A large company has two divisions D1 and D2.[14] The company supplies retailers with oil and spirit.
It is desired to allocate each retailer to either division D1 or division D2. This division will be the retailer’s supplier. As far as possible this division must be made so that D1 controls 40% of the market and D2 the remaining 60%. The retailers are listed below as M1 to M23. Each retailer has an estimated market for oil and spirit. Retailers M1 to M8 are in region 1; retailers M9 to M18 are in region 2; retailers M19 to M23 are in region 3. Certain retailers are considered to have good growth prospects and categorized as group A and the others are in group B. Each retailer has a certain number of delivery points as given below. It is desired to make the 40/60 split between D1 and D2 in each of the following respects:
-
Total number of delivery points
-
Control of spirit market
-
Control of oil market in region 1
-
Control of oil market in region 2
-
Control of oil market in region 3
-
Number of retailers in group A
-
Number of retailers in group B
Table 13.1:
|
Oil market |
Delivery |
Spirit market |
Growth |
||
|
Retailer |
( |
points |
( |
category |
|
|
M1 |
9 |
11 |
34 |
A |
|
|
M2 |
13 |
47 |
411 |
A |
|
|
M3 |
14 |
44 |
82 |
A |
|
|
Region 1 |
M4 |
17 |
25 |
157 |
B |
|
M5 |
18 |
10 |
5 |
A |
|
|
M6 |
19 |
26 |
183 |
A |
|
|
M7 |
23 |
26 |
14 |
B |
|
|
M8 |
21 |
54 |
215 |
B |
|
|
M9 |
9 |
18 |
102 |
B |
|
|
M10 |
11 |
51 |
21 |
A |
|
|
M11 |
17 |
20 |
54 |
B |
|
|
M12 |
18 |
105 |
0 |
B |
|
|
Region 2 |
M13 |
18 |
7 |
6 |
B |
|
M14 |
17 |
16 |
96 |
B |
|
|
M15 |
22 |
34 |
118 |
A |
|
|
M16 |
24 |
100 |
112 |
B |
|
|
M17 |
36 |
50 |
535 |
B |
|
|
M18 |
43 |
21 |
8 |
B |
|
|
M19 |
6 |
11 |
53 |
B |
|
|
M20 |
15 |
19 |
28 |
A |
|
|
Region 3 |
M21 |
15 |
14 |
69 |
B |
|
M22 |
25 |
10 |
65 |
B |
|
|
M23 |
39 |
11 |
27 |
B |
There is a certain flexibility in that any share may vary by 
%. That is, the share can vary between the limits 35/65 and 45/55.
The primary aim is to find a feasible solution. If, however, there is some choice, then possible objectives are (i) to minimize the sum of the percentage deviations from the 40/60 split and (ii) to minimize the maximum such deviation.
Build a model to see if the problem has a feasible solution and if so find the optimal solutions.
The numerical data are given in Table 13.1.