2-1

Scheduling Constraints

2-2

Outline

• Activities

• Temporal constraints

• Resources

• Resource constraints (mono-activity)

• Resource constraints (two activities)

2-3

Activities

• Interval (block) activities

• Splittable activities (with interruption cost?)

A

time

A

time

A A A

2-4

Temporal constraints

• Earliest and latest start and end times

• Fixed or variable durations

• Precedence constraints

• Precedence constraints with minimal delays

• Precedence constraints with fixed delays

• Maximal delays

2-5

Resources

• Unary resources

• Discrete resources

• State resources

• Energetic resources

2-6

Unary resources

• Main resources in 4/10 problems

• Example: one individual machine or person

A B

C

time

2-7

Discrete resources

• Main resources in 5/10 problems

• Example: group of identical machines

A

B

Dtime

C

2-8

State resources

• Main resources in 1/20 problems

• Example: oven with different temperatures

D

time

A

B

C

2-9

Energetic resources

• Main resources in 1/20 problems

• Example: number of man-days per week

A B

timeC

2-10

Resource constraints

• Mono-activity– Resource requirement– Resource provision– Resource consumption/production– Periods during which a resource is not (or not

fully) available– Optional or variable requirements– Transition times

2-11

Resource requirement

• Unary resource: A requires R• Discrete resource: A requires c units of R• State resource: A requires R ...

– in a given state s

– in any of a given set of states {s1 ... sn}

– not in a given state s

– not in any of a given set of states {s1 ... sn}

• Energetic resource: A requires w units of R per time period (e.g., 4 man-months per month)

2-12

Resource provision

• Similar to resource requirement when part of the problem is to make the resources available for the requiring activities

• The required capacity can never exceed the provided capacity

2-13

Resource consumption

• Unary resource: A consumes R• Discrete resource: A consumes c units of R

• The consumed capacity is no longer available for any other activity

• The resource is required from the start time of the activity up to the end of the schedule

2-14

Resource production

• Unary resource: A produces R• Discrete resource: A produces c units of R

• The produced capacity becomes available when the activity ends

• The resource is provided from the end time of the activity up to the end of the schedule

2-15

Capacity constraints

• Periods during which a resource is not fully available– Maintenance periods– Vacations– Forbidden states (at night)

• Periods during which some minimal amount must be required or provided

2-16

Optional or variable requirements

• Optional activities– Resource alternatives– Sub-contracts

2-17

Optional or variable requirements

• Time versus capacity tradeoffs

2 people during 3 days

or 3 people during 2 days

2-18

Optional or variable requirements

• Variable requirement over timeExample: 8 person-days with either 2 or 3 people at any time

2-19

Transition times

• Tool setup between two tasks on the same machine

• State change (oven temperature, color used in a painting shop)

• Cleaning

2-20

Resource constraints

• Two activities– Percentage constraints– Synchronization constraints

2-21

Percentage constraints

• % of activity A on resource RA must be done before (or after):– B starts– B ends

– % of activity B on resource RB are done

• Often complements variable requirements over time

2-22

Synchronization constraints

• When A executes, B requires (at least, at most) c1 units of resource R

• When A does not execute, B requires (at least, at most) c2 units of resource R