Performance of 24 non-life insurance companies in Taiwan.

## Abstract

Data envelopment analysis (DEA) models assess decision-making units (DMUs), which directly convert multiple inputs into multiple outputs. Network DEA models have been studied extensively. However, the performance indices that link the two stages are assumed to be fixed or non-discretionary; their values are not adjustable. These models only assumed that the reductions on the inputs and additions on the outputs would improve the overall efficiency. But in the real world, the link is always adjustable. “Free links” means that the intermediate items are adjustable or discretionary, and each DMU can be increased or decreased from the observed one. The current chapter introduces a two-phase procedure with free links to assess system performance, Phase-I is a proposed slack-based measurement (SBM) model to partition the links into two sets: as-input and as-output. Phase-II is a modified SBM model to determine the slack of each input, as-input link, output and as-output link. This proposed model counts the slacks associated with the intermediate items in the efficiency scores and determines the entire system performance by the directional distance function. It is validated using network procedure and assesses the performance of supply chain management system.

### Keywords

- data envelopment analysis
- performance measure
- directional distance function
- network DEA
- slack-based measure

## 1. Introduction

The data envelopment analysis (DEA) models assess a set of homogeneous decision-making units (DMUs) that convert inputs into outputs. Fewer input values and more output values are desired and DMUs may be classified as being either efficient or inefficient. Tone and Tsutsui [1, 2] introduce network and dynamic DEA and categorize links into two types—“fixed links” and “free links”. The free links mean the links are adjustable; each DMU can be increased or decreased from the observed one and identifies the improvement target of each inefficient DMU on the frontier that is constructed by the efficient DMUs.

Seiford and Zhu [3] and Zhu [4] have introduced a two-stage process to measure the profitability and marketability of 55 US commercial banks and top Fortune 500 companies, respectively. They propose the effect of bank size on profitability and marketability through evaluating both technical and scale efficiencies. Sexton and Lewis [5] use a two-stage approach to evaluate the scores of American Major League Baseball teams. There are many other cases in which the whole operation is separated into more than two processes. These may have a series structure, a parallel structure, or a mixture of these. These structures are generally called network structures and the DEA technique to measure the efficiency of systems with a network structure is called network DEA (Färe & Grosskopf [6]). Färe and Whittaker [7] and Färe and Grosskopf [8] introduce models to compute the efficiency scores of sub-processes in network-structured DEA problems. Lewis and Sexton [9] introduce a network DEA model which focuses efficiency-enhancing strategies on individual stages of the production process. Kao and Hwang [10] introduce a framework for breaking down the efficiency of the entire process into the product of the efficiencies of the two-stage process. It assumes that the weights on the links are the same for the two stages, that is, the weights on the outputs in the first stage are assumed to be equal to the weights on the inputs in the second stage. In the real world, the relative weight of each stage is determined corresponding to its importance. Thus, the different weights in the entire system are mentioned in recent studies. Chen et al. [11] mentions that the overall efficiency scores resulting from Kao and Hwang [10] are not direct indicators of potential input reductions or output increases not realized by the inefficient DMUs. They develop an approach to determine the DEA frontier or DEA projections for inefficient DMUs. Chen et al. [12] note that the envelopment-based network DEA model should be used for determining the frontier projection for inefficient DMUs, whereas the multiplier-based network DEA model should be used for determining the divisional efficiency because it does not account for the intermediate links. Kao [13] proposes a dynamic DEA model to measure system and period efficiencies at the same time for multi-period systems. Chang et al. [14] take into account the ownership structure of networks in constructing effective network DEA models and accordingly develop three ownership-specified (centralized, distributed, and hybrid) network DEA models. Huang et al. [15] proposed a two-stage network model with bad outputs and supper efficiency (US-NSBM). Empirical comparisons show that the US-NSBM may be promising and practical for taking the nonperforming loans into account and being able to rank all samples.

However, these approaches do not count the slacks associated with the intermediate items in the efficiency scores. Consequently, the efficiency scores are greater than the actual efficiency. In addition, there is no DMU with an efficiency score equal to 1 because the properties of intermediate performance evaluation items would lead to conflicts. For instance, in the two-stage process problem, Stage-2 may have to reduce inputs (links) to achieve an efficient status. However, doing so would lead to a reduction in outputs in Stage-1, thereby reducing the efficiency of Stage-1. In other words, there are still two efficiency frontiers for the two sub-processes. One may desire a single frontier for the entire production system.

“Link” cannot be adjusted freely in a radial model which adjusts the inputs and outputs by the efficiency scores in a two-stage process. For this model, the entire system efficiency cannot be improved by adjusting links, see Kao and Hwang [10] and Lewis and Sexton [9]. “Link” that applies in a non-radial model has been discussed in recent years. Tone and Tsutsui [1] introduce a network DEA and categorize links into two types—“fixed links” and “free links.” “Free links” means the intermediate items are adjustable or discretionary; each DMU can be increased or decreased from the observed one and is free to assign each individual link to one of the three characteristics: as-input, as-output, or non-discretionary so that the entire system efficiency could be maximized. “Fixed links” means the intermediate products are beyond the control of DMUs. In the radial model, “links” cannot be adjusted freely, which adjust the inputs and outputs by the efficiency scores in a two-stage process. Tone and Tsutsui [2] introduce the dynamic slack-based measure (DSBM) model and the incorporation of slacks with *free* and *fixed* links into the efficiency score. They categorize the links into four types: desirable, undesirable, discretionary (free), and non-discretionary (fixed). The article incorporates the slacks of free links into the efficiency score in two ways: an ex-post approach (adjusted score) and incorporation through 0–1 MIP. The ex-post approach includes a two-phase procedure. Tone and Tsutsui [1] introduce the links are discretionary regarding their status, as-input or as-output. Liu and Liu [16, 17] adopt VGM and GBM models to assess the performance of supply chain management.

Chambers et al. [18] introduced the directional distance function (DDF) based on the Luenberger benefit function to obtain the technical efficiency by increasing the outputs and reducing the inputs simultaneously. Later, Chambers et al. [19] introduced the DDF of DEA to measure the technical efficiency. This chapter develops a model for an improved efficiency measure through directional distance formulation of data envelopment analysis.

The contribution and innovative progress for this chapter are (1) creating a new SBM model and converting multi-efficiency frontiers for the separation processes to an aggregation efficiency frontier for the entire production system and (2) adopting free links application and introducing DDF with a virtual gap diagram to assess the performance of the entire system. The rest of this chapter is organized as follows. The proposed two-phase two-stage performance evaluation models and DDF are presented in Section 2. Because the uniqueness of the optimal solution is important, we report an experiment on this subject using a real-world bank performance assessment in Section 3. We conclude this chapter in the last section.

## 2. The proposed two-phase two-stage performance evaluation

*J* denotes the set of homogeneous decision-making units of a network process that are evaluated by a set of inputs, *I*, a set of free links, *R*. *DMUo* represents the DMU under evaluation. To maximize the system efficiency score of *DMU*_{o}, each link in set *I* and *R* and

### 2.1. Two stages: Phase-I

Envelopment via the SBM fractional programming model [M1] is used to measure the relative performance of *DMUo*. The decision variables of slack are the values to be subtracted at the *i*th input and the value to be added at the *r*th output, respectively. The decision variables denote the weights of *DMUj* at Stage-1 and Stage-2, respectively. The right-hand side of constraints (1.2) and (1.3) expresses the targets of the inputs and links (outputs) at Stage-1. Each link *d* in set *Dfree* is free to increase the slack *d* in set *d* in set *d* of *DMUo* is assigned to sets *DMUo*. The first task is to assign all elements in the set to either *d* in sets and add them to its aggregate performance score. Phase-II of our solving procedure addresses the first task.

[M1]

For the two-phase procedure which is depicted in Figure 1, Phase-I is to determine the maximum slack values on each input and output; [M1] presents this purpose and adopts Eq. (1.3) and (1.4) to distinguish links to be as-input, discretionary, and as-output, which express as three subsets, *DMUj* in evaluating *DMUo*.

[M2]

The solution of [M2] for each link *d* in set *d* is assigned to set

### 2.2. Two stages: Phase-II

The results of Phase-I indicate that *DMUo* already partitioned set *I* *R*

[M3]

The dual form of [M3] is expressed as [M4]. The decision variables of [M4] possess properties*i*th input and the *r*th output, respectively. The terms *d* in sets

[M4]

Inequality (4.2) may be revised such that *DMUj* is not greater than 1.

### 2.3. Proposed directional distance function approach

The directional distance function (DDF) measures the distance from a certain operation point (e.g., *DMUo*) to the efficient frontier of the technology along the positive semi-ray defined by vector *g*. Given a directional vector *X* = *Y* = *g*. The objective function (4.1) can be converted to (4.9) which is to minimize the virtual input and maximize the virtual output to reach the efficient frontier.

The graph technology can be represented by *x*-axis (*y*-axis (*DMUo* to origin. To evaluate different *DMUo*, one may directly compare their vectors of weights; virtual gap, *x*-axis denotes the virtual input and *y*-axis denotes the virtual output.

### 2.4. Overall stage efficiencies

Similar to the SBM non-oriented models of Tone and Tsutsui [20], the solutions of Phase-II provide a reference set of DMUs for *DMUo*. The target for the performance items in sets *I,**R* can be obtained using [E1]. The measured performance value *DMUo* in the overall two-stage process which is expressed as [E2].

These points are the projection of *DMUo* on the frontier.

The results of Phase-II,

If set

If set

From Eqs. [E3] and [E4], we obtain the performance scores of Stage-1 and Stage-2, respectively, which identify the performance of each stage.

### 2.5. To extend two-stage to network process

Liu and Liu [16] extend the two-stage to network process. The network contains a set of sub-processes (nodes), *H*. The nodes are assigned ordinal numbers 1, 2, 3,…, *n*. Let A denote the set of network links. There are *n* homogeneous DMUs in set *J*, named *DMU*_{1}, *DMU*_{2},…, and *DMU*_{n}, which are randomly processed by the sub-processes in set *H*. The network structure is depicted in Figure 3.

#### 2.5.1. Inputs and outputs

At each sub-process *h*, there is a set of input measures *j* in set *J*, let *i*th input measure and the *r*th output measure at the sub-process *h*, respectively. Let *i*th input and the *r*th output at sub-process *h*, respectively.

#### 2.5.2. Links

Each sub-process may have links to other sub-processes. Let (*h*, *k*) denote the link between sub-processes *h* and *k*, *h* > *k*. Let *h, k*).*d*th link in set *DMUo* that is under evaluation. The volume of link *d* on link (*h*, *k*) could be increased or decreased with a slack to improve the efficiency of *DMUo* as well.

## 3. Illustrative examples

This study adopts a dataset covering 24 non-life insurance companies in Taiwan from Kao and Hwang [10] to illustrate the proposed two-phase procedure. Table 1 summarizes the performance datasheet of 24 non-life insurance companies in Taiwan.

**The inputs of the system:**

Operation expenses (*x*_{1}): salaries of the employees and various types of costs incurred in daily operation and.

Insurance expenses (*x*_{2}): expenses paid to agencies, brokers, and solicitors and other expenses associated with marketing the service of insurance.

**The links of the system:**

Direct written premiums (*z*_{1}): premiums received from insured clients.

Reinsurance premiums (*z*_{2}): premiums received from ceding companies.

**The outputs of the system:**

Under-writing profit (*y*_{1}): profit earned from the insurance business.

Investment profit (*y*_{2}): profit earned from the investment portfolio.

### 3.1. Phase-I

Table 2 summarizes the results of Phase-I and Phase-II. In the Phase-I column, for example, when *DMU1* is being evaluated, *DMUo* = *DMU1* and the optimal solution of [M1] is

Banks | DMUj | Operation expenses | Insurance expenses | Direct written premiums | Reinsurance premiums | Underwriting profit | Investment profit |
---|---|---|---|---|---|---|---|

(x1j) | (x2j) | (z1j) | (z2j) | (y1j) | (y2j) | ||

Taiwan Fire | 1 | 1,178,744 | 673,512 | 7,451,757 | 856,735 | 984,143 | 681,687 |

Chung Kuo | 2 | 1,381,822 | 1,352,755 | 10,020,274 | 1,812,894 | 1,228,502 | 834,754 |

Tai Ping | 3 | 1,177,494 | 592,790 | 4,776,548 | 560,244 | 293,613 | 658,428 |

China Mariners | 4 | 601,320 | 594,259 | 3,174,851 | 371,863 | 248,709 | 177,331 |

Fubon | 5 | 6,699,063 | 3,531,614 | 37,392,862 | 1,753,794 | 7,851,229 | 3,925,272 |

Zurich | 6 | 2,627,707 | 668,363 | 9,747,908 | 952,326 | 1,713,598 | 415,058 |

Taian | 7 | 1,942,833 | 1,443,100 | 10,685,457 | 643,412 | 2,239,593 | 439,039 |

Ming Tai | 8 | 3,789,001 | 1,873,530 | 17,267,266 | 1,134,600 | 3,899,530 | 622,868 |

Central | 9 | 1,567,746 | 950,432 | 11,473,162 | 546,337 | 1,043,778 | 264,098 |

The First | 10 | 1,303,249 | 1,298,470 | 8,210,389 | 504,528 | 1,697,941 | 554,806 |

KuoHua | 11 | 1,962,448 | 672,414 | 7,222,378 | 643,178 | 1,486,014 | 18,259 |

Union | 12 | 2,592,790 | 650,952 | 9,434,406 | 1,118,489 | 1,574,191 | 909,295 |

Shingkong | 13 | 2,609,941 | 1,368,802 | 13,921,464 | 811,343 | 3,609,236 | 223,047 |

South China | 14 | 1,396,002 | 988,888 | 7,396,396 | 465,509 | 1,401,200 | 332,283 |

Cathay Century | 15 | 2,184,944 | 651,063 | 10,422,297 | 749,893 | 3,355,197 | 555,482 |

Allianz President | 16 | 1,211,716 | 415,071 | 5,606,013 | 402,881 | 854,054 | 197,947 |

Newa | 17 | 1,453,797 | 1,085,019 | 7,695,461 | 342,489 | 3,144,484 | 371,984 |

AIU | 18 | 757,515 | 547,997 | 3,631,484 | 995,620 | 692,731 | 163,927 |

North America | 19 | 159,422 | 182,338 | 1,141,951 | 483,291 | 519,121 | 46,857 |

Federal | 20 | 145,442 | 53,518 | 316,829 | 131,920 | 355,624 | 26,537 |

Royal & Sunalliance | 21 | 84,171 | 26,224 | 225,888 | 40,542 | 51,950 | 6491 |

Aisa | 22 | 15,993 | 10,502 | 52,063 | 14,574 | 82,141 | 4181 |

AXA | 23 | 54,693 | 28,408 | 245,910 | 49,864 | 0.10 | 18,980 |

Mitsui Sumitomo | 24 | 163,297 | 235,094 | 476,419 | 644,816 | 142,370 | 16,976 |

When *DMU4* is being evaluated, *DMUo* = *DMU4* and the optimal solution of [M1] is *d* = 1, acts as an “as-input” item and may have a better solution. Therefore, 1(3,174,850) is recorded under the*DMU7* and *DMU18* have solution processes that are similar to *DMU4*.

### 3.2. Phase-II

Because each link may be “as-input” or “as-output”, the two links may have four possible combinations of *DMU*_{4} in Category A treats the first links as “as-input” (slack = 1,072,937) and is an undesirable output with respect to Stage-1 and a desirable input with respect to Stage-2. Meanwhile, the second set of links is “as-output” (slack = 135,818) and represents a desirable output with respect to Stage-1 but an undesirable input with respect to Stage-2.

Proceeding to Phase-II, which employs [M5], the optimal solutions for the evaluated DMU are listed in Table 4. The second column presents four efficiency scores obtained from (M5), Eqs. [E3], [E4], and [E2], which identify the Stage-1 efficiency (*DMUj* of each evaluated *DMUo*. The projection points of *DMUo* on the frontier which obtain from

DMUj | Reference DMU | Projected Points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

( | ( | ( | ( | ( | ( | |||||

1 | 1.000 | 1.000 | 1.000 | 1 | 1,178,744 | 673,512 | 7,451,757 | 856,735 | 984,143 | 681,687 |

2 | 1.000 | 1.000 | 1.000 | 2 | 1,381,822 | 1,352,755 | 10,020,274 | 1,812,894 | 1,228,502 | 834,754 |

3 | 1.000 | 1.000 | 1.000 | 3 | 1,177,494 | 592,790 | 4,776,548 | 560,244 | 293,613 | 658,428 |

4 | 0.565 | 0.144 | 0.168 | 22 | 601,320 | 386,832 | 2,101,914 | 507,681 | 2,842,519 | 177,331 |

5 | 1.000 | 1.000 | 1.000 | 5 | 6,699,063 | 3,531,614 | 37,392,862 | 1,753,794 | 7,851,229 | 3,925,272 |

6 | 0.654 | 0.316 | 0.437 | 12,22 | 1,384,733 | 668,363 | 4,734,713 | 976,808 | 4,422,004 | 415,058 |

7 | 0.300 | 0.212 | 0.296 | 22 | 1,679,395 | 1,102,796 | 5,467,038 | 1,530,389 | 8,625,473 | 439,039 |

8 | 0.333 | 0.223 | 0.316 | 22 | 2,382,571 | 1,564,544 | 7,756,129 | 2,171,174 | 12,237,025 | 622,868 |

9 | 0.322 | 0.112 | 0.213 | 22 | 1,010,217 | 663,372 | 3,288,623 | 920,585 | 5,188,537 | 264,098 |

10 | 0.308 | 0.395 | 0.423 | 2,22 | 1,303,249 | 1,057,171 | 6,739,349 | 1,437,967 | 4,040,010 | 554,806 |

11 | 1.000 | 1.000 | 1.000 | 11 | 1,962,448 | 672,414 | 7,222,378 | 643,178 | 1,486,014 | 18,259 |

12 | 1.000 | 1.000 | 1.000 | 12 | 2,592,790 | 650,952 | 9,434,406 | 1,118,489 | 1,574,191 | 909,295 |

13 | 0.976 | 0.020 | 0.345 | 5,22 | 2,558,630 | 1,368,802 | 13,923,383 | 769,865 | 3,609,236 | 1,449,235 |

14 | 0.310 | 0.206 | 0.284 | 22 | 1,396,002 | 916,702 | 4,544,491 | 1,272,140 | 7,169,973 | 364,952 |

15 | 0.591 | 0.516 | 0.702 | 12,22 | 1,721,308 | 651,063 | 6,056,436 | 1,001,547 | 3,489,394 | 555,482 |

16 | 0.460 | 0.210 | 0.332 | 12,22 | 712,644 | 415,071 | 2,369,909 | 586,830 | 3,069,552 | 197,947 |

17 | 0.215 | 0.254 | 0.343 | 22 | 1,422,899 | 934,363 | 4,632,050 | 1,296,650 | 7,308,093 | 371,984 |

18 | 0.820 | 0.201 | 0.240 | 22 | 757,515 | 497,432 | 2,465,985 | 690,305 | 3,890,642 | 198,035 |

19 | 0.582 | 0.312 | 0.474 | 5,22 | 159,422 | 102,620 | 556,110 | 134,910 | 755,528 | 46,857 |

20 | 1.000 | 1.000 | 1.000 | 20 | 145,442 | 53,518 | 316,829 | 131,920 | 355,624 | 26,537 |

21 | 0.458 | 0.265 | 0.286 | 22 | 24,829 | 16,304 | 80,827 | 22,626 | 127,524 | 6491 |

22 | 1.000 | 1.000 | 1.000 | 22 | 15,993 | 10,502 | 52,063 | 14,574 | 82,141 | 4181 |

23 | 1.000 | 1.000 | 1.000 | 23 | 54,693 | 28,408 | 245,910 | 49,864 | 0 | 18,980 |

24 | 0.504 | 0.073 | 0.177 | 22 | 163,297 | 107,231 | 531,590 | 148,806 | 838,704 | 42,690 |

The nine efficient DMUs, 1, 2, 3, 5, 11, 12, 20, 22, and 23, are consistent, with all of their performance scores in Stage-1 and Stage-2 being equal to one. The efficiency scores for the inefficient DMUs for both Stage-1 and Stage-2 are less than 1. For instance, *DMU4* has scores of 0.565 and 0.144 in Stage-1 and Stage-2. An obvious means of improving overall efficiency is to focus on Stage-2.

The virtual weight is expressed as *DMU4* has scores of 0.565 and 0.144 in Stage-1 and Stage-2. An obvious means of improving overall efficiency is to focus on Stage-2 output item

DMUj | Input Items | Intermediate Items | Output Items | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 1.000 | 76% | 15% | 9% | — | — | 10% | 7% | 83% |

2 | 1.000 | 84% | 8% | 8% | — | — | 8% | 8% | 84% |

3 | 1.000 | 4% | 96% | — | — | 19% | 2% | 0% | 78% |

4 | 0.168 | 53% | 24% | 24% | — | — | 8% | 8% | 85% |

5 | 1.000 | 78% | 14% | 8% | — | — | 4% | 10% | 86% |

6 | 0.437 | 26% | 48% | 26% | — | — | 20% | 20% | 60% |

7 | 0.296 | 33% | 33% | 33% | — | — | 0% | 28% | 72% |

8 | 0.316 | 33% | 33% | 33% | — | — | 73% | 10% | 17% |

9 | 0.213 | 33% | 33% | 33% | — | — | 33% | 33% | 33% |

10 | 0.423 | 59% | 21% | 21% | — | — | 2% | 2% | 97% |

11 | 1.000 | 1% | 99% | — | — | 91% | 1% | 8% | 1% |

12 | 1.000 | 7% | 73% | 20% | — | — | 48% | 3% | 49% |

13 | 0.345 | 92% | 4% | — | 4% | 88% | — | 8% | 4% |

14 | 0.284 | 33% | 33% | 33% | — | — | 70% | 9% | 21% |

15 | 0.702 | 21% | 58% | 21% | — | — | 17% | 26% | 57% |

16 | 0.332 | 28% | 44% | 28% | — | — | 20% | 20% | 61% |

17 | 0.343 | 33% | 33% | 33% | — | — | 33% | 11% | 55% |

18 | 0.240 | 25% | 25% | 25% | 25% | — | — | 50% | 50% |

19 | 0.474 | 25% | 25% | 25% | 25% | — | — | 44% | 56% |

20 | 1.000 | 2% | 96% | — | 2% | 45% | — | 40% | 15% |

21 | 0.286 | 25% | 25% | 25% | 25% | — | — | 40% | 60% |

22 | 1.000 | 55% | 45% | — | — | 23% | 23% | 32% | 23% |

23 | 1.000 | 4% | 96% | — | — | — | 100% | 0% | 0% |

24 | 0.177 | 33% | 33% | — | 33% | 44% | — | 23% | 33% |

## 4. Discussion and conclusions

The objective of efficiency assessment is to identify weaknesses such that the appropriate steps to improve the entire system’s performance. This chapter introduces a two-phase procedure to evaluate the two-stage and network models with “free” links. This new model adopts SBM and considers not only the input and output slacks in the objective function but also the slacks of links. The resultant DEA scores provide completely information on how to project inefficient DMUs onto the DEA frontier for specific two-stage processes. Instead of the two conflicting roles that each link plays in existing models, each link plays a single role in the proposed two-phase process system in that it is either desirable or undesirable. The SBM model in this chapter counts the slacks associated with links in the efficiency scores, overcoming the hurdle. The bank case study takes the example on adjustment in the slacks and defines the best practice performance that the DMU under evaluation will need to attain to achieve the best efficiency. To achieve the best-practice efficiency, each DMU determines a set of weights for input, output, and link, where the links are designated as either “as-input” or “as-output”. Input and as-input measures reduce slacks, while output and as-output measures increase slacks to reach their targets on the production frontier. This study only introduces a two-stage procedure to assess the entire system. It also can be extended to more complex network processes, applied in series multistage, share resource (Chen et al. [21] and Liang et al. [22]), dynamic network DEA (Tone & Tsutsui [2] and Kao [13]), assurance region (Thompson et al. [23]), cone ratio model (Charnes et al. [24]), and virtual weight analysis models (Sarrico & Dyson [25]) in future research.

## Acknowledgments

This research is supported by National Science Council of Taiwan, Republic of China, under the project NSC100-2221-E-009-065-MY3.

## Conflict of interests

The authors declare that there is no conflict of interest regarding the publication of this paper.