DMU, it is its Weak BCC projection that always exhibits the same RTS nature as itself while its strong BCC projection is not in some cases. In addition, Weak BCC projection is the most representative point among all frontier points for Weak BCC inefficient DMU. Therefore the projection should be Weak BCC projection, not strong BCC projection, when estimating RTS of Weak BCC inefficient DMUs.

Appendix. Proof of Lemma 1 to Lemma 6

Proof of Lemma 1

Since DMU₀ is Weak BCC inefficient, it is easily known that lemma 1 holds.□

Proof of Lemma 2

Since it is Weak BCC inefficient, DMU₀ does not belong to E _BCC . Also it is obvious that ${\mathrm{D}\widehat{\mathrm{M}}\mathrm{U}}_0^{\mathrm{w}}$ must superpose with an existing point of the Weak BCC frontier. Then the existing point

• (1) does not belong to E _BCC . Hence E _BCC = E ^{w − single} _BCC Thus (i) holds.

• (2) belongs to E _BCC . In this case there is no new element added to E _BCC . Hence E _BCC = E ^{w − single} _BCC . Thus (ii) holds.

Proof of Lemma 3

First as we mention above, E _CCR , E _BCC , and E _NIRS are the set E of Original dataset under CCR, BCC, and NIRS model respectively. And E ^{w − single} _CCR , E ^{w − single} _BCC and E ^{w − single} _NIRS are the set E of Single-Weak-Projection dataset under CCR, BCC, and NIRS model respectively. From the proof of lemma 2, we know that E _BCC = E ^{w − single} _BCC holds. In a similar way, E _CCR = E ^{w − single} _CCR and E _NIRS = E ^{w − single} _NIRS hold.

Proof of Lemma 4 and Lemma 5

It is similar to the proof of lemma 2. Thus it is omitted.

Proof of Lemma 6

Let X ^∗ be an optimal solution of (i), then it is also an optima of (ii). Otherwise, there will be X ^∗₁ such that k(C ^T X ^∗₁ ) < k(C ^T X ^∗). Hence we have C ^T X ^∗₁  < C ^T X ^∗. Note that (i) and (ii) have the same constrains, thus X ^∗ is not an optimal solution of (i), which is a contradiction. And vice versa.