Illustrative example

A manager of a retail company governs eight branches and periodically assesses them by observing four performance indices: number of Employees, Cost, Turnover, and Profit, as depicted in Table 11. Employees and Cost are treated as input indices, while Turnover and Profit are the output indices. Lower inputs and higher outputs are preferred to generate a higher efficiency score. Different to the first example, the characteristic of a large scale in the value is across indices, not UOAs (branches). In the following subsections, we illustrate how to obtain the preferable ranking and robust ranking for the manager.

Table 11. The indices data in illustrative example

Branch j Input index Output index

Employee x_1j Cost x_2j Turnover y_1j Profit y_2j

A 20 6583 7929 419 B 21 7713 8414 406 C 18 6980 8020 359 D 24 8273 9947 373 E 28 8566 9741 412 F 23 8397 9408 500 G 29 7011 7890 621 H 26 8680 9701 705

3.6.2.1 Preferable ranking

In order to discuss the proportion of each index in different models, we assess these

branches by using DEA (CCR input-oriented model), VWR-DEA (CCR input-oriented model with virtual weights restrictions), CWA and VWR-CWA models. The general form of virtual weights restrictions Eq. (3-22) can be rewritten as Eq. (3-46) for the current numerical example, with two input and two output indices:

, , , ,

2 0

2 2 1 1 1 2 2 2 1 1

1x _jV +α x _jV +β y _jU +β y _jU ≥ j= AK H

α (3-46)

If the manager has the preference that the proportion of Profit is no less than half of Turnover, then the parameters (α , ₁ α , ₂ β , ₁ β ) are substituted by (0, 0, -1, 2). Eq. (3-46) is further ₂ rewritten as Eq. (3-47) for all branches j:

, , , ,

, 0 2 _{2 2}

1

1U y U j A H

y

- _j + _j ≥ = K (3-47)

The proportion allocation of each index obtained from the original DEA model, as depicted in column (1) of Table 12, is extremely disproportional in most branches, even though we add the virtual weights restriction Eq. (3-47) in the DEA model (VWR-DEA), as depicted in column (2) of Table 12. For instance, despite the preferable virtual weights restriction Eq. (3-47), branches G and H still choose their favorable weight to create a feasible disproportion in Turnover (0%) and Profit (100%). Besides, comparing DEA with CWA, as depicted in column (1) of Tables 12 and Table 13, the proportion allocation in the DEA model is more unstable than the CWA model, without large variation in all branches. The comparison between VWR-DEA and VWR-CWA, as depicted in column (2) of Tables 12 and 13, would have similar results. These results imply that the proportion allocation obtained, whether in the DEA or VWR-DEA models, cannot reflect the manager’s preference altogether.

CWA provided the assessment results in column (1) of Table 13. They show that branch A and B are the best and worst, respectively. Following these common weights (V₁^*, V₂^*, U₁*, U₂^*) = (1.00, 1.27, 1.00, 1.00) used in CWA, as depicted in column (1) of Table 13, the manager would observe a large difference in relative proportion, whether between the virtual inputs (P₁^I_j,P₂^I_j) or outputs (P₁^O_j,P₂^O_j); for instance, in the row of branch A, Employee (0.02%) vs. Cost (99.8%) and Turnover (95.0%) vs. Profit (5.0%).

From a managerial scenario, it reveals that the input index Cost and output index Turnover take a considerably large proportion of branch A’s rating. The other branches appear to be in a similar situation. This kind of extreme disproportion may not be accepted under

specific practical exercises, even though the manager expects quick business development. In fact, in any case, Profit still plays an important role in rating. The virtual assurance region can assist the manager in easily adding his preference in Profit.

Table 12. The proportion results of DEA and VWR-DEA in illustrative example

Branch j

(1) DEA (2) VWR-DEA with Eq. (3-47)

I

P₁j P₂^I_j P₁^O_j P₂^O_j P₁^I_j P₂^I_j P₁^O_j P₂^O_j A 35.2% 64.8% 95.1% 4.9% 13.0% 87.0% 58.7% 41.3%

B 45.8% 54.2% 82.6% 17.4% 100.0% 0.0% 60.8% 39.2%

C 65.4% 34.6% 78.1% 21.9% 100.0% 0.0% 62.6% 37.4%

D 34.1% 65.9% 96.5% 3.5% 100.0% 0.0% 66.6% 33.4%

E 0.0% 100.0% 100.0% 0.0% 0.0% 100.0% 63.9% 36.1%

F 46.0% 54.0% 81.2% 18.8% 11.7% 88.3% 58.4% 41.6%

G 10.1% 89.9% 72.4% 27.6% 30.1% 69.9% 0.0% 100.0%

H 7.5% 92.5% 73.9% 26.1% 100.0% 0.0% 0.0% 100.0%

Table 13. The assessment results of CWA and VWR-CWA in illustrative example Branch

j

(1) CWA (2) VWR-CWA with Eq. (3-47) (V₁^*,V₂^*,U₁^*,U₂^*) = (1.00, 1.27, 1.00, 1.00) (V₁^*,V₂^*,U₁^*,U₂^*) = (94.25, 1.92, 1.00, 13.30)

∗

ζj Ranking P₁^I_j P₂^I_j P₁^O_j P₂^O_j ξ^∗_j Ranking P₁^I_j P₂^I_j P₁^O_j P₂^O_j A 1.000 1 0.02% 99.8% 95.0% 5.0% 1.000 1 13.0% 87.0% 58.7% 41.3%

B 0.902 8 0.02% 99.8% 95.4% 4.6% 0.935 6 11.8% 88.2% 60.9% 39.1%

C 0.947 4 0.02% 99.8% 95.7% 4.3% 0.996 3 11.2% 88.8% 62.6% 37.4%

D 0.984 2 0.02% 99.8% 96.4% 3.6% 0.964 5 12.5% 87.5% 66.6% 33.4%

E 0.934 6 0.03% 99.7% 95.9% 4.1% 0.878 7 13.8% 86.2% 63.9% 36.1%

F 0.931 7 0.02% 99.8% 95.0% 5.0% 0.975 4 11.9% 88.1% 58.5% 41.5%

G 0.956 3 0.03% 99.7% 92.7% 7.3% 0.875 8 16.9% 83.1% 48.8% 51.2%

H 0.945 5 0.02% 99.8% 93.2% 7.8% 0.998 2 12.8% 87.2% 50.8% 49.2%

The manager reassesses these branches using the VWR-CWA model. The assessment results of VWR-CWA are arranged in column (2) of Table 13 by using the other common weights (V₁^*, V₂^*, U₁^*, U₂^*) = (94.25, 1.92, 1.00, 13.3). Focusing on the row of branch A in Table 13, the proportion of Turnover (P₁^O_j ) vs. Profit (P₂^O_j ) changes from the CWA disproportion 95.0% vs. 5.0% to the 58.7% vs. 41.3% in VWR-CWA. Similar changes also can be seen in other branches. The rankings of the eight branches under CWA and VWR-CWA are completely different. However, the ranking obtained from VWR-CWA is more preferable and reliable to the manager because its preference is considered.

Obviously, the virtual restriction Eq. (3-47) has an influence on the final ranking of the branches. In the above case, Eq. (3-47) is one of general form Eq. (3-22) with the parameter

W = 1. In order to strengthen the preference for the manager, they can add more restrictions to obtain its most preferable ranking for all branches in VWR-CWA. In addition, the different preferable constraints also can be only assigned to certain UOAs to keep the original characteristics in performance indices of each UOA.

3.6.2.2 Robust ranking

Column (2) of Table 13 shows the single preference that the manager assigned. It is common that there exists a situation that the manager has no preference about the relationship among indices. What they concerned is one acceptable and feasible proportion of virtual inputs and virtual outputs in the same measure. The manager can determine the acceptable interval [B_r^OL, B_r^OU ] and [B , _i^IL B ] for _i^IU P_rj^O and P_ij^I , respectively. For the current numerical example with two inputs (m = 2) and two outputs (s = 2), all the values of δ_r^-, δ_r⁺,

-τi and τ_i⁺ are set within 0 and 1 to ensure that P_rj^O and P_ij^I are between 0 and 1. For the purposes of clearly illustrating our approach, we set the lower bound B = 0.4 and upper _r^OL bound B_r^OU = 0.6, respectively. In other words, P_rj^O would be limited within the interval [40%, 60%]. If a larger interval is allowed, one may set the interval [20%, 80%].

From a managerial scenario, while managers desire to understand the ranking of branches under variant kinds of limitations for P_rj^O and P_ij^I , Eq. (3-30) and Eq. (3-31) provide one systematic setting of lower bound and upper bound. For the cases where δ_r^- and

+

δr are set at five levels 0.2, 0.4, 0.6, 0.8, and 1.0, P_rj^O would be limited in the gradually wider intervals [40%, 60%], [30%, 70%], [20%, 80%], [10%, 90%] and [0%, 100%], respectively. With the same setting for τ_i⁻ and τ_i⁺, P_ij^I would have the same limitations as above.

Table 14. The 25 combinations of interval limitation for P_rj^O and P_ij^I Combination symbol [B_i^IL ,B_i^IU]

[0%, 100%] [10%, 90%] [20%, 80%] [30%, 70%] [40%, 60%]

[B_r^OL ,B_r^OU ]

[0%, 100%] C1 C2 C3 C4 C5

[10%, 90%] C6 C7 C8 C9 C10

[20%, 80%] C11 C12 C13 C14 C15

[30%, 70%] C16 C17 C18 C19 C20

[40%, 60%] C21 C22 C23 C24 C25

As depicted in Table 14, there are 25 combinations of interval limitation for P_rj^O and

I

Pij . Obviously, different interval limitations for P_rj^O and P_ij^I may have different assessment results for the ranking. In this numerical example, we can employ the VWR-CWA model in carrying out an assessment for each combination with corresponding intervals [B_i^IL ,B_i^IU] and [B_r^OL ,B_r^OU]. For instance, the results for C12 and C22 are depicted in Table 15. For the combination C12, the general virtual weights restrictions Eq. (3-44) can be rewritten as Eq.

(3-48) to Eq. (3-51) by removing four of the same and repeatable restrictions for all branches from the parameters setting in Eq. (3-34) to Eq. (3-37):

, , , ,

0 2

0 8

0. y₁ U₁ - . y₂ U₂ j A H

_{j j} ≥ = K (3-48)

, , , ,

0 8

0 2

0. y₁ U₁ . y₂ U₂ j A H

- _j + _j ≥ = K

(3-49) ,

, , ,

0 1

0 9

0. x₁V₁- .x₂ V₂ j A H

_{j j} ≥ = K

(3-50) ,

, , ,

0 9

0 1

0.x₁V₁ . x₂ V₂ j A H

- _j + _j ≥ = K

(3-51) As Table 15 depicted, the ranking is inconsistent between the two combinations C12 and

C22. For managers, it is expected that more outcomes form all kinds of combinations that can help them make more accurate and robust judgments in the ranking of branches.

Table 15. The assessment results in VWR-CWA of C12 and C22

(1) VWR-CWA of C12 (2) VWR-CWA of C22

(V₁,V₂,U₁,U₂) = (191.30, 1.09, 1.00, 6.67) (V₁,V₂,U₁,U₂) = (121.38, 2.20, 1.00, 17.78) Branch

j

∗

ξj Ranking P₁^I_j P₂^I_j P₁^O_j P₂^O_j ξ^∗_j Ranking P₁^I_j P₂^I_j P₁^O_j P₂^O_j A 0.975 2 35% 65% 74% 26% 0.909 3 14% 86% 52% 48%

B 0.895 7 32% 68% 76% 24% 0.801 6 13% 87% 54% 46%

C 0.942 3 31% 69% 77% 23% 0.821 5 12% 88% 56% 44%

D 0.914 5 34% 66% 80% 20% 0.785 7 14% 86% 60% 40%

E 0.850 8 36% 64% 78% 22% 0.767 8 15% 85% 57% 43%

F 0.940 4 32% 68% 74% 26% 0.860 4 13% 87% 51% 49%

G 0.912 6 42% 58% 66% 34% 1.000 1 19% 81% 42% 58%

H 1.000 1 34% 66% 67% 33% 0.998 2 14% 86% 44% 56%

While compiling statistics from 25 combinations, we obtained the percentage of occurrence frequency in each ranking, as depicted in Table 16. It is not hard to observe that except for branch G, the high occurrence frequency centralizes in a few rankings for other

branches. For instance, branch H is only ranked 1^st and 2^nd. For branch E, the ranking of 7^th and 8^th occurs in all combinations. Undoubtedly, branch H is always better than branch E. If managers choose the highest occurrence frequency as the representative branch of each ranking level, the ranking list for 1^st to 8^th is H, A, C, F, D, B, E and G.

Table 16. The summary of the 25 ranking results with C1 to C25 Ranking

Branch j 1^st 2^nd 3^rd 4^th 5^th 6^th 7^th 8^th

A 10 8 7 0 0 0 0 0

B 0 0 0 0 0 18 7 0

C 0 0 15 1 9 0 0 0

D 0 0 0 0 15 3 7 0

E 0 0 0 0 0 0 10 15

F 0 0 1 24 0 0 0 0

G 2 5 2 0 1 4 1 10

H 13 12 0 0 0 0 0 0

Total 25 25 25 25 25 25 25 25

Robust Ranking H A C F D B E G

Table 17. The ranking of branch G in the 25 ranking results with C1 to C25 Ranking of Branch G [B_i^IL ,B_i^IU]

[0%, 100%] [10%, 90%] [20%, 80%] [30%, 70%] [40%, 60%]

[B_r^OL ,B_r^OU]

[0%, 100%] 8^th 8^th 8^th 8^th 8^th

[10%, 90%] 8^th 8^th 8^th 8^th 8^th

[20%, 80%] 6^th 6^th 6^th 6^th 7^th

[30%, 70%] 2^nd 2^nd 2^nd 3^rd 5^th

[40%, 60%] 1^st 1^st 2^nd 2^nd 3^rd

Under the above ranking rule, the ranking of branch G is debatable due to its average occurrence in multiple ranking levels. In other words, branch G’s ranking varies largely under different combinations. We further observe the ranking status of branch G in all combinations, as depicted in Table 17. While fixing the interval [B , _r^OL B_r^OU] with [0%, 100%] or [10%, 90%] for P , branch G is ranked the last of all branches, whatever the interval [_rG^O B , _i^IL B ] _i^IU for P . On the contrary, while we shorten the interval [_iG^I B , _r^OL B_r^OU] step by step from [0%, 100%] to [40%, 60%] for P , fixing the interval [_rG^O B , _i^IL B ] at [0%, 100%] for _i^IU P , branch _iG^I G can reach the best one of all branches.

Following the above observation, we understand that the ranking of branch G is deeply affected by the variation of interval [B , _r^OL B_r^OU]. If the manager is asked to only select some

combinations as the reference of assessment, they should concentrate more attention in determining the appropriate interval [B , _r^OL B_r^OU]. Branch G will obtain a different ranking while the manager determines a different interval [B , _r^OL B_r^OU]. As for the determination of interval [B , _i^IL B ], in this case it is not necessary for the manager to cost more effort _i^IU because these combinations show the same ranking while the interval [B , _i^IL B ] varies. _i^IU

In order to explore the cause of the above phenomenon, we observe the relationship between the ranking variations and proportion variations of branch G while varying interval [B , _i^IL B ] or [_i^IU B , _r^OL B_r^OU], as Table 18 and Table 19 depicted. It is obvious that the values of P₁^I_G(54%) and P₂^I_G(46%) obtained in C1 are simultaneously satisfied with a narrower interval [B , _i^IL B ] in C2, C3, C4, and C5. Therefore, as depicted in Table 18, while fixing _i^IU the interval [B , _r^OL B_r^OU] at [0%, 100%] and shortening the interval [B , _i^IL B ], we still _i^IU obtain the invariant values of proportion and ranking for branch G. However, as depicted in Table 19, P₁^O_G(74%) and P₂^O_G(26%) obtained in C1 are not satisfied with the narrower interval [B , _r^OL B_r^OU] in C6, C11, C16, and C21. In order to satisfy narrower intervals [B , _r^OL

OU

Br ], the smaller P₁^O_G and P₂^O_G are necessary. Therefore, the above variation in interval [B , _r^OL B_r^OU] easily results in the variations of P_1G^I , P₂^I_G and ranking.

Table 18. The proportion variations of indices of branch G while varying [B , _i^IL B ] _i^IU Combination Ranking [B_i^IL ,B_i^IU] P₁^I_G P₂^I_G [B_r^OL ,B_r^OU] P₁^O_G P₂^O_G

C1 8^th [0%, 100%] 54% 46% [0%, 100%] 74% 26%

C2 8^th [10%, 90%] 54% 46% [0%, 100%] 74% 26%

C3 8^th [20%, 80%] 54% 46% [0%, 100%] 74% 26%

C4 8^th [30%, 70%] 54% 46% [0%, 100%] 74% 26%

C5 8^th [40%, 60%] 54% 46% [0%, 100%] 74% 26%

Table 19. The proportion variations of indices of branch G while varying [B , _r^OL B_r^OU] Combination Ranking [B_i^IL ,B_i^IU] P₁^I_G P₂^I_G [B_r^OL ,B_r^OU] P₁^O_G P₂^O_G

C1 8^th [0%, 100%] 54% 46% [0%, 100%] 74% 26%

C6 8^th [0%, 100%] 54% 46% [10%, 90%] 74% 26%

C11 6^th [0%, 100%] 42% 58% [20%, 80%] 66% 34%

C16 2^nd [0%, 100%] 23% 77% [30%, 70%] 53% 47%

C21 1^st [0%, 100%] 19% 81% [40%, 60%] 42% 58%

Following the above discussion, we conclude that given the fixed interval [B , _r^OL B_r^OU], if the value of P_1G^I and P₂^I_G obtained in C1 is feasible in the narrowest interval [B , _i^IL B ] _i^IU of C5, then the values of P₁^I_G and P₂^I_G are also feasible in C2, C3, and C4. Most importantly, the ranking is invariant with the same proportion in these combinations. If the manager needs to complete all combinations, it is helpful for them to deduce the times of assessment by omitting C2, C3, and C4 while fixing the interval [B , _r^OL B_r^OU] at [0%, 100%].

在文檔中 以多指標評比受評單位時之共同權重分析 (頁 50-58)