Closest target setting with minimum improvement costs considering demand and resource mismatches

Abstract

Data envelopment analysis models can be used to measure efficiency performance and yield an improvement target for the evaluated decision-making units. However, such models have not considered market factors. Demand fulfillment and resource management also matter in production. Sales losses due to insufficient stock or inventory holding costs happen when the production output of a factory is lower or higher than the market demand. Moreover, the cost of purchasing to cover shortages of a necessary resource or disposing of a surplus resource happens when the resource amount is lower or higher than the level required for production. In this study, we adopt the definition of penalized output used to quantify the mismatch between demand level and actual output, and we propose the concept of penalized input to deal further with mismatches between owned resources and actual input. We then develop an extended closest target setting model with both penalized input and penalized output to find a projection on the demand-truncated frontier with minimum improvement costs. Finally, two simple numerical examples are used to demonstrate the applicability and practicality of the proposed approach.

Appendices

Appendix A

Charnes et al. (1985) proposed the additive model (ADD model), a third DEA model, to actually deal with least-norm projections to the frontier.

$$\begin{aligned} & {\text{max}}\quad { }\mathop \sum \limits_{i = 1}^{m} s_{ik}^ - + \mathop \sum \limits_{r = 1}^{s} s_{rk}^{ + } \\ & {\text{s}}.{\text{t}}.\quad { }\mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{ij} + s_{ik}^ - = x_{ik} \quad i = 1, \ldots ,m \\ & \qquad \mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} - s_{rk}^{ + } = y_{rk} \quad r = 1, \ldots ,s \\ & \qquad \mathop \sum \limits_{j = 1}^{n} \lambda_{j} = 1,\lambda_{j} \ge 0,\quad j = 1, \ldots ,n \\ & \qquad s_{ik}^ - \ge 0\quad i = 1, \ldots ,m \\ & \qquad s_{rk}^{ + } \ge 0\quad r = 1, \ldots ,s \\ \end{aligned}$$

The ADD model aims to maximize the sum of slacks of inputs and outputs thus to find a target on the frontier with the greatest improvement in both inputs and outputs. Letting $\left({\lambda }_{j}^{*},{s}_{i}^{-*},{s}_{r}^{+*}\right)$ be the optimal solution of above model when ${\text{DMU}}_{k}$ is evaluated, then we have:

Definition A1:

${\text{DMU}}_{k}$ is Pareto–Koopmans efficient if and only if ${s}_{ik}^{-*}=0$ and ${s}_{rk}^{+*}=0$ for $i=1,\dots ,m$ and $r=1,\dots ,s$.

Appendix B: Closest target setting in FDH technologies

Based on mADD model, which is formulated by using the non-convexity axiom, model (13) in our study is expressed as output-oriented formulation. In many practical situations, however, it is desirable to use measures of efficiency that are non-oriented and non-radial in character (Silva et al., 2003; Kerstens and Van de Woestyne, 2021). So here, we develop a second non-oriented and non-radial model to find closest target on demand-truncated frontier in Free Disposal Hull (FDH) technologies.

We add an additional constraint ${\lambda }_{j}=\{\text{0,1}\}$ to exhibit non-convexity and variable returns to scale (VRS). The objective function is still to minimize the overall improvement costs to demand-truncated frontier. And penalized inputs and penalized outputs of evaluated DMU are still used to incorporate supply and demand information.

$$\begin{aligned} & \min \quad z = \mathop \sum \limits_{i = 1}^{m} c_{i}^{I} s_{ik}^ - + \mathop \sum \limits_{r = 1}^{s} c_{r}^{O} s_{rk}^{ + } \\ & {\text{s}}{\text{.t}}{.}\quad \mathop \sum \limits_{j \in E} \lambda_{j} x_{ij} = x_{ik}^{P} - s_{ik}^ - \quad i = 1, \ldots ,m \\ & \qquad \mathop \sum \limits_{j \in E} \lambda_{j} y_{rj} = y_{rk}^{P} + s_{rk}^{ + } \le D_{rk} \quad r = 1, \ldots ,s \\ & \qquad \mathop \sum \limits_{j \in E} \lambda_{j} = 1 \\ & \qquad - \mathop \sum \limits_{i = 1}^{m} v_{i} x_{ij} + \mathop \sum \limits_{r = 1}^{s} u_{r} y_{rj} + u_{0} + d_{j} = 0 \quad j \in E \\ & \qquad v_{i} \ge 1 \quad i = 1, \ldots ,m \\ & \qquad u_{r} \ge 1 \quad r = 1, \ldots ,s \\ & \qquad d_{j} \le Mb_{j} \quad j \in E \\ & \qquad \lambda_{j} \le M\left( {1 - b_{j} } \right) \quad j \in E \\ & \qquad \lambda_{j} = \left\{ {0,1} \right\}\quad j \in E \\ & \qquad b_{j} \in \left\{ {0,1} \right\} \quad j \in E \\ & \qquad d_{j} \ge 0,\lambda_{j} \ge 0 \quad j \in E \\ & \qquad s_{ik}^ - \ge 0 \quad i = 1, \ldots ,m \\ & \qquad s_{rk}^{ + } \ge 0 \quad r = 1, \ldots ,s \\ \end{aligned}$$

(15)

Here, E still represents the set of efficient DMUs and the marginal cost vectors for reducing input and increasing output are ${\left({c}_{1}^{I},\dots ,{c}_{m}^{I}\right)}^{T}>{0}_{m}$ and ${\left({c}_{1}^{o},\dots ,{c}_{s}^{o}\right)}^{T}>{0}_{{\varvec{s}}}$, respectively. The target projection is limited under demand lines, that is, the output of target is no greater than its corresponding demand level.