Production Scheduling Nahmias, Chapter 8 ( D t lj l i ) (sv ...

Scheduling Problems in General are Hard
• Dilemma:
– NP hard combinatorial problems
– Too hard to find optimal solutions!
– NNeed dsomething thi anyway.
9E+22
8E+22
7E+22
6E+22
5E+22 5
• Classifying “Hardness”:
3E+22
– As the number of variables (n) grow
– Class P: has a polynomial solution.
– Class NP: has no polynomial solution.
2E+22
1E+22
0
• Sequencing problems are NP hard and complexity
grow as n!.
C /10000 d 10000 10
– Compare en /10000 and 10000n10 .
4E+22
56 57 58 59 60 61 62 63
• At n = 40, e n /10000 = 2.4 × 10 13 , 10000n 10 = 1.0 × 10 20
• At n = 80, e n /10000 = 5.5 × 10 30 , 10000n 10 = 1.1 × 10 23
– 3! = 6, 4! = 24, 5! = 120, 6! = 720, … 10! =3,628,800, while
13! = 6,227,020,800
25!= 15,511,210,043,330,985,984,000,000
Example: Computation Times
• A computer can examine 1,000,000 sequences per second and we wish
to build a scheduling system that has response time of no longer than
one minute. How many jobs can we sequence optimally?
Number of Jobs Computer Time
5 0.12 millisec
6 0.72 millisec
7 5.04 millisec
8 40.32 millisec
9 0.36 sec
10 3.63 sec
11 39.92 sec
12 798 7.98 min i
13 1.73 hr
14 24.22 hr
15 15.14 day
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20 77,147 years
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