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 |
(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.