Families of linear efficiency programs based on Debreu’s loss function

Abstract

Gerard Debreu introduced a well known radial efficiency measure which he called a “coefficient of resource utilization.” He derived this scalar from a much less well known “dead loss” function that characterizes the monetary value sacrificed to inefficiency, and which is to be minimized subject to a normalization condition. We use Debreu’s loss function, together with a variety of normalization conditions, to generate several popular families of linear efficiency programs. Our methodology also can be employed to generate entirely new families of linear efficiency programs.

Appendix

Proof of Proposition 1

Let $\left( {c^{*} ,p^{*} ,\alpha^{*} } \right)$ be an optimal solution of program A3. Then, since $\left( {c^{*} ,p^{*} ,\alpha^{*} } \right) \in SH\left( T \right)$, there exists a vector $\left( {x^{*} ,y^{*} } \right) \in \partial^{W} \left( T \right)$ such that $\sum\nolimits_{r = 1}^{s} {p_{r}^{*} y_{r}^{*} } - \sum\nolimits_{i = 1}^{m} {c_{i}^{*} x_{i}^{*} } = \alpha^{*}$ and $\sum\nolimits_{r = 1}^{s} {p_{r}^{*} y_{r}^{*} } - \sum\nolimits_{i = 1}^{m} {c_{i}^{*} x_{i}^{*} } \ge \sum\nolimits_{r = 1}^{s} {p_{r}^{*} v_{r} } - \sum\nolimits_{i = 1}^{m} {c_{i}^{*} u_{i} }$, $\forall \left( {u,v} \right) \in T$. Hence, by definition, $\left( {c^{*} ,p^{*} } \right) \in Q\left( {x^{*} ,y^{*} } \right)$. Now, we observe that $\left( {x^{*} ,y^{*} ;c\left( {x^{*} ,y^{*} } \right),p\left( {x^{*} ,y^{*} } \right)} \right)$, with $c\left( {x^{*} ,y^{*} } \right) = c^{*}$ and $p\left( {x^{*} ,y^{*} } \right) = p^{*}$, is a feasible solution of A2. Finally, it is easy to prove that $\left( {x^{*} ,y^{*} ;c\left( {x^{*} ,y^{*} } \right),p\left( {x^{*} ,y^{*} } \right)} \right)$ is also an optimal solution of A2 and, in fact, program A2 has the same optimal value as program A3. □

Proof of Proposition 2

It is apparent from the structure of program A3 and the fact that if $\sum\nolimits_{r = 1}^{s} {p_{r} y_{rj} } - \sum\nolimits_{i = 1}^{m} {c_{i} x_{ij} } - \alpha \le 0$, $\forall j = 1, \ldots ,n$, then $\sum\nolimits_{r = 1}^{s} {p_{r} v_{r} } - \sum\nolimits_{i = 1}^{m} {c_{i} u_{i} } - \alpha \le 0$, $\forall \left( {u,v} \right) \in T$. Also, it is easy to prove that if $\left( {c^{*} ,p^{*} ,\alpha^{*} } \right)$ is an optimal solution of program A4, then $\left( {c^{*} ,p^{*} ,\alpha^{*} } \right) \in SH\left( T \right)$. □

We now prove Programs 1, 5 and 7. Proofs of the remaining programs are trivial.

Program 1. The BCC input-oriented program

Consider the linear loss function program A4 with linear normalization condition LNC1. As a consequence of LNC1 the objective function of Program 1 is equivalent to $1 + \min \left\{ { - \sum\nolimits_{r = 1}^{s} {p_{r} y_{r0} } + \alpha } \right\}$ and to $1 - \max \left\{ {\sum\nolimits_{r = 1}^{s} {p_{r} y_{r0} } - \alpha } \right\}$, which yields

$$ 1 - L\left( {x_{0} ,y_{0} ;{\text{NC}}1} \right) = \begin{aligned}{*{20}c} {\max_{c,p,\alpha } } \hfill & {\sum\limits_{r = 1}^{s} {p_{r} y_{r0} } - \alpha } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\sum\limits_{r = 1}^{s} {p_{r} y_{rj} } - \sum\limits_{i = 1}^{m} {c_{i} x_{ij} } - \alpha \le 0,\quad \forall j} \hfill \\ {} \hfill & {c \ge 0_{m} ,p \ge 0_{s} } \hfill \\ {} \hfill & {\sum\limits_{i = 1}^{m} {c_{i} x_{i0} } = 1\quad (LNC1)} \hfill \\ \end{aligned} $$

This program is exactly the multiplier form of the BCC input-oriented program. Being linear duals, the optimal value of the envelopment form equals the optimal value of the multiplier form. Therefore $1 - L\left( {x_{0} ,y_{0} ;{\text{LNC}}1} \right) = \theta^{*}$.

Program 5. The input-oriented Russell program

This program assumes that $x_{0} > 0_{m}$. By means of the change of variables $\theta_{i} = {\frac{{x_{i0} - s_{i0}^{ - } }}{{x_{i0} }}} = 1 - {\frac{{s_{i0}^{ - } }}{{x_{i0} }}}$, $i = 1, \ldots ,m$, we get that Program 5 is equivalent to

$$ \begin{aligned}{*{20}c} {1 - \max_{{\lambda ,s_{0}^{ - } }} } \hfill & {\sum\limits_{i = 1}^{m} {{\frac{{s_{i0}^{ - } }}{{mx_{i0} }}}} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} } = x_{i0} - s_{i0}^{ - } ,\quad \forall i} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} } \ge y_{r0} ,\quad \forall r} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } = 1} \hfill \\ {} \hfill & {\lambda \ge 0_{n} ,s_{0}^{ - } \ge 0_{m} } \hfill \\ \end{aligned} $$

In words, the input-oriented Russell program is equivalent to 1 minus a weighted additive program with weights $w_{i}^{ - } = {1 \mathord{\left/ {\vphantom {1 {mx_{i0} }}} \right. \kern-\nulldelimiterspace} {mx_{i0} }}$, $i = 1, \ldots ,m$, and $w_{r}^{ + } = 0$, $r = 1, \ldots ,s$. Finally, thanks to Program 4, we have that $\left\{ {c_{i} \ge {1 \mathord{\left/ {\vphantom {1 {mx_{i0} ,i = 1, \ldots ,m}}} \right. \kern-\nulldelimiterspace} {mx_{i0} ,i = 1, \ldots ,m}}} \right\}$ are the normalization conditions for Program 5 and, at optimum, $1 - L\left( {x_{0} ,y_{0} ;{\text{LNC}}5} \right) = \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\theta_{i}^{*} }$.

Program 7. The enhanced Russell graph program

Consider the linear loss function program A4 with linear normalization condition LNC7. This program assumes that $x_{0} > 0_{m}$ and $y_{0} > 0_{s}$. This program is equivalent (at its optimal solutions) to another program with the same constraints and objective function $\max \left\{ {1 - \left( { - \sum\nolimits_{r = 1}^{s} {p_{r} y_{r0} } + \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } + \alpha } \right)} \right\}$. Performing the change of variables $\omega = 1 - \left( { - \sum\nolimits_{r = 1}^{s} {p_{r} y_{r0} } + \sum\nolimits_{i = 1}^{m} {c_{i} x_{i0} } + \alpha } \right)$ leads to the following equivalent reformulation (at its optimal solutions).

$$ \begin{aligned}{*{20}c} {\max_{c,p,\alpha ,\omega } } \hfill & \omega \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\omega = 1 - \sum\limits_{i = 1}^{m} {c_{i} x_{i0} } + \sum\limits_{r = 1}^{s} {p_{r} y_{r0} } - \alpha } \hfill \\ {} \hfill & { - \sum\limits_{i = 1}^{m} {c_{i} x_{ij} } + \sum\limits_{r = 1}^{s} {p_{r} y_{rj} } - \alpha \le 0,\quad \forall j} \hfill \\ {} \hfill & { - c_{i} \le - {\frac{1}{{mx_{i0} }}},\quad \forall i} \hfill \\ {} \hfill & {{\frac{\omega }{{sy_{r0} }}} - p_{r} \le 0,\quad \forall r} \hfill \\ {} \hfill & {c \ge 0_{m} ,p \ge 0_{s} } \hfill \\ \end{aligned} $$

The first added restriction is just the definition of ω. The final set of restrictions has been reordered so as to have all the variables on the same side. The linear dual of the reformulated program is

$$ \begin{aligned}{*{20}c} {\min_{{\beta ,\mu ,t_{0}^{ - } ,t_{0}^{ + } }} } \hfill & {\beta - \frac{1}{m}\sum\limits_{i = 1}^{m} {{\frac{{t_{i0}^{ - } }}{{x_{i0} }}}} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\beta + \frac{1}{s}\sum\limits_{r = 1}^{s} {{\frac{{t_{r0}^{ + } }}{{y_{r0} }}}} = 1} \hfill \\ {} \hfill & {\beta x_{i0} - \sum\limits_{j = 1}^{n} {\mu_{j} x_{ij} } - t_{i0}^{ - } \ge 0,\quad \forall i} \hfill \\ {} \hfill & { - \beta y_{r0} + \sum\limits_{j = 1}^{n} {\mu_{j} y_{rj} } - t_{r0}^{ + } \ge 0,\quad \forall r} \hfill \\ {} \hfill & {\beta - \sum\limits_{j = 1}^{n} {\mu_{j} } = 0,} \hfill \\ {} \hfill & {\mu \ge 0_{n} ,t_{0}^{ - } \ge 0_{m} ,t_{0}^{ + } \ge 0_{s} } \hfill \\ \end{aligned} . $$

Making a second change of variables $t_{i0}^{ - } = \beta s_{i0}^{ - }$, $i = 1, \ldots ,m$, $t_{r0}^{ + } = \beta s_{r0}^{ + }$, $r = 1, \ldots ,s$, $\mu_{j} = \beta \lambda_{j}$, $j = 1, \ldots ,n$, generates

$$ \begin{aligned}{*{20}c} {\min_{{\beta ,\lambda ,s_{0}^{ - } ,s_{0}^{ + } }} } \hfill & {\beta \left( {1 - \frac{1}{m}\sum\limits_{i = 1}^{m} {{\frac{{s_{i0}^{ - } }}{{x_{i0} }}}} } \right)} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\beta \left( {1 + \frac{1}{s}\sum\limits_{r = 1}^{s} {{\frac{{s_{r0}^{ + } }}{{y_{r0} }}}} } \right) = 1} \hfill \\ {} \hfill & {\beta \left( {x_{i0} - \sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} } - s_{i0}^{ - } } \right) \ge 0,\quad \forall i} \hfill \\ {} \hfill & { - \beta \left( {y_{r0} - \sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} } + s_{r0}^{ + } } \right) \ge 0,\quad \forall r} \hfill \\ {} \hfill & {\beta \left( {1 - \sum\limits_{j = 1}^{n} {\lambda_{j} } } \right) = 0,} \hfill \\ {} \hfill & {\lambda \ge 0_{n} ,\,s_{0}^{ - } \ge 0_{m} ,\,s_{0}^{ + } \ge 0_{s} } \hfill \\ \end{aligned} . $$

The first restriction tells us two things. First that $\beta = \left( {1 + \frac{1}{s}\sum\nolimits_{r = 1}^{s} {{\frac{{s_{r0}^{ + } }}{{y_{r0} }}}} } \right)^{ - 1} > 0$, which means that the objective function can be rewritten as shown below, and second, as a consequence, that all restrictions but the first can be simplified by deleting β. Therefore this nonlinear program can be rewritten as

$$ \begin{aligned}{*{20}c} {\min_{{\lambda ,s_{0}^{ - } ,s_{0}^{ + } }} } \hfill & {{{\left( {1 - \frac{1}{m}\sum\limits_{i = 1}^{m} {{\frac{{s_{i0}^{ - } }}{{x_{i0} }}}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \frac{1}{m}\sum\limits_{i = 1}^{m} {{\frac{{s_{i0}^{ - } }}{{x_{i0} }}}} } \right)} {\left( {1 + \frac{1}{s}\sum\limits_{r = 1}^{s} {{\frac{{s_{r0}^{ + } }}{{y_{r0} }}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \frac{1}{s}\sum\limits_{r = 1}^{s} {{\frac{{s_{r0}^{ + } }}{{y_{r0} }}}} } \right)}}} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {x_{i0} - \sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} } - s_{i0}^{ - } \ge 0,\quad \forall i} \hfill \\ {} \hfill & {y_{r0} - \sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} } + s_{r0}^{ + } \le 0,\quad \forall r} \hfill \\ {} \hfill & {1 - \sum\limits_{j = 1}^{n} {\lambda_{j} } = 0,} \hfill \\ {} \hfill & {\lambda \ge 0_{n} ,s_{0}^{ - } \ge 0_{m} ,s_{0}^{ + } \ge 0_{s} } \hfill \\ \end{aligned} . $$

The restrictions are exactly the restrictions of the additive program. The first two sets of restrictions can be equivalently written as equalities. Therefore, if we perform a third change of variables $\theta_{i} = 1 - {\frac{{s_{i0}^{ - } }}{{x_{i0} }}},i = 1, \ldots ,m,\phi_{r} = 1 + {\frac{{s_{r0}^{ + } }}{{y_{r0} }}},r = 1, \ldots ,s$, we finally get

$$ \begin{aligned}{*{20}c} {\min_{\lambda ,\theta ,\phi } } \hfill & {{{\frac{1}{m}\sum\limits_{i = 1}^{m} {\theta_{i} } } \mathord{\left/ {\vphantom {{\frac{1}{m}\sum\limits_{i = 1}^{m} {\theta_{i} } } {\frac{1}{s}\sum\limits_{r = 1}^{s} {\phi_{r} } }}} \right. \kern-\nulldelimiterspace} {\frac{1}{s}\sum\limits_{r = 1}^{s} {\phi_{r} } }}} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} } \le \theta_{i} x_{i0} ,\quad \forall i} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} } \ge \phi_{r} y_{r0} ,\quad \forall r} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } = 1} \hfill \\ {} \hfill & {\lambda \ge 0_{n} ,\theta \le 1_{m} ,\phi \ge 1_{s} } \hfill \\ \end{aligned} $$

which is, exactly, the enhanced Russell graph program of Pastor et al. (1999), also known as the SBM (Slacks-Based Measure) (Tone 2001).

Finally, considering all the above steps, we have at optimum

$$ 1 - L\left( {x_{0} ,y_{0} ;{\text{LNC}}7} \right) = {{\frac{1}{m}\sum\limits_{i = 1}^{m} {\theta_{i}^{*} } } \mathord{\left/ {\vphantom {{\frac{1}{m}\sum\limits_{i = 1}^{m} {\theta_{i}^{*} } } {\frac{1}{s}\sum\limits_{r = 1}^{s} {\phi_{r}^{*} } }}} \right. \kern-\nulldelimiterspace} {\frac{1}{s}\sum\limits_{r = 1}^{s} {\phi_{r}^{*} } }} $$