Ranking Universities using Data Envelopment Analysis

Transcription

1 Ranking Universities using Data Envelopment Analysis Bronwen Edge September 1, 2016 Bronwen Edge Data Envelopment Analysis September 1, / 21

2 Outline 1 Introduction What is DEA CCR Model BCC Model 2 The Data Overview 3 Scenario 4 Examples Adams CRS Adams VRS Adams Scottish Universities Choosing inputs and outputs 5 Discussion Advantages of DEA Limitations of DEA 6 Questions Bronwen Edge Data Envelopment Analysis September 1, / 21

3 Introduction Each year university ranking tables are released, A universities place in such tables greatly influences a student s decision. Can a personal ranking be made to suit a students individual needs? DEA a suitable method for this problem? Bronwen Edge Data Envelopment Analysis September 1, / 21

4 What is DEA Introduced by Charnes, Cooper and Rhodes in It is a benchmarking technique used to evaluate firms and assess their performance against their peers. DEA creates a frontier based on the most efficient firms which envelopes the inefficient firms. E and F are inefficient. Bronwen Edge Data Envelopment Analysis September 1, / 21

5 What is DEA Figure 1: To improve F efficiency a proportional decrease in input usage should be made to place F on the frontier while still producing the same outputs. E is the projected point for optimum efficiency for firm E. DEA involves constructing a hypothetical best case firm out of the existing firms, if any of firms are not as efficient as this best case one, they are inefficient. Collecting all the efficient cases together makes up the frontier, which then envelops the inefficient firms. Bronwen Edge Data Envelopment Analysis September 1, / 21

6 Notation DMU s Inputs X Outputs Q... DMU n x n q n Notation: There are m inputs, s outputs for each of the N firms, for the nth firm the inputs and outputs are represented by vectors x n and q n. Efficiency = Outputs Inputs Bronwen Edge Data Envelopment Analysis September 1, / 21

7 CCR Model Ratio Form Charnes, Cooper and Rhodes (1978) proposed a CCR model that had an input orientation and assumed constant returns to scale. Input orientation means that when trying to improve efficiency, inputs are reduced while outputs remain constant. As the returns are constant an increase/decrease in inputs will lead to an equivalent increase/decrease in outputs. Bronwen Edge Data Envelopment Analysis September 1, / 21

8 CCR Model Ratio Form We would like to measure the ratio of all outputs over all inputs. max u,v (u q n /v x n ), st u q j v x j 0, u, v 0, Set v x n = 1 to avoid an infinite amount of solutions. max u,v (u q n ), st v x n = 1 u q j v x j 0, u, v 0, j = 1, 2,..., N u =output weights v =input weights Bronwen Edge Data Envelopment Analysis September 1, / 21

9 CCR Model Envelopment Form Next we use the duality in linear programming, to derive the Envelopment form for this problem: min θ,λ θ, st q n + Qλ 0, θx n Xλ 0, λ 0. λ the constraint vector, θ n is the efficiency score of the nth firm. Involves fewer constraints. θ = 1 a DMU is efficient and is added to the frontier. Bronwen Edge Data Envelopment Analysis September 1, / 21

10 BCC Model The CCR model is used when an increase in input results in the same proportional increase in output. However most of the time this doesn t happen. In 1984, Banker, Charnes and Cooper suggested adjusting the CRS model to account for this, the Variable Returns to Scale (BCC) model. min θ,λ θ, st q i + Qλ 0, θx i Xλ 0, N1 λ = 1 λ 0. This model envelopes the data tighter and can tell you if a DMU is too large or small to be efficient. It is a way of only comparing DMU s of a similar size. Bronwen Edge Data Envelopment Analysis September 1, / 21

11 The Data Overview Choose to look at 69 university mathematics departments from across UK. Missing data. Different units. Massive variation in institutions. I sourced the data from: Bronwen Edge Data Envelopment Analysis September 1, / 21

12 Scenario Adam is a Scottish mathematics student, he will be choosing his options for university soon. Adam is very academic and has been advised to apply for Oxbridge but doesn t really feel like they re for him. He is interested in exploring other options before he makes his decision. Adam doesn t want to accumulate a muscular amount of debt so hes looking to reduce his living costs. He s interested in value for money so studying at a Scottish university is attractive and high teaching standards is important. Adam is interested in working rather than further study. Adams hobbies include: Gunplay. Bronwen Edge Data Envelopment Analysis September 1, / 21

13 Examples 1 Inputs: Entry Requirements with teaching score Average annual fee Cost of private accommodation Outputs: Percentage of students with a good degree Average salary after 6 months Percentage of students working in a professional job after 6 months Bronwen Edge Data Envelopment Analysis September 1, / 21

14 Adams CRS Entry requirements with teaching score, average annual fee, cost of private accommodation against percentage of students with a good degree, average salary after 6 months and percentage of students working in a professional job after 6 months. Bronwen Edge Data Envelopment Analysis September 1, / 21

15 Adams VRS Entry requirements with teaching score, average annual fee, cost of private accommodation against percentage of students with a good degree, average salary after 6 months and percentage of students working in a professional job after 6 months. Bronwen Edge Data Envelopment Analysis September 1, / 21

16 Examples:Scottish Universities CRS Entry requirements with teaching score, average annual fee, cost of private accommodation against percentage of students with a good degree, average salary after 6 months and percentage of students working in a professional job after 6 months. Bronwen Edge Data Envelopment Analysis September 1, / 21

17 Adams Scottish Universities VRS Entry requirements with teaching score, average annual fee, cost of private accommodation against percentage of students with a good degree, average salary after 6 months and percentage of students working in a professional job after 6 months. Bronwen Edge Data Envelopment Analysis September 1, / 21

18 Choosing inputs and outputs Things to think about when selecting inputs and outputs: Fitting inputs and outputs to a students personal gives a better model. Try not to introduce bias, be objective! Try and avoid double counting, Avoid mutually exclusive outputs. Bronwen Edge Data Envelopment Analysis September 1, / 21

19 Advantages of DEA DEA can handle multiple input and output models, Peers are directly compared against peers, Inputs and outputs can have very different units. Works very well for objective inputs and outputs. Bronwen Edge Data Envelopment Analysis September 1, / 21

20 Limitations of DEA As DEA is a extreme point technique it is skewed by extreme values or outliers. Large problems can be computationally intensive. Choosing the right inputs/outputs/environmental factors is tricky when considering universities. Some may lead to double counting as they are not independent. Not convinced that DEA is suitable for this problem. Bronwen Edge Data Envelopment Analysis September 1, / 21

21 Questions Thanks for listening, Any Questions? Bronwen Edge Data Envelopment Analysis September 1, / 21