Other approaches to restrict multipliers

Transkriptio

1 Other approaches to restrict multipliers Heikki Tikanmäki Optimointiopin seminaari

2 Contents Short revision (6.2) Another Assurance Region Model (6.3) Cone-Ratio Method (6.4) An Application of the Cone-Ratio Model (6.5)

3 Assurance Region Model (revision) Last week we restrictid the weights by adding constraints like: Loosely speaking, this means that the order of magnitude of the relative weights is fixed

4 Assurance Region Global Model Modified Assurance Region Model Weight restrictions described by an example: The bounds for inputs are analogous

5 Motivation The traditional Assurance Region Model bounds can be written as The relative weight of input 1 depends on the input value of that particular DMU. In AR Global Model the bounds for relative weights are common to all DMU s

6 Cone-Ratio Model Cone-Ratio Model is a generalisation of the Assurance Region Method The admissible regions of weights are polyhedral convex cones

7 Admissible Regions of Weights Let j=1,,k be direction vectors and the feasible region of v be a convex cone spanned by them. The conditions for the outputs can be written analogously

8 Example (1/2) Assurance Region method as a special case of Cone-Ratio method This region is spanned by two vectors

9 Example (2/2)

10 CCR Formulation (1/3) The CCR Model written in this case: This is the ordinary CCR model if and

11 CCR Formulation (2/3) V spanned by by and U spanned

12 CCR Formulation (3/3) The dual problem can be expressed as

13 The Hospital Example (1/2)

14 The Hospital Example (2/2) The data set (X,Y) as before. Define Then The transformet data set For Hospital H1:

15 How to Choose admissible directions Several ways to choose and Use the knowledge of experts First solve the original CCR model and then select preferable DMU s among the efficient ones. Use the set of the optimal weights for the preferable DMU s as the admissible directions Combine these two methods (This approach is chosen in the following application)

16 An Application of the Cone-Ratio Method (1/6) We consider the banks in Texas Year 1984 was the last good year the amount of bank failures started to increase. The following years were even worse The approach used by Texas Banking Department failed to catch this development and this was considered to be a fair test for DEA

17 Application (2/6)

18 Application (3/6) Inputs chosen by experts Interst Expenses Non-Interest Expenses Provision for Loan Loses Total Deposits Outputs chosen by the experts Interest Incomes Total Non-Interst Income Allowance for Loan Loses Total Loans

19 Application (4/6) Besides, the Texas Banks some external excellent banks were included in the study All the banks were submitted to DEA efficiency evaluation Dual multiplier values for inefficient excellent banks were not used

20 Application (5/6)

21 Application (6/6) Some excellent banks improved their efficiency All of the Texas Banks lose efficiency status in 1985 Many Texas banks lose the status of full efficiency from 1984 to 1985

22 Summary Different methods for restricting multipliers: Assurance Region Model Assurance Region Model Global Cone-Ratio Method Examples General Hospital example with CRM Real world study of Texas banks using CRM

23 References Cooper W.W., Seiford L.M., Tone K., Data Envelopment Analysis, Springer.

24 Exercise How are Assurance Region Method, Assurance Region Global Model, Cone- Ratio Method and CCR Model related to each others?