Simulation. Stochastic scheduling example: Can we get the work done in time?

Transcription

1 Simulation Stochastic scheduling example: Can we get the work done in time? Example of decision making under uncertainty, combination of algorithms and probability distributions 1

2 Example study planning Job p d 1. Modeling assignment simulation 2. Algorithms and networks assignment 3. Game engine programming assignment 1 4. Midterm exam algorithms and networks Simulation assignment Game engine programming assignment Assumption: No deadline extensions You cannot do everything, decide on what you do NOW 2

3 Minimize number of tardy obs on a single machine Single machine continuously available from time zero onwards n obs have to be processed Known processing time p No preemption Known due date d Only reward (fixed amount) if ob is completed in time We cannot complete everything before its due date Obective maximize reward Decision to make now: accept or reect 3

4 Moore-Hodgson 1. Number the obs in Earliest Due Date (EDD) order 2. Let S denote the EDD schedule 3. Find the first ob not on time in S (suppose this is ob ) 4. Remove from S the largest available ob from obs 1,, 5. Continue with Step 3 for this new schedule S until all obs are on time 4

5 Resulting schedule Observations First the on time obs On time obs in EDD order Forget about the late obs Knowing the on time set is sufficient 5

6 Dominance rule Let S 1 and S 2 be two schedules In these schedule let E 1 and E 2 be the subsets of obs 1,, All obs in E 1 and E 2 are on time (feasible) Cardinality of E 1 and E 2 is equal The total processing time of the obs in E 2 is more than the total processing time of the obs in E 1 Then subset E 2 can be discarded. 6

7 Proof (sketch) Take an optimal schedule starting with E 2 (remainder: obs from +1,, n) E 2 remainder 0 time E 1 remainder 7

8 Dynamic programming Will be useful for stochastic processing times. Jobs are numbered in EDD order Find E * (k): feasible subset of obs 1,, with cardinality k (so k on-time obs) and minimum total processing time Use state variables f (k) equal to p(e * (k)) Define z as maximum number of on time obs from obs 1,, 8

9 Recurrence relation Initialization = 0: f 0 (k)=0 for k=0 (and + otherwise), z 0 = 0 Recurrence: f +1 (0)=0 f +1 (k)=min{f (k),f (k-1)+p +1 } (k=1,,z ) If f (z )+p +1 d +1 then z +1 =z +1 and f +1 (z +1 )=f (z )+p +1 ; else z +1 =z. Final answer z n 9

10 Moore-Hodgson Revisited 1. Number the obs in EDD order 2. Compute the values f (z ): If f (z )+p +1 d +1 then z +1 =z +1 and f +1 (z +1 )=f (z )+p +1 i.e. J +1 is added else z +1 = z and f +1 (z +1 ) = min{f (z ),f (z -1)+p +1 } i.e. largest ob is removed 10

11 Stochastic processing times Completion times are uncertain Decision about accept or reect must be made before running the schedule When do you consider a ob on time? 11

12 On time stochastically Work with a sequence of on time obs (instead of a set of completion times) Compute the probability that a ob is ready on time If this probability is large enough (at least equal to the minimum success probability msp) then accept it as on time 12

13 Classes of processing times Gamma distribution Negative binomial distribution Equally disturbed processing times p Normal distribution Jobs must be independent 13

14 Class 1: Gamma distribution Stochastic processing time P follows Gamma distribution with parameters a and b (common) If X 1 and X 2 follow the Gamma distribution and are independent, then X 1 +X 2 is Gamma distributed with parameters a 1 +a 2 and b 14

15 More gamma Define S as the set of ob and all its predecessors in the schedule Define p(s) as the sum of all processing times of obs in S What is the distribution of C = p(s)? Then completion time C =p(s) follows a gamma distribution with parameters a(s) and b. 15

16 Even more gamma Denote the msp of ob by y Job is on time if PC d y PC d Is only determined by a(s) and does not depend on which or how many obs are in S is decreasing in a(s) You can compute the maximum value of a(s) such that P Hence C d y P say a(s) = D C d y a( S) D What does this tell you about a solution algorithm? 16

17 Last of Gamma Treat D as ordinary due dates Treat a as ordinary deterministic processing times Then the dominance rule still holds You can use Moore-Hodgson! 17

18 Machine failures No work lost because of failures Job proceeds at point where it was left before the failure 18

19 Machine failures: continuous time Time-to-failure exponential distribution: f(x) = λe -λx Deterministic processing times and reparation times B Again, use S to denote ob and its predecessors in the schedule; the total processing time of S is p(s) For a given schedule: P(C d ) e p(s) dp(s) B k0 ( p(s)) k! k P(C d Compute D Moore-Hodgson!!!! ) only depends on p(s) and is decreasing in p(s) as the maximum p(s) s.t. P(C d ) y 19

20 Machine failures: combine with stochastic processing times Time-to-failure exponential distribution: f(x) = λe -λx Stochastic processing times: P follows Gamma(p /b, b) Again P(C d ) only depends on p(s) and is decreasing in p(s) Compute D as the maximum p(s) s.t. P( C d ) y Use simulation to compute D!!!!! Moore Hodgson solves it. 20

21 Conclusion Moore-Hodgson = Dynamic Programming DP is applicable in a stochastic environment Stochastic on time: work with the minimum success probability EDD sequence optimal for the on time set References: Maran van den Akker and Han Hoogeveen (2008). Minimizing the number of late obs in a stochastic setting using a chance constraint. Journal of Scheduling Volume 11, number 1, pp: Thesis Adriaan Schipper. Stochastic Single-Machine Scheduling with Breakdowns 21