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Branch and Bound

- Enumerative method
- Guarantees finding the best schedule
- Basic idea:
- Look at a set of schedules
- Develop a bound on the performance
- Discard (fathom) if bound worse than best schedule found before (incumbent)

ISE480 Sequencing and Scheduling

OK, now what?

- So we’ve got a IP formulation of the problem, how do we solve it?
- Using standard IP solution techniques such as branch-and-bound
- Doesn’t mean the problem is easy
- Will now talk about branch-and-bound (B&B) which can be used to solve IPs and other hard problems

ISE480 Sequencing and Scheduling

Branch-and-Bound

- Idea
- Systematically search through possible variable values
- Use heuristics to pick a decision to try (“branch”)
- Use lower bounds on solutions to “bound” the search (fathom)
- Creates a search tree

ISE480 Sequencing and Scheduling

Branch

a = 0

a = 1

b = 0

b = 1

b = 0

b = 1

c = 0

c = 1

c = 0

c = 1

c = 0

c = 1

c = 0

c = 1

B&B Search Tree: Branching- Imagine a problem with 3 variables
- a, b, c є{0, 1}

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ISE480 Sequencing and Scheduling

a = 0

a = 1

b = 0

b = 1

b = 0

c = 0

c = 1

c = 0

Bound

B&B Search Tree: Bounding- Imagine I have a way to calculate a lower bound on the cost at each node

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ISE480 Sequencing and Scheduling

B&B

- Branch: assign a heuristic value to a variable
- Creates two subproblems
- Bound: compare lower bound at node with best known solution
- If LB > best, you can backtrack right away

ISE480 Sequencing and Scheduling

B&B for IP

- Usually lower bound is found by solving the linear relaxation of the IP
- LP formed by ignoring integral constraints
- Branch on one of the integer variables with a non-integer value to be:
- greater than or equal to the next highest integer, or
- Less than or equal to the next lowest integer

ISE480 Sequencing and Scheduling

Classic Result

- The EDD rule is optimal for
- If jobs have different release dates, which we denote

then the problem is NP-Hard

- What makes it so much more difficult?

ISE480 Sequencing and Scheduling

Delay Schedule

EDD

Add a delay

1

3

2

0 5 10 15

What makes this problem hard is thatthe optimal schedule is not necessarilya non-delay schedule

ISE480 Sequencing and Scheduling

Final Classic Result

- The preemptive EDD rule is optimal for the preemptive (prmp) version of the problem
- Note that in the previous example, the preemptive EDD rule gives us the optimal schedule
- Note that if rj = 0, then the above does not hold.

ISE480 Sequencing and Scheduling

Branch and Bound

- The problem

cannot be solved using a simple dispatching rule so we will try to solve it using branch and bound

- To develop a branch and bound procedure:
- Determine how to branch
- Determine how to bound

ISE480 Sequencing and Scheduling

Data

ISE480 Sequencing and Scheduling

Branch and bound

- Enumeration in asearchtree
- each node is a partial solution, i.e. a part of the solution space

Level 0

root node

Level 1 (n nodes)

child nodes

...

Level 2 (n*(n-1))

child nodes

...

ISE480 Sequencing and Scheduling

Branching

(•,•,•,•)

(1,•,•,•)

(2,•,•,•)

(3,•,•,•)

(4,•,•,•)

Discard immediately because

ISE480 Sequencing and Scheduling

Branching

(•,•,•,•)

(1,•,•,•)

(2,•,•,•)

(3,•,•,•)

(4,•,•,•)

Need to develop lower bounds on

these nodes and do further branching.

ISE480 Sequencing and Scheduling

Bounding (in general)

- Typical way to develop bounds is to relax the original problem to an easily solvable problem
- Three cases:
- If there is no solution to the relaxed problem there is no solution to the original problem
- If the optimal solution to the relaxed problem is feasible for the original problem then it is also optimal for the original problem
- If the optimal solution to the relaxed problem is not feasible for the original problem it provides a bound on its performance

ISE480 Sequencing and Scheduling

Relaxing the Problem

- The problem

is a relaxation to the problem

- Not allowing preemption is a constraint in the original problem but not the relaxed problem
- We know how to solve the relaxed problem (preemptive EDD rule)

ISE480 Sequencing and Scheduling

Bounding

- Preemptive EDD rule optimal for the preemptive version of the problem
- Thus, solution obtained is a lower bound on the maximum delay
- If preemptive EDD results in a non-preemptive schedule all nodes with higher lower bounds can be discarded.

ISE480 Sequencing and Scheduling

Lower Bounds

- Start with (1,•,•,•):
- Job with EDD is Job 4 but
- Second earliest due date is for Job 3

Job 1

Job 2

Job 3

Job 4

0 10 20

ISE480 Sequencing and Scheduling

Branching

(•,•,•,•)

(1,•,•,•)

(2,•,•,•)

(3,•,•,•)

(4,•,•,•)

(1,2,•,•)

(1,3,•,•)

(1,3,4,2)

ISE480 Sequencing and Scheduling

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