solve useing dynamic programming A total of 6 units of water is to be allocated
ID: 3199016 • Letter: S
Question
solve useing dynamic programming
A total of 6 units of water is to be allocated optimally to three users. The allocations is made in discrete steps of one unit ranging from 0 to 6. With the three users denoted as user 1, user 2, and user 3 respectively. The returns obtained from the users for a given allocation are given in the following table. Find allocations to the three users so that the total teturn is maximized.
Answer: x1 = 2, x2 = 1, x3 = 3 and Maximun return = 28
amount of water allocated return from return from return from user 1 user 2 user 3 B1(x) B2(x) B3(x) 0 0 0 0 1 5 5 7 2 8 6 12 3 9 3 15 4 8 -4 16 5 5 -15 15 6 0 -30 12Explanation / Answer
amount of water allocated
return from
return from
return from
User 1
User 2
User 3
B1(x)
B2(x)
B3(x)
0
0
0
0
1
5
5
7
2
8
6
12
3
9
3
15
4
8
-4
16
5
5
-15
15
6
0
-30
12
Each user is to be allocated between 1 and 6
So maximum a user can be allocated is 4 with other user being allocated each 1.
So rearranging the matrix we get
amount of water allocated
return from
return from
return from
user 1
user 2
user 3
B1(x)
B2(x)
B3(x)
0
0
0
0
1
5
5
7
2
8
6
12
3
9
3
15
4
8
-4
16
Possible allocations are as follows
B1
B2
B3
Total Return
Allocation
1
1
4
Return
5
5
16
26
Allocation
1
2
3
Return
5
6
15
26
Allocation
2
1
3
Return
8
5
15
28
Allocation
1
3
2
Return
5
3
12
20
Allocation
3
1
2
Return
9
5
12
Allocation
2
2
2
Return
8
6
12
26
Allocation
3
2
1
Return
9
6
7
22
Allocation
2
3
1
Return
8
3
7
17
Allocation
2
2
2
Return
8
6
12
26
Allocation
1
4
1
Return
5
-4
7
8
Allocation
4
1
1
Return
8
5
7
20
Best allocation is
B1
B2
B3
2
1
3
and maximum return is 28
amount of water allocated
return from
return from
return from
User 1
User 2
User 3
B1(x)
B2(x)
B3(x)
0
0
0
0
1
5
5
7
2
8
6
12
3
9
3
15
4
8
-4
16
5
5
-15
15
6
0
-30
12
Each user is to be allocated between 1 and 6
So maximum a user can be allocated is 4 with other user being allocated each 1.
So rearranging the matrix we get
amount of water allocated
return from
return from
return from
user 1
user 2
user 3
B1(x)
B2(x)
B3(x)
0
0
0
0
1
5
5
7
2
8
6
12
3
9
3
15
4
8
-4
16
Possible allocations are as follows
B1
B2
B3
Total Return
Allocation
1
1
4
Return
5
5
16
26
Allocation
1
2
3
Return
5
6
15
26
Allocation
2
1
3
Return
8
5
15
28
Allocation
1
3
2
Return
5
3
12
20
Allocation
3
1
2
Return
9
5
12
Allocation
2
2
2
Return
8
6
12
26
Allocation
3
2
1
Return
9
6
7
22
Allocation
2
3
1
Return
8
3
7
17
Allocation
2
2
2
Return
8
6
12
26
Allocation
1
4
1
Return
5
-4
7
8
Allocation
4
1
1
Return
8
5
7
20
Best allocation is
B1
B2
B3
2
1
3
and maximum return is 28
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