案例分析:航线效率

数据准备

本案例中,有13家航空公司(DMUs)以3个输入和2个输出来估计效率。输入数据包括载客量、燃油和员工数量,输出数据包括旅客人数和货运数量

DMU Aircraft
(fleet size)
Fuel
(gallons)
Employee
(units)
Passenger
(passenger-miles)
Freight
(ton-miles)
A 109 392 8259 23756 870
B 115 381 9628 24183 1359
C 767 2673 70923 163483 12449
D 90 282 9683 10370 509
E 461 1608 40630 99047 3726
F 628 2074 47420 128635 9214
G 81 75 7115 11962 536
H 153 458 10177 32436 1462
I 455 1722 29124 83862 6337
J 103 400 8987 14618 785
K 547 1217 34680 99636 6597
L 560 2532 51536 135480 10928
M 423 1303 32683 74106 4258

编程环境

Python 3.7.4 + PuLP 2.0

Execute

Step. 1

引入必要的包

from pulp import LpProblem, LpMinimize, LpVariable, lpSum, value

Step. 2

构建数据集

K = ["A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M"]
I = ["Aircraft", "Fuel", "Employee"]
J = ["Passenger", "Freight"]

Step. 3

引入数据

X={'Aircraft': {'A': 109.0,
  'B': 115.0,
  'C': 767.0,
  'D': 90.0,
  'E': 461.0,
  'F': 628.0,
  'G': 81.0,
  'H': 153.0,
  'I': 455.0,
  'J': 103.0,
  'K': 547.0,
  'L': 560.0,
  'M': 423.0},
 'Fuel': {'A': 392.0,
  'B': 381.0,
  'C': 2673.0,
  'D': 282.0,
  'E': 1608.0,
  'F': 2074.0,
  'G': 75.0,
  'H': 458.0,
  'I': 1722.0,
  'J': 400.0,
  'K': 1217.0,
  'L': 2532.0,
  'M': 1303.0},
 'Employee': {'A': 8259.0,
  'B': 9628.0,
  'C': 70923.0,
  'D': 9683.0,
  'E': 40630.0,
  'F': 47420.0,
  'G': 7115.0,
  'H': 10177.0,
  'I': 29124.0,
  'J': 8987.0,
  'K': 34680.0,
  'L': 51536.0,
  'M': 32683.0}}

Y={'Passenger': {'A': 23756.0,
  'B': 24183.0,
  'C': 163483.0,
  'D': 10370.0,
  'E': 99047.0,
  'F': 128635.0,
  'G': 11962.0,
  'H': 32436.0,
  'I': 83862.0,
  'J': 14618.0,
  'K': 99636.0,
  'L': 135480.0,
  'M': 74106.0},
 'Freight': {'A': 870.0,
  'B': 1359.0,
  'C': 12449.0,
  'D': 509.0,
  'E': 3726.0,
  'F': 9214.0,
  'G': 536.0,
  'H': 1462.0,
  'I': 6337.0,
  'J': 785.0,
  'K': 6597.0,
  'L': 10928.0,
  'M': 4258.0}}

Step. 4

使用 CRS DEA 模型的输入导向型模型的对偶形式.

model += theta_r #Dual formulation

Step. 5

使用VRS DEA模型输入导向型的对偶形式

theta_r = LpVariable(f'theta_r')
lambda_k = LpVariable.dicts(f'lambda_k', lowBound=0, indexs = K)
model += theta_r #Dual formulation
for i in I:
        model += lpSum([
                lambda_k[k] * X[i][k]
            for k in K]) <= theta_r * float(X[i][K[r]])
    for j in J:
        model += lpSum([
                lambda_k[k] * Y[j][k]
            for k in K]) >= float(Y[j][K[r]])
    model += lpSum([ lambda_k[k] for k in K]) == 1 #Convex Combination for r'

Step. 6

调用函数获取输出

OE_outputText = 'These are OE of all DMUs\n-------------\n'
TE_outputText = 'These are TE of all DMUs\n-------------\n'
SE_outputText = 'These are SE of all DMUs\n-------------\n'

for k in range(len(K)):
    OE_text, OE_val = getOverallEfficiency(k)
    TE_text, TE_val = getTechnicalEfficiency(k)
    OE_outputText += OE_text
    TE_outputText += TE_text
    SE_outputText += f'{K[k]}{round(OE_val / TE_val, 3)}\n'
print(OE_outputText)
print(TE_outputText)
print(SE_outputText)

结果

我们发现研究C、G H,我,K和L都处于一个更好的效率状态,D J和M从这个表处于低效率状态。可以将规模效应SE与输入和输出数据分解来理解和提高低效率状态。

DMU OE TE SE
A 0.978 1 0.978
B 0.968 1 0.968
C 1 1 1
D 0.537 0.9 0.597
E 0.969 0.996 0.973
F 0.978 1 0.978
G 1 1 1
H 1 1 1
I 1 1 1
J 0.619 0.886 0.698
K 1 1 1
L 1 1 1
M 0.835 0.849 0.984

总结

当我想知道如何对不同的航空公司或银行分支机构进行基准测试时,DEA是一种评估效率的可靠技术。该技术适用于任何需要评估同质单元效率的行业的执行人员或分析师,特别是在制造业。在应用DEA模型时,各单元之间可以通过相对效率进行比较,但不能告诉我们如何提高效率。因此,由于DEA只是一个评估各单元相对效率的工具,因此需要采用另一个途径来寻找改进的解决方案。

参考文献

[1] Coelli, T. J., Rao, D. S. P., O'Donnell, C. J., & Battese, G. E. (2005). An introduction to efficiency and productivity analysis. Springer Science & Business Media.
[2] Data envelopment analysis. (2020,June 9). In Wikipedia, the free encyclopedia. Retrieved June 20, 2020, from https://en.wikipedia.org/wiki/Data_envelopment_analysis
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