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An integrated approach based on DEA and AHP

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Abstract

This research proposes a theoretical framework to assess the performance of Decision Making Units (DMUs) by integrating the Data Envelopment Analysis (DEA) and Analytic Hierarchy Process (AHP) methodologies. According to this, we consider two sets of weights of inputs and outputs under hierarchical structures of data. The first set of weights, represents the best attainable level of efficiency for each DMU in comparison to other DMUs. This level of efficiency can be less than or equal to that of obtaining from a traditional DEA model. The second set of weights reflects the priority weights of inputs and outputs for all DMUs, using AHP, in the DEA framework. We assess the performance of each DMU in terms of the relative closeness to the priority weights of inputs and outputs. For this purpose, we develop a parametric distance model to measure the deviations between the two sets of weights. Increasing the value of a parameter in a defined range of efficiency loss, we explore how much the deviations can be improved to achieve the desired goals of the decision maker. This may result in various ranking positions for each DMU in comparison to the other DMUs. To highlight the usefulness of the proposed approach, a case study for assessing the financial performance of eight listed companies in the steel industry of China is carried out.

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Correspondence to Mohammad Sadegh Pakkar.

Appendices

Appendix A

See Table 8.

Table 8 The measure of relative closeness to AHP weights [\(\Delta _{k}(\theta )\)], versus efficiency loss [\(\theta \)] for each DMU

Appendix B. Glossary of modeling symbols

  • \(\hat{x}_{ij} \)is the value of input \(i(i = 1, 2, ..., m)\) for DMU \(j (j = 1, 2, ..., n)\).

  • \(\hat{y}_{rj}\) is the value of output \(r(r = 1, 2.., s)\) for DMU \( j (j = 1, 2, ...,n)\).

  • \(\hat{x}_{i(\min )} \) is the minimum value of input \(i\) for all DMUs.

  • \(\hat{x}_{i(\max )} \) is the maximum value of input \(i\) for all DMUs.

  • \(\hat{y}_{r(\min )} \) is the minimum value of output \(r\) for all DMUs.

  • \(\hat{y}_{r(\max )} \) is the maximum value of output \(r\) for all DMUs.

  • \(x_{ij}\) is the normalized value of input \(i\) for DMU \(j\).

  • \(y_{rj}\) is the normalized value of output \(r\) for DMU \(j\).

  • \(k\) is the index for the DMU under assessment where \(k (k = 1, 2, ..., n)\).

  • \(E_{k}\) is the relative efficiency of DMU under assessment in the CCR model.

  • \(v_{i}\) is the weight of input \(i(i = 1, 2, ..., m)\) in the CCR model after normalizing the original data.

  • \(u_{r}\) is the weight of output \(r (r = 1, 2.., s)\) in the CCR model after normalizing the original data.

  • \(x_{hij}\) is the value of input \(i(i = 1, 2, ..., m)\) of input category \(h(h = 1, 2, ..., M)\) for DMU \(j (j = 1, 2, ..., n)\) after normalizing the original data.

  • \(y_{lrj}\) is the value of output \(r(r = 1, 2.., s)\) of output category \(l(l = 1, 2, ..., S)\) for DMU \(j (j = 1, 2, ..., n)\) after normalizing the original data.

  • \(v_{hi}\) is the internal weight of input \(i\) in input category \(h\).

  • \(u_{lr}\) is the internal weight of output \(r\)in output category \(l\).

  • \(q_{h}\) is the weight of input category \(h\).

  • \(p_{l}\) is the weight of output category \(l\).

  • \(v^{\prime }_{hi}\) is the weight of input \(i\) of input category \(h,v^{\prime }_{hi} = q_{h}v_{hi}\).

  • \(u^{\prime }_{lr}\) is the weight of output \(r\) of output category \(l,u^{\prime }_{lr} = p_{l}u_{lr}\).

  • \(E_{k}^{*}\) is the optimal efficiency value of the DMU under assessment, \(k(k = 1, 2, ..., n)\), in the CCR model.

  • \(E^{\prime }_{k}\) is the efficiency of the DMU under assessment in the two-level DEA model.

  • \(\eta \) is the minimum efficiency loss in the two-level DEA model.

  • \(a_{ho}\) is the \(h - o\) entry of the pairwise comparison matrix, \(A\), with \(M\) input categories.

  • \(b_{hit}\) is the \(l - t\) entry of a pairwise comparison matrix, \(B\), with \(m\) inputs under an input category.

  • \(w_{h}\) is the priority weight of criterion (input category) \(h(h = 1, ..., M)\) in AHP.

  • \(w^{\prime }_{l}\) is the priority weight of criterion (output category) \(l(l = 1, ..., \hbox {S})\) in AHP.

  • \(e_{hi}\) is the internal priority weight of sub-criterion \(i(i = 1, ..., m)\) under criterion \(h\) in AHP.

  • \(e^{\prime }_{lr}\) is the internal priority weight of sub-criterion \(r(r = 1, ..., s)\) under criterion \(l\) in AHP.

  • \(\bar{{v}}_{hi} \) is the weight bound for the weight of input \(i\)under input category \(h\).

  • \(\bar{{u}}_{lr} \) is the weight bound for the weight of output \(r\) under output category \(l\).

  • \(\lambda _{\max }\) is the largest eigenvalue.

  • \(R.I\). is the average random consistency index

  • \(C.R\). is the random consistency ratio.

  • \(N\) is the size of a pairwise comparison matrix.

  • \(\kappa \)(kappa) is the maximum efficiency loss.

  • \(\alpha \) is a scaling factor for the weight of output \(r\) under output category \(l\).

  • \(\beta \) is a scaling factor for the weight of input \(i\) under input category \(h\).

  • \(\theta \) is the parameter of efficiency loss.

  • \({\hat{u}}_{lr}^{\prime *} (\kappa )\) is the optimal priority weight of output \(r\) of output category \(l\) with the maximum efficiency loss, \(\kappa \).

  • \({\hat{v}}_{hi}^{\prime *} (\kappa )\) is the optimal priority weight of input \(i\) under input category \(h\) with the maximum efficiency loss, \(\kappa \).

  • \(Z_{k}^{*}(\theta )\) is the optimal value of the objective function in the parametric distance model for the DMU under assessment where \(\eta \le \theta \le \kappa \).

  • \(\Delta _{k}(\theta )\) is the measure of closeness for the DMU under assessment.

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Pakkar, M.S. An integrated approach based on DEA and AHP. Comput Manag Sci 12, 153–169 (2015). https://doi.org/10.1007/s10287-014-0207-9

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