Measuring the Efficiency of Decision-making Units : Applied Data Envelopment Analysis
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Measuring the Efficiency of Decision-making Units : Applied Data Envelopment Analysis. 報告人:陳明山 學 號: 917803. 1. A. Boussofiane, R. G. Dyson, E. Thanassoulis, “Applied data envelopment analysis,” European Journal of Operations Research (1991) 1-15.

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917803

Measuring the Efficiency of Decision-making Units : Applied Data Envelopment Analysis

報告人:陳明山

學 號:917803


917803

1.

A. Boussofiane, R. G. Dyson, E. Thanassoulis, “Applied data envelopment analysis,” European Journal of Operations Research (1991) 1-15

A. Charnes, W. W. Cooper, E. Rhodes, “Measuring the efficiency of decision making,” European Journal of Operations Research (1978) 429-444

2.

3.

Chiang Kao, Yong Chi Yang, “Reorganization of forest districts via efficiency measurement,” European Journal of Operations Research (1992) 356-362

4.

5.

高強, 黃旭男, Toshiyuki Sueyoshi “管理績效評估 -資料包絡分析法,” 華泰文化事業公司 (2003) 97-117

Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398

Reference


917803

Outline

■Deterministic & Stochastic DEA

■Introduction for DEA

■Deterministic Model

■Case Study on Deterministic Model

■Drawbacks for Deterministic Model

■Stochastic Model

■Illustration

■Case Study

■Conclusion


917803

Deterministic & Stochastic DEA o

一般傳統DEA模式僅將過去確定的資訊納入DEA的運算架構中,稱為Deterministic DEA。若將未來不確定性資訊納入DEA的運算架構中,並以隨機過程予以描述,則稱為Stochastic DEA

o高強, 黃旭男, Toshiyuki Sueyoshi “管理績效評估 -資料包絡分析法,” 華泰文化事業公司 (2003) 97


917803

Introduction for DEA1

In the simplest case where a unit has a single input and a single output, efficiency is defined simply as:

output

Efficiency =

input

1 A. Boussofiane, R. G. Dyson, E. Thanassoulis, “Applied data envelopment analysis,” European Journal of Operations Research (1991) 1-15


917803

Introduction for DEA2

More typically organizational units have multiple inputs and outputs, defining the efficiency as:

weighted sum of output

Efficiency =

weighted sum of input

2 A. Boussofiane, R. G. Dyson, E. Thanassoulis, “Applied data envelopment analysis,” European Journal of Operations Research (1991) 1-15


917803

Deterministic Model 3

where

n= the number of units

s= the number of outputs

m= the number of inputs

subject to

ur = the weight given to output r

vi = the weight given to input i

yrj = amount of output r from unit j

xij = amount of input i from unit j

ur, vi

ε; r=1, 2, …, s; i=1, 2, …, m

j=1, 2, …, n

3 A. Charnes, W. W. Cooper, E. Rhodes, “Measuring the efficiency of decision making,” European Journal of Operations Research (1978) 429-444


917803

Deterministic Model 4

where

n= the number of units

subject to

s= the number of outputs

m= the number of inputs

ur = the weight given to output r

vi = the weight given to input i

yrj = amount of output r from unit j

xij = amount of input i from unit j

ur, vi

ε; r=1, 2, …, s; i=1, 2, …, m

j=1, 2, …, n

4 CS Sarrico, RG Dyson“Using DEA for planning in UK university,” Journal of the Operations Research Society (2000) 789-800


917803

Case study5 on deterministic model

Input

Output

原始林

木蓄積

森林

遊樂

木材

生產

平均

蓄積

預算

勞力

面積

67.55

85.78

80.33

205.92

51.28

82.09

123.02

71.77

61.95

25.83

27.87

72.60

84.83

82.83

123.98

104.65

183.49

117.51

104.94

82.44

88.16

99.77

105.80

107.60

132.73

104.28

44.37

55.13

53.30

144.16

32.07

46.51

87.35

69.19

33.00

9.51

14.00

44.67

159.12

60.85

108.46

79.06

59.66

84.50

127.28

98.80

123.14

86.37

227.20

146.43

173.48

171.11

26.04

43.51

27.28

14.09

46.20

44.87

43.33

44.83

45.43

19.40

25.47

5.55

11.53

85.00

173.93

132.49

196.29

144.99

108.53

125.84

74.54

79.60

120.09

131.79

135.65

110.22

23.95

6.45

42.67

16.15

0.00

0.00

404.69

6.14

1252.62

0.00

0.00

24.13

49.09

文山

竹東

大甲

大雪山

埔里

巒大

玉山

楠濃

恆春

關山

玉里

木瓜

蘭陽

資料來源:1978~1988年林務局統計資料

5 Chiang Kao, Yong Chi Yang, “Reorganization of forest districts via efficiency measurement,” European Journal of Operations Research (1992) 356-362


917803

Drawbacks for Deterministic Model

傳統DEA模式將過去確定的資訊納入DEA的運算架構,作為未來決策考量,從現實觀點較不實際。然未來產出通常受外在經濟或其他變動因子影響,在未來績效進行預測時,將未來產出視為隨機性變數,較使用過去資料來預估為適切


917803

Case Study 6

日本連鎖便利商店實際投入與產出估計值

2004年 (產出預估值)

2003年 (投入)

投入及產出

估計值

資本

營業額(億日圓)

顧客數(人/時)

員工人數

分店數

便利商店

(萬日圓)

1720000

3350

7780

Seven-Eleven

1658500

1777

6531

Family Mart

346295

322

664

HOTSPAR

2800

22

54

Apple Mart

6000

85

102

Everyone

7150

36

67

Caramel Mart

4000

162

866

Coco Store

6高強, 黃旭男, Toshiyuki Sueyoshi “管理績效評估 -資料包絡分析法,” 華泰文化事業公司 (2003) 112-117


917803

Case Study 6

日本連鎖便利商店實際投入與產出估計值

2004年 (產出預估值)

2003年 (投入)

投入及產出

估計值

資本

營業額(億日圓)

顧客數(人/時)

員工數

分店數

便利商店

(萬日圓)

OP ML PE

OP ML PE

1720000

3350

7780

Seven-Eleven

20341 19661 18981

411410 329128 246846

1658500

1777

6531

Family Mart

8978 8397 7816

203475 162378 119107

1058 883 708

21866 16839 11310

346295

322

664

HOTSPAR

73 64 55

1866 1409 933

2800

22

54

Apple Mart

235 175 114

5372 4263 3070

6000

85

102

Everyone

110 84 58

1968 1443 944

7150

36

67

Caramel Mart

1399 1288 1177

24245 18221 11673

4000

162

866

Coco Store

OP:最樂觀估計值, ML:最可能估計值, PE:最悲觀估計值

6高強, 黃旭男, Toshiyuki Sueyoshi “管理績效評估 -資料包絡分析法,” 華泰文化事業公司 (2003) 112-117


917803

Case Study 6

根據經驗統計,樂觀估計值、最可能估計值與最悲觀估計值之機率分配為beta分配,可得到下列估計值:

6高強, 黃旭男, Toshiyuki Sueyoshi “管理績效評估 -資料包絡分析法,” 華泰文化事業公司 (2003) 112-117


917803

表效率值之最大期望水準

表效率值大於 之可容忍誤差程度

ur, vi

ε; r=1, 2, …, s; i=1, 2, …, m;

j=1, 2, …, n

Stochastic Model 7

where

n= the number of units

s= the number of outputs

subject to

m= the number of inputs

ur = the weight given to output r

vi = the weight given to input i

yrj = amount of output r from unit j

xij = amount of input i from unit j

7 Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398


917803

Stochastic Model

Assume


917803

表效率值之期望水準

表效率值大於 之可容忍誤差程度

ur, vi

ε; r=1, 2, …, s; i=1, 2, …, m;

j=1, 2, …, n

Stochastic Model 7

where

n= the number of units

s= the number of outputs

subject to

m= the number of inputs

ur = the weight given to output r

vi = the weight given to input i

yrj = amount of output r from unit j

xij = amount of input i from unit j

7 Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398


917803

Illustration

output

inputs

output

y

x1

x2

OP ML PE

2

6

1

2

3

2

U0

U1

2

5

1.5

2

2.5

2

→ F-1(0.9)=1.282

Max hUO = 2u

Subject to

2v1+6v2 = 1

hU0=0.06

u= 0.03

v1= 0.5

v2≒0

hU1=0.22

u= 0.11

v1= 0

v2≒0.2

1.6 v1+ 4.8 v2-{2u+ *1.282}

1.6 v1 - 4 v2 -{2u+ *1.282}

u, v1, v2

10-4


917803

Case study 8

自第2次世界大戰以後,為確保供油穩定,日本石油產業一直受日本政府的保護,因此產油成本一直高居不下。日本政府立法通過自1997年4月開放民間營運石油產業,為因應市場開放及提高營運效率,是以日本石油公司於1997年委託進行本項研究,以作為未來營運策略參考

8 Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398


917803

Case study 8

INPUT (1998)

OUTPUT ESTIMATE (1999)

No of

employees

Size of

station

Operation

cost

Group

Station

Gasoline

PE ML OP

Petrol

PE ML OP

420 480 530

.

.

.

500 540 600

170 200 220

.

.

.

120 140 155

10

.

.

.

9

958

.

.

.

1087

5203

.

.

.

1087

Large

1

.

.

.

20

Medium

21

.

.

.

40

5

.

.

.

7

513

.

.

.

628

3028

.

.

.

3634

140 180 210

.

.

.

230 250 280

45 60 70

.

.

.

100 115 135

Small

41

.

.

.

60

75 85 100

.

.

.

65 80 90

20 30 35

.

.

.

25 35 40

3

.

.

.

4

287

.

.

.

326

1307

.

.

.

1453

8 Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398


917803

Case study 8

β=1.0

efficiency

Group

Station

α=0.05

α=0.1

α=0.5

α=0.9

α=0.95

Large

1

.

20

mean

S dev.

89.77

.

95.19

87.92

5.30

91.06

.

96.21

89.12

5.32

95.95

.

100.00

93.63

5.38

101.41

.

104.43

98.69

5.52

103.08

.

105.77

100.21

5.56

94.08

.

100.00

93.71

5.55

Med.

21

.

40

mean

S dev.

60.51

.

70.77

67.61

8.60

59.68

.

69.65

66.64

8.45

63.61

.

75.02

71.28

9.17

67.02

.

79.82

75.39

9.85

68.03

.

81.29

76.64

10.05

65.35

.

73.62

70.89

8.90

Small

41

.

60

mean

S dev.

62.17

.

53.57

54.40

8.41

63.01

.

54.34

55.21

8.53

66.17

.

57.30

58.31

8.99

69.68

.

60.62

61.80

9.51

70.74

.

61.63

62.87

9.67

69.88

.

61.40

59.61

8.09

8 Toshiyuki Sueyoshi, “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company,” The International Journal of Management Science (2000) 385-398


917803

4. 在隨機性模式假設隨機變數為常態分配(

,是否有其他分配更為合適,未來可進一步研究

Conclusion

  • 上述日本油業案例,隨機性模式與確定性模式結果相似,主要由於在隨機性模式之估計準確

2. 由敏感性分析結果可獲知各條件效率趨勢應為一致

3. 大型油站顯然較中小型油站有效率,而中型油站較小型油站有效率,建議中小型油站應著眼於進一步整併以提高營運績效


917803

Thanks for your attention


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