1. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 1 Outline Questions? Comments? Quiz Introduction to scheduling Notation Measures Optimality Equivalency Formal routine for active and non-delay schedules Next quiz on 10/1 First midterm on 10/8, review on 10/6, covers everything up to and including material covered on 10/1

2. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 2 Kinds of scheduling Taking sequences and placing them in a schedule is called time tabling (creating a Gantt chart) Semiactive - process each job as soon as it can be (slide to the left on the chart) Active - No operation can be started earlier without delaying some other operation Non-delay - no machine is kept idle Non-feasible - does not meet Technological Constraints Number of possible schedules (including non-feasible ones) = (n!) m

3. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 3 Notations and definitions d i - due date of Job i r i - ready time of Job i a i - allowance = d i - r i s i - slack = d i - remaining operations p ij - the time required to process o ij W ik - the waiting time of Job i preceding its k th operation (not the work on M k ): J 1 ’s time = W 11 +p 11 +W 12 +p 13 +W 13 +p 12 +W 14 +p 1 17, if the TC is M1 to M3 to M2 to M17 We designate the kth operation as o ij(k)

4. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 4 Notations and definitions (continued) C i is the completion time of Job i F i is the flow time of Job i = C i - r i Even though the English words have an identical meaning, we distinguish between Lateness and Tardiness Lateness L i =C i - d i, therefore maybe positive or negative depending on whether we complete a job before or after its due date

5. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 5 Notations and definitions (continued) Tardiness is non- zero only if the job is completed after its due date: T i = max{L i, 0} We also define Earliness as E i = Max{-L i, 0} The weight or importance of a job is indicated either by w i or Some of our definitions refer to instants in time Completion, readiness Others refer to elapsed time Processing, Waiting, Flow

6. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 6 Notations and definitions (continued) Scheduling - the ordering of operations subject to restrictions and providing start and finishing times for each operation Closed Shop - serves customers from inventory (make to stock) Open shops - Jobs are made to order

7. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 7 W i m O i j(2) W i 2 Notations and definitions (continued)- Schematic M1M1 M2 MmMm Processing Order - M m-1, M j, M m …..M 1, M 2 MjMj M m-1 O i m-1(1) O i2(m) O i1(m-1) CiCi riri O i m(3) p i m-1(1)

8. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 8 Measures Optimality or goodness of schedules only makes sense if we define the measure under which we are considering optimality or goodness. There are three broad categories of measures: Completion time Due dates Inventory or utilization We also define a general class of measures called regular measures

9. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 9 Measures (continued) Based on completion Time Maximum Flow time Maximum Completion time Average Flow time Average Completion time Weighted Average Completion time

10. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 10 Measures (continued) Based on due dates Average Lateness Maximum Lateness Average Tardiness Maximum Tardiness Number of tardy jobs

11. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 11 Measures (continued) Based on Inventory or Utilization Average number of waiting jobs Average number of unfinished jobs (WIP) Average number of completed jobs (finished goods) Average Idle time Maximum Idle time

12. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 12 Measures (continued) Regular measures Always are minimized Are non decreasing in completion times Examples Average and maximum completion time Average and maximum flow time Average and maximum lateness Average and maximum tardiness Number of tardy jobs

13. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 13 Measures (continued) Given two sets of completions times obtained under two schedules generated for the same problem: C and C’ if C i <= C i ’ implies that R(C)<=R(C’) then R is regular

14. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 14 Classification notation All problems can be classified as n/m/A/B where n - number of jobs m - number of machines A - pattern F - Flow Shop P - Permutation G - General Job Shop B - Measure C max, F max etc.

15. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 15 Classification notation (continued) for example n/3/F/Fbar means any number of jobs on three machines in a flow shop being measured on the basis of average flow time

16. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 16 Some further definitions Jobs and ready times fixed = Static Parameters known and fixed = Deterministic Random arrival of jobs= Dynamic Uncertain processing times= stochastic

17. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 17 Optimality Since for any given problem there are a countable number of possible schedules (as long as we do not allow preemption or unnecessary delays) there must be an optimum (or optima) because we can (theoretically) compare all possible schedules and select the best one If we look at the space that contains our schedules and attempt to locate the optimum we find that:

18. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 18 Optimality (continued) Optimal all possible feasible semi active Active non - delay

19. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 19 Equivalent measures Two measures are equivalent if a schedule optimal with respect to one is also optimal with respect to the other and vice versa. Cbar, Fbar, Wbar, Lbar are equivalent Note that a schedule optimal to L max is optimal with T max, but not vice versa C max, Nbar p and I bar are also equivalent Cbar, Fbar, Lbar, Nbar u, Nbar w are equivalent for one machine The choice of measure depends on the circumstances

20. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 20 Schedule Generation As a start, we will define a routine that will generate an active schedule A semiactive schedule is one that starts every job as soon as it can, while obeying the technological and scheduling sequences. Also, the set of all semiactive schedules for a problem contains the optimal schedule Fortunately, the set of active schedules also contains the optimum and is a smaller set. We can forget about generating semiactive schedules

21. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 21 Active scheduling For a given problem there will be many active schedules The routine we will use generates only one and we will have to make frequent choices. Were we to follow each of these decision paths, we would generate all the active schedules and find the optimum However, our purpose here is to make those choices as intelligently as possible, even though it is difficult to foresee their eventual consequence An active schedule is one in which no operation could be started earlier without delaying another operation or violating the technological constraints

22. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 22 Definitions First we will define some terminology useful for our routine: Class of problems - n/m/G/B with no restrictions Stage - step in the routine that places an operation into the schedule - there are therefore nm stages t - counter for stages P t - partial schedule at stage t Schedulable operation - an operation with all its predecessors in P t S t - set of schedulable operations at stage t

23. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 23 Definitions (continued) sigma k - the earliest time an operation o k in S t could be started phi k - the earliest time that o k in S t could be finished phi k = sigma k + p k

24. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 24 Routine by Giffler and Thompson 1. t = 1, S 1 is the set of first operations in all jobs 2. Find min{phi k in S t } and designate it phi* Designate M on which phi* occurs as M* (could be arbitrary) 3. Choose o j in S t such that it satisfies these conditions: a. It uses M* b. sigma j < phi* 4. a. Add o j to P t, which now becomes P t+1 b. Delete o j from S t which now becomes S t+1 c. Add the operation that follows o j in the same job to S t+1 d. Increment t by 1

25. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 25 Routine by Giffler and Thompson (continued) 5. If there are operations left to schedule, go to step 2, else stop Note well that at step 3b. sigma j < phi*, we will often have several choices. We always have at least one, namely, phi* These choices are an extensive topic that we will cover later Follow the example I have taken from French Generating these schedules is tedious work, so leave yourselves some extra time for that homework.

26. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 26 Non-delay schedules Non-delay schedules are a smaller set than the active schedules and therefore are a tempting set to explore Unfortunately, they do not always contain the optimum We will not let that deter us, because non-delay schedules have been found to be usually very good, if not optimal A non-delay schedule is one where every operation is started as soon as it can be

27. Session 10 University of Southern California ISE514 September 24, 2015 Geza P. Bottlik Page 27 Non-delay schedules (continued) We change two steps in the procedure for active schedules to obtain a non-delay procedure: Step 2. instead of phi, we select sigma Find min{sigma k in S t } and designate it sigma* Designate M on which sigma* occurs as M* (could be arbitrary) Step 3 b. sigma j = sigma*