스케줄 이론 Single Machine Independent Jobs – part1 Problems without due dates

1. 스케줄 이론 Single Machine Independent Jobs – part1 Problems without due dates

2. 1. Notational Conventions • Definition of the problems • (Method shown in Pinedo's text book 1st Ed.):  A/B/C/D • A) Job arrival pattern (Static=number of jobs, Dynamic=arrival distribution) • B) Number of machines • C) Flow pattern (Flow shop, Job shop, General Shop-either Flow shop or Job shop) • D) Evaluation Criteria • (예) Static n job / 2 Machine / Flow shop / Min. Completion Time Problem  n/2/F/Cmin • (예) n/m/G/Tmin • I.2.2 Variables • Problem Variables (Job Descriptors): small letters • Decision Variables: Capital letters

3. Jobs and machines || • (Method shown in Pinedo's text book 2st Ed.):  • Data (assumed to be given)

4. Describing a scheduling problem || Machine environment Objective (to be minimized) Process characteristics and constraints

5. Machine environment a • Single machine and machines in parallel

6. Machine environment a (2) • Machines in series

7. Processing characteristics and constraints b

8. Processing characteristics and constraints b (2)

9. Objectives g • Performance measures of individual jobs

10. Objectives g (2) • Functions to be minimized

11. 1. Notational Conventions • Problem Variables • a. Jobs        sometimes    • b. Machines sometimes   • c. Processing Time ( or ): processing time of Job j                                ( or   ): i-th operation of j-th Job • d. Ready Time ( ): the earliest time that processing of the first operation of Job j could begin • e. Due Date ( ): the time by which the processing of the last operation is due to be completed • f. Allowed Time ( ): allowed time in the shop (= )

12. 1. Notational Conventions • Decision Variables • a. Completion Time ( ): Absolute time • b. Flow Time ( ): Time spent in the shop =  • c. Lateness( ) = • Negative Lateness is possible     5 ( ) - 10 ( ) = - 5 • Any Good Points in having Negative ? • d. Tardiness( ) = • e. Earliness( ) = • Tardiness and earliness are all positive numbers. • f. Waiting Time( ) = • (Digression) • We may think of , the waiting time for the i-th operation of Job j • (?)  A schedule is completely described by a set of • ⇔ 2 Schedules ( ∧ & ~ ) are identical / equivalent (w.r.t. some  performance criteria) iff        element-wise

13. 1. Notational Conventions • Performance Measures • ↓MFT (Mean Flow Time) =  • ↓Mean Tardiness =           • ↓Max Flow Time  =         • ↓Max Tardiness  =         • ↓# of Tardy Jobs = 

14. 1. Notational Conventions • (Definition) Regular Measure (Z) of Performance refer to a performance measure for the following case:         • ① Scheduling objective is to minimize Z. • ② Performance measure(Z) is a function of completion time. 즉 • ③ Increases only if one of increases. Namely, we have   such that • (Example) Quantities that are not Regular Measure • Average Earliness • Max Earliness • Difference between the largest to second largest completion times • We need to consider only  Dominant Schedule Sets

15. 1. Notational Conventions • (Definition) Dominant Set • A concept to reduce solution space from complete enumeration • Reasoning Procedures • 1. Consider an arbitrary schedule (S is a string consisting of ) where D is a set of a certain class of schedules • 2. Show that ∃ a schedule    where for all j. • 3. For regular measures, above implies • 4. It is sufficient to consider schedules in D only. • (Example) Suppose (Mean Tardiness) is the measure of performance in single m/c scheduling problem. Now suppose there exists a job k that satisfies ,  then there exists an optimal sequence in which job k is assigned the last. • ⇒  We can consider only n-1 jobs excluding job k.

16. 2. Introduction • Single Machine is Not that Restrictive in Real Applications • Chemical Process Industry: whole facility can be regarded as one M/C. • Bottleneck Process in Process Industries as well as Machine Industries (Temporary Bottleneck) • Single Processor Computing System • Tape Drive, etc. • Basic Single-machine Assumptions • 1. We have n independent, single-operation jobs. • 2. Sequence independent set-up time can be included in each processing time. • 3. Job descriptors(    ) are completely known. • 4. No idle time. • 5. No interruption once a job is started. • (Example) Process industry.  Is assumption 2 reasonable? • We need consider only Permutation Schedules (The total number of possible Schedules: n! )

17. 2. Introduction • Permutation Schedule • Schedules are completely specified by giving processing order (n!). •  So we call these sequencing problems. •  So Performance measures such as “Max flow time”, “Max # of tardy jobs” are irrelevant. • Use Bracket to indicate position in sequence • [5]=2 • d[1] = ?

18. 2. Introduction • Theorem 2.1 With above assumptions, schedules without inserted idle time constitute a dominant set. • Proof • Obvious but we need a formal proof. • (Hint)  Consider two schedules S and S'.  • Theorem 2.2 With above assumptions, schedules without preemption constitute a dominant set. • Proof

19. 3. Problems Without Due Dates • The relationship between FLOW TIME and INVENTORY • Are they proportional? Let us prove it in two cases (Static and Dynamic) a. Static Case • Let  J(t) = # of Jobs in System at time t, V(t) = Inventory Level at time t • By rearranging for A, we get • 즉 minimizing     is directly proportional to minimizing

20. 3. Problems Without Due Dates

21. 3. Problems Without Due Dates • b. Dynamic Case  (Say, Job arrival is ) • Case 1. Assume .    즉 Jobs are completed in arriving order. • Also assume everything completed by . • Now consider

22. 3. Problems Without Due Dates • So

23. 3. Problems Without Due Dates • Case 2. Jobs are Completed In Random Order • Now consider the case when jobs may finish in random.

24. 3. Problems Without Due Dates

25. 3. Problems Without Due Dates c. Discussion ① Under steady state assumption (Rate of completion  Rate of arrival) above holds :  1) Static or Dynamic                        2) No matter how we schedule ( FIFO or Whatever )                     3) Even with weighted inventory costs ② Also think Above implies in Steady State A schedule which minimizes MFT also minimizes inventory (i.e. # of jobs in system),  mean lateness, mean waiting time.

26. Proof • Proof of Theorem 2.1

27. Proof • Proof of Theorem 2.2

28. Proof

29. 3. Problems Without Due Dates • Theorem 2.3  SPT (Shortest Processing Time) sequencing minimizes Mean Flow Time. • Proof ① ②   Graphical Proof of Baker's. ③  ④  Formal (?)    S and S'  ..

30. 3. Problems Without Due Dates • (Discussion) • Theorem 2.4 WSPT(Weighted SPT) sequencing minimizes WMFT (Weighted Mean Flow Time). Proof : Follow the reasoning of ④.