1. Online Scheduling with Known Arrival Times Nicholas G Hall (Ohio State University) Marc E Posner (Ohio State University) Chris N Potts (University of Southampton) 14 May 2008, CIRM, Marseille

2. 2 Motivation Orders are accepted, for example, weekly. Between two orders, no new jobs become available. Until an order arrives, its details (for example, its processing time and value) are not known. Meanwhile, scheduling decisions must be made for the available jobs. Consider a typical make-to-order production system that uses a periodic ordering and scheduling process. In order to keep resources available to process new jobs, idle time may be inserted.

3. 3 A New Scheduling Environment Data for currently available jobs are known. The potential job arrival times effectively define planning periods. New jobs may arrive only at known future times. There is no restriction on the data for the new jobs, which become known only on arrival. Therefore, we consider our problem to be an online planning period scheduling problem.

4. 4 Online vs. Offline Scheduling Classical offline: all data are known at the start of the planning horizon. If the number of potential job arrival times is large and uniformly distributed, then the new environment approaches the classical online environment as a limiting case. Classical online: new jobs may arrive at any time, and data become known only on arrival. Our new environment interpolates between the classical offline and classical online environments.

5. 5 Problem Definition Consider a single machine scheduling problem environment. Each job j has a release date r j where r j  {t 0, t 1,…, t T }, a processing time p j and a weight w j. Let 0 = t 0 < t 1 < … < t T denote known potential job arrival times. The goal is to minimize  w j C j, the total weighted completion time of the jobs, a widely used measure of customer service.

6. 6 Related Online Literature (1 of 2) Several studies (e.g. Hoogeveen and Vestjens, 1996) have established that no online algorithm can have a competitive ratio that is better than 2. Consider the classical online version of our problem. The competitive ratio compares the solution value  w j C j given by the online algorithm to the optimal offline solution value C*. Anderson and Potts (2004) propose a “best possible” online algorithm with a competitive ratio of 2.

7. 7 Related Online Literature (2 of 2) The following instance provides a lower bound on the competitive ratio. If t  1, then no other job arrives and therefore  w j C j /C * = t+1  2. Job 1 arrives at time zero: r 1 = 0, p 1 = 1, w 1 = . Suppose that job 1 starts processing at time t. If t 2.

8. 8 Research Question The best possible algorithm for the classical online problem has a competitive ratio of 2. However, in our problem, we have more information than in the classical online case: we know times when jobs cannot arrive. Consequently, the best possible competitive ratio may be smaller than 2.

9. 9 Main Result We describe a simple online scheduling algorithm which runs in O(n log n + T 2 ) time, and which achieves a competitive ratio of R* = max { min {R s (v)}} where R s (v) = (t s+1 + t s +  (t s+1 – t s ) 2 + 4t v t s+1 )/2t s+1 This ratio is best possible for the problem. Note that R* lies between (1+  5)/2 and 2, depending on the precise values of t 1,…,t T. v=1,…,T s=0,…,v  1

10. 10 Lower Bound for the Offline Problem The job splitting lower bound of Belouadah, Posner and Potts (1992) provides a lower bound z L on the total weighted completion time of an optimal offline schedule, which we use in our analysis. Choose an available job j with the largest w j /p j and schedule it to start as early as possible. If no job k with w k /p k > w j /p j becomes available during the processing of job j, then process job j to completion. Otherwise, choose the first job k with w k /p k > w j /p j to become available during the processing of job j and split job j at time r k.

11. 11 Mathematical Program for Lower Bound Inequality (1) models a situation where job j with r j = t 0 starts processing at time t s and is still in process at time t s+1 when another job with large weight and small processing time arrives. Maximize R subject to (t s + p)/t s+1 ≥ R, s = 0,…,v  1 (1) (t v + p)/(t 0 + p) ≥ R (2) p, R ≥ 0 Inequality (2) models a situation where job j with r j = t 0 starts at time t v, and no other job is released. We introduce the following problem P(v):

12. 12 Lower Bound Theorem Theorem. The competitive ratio of any online algorithm is at least R* = max 0<v≤T {R(v)}. Proof. Let R* = R(v), and let p* denote the optimal value of p in problem P(v). Consider an instance with 1 or 2 jobs, where r 1 = 0, p 1 = p* and w 1 = 1. Let the algorithm start job 1 at time t s, where 0 ≤ s ≤ T. If s < v, then an adversary releases a second job with large weight at time t s+1, generating a performance ratio of (t s + p*)/t s+1 ≥ R*. If s ≥ v, then no other job is released, generating a performance ratio of (t s + p*)/(t 0 + p*) ≥ R*. □

13. 13 Optimality Conditions for P(v) Lemma. In an optimal solution to problem P(v), constraint (2) and at least one constraint of (1) are satisfied at equality. Proof. Let (p,R) denote an optimal solution to P(v). If each constraint (1) is satisfied as a strict inequality, then the solution (p , R +  t v /p(p  )) is feasible for problem P(v). If constraint (2) is satisfied as a strict inequality, then the solution (p  + , R +  /t v ) is feasible for problem P(v). In both cases, the optimality of (p,R) is contradicted. □

14. 14 Value of R* Theorem. R* = max { min {R s (v)}}, where R s (v) = (t s+1 + t s +  (t s+1 – t s ) 2 + 4t v t s+1 )/2t s+1 We choose R(v) = min s = 0,…,v-1 {R s (v)} since other values of s give infeasible values of p and R. □ v=1,…,T s=0,…,v  1 Proof. The previous result shows that p(v) and R(v) are obtained by solving (t s + p)/t s+1 = R and (t v + p)/p = R for some s, which provides the value R s (v).

15. 15 Design of Online Algorithm j tCjCj Need to protect against the arrival of a short job with very large weight arriving at time t. Thus, only process job j if C j /t ≤ R*.

16. 16 Online Algorithm Algorithm CSWPT 1. Let j = an available job with largest w j /p j ratio. Let t u = the earliest future potential job arrival time. 2. If job j cannot complete by time R*t u, then insert idle time up to time t u and go to Step 4. 0. Compute the lower bound R* on the competitive ratio. 3. Schedule job j. If it completes processing before time t u, then go to Step 1. 4. If u < T, or there are available jobs, then go to Step 1.

17. 17 Analysis of Algorithm CSWPT Restrict the instances to be considered to those with w j /p j = 0 or 1. Thus, each job is classified as type 0 or type 1. Establish that the completion time of any type 1 job under Algorithm CSWPT is no more than R* times its contribution to the job splitting lower bound. z C denotes the cost of the schedule σ C delivered by Algorithm CSWPT. z L denotes the value of the lower bound provided by schedule σ L created by the job splitting procedure.

18. 18 Characterizing a Worst-Case Instance Lemma. Given any instance, there exists another instance for which z C /z L is at least as large and w j /p j = 0 or 1. Proof. Suppose that there is more than one distinct nonzero w j /p j ratio. Choose a set of jobs with nonzero w j /p j ratio such that this ratio is not the largest. If the value of z C /z L for jobs restricted to this set is larger than the value of z C /z L for all of the jobs, then increase the weight of all jobs in this set so that they still have equal w j /p j values.

19. 19 Characterizing a Worst-Case Instance Otherwise, similarly decrease the weight of jobs in this set. Repeating this argument reduces the number of distinct nonzero w j /p j ratios to one and does not decrease z C /z L. A rescaling of the weights allows us to achieve w j /p j = 0 or 1 without affecting the value of z C /z L. □

20. 20 Properties of Restricted Instances In the job splitting procedure, only type 0 jobs are split. Thus, each type 1 job j contributes C j (σ L ) to the lower bound. Type 1 jobs are sequenced in the same order by Algorithm CSWPT and by the job splitting procedure. For any type 1 job j, C j (σ L ) ≤ C j (σ C ).

21. 21 Type 1 Job j Starts after Type 0 Job i Lemma. If a type 1 job j starts processing in σ C when a type 0 job i completes, then C j (σ C )/C j (σ L ) ≤ R*. Proof. Let job i start at time t l-1 or later but before time t l. Then C j (σ C ) = C i (σ C ) + p j ≤ R*t l + p j ≤ R*(r j + p j ) since job j cannot have been released when job i started. Therefore, C j (σ C )/C j (σ L ) ≤ R*. □

22. 22 Type 1 Job j Starts after Type 1 Job i Lemma. If a type 1 job j starts processing in σ C when another type 1 job i completes, then C j (σ C )/C j (σ L ) ≤ C i (σ C )/C i (σ L ). Proof. C j (σ C )/C j (σ L ) ≤ (C i (σ C ) + p j )/(C i (σ L ) + p j ) ≤ C i (σ C )/C i (σ L ). □

23. 23 Type 1 Job j Preceded by Idle Time Lemma. If a type 1 job j starts processing in σ C immediately after a period of idle time, then C j (σ C )/C j (σ L ) ≤ R*.

24. 24 Main Result Theorem. Algorithm CSWPT has the best possible competitive ratio of R*. Proof. The upper bound follows from the three previous lemmas. From the lower bound theorem, this result is best possible. □

25. 25 Special Case: Equally Spaced t u Values Suppose that t u+1 – t u is constant, for u = 0,…,T  1. Then R * = (t T + t T-1 +  (5t T 2 + t T-1 2 - 2t T-1 t T ))/2t T Moreover, as T → ∞ with t T fixed, R* → 2. Thus, we obtain the classical online result.

26. 26 Conclusions (1 of 2) We consider an online planning period scheduling problem, which falls within a wider class of online planning period problems. For the single machine total weighted completion time problem, a lower bound on the competitive ratio is given by the optimal value of a mathematical program. We provide a closed form expression for this value, which is between (1 + √5)/2 and 2, and depends on the potential job arrival times. We also describe a fast online algorithm with a competitive ratio that matches this lower bound, and which is therefore best possible.

27. 27 Conclusions (2 of 2) For the case of equally spaced potential job arrival times, as T → ∞ the competitive ratio becomes arbitrarily close to 2. There are many challenges for future research in online planning period (scheduling) problems.

28. Copies of the related paper are available by request at: C.N.Potts@soton.ac.uk Thank you for your attention! This work is supported by The National Science Foundation under grant DMI-0421823, and by EPSRC under grant EP/D060931/1.