Industrial Systems Engineering Dept. İzmir University of Economics Scheduling 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

Scheduling in High-Volume Systems 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: Product and product design Preventive maintenance Rapid repair when breakdowns occur Optimal product mixes Minimization of quality problems Reliability and timing of supplies

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.

Demand Setup cost production rate production Quantity 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 usage rate Holding 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

Scheduling in Low-Volume Systems (Job-shop Scheduling) Gantt charts: Organize and visually display the actual or intended use of resources in a time framework.

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 D 10 17 E 5 15 F 12 18

EXAMPLE 2 FCFS solution: A-B-C-D-E-F Makespan = 41 days Seq. Proc. Time Due Date Flow Time Days Tardy A 2 7 B 8 16 10 C 4 14 D 17 24 E 5 15 29 F 12 18 41 23 120 54 Makespan = 41 days Average Flow Time = 120/6=20 days Average Tardiness = 54/6=9 days Average Number of Jobs = 120/41=2.93 jobs/day

EXAMPLE 2 SPT solution: A-C-E-B-D-F Makespan = 41 days Seq. Proc. Time Due Date Flow Time Days Tardy A 2 7 C 4 6 E 5 15 11 B 8 16 19 3 D 10 17 29 12 F 18 41 23 108 40 Makespan = 41 days Average Flow Time = 108/6=18 days Average Tardiness = 40/6=6.67 days Average Number of Jobs = 108/41=2.63 jobs/day

EXAMPLE 2 EDD solution: C-A-E-B-D-F Makespan = 41 days Seq. Proc. Time Due Date Flow Time Days Tardy C 4 A 2 7 6 E 5 15 11 B 8 16 19 3 D 10 17 29 12 F 18 41 23 110 38 Makespan = 41 days Average Flow Time = 110/6=18.33 days Average Tardiness = 38/6=6.33 days Average Number of Jobs = 110/41=2.68 jobs/day

EXAMPLE 2 CR solution: C-F-A-E-B-D Makespan = 41 days Seq. Proc. Time Due Date Flow Time Days Tardy C 4 F 12 18 16 A 2 7 11 E 5 15 23 8 B 31 D 10 17 41 24 133 58 Makespan = 41 days Average Flow Time = 133/6=22.17 days Average Tardiness = 58/6=9.67 days Average Number of Jobs = 133/41=3.24 jobs/day

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. Schedule S i k t t + pi + pk Schedule S` k i t t + pi + pk

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: Job time must be known and constant for each job at each work center Job times must be independent of the job sequence All jobs must follow the same two-step work sequence Job priorities can not be used 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 List the jobs and their times at each work center 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. Eliminate the job and its time from further consideration Repeat steps 2 and 3, working toward the center of the sequence, until all jobs have been sequenced.

EXAMPLE 4 D 2nd 3rd 4th 5th 6th Processing times (hours) Job Work Center 1 Work Center 2 A 5 B 4 3 C 8 9 D 2 7 E 6 F 12 15 D 2nd 3rd 4th 5th 6th

EXAMPLE 4 D 2nd 3rd 4th 5th B Processing times (hours) Job Work Center 1 Work Center 2 A 5 B 4 3 C 8 9 E 6 F 12 15 D 2nd 3rd 4th 5th B

EXAMPLE 4 D 2nd 3rd 4th A B Processing times (hours) Job Work Center 1 5 C 8 9 E 6 F 12 15 D 2nd 3rd 4th A B

EXAMPLE 4 D E 3rd 4th A B Processing times (hours) Job Work Center 1 8 9 E 6 F 12 15 D E 3rd 4th A B

EXAMPLE 4 D E C 4th A B Processing times (hours) Job Work Center 1 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 6 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 6 5 Worker1

2. Subtract 1 from each day’s requirement, except for the circled days 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 6 5 Worker1

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.