Chapter 17: Operations Scheduling• Scheduling is the allocation of resources over time to

perform a collection of tasks • Resources

– Workers, Machines, Tools• Tasks

– Operations that bring some physical changes to material in order to eventually manufacture products

– Setups such as walking to reach the workplace, obtaining and returning tools, setting the required jigs and fixtures, positioning and inspecting material, cleaning etc.

Job Shop

Batch production

ShopFlow C

usto

miz

atio

n

High

LowHighLow

Volume

Project

Production Systems

Continuous production

Job Shop

Batch production

ShopFlow C

usto

miz

atio

n

High

LowHighLow

Volume

Project

Production SystemsAircraft

Automobile

Oil refinery

Books

Custom-made machines and parts

Continuous production

Job Shop

Continuous production

Batch production

ShopFlow C

usto

miz

atio

n

High

LowHighLow

Volume

Production SystemsProject

Labor intensive

Capital intensive

Job Shop

Batch production

ShopFlow C

usto

miz

atio

n

High

LowHighLow

Volume

Production SystemsProject

More frequent Rescheduling

Less frequent Rescheduling

Continuous production

Infinite/Finite Scheduling

• Infinite - assumes infinite capacity– Loads without regard to capacity, levels the

load and sequence jobs– Job shop/batch production

• Finite - assumes finite (limited) capacity– Sequences jobs as part of the loading decision,

resources are never loaded beyond capacity– Flow shop/continuous production

Scheduling Job Shop/Batch Production

• Batch Production– many planning steps

• aggregate planning• master scheduling• material requirements planning (MRP)• capacity requirements planning (CRP)

• Scheduling determines– machine/worker/job assignments– resource/requirement matchings

Objectives in Scheduling• Conformance to prescribed deadlines

– Meet customer due dates, minimize job lateness, minimize maximum lateness, minimize number of tardy jobs

• Response time or lead time– Minimize mean completion time, minimize average time in

the system• Efficient utilization of resources

– Maximize machine or labor utilization, minimize idle time, maximize throughput, minimize the length of time the shop is open

• Costs– Minimize work-in-process inventory, minimize overtime

Responsibilities of Production Control Department

• Loading – Allocate orders to workers and machines, worker and

machines to work centers etc.• Sequencing

– Release work orders to shop & issue dispatch lists for individual machines

• Monitoring – Maintain progress reports on each job until it is complete

Loading

• Allocate orders to workers and machines, workers and machines to work centers, etc.

• Perform work on most efficient resources

• Use assignment method of linear programming to determine allocation

Assignment Method1. Perform row reductions

– Subtract minimum value in each row from all other row values

2. Perform column reductions– Subtract minimum value in each column from all other

column values3. Line Test

– Cross out all zeros in matrix using minimum number of horizontal & vertical lines. If number of lines equals number of rows in matrix, optimum solution has been found, stop.

4. Matrix Modification– Subtract minimum uncrossed value from all uncrossed values

& add it to all cells where two lines intersect. Go to Step 3.

Assignment ExampleCooker

Food 12 34Beans 105 610Peaches 62 46Tomatoes 76 56Corn 95 410

Row reduction Column reduction Line Test5 0 1 5 3 0 1 4 3 0 1 44 0 2 4 2 0 2 3 2 0 2 32 1 0 1 0 1 0 0 0 1 0 05 1 0 6 3 1 0 5 3 1 0 5

Number lines <> number of rows so modify matrix

Assignment Example

Cooker Food 12 34Beans 10 12 Peaches 00 21 Tom 03 20Corn 11 03

Modify matrix Line Test

1 0 1 2 1 0 1 20 0 2 1 0 0 2 10 3 2 0 0 3 2 01 1 0 3 1 1 0 3

# lines = # rows so at optimal solution

Cooker

Food 1 2 3 4Beans 10 5 6 10Peaches 6 2 4 6Tomatoes 7 6 5 6Corn 9 5 4 10

Orders completed in 6 hoursTotal number of hours = 21

Sequencing

• Prioritize jobs assigned to a resource• If no order specified use first-come first-

served (FCFS)• Many other sequencing rules exist• Each attempts to achieve to an objective

Sequencing Rules

• FCFS - first-come, first-served• SOT - shortest operating time• DDATE - earliest due date• STR - slack time remaining

– (due date - today’s date) - (remaining processing time)

• LCFS - last come, first served

Critical Ratio Rule

• CR = time remaining / work remaining

Jobs with the smallest CR are run first.

due date - today’s date remaining processing time

=

Performance Measures• Completion time

– Epoch at which the job is completed• Makespan

– Completion time of the last job processed• Lateness

– Lateness of job j – = (completion time of job j)-(due date of job j)

• Tardiness– Tardiness of job j = max(0, lateness of job j)

Sequencing Rule Example

A 510(10-1) - 5 = 4(10-1)/5 = 1.80B 1015(15-1)-10 = 4(15-1)/10 = 1.40C 25(5-1)-2 = 2 (5-1)/2 = 2.00D 812(12-1)-8 = 3 (12-1)/8 = 1.37E 68(8-1)-6 = 1 (8-1)/6 = 1.16

120 possible sequences for 5 jobs

Processing Due CriticalJob Time Date Slack Ratio

First-Come First-Served

A 5 10B 10 15C 2 5D 8 12E 6 8

Average

Start Processing Completion DueSequenceTime Time Time Date Tardiness

Earliest Due Date

C 2 5E 6 8A 5 10D 8 12E 10 15

Average

Start Processing Completion DueSequenceTime Time Time Date Tardiness

Slack Time Remaining

E 6 8C 2 5D 8 12A 5 10B 10 15

Average

Slack for each job A - 4, B - 4, C - 2, D - 3, E - 1

Start Processing Completion DueSequenceTime Time Time Date Tardiness

Critical Ratio

E 6 8D 8 12B 10 15A 5 10C 2 5

Average

CR for each job A - 1.80, B - 1.40, C - 2.00, D - 1.37, E - 1.16

Start Processing Completion DueSequenceTime Time Time Date Tardiness

Shortest Operating Time

C 2 5A 5 10E 6 8D 8 12B 10 15

Average

Start Processing Completion DueSequenceTime Time Time Date Tardiness

Summary

FCFS 18.60 9.6 3 23DDATE 15.00 5.6 316STR 16.40 6.8 416CR 20.80 11.2 426SOT 14.80 6.0 316

* Best values** Guaranteed best values

Average Average No. of MaximumRule Completion Time Tardiness Jobs Tardy Tardiness

** **

*

**

*

**

Johnson’s Rule: Sequencing Jobs Through a Two Machine Flow Shop to Minimize Makespan

1. List time required to process each job at each machine. Set up a one-dimensional matrix to represent desired sequence with # of slots equal to # of jobs.

2. Select smallest processing time at either machine. If that time is on machine 1, put the job as near to beginning of sequence as possible.

3. If smallest time occurs on machine 2, put the job as near to the end of the sequence as possible.

4. Remove job from list.5. Repeat steps 2-4 until all slots in matrix are filled & all jobs are

sequenced.

Johnson’s Rule Example

A 6 8B 11 6C 7 3D 9 7E 5 10

Machine Machine Job Center 1 Center 2

Johnson’s Rule Example

A 6 8B 11 6

D 9 7E 5 10

Machine Machine Job Center 1 Center 2

C

Johnson’s Rule Example

A 6 8B 11 6

D 9 7

Machine Machine Job Center 1 Center 2

CE

Johnson’s Rule Example

B 11 6

D 9 7

Machine Machine Job Center 1 Center 2

CAE

Johnson’s Rule Example

D 9 7

Machine Machine Job Center 1 Center 2

CBAE

Johnson’s Rule Example Machine Machine

Job Center 1 Center 2

CBDAE

Johnson’s Rule Example

E 5 10A 6 8D 9 7B 11 6C 7 3

Machine Machine Job Center 1 Center 2

CBDAEEach triplet above shows the start, processing, and stopping times of an operation. Johnson’s rule guarantees that the above schedule gives the best value (41) of makespan.

Monitoring with Gantt Chart

1 2 3 4 5 6 8 9 10 11 12 Days

1

2

3

Today’s Date

Job 32B

Job 23C

Job 11C Job 12A

Faci

lity

Key:

PlannedActivity

CompletedActivity

Behind schedule

Ahead of schedule

On schedule

Gantt Chart shows both planned and completed activities against a time scale

Employee Scheduling

• Labor is very flexible resource• Scheduling workforce is a complicated

repetitive task• An objective is to create a consecutive days

off schedule that uses the fewest workers. • Heuristics are commonly used.

Employee Scheduling Heuristic

1. Consider the first worker and the unassigned workload.

2. Assign the worker to all days except the two consecutive days with the lowest unassigned workload.

3. If there are unassigned workload, consider the next worker and repeat step 2. Else, stop.

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1Worker 4 2 1 2 1 1 2 1

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1Worker 4 2 1 2 1 1 2 1

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1Worker 4 2 1 2 1 1 2 1Worker 5 1 0 1 1 1 1 0

Employee Scheduling Example

Number of Workers RequiredM Tu W Th F S Su

3 3 4 3 4 5 3Worker 1 3 3 4 3 4 5 3Worker 2 3 3 3 2 3 4 2Worker 3 2 2 3 2 2 3 1Worker 4 2 1 2 1 1 2 1Worker 5 1 0 1 1 1 1 0

Employee Scheduling Example

Remark:The resulting schedule provides consecutive days

off to 4 workers among 5. Still, it’s better than two schedules shown next.

Employee Scheduling ExampleDay of week M T W Th F Sa SuMin # workers 3 3 4 3 4 5 3Worker 1 O X X 0 X X XWorker 2 O X X 0 X X XWorker 3 X O X X O X XWorker 4 X O X X X X OWorker 5 X X O X X X OIMPROVED SCHEDULEDay of week M T W Th F Sa SuMin # workers 3 3 4 3 4 5 3Worker 1 O O X X X X XWorker 2 O O X X X X XWorker 3 X X O O X X XWorker 4 X X X O X X OWorker 5 X X X X O X O

Reading

• Reading: Chapter 17 pp. 668-74, 678-81• Chapter end problems: 4,6,12,13