Effective production: measuring of the sales effect using data envelopment analysis

Abstract

Sales fluctuations lead to variations in the output levels affecting technical efficiency measures of operations when units sold are used at an output measure. The present study uses the concept of “effective production” and “effectiveness” to account for the effect of sales on operational performance measurements in a production system. The effectiveness measure complements the efficiency measure which does not account for the sales effect. The Malmquist productivity index is used to measure the sales effects characterized as the difference between the production function associated with efficiency and the sales-truncated production function associated with effectiveness. The proposed profit effectiveness is distinct from profit efficiency in that it accounts for sales. An empirical study of US airlines demonstrates the proposed method which describes the strategic position of a firm and a productivity-change analysis. These results demonstrate the concept of effectiveness and quantifies the effect of using sales as output.

Appendices

Appendix 1: Cost function estimation

The sign-constrained convex nonparametric least squares (CNLS) technique is used to estimate the cost function and marginal cost. CNLS can be traced to the seminal work of Hildreth (1954) and was popularized by Kuosmanen (2008) as a powerful tool for describing the average behavior of observations. CNLS avoids strong prior assumptions regarding function form while maintaining the standard regularity conditions from microeconomic theory for production functions, namely continuity, monotonicity, and concavity. Kuosmanen and Johnson (2010) demonstrated that inefficiency estimated by the sign-constrained CNLS is equivalent to that estimated by DEA. This study imposes the axioms of monotonicity and convexity on cost function and estimates it by sing-constrained CNLS to obtain marginal cost estimates (Kuosmanen 2012).

Let $C_k$ be the total cost equal to fuel expenses plus salaries and benefits expenses of firm k. $\varepsilon _k$ be the inefficiency term of firm k. Let index h be an alias of index k, $\alpha _k$ be the intercept coefficient, and $\beta _{ kj}$ be the slope coefficient of the $j\hbox {th}$ output of $k\hbox {th}$ firm. In particular, $\beta _{ kj}$ is the coefficients of the tangent hyperplanes to the piece-wise linear cost frontier which can be interpreted as the marginal cost of outputs. We obtain the marginal cost estimate $\beta _{ kj}$ of firm k by solving the following sign-constrained CNLS.

$$\begin{aligned} \begin{array}{l} \min \sum \nolimits _{k} \varepsilon _{k}^{2} \\ \hbox {s.t}.\ln C_{k} =\ln \left( {\alpha _k + \sum \nolimits _{j} \beta _{ kj} Y_{ kj} } \right) +\varepsilon _{k}, \forall k \\ \qquad \alpha _k + \sum \nolimits _{j} \beta _{ kj} Y_{ kj} \ge \alpha _h + \sum \nolimits _j \beta _{ hj} Y_{ kj}, \forall k, \forall h \\ \qquad \beta _{ kj} \ge 0, \forall j,k \\ \qquad \varepsilon _k \ge 0, \forall k \end{array} \end{aligned}$$

(5)

Next, the marginal price for passenger-miles is a fixed mark-up of marginal cost by operating margin of all firms (i.e., the industry average). Operating margin data is available from Airlinefinancials.com (2014).

Appendix 2: Dataset

See Table 6.

Table 6 Dataset in the U.S. airlines

Appendix 3: Productivity analysis with marginal price

See Tables 7 and 8.

Table 7 Productivity-level analysis of US airlines firms

Table 8 Productivity-change analysis of US airlines firms

Appendix 4: Productivity analysis with average price/perfect competition assumption

The average price for APO and RPS is calculated by total passenger revenue over scheduled and nonscheduled passenger-miles. Note that, under the perfection competition assumption, RPS is exogenous and no airlines have market power to change the price since a significant time delay by passing information between marketing department and operations department.

The results are shown in Tables 9 and 10. The price change only affects the profit efficiency/effectiveness and allocative efficiency/effectiveness. In general, the result is consistent with the one shown in Sect. 6 under imperfect competition; however, the difference between efficiency and effectiveness is diminished. For example, in productivity level analysis, we claim that the profit efficiency is larger than the profit effectiveness in industry level in 2008: $\textit{PE}=0.73$ and $\textit{PE}^{E}=0.35$ in Sect. 6.2; however, $\textit{PE}=0.77$ and $\textit{PE}^{E}=0.60$ in this “Appendix”. Similar conclusions hold for change in profit efficiency and change in profit effectiveness. Take the economic crisis between 2007 and 2008 as an example, Sect. 6.3 showed $\textit{CPE}=0.85$ and $\textit{CPE}^{E}=0.59$; however, $\textit{CPE}=1.03$ and $\textit{CPE}^{E}=0.87$ in this “Appendix”. The perfection competition case here also validated the effectiveness which complements the efficiency since $\textit{CPE}^{E}=0.87$ justified the 2008 economic crisis rather than the progress by $\textit{CPE}=1.03$.

Table 9 Productivity-level analysis of US airlines firms

Table 10 Productivity-change analysis of US airlines firms