Here, we consider the overall profit efficiency and overall profit MPI of DMUo,o=1,...,n when input, output, and input-output prices are uncertain and also define an interval for the overall profit MPI of DMUo. We reiterate that there are n DMUs under consideration.
Assume that \(\left [x^{pL}_{ij}, x^{pU}_{ij}\right ]\) and \(\left [y^{pL}_{kj}, y^{pU}_{kj}\right ]\) are the intervals of input i and output k of DMUj,(j∈J) in period p, respectively. Additionally, \(\left [c^{pL}_{io}, c^{pU}_{io}\right ]\), and \(\left [r^{pL}_{ko}, r^{pU}_{ko}\right ]\) are the intervals of the input-output prices of input i and output k of DMUo,o=1,...,n in period p, respectively. Models (3) and (4) can be extended to the overall profit efficiency models (5) and (6) with data uncertainty, respectively:
$$ \begin{array}{llll} {Within-period\ time} & & & \\ B_{o}^{p}\left(x_{o}^{p}, y_{o}^{p}|p=t, t+1\right)=&\max&\ \frac{\left(r_{o}^{p}\right)^{T}y}{\left(r_{o}^{p}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{p}\right)^{T}x}{\left(c_{o}^{p}\right)^{T}x^{p}_{o}}\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{p}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{p}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&c^{p}_{io}\in\left[c^{pL}_{io}, c^{pU}_{io}\right],&i=1,...,m\\ &&r^{p}_{ko}\in\left[r^{pL}_{ko}, r^{pU}_{ko}\right],&k=1,...,s\\ &&x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right],&i=1,...,m\\ &&y^{p}_{kj}\in\left[y^{pL}_{kj}, y^{pU}_{kj}\right],&k=1,...,s\\ &&\lambda_{j}\geq0,&j\in J.\\ \end{array} $$
(5)
$$ \begin{array}{llllc} Adjacent-period time & \\ D_{o}^{q}\left(x_{o}^{p}, y_{o}^{p}|p,q=t, t+1, p\neq q\right)=&\max&\ \frac{\left(r_{o}^{q}\right)^{T}y}{\left(r_{o}^{q}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{q}\right)^{T}x}{\left(c_{o}^{q}\right)^{T}x^{p}_{o}}\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{q}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{q}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&c^{q}_{io}\in\left[c^{qL}_{io}, c^{qU}_{io}\right],&i=1,...,m\\ &&r^{q}_{ko}\in\left[r^{qL}_{ko}, r^{qU}_{ko}\right],&k=1,...,s\\ &&x^{q}_{ij}\in\left[x^{qL}_{ij}, x^{qU}_{ij}\right],&i=1,...,m\\ &&y^{q}_{kj}\in\left[y^{qL}_{kj}, y^{qU}_{kj}\right],&k=1,...,s\\ &&x^{p}_{io}\in\left[x^{pL}_{io}, x^{pU}_{io}\right],&i=1,...,m\\ &&y^{p}_{ko}\in\left[y^{pL}_{ko}, y^{pU}_{ko}\right],&k=1,...,s.\\ &&\lambda_{j}\geq0,&j\in J\\ \end{array} $$
(6)
It can be observed that models (5) and (6) are nonlinear programming programs because of data uncertainty.
Of particular importance is how to solve the newly constructed profit efficiency models with data uncertainty in (5) and (6). To illustrate these issues, we introduce the following definitions, which are similar to those of Park (2001).
Definition 5
(Potential profit efficiency in the within-period time) The DMUo to be evaluated is potentially profit efficient in the within-period if and only if there exists at least one set of prices \(c^{p}_{o}\in \left [c^{pL}_{o}, c^{pU}_{o}\right ]\) and \(r^{p}_{o}\in \left [r^{pL}_{o}, r^{pU}_{o}\right ]\) and at least one set of input-output data satisfying \(x^{p}_{ij}\in \left [x^{pL}_{ij}, x^{pU}_{ij}\right ]\) and \(y^{p}_{kj}\in \left [y^{pL}_{kj}, y^{pU}_{kj}\right ] (j\in J)\), so that \(B^{p*}_{o}=0\)Footnote 1 in model (5).
Definition 6
(Perfect profit efficiency in the within-period time) The DMUo to be evaluated is perfectly profit efficient in the within-period if and only if, for all \(c^{p}_{o}\in \left [c^{pL}_{o}, c^{pU}_{o}\right ]\) and \(r^{p}_{o}\in \left [r^{pL}_{o}, r^{pU}_{o}\right ]\) and all input-output data, \(x^{p}_{ij}\in \left [x^{pL}_{ij}, x^{pU}_{ij}\right ]\) and \(y^{p}_{kj}\in \left [y^{pL}_{kj}, y^{pU}_{kj}\right ] (j\in J), B^{p*}_{o}=0\) are satisfied in model (5).
Definition 7
(Potential profit efficiency in the adjacent period) The DMUo to be evaluated is potentially profit efficient in the adjacent period if and only if there exists at least one set of prices \(c^{q}_{o}\in \left [c^{qL}_{o}, c^{qU}_{o}\right ]\) and \(r^{q}_{o}\in \left [r^{qL}_{o}, r^{qU}_{o}\right ]\) and at least one set of input-output data satisfying \(x^{q}_{ij}\in \left [x^{qL}_{ij}, x^{qU}_{ij}\right ], y^{q}_{kj}\in \left [y^{qL}_{kj}, y^{qU}_{kj}\right ] (j\in J), x^{p}_{io}\in \left [x^{pL}_{io}, x^{pU}_{io}\right ]\) and \(y^{p}_{ko}\in \left [y^{pL}_{ko}, y^{pU}_{ko}\right ] (p, q=t, t+1, p\neq q)\), so that \(D^{q*}_{o}=0\) in model (6).
Definition 8
(Perfect profit efficiency in the adjacent period) The DMUo to be evaluated is perfectly profit efficient in the adjacent-period time if and only if, for all \(c^{q}_{o}\in \left [c^{qL}_{o}, c^{qU}_{o}\right ]\) and \(r^{q}_{o}\in \left [r^{qL}_{o}, r^{qU}_{o}\right ]\) and all input-output data, \(x^{q}_{ij}\in \left [x^{qL}_{ij}, x^{qU}_{ij}\right ], y^{q}_{kj}\in \left [y^{qL}_{kj}, y^{qU}_{kj}\right ] (j\in J), x^{p}_{io}\in \left [x^{pL}_{io}, x^{pU}_{io}\right ]\) and \(y^{p}_{ko}\in \left [y^{pL}_{ko}, y^{pU}_{ko}\right ] (p, q=t, t+1, p\neq q)\) are satisfied so that \(D^{q*}_{o}=0\) in model (6).
In definitions 5 and 7, the profit efficiency of DMUo is measured for some data, while definitions 6 and 8 refer to the profit efficiency of DMUo for all data. Therefore, perfect profit efficiency is measured in a more rigid manner than potential profit efficiency. In the spirit of Park (2001), we can represent these definitions using the following mathematical formulations, where term UPEW-S (UPEW-P) refers to the uncertain profit efficiency of \(DMU_{o}^{p}=\left (x_{o}^{p}, y_{o}^{p}\right)\) with the technology and prices at time p, the within period, for some (for perfect (all)). Additionally, term UPEA-S (UPEA-P) refers to uncertain profit efficiency of \(DMU_{o}^{p}=\left (x_{o}^{p}, y_{o}^{p}\right)\) with technology and prices at time q, the adjacent period, for some (for perfect (all)) (p,q=t,t+1, p≠q):
$$ \begin{array}{rlclc} &&\text{The UPEW-S model:}\\ \overline{B}_{o}^{p}\left(x_{o}^{p}, y_{o}^{p}|p=t, t+1\right)=&\max&\ \frac{\left(r_{o}^{p}\right)^{T}y}{\left(r_{o}^{p}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{p}\right)^{T}x}{\left(c_{o}^{p}\right)^{T}x^{p}_{o}}\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{p}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{p}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&\text{for some}\{c^{p}_{io}\in\left[c^{pL}_{io}, c^{pU}_{io}\right],&i=1,...,m\}\\ &&\text{for some}\{r^{p}_{ko}\in\left[r^{pL}_{ko}, r^{pU}_{ko}\right],&k=1,...,s\}\\ &&\text{for some}\{x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right],&i=1,...,m\}\\ &&\text{for some}\{y^{p}_{kj}\in\left[y^{pL}_{kj}, y^{pU}_{kj}\right],&k=1,...,s\}\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(7)
$$ \begin{array}{rlclc} &&\text{The UPEW-P model:}\\ &&\text{Model (7) with } \underline{B}_{o}^{p} \text{ in place of } \overline{B}_{o}^{p} \text{ and }\prime\prime \text{ for all}\prime\prime\\ &&\text{in place of }\prime\prime \text{ for some}.\prime\prime\\ \end{array} $$
(8)
$$ \begin{array}{rlclc} &&\text{The UPEA-S model:}\\ \overline{D}_{o}^{q}\left(x_{o}^{p}, y_{o}^{p}|p,q=t, t+1, p\neq q\right)=&\max&\ \frac{\left(r_{o}^{q}\right)^{T}y}{\left(r_{o}^{q}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{q}\right)^{T}x}{\left(c_{o}^{q}\right)^{T}x^{p}_{o}}\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{q}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{q}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&\text{for some } \left\{c^{q}_{io}\in[c^{qL}_{io}, c^{qU}_{io}],\quad i=1,...,m\right\}\\ &&\text{for some } \left\{r^{q}_{ko}\in [r^{qL}_{ko}, r^{qU}_{ko}],\quad k=1,...,s\right\}\\ &&\text{for some } \left\{x^{q}_{ij}\in[x^{qL}_{ij}, x^{qU}_{ij}],\quad i=1,...,m\right\}\\ &&\text{for some } \left\{y^{q}_{kj}\in [y^{qL}_{kj}, y^{qU}_{kj}],\quad k=1,...,s\right\}\\ &&\text{for some } \left\{x^{p}_{io}\in[x^{pL}_{io}, x^{pU}_{io}],\quad i=1,...,m\right\}\\ &&\text{for some } \left\{y^{p}_{ko}\in [y^{pL}_{ko}, y^{pU}_{ko}],\quad k=1,...,s\right\}\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(9)
$$ \begin{array}{rlclc} &&\text{The UPEA-P model:}\\ &&\text{Model (9) with } \underline{D}^{q}_{o} \text{ in place of } \overline{D}_{o}^{q} \text{ and }\prime\prime \text{for all}\prime\prime\\ &&\text{in place of }\prime\prime \text{for some}.\prime\prime\\ \end{array} $$
(10)
Clearly, \(\overline {B}_{o}^{p}\geq \underline {B}_{o}^{p}\) (\(\overline {D}_{o}^{q}\geq \underline {D}_{o}^{q}\)) because the feasible region of model (8) (10) is always contained within the feasible region of model (7) (9).
Using models (7) and (8), we can obtain the interval of profit efficiency of DMUo in the within-period as \(\left [\underline {B}_{o}^{p}, \overline {B}_{o}^{p}\right ]\). Additionally, by models (9) and (10), the interval of profit efficiency of DMUo in the adjacent period can be obtained as \(\left [\underline {D}_{o}^{q}, \overline {D}_{o}^{q}\right ]\). \(\underline {B}_{o}^{p} \left (\underline {D}_{o}^{q}\right)\) is the lower bound of the interval overall profit efficiency of DMUo from the pessimistic viewpoint in the within period (adjacent period) and \(\overline {B}_{o}^{p} \left (\overline {D}_{o}^{q}\right)\) is the upper bound of the interval overall profit efficiency of DMUo from the optimistic viewpoint in the within period (adjacent period).
Remarks 2
It is clear that model (5) is equivalent to the UPEW-S model (7) and model (6) is equivalent to the UPEA-S model (9).
Because of the notion of maximization “for some” and “for all” in the permissible data, the UPEW-S (7), UPEW-P (8), UPEA-S (9), and UPEA-P models (10) are equivalent to the two-level mathematical programs (11), (12), (13), and (14), respectively:
$$ \begin{array}{llllc} \overline{B}_{o}^{p}=\max&\max&\ \frac{\left(r_{o}^{p}\right)^{T}y}{\left(r_{o}^{p}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{p}\right)^{T}x}{\left(c_{o}^{p}\right)^{T}x^{p}_{o}}\\ c^{p}_{io}\in\left[c^{pL}_{io}, c^{pU}_{io}\right]\\ r^{p}_{ko}\in \left[r^{pL}_{ko}, r^{pU}_{ko}\right]\\ x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right]\\ y^{p}_{kj}\in\left[y^{pL}_{kj}, y^{pU}_{kj}\right]\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{p}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{p}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(11)
$$ \begin{array}{rlclc} &&\text{Model (11) but with} \prime\prime \underline{B}_{o}^{p}=\text{min}\prime\prime \text{in place of }\prime\prime\overline{B}_{o}^{p}=\text{max}\prime\prime.\\ \end{array} $$
(12)
$$ \begin{array}{llllc} \overline{D}_{o}^{p}=\max&\max&\ \frac{\left(r_{o}^{q}\right)^{T}y}{\left(r_{o}^{q}\right)^{T}y^{p}_{o}}-\frac{\left(c_{o}^{q}\right)^{T}x}{\left(c_{o}^{q}\right)^{T}x^{p}_{o}}\\ c^{q}_{io}\in\left[c^{qL}_{io}, c^{qU}_{io}\right]\\ r^{q}_{ko}\in \left[r^{qL}_{ko}, r^{qU}_{ko}\right]\\ x^{q}_{ij}\in\left[x^{qL}_{ij}, x^{qU}_{ij}\right]\\ y^{q}_{kj}\in \left[y^{qL}_{kj}, y^{qU}_{kj}\right]\\ x^{p}_{io}\in\left[x^{pL}_{io}, x^{pU}_{io}\right]\\ y^{p}_{ko}\in \left[y^{pL}_{ko}, y^{pU}_{ko}\right]\\ &s.t.&-\sum_{j\in J}\lambda_{j}y^{q}_{j}+y\leq0\\ &&\sum_{j\in J}\lambda_{j}x^{q}_{j}-x\leq0\\ &&\sum_{j\in J}\lambda_{j}=1\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(13)
$$ \begin{array}{rlclc} &&\text{Model (13) but with } \prime\prime\underline{D}_{o}^{p}=\text{min}\prime\prime \text{ in place of } \prime\prime\overline{D}_{o}^{p}=\text{max}\prime\prime. \end{array} $$
(14)
In model (11), the inner program calculates the overall profit efficiency for each given set of \(\left (x^{p}_{o}, y^{p}_{o}\right)\) for a price vector \(\left (r_{o}^{p}, c_{o}^{p}\right)\) defined in the outer program, using the technology of period p, while the outer program determines the set of \(\left (x^{p}_{o}, y^{p}_{o}\right)\) and price vector \(\left (r_{o}^{p}, c_{o}^{p}\right)\) that generate the highest overall profit efficiency. Additionally, in model (13), the inner program calculates the overall profit efficiency for each given set of \(\left (x^{p}_{o}, y^{p}_{o}\right)\) for a given price vector \(\left (r_{o}^{q}, c_{o}^{q}\right)\) using the technology of period q (p,q=t,t+1,p≠q), defined in the outer program, while the outer program determines the set of \(\left (x^{p}_{o}, y^{p}_{o}\right)\) and price vector \(\left (r_{o}^{q}, c_{o}^{q}\right)\) that generate the highest overall profit efficiency. A similar explanation can be provided for models (12) and (14).
By duality, models (11)-(14) are equivalent to the following models:
$$ \begin{array}{llllc} \overline{B}_{o}^{p}=\max&\min&\ \mu_{o}\\ c^{p}_{io}\in\left[c^{pL}_{io}, c^{pU}_{io}\right]\\ r^{p}_{ko}\in \left[r^{pL}_{ko}, r^{pU}_{ko}\right]\\ x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right]\\ y^{p}_{kj}\in \left[y^{pL}_{kj}, y^{pU}_{kj}\right]\\ &s.t.&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{p}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\mu^{p}_{k}=\frac{r^{p}_{ko}}{\left(r^{p}_{o}\right)^{T}y^{p}_{o}},&k=1,...,s&\\ &&v^{p}_{i}=\frac{c^{p}_{io}}{\left(c^{p}_{o}\right)^{T}x^{p}_{o}},&i=1,...,m&\\ &&\mu^{p}_{k}\geq0,&k=1,...,s\\ &&v^{p}_{i}\geq0,&i=1,...,m\\ &&\mu_{0}&free,\\ \end{array} $$
(15)
$$ \begin{array}{rlclc} &&\text{Model (15) but with }\prime\prime\underline{B}_{o}^{p}=\text{min}\prime\prime \text {in place of } \prime\prime\overline{B}_{o}^{p}=\text{max}\prime\prime.\\ \end{array} $$
(16)
$$ \begin{array}{llllc} \overline{D}_{o}^{p}=\max&\min&\ \mu_{o}\\ c^{q}_{io}\in\left[c^{qL}_{io}, c^{qU}_{io}\right]\\ r^{q}_{ko}\in \left[r^{qL}_{ko}, r^{qU}_{ko}\right]\\ x^{q}_{ij}\in\left[x^{qL}_{ij}, x^{qU}_{ij}\right]\\ y^{q}_{kj}\in \left[y^{qL}_{kj}, y^{qU}_{kj}\right]\\ x^{p}_{io}\in\left[x^{pL}_{io}, x^{pU}_{io}\right]\\ y^{p}_{ko}\in \left[y^{pL}_{ko}, y^{pU}_{ko}\right]\\ &s.t.&\sum_{k=1}^{s}\mu^{q}_{k}y^{q}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{q}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\mu^{q}_{k}=\frac{r^{q}_{ko}}{\left(r^{q}_{o}\right)^{T}y^{p}_{o}},&k=1,...,s&\\ &&v^{q}_{i}=\frac{c^{q}_{io}}{\left(c^{q}_{o}\right)^{T}x^{p}_{o}},&i=1,...,m&\\ &&\mu^{q}_{k}\geq0,&k=1,...,s\\ &&v^{q}_{i}\geq0,&i=1,...,m\\ &&\mu_{0}&free.\\ \end{array} $$
(17)
$$ \begin{array}{rlclc} &&\text{Model (17) but with } \prime\prime\underline{D}_{o}^{p}=\text{min}\prime\prime \text{ in place of } \prime\prime\overline{D}_{o}^{p}=\text{max}\prime\prime. \end{array} $$
(18)
First, we proceed to models (16) and (18). Their inner and outer programs have the same objective of minimization. Therefore, they can be combined into a one-level model by considering all constraints of the two programs simultaneously. The one-level models equivalent to (16) and (18) are (19) and (20), respectively:
$$ \begin{array}{llllc} \underline{B}_{o}^{p}=\min&\ \mu_{o}\\ &s.t.&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{p}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\mu^{p}_{k}=\frac{r^{p}_{ko}}{\left(r^{p}_{o}\right)^{T}y^{p}_{o}},&k=1,...,s&\\ &&v^{p}_{i}=\frac{c^{p}_{io}}{\left(c^{p}_{o}\right)^{T}x^{p}_{o}},&i=1,...,m&\\ &&c^{p}_{io}\in\left[c^{pL}_{io}, c^{pU}_{io}\right]\\ &&r^{p}_{ko}\in \left[r^{pL}_{ko}, r^{pU}_{ko}\right]\\ &&x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right]\\ &&y^{p}_{kj}\in \left[y^{pL}_{kj}, y^{pU}_{kj}\right]\\ &&\mu^{p}_{k}\geq0,&k=1,...,s\\ &&v^{p}_{i}\geq0,&i=1,...,m\\ &&\mu_{o}&free\\ &&\lambda_{j}\geq0,\\ \end{array} $$
(19)
$$ \begin{array}{llllc} \underline{D}_{o}^{p}=\min& \mu_{o}\\ &s.t.&\sum_{k=1}^{s}\mu^{q}_{k}y^{q}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{q}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\mu^{q}_{k}=\frac{r^{q}_{ko}}{\left(r^{q}_{o}\right)^{T}y^{p}_{o}},&k=1,...,s&\\ &&v^{q}_{i}=\frac{c^{q}_{io}}{(c^{q}_{o})^{T}x^{p}_{o}},&i=1,...,m&\\ &&c^{q}_{io}\in\left[c^{qL}_{io}, c^{qU}_{io}\right]\\ &&r^{q}_{ko}\in \left[r^{qL}_{ko}, r^{qU}_{ko}\right]\\ &&x^{q}_{ij}\in\left[x^{qL}_{ij}, x^{qU}_{ij}\right]\\ &&y^{q}_{kj}\in \left[y^{qL}_{kj}, y^{qU}_{kj}\right]\\ &&x^{p}_{io}\in\left[x^{pL}_{io}, x^{pU}_{io}\right]\\ &&y^{p}_{ko}\in \left[y^{pL}_{ko}, y^{pU}_{ko}\right]\\ &&\mu^{q}_{k}\geq0,&k=1,...,s\\ &&v^{q}_{i}\geq0,&i=1,...,m\\ &&\mu_{o}&free\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(20)
Evidently the above models (19) and (20) are nonlinear programming problems and thus difficult to solve. To linearize model (19), we introduce variables zp and τp, defined by:
\(z^{p}=\frac {1}{\left (c^{p}_{o}\right)^{T}x^{p}_{o}}\),
\(\tau ^{p}=\frac {1}{\left (r^{p}_{o}\right)^{T}y^{p}_{o}}\),
so that
\(v^{p}_{i}=c^{p}_{io}z^{p}\Longleftrightarrow c^{pL}_{io}z^{p}\leq v^{p}_{i}\leq c^{pU}_{io}z^{p}\),
\(\mu ^{p}_{k}=r^{p}_{ko}\tau ^{p}\Longleftrightarrow r^{pL}_{ko}\tau ^{p}\leq \mu ^{p}_{k} \leq r^{pU}_{ko}\tau ^{p}\),
\(\sum _{k=1}^{s}r^{p}_{ko}\tau ^{p}y^{p}_{ko}=1\Longleftrightarrow \sum _{k=1}^{s}\mu ^{p}_{k}y^{p}_{ko}=1 \),
\(\sum _{i=1}^{m}c^{p}_{io}z^{p}x^{p}_{io}=1\Longleftrightarrow \sum _{i=1}^{m}v^{p}_{i}x^{p}_{io}=1 \).
Similarly, to the linearization of model (20), we introduce variables zq and τq, defined by:
\(z^{q}=\frac {1}{\left (c^{q}_{o}\right)^{T}x^{p}_{o}}\),
\(\tau ^{q}=\frac {1}{\left (r^{q}_{o}\right)^{T}y^{p}_{o}}\),
so that
\(v^{q}_{i}=c^{q}_{io}z^{q}\Longleftrightarrow c^{qL}_{io}z^{q}\leq v^{q}_{i}\leq c^{qU}_{io}z^{q}\),
\(\mu ^{q}_{k}=r^{q}_{ko}\tau ^{q}\Longleftrightarrow r^{qL}_{ko}\tau ^{q}\leq \mu ^{q}_{k} \leq r^{qU}_{ko}\tau ^{q}\),
\(\sum _{k=1}^{s}r^{q}_{ko}\tau ^{q}y^{p}_{ko}=1\Longleftrightarrow \sum _{k=1}^{s}\mu ^{q}_{k}y^{p}_{ko}=1 \),
\(\sum _{i=1}^{m}c^{q}_{io}z^{q}x^{p}_{io}=1\Longleftrightarrow \sum _{i=1}^{m}v^{q}_{i}x^{p}_{io}=1 \),
where p,q=t,t+1,p≠q.
Using the above variable alterations, models (19) and (20) can be converted into the following programming problems, whose optimal objective values coincide with those of (19) and (20), respectively:
$$ \begin{array}{llllc} \underline{B}_{o}^{p}=\min&\ \mu_{o}\\ &s.t.&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{p}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{ko}=1\\ &&\sum_{i=1}^{m}v^{p}_{i}x^{p}_{io}=1\\ &&c^{pL}_{io}z^{p}\leq v^{p}_{i}\leq c^{pU}_{io}z^{p}\\ &&r^{pL}_{ko}\tau^{p}\leq \mu^{p}_{k} \leq r^{pU}_{ko}\tau^{p}\\ &&x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right]\\ &&y^{p}_{kj}\in \left[y^{pL}_{kj}, y^{pU}_{kj}\right]\\ &&z^{p}\geq0\\ &&\tau^{p}\geq0\\ &&\mu^{p}_{k}\geq0,&k=1,...,s\\ &&v^{p}_{i}\geq0,&i=1,...,m\\ &&\mu_{o}&free\\ &&\lambda_{j}\geq0,\\ \end{array} $$
(21)
$$ \begin{array}{llllc} \underline{D}_{o}^{p}=\min& \mu_{o}\\ &s.t.&\sum_{k=1}^{s}\mu^{q}_{k}y^{q}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{q}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\sum_{k=1}^{s}\mu^{q}_{k}y^{p}_{ko}=1\\ &&\sum_{i=1}^{m}v^{q}_{i}x^{p}_{io}=1\\ &&c^{qL}_{io}z^{q}\leq v^{q}_{i}\leq c^{qU}_{io}z^{q}\\ &&r^{qL}_{ko}\tau^{q}\leq \mu^{q}_{k} \leq r^{qU}_{ko}\tau^{q}\\ &&x^{q}_{ij}\in\left[x^{qL}_{ij}, x^{qU}_{ij}\right]\\ &&y^{q}_{kj}\in \left[y^{qL}_{kj}, y^{qU}_{kj}\right]\\ &&x^{p}_{io}\in\left[x^{pL}_{io}, x^{pU}_{io}\right]\\ &&y^{p}_{ko}\in \left[y^{pL}_{ko}, y^{pU}_{ko}\right]\\ &&z^{q}\geq0\\ &&\tau^{q}\geq0\\ &&\mu^{q}_{k}\geq0,&k=1,...,s\\ &&v^{q}_{i}\geq0,&i=1,...,m\\ &&\mu_{o}&free\\ &&\lambda_{j}\geq0.\\ \end{array} $$
(22)
Similar to Lemma (1) of (Podinovski 2001), we propose the following Lemma, which refers to the general weight bound problem:
Lemma 1
Imposing the absolute bounds of \(c^{pL}_{io}z^{p}\leq v^{p}_{i}\leq c^{pU}_{io}z^{p}, (i=1,...,m, z^{p}\geq 0)\) and \(r^{pL}_{ko}\tau ^{p}\leq \mu ^{p}_{k} \leq r^{pU}_{ko}\tau ^{p}, (k=1,...,s, \tau ^{p}\geq 0)\) is equivalent to imposing bounds on the ratios of the weights of the following form:
\(\frac {c^{zL}_{io}}{c^{zU}_{ko}}\leq \frac {v^{z}_{i}}{v^{z}_{k}} \leq \frac {c^{zU}_{io}}{c^{zL}_{ko}}~~~~~ i, k=1...,m,~~k>i, z=p, q\),
\(\frac {r^{zL}_{lo}}{r^{zL}_{\kappa o}}\leq \frac {\mu ^{z}_{l}}{\mu ^{z}_{\kappa }}\leq \frac {r^{zU}_{lo}}{r^{zL}_{\kappa o}}~~~~ l, \kappa =1,...,s,~~~\kappa >l, z=p, q\).
Let \(y_{kj}=\alpha _ky^{U}_{kj}+(1-\alpha _k)y^{L}_{kj}\) and \(x_{ij}=\beta _ix^{U}_{ij}+(1-\beta _i)x^{L}_{ij}\) for some αk∈[0,1] and βi∈[0,1]. It is easy to show that the first three constraints of models (21) and (22) can be written as (23) and (24), respectively:
$$ {}\begin{aligned} &\sum_{k=1}^{s}\gamma_{k}\left(y^{Up}_{kj}-y^{Lp}_{kj}\right)+\sum_{k=1}^{s}\mu^{p}_{k}y^{Lp}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{Lp}_{ij}-\sum_{i=1}^{m}\omega_{i}\left(x^{Up}_{ij}-x^{Lp}_{ij}\right)+\mu_{o}\geq0,&j=1,...n\\ &\sum_{k=1}^{s}\gamma_{k}\left(y^{Up}_{ko}-y^{Lp}_{ko}\right)+\sum_{k=1}^{s}\mu^{p}_{k}y^{Lp}_{ko}=1\\ &\sum_{i=1}^{m}\omega_{i}\left(x^{Up}_{io}-x^{Lp}_{io}\right)+\sum_{i=1}^{m}v^{p}_{i}x^{Lp}_{io}=1,\\ \end{aligned} $$
(23)
$$ {}\begin{aligned} &\sum_{k=1}^{s}\gamma_{k}\left(y^{Uq}_{kj}-y^{Lq}_{kj}\right)+\sum_{k=1}^{s}\mu^{q}_{k}y^{Lq}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{Lq}_{ij}-\sum_{i=1}^{m}\omega_{i}\left(x^{Uq}_{ij}-x^{Lq}_{ij}\right)+\mu_{o}\geq0,&j=1,...n\\ &\sum_{k=1}^{s}\gamma_{k}\left(y^{Up}_{ko}-y^{Lp}_{ko}\right)+\sum_{k=1}^{s}\mu^{q}_{k}y^{Lp}_{ko}=1\\ &\sum_{i=1}^{m}\omega_{i}\left(x^{Up}_{io}-x^{Lp}_{io}\right)+\sum_{i=1}^{m}v^{q}_{i}x^{Lo}_{io}=1,\\ \end{aligned} $$
(24)
where γk=μkαk and ωi=viβi for each i and k.
By applying Lemma 1 and constraints (23) and (24) to models (21) and (22), we obtain the equivalent linear formulations of models (21) and (22), called the CR-DEA models, as follows:
$$\begin{aligned} \underline{B}_{o}^{p}=\min \ \mu_{o}\\ s.t. \ \ \ \ \ &\sum_{k=1}^{s}\gamma_{k}(y^{Up}_{kj}-y^{Lp}_{kj})+\sum_{k=1}^{s}\mu^{p}_{k}y^{Lp}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{Lp}_{ij}-\sum_{i=1}^{m}\omega_{i}(x^{Up}_{ij}-x^{Lp}_{ij})+\mu_{o}\geq0,&j=1,...n\\ &\sum_{k=1}^{s}\gamma_{k}(y^{Up}_{ko}-y^{Lp}_{ko})+\sum_{k=1}^{s}\mu^{p}_{k}y^{Lp}_{ko}=1\\ &\sum_{i=1}^{m}\omega_{i}(x^{Up}_{io}-x^{Lp}_{io})+\sum_{i=1}^{m}v^{p}_{i}x^{Lp}_{io}=1\\ &\frac{c^{pL}_{io}}{c^{pU}_{ko}}\leq\frac{v^{p}_{i}}{v^{p}_{k}} \leq\frac{c^{pU}_{io}}{c^{pL}_{ko}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i, k=1...,m,k>i\\ &\frac{r^{pL}_{lo}}{r^{pL}_{\kappa o}}\leq\frac{\mu^{p}_{l}}{\mu^{p}_{\kappa}}\leq\frac{r^{pU}_{lo}}{r^{pL}_{\kappa o}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ l, \kappa=1,...,s, \kappa>l&\\ \end{aligned} $$
$$ \begin{aligned} &0\leq \gamma_{k}\leq \mu^{p}_{k}\\ &0\leq \omega_{i}\leq v^{p}_{i}\\ &\mu^{p}_{k}\geq0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~k=1,...,s&\\ &v^{p}_{i}\geq0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i=1,...,m&\\ &\mu_{0}, free~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, \end{aligned} $$
(25)
$$ \begin{aligned} \underline{D}_{o}^{p}=\min \ \mu_{o}\\ s.t.&\sum_{k=1}^{s}\gamma_{k}\left(y^{Uq}_{kj}-y^{Lq}_{kj}\right)+\sum_{k=1}^{s}\mu^{q}_{k}y^{Lq}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{Lq}_{ij}-\sum_{i=1}^{m}\omega_{i}\left(x^{Uq}_{ij}-x^{Lq}_{ij}\right)+\mu_{o}\geq0,&j=1,...n\\ &\sum_{k=1}^{s}\gamma_{k}\left(y^{Up}_{ko}-y^{Lp}_{ko}\right)+\sum_{k=1}^{s}\mu^{q}_{k}y^{Lp}_{ko}=1\\ &\sum_{i=1}^{m}\omega_{i}\left(x^{Up}_{io}-x^{Lp}_{io}\right)+\sum_{i=1}^{m}v^{q}_{i}x^{Lp}_{io}=1\\ &\frac{c^{qL}_{io}}{c^{qU}_{ko}}\leq\frac{v^{q}_{i}}{v^{q}_{k}} \leq\frac{c^{qU}_{io}}{c^{qL}_{ko}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i, k=1...,m,k>i&\\ &\frac{r^{qL}_{lo}}{r^{qL}_{\kappa o}}\leq\frac{\mu^{q}_{l}}{\mu^{q}_{\kappa}}\leq\frac{r^{qU}_{lo}}{r^{qL}_{\kappa o}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ l, \kappa=1,...,s, \kappa>l&\\ &0\leq \gamma_{k}\leq \mu^{q}_{k}\\ &0\leq \omega_{i}\leq v^{q}_{i}\\ &\mu^{q}_{k}\geq0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~k=1,...,s&\\ &v^{q}_{i}\geq0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i=1,...,m&\\ &\mu_{o},~~~~~~free.\\ \end{aligned} $$
(26)
Now, we proceed to models (15) and (17). By applying the mentioned variable alterations to (15) and (17), we have:
$$ \begin{array}{llllc} \overline{B}_{o}^{p}=\max&\min&\ \mu_{o}\\ \frac{v^{p}_{i}}{v^{p}_{k}}\in\Big[\frac{c^{pL}_{io}}{c^{pU}_{ko}},\frac{c^{pU}_{io}}{c^{pL}_{ko}}\Big]\\ \frac{\mu^{p}_{l}}{\mu^{p}_{\kappa}}\in\Big[\frac{r^{pL}_{lo}}{r^{pL}_{\kappa o}},\frac{r^{pU}_{lo}}{r^{pL}_{\kappa o}}\Big]\\ x^{p}_{ij}\in\left[x^{pL}_{ij}, x^{pU}_{ij}\right]\\ y^{p}_{kj}\in \left[y^{pL}_{kj}, y^{pU}_{kj}\right]\\ &s.t.&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{kj}-\sum_{i=1}^{m}v^{p}_{i}x^{p}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &&\sum_{k=1}^{s}\mu^{p}_{k}y^{p}_{ko}=1\\ &&\sum_{i=1}^{m}v^{p}_{i}x^{p}_{io}=1\\ &&\mu^{p}_{k}\geq0,&k=1,...,s\\ &&v^{p}_{i}\geq0,&i=1,...,m\\ &&\mu_{0}&free,\\ \end{array} $$
(27)
$$\begin{array}{llllc} \overline{D}_{o}^{p}=\max&\min&\ \mu_{o}\\ \frac{\mu^{q}_{l}}{\mu^{q}_{\kappa}}\in\Big[\frac{r^{qL}_{lo}}{r^{pL}_{\kappa o}},\frac{r^{qU}_{lo}}{r^{qL}_{\kappa o}}\Big]\\ \frac{\mu^{q}_{l}}{\mu^{q}_{\kappa}}\in\Big[\frac{r^{qL}_{lo}}{r^{qL}_{\kappa o}},\frac{r^{qU}_{lo}}{r^{qL}_{\kappa o}}\Big]\\ x^{q}_{ij}\in[x^{qL}_{ij}, x^{qU}_{ij}]\\ y^{q}_{kj}\in [y^{qL}_{kj}, y^{qU}_{kj}]\\ x^{p}_{io}\in[x^{pL}_{io}, x^{pU}_{io}]\\ y^{p}_{ko}\in [y^{pL}_{ko}, y^{pU}_{ko}]\\ \end{array} $$
$$ \begin{aligned} s.t.\ \ \ \ \ \ &\sum_{k=1}^{s}\mu^{q}_{k}y^{q}_{kj}-\sum_{i=1}^{m}v^{q}_{i}x^{q}_{ij}+\mu_{o}\geq0,&j=1,...n\\ &\sum_{k=1}^{s}\mu^{q}_{k}y^{p}_{ko}=1\\ &\sum_{i=1}^{m}v^{q}_{i}x^{p}_{io}=1\\ &\mu^{q}_{k}\geq0,&k=1,...,s\\ &v^{q}_{i}\geq0,&i=1,...,m\\ &\mu_{0}&free. \end{aligned} $$
(28)
Models (27) and (28) are two-level models. Several authors have proposed methods for solving two-level programs (see, e.g., (Bialas and Karwan 1984; Vicente and Calamai 1994)). However, due to the special structure of models (27) and (28), we introduce another solution in “Computational aspects” section.
We summarize the facts in the above propositions as follows:
- 1.
If we enclose uncertain data into the profit efficiency model (3) as within-period model (5), this model measures the upper bound on the profit efficiency of DMUo in time periods t and t+1. If we enclose the same uncertain data into the CR-DEA DEA model in the form of weight restrictions as model (25), this model measures the lower bound on the profit efficiency of DMUo in time periods t and t+1.
- 2.
If we enclose uncertain data into the profit efficiency model (4) as adjacent-period model (6), this model measures the upper bound on the profit efficiency of DMUo at time t (t+1) relative to the frontier at time t+1 (t). If we enclose uncertain data into the CR-DEA DEA model in the form of weight restrictions as model (26), this model measures the lower bound on the profit efficiency of DMUo at time t (t+1) relative to the frontier at time t+1 (t).
- 3.
Profit efficiency model (5), model UPEW-S (7), and two-level model (27) yield the same profit efficiency as potential profit efficiency or the upper bound of profit efficiency in the within period. Model (5) incorporates uncertain data directly into the envelopment model. Model (7) measures the same profit efficiency for the most optimistic viewpoint. A similar argument can be put forward for models (6), (9), and (28).
- 4.
UPEW-P model (8), two-level model (12), and CR-DEA model (25) yield the same efficiency as perfect efficiency or the lower bound of the profit efficiency of DMUo at time periods t and t+1 (in the within period). Model (8) calculates the same profit efficiency of DMUo from the pessimistic viewpoint. A similar argument can be put forward for model UPEA-P (10), two-level model (14), and CR-DEA model (26) (in the adjacent period).
Definition 9
The lower and upper bounds of the overall profit MPIs are obtained as follows:
\(\overline {M}=\sqrt {\frac {\overline {\rho }^{t+1}_{t}}{\underline {\rho }^{t}_{t}}\times \frac {\overline {\rho }^{t+1}_{t+1}}{\underline {\rho }^{t}_{t+1}}}\),\(\underline {M}=\sqrt {\frac {\underline {\rho }^{t+1}_{t}}{\overline {\rho }^{t}_{t}}\times \frac {\underline {\rho }^{t+1}_{t+1}}{\overline {\rho }^{t}_{t+1}}}\), where \(\overline {\rho }^{p}_p~\left (\underline {\rho }^{p}_{p}\right), p=t, t+1\) represents the optimistic (pessimistic) efficiency in the within period and is computed by model (7)(8) and definition 3. Additionally, \(\overline {\rho }^{p}_{q} ~\left (\underline {\rho }^{p}_{q}\right), p,q=t, t+1, p\neq q\) represents the optimistic (pessimistic) efficiency in the adjacent-period time, and is computed by model (9)(10) and definition 3.
Theorem 3
Any \(\underline {M}\leq M\leq \overline {M}\) can be considered as the overall profit MPI for DMUo.
Proof
See (Emrouznejad et al. 2011). □
Emrouznejad et al. (2011) divided the overall MPI of any DMUo into six classes, as follows:
No change in productivity class. This class includes all the DMUs with constant productivity, that is, \(E_o=\{DMU_j:\underline {M}_j=\overline {M}_j=1\}\).
Fully increasing productivity class. This class includes all the DMUs with increasing productivity and observed progress under the pessimistic viewpoint, that is, \(E^{++}=\{DMU_j:1<\underline {M}_{j}\leq \overline {M}_{j}\}\).
Fully decreasing productivity class. This class includes all the DMUs with decreasing productivity and observed regress under the optimistic viewpoint, that is, \(E^{--}=\{DMU_j:\underline {M}_{j}\leq \overline {M}_j<1\}\).
Partially increasing productivity class. This class includes all the DMUs with increasing productivity under the optimistic viewpoint and no change in productivity under the pessimistic viewpoint, that is, \(E^{+}=\{DMU_j:\underline {M}_j=1, \overline {M}_j>1\}\).
Partially decreasing productivity class. This class includes all the DMUs with decreasing productivity under the pessimistic viewpoint and no change in productivity under the optimistic viewpoint, that is, \(E^{-}=\{DMU_j:\underline {M}_j<1, \overline {M}_j=1\}\).
Partially increasing–decreasing productivity class. This class includes all the DMUs with increasing productivity under the optimistic viewpoint and decreasing productivity under the pessimistic viewpoint, that is, \(E=\{DMU_j:\underline {M}_j<1<\overline {M}_{j}\}\).