One approach is to do brute-force by generating all possibilities. For the second part is seems you will need CPM or PERT diagram.

Dear Aung Pyae,

It seems that the second question is incomplete or wrong. Because the arrangement of tasks to complete the project have produced a simple path (it is linear) and based on each friends-to-tasks assignment it takes 20 days to complete the project.

But about the first question, due to the help of two friends, the number of helpers which we require are as follows:

9-10 : 2, 10-11 : 2, 11-12 : 5, 12-13 : 7, 13-14 : 4, 14-15 : 2, 15-16 : 1, 16-17 : 2.

Then, define decision variables as follows:

Xj := Number of part-time helpers who begin their works at 'j' o'clock and do it for a 3-hours period. (j=9, 10, ..., 14).

Then, solve the following ILP problem:

Min 30 X9 + 30 X10 + ... + 30 X14

s.t.

X9 >= 2,

X9 + X10 >= 2,

X9 + X10 + X11 >=5,

X10 + X11 + X12 >= 7,

X11 + X12 + X13 >= 4,

X12 + X13 + X14 >= 2,

X13 + X14 >= 1

X14 >= 2

X9, X10, ..., X14 >=0 and integer

Dear Aung Pyae,

Please see the simplex method in the following paper:

Article Practical application of simplex method for solving linear p...

Best regards,

Aleksandar Velinov

The most regular approach is formulate a mathematical optimization model and try to implement it in some optimization-framework. The problem you are showing are not complicated at all, so this regular\general approach surely will give you the optimal solution (if this exists).