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This research proposes an integrated approach to the Data Envelopment Analysis (DEA) and Analytic Hierarchy Process (AHP) methodologies for ratio analysis. According to this, we compute two sets of weights of ratios in the DEA framework. All ratios are treated as outputs without explicit inputs. The first set of weights represents the most attainable efficiency level for each Decision Making Unit (DMU) in comparison to the other DMUs. The second set of weights represents the relative priority of output-ratios using AHP. We assess the performance of each DMU in terms of the relative closeness to the priority weights of output-ratios. For this purpose, we develop a parametric goal programming 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. An illustrated example of eight listed companies in the steel industry of China is used to highlight the usefulness of the proposed approach.

Ratio analysis is a commonly used analytical tool for measuring the relative performance of a Decision Making Units (DMUs) by focusing on one input/output at a time [

DEA is a data-oriented approach for assessing the relative efficiency of Decision Making Units (DMUs) that use multiple inputs to produce multiple outputs. The standard DEA models are formulated using absolute nu- merical input and output data [

However, the weights provided by (ratio-based) DEA models are often unrealistic. Assigning the extremely large or small weights to certain (ratio) data may be undesirable for some Decision Makers (DMs) because it may categorize a DMU as efficient for unlikely weight combinations. Hence, there is a point to argue that the relative priority of (ratio) data should be taken into account in efficiency assessments.

On the other hand, AHP is a multi-attribute decision-making method that can reflect a priori information about the relative priority of (ratio) data in efficiency assessments. AHP usually involves three stages for gene- rating weights. These stages are the decomposition into a hierarchy, comparative judgments, and synthesis of priorities [

• Absolute weight restrictions. These restrictions directly impose upper and (or) lower bounds on the weights of inputs (outputs) using AHP [

• Relative weight restrictions. These restrictions limit the relationship between the weights of inputs (outputs) using AHP [

• Virtual weight restrictions. A single virtual input (output) is defined as the weighted sum of all inputs (out- puts). We refer to the proportion of each component of such sum as the “virtual weight” of an input (output). These restrictions limit virtual weights using AHP [

• Restrictions on input (output) units. These restrictions impose bounds on changes of inputs (outputs) while the relative importance of such changes is computed using AHP [

There are a number of other methods that do not necessarily apply additional restrictions to a DEA model. Such as converting the qualitative data in DEA to the quantitative data using AHP [

Recently the author has applied AHP weights as weight bounds into a two-level DEA model [

This research has been organized to proceed along the following stages (

1) Computing the efficiency of each DMU using ratio-based DEA Model (3) without explicit inputs. The computed efficiencies are applied in Model (7).

2) Computing the priority weights of output-ratios for all DMUs using AHP, which impose weight bounds into Model (7).

An integrated approach to DEA, AHP and ratio-analysis

3) Obtaining an optimal set of weights for each DMU using ratio-based DEA Model (7) (minimum efficiency loss η).

4) Obtaining an optimal set of weights for each DMU using Model (7) bounded by AHP (maximum efficiency loss κ). Note that if the AHP weights are added to (7), we obtain Model (9).

5) Measuring the performance of each DMU in terms of the relative closeness to the priority weights of out- put-ratios. For this purpose, we develop a parameter goal programming model. Increasing a parameter in a de- fined range of efficiency loss, we explore how much a DM can achieve its goals. This may result in various ranking positions for a DMU in comparison to the other DMUs.

A ratio-based DEA model can be formulated similar to a classical DEA model without explicit inputs [

where

where

Model (3) looks like a DEA model without inputs that combines the standard DEA methodology with ratio analysis.

There are basically three steps for considering decision problems by AHP namely: 1) Decomposition; 2) Pair- wise Comparisons and Judgment Matrix; and 3) Synthesis [

Step 1: Decomposition. This step includes decomposition of the decision problem into elements according to their common characteristics and the formation of a hierarchical model having different levels. In this study, the AHP hierarchical model has three levels: problem objective, criteria and sub-criteria. The problem objective is to prioritize the output-ratios, the criteria are the categories of output-ratios, and the sub-criteria are various out- put-ratios that are organized into these categories (

Step 2: Pairwise Comparisons and Judgment Matrix. In this step, a decision maker makes a pairwise compari- son matrix of different criteria, denoted by

Step 3: Synthesis. In this step, the AHP method obtains the priority weights of criteria by computing the ei- genvector of matrix

In a similar way, the priority weights of sub-criteria under each criterion is obtained by computing the eigen- vector of matrix

To determine whether or not the inconsistency in a comparison matrix is reasonable the random consistency ratio,

where

We develop our formulation based on a simplified version of the generalized distance model (see for example [

The AHP hierarchical model for prioritizing ratios

Model (7) identifies the minimum efficiency loss η (eta) needed to arrive at an optimal set of weights. The first constraint ensures that each DMU loses no more than η of its best attainable efficiency,

To obtain the weight bounds for the weights of output-ratios in Model (7), this study aggregates the priority weights of two different levels in AHP as follows:

where

In order to estimate the maximum efficiency loss

The first set of constraints changes the priority weights of output-ratios to weights for the new system by means of a scaling factor

In this stage, we develop a parametric goal programming model that can be solved repeatedly to generate the various sets of weights for the discrete values of the parameter

where _{k}. The first set of equations indicates the goal equations whose right-hand sides are the priority weights of output-ratios adjusted by a scaling variable.

Because the range of deviations computed by the objective function is different for each DMU, it is necessary to normalize it by using relative deviations rather than absolute ones [

where

In this section we present the application of the proposed approach to assess the financial performance of eight listed companies in the steel industry of China. The companies’ financial ratios, adopted from [

. There are five categories of financial ratios at the criteria level. Each one includes three different ratios at the sub-criteria level

On the other hand, solving Model (9) for the DMU under assessment, we adjust the priority weights of out- put-ratios obtained from AHP in such a way that they become compatible with the weights’ structure in the ratio- based DEA models.

. Financial market data (ratios) for eight listed companies in China’s steel industry

Outputs (Ratios) | DMU1 | DMU2 | DMU3 | DMU4 | DMU5 | DMU6 | DMU7 | DMU8 |
---|---|---|---|---|---|---|---|---|

0.544 | 0.622 | 0.673 | 0.737 | 0.659 | 0.610 | 0.616 | 0.768 | |

0.520 | 0.182 | 0.494 | 0.403 | 0.345 | 0.641 | 0.407 | 0.430 | |

1.152 | 0.459 | 0.911 | 0.644 | 1.147 | 1.098 | 0.905 | 0.710 | |

5.416 | 7.042 | 4.070 | 8.867 | 11.813 | 6.075 | 5.714 | 9.489 | |

1.423 | 1.061 | 1.138 | 1.664 | 2.281 | 2.568 | 1.470 | 1.784 | |

−0.177 | −0.307 | 0.187 | 0.924 | −0.551 | −0.218 | −0.186 | 0.016 | |

0.016 | 0.071 | 0.021 | 0.018 | 0.007 | 0.004 | 0.003 | 0.005 | |

0.023 | 0.075 | 0.024 | 0.030 | 0.015 | 0.011 | 0.005 | 0.009 | |

0.043 | 0.371 | 0.053 | 0.075 | 0.042 | 0.018 | 0.012 | 0.020 | |

0.044 | 0.129 | 0.306 | 0.155 | 0.004 | 0.026 | 0.165 | 0.176 | |

−1.834 | −1.877 | −3.242 | −2.724 | −3.759 | −5.427 | −5.816 | −5.009 | |

−0.372 | −0.270 | 0.076 | −0.250 | −0.474 | −0.374 | −0.436 | −0.490 | |

13.229 | 9.357 | 6.869 | 24.910 | 43.117 | 16.830 | 15.505 | 26.141 | |

4.337 | 3.529 | 2.235 | 4.193 | 6.437 | 2.590 | 4.312 | 3.682 | |

15.258 | 7.221 | 18.143 | 8.060 | 20.385 | 39.809 | 60.400 | 35.571 |

. The AHP hierarchical model^{*}

Objective level | Criteria level | Sub-criteria level | |
---|---|---|---|

Prioritizing the financial ratios | |||

^{*} = Criterion

. Optimal weights of output-ratios of Model (9) for DMU5

0.160 | 0.070 | 0.031 | 0.039 | 0.073 | 0.014 | 0.283 | 0.156 |

0.086 | 0.057 | 0.214 | 0.090 | 0.335 | 0.141 | 0.532 | 2.2788 |

The maximum efficiency loss for the DMU under assessment to achieve the corresponding weights in Model (9) is equal to

Going one step further to the solution process of the parametric goal programming model in (10) we proceed to the estimation of total deviations from the AHP weights for each DMU while the parameter

. Minimum and maximum efficiency loss

Efficiency Loss | DMU1 | DMU2 | DMU3 | DMU4 | DMU5 | DMU6 | DMU7 | DMU8 |
---|---|---|---|---|---|---|---|---|

η | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

κ | 0.34 | 0.066 | 0.311 | 0.146 | 0 | 0.276 | 0.143 | 0.119 |

. The ranking position of each DMU based on the minimum distance to priority weights of output-ratios

θ | DMU1 | DMU2 | DMU3 | DMU4 | DMU5 | DMU6 | DMU7 | DMU8 |
---|---|---|---|---|---|---|---|---|

Z^{*}(0) | 0.6744 | 0.0693 | 0.3767 | 0.1462 | 0 | 0.3863 | 0.1884 | 0.2122 |

Rank | 8 | 2 | 6 | 3 | 1 | 7 | 4 | 5 |

The relative closeness to the priority weights of ratios [∆(θ)], versus efficiency loss (θ) for each DMU

We develop an integrated approach based on DEA and AHP methodologies for ratio analysis. We define two sets of weights of ratios in the DEA framework. All ratios are treated as outputs without implicit inputs. The first set represents the weights of output-ratios with minimum efficiency loss. The second set represents the correspond- ing priority weights using AHP with maximum efficiency loss. We assess the performance of each DMU in comparison to the other DMUs based on the relative closeness of the first set of weights to the second set of weights. Improving the measure of relative closeness in a defined range of efficiency loss, we explore the vari- ous ranking positions for the DMU under assessment in comparison to the other DMUs.