You are given some tasks of size xK, yK and zK respectively and there are at most 9 tasks each. Determine the best that you could achieve if you are to fill up a hole of size SK, for the following values of x, y, z and S, by maximizing the space used (up to S), i.e. minimizing the used space, if any. Show how many tasks of size x, how many tasks of size y and how many tasks of size z are used in each of the three cases. (g) Now if we relax the requirement so that there are no upper limits on the number of tasks for each type, determine the maximal space usage and the task mix. (h) If we tighten the requirement so that there are still at most 9 tasks of each size, but we also require that each type of tasks must be used at least once, determine maximal space usage and the task mix. Case 1 2 3 X 22 26 28 y 33 43 65 50 77 74 S 488 556 777 You could complete the following table. Fill in the number of tasks inside the brackets and the corresponding maximal usage. (At most 9 tasks each (g) Unlimited number of tasks (h) Strictly 1 to 9 tasks each Case 2 1 3 1 2 3 2 1 3 22×( ) 26×( ) 28x() 22×( ) 26x() 28×( ) 22x() 26x() 28×( ) 33x() 43x() 65x() 33x() y 43x() 65x() 33x() 43x() 65x() 50x() 77x() 74x() 50x() 77x() 74x() 50x() 77x() 74x() S 556 488 777 488 556 488 777 777 556 X Z Max usage
You are given some tasks of size xK, yK and zK respectively and there are at most 9 tasks each. Determine the best that you could achieve if you are to fill up a hole of size SK, for the following values of x, y, z and S, by maximizing the space used (up to S), i.e. minimizing the used space, if any. Show how many tasks of size x, how many tasks of size y and how many tasks of size z are used in each of the three cases. (g) Now if we relax the requirement so that there are no upper limits on the number of tasks for each type, determine the maximal space usage and the task mix. (h) If we tighten the requirement so that there are still at most 9 tasks of each size, but we also require that each type of tasks must be used at least once, determine maximal space usage and the task mix.
Case
1
2
3
X
22
26
28
y
33
43
65
50
77
74
S
488
556
777
You could complete the following table. Fill in the number of tasks inside the brackets and the corresponding maximal usage.
(At most 9 tasks each
(g) Unlimited number of tasks
(h) Strictly 1 to 9 tasks each
Case
2
1
3
1
2
3
2
1
3
22×( )
26×(
)
28x() 22×(
)
26x()
28×(
)
22x()
26x()
28×( )
33x()
43x()
65x() 33x()
y
43x()
65x() 33x()
43x()
65x()
50x()
77x()
74x() 50x()
77x()
74x() 50x()
77x()
74x()
S
556
488
777
488
556
488
777
777
556
X
Z
Max usage
Step by step
Solved in 6 steps
