1. Scheduling THE SCHEDULING ACTIVITY AT MACHINE SHOP

2. 2 As it was previously mentioned, there are three primary objectives (goals) which apply to all scheduling problems: Job completions should not be “late” (about due dates). The the length of time during which a job stays in the system should be “minimum” (about flow time) It is desired to fully utilize the capacities of work centers (about work center utilization). Objectives (goals) of Scheduling

3. 3 Kinds of Scheduling Problems Studies about scheduling activity point out that there are two fundamental kinds of scheduling problems. Static Scheduling Problems Dynamic Scheduling Problems where, problem consists of a fixed set of jobs which are to be all completed. The number of jobs does not change. deals with scheduling of a continuous situation, which means that, new jobs are continually added to the production environment.

4. 4  The static scheduling problem contains a fixed set of jobs which are to be completed. which are to be completed.  The typical basic assumptions in this kind of problem can be stated as follows; be stated as follows;... entire set of jobs arrive simultaneously, and... entire set of jobs arrive simultaneously, and all respective work centers are available at that time. all respective work centers are available at that time. Majority of the static scheduling problems use a criterion called “minimum make span” which can be defined as the minimum total time to process all subject jobs. This criterion is related with the scheduling objective which is about “flow time”. Majority of the static scheduling problems use a criterion called “minimum make span” which can be defined as the minimum total time to process all subject jobs. This criterion is related with the scheduling objective which is about “flow time”. Static Scheduling Approach

5. 5 Static Scheduling Problems use known and nonvarying process times. use process times which are subject to random variations. Deterministic Solutions Stochastic Solutions Methods Producing Optimum Results Methods Utilizing Heuristic Procedures called dispatching or sequencing rules. scheduling decisions are made sequentially rather than at once only applicable to relatively small problems.

6. 6 Dynamic Scheduling Approach Analytical approaches had been developed which are “based on queuing models that provide expected steady state conditions for certain kinds of situations and time distributions”. Dynamic Scheduling Problems Deterministic Solutions Stochastic Solutions To find a solution to dynamic scheduling problems, usually the approach is, to use different “dispatching rules” at the work centers. Typically, the criteria used in such queuing cases are; Avarage Flow Time Average Work-in-process or number of jobs in the system and Work/ machine center utilization.

7. 7 Job Shop Dispatching and Common Priority Sequencing Rules At any given point in time, there exists a set of “n” jobs to be scheduled on “m” machines. This means, at any time there are (n!) m possible ways to schedule the jobs.

8. 8 Job Shop Dispatching and Common Priority Sequencing Rules One method to generate schedules in job shops is with “dispatching” which allows the schedule for a work station to evolve over a period of time. The decision about which job to process next is made with simple priority rules whenever the work station becomes available for further processing. Major advantage of “dispatching” method is that latest information on operating conditions of the work center can be incorporated into the schedule as it evolves.

9. 9 Job Shop Dispatching and Common Priority Sequencing Rules Common Priority Sequencing Rules Earliest Due Date (EDD) (single dimension) Shortest Processing Time (SPT) (single dimension) First Come First Served (FCFS) (single dimension) Slack per Remaining Operations(S/OP) (multiple dimension) Critical Ratio (CR) (multiple dimension) MULTIPLE DIMENSION RULES SINGLE DIMENSION RULES

10. 10 Job Shop Dispatching and Common Priority Sequencing Rules They base the priority on single aspect of the job, such as, arrival time, the due date, or the processing time at the current work center. They base the priority on single aspect of the job, such as, arrival time, the due date, or the processing time at the current work center. SINGLE DIMENSION RULES MULTIPLE DIMENSION RULES They incorporate information about the remaining work centers at which the job must be processed, in addition to the processing time at the present work station or the due date considered by single dimension rules. They incorporate information about the remaining work centers at which the job must be processed, in addition to the processing time at the present work station or the due date considered by single dimension rules.

11. 11 Job Shop Dispatching and Common Priority Sequencing Rules The critical ratio is calculated by dividing “the time remaining until the due date of the job” by the “remaining total shop time for the job”. CR = [Due date of the job-Today’s date] / Σ remaining shop time CR = [Due date of the job-Today’s date] / Σ remaining shop time  CR < 1.0 implies that the job is behind schedule, CR = 1.0 implies that the job is on schedule CR > 1.0 implies that the job is ahead of schedule. CR = 1.0 implies that the job is on schedule CR > 1.0 implies that the job is ahead of schedule. The job with the lowest CR is scheduled next. The job with the lowest CR is scheduled next. Critical Ratio (CR) (multiple dimension)

12. 12 Job Shop Dispatching and Common Priority Sequencing Rules There may be different ways to set the due dates of jobs. For example, a due date might have been calculated by computerized methods as in MRP or might have been simply determined by the customer. The job with the earliest due date is the job which is to be scheduled next. The job with the earliest due date is the job which is to be scheduled next. Earliest Due Date (EDD) (single dimension)

13. 13 Job Shop Dispatching and Common Priority Sequencing Rules The job which arrives the work station earlier than the others has the highest priority and is to be scheduled next. The job which arrives the work station earlier than the others has the highest priority and is to be scheduled next. First Come First Served (FCFS) (single dimension)

14. 14 Job Shop Dispatching and Common Priority Sequencing Rules Slack is the difference between “the time remaining until the due date of the job” and “remaining total shop time for the job” including that of the operation being scheduled. Slack is the difference between “the time remaining until the due date of the job” and “remaining total shop time for the job” including that of the operation being scheduled. The job’s priority is determined by dividing the slack by the number of the operations that remain including the operation which is being scheduled. The job’s priority is determined by dividing the slack by the number of the operations that remain including the operation which is being scheduled. S/RO = [(Due date – Today’s date)-Total shop time remaining]/ Number of remaining operations S/RO = [(Due date – Today’s date)-Total shop time remaining]/ Number of remaining operations Slack per Remaining Operations(S/OP) (multiple dimension)

15. 15 Job Shop Dispatching and Common Priority Sequencing Rules King of the priority sequencing rules! King of the priority sequencing rules! The job which has the shortest processing time is scheduled next. The job which has the shortest processing time is scheduled next. Shortest Processing Time (SPT) (single dimension)

16. 16 Job Shop Dispatching and Common Priority Sequencing Rules The principle is picking any job in the queue with equal probability. This rule is often used as a benchmark for other rules. In other words, it is used for evaluation of other rules with respect to each other by using reference values obtained from using R rule. Random (R) (single dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

17. 17 Job Shop Dispatching and Common Priority Sequencing Rules It is an extension of SPT rule in the sense that it considers all processing time remaining until the job is completed. This means that, the next job to be processed is the one which has least work remaining along its routing including the work in current station. Least Work Remaining (LWR) (multiple dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

18. 18 Job Shop Dispatching and Common Priority Sequencing Rules It considers the number of successive operations. This rule is again another variant of SPT rule. It considers the number of successive operations. The next job to be picked is the one which has smallest number of successive operations including the operation being scheduled. The next job to be picked is the one which has smallest number of successive operations including the operation being scheduled. Fewest Operations Remaining (FOR) (multiple dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

19. 19 Job Shop Dispatching and Common Priority Sequencing Rules It is a variant of EDD rule. It subtracts “remaining total shop time for the job” including that of the operation being scheduled from the “time remaining until the due date”. The resulting value is called “slack”. It is a variant of EDD rule. It subtracts “remaining total shop time for the job” including that of the operation being scheduled from the “time remaining until the due date”. The resulting value is called “slack”. Slack time =(due date - today’s date) - (remaining processing time) Slack time =(due date - today’s date) - (remaining processing time) Jobs are run in the order of smallest amount of slack. Jobs are run in the order of smallest amount of slack. Slack Time (ST) (multiple dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

20. 20 Job Shop Dispatching and Common Priority Sequencing Rules This is a different kind of rule which is based on machine utilization. This is a different kind of rule which is based on machine utilization. It considers the “next queues” at each of the succeeding work centers to which the jobs will go. It considers the “next queues” at each of the succeeding work centers to which the jobs will go. It selects the job which is going to the smallest queue. Here, the measure of the queue might be either in hours or perhaps in number of jobs. It selects the job which is going to the smallest queue. Here, the measure of the queue might be either in hours or perhaps in number of jobs. Next Queue (NQ) (multiple dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

21. 21 Job Shop Dispatching and Common Priority Sequencing Rules This is another rule for which the objective is to maximize machine (capacity) utilization. This is another rule for which the objective is to maximize machine (capacity) utilization. It selects the job which has minimum setup time on the machine. It selects the job which has minimum setup time on the machine. It should be noted that, this rule clearly considers dependencies between setup times and job sequence (setup time for a specific job may change according to sequence). It should be noted that, this rule clearly considers dependencies between setup times and job sequence (setup time for a specific job may change according to sequence). Least Setup (LSU) (single dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

22. 22 Job Shop Dispatching and Common Priority Sequencing Rules is the difference between a late job’s due date and its completion time is the difference between a late job’s due date and its completion time TARDINESS (single dimension) There are nalso some other rules most of which can be considered as variants of the already mentioned ones ; There are nalso some other rules most of which can be considered as variants of the already mentioned ones ;

23. 23 Job Shop Dispatching and Common Priority Sequencing Rules Altough priority sequencing rules seem to be simple, the actual task of scheduling hundreds of jobs through hundreds of work centers requires intensive data gathering and manipulation. The scheduler needs information on each job’s processing requirements; “the job’s due date”, “its routing”, “the standard setup at each operation”, “the processing time at each op.”, “the expected waiting time at each op”; “whether alternative work centers could be used at each operation”, “the components and raw materials required at each operation”, and “the current status of the job”. Computers are needed to track the data and to maintain valid priorities. Any priority sequencing rule can be used to schedule any number of work centers with the dispatching procedure.

24. 24 Single (One)-Machine Case Machine A Job 1 Job 2 Job n Priority sequencing rule “single-machine” environment is the simple and special case of all other or more complex environments. The results of single-machine problems not only provide insight to the single-machine environment, but also, provide a basis for heuristics of more complex environments. In most of the practical applications, scheduling problems in complex environments are broken down into smaller sub problems each of which becomes a single-machine case. Single (One) Machine Case

25. 25 Single (One)-Machine Case Problem Definition for Single-Machine Case Problem Definition for Single-Machine Case Majority of the studies on single-machine scheduling is based on the with the assumptions that; Majority of the studies on single-machine scheduling is based on the static problem of how to best schedule a fix set of jobs through a single machine with the assumptions that; Machine A Job 1 Job 2 Job n Priority sequencing rule Fixed set of jobs are available at the start of the scheduling period setup times are independent of the sequence. Machine A is available when the fixed set of jobs arrive.

26. 26 Single (One)-Machine Case What is meant by saying “setup times are independent of sequence”? What is meant by saying “setup times are independent of sequence”? Simply, it means, setup times does not alter the total make-span of the entire set of jobs no matter in which sequence the jobs are processed. Simply, it means, setup times does not alter the total make-span of the entire set of jobs no matter in which sequence the jobs are processed. In this case, the total make-span equals the sum of all setup and run times for any sequence of jobs In this case, the total make-span equals the sum of all setup and run times for any sequence of jobs

27. 27 Single (One)-Machine Case What is meant by saying “setup times are independent of sequence”? What is meant by saying “setup times are independent of sequence”? A DRILL MACHINE CASE

28. 28 Single (One)-Machine Case What is meant by saying “setup times are independent of sequence”? What is meant by saying “setup times are independent of sequence”? Where setup times are sequence dependent, the total make-span is altered with a change in sequence of jobs. Where setup times are sequence dependent, the total make-span is altered with a change in sequence of jobs. A FURNACE CASE

29. 29 Single (One)-Machine Case What is meant by saying “setup times are independent of sequence”? What is meant by saying “setup times are independent of sequence”? As a result one may come up with the conclusion that development of analytical models for the case of single-machine problems where setup is sequence dependent is rather complex. As a result one may come up with the conclusion that development of analytical models for the case of single-machine problems where setup is sequence dependent is rather complex. In this respect, it is fortunate that, in most of the single-machine environments make-span does not depend on the sequence (sequence independent). In this respect, it is fortunate that, in most of the single-machine environments make-span does not depend on the sequence (sequence independent).

30. 30 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) If the objective is to “minimize the make-span” (minimize the total time to run the entire set of jobs), it is clear that make-span will be equal to sum of all “setup+run” times and will be same for all sequencing options. If the objective is to “minimize the make-span” (minimize the total time to run the entire set of jobs), it is clear that make-span will be equal to sum of all “setup+run” times and will be same for all sequencing options. However, if the objective is to “minimize the average time each job spends at the machine”, it can be shown that this can be achieved by sequencing jobs in ascending order according to their total processing time (setup+run time). However, if the objective is to “minimize the average time each job spends at the machine”, it can be shown that this can be achieved by sequencing jobs in ascending order according to their total processing time (setup+run time).

31. 31 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) We can show this fact by the following example : We can show this fact by the following example : Average Flow Time = Sum of flow times / Number of jobs where, Sum of Flow Times = Σ [waiting time+(setup time+processing time)] values for each job in the system

32. 32 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) In this example : In this example : “average time in the system” “minimize the average time each job spends at the machine”, SPT The Criterion was... The Objective was... The Rule used was... The Measure used was... Average Flow Time = Sum of flow times / Number of jobs The average time in the system will always be minimized by processing the next job which has the shortest processing time at the current operation.

33. 33 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) SPT also proves to be very usefull when the objective is “minimizing the average number of jobs in the system” in a single-machine case. SPT also proves to be very usefull when the objective is “minimizing the average number of jobs in the system” in a single-machine case. This time, the measure which we are going to use is “average work-in-process (wip) inventory” for which the formulation is; This time, the measure which we are going to use is “average work-in-process (wip) inventory” for which the formulation is; Avarage wip inventory = Sum of flow times / Make-span Avarage wip inventory = Sum of flow times / Make-span

34. 34 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Example : Example :

35. 35 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Here, at this point, we should also note that “average wip inventory” and “average flow time” measures are directly related with each other. That is, when the value of one decreases, the value of the other also decreases and vice versa. Here, at this point, we should also note that “average wip inventory” and “average flow time” measures are directly related with each other. That is, when the value of one decreases, the value of the other also decreases and vice versa. Compare the results in previous example and following example. Compare the results in previous example and following example.

36. 36 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Example : Example :

37. 37 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) When the objective is to “minimize the average job lateness”, again SPT seems to yield better overall results in sequencing jobs for the single-machine case. When the objective is to “minimize the average job lateness”, again SPT seems to yield better overall results in sequencing jobs for the single-machine case. Ofcourse, to introduce the lateness criterion, we must first establish due dates for the jobs. Ofcourse, to introduce the lateness criterion, we must first establish due dates for the jobs. We can again demonstrate what is meant by “better overall results” by comparing the following examples We can again demonstrate what is meant by “better overall results” by comparing the following examples

38. 38 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Example : ( we have added 2 new columns and 3 new measures) Example : ( we have added 2 new columns and 3 new measures) where; Average Minutes Early = (Sum of Minutes Early) / Number of Jobs Average Minutes Past Due = (Sum of Minutes Past Due) / Number of Jobs Average Total Inventory=(Sum of Time in System with respect to actual pickup time)/Makespan

39. 39 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Example (continued): Example (continued): “Average Total Inventory” will be equal to “Average Wip Inventory” of the system when sum of time in system is equal to sum of flow times. That is, when individual actual pickup times are equal to individual flow times.

40. 40 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Example (continued): Example (continued):  SPT rule is overweighing values with respect to average flow time, average minutes early, average wip inventory and average total inventory.  On the other hand, EDD rule provides lower average minutes past due and lower maximum minutes past due (EDD: 25, SPT: 45).

41. 41 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Summary of research studies for EDD rule: Summary of research studies for EDD rule: EDD rule performs well with respect to EDD rule performs well with respect to the percentage of jobs past due and the percentage of jobs past due and the variance of hours past due. the variance of hours past due. easy adjustment of schedules when due dates change. easy adjustment of schedules when due dates change. It does not perform well with It does not perform well with flow time, flow time, wip inventory or wip inventory or work center utilization. work center utilization.

42. 42 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Summary of research studies for SPT rule: Summary of research studies for SPT rule: SPT rule tends to SPT rule tends to minimize the average flow time minimize the average flow time minimize the wip inventory minimize the wip inventory minimize the percentage of jobs past due minimize the percentage of jobs past due maximize work center (and hence shop) utilization maximize work center (and hence shop) utilization

43. 43 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Summary of research studies for SPT rule: Summary of research studies for SPT rule: It has disadvanteges of It has disadvanteges of tendency to produce a large variance in past due hours because the larger jobs might have to wait a long time for processing. tendency to produce a large variance in past due hours because the larger jobs might have to wait a long time for processing. providing no opportunity to adjust schedules when due dates changed. providing no opportunity to adjust schedules when due dates changed. possibility of increasing total inventory since it tends to push all work to the finished state. possibility of increasing total inventory since it tends to push all work to the finished state.

44. 44 Application of Single-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) FCFS rule performs poorly with respect to all performance measures. However, this result is something expected since it does not consider any of the job characteristics. FCFS rule performs poorly with respect to all performance measures. However, this result is something expected since it does not consider any of the job characteristics.

45. 45 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Priority sequencing rules such as CR and S/RO, also consider information about the remaining work centers at which the job must be processed in addition to the processing time at the current work center or the due date which is considered by single-dimension rules. Priority sequencing rules such as CR and S/RO, also consider information about the remaining work centers at which the job must be processed in addition to the processing time at the current work center or the due date which is considered by single-dimension rules. Thus, since they apply more than one aspect of the job, they are multiple-dimension rules. Thus, since they apply more than one aspect of the job, they are multiple-dimension rules. In order to see how multiple dimension rules are used and to compare their results against the results of SPT, FCFS and EDD rules, one can review the following case study. In order to see how multiple dimension rules are used and to compare their results against the results of SPT, FCFS and EDD rules, one can review the following case study.

46. 46 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Case Study: NC Milling Machine at Taylor Machine Shop Taylor Machine Shop processes engine blocks for automotive industry. These blocks are manufactured from blank casting material and the initial metal removal process is performed on the NC Milling Machine. Taylor Machine Shop processes engine blocks for automotive industry. These blocks are manufactured from blank casting material and the initial metal removal process is performed on the NC Milling Machine. After the nc milling operation, according to its type and size, each engine block ondergoes a series of operations successively at different work centers. After the nc milling operation, according to its type and size, each engine block ondergoes a series of operations successively at different work centers. Customers request to pickup their engine blocks on prescheduled due dates. Customers request to pickup their engine blocks on prescheduled due dates.

47. 47 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Case Study: NC Milling Machine at Taylor Machine Shop Currently there are six different blank castings at the NC Milling Machine and the machine is available for processing respective jobs. Currently there are six different blank castings at the NC Milling Machine and the machine is available for processing respective jobs. Setup times for each engine block is identical, so they are considered to be independent of the sequence and included in the respective processing time. Setup times for each engine block is identical, so they are considered to be independent of the sequence and included in the respective processing time. At Taylor Machine Shop, the current M-day is 1400 and the net daily work is 7,5 hrs. At Taylor Machine Shop, the current M-day is 1400 and the net daily work is 7,5 hrs.

48. 48 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Case Study: NC Milling Machine at Taylor Machine Shop What we want to do is, to obtain the results of CR, S/OP, SPT, FCFS and EDD sequencing rules for these six jobs and compare them to make a decision on which sequence to use. What we want to do is, to obtain the results of CR, S/OP, SPT, FCFS and EDD sequencing rules for these six jobs and compare them to make a decision on which sequence to use.

49. 49 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution is to calculate the CR and S/OP values for each job by using the given data as in below: The first step is to calculate the CR and S/OP values for each job by using the given data as in below: CR = [Time remaining until due date]/[Shop time remaining] = 15/6 = 2.33 and S/OP = [Time remaining until the due date - Shop time remaining] / # of operations remaining = [14-6]/10 = 8/10 = 0.80 for Job 1.

50. 50 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution The second step is, to squence six jobs at the NC Milling Machine according to five specified priority sequencing rules The second step is, to squence six jobs at the NC Milling Machine according to five specified priority sequencing rules OR

51. 51 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution The second step is, to squence six jobs at the NC Milling Machine according to five specified priority sequencing rules The second step is, to squence six jobs at the NC Milling Machine according to five specified priority sequencing rules

52. 52 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution When sequencing the jobs with respect to EDD rule, we might expect to see the results of two separate sequencing cases and compare them with each other. When sequencing the jobs with respect to EDD rule, we might expect to see the results of two separate sequencing cases and compare them with each other. In the first case, we can use the “customer due date”s of the jobs (which are usually completion dates of the respective shop orders). In the first case, we can use the “customer due date”s of the jobs (which are usually completion dates of the respective shop orders). In the second case, we can estimate pickup (due) times for each job (which are usually the completion times of the nc milling operation as calculated by Shop Floor Control Module). In the second case, we can estimate pickup (due) times for each job (which are usually the completion times of the nc milling operation as calculated by Shop Floor Control Module).

53. 53 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution The approach which we may use in estimating the subject pickup times from NC Milling Machine is given as in below: The approach which we may use in estimating the subject pickup times from NC Milling Machine is given as in below:

54. 54 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution is to calculate “job flow time”, “hours early” and “hours past due” values and to make an assumption of “actual pickup hours” of each job with respect to each priority sequencing rule. The results of this step are given as follows: The third step is to calculate “job flow time”, “hours early” and “hours past due” values and to make an assumption of “actual pickup hours” of each job with respect to each priority sequencing rule. The results of this step are given as follows:

55. 55 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution... The third step...

56. 56 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution... The third step...

57. 57 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution... The third step...

58. 58 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution... The third step...

59. 59 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution... The third step...

60. 60 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) Solution is to calculate the values of the measures for each sequencing rule. The fourth step is to calculate the values of the measures for each sequencing rule. The average time spent (avg flow time), the average lateness (avg past due hours), the average number of jobs in the system (avg wip inventory) and the average total number of jobs in the system (avg total inventory) will be all minimized if we use the SPT rule. On the other hand, both multiple-dimension rules (CR and S/OP) yield better results with respect to average early time when compared to EDD rule by maximizing it. Also, variance in average past due hours (average lateness) for both of these rules are lower than the other rules and S/OP rule minimized this variance.

61. 61 Application of Multiple-Dimension Rules (Analysis of Single- Machine Case where setup times are independent of sequence) END OF SINGLE MACHINE CASE

62. 62 The Two-Machine (flow shop) Case Development of a scheduling procedure for the two- machine case is more complex than for single-machine systems. Development of a scheduling procedure for the two- machine case is more complex than for single-machine systems. In the two machine case, the schedule which involves both machines must satisfy the chosen objective. The following basic assumptions are valid for this case; In the two machine case, the schedule which involves both machines must satisfy the chosen objective. The following basic assumptions are valid for this case; Each job always goes from a particular machine to another machine (i.e. we consider routings), Each job always goes from a particular machine to another machine (i.e. we consider routings), All jobs are available at the start of the schedule, and All jobs are available at the start of the schedule, and Setup times are independent. Setup times are independent.

63. 63 The Two-Machine (flow shop) Case A set of rules has been developed to minimize the make- span in the two-machine case. A set of rules has been developed to minimize the make- span in the two-machine case. Note that, while the minimum make-span does not depend on job sequencing in the one-machine case, this is not true for the two-machine case. Note that, while the minimum make-span does not depend on job sequencing in the one-machine case, this is not true for the two-machine case. It should be also noted that, if total time to run the entire batch of jobs is to be minimized, this does not ensure either the average time each job spends in the system or the average number of jobs in the system will also be minimized. It should be also noted that, if total time to run the entire batch of jobs is to be minimized, this does not ensure either the average time each job spends in the system or the average number of jobs in the system will also be minimized.

64. 64 The Two-Machine (flow shop) Case the make-span has two basic advantages; Minimizing the make-span has two basic advantages; The group of jobs is completed in the minimum time The group of jobs is completed in the minimum time The utilization of the two-machine flow shop is maximized. Utilizing the first machine continuously until it processes the last job, minimizes the idle time on the second machine. The utilization of the two-machine flow shop is maximized. Utilizing the first machine continuously until it processes the last job, minimizes the idle time on the second machine.

65. 65 The Two-Machine (flow shop) Case JOHNSON’s RULE “Johnson’s Rule” is a procedure that minimizes make- span in scheduling a group of jobs on two machines. “Johnson’s Rule” is a procedure that minimizes make- span in scheduling a group of jobs on two machines. The sequence of jobs on the two machines should be identical and that the priority assigned to a job should therefore be the same at both. The sequence of jobs on the two machines should be identical and that the priority assigned to a job should therefore be the same at both. Is based on the assumption of a known set of jobs, each with a known processing time and available to begin processing on the first machine. Is based on the assumption of a known set of jobs, each with a known processing time and available to begin processing on the first machine.

66. 66 The Two-Machine (flow shop) Case JOHNSON’s RULE (Procedure) Step 1: Select the job with the minimum processing time on either machine 1 or machine 2. If this time is associated with machine 1, schedule this job first. If it is for machine 2, schedule this job last in the series of jobs to be run. Remove this job from further consideration.  Step 2: Select the job with next smallest processing time and proceed as in step 1 (if for machine 1, schedule it next, if for machine 2, schedule it as near to last as possible). Any ties can be broken randomly.   Step 3: Continue this process until all of the jobs have been scheduled.

67. 67 The Two-Machine (flow shop) Case EXAMPLE (Problem) Five motors will be repaired at two work centers in the following manner; Work Center 1: Dismantle the motor and clean the parts Work Center 2: Replace the parts as necessary, and test the motor. The customer’s shop can not operate until all the motors have been repaired. For this reason the plant manager of Morris Machine Company wants to develop a schedule which minimizes the make-span. The estimated times to repair each motor is given as in following Figure 47a.

68. 68 The Two-Machine (flow shop) Case EXAMPLE (Problem) The estimated times to repair each motor is given as in following Figure.

69. 69 The Two-Machine (flow shop) Case EXAMPLE (Solution)

70. 70 EXAMPLE (Solution) No other sequence of respective jobs will produce a lower make-span. This schedule minimizes the idle time of work center 2 and gives the fastest repair time for all five motors. Note that the schedule recognizes that a job can not begin at work center 2 until it has been completed at work center 1. The Two-Machine (flow shop) Case

71. 71   The rules applied for two-machine case can be used in larger flow shop scheduling problems.  Campbell, Dudek and Smith (CD&S) have developed an efficient heuristic procedure for this purpose.   This procedure uses the Johnson’s algorithm to solve a series of two-machine approximations for the actual problem of having m-machines and is called “CD&S Algorithm for m-machines”. The m-Machine (flow shop) Case

72. 72 CD&S ALGORITHM FOR m-machines  Step 1: Solve the first problem considering only machine 1 and machine m ignoring the intervening m-2 machines.   Step 2: Solve the second problem by pooling the first two machines (1 and 2) and the last two machines (m-1 and m) to form two dummy machines. Processing time at the first dummy machine is the sum of the processing time on machines 1 and 2 for each order. Processing time at the second dummy machine is the sum of the processing time at machines m-1 and m for each order. The m-Machine (flow shop) Case

73. 73 CD&S ALGORITHM FOR m-machines  Step 3: Continue in this manner until m-1 problems are solved. In the final problem, the first dummy machine contains machines 1 through m-1, and the second dummy machine contains machines 2 through m.   Step 4: Compute the make-span for each problem solved and select the best sequence. The m-Machine (flow shop) Case

74. 74 Example for m-Machines (Problem)  Let’s assume that we have four jobs which has identical routings according to which jobs should flow through five work centers in the same order.   Again let’s assume that all five jobs are available at the start of the schedule and their setup times are independent.  The respective processing time of each job for each work center is given as follows; The m-Machine (flow shop) Case

75. 75 Example for m-Machines (Problem)  We want to find the schedule which minimizes the make-span required to process all these jobs. The m-Machine (flow shop) Case

76. 76 Example for m-Machines (Solution) Step 1 : The m-Machine (flow shop) Case

77. 77 Example for m-Machines (Solution) Step 2 : The m-Machine (flow shop) Case

78. 78 The m-Machine (flow shop) Case Example for m-Machines (Solution) Step 3 :

79. 79 The m-Machine (flow shop) Case Example for m-Machines (Solution) Step 4:

80. 80 Example for m-Machines (Solution) Step 5 :   Solution of step 1 : Job 2 - Job 1 - Job 4 - Job 3 Solution of step 2 : Job 1 - Job 4 - Job 2 - Job 3 Solution of step 3 : Job 1 - Job 4 - Job 3 - Job 2 Solution of step 4 : Job 2 - Job 1 - Job 4 - Job 3 The m-Machine (flow shop) Case

81. 81 The m-Machine (flow shop) Case Example for m-Machines (Solution)

82. 82 Some notes for multiple machine flow shop scheduling problems The size of the problems which can be treated with analytical methods is small and have limited applicability in the “real world”. Computer time required to solve scheduling problems with analytical methods grows exponentially as the number of jobs or machines increases. Minimizing the make-span, is not the same as “minimizing average time in the system” or “minimizing the average number of jobs in the system”. The static scheduling assumptions (beginning with all machines idle and all jobs available, and ending with all jobs processed and all machines idle) clearly influence the results. The static scheduling assumptions (beginning with all machines idle and all jobs available, and ending with all jobs processed and all machines idle) clearly influence the results. The m-Machine (flow shop) Case

83. 83 Some notes for multiple machine flow shop scheduling problems   The processing times of machines do not reflect any randomness, which could reduce the applicability of the techniques used for solution. It is important to note that, the two-machine scheduling rules utilize the “shortest processing time” logic.The SPT application in the two-machine case is not exactly the same as it was in the single-machine case. It is important to note that, the two-machine scheduling rules utilize the “shortest processing time” logic.The SPT application in the two-machine case is not exactly the same as it was in the single-machine case. The m-Machine (flow shop) Case

84. 84 The m-Parallel Machines Case From the practical point of view, it is important because in real world the occurrance of parallel formation of machines (resources) is common. From the practical point of view, it is important because in real world the occurrance of parallel formation of machines (resources) is common. In this type of formation, all of the individual machines are capable of performing the same process on the available set of jobs. In this type of formation, all of the individual machines are capable of performing the same process on the available set of jobs.

85. 85 The m-Parallel Machines Case Also, techniques for parallel machines are often used in decomposition procedures for multistage systems. Also, techniques for parallel machines are often used in decomposition procedures for multistage systems. When dealing with parallel machines, generally three primary objectives are considered. These objectives are; When dealing with parallel machines, generally three primary objectives are considered. These objectives are; minimization of the make-span,minimization of the make-span, total completion time, andtotal completion time, and maximum lateness.maximum lateness.

86. 86 The m-Parallel Machines Case When dealing with machines in parallel, the make-span becomes an objective of significant interest in comparision to single-machine case. When dealing with machines in parallel, the make-span becomes an objective of significant interest in comparision to single-machine case. In practice, the scheduler whose job is to deal with balancing the work load on machines which operate in parallel is to minimize the make-span. This will ensure a good balance. In practice, the scheduler whose job is to deal with balancing the work load on machines which operate in parallel is to minimize the make-span. This will ensure a good balance.

87. 87 The m-Parallel Machines Case We may consider the scheduling on parallel machines as a two step procedure, where; We may consider the scheduling on parallel machines as a two step procedure, where; the first step is to determine which jobs are to be allocated to which machine, andthe first step is to determine which jobs are to be allocated to which machine, and the second step is the determination of priority sequence of the jobs allocated to each machine.the second step is the determination of priority sequence of the jobs allocated to each machine.

88. 88 The m-Parallel Machines Case On parallel machines, “preemption”s play an important role. On parallel machines, “preemption”s play an important role. Here, the word “preemption” refers to the interruption of the processing of a job (preempt) at any time and put a different job on the machine. In such a case, the amount of processing on the preempted job is not lost. When a preempted job is put back on a machine it only needs the remaining proceesing time.Here, the word “preemption” refers to the interruption of the processing of a job (preempt) at any time and put a different job on the machine. In such a case, the amount of processing on the preempted job is not lost. When a preempted job is put back on a machine it only needs the remaining proceesing time. In the single-machine case the role of preemption is important only when jobs are released at different times. However, in the case where machines are in parallel preemptions are important even if all the jobs are released at the same time.In the single-machine case the role of preemption is important only when jobs are released at different times. However, in the case where machines are in parallel preemptions are important even if all the jobs are released at the same time.

89. 89 “m” Parallel Identical Machines : The Make-span Without Preemptions to be considered in this group has the following characteristics; The first priority sequencing problem to be considered in this group has the following characteristics; There are “m” identical machines in parallel which are all available at the start.There are “m” identical machines in parallel which are all available at the start. There are “n” jobs each requiring a single operation which can be performed on any one of the “m” machines, and they are all available at the start. Preemptions are not allowed.There are “n” jobs each requiring a single operation which can be performed on any one of the “m” machines, and they are all available at the start. Preemptions are not allowed. Make-span is equal to the completion (flow) time of the last job to leave the system and,Make-span is equal to the completion (flow) time of the last job to leave the system and, The objective is to minimize the make-span,The objective is to minimize the make-span, that is balancing the work load on “m” parallel identical machines.

90. 90 “m” Parallel Identical Machines : The Make-span Without Preemptions One of the heuristic methods uses the “longest processing time first” (LPT) rule for the solution of this problem. One of the heuristic methods uses the “longest processing time first” (LPT) rule for the solution of this problem. LPT rule allocates “m” longest jobs to “m” machines starting from the job which has the longest processing time. After this, whenever a machine is freed, the largest of the remaining (unscheduled) jobs is loaded to that machine.LPT rule allocates “m” longest jobs to “m” machines starting from the job which has the longest processing time. After this, whenever a machine is freed, the largest of the remaining (unscheduled) jobs is loaded to that machine. Thus, LPT rule tries to place the shorter jobs toward the end of the schedule where tey can be used for balancing the work load distribution on the respective “m” machines.Thus, LPT rule tries to place the shorter jobs toward the end of the schedule where tey can be used for balancing the work load distribution on the respective “m” machines.

91. 91 “m” Parallel Identical Machines : The Make-span Without Preemptions The is treated as the “ ” for the respective problem. Then the following argument follows; The make-span obtained by LPT rule is treated as the “worst case scenario” for the respective problem. Then the following argument follows; If make-span obtained by LPT schedule is, then there should exist at least one make- span (better solution) which is possibly unknown and can be called as “”. Ofcourse, the so called “optimal make-span” should be smaller than the “LPT Make-span”.If make-span obtained by LPT schedule is assumed to be the worst case, then there should exist at least one make- span (better solution) which is possibly unknown and can be called as “optimal make-span”. Ofcourse, the so called “optimal make-span” should be smaller than the “LPT Make-span”. If we can not determine a smaller make-span, the LPT Make-span will be assumed to be the optimal solution to the problem.If we can not determine a smaller make-span, the LPT Make-span will be assumed to be the optimal solution to the problem.

92. 92 “m” Parallel Identical Machines : The Make-span Without Preemptions This heuristic method for this specific type of scheduling problem, has some properties which can be shown by following inequalities; This heuristic method for this specific type of scheduling problem, has some properties which can be shown by following inequalities; MakeSpan(LPT)/MakeSpan(OPT) ≤ (4/3)-(1/3m) ; This inequality tells us what should be the maximum value of the ratio of “worst case make-span” to “optimal make-span” by considering the number of machines. If the value of this ratio is “1”, we can conclude that “worst case make-span” is equal to “optimal make- span”. If the value is more than “1”, we can conclude that there is an “optimal make-span” which is smaller than the “worst case make- span” and the value of the “optimal make-span” is; MakeSpan(OPT) ≥ [ MakeSpan(LPT) / ((4/3)-(1/3m))]

93. 93 “m” Parallel Identical Machines : The Make-span Without Preemptions This heuristic method for this specific type of scheduling problem, has some properties which can be shown by following inequalities; This heuristic method for this specific type of scheduling problem, has some properties which can be shown by following inequalities; MakeSpan(OPT) ≥ (Σn p j )/m, where p j is the processing time of individual job. This inequality also gives an idea about the probable value of the “optimal make-span”. Right hand side of the inequality simply divides the sum of all the processing times of “n” jobs by the number of available identical parallel machines which is denoted by “m”.

94. 94 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Problem) Let’s assume that, we have 9 jobs for which the processing times are given as follows and 4 parallel identical machines. Also assume that no preemptions are allowed. Our problem is to find a schedule which will provide us the minimum make- span to process all the 9 jobs by utilizing the 4 parallel and identical machines.

95. 95 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Solution) We are going to use “longest processing time first” (LPT) rule. According to this rule, as the first step, we will allocate “4” longest jobs to “4” machines starting from the job which has the longest processing time...

96. 96 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Solution) Thus, we obtain the LPT Make- span which is 15 hours.... after that, whenever a machine is freed, the largest of the remaining (unscheduled) jobs is loaded to that machine as shown in following Figure. Thus, we obtain the LPT Make- span which is 15 hours.

97. 97 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Solution) In the third step, we can make use of the previously given inequalities for the analysis of the constructed LPT schedule; In the third step, we can make use of the previously given inequalities for the analysis of the constructed LPT schedule; In our example case, MakeSpan(LPT)= 15 hrs and the number of parallel identical machines is m=4.In our example case, MakeSpan(LPT)= 15 hrs and the number of parallel identical machines is m=4. Thus, the value of the ratio;Thus, the value of the ratio; MakeSpan(LPT)/MakeSpan(OPT) ≤1.33 – (1/12) MakeSpan(LPT)/MakeSpan(OPT) ≤1.33 – (1/12) ≤ 1.33-0.083 ≤ 1.33-0.083 ≤ 1,247 ≤ 1,247 Therefore;Therefore; MakeSpan(OPT) ≥ 15/1.247 MakeSpan(OPT) ≥ 15/1.247 ≥ 12.03 hrs ≥ 12.03 hrs

98. 98 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Solution) In the third step, we can make use of the previously given inequalities for the analysis of the constructed LPT schedule; In the third step, we can make use of the previously given inequalities for the analysis of the constructed LPT schedule; Also for inequality,Also for inequality, MakeSpan(OPT) ≥ (Σn pj)/m the value of (Σn pj)/m=48/4=12 hrs the value of (Σn pj)/m=48/4=12 hrs

99. 99 “m” Parallel Identical Machines : The Make-span Without Preemptions Example (Solution)  Thus, the results of the analysis tells us that there is a better schedule for the problem and the make-span of this schedule is ≥12hrs. By trial and error this better schedule with optimal make-span can be constructed as follows;

100. 100 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions  The second priority sequencing problem to be considered in this group has the same characteristics as previously described in this section but,  The second priority sequencing problem to be considered in this group has the same characteristics as previously described in this section but, this time the objective is to minimize the total completion (flow) time.   Studies on this type of problem has shown that SPT rule provides an optimal solution for this type of priority sequencing problem, but, it is not the only rule which will minimize the “total completion time”.

101. 101 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions  Solution method : Consider that we have n jobs and m identical parallel machines where preemptions are not allowed. According to the SPT rule, at the start, the smallest job has to go on machine 1, the second smallest on machine 2 and so on. The (m+1)th smallest job follows the smallest job on machine 1, the (m+2)th smallest job follows the second smallest job on machine 2 and so on.

102. 102 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions Example (Problem)   Assume that we have the previous case of 9 jobs which are given as follows and the same number of parallel identical machines which is 4.   Also assume that this time our objective is to minimize the total completion (flow) time.

103. 103 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions Example (Solution)   Since our objective is to minimize the total completion time, we are going to use the SPT rule for priority sequencing of the subject 9 jobs over 4 parallel identical machines.  The next smallest job is job 5 and is assigned to machine 4. Remaining smallest jobs are job 6 ≤ job 3 ≤ job 4 ≤ job 1 ≤ job2. So, we assign job 6 as the second job for machine 1, job 3 as the second job for machine 2, job 4 as the second job for machine 3, job 1 as the second job for machine 4 and job 2 as the third job for machine 1.  First, since the jobs 7, 8 and 9 have the same smallest processing time, we assign job 7 to machine 1, job 8 to machine 2 and job 9 to machine 3. The next smallest job is job 5 and is assigned to machine 4. Remaining smallest jobs are job 6 ≤ job 3 ≤ job 4 ≤ job 1 ≤ job2. So, we assign job 6 as the second job for machine 1, job 3 as the second job for machine 2, job 4 as the second job for machine 3, job 1 as the second job for machine 4 and job 2 as the third job for machine 1.

104. 104 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions Example (Solution)   The resultant schedule will be as follows; As can be easily verified from the above Figure, the make-span of the SPT schedule for this problem is 16 hours. Also, the total completion time of the schedule is 74 hours.

105. 105 “m” Parallel Identical Machines : The Total Completion Time Without Preemptions Example (Solution)   The comparision of total completion (flow) times;

106. 106 “m” Parallel Identical Machines : The Make-span With Preemptions   The third priority sequencing problem which we will consider is the minimization of make-span problem where preemptions are allowed. Studies on about the subject problem showed that, allowing preemptions usually (but not always) simplifies the analysis of the problem.

107. 107 “m” Parallel Identical Machines : The Make-span With Preemptions   The following three step method was developed to construct a schedule which minimizes the make-span; Step 1: Take the “n” jobs and process them one after another on a single-machine in any sequence. The make-span is then equal to the sum of the “n” processing times. Step 2: Take this single-machine schedule and cut it into “m” intervals (parts).Step 2: Take this single-machine schedule and cut it into “m” intervals (parts). Step 3: Take the processing sequence of the first interval as the schedule for machine 1; the processing sequence of the second interval as the schedule for machine 2, and so on.Step 3: Take the processing sequence of the first interval as the schedule for machine 1; the processing sequence of the second interval as the schedule for machine 2, and so on.

108. 108 “m” Parallel Identical Machines : The Make-span With Preemptions Example (Problem)   Assume that we have 5 jobs as given below and 3 parallel identical machines. We can unload any of the respective jobs from one machine before they are completed and reload them at a later time to perform the remaining processing. Our objective is to establish a schedule which minimizes the make-span.

109. 109 “m” Parallel Identical Machines : The Make-span With Preemptions Example (Solution)   Step 1: We construct a single-machine schedule where make-span is equal to 26 hrs.

110. 110 “m” Parallel Identical Machines : The Make-span With Preemptions Example (Solution)   Step 2: We divide the single-machine schedule into 3 intervals where 3=m (the number of machines).

111. 111 “m” Parallel Identical Machines : The Make-span With Preemptions Example (Solution)  We assign the first interval to machine 1, the second interval to machine 2, and the third interval to machine 3.  Step 3: We assign the first interval to machine 1, the second interval to machine 2, and the third interval to machine 3.