Why scheduling problem comes under N-P Hard problems.pptx

Why scheduling problem comes under

N-P Hard problems?.

P-Problem

Polynomial algorithms are sometimes called efficient or good. The class of all polynomially

solvable problems is called class P

Polynomial problem

1 Linear method

2 Branching method etc.

If function is polynomial function we can say that the algorithm has complexity

O(N)^KWhere

N Problem parameter like size

K order of polynomial

NP Hard-Problem◦ Another class of optimization problems is known as NP-hard problems. For such problems, no

polynomial-time algorithms are known and it is generally believed that these problems cannot be solved in polynomial time

To solve NP-hard problem

1 Restrict the range of inputs use

2 limit the size of inputs

3 settle for approximate outputs so on

NP HARD

TSP Traveling Salesman Problem

what is the shortest possible route that visits each city exactly once and returns to the origin city

no of possible outcome (N-1)!For 5 city no of possible outcome is 4!For 100 city no of possible outcome is 99!

In order to find a “good” solution within an acceptable amount of time, two types of algorithms can be

developed:

1 approximation algorithms.

we try to solve it analytically

2 heuristic algorithms.

Heuristic algorithm is usually analyzed experimentally

Scheduling problems All scheduling problems are deterministic

The final solution must accomplish all the problem constraints.

adjusting to certain criteria as minimum cost, lateness, process time, inventory time, etc.

1. Single machine problem

2. Flow shop scheduling problem

3. Job shop scheduling we have to consider sequence and schedule so for 10 machine 10 operation it is highly difficult to solve it by simple polynomial due to time constrain

There is (10!)^10 possible ways

HeuristicIf problem is exponential and non polynomial it is hard to solve such problem

Using some rules we simplify the problem

to get good solution which is may or may not be exact

To solve problem in polynomial time

Some heuristic used in scheduling problems

1 palmer

2 cds algorithm

3 Johnsons algorithm

META heuristic

NP-Complete What is NP-Complete?

A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly (ie. in polynomial time) transformed into x. In other words:

x is in NP, and

Every problem in NP is reducible to x

So what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly then all NP problems can be solved quickly

LiSA

LiSA - A Library of Scheduling Algorithms is a software package for solving deterministic

shop scheduling problems

Also It is use check of the complexity status of a

problem.

1|prec|∑Cj is NP-hard

in the strong sense

Conclusion

As the traditional method unable to solve scheduling problem in polynomial time we requires advance methods like Heuristic and Meta Heuristic to solve such hard problems to get optimum solution.

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