Nonradial plant capacity concepts: proposals and attainability

Abstract

This contribution observes that plant capacity notions based on traditional radial efficiency measures may leave substantial amounts of slacks or unmeasured inefficiency. These unmeasured inefficiencies can result in inaccurate assessments of production capabilities, potentially leading to suboptimal operational and strategic decisions. To remedy this problem, we define new nonradial output-oriented and input-oriented plant capacity concepts based on nonradial Färe-Lovell efficiency measures. By leveraging nonradial measures, our approach captures multidimensional inefficiencies, providing a more nuanced and accurate evaluation of production performance across various input and output dimensions. Furthermore, we also explore how the introduction of nonradial attainability levels can render the attainable output-oriented plant capacity concept more flexible. This flexibility allows for the incorporation of realistic operational constraints, ensuring that capacity assessments are both practical and adaptable to diverse production environments. An empirical illustration on a secondary data set illustrates the pertinent differences between radial and nonradial plant capacity notions. Our empirical analysis demonstrates that nonradial measures offer a more detailed understanding of capacity utilization. In particular, it shows that nonradial plant capacity concepts are especially important on a nonconvex technology.

Appendices

Appendices: Supplementary material

Proofs

Proposition A.1

The maximal output capacity $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}$ has the following properties:

• (i) It belongs to the isoquant of $P^f(\varvec{x^f})$, i.e., $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P^f(\varvec{x^f}).$

• (ii) It belongs to the isoquant of $P(\varvec{x^f},+\infty )$, i.e., ${\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P(\varvec{x^f},+\infty )}$.

Proof

First, suppose $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\notin \text {Isoq }P^f(\varvec{x^f})$, then there exists $\theta \in (1,\infty )$, such that $\theta \varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in P^f(\varvec{x^f})$. As $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}=DF_o^f(\varvec{x^f},\varvec{y})\varvec{y}$ (see Definition 3.5), we obtain $\theta DF_o^f(\varvec{x^f},\varvec{y})\varvec{y}\in P^f(\varvec{x^f})$. Let $\theta ^*=\theta DF_o^f(\varvec{x^f},\varvec{y})>DF_o^f(\varvec{x^f},\varvec{y})$, $\theta ^*y\in P^f(\varvec{x^f})$ holds. As a consequence, there exists a feasible solution $\theta ^*(>DF_o^f(\varvec{x^f},\varvec{y}))$ to Program (8). Therefore, $DF_o^f(\varvec{x^f},\varvec{y})$ is not the optimal solution of Program (8), which contradicts to the the definition of $DF_o^f(\varvec{x^f},\varvec{y})$ as shown in (8). Hence, $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P^f(\varvec{x^f})$.

Second, to prove $\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P(\varvec{x^f},+\infty )$, we only need to prove $\text {Isoq }P^f(\varvec{x^f})=\text {Isoq }P(\varvec{x^f},+\infty )$ as ${\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P^f(\varvec{x^f})}$. Recall that $P(\varvec{x^f},+\infty )=\{\varvec{y}\mid (\varvec{x^f},+\infty ,\varvec{y})\in T\}$ and $T^f=\{(\varvec{x^f},\varvec{y})\mid (\varvec{x^f},\varvec{x^v},\varvec{y}) \in T\}$, $P(\varvec{x^f},+\infty )$ can be reformulated as $P(\varvec{x^f},+\infty )=\{\varvec{y}\mid (\varvec{x^f},\varvec{y})\in T^f\}$ because of free disposability in variable inputs (i.e., $\varvec{x^v}<+\infty$). Combining $P^f(\varvec{x^f})=\{\varvec{y}\mid (\varvec{x^f},\varvec{y})\in T^f\}$, we have $P(\varvec{x^f},+\infty )=P^f(\varvec{x^f})$. Consequently, $\text {Isoq }P(\varvec{x^f},+\infty )=\text {Isoq }P^f(\varvec{x^f})$. Thus, ${\varvec{y}_{o,(\varvec{x^f}, \varvec{y})}\in \text {Isoq }P(\varvec{x^f},+\infty )}$. $\square$

Proposition A.2

The minimal input capacity $\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}$ with the fixed inputs $\varvec{x^f}$ belongs to the isoquant of $L(\epsilon )$, i.e., $(\varvec{x^f},\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in \text {Isoq }L(\epsilon )$.

Proof

Suppose $(\varvec{x^f},\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\notin \text {Isoq }L(\epsilon )$, then there exists $\beta \in [0,1)$ such that $\beta (\varvec{x^f},\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in L(\epsilon )$. By the assumption of free disposability in (variable) inputs, we have $(\varvec{x^f},\beta \varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}\in L(\epsilon )$ because of $\varvec{x^f}>\beta \varvec{x^f}$. As $\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}=DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\varvec{x^v}$ (see Definition 3.6), we obtain $(\varvec{x^f},\beta DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\varvec{x^v}\in L(\epsilon ))$. Let $\beta ^*=\beta DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )<DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$,$(\varvec{x^f},\beta ^*\varvec{x^v})\in L(\epsilon )$ holds. Consequently, there exists a feasible solution $\beta ^*<DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ to Program (7). Hence, $DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ is not the optimal solution of Program (7) when $\varvec{y}=\epsilon$, which contradicts to the definition of $DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ as shown in (7). Therefore, $(\varvec{x^f},\varvec{x^v}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in \text {Isoq }L(\epsilon )$. $\square$

Proposition A.3

The optimal output capacity $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}$ has the following properties:

• (i) It pertains to the efficient subset of $P^f(\varvec{x^f})$, i.e., $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}\in \text {Eff } P^f(\varvec{x^f})$.

• (ii) It pertains to the efficient subset of $P(\varvec{x^f},+\infty )$, i.e., $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}\in \text {Eff } P(\varvec{x^f},+\infty )$.

Proof

First, suppose $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}\notin \text {Eff } P^f(\varvec{x^f})$, then there exists $\varvec{y'}\ge \varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})},\varvec{y'}\ne \varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}$, such that $\varvec{y'}\in P^f(\varvec{x^f})$. Since $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}=\theta ^*\odot \varvec{y}$ (see Definition 3.9), we obtain $\theta ^*\odot \varvec{y}+\varvec{z}\in P^f(\varvec{x^f})$ where $\varvec{z}\in {\mathbb {R}}_+^r=\varvec{y'}-\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}$. Suppose the tth element of vector $\varvec{z}$ denoted by $z_t,t=1,...,s$ is strictly greater than zero while other elements are equal to zero, then we get $(\theta _1^*y_1,...,\theta _t^*y_t+z_t,...,\theta _s^*y_s)\in P^f(\varvec{x^f})$. Let $\theta '_t=\frac{z_t}{y_t}>0$ and $\theta _t^{**}=\theta '_t+\theta _t^*>\theta _t^*$, we have $(\theta _1^*y_1,...,\theta _t^{**}y_t,...,\theta _s^*y_s)\in P^f(\varvec{x^f})$. Thus, $(\theta _1^*,...,\theta _t^{**},...,\theta _s^*)$ is a feasible solution to Program (19), the corresponding value of objective function is $\sum \limits _{r=1,r\ne t}^s\mu _r\theta _r^*+\mu _t\theta _t^{**}$. Since $\sum \limits _{r=1,r\ne t}^s\mu _r\theta _r^*+\mu _t\theta _t^{**}> \sum \limits _{r=1,r\ne t}^s\mu _r\theta _r^*+\mu _t\theta _t^{**}=\sum \limits _{r=1}^s\mu _r\theta _r^*$, $(\theta _1^*,...,\theta _t^{*},...,\theta _s^*)$ is not the optimal solution of Program (19), which contracts to the definition of $WNDF_o^f(\varvec{x^f},\varvec{y})$ as shown in (19). Hence, $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}\in \text {Eff } P^f(\varvec{x^f})$.

Second, as $P(\varvec{x^f},+\infty )=P^f(\varvec{x^f})$ (see the proof of Proposition 3.1), it is obvious that $\text {Eff }P(\varvec{x^f},+\infty )$ = Eff $P^f(\varvec{x^f})$. As a consequence, $\varvec{y}^{WN}_{o,(\varvec{x^f},\varvec{y})}\in \text {Eff } P(\varvec{x^f},+\infty )$. $\square$

Proposition A.4

The generalized framework for the biased O-O PCU measure is defined as:

$$\begin{aligned} GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=\max \{\sum _{r=1}^s\mu _r\theta _r\mid \theta \odot \varvec{y}\in P^f(\varvec{x^f}),\mu \in \Lambda ,\theta \in \Gamma \}. \end{aligned}$$

(A.1)

whereby:

• (i) $\Lambda =\Lambda ^1=\{\mu \mid \mu _1=\mu _2=\cdot \cdot \cdot =\mu _s=\frac{1}{s}\}$, $\Gamma =\Gamma ^1=\{\theta \mid \theta _1=\theta _2=\cdot \cdot \cdot =\theta _s\ge 1\}\Rightarrow GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )$=$DF_o^f(\varvec{x^f},\varvec{y})$;

• (ii) $\Lambda =\Lambda ^2=\{\mu \mid \mu _r=1,\mu _{-r}=0\}$, $\Gamma =\Gamma ^2=\{\theta \mid \theta _r\ge 1,\theta _{-r}=1\}\Rightarrow GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$;

• (iii) $\Lambda =\Lambda ^3=\{\mu \mid \sum \limits _{r=1}^s\mu _r=1, \mu _r>0,r=1,...,s\}$, $\Gamma =\Gamma ^3=\{\theta \mid \theta \ge 1\}\Rightarrow GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=WNDF_o^f(\varvec{x^f},\varvec{y})$;

• (iv) $\Lambda =\Lambda ^1=\{\mu \mid \mu _1=\mu _2=\cdot \cdot \cdot =\mu _s=\frac{1}{s}\}$, $\Gamma =\Gamma ^3=\{\theta \mid \theta \ge 1\}\Rightarrow GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=NDF_o^f(\varvec{x^f},\varvec{y})$;

Proof

First, when $\mu _1=\mu _2=\cdot \cdot \cdot =\mu _s=\frac{1}{s}$, let $\bar{\theta }=\theta _1=\theta _2=\cdot \cdot \cdot =\theta _s\ge 1$, we have $GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=\max \{\bar{\theta }\mid \bar{\theta }\varvec{y}\in P^f(\varvec{x^f}),\bar{\theta }\in [1,+\infty )\}$=$DF_o^f(\varvec{x^f},y)$.

Second, when $\mu _r=1,\mu _{-r}=0$, and $\theta _r\ge 1,\theta _{-r}=1$, we have $GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=\max \{\theta _r\mid (\theta _r y_r,\varvec{y_{-r}})\in P^f(\varvec{x^f}),\theta _r\in [1,+\infty ) \}$=$DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$.

Third, when $\sum \limits _{r=1}^s\mu _r=1, \mu _r>0,r=1,...,s$ and $\theta \ge 1$, $GDF_o^f(\varvec{x^f}, \varvec{y}\mid \Lambda ,\Gamma )=WNDF_o^f(\varvec{x^f},\varvec{y})$ by (19).

Fourth, when $\mu _1=\mu _2=\cdot \cdot \cdot =\mu _s=\frac{1}{s}$, $GDF_o^f(\varvec{x^f},\varvec{y}\mid \Lambda ,\Gamma )=NDF_o^f(\varvec{x^f},\varvec{y})$ by (16). $\square$

Proposition A.5

The following linkages can be established among biased radial O-O PCU measure, partial O-O PCU measure, and Färe-Lovell O-O PCU measure ($s\ge 1$):

$$\begin{aligned} 1\le DF_o^f(\varvec{x^f},\varvec{y})\le DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})\le NDF_o^f(\varvec{x^f},\varvec{y}),r=1,...,s. \end{aligned}$$

(A.2)

In particular,

(i) a sufficient condition for $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})< NDF_o^f(\varvec{x^f},\varvec{y}),r=1,...,s$ is that $\varvec{y}\notin \text {Eff } P^f(\varvec{x^f})$, i.e., $NDF_o^f(\varvec{x^f},\varvec{y})>1$;

(ii) a sufficient condition for $DF_o^f(\varvec{x^f},\varvec{y})= NDF_o^f(\varvec{x^f},\varvec{y})=DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$ is that output is a singleton, i.e., $s=1$.

Proof

First, $DF_o^f(\varvec{x^f},\varvec{y})\ge 1$ is satisfied by definition (see measure (8)).

Second, let $\theta ^*\ge 1$ be the optimal solution to program $\max \{\theta \mid \theta \varvec{y}\in P^f(\varvec{x^f}),\theta \in [1,+\infty )\}$, i.e., $\theta ^*=DF_o^f(\varvec{x^f},\varvec{y})$, then we have $(\theta ^*y_r,\theta ^*\varvec{y_{-r}})\in P^f(\varvec{x^f})$. By the assumption of free disposability in outputs, we obtain $(\theta ^*y_r,\varvec{y_{-r}})\in P^f(\varvec{x^f})$ due to $\varvec{y_{-r}}\le \theta ^*\varvec{y_{-r}}$. Thus, $\theta ^*$ is a feasible solution to program $\max \{\theta _r\mid (\theta _r y_r,\varvec{y_{-r}})\in P^f(\varvec{x^f}),\theta _r\in [1,+\infty )\},r=1,...,s$, thereby $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})\ge \theta ^*$ (see Definition 3.2), i.e., $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})\ge DF_o^f(\varvec{x^f},\varvec{y})$.

Third, to prove $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})\le NDF_o^f(\varvec{x^f},\varvec{y}),r=1,...,s$, we should consider the following two cases: (i) $NDF_o^f(\varvec{x^f},\varvec{y})=1$ and (ii) $NDF_o^f(\varvec{x^f},\varvec{y})>1$. In the former case, we have $\theta _1^*=...=\theta _s^*=1$ where $\theta ^*$ is the optimal solution of Program (16) whose component is $\theta _r^*,r=1,...,s$. Clearly, $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})=1$ must be satisfied according Definition 3.2, which can be proven by contradiction as follows. Suppose $\theta _r^*=1$ is not the optimal solution of Program (11), then there exists $\theta _r^{**}>1$ such that $(\theta _r^{**}y_r,\varvec{y_{-r}})\in P^f(\varvec{x^f})$. Thus, $(1_1,...,\theta _r^{**},...,1_s)$ is a feasible solution to Program (16). The corresponding value of objective function is $\frac{s-1+\theta _r^{**}}{s}>1$ because of $\theta _r^{**}>1,$ thereby $\theta _1^*=...=\theta _s^*=1$ is not the optimal of Program (16) and $NDF_o^f(\varvec{x^f},\varvec{y})\ne 1$, which contradicts to $NDF_o^f(\varvec{x^f},\varvec{y})=1$. Hence, we obtain $NDF_o^f(\varvec{x^f},\varvec{y})=DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}}),r=1,...,s$. In the latter case, we consider the following two subcases (ii-1) $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})=1$ and (ii-2) $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})>1$. In sub-case (ii-1), $NDF_o^f(\varvec{x^f},\varvec{y})>DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$ holds clearly. In sub-case (ii-2), suppose $\theta _r'$ is the optimal solution of Program (11), i.e., $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})=\theta _r'$. Then $(1_1,...,\theta _r',...,1_s)$ is a feasible solution to Program (16), from which we obtain $\frac{s-1+\theta _r'}{s}\le NDF_o^f(\varvec{x^f},\varvec{y})$. Reformulate the formula above, we get $\frac{NDF_o^f(\varvec{x^f},\varvec{y})-1}{\theta _r'-1}\ge \frac{1}{s}$. Therefore, $\frac{NDF_o^f(\varvec{x^f},\varvec{y})-1}{DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})-1}>0$, i.e., $NDF_o^f(\varvec{x^f},\varvec{y})>DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$. Wrapping up, $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})\le NDF_o^f(\varvec{x^f},\varvec{y}),r=1,...,s$.

Fourth, when $NDF_o^f(\varvec{x^f},\varvec{y})>1$, $DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})< NDF_o^f(\varvec{x^f},\varvec{y}),r=1,...,s$ have been proven above (see subcases (ii-1) and (ii-2)).

Finally, when the output is a singleton, $DF_o^f(\varvec{x^f},\varvec{y})= NDF_o^f(\varvec{x^f},\varvec{y})=DF_{o(r)}^f(\varvec{x^f},y_r,\varvec{y_{-r}})$ always holds by definition. $\square$

Proposition A.6

The optimal input capacity $\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}$ with the fixed inputs $\varvec{x^f}$ belongs to the isoquant of $L(\epsilon )$, i.e., $(\varvec{x^f},\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in \text {Isoq } L(\epsilon )$.

Proof

Suppose $(\varvec{x^f},\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\notin \text {Isoq } L(\epsilon )$, then there exists $\beta \in [0,1)$ such that $\beta (\varvec{x^f},\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in L(\epsilon )$. By the assumption of free disposability in (variable) inputs, we can induce that $(\varvec{x^f},\beta \varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}\in L(\epsilon )$ due to $\varvec{x^f}>\beta \varvec{x^f}$. Since $\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )}=\beta ^*\odot \varvec{x^v}$ (see Definition 3.12), we obtain $(\varvec{x^f},\beta \cdot \beta ^*\odot \varvec{x^v})\in L(\epsilon )$. Let $\beta ^{**}=\beta \cdot \beta ^*<\beta ^*$, then $(\varvec{x^f},\beta ^{**}\odot \varvec{x^v})\in L(\epsilon )$. As a consequence, there exists a feasible solution $\beta ^{**}<\beta ^*$ to Program (29) whose corresponding value of objective function is $\sum \limits _{i=1}^{m^v}\eta _i\beta ^{**}$. $\beta ^*$ is not the optimal solution of Program (29) because of $\sum \limits _{i=1}^{m^v}\eta _i\beta ^{**}<\sum \limits _{i=1}^{m^v}\eta _i\beta ^{*}$, contradicting to the definition of $WNDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ as shown in (29). Therefore, $(\varvec{x^f},\varvec{x}^{v,WN}_{i,(\varvec{x^f},\varvec{x^v},\epsilon )})\in \text {Isoq } L(\epsilon )$. $\square$

Proposition A.7

The generalized framework for the biased I-O PCU measure is defined as:

$$\begin{aligned} GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi ) =\min \{\sum \limits _{i=1}^{m^v}\eta _i\beta _i\mid (\varvec{x^f},\beta \odot \varvec{x^v})\in L(\epsilon ),\eta \in \Upsilon ,\beta \in \Phi \}, \end{aligned}$$

(A.3)

where

• (i) $\Upsilon =\Upsilon ^1=\{\eta \mid \eta _1=\eta _2 =\cdot \cdot \cdot =\eta _{m^v}=\frac{1}{m^v}\},\Phi =\Phi ^1=\{\beta \mid \ 0\le \beta _1=\beta _2 =\cdot \cdot \cdot =\beta _{m^v}\le 1\}\Rightarrow$ $GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi ) =DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$;

• (ii) $\Upsilon =\Upsilon ^2=\{\eta \mid \sum \limits _{i=1}^{m^v}\eta _i=1,\eta _i>0,i=1,...,m^v\}, \Phi =\Phi ^2=\{\beta \mid 0\le \beta \le 1\}\Rightarrow GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi ) =WNDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$.

• (iii) $\Upsilon =\Upsilon ^1=\{\eta \mid \eta _1=\eta _2=\cdot \cdot \cdot =\eta _{m^v}=\frac{1}{m^v}\},\Phi =\Phi ^2=\{\beta \mid 0\le \beta \le 1\}\Rightarrow GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi ) =NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$.

Proof

First, when $\eta _1=\eta _2=\cdot \cdot \cdot =\eta _{m^v}=\frac{1}{m^v}$, let $\bar{\beta }=\beta _1=\beta _2=\cdot \cdot \cdot =\beta _{m^v}$, we have $GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi )=\min \{\bar{\beta }\mid (\varvec{x^f},\bar{\beta }\varvec{x^v})\in L(\epsilon ),\bar{\beta }\in [0,1]\}=DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$.

Second, when $\sum \limits _{i=1}^{m^v}\eta _i=1,\eta _i>0,i=1,...,m^v$ and $0\le \beta \le 1$, $GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi )=WNDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ holds by (29).

Third, when $\eta _1=\eta _2=\cdot \cdot \cdot =\eta _{m^v}=\frac{1}{m^v}$, $GDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon \mid \Upsilon ,\Phi )=NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ holds by (25).

Proposition A.8

The following linkage can be established between the biased (sub-vector) radial I-O PCU measure and the biased Färe-Lovell I-O PCU measure ($m^v\ge 1$):

$$\begin{aligned} NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\le DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\le 1. \end{aligned}$$

(A.4)

In particular, a sufficient condition for $NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )=DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ is that the variable input is a singleton, i.e., $m^v=1$.

Proof

First, $DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\le 1$ is satisfied by setting $\varvec{y}=\epsilon$ in measure (7).

Second, let $\beta ^*\in [0,1]$ be the optimal solution of program $\min \{\beta \mid (\varvec{x^f},\beta \varvec{x^v})\in L(\epsilon ), \beta \in [0,1]\},$ i.e., $\beta ^*=DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$, then we get $\beta _1=\beta _2=\cdot \cdot \cdot =\beta _{m^v}=\beta ^*$ is a feasible solution to program $\min \{\frac{1}{m^v}\sum \limits _{i=1}^{m^v}\beta _i\mid (\varvec{x^f},\beta \odot \varvec{x^v})\in L(\epsilon ),\beta _i\in [0,1]\}$. The corresponding value of objective function is $\beta ^*$. Hence, $\beta ^*\ge NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$ (see measure (25)), i.e., $NDF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )\le DF_i^{SR}(\varvec{x^f},\varvec{x^v},\epsilon )$.

Proposition A.9

There exists a set $\Delta$, such that for any $\hat{\xi }\in \Delta$, we have:

• (i) $\forall \xi \ge \hat{\xi },$ we have $ANPCU_o(\varvec{x},\varvec{x^f},\varvec{y},\xi )=NPCU_o(\varvec{x},\varvec{x^f},\varvec{y})$.

• (ii) $\forall \xi \le \hat{\xi }$ and $\xi \ne \hat{\xi },$ we have $ANPCU_o(\varvec{x},\varvec{x^f},\varvec{y},\xi )>NPCU_o(\varvec{x},\varvec{x^f},\varvec{y})$.

Proof

$ANDF_o^f(\varvec{x^f},\varvec{y},\hat{\xi })$ is non-decreasing with $\hat{\xi }$ because the feasible region of Model (B.9) is enlarged as $\hat{\xi }$ increases. Combined with $ANDF_o^f(\varvec{x^f},\varvec{y},\hat{\xi })\le NDF_o^f(\varvec{x^f},\varvec{y})$, we clearly obtain that there exists a set $\Delta$, such that for any $\hat{\xi }\in \Delta$, we have (i) $\forall \xi \ge \hat{\xi },$ $ANDF_o(\varvec{x^f},\varvec{y},\xi )=NDF_o(\varvec{x^f},\varvec{y})$; and (ii) $\forall \xi \le \hat{\xi }$ and $\xi \ne \hat{\xi },$ $ANDF_o(\varvec{x^f},\varvec{y},\xi )<NDF_o(\varvec{x^f},\varvec{y})$. By the definitions of $NPCU_o(\varvec{x},\varvec{x^f},\varvec{y})$ and $ANPCU_o(\varvec{x},\varvec{x^f},\varvec{y},\bar{\xi })$ (see Definitions 3.7 and 3.15), we can induce that for any $\hat{\xi }\in \Delta$, we have (i)$\forall \xi \ge \hat{\xi },$ $ANPCU_o(\varvec{x},\varvec{x^f},\varvec{y},\xi )=NPCU_o(\varvec{x},\varvec{x^f},\varvec{y})$; and (ii) $\forall \xi \le \hat{\xi }$ and $\xi \ne \hat{\xi },$ $ANPCU_o(\varvec{x},\varvec{x^f},\varvec{y},\xi )>NPCU_o(\varvec{x},\varvec{x^f},\varvec{y})$.

Computing nonradial plant capacity notions

This appendix presents the estimation of various capacity concepts’ components within a non-parametric frontier framework, assuming VRS. To formulate the models, we will first review the notations introduced in this contribution. The vector of m inputs, denoted as $\varvec{x} \in {\mathbb {R}}^m_+$, is capable of generating a vector of s outputs, denoted as $\varvec{y} \in {\mathbb {R}}_+^s$. The input vector $\varvec{x}$ can be divided into two parts: a fixed component ($\varvec{x^f}$) and a variable component ($\varvec{x^v}$), represented as $\varvec{x} = (\varvec{x^f}, \varvec{x^v})$. For each observed production unit k under evaluation, and the corresponding output vector ($\varvec{y_k}$) are known. The fixed and variable input components for the unit are denoted as $\varvec{x^f_{k}}$ and $\varvec{x^v_{k}}$, respectively. Lastly, since non-parametric frontier technologies are based on activity analysis, we require a vector of activity variables, $\lambda = (\lambda _1, \dots , \lambda _n)$, which indicates the intensity levels at which each of the n observed activities is conducted.

Note that to save space, we only present linear programs of efficiency and plant capacity measures under C. The efficiency and plant capacity measures under NC can be computed by adding the binary integer constraint $\lambda _j\in \{0,1\}$ to the linear programs under C.

Using nonparametric frontier technologies, one can obtain the weighted Färe-Lovell output efficiency measure relative to production correspondence $P(\varvec{x})$ for an evaluated observation ($\varvec{x_k},\varvec{y_k}),k=1,...,n$ as

$$\begin{aligned} \begin{array}{llll} WNDF_o(\varvec{x_k},\varvec{y_k})=& \max \limits _{\theta _r,\lambda _j}& \sum \limits _{r=1}^s\mu _r\theta _r \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}\le x_{ik},i=1,...,m,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \theta _r y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & \theta _r\ge 1,\lambda _j\ge 0,r=1,...,s,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.1)

By analogy, the weighted Färe-Lovell output efficiency measure relative to production correspondence $P^f(\varvec{x^f})$ observation ($\varvec{x_k},\varvec{y_k}),k=1,...,n$ is computed as

$$\begin{aligned} \begin{array}{llll} WNDF_o^f(\varvec{x_k^f},\varvec{y_k})=& \max \limits _{\theta _r,\lambda _j,x_{k}^v}& \sum \limits _{r=1}^s\mu _r\theta _r \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le x_{k}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \theta _r y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & x_{k}^v\ge 0,\theta _r\ge 1,\lambda _j\ge 0,r=1,...,s,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.2)

The Färe-Lovell output efficiency measures relative to production correspondence $P(\varvec{x})$ for observation ($\varvec{x_k},\varvec{y_k}),k=1,...,n$ as

$$\begin{aligned} \begin{array}{llll} NDF_o(\varvec{x_k},\varvec{y_k})=& \max \limits _{\theta _r,\lambda _j}& \frac{1}{s}\sum \limits _{r=1}^s\theta _r \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}\le x_{ik},i=1,...,m,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \theta _r y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & \theta _r\ge 1,\lambda _j\ge 0,r=1,...,s,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.3)

By analogy, the Färe-Lovell output efficiency measure relative to production correspondence $P^f(\varvec{x^f})$ observation ($\varvec{x_k},\varvec{y_k}),k=1,...,n$ is computed as

$$\begin{aligned} \begin{array}{llll} NDF_o^f(\varvec{x_k^f},\varvec{y_k})=& \max \limits _{\theta _r,\lambda _j,x_{k}^v}& \frac{1}{s}\sum \limits _{r=1}^s\theta _r \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le x_{k}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \theta _r y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & x_{k}^v\ge 0,\theta _r\ge 1,\lambda _j\ge 0,r=1,...,s,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.4)

We can obtain the sub-vector weighted Färe-Lovell input efficiency measure relative to input correspondence $L(\varvec{y})$ for observation $(\varvec{x_k},\varvec{y_k})$ as:

$$\begin{aligned} \begin{array}{llll} WNDF_i^{SR}(\varvec{x_k^f},\varvec{x_k^v},\varvec{y_k})=& \min \limits _{\beta _i,\lambda _j}& \sum \limits _{i=1}^{m^v}\eta _i\beta _i \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le \beta _i x_{ik}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & 0\le \beta _i \le 1,\lambda _j\ge 0,i=1,...,m^v,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.5)

By analogy, the sub-vector weighted Färe-Lovell input efficiency measure relative to input correspondence $L(\epsilon )$ is computed as:

$$\begin{aligned} \begin{array}{llll} WNDF_i^{SR}(\varvec{x_k^f},\varvec{x_k^v},\epsilon )=& \min \limits _{\beta _i,\lambda _j}& \sum \limits _{i=1}^{m^v}\eta _i\beta _i \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le \beta _i x_{ik}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \epsilon ,r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & 0\le \beta _i \le 1,\lambda _j\ge 0,i=1,...,m^v,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.6)

One can obtain the sub-vector Färe-Lovell input efficiency measure relative to input set $L(\varvec{y})$ for observation $(\varvec{x_k},\varvec{y_k})$ as:

$$\begin{aligned} \begin{array}{llll} NDF_i^{SR}(\varvec{x^f_k},\varvec{x^v_k},\varvec{y_k})=& \min \limits _{\beta _i,\lambda _j}& \frac{1}{m^v}\sum \limits _{i=1}^{m^v}\beta _i \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le \beta _i x_{ik}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge y_{rk},r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & 0\le \beta _i \le 1,\lambda _j\ge 0,i=1,...,m^v,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.7)

By analogy, the biased Färe-Lovell I-O PCU measure for observation $(\varvec{x_k},\varvec{y_k})$ is computed as:

$$\begin{aligned} \begin{array}{llll} NDF_i^{SR}(\varvec{x^f_k},\varvec{x^v_k},\epsilon )=& \min \limits _{\beta _i,\lambda _j}& \frac{1}{m^v}\sum \limits _{i=1}^{m^v}\beta _i \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le \beta _i x_{ik}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \epsilon ,r=1,...,s,\\ & & \sum \limits _{j=1}^n\lambda _j=1\\ & & 0\le \beta _i \le 1,\lambda _j\ge 0,i=1,...,m^v,\\ \end{array} \end{aligned}$$

(B.8)

We can model the attainable Färe-Lovell output efficiency measure at level $\bar{\xi }$ as

$$\begin{aligned} \begin{array}{llll} ANDF_o^f(\varvec{x_k^f},\varvec{y_k},\bar{\xi })=& \max \limits _{\theta _r,\lambda _j}& \frac{1}{s}\sum \limits _{r=1}^s\theta _r \\ & s.t& \sum \limits _{j=1}^n\lambda _j x_{ij}^f\le x_{ik}^f,i=1,...,m^f,\\ & & \sum \limits _{j=1}^n\lambda _j x_{ij}^v\le x_{i}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j y_{rj}\ge \theta _r y_{rk},r=1,...,s,\\ & & x_{i}^v\le \bar{\xi }_i x_{ik}^v,i=1,...,m^v,\\ & & \sum \limits _{j=1}^n\lambda _j=1,\\ & & \theta _r\ge 0,\lambda _j\ge 0,r=1,...,s,j=1,...,n.\\ \end{array} \end{aligned}$$

(B.9)

The constraint $x_{i}^v\le \bar{\xi }_i x_{ik}^v,i=1,...,m^v$ establishes a link between the observed ith variable input and the decision variable $x_{i}^v$ via $\bar{\xi }_i$.