模型评价模型13 数据包络分析(四)

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.

构建模型

model = LpProblem('CRS_model', LpMinimize)

构建决策变量$\theta_r$和$\lambda_k$.

theta_r = LpVariable(f'theta_r')
lambda_k = LpVariable.dicts(f'lambda_k', lowBound=0, indexs=K)

设置目标函数 $$\operatorname{Min} E_{r}^{C R S}=\theta_{r}$$

model += theta_r #Dual formulation

设置约束条件 $$ \begin{gathered} \text { s.t. } \sum_{k=1}^{K} \lambda_{k} X_{k i} \leq \theta_{r} X_{r i}, \forall i \\ \sum_{k=1}^{K} \lambda_{k} Y_{k j} \geq Y_{r j}, \forall j \\ \lambda_{k} \geq 0, \forall k \end{gathered} $$

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]])

求解模型
```
model.solve()
```

Step. 5¶

使用VRS DEA模型输入导向型的对偶形式

构建模型

model = LpProblem('VRS_model', LpMinimize)

构建决策变量$\theta_r$和$\lambda_k$.

theta_r = LpVariable(f'theta_r')
lambda_k = LpVariable.dicts(f'lambda_k', lowBound=0, indexs = K)

构建目标函数 $$\operatorname{Min} E_{r}^{V R S}=\theta_{r}$$

model += theta_r #Dual formulation

设置约束条件 $$ \begin{gathered} \text { s.t. } \sum_{k=1}^{K} \lambda_{k} X_{k i} \leq \theta_{r} X_{r i}, \forall i \\ \sum_{k=1}^{K} \lambda_{k} Y_{k j} \geq Y_{r j}, \forall j \\ \sum_{k=1}^{K} \lambda_{k}=1 \\ \lambda_{k} \geq 0, \forall k \end{gathered} $$

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'

求解模型
```
model.solve()
```

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
[3] Lee, C. Y., & Johnson, A. L. (2012). Two-dimensional efficiency decomposition to measure the demand effect in productivity analysis. European Journal of Operational Research, 216(3), 584-593.
[4] Returns to scale (2020, April 16). In Wikipedia, the free encyclopedia. Retrieved June 22, 2020, from https://en.wikipedia.org/wiki/Returns_to_scale
[5] Vörösmarty, G., & Dobos, I. (2020). A literature review of sustainable supplier evaluation with Data Envelopment Analysis. Journal of Cleaner Production, 121672.