SchedulingScheduling

Mahmut Ali GÖKÇE

Industrial Systems Engineering Dept.

İzmir University of Economics

Topics

• Scheduling– Scheduling in high-volume systems– Scheduling in intermediate-volume systems– Scheduling in low-volume systems

• Gantt charts• Sequencing

– Priority rules and performance measures– Single machine scheduling– Proof that WSPT is optimal for minimizing weighted

completion time– Sequencing jobs through two work centers

– Cyclic scheduling in service systems

Scheduling

• Establishing the timing of the use of resources such as equipment, facilities, and human activities.

Manufacturers

• Schedule production, developing schedules for workers, equipment, purchases, maintenance, etc.

• Effective scheduling can yield cost savings and increases in productivity

Hospitals

• Schedule admissions, surgery, nursing assignments, support services such as meal preparation, security, maintenance, cleaning.

• Effective scheduling can save lives and improve patient care.

Educational Institutions

• Schedule classrooms, instruction and students.

• Effective scheduling can reduce the need for expansion of facilities and satisfy constraints of people.

Scheduling Decisions

• Dependent on earlier decisions such as– Capacity of the system– Equipment selections– Selection and training of workers– Design of products and services

• May aim at achieving a tradeoff among conflicting goals: Efficient utilization vs. minimization of waiting times, inventory, process times.

Scheduling Operations

• High-volume systems (flow systems)

• Intermediate-volume systems

• Low volume systems (job shop)

Scheduling in High-Volume Systems

• Standardized equipment and activities that provide identical or highly similar operations on customers or products as they pass through the system.

• Perform best with a high, uniform output.

Scheduling in High-Volume Systems

• Goal: Smooth rate of flow of goods or customers– Production Ex: Production of autos, personal

computers, radios and television, toys, appliances,

– Process Ex: Petroleum refining, sugar refining, waste treatment

– Service Ex: Cafeteria lines

• Line balancing: Allocating the required tasks to workstations so that they satisfy sequencing constraints and are balanced wrt equal work times among stations. (Ch 6)

Scheduling in High-Volume Systems

• Goals: Maximum utilization of equipment and personnel, highest possible rate of output

• Handling disruptions can reduce output: Equipment failures, material shortages, accidents, absences.

Scheduling in High-Volume Systems

• For success:1. Product and product design

2. Preventive maintenance

3. Rapid repair when breakdowns occur

4. Optimal product mixes

5. Minimization of quality problems

6. Reliability and timing of supplies

Scheduling in High-Volume Systems

Scheduling in Intermediate-Volume Systems

• Between standardized type of output of flow-shop and made-to-order output of job shops.

• Periodically shift from one job to another• Run sizes are larger than job shop.• Ex: Production of canned foods, baked

goods, paint, cosmetics• Issues: Run size and timing of jobs and

the sequence in which jobs should be processed.

• Setup cost may depend on the order in which jobs are processed: similar jobs may require less setup change between them.

• Makes sequencing more complex, since setup costs have to be observed/estimated for every sequence combination

up

p

h

DKQ

20

production rate

usage rate

production Quantity

Holding cost

Demand Setup cost

Scheduling in Low-Volume Systems(Job-shop Scheduling)

• Products are made to order

• Orders usually differ considerably in terms of processing requirements, materials needed, processing time, and processing sequence and setup.

• Issues: – How to distribute the load among work

centers (loading)– What job processing to use

• Gantt charts: Organize and visually display the actual or intended use of resources in a time framework.

Scheduling in Low-Volume Systems(Job-shop Scheduling)

Gantt Charts

• Ex: Load chart: Helps a manager rework loading assignments to better utilize work centers

Gantt Charts

• Ex: Schedule chart: Shows which jobs are on schedule and which are behind or late.

Infinite Loading vs. Finite Loading

• Infinite loading: Assigns jobs to work centers without regard to the capacities of the work centers.– Queues may form

• Finite loading: Considers capacities of the work centers.– More complex

General Approaches to Scheduling

• Forward scheduling: Scheduling ahead, from some point in time.– “How long will it take to complete this job?”

• Backward scheduling: Scheduling by working backwards from the due dates.– “When is the latest time the job can be

started and still be completed by the due date?”

::: Skip the assignment method :::

Sequencing

• Loading determines the machines or work centers for jobs, but not the order in which they are processed.

• Sequencing is concerned with determining job processing order.

Priority Rules

• Priority rules are simple heuristics used to select the order (sequence) in which the jobs will be processed.

• Typical assumption: Job setup cost and time are independent of the processing sequence.

• Job processing times and due dates are important pieces of information.

• Due dates may be the result of – Delivery times promised to customers

– MRP processing

– Managerial decisions

Priority Rules

• FCFS (first come, first served)

• SPT (shortest processing time)

• EDD (earliest due date)

• CR (critical ratio)

• S/O (slack per operation)

• Rush

Performance Measures

• Job flow time

• Job lateness

• Tardiness

• Makespan: – Total time needed to complete a group of

jobs

• Average number of jobs

EXAMPLE 2

• Determine the sequence and performance measures for the given group of jobs, arriving in the given sequence:

Job Sequence Processing Time Due Date

A 2 7

B 8 16

C 4 4

D 10 17

E 5 15

F 12 18

EXAMPLE 2

Seq. Proc. Time Due Date Flow Time Days Tardy

A 2 7 2 0

B 8 16 10 0

C 4 4 14 10

D 10 17 24 7

E 5 15 29 14

F 12 18 41 23

41 120 54

• FCFS solution: A-B-C-D-E-F

Average Flow Time = 120/6=20 daysAverage Tardiness = 54/6=9 days

Makespan = 41 days

Average Number of Jobs = 120/41=2.93 jobs/day

EXAMPLE 2

Seq. Proc. Time Due Date Flow Time Days Tardy

A 2 7 2 0

C 4 4 6 2

E 5 15 11 0

B 8 16 19 3

D 10 17 29 12

F 12 18 41 23

41 108 40

• SPT solution: A-C-E-B-D-F

Average Flow Time = 108/6=18 daysAverage Tardiness = 40/6=6.67 days

Makespan = 41 days

Average Number of Jobs = 108/41=2.63 jobs/day

EXAMPLE 2

Seq. Proc. Time Due Date Flow Time Days Tardy

C 4 4 4 0

A 2 7 6 0

E 5 15 11 0

B 8 16 19 3

D 10 17 29 12

F 12 18 41 23

41 110 38

• EDD solution: C-A-E-B-D-F

Average Flow Time = 110/6=18.33 daysAverage Tardiness = 38/6=6.33 days

Makespan = 41 days

Average Number of Jobs = 110/41=2.68 jobs/day

EXAMPLE 2

Seq. Proc. Time Due Date Flow Time Days Tardy

C 4 4 4 0

F 12 18 16 0

A 2 7 18 11

E 5 15 23 8

B 8 16 31 15

D 10 17 41 24

41 133 58

Average Flow Time = 133/6=22.17 daysAverage Tardiness = 58/6=9.67 days

Makespan = 41 days

Average Number of Jobs = 133/41=3.24 jobs/day

• CR solution: C-F-A-E-B-D

EXAMPLE 2

• Comparison of the 4 rules:

Rule Average Flow Time

Average Tardiness

Average Number of Jobs

FCFS 20.00 9.00 2.93

SPT 18.00 6.67 2.63

EDD 18.33 6.33 3.68

CR 22.17 9.67 3.24

SPT

• SPT is always superior (optimal) in terms of– Minimizing flow time– Minimizing the average number of jobs (and

work-in-process inventory)– Completion time

• Disadvantage: Makes long jobs wait

FCFS

• FCFS is widely used in service systems since it is– Fair– Simple– And since we typically can not estimate

processing times of individual jobs.

EDD & CR

• EDD typically performs good in minimizing lateness– Disadvantage: Does not consider job

processing times.

• CR typically performs good in minimizing job tardiness

Theorem: For 1||wjCj the WSPT rule is optimal.

• Proof (by contradiction):Suppose a schedule S, which is not WSPT, is optimal. In S, there must be at least two adjacent jobs, say job i followed by job k, such that

wi / pi < wk / pk . • Assume job i starts its processing at time t.• Consider a new schedule S` where jobs i and k

are exchanged and all other jobs remain in their original position. The total weighted completion times of the jobs processed before and after i and k are unchanged in S`.

Theorem: For 1||wjCj the WSPT rule is optimal.

k

ki

i

t t + pi + pk

t t + pi + pk

Schedule S

Schedule S`

Proof (by contradiction) continued…

• Total weighted completion times of jobs i and k :

• Under S: (t+pi)wi + (t+pi+pk)wk

>

• Under S`: (t+pk)wk + (t+pk+pi)wi

• Contradicts the optimality of S and proves the theorem.

Sequencing Jobs through Two Work Centers

Sequencing Jobs through Two Work Centers

• Johnson’s rule– Minimizes makespan for a group of jobs to be

processed on two machines or at two work centers

• Assumptions:1. Job time must be known and constant for each job

at each work center2. Job times must be independent of the job sequence3. All jobs must follow the same two-step work

sequence4. Job priorities can not be used5. All units of a job must be completed at the first

work center before a job moves on to the second work center

Johnson’s Rule

1. List the jobs and their times at each work center

2. Select the job with the shortest time. If the shortest time is at the first work center, schedule that job first; if the time is at the second work center, schedule the job last. Break ties arbitrarily.

3. Eliminate the job and its time from further consideration

4. Repeat steps 2 and 3, working toward the center of the sequence, until all jobs have been sequenced.

EXAMPLE 4Processing times (hours)

Job Work Center 1 Work Center 2

A 5 5

B 4 3

C 8 9

D 2 7

E 6 8

F 12 15

D 2nd 3rd 4th 5th 6th

EXAMPLE 4Processing times (hours)

Job Work Center 1 Work Center 2

A 5 5

B 4 3

C 8 9

E 6 8

F 12 15

D 2nd 3rd 4th 5th B

EXAMPLE 4Processing times (hours)

Job Work Center 1 Work Center 2

A 5 5

C 8 9

E 6 8

F 12 15

D 2nd 3rd 4th A B

EXAMPLE 4Processing times (hours)

Job Work Center 1 Work Center 2

C 8 9

E 6 8

F 12 15

D E 3rd 4th A B

EXAMPLE 4Processing times (hours)

Job Work Center 1 Work Center 2

C 8 9

F 12 15

D E C 4th A B

D E C F A B

Cyclical Scheduling in Service Systems

• Employees must be assigned to work shifts or time slots, and have days off, on a repeating (cyclic) basis.

Day Mon Tue Wed Thu Fri Sat Sun

Staff needed

2 4 3 4 6 5 5

Cyclical Scheduling in Service Systems

• ALGORITHM:

1. Make the first worker’s assignment such that the two days with the lowest need are designated as days off. Circle those days. In case of a tie, pick the pair with the lowest adjacent requirement.

Day Mon Tue Wed Thu Fri Sat Sun

Staff needed

2 4 3 4 6 5 5

Worker1 2 4 3 4 6 5 5

2. Subtract 1 from each day’s requirement, except for the circled days. Assign the next employee, again using the two lowest consecutive days as days off:

Day Mon Tue Wed Thu Fri Sat Sun

Staff needed

2 4 3 4 6 5 5

Worker1 2 4 3 4 6 5 5

Worker1 2 4 2 3 5 4 4

3. Repeat the preceding step for each additional worker until all staffing requirements have been met. Don’t subtract from a value of zero.

Gant chart is a graphical representation of tasks over a specific period of time.  In its correct form, the Gant chart should represent the tasks, resources, time frame and required founding needed by the project manager to complete its intended purpose.