Yingchao ZHAO, Caritas Institute of Higher Education Xiangtong QI, HKUST Minming LI, City University of Hong Kong

 Traditional scheduling objective ◦ minimizing makespan ◦ minimizing total completion time ◦ minimizing total flow time.  Wan and Qi (Naval Research Logistics) proposed the model that considers both traditional objective and time slot cost.  Kulkarni and Munagala (WAOA 2012)studied online algorithms to minimize the total weighted completion time plus time slot costs.  Leonardi et al. (MAPSP 2013) designed a PTAS for the offline version of the problem.

 Increasing ◦ Trivial ◦ Execute the jobs as early as possible  Decreasing ◦ Linear decreasing ◦ Accelerative decreasing ◦ Decelerative decreasing ◦ General decreasing  Objective ◦ Minimize: total weighted completion time + time slot costs

π(t) t

 For each job i: ◦ Processing time p i ◦ Weight w i  The time slot cost π satisfies π k - π k+1 =ε, ε>0  We give a polynomial time algorithm to get the optimal schedule for this problem. ◦ Given two jobs, which should be processed earlier? ◦ Given a job, when should it be processed?

If the time slot cost decreases as a linear function with respect to time k, then there exists an optimal schedule to the problem in which jobs are scheduled in WSPT (Weighted Shortest Processing Time) order.

For a job with processing time p and weight w  If w/p>ε, process this job as early as possible  If w/p<ε, process this job as late as possible  If w/p=ε, the job has the same cost no matter when it is processed

t 0K 0 t K t K0w/p>εw/p<εw/p=ε  Sort the jobs according to decreasing w/p.  Partition jobs into three sets: J 1, J 2, J 3  Process J 1 and J 2 from time zero, and process J 3 as late as possible. Solved in polynomial time!

π(t) t

 The time slot cost π satisfies π k - π k+1 > π k-1 - π k, for 1<k<K  For each job ◦ Processing time p ◦ Weight w ◦ Worst starting time k’, π k’+1 - π k’+p+1 > w > π k’ - π k’+p  There is only one possible idle time interval in an optimal schedule. Every job has a worst starting time

 If a job starts before its worst starting time k’, then there is no idle time before it in the optimal schedule.  If a job starts after its worst starting time k’, then there is no idle time after it in the optimal schedule.  If a job starts at its worst starting time k’, then there is no idle time before and after it in the optimal schedule. k’ 0 K

 In an optimal schedule, the jobs are partitioned into two parts. ◦ The first part starts from time zero. ◦ The second part ends at time K.  In each part, jobs are processed in WSPT order. 0K How to partition jobs?

0xy Where is job m?  Job m is in the last job of in the first part  Job m is in the last job of in the second part jobs in the same part are scheduled in WSPT order f(m,x,y) denotes the minimum cost for jobs 1 to m, where the first part ends at x, and the second part starts at y. need to know f(m-1, x-p m, y) need to know f(m-1, x, y) O(nP 2 )

π(t) t

 The time slot cost π satisfies π k - π k+1 < π k-1 - π k, for 1<k<K  For each job ◦ Processing time p ◦ Weight w ◦ Preferred starting time: k*, π k*+1 – π k*+p+1 > w > π k* - π k*+p ◦ Preferred processing interval: [k*, k*+p]

k* k*+p 0 K  If a job starts before its preferred starting time k*, the earlier it is processed, the more cost it will have.  If a job starts after its preferred starting time k*, the later it is processed, the more cost it will have.  If a job starts at its preferred starting time k*, it will have the least possible cost.

Job i Job j If two jobs’ preferred processing intervals have no overlap, then the job with earlier preferred starting time has greater ratio of w/p than the other job. w i /p i w j /p j > If two jobs’ preferred processing intervals have overlap, but not completely contain each other, then the job with earlier preferred starting time has greater ratio of w/p than the other job. Job i Job j w i /p i w j /p j >

 The jobs are scheduled in WSPT order in the optimal solution.  If two jobs are adjacent in WSPT order and their preferred processing intervals have overlap, then there exists no idle time slot between these two jobs in the optimal solution. Job i Job j (w i, p i) (w j, p j ) Job i’ (w i + w j, p i +p j ) Algorithm: 1.sort by WSPT 2.merge and replace 3.assign to preferred intervals O(n 2 )

π(t) t

 Theorem 1. The problem with arbitrary non- increasing time slot cost is NP-hard in the strong sense. 3-Partition Instance:  n=3q integers a 1, a 2, …, a 3q  integer b with b/4<a j <b/2  sum(a j )=q*b Scheduling Instance:  n=3q jobs  processing time a j, weight ε*a j  K=q(b+1)

 Polynomial time algorithm for the accelerative decreasing case  More classes of decreasing cost  Other objectives besides total weighted completion time  Multiprocessor

 J. Kulkarni and K. Munagala, Algorithms for cost aware scheduling, in Proceedings of 10th International Workshop on Approximation and Online Algorithms (WAOA 2012),  S. Leonardi, N. Megow, R. Rischke, L. Stougie, C. Swamy and J. Verschae, Scheduling with time-varying cost: deterministic and stochastic models, in 11th Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2013).  G. Wan and X. Qi, Scheduling with variable time slot costs, Naval Research Logistics, 57 (2010)