Problem Statement

A car manufacturer wants to evaluate the efficiencies of different garages who have received a franchise to sell its cars.[23] The method to be used is Data Envelopment Analysis (DEA). References to this technique are given in Section 3.2. Each garage has a certain number of measurable ‘inputs’. These are taken to be: Staff, Showroom Space, Catchment Population in different economic categories and annual Enquiries for different brands of car. Each garage also has a certain number of measurable ‘outputs’. These are taken to be: Number Sold of different brands of car and annual Profit. Table 23.1 gives the inputs and outputs for each of the 28 franchised garages.

A central assumption of DEA (although modified models can be built to alter this assumption) is that constant returns to scale are possible, i.e. doubling a garage’s inputs should lead to a doubling of all its outputs. A garage is deemed to be efficient if it is not possible to find a mixture of proportions of other garages whose combined inputs do not exceed those of the garage being considered, but whose outputs are equal to, or exceed, those of the garage. Should this not be possible then the garage is deemed to be inefficient and the comparator garages can be identified.

Table 23.1:  

   

Inputs

 

Outputs

     

Show-

   

Enq.

Enq.

       
     

room

Popn.

Popn.

Alpha

Beta

 

Alpha

Beta

 
   

Staff

space

cat. 1

cat. 2

model

model

 

sales

sales

Profit

Garage

 

(100 m$^2$)

(1000s)

(1000s)

(100s)

(100s)

 

(1000s)

(1000s)

(millions)

1

Winchester

7

8

10

12

8.5

4

 

2

0.6

1.5

2

Andover

6

6

20

30

9

4.5

 

2.3

0.7

1.6

3

Basingstoke

2

3

40

40

2

1.5

 

0.8

0.25

0.5

4

Poole

14

9

20

25

10

6

 

2.6

0.86

1.9

5

Woking

10

9

10

10

11

5

 

2.4

1

2

6

Newbury

24

15

15

13

25

1.9

 

8

2.6

4.5

7

Portsmouth

6

7

50

40

8.5

3

 

2.5

0.9

1.6

8

Alresford

8

7.5

5

8

9

4

 

2.1

0.85

2

9

Salisbury

5

5

10

10

5

2.5

 

2

0.65

0.9

10

Guildford

8

10

30

35

9.5

4.5

 

2.05

0.75

1.7

11

Alton

7

8

7

8

3

2

 

1.9

0.70

0.5

12

Weybridge

5

6.5

9

12

8

4.5

 

1.8

0.63

1.4

13

Dorchester

6

7.5

10

10

7.5

4

 

1.5

0.45

1.45

14

Bridport

11

8

8

10

10

6

 

2.2

0.65

2.2

15

Weymouth

4

5

10

10

7.5

3.5

 

1.8

0.62

1.6

16

Portland

3

3.5

3

20

2

1.5

 

0.9

0.35

0.5

17

Chichester

5

5.5

8

10

7

3.5

 

1.2

0.45

1.3

18

Petersfield

21

12

6

6

15

8

 

6

0.25

2.9

19

Petworth

6

5.5

2

2

8

5

 

1.5

0.55

1.55

20

Midhurst

3

3.6

3

3

2.5

1.5

 

0.8

0.20

0.45

21

Reading

30

29

120

80

35

20

 

7

2.5

8

22

Southampton

25

16

110

80

27

12

 

6.5

3.5

5.4

23

Bournemouth

19

10

90

22

25

13

 

5.5

3.1

4.5

24

Henley

7

6

5

7

8.5

4.5

 

1.2

0.48

2

25

Maidenhead

12

8

7

10

12

7

 

4.5

2

2.3

26

Fareham

4

6

1

1

7.5

3.5

 

1.1

0.48

1.7

27

Romsey

2

2.5

1

1

2.5

1

 

0.4

0.1

0.55

28

Ringwood

2

3.5

2

2

1.9

1.2

 

0.3

0.09

0.4


A linear programming model can be built to identify efficient and inefficient garages and their comparators.



[23] Reproduced with permission of John Wiley & Sons Ltd. (Williams 1999, pp. 253–255).