On Scheduling with Non-increasing Time Slot Cost to Minimize

Total Weighted Completion TimeYingchao ZHAO, Caritas Institute of Higher EducationXiangtong 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.

History

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

Time slot cost

Linear Decreasing Time Slot Cost

π(t)

t

For each job i:◦ Processing time pi◦ Weight wi

The time slot cost π satisfiesπk- πk+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?

Linear Decreasing Time Slot Cost

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 wIf w/p>ε, process this job as early as possibleIf w/p<ε, process this job as late as possibleIf w/p=ε, the job has the same cost no matter when it is processed

Optimal Scheduling for linear cost

t0 K

0 tK

tK0 w/p>ε w/p<εw/p=ε

Sort the jobs according to decreasing w/p.Partition jobs into three sets: J1, J2, J3Process J1 and J2 from time zero, and process J3 as late as possible.

Solved in polynomial time!

Accelerative Decreasing Time Slot

Costπ(t)

t

The time slot cost π satisfiesπk- πk+1> πk-1- πk, for 1<k<K

For each job ◦ Processing time p◦ Weight w◦ Worst starting time k’,

πk’+1 - πk’+p+1> w > πk’ - πk’+p

Accelerative Decreasing Time Slot Cost

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.

Optimal Scheduling for Accelerative Decreasing Cost

0 K

How to partition jobs?

Dynamic programming

0 x y

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-pm, y)

need to know f(m-1, x, y)O(nP2)

Decelerative Decreasing Time Slot

Costπ(t)

t

The time slot cost π satisfiesπk- πk+1< πk-1- πk, for 1<k<K

For each job ◦ Processing time p◦ Weight w◦ Preferred starting time: k*,

πk*+1 – πk*+p+1> w > πk* - πk*+p

◦ Preferred processing interval: [k*, k*+p]

Decelerative Decreasing Time Slot Cost

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.

wi/pi wj/pj>

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

wi/pi wj/pj>

Optimal Scheduling for Decelerative Decreasing Cost 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

(wi, pi) (wj, pj)

Job i’

(wi +wj, pi+pj)

Algorithm:1. sort by WSPT2. merge and

replace3. assign to

preferred intervalsO(n2)

General Decreasing Time Slot Cost

π(t)

t

Theorem 1. The problem with arbitrary non-increasing time slot cost is NP-hard in the strong sense.

General Decreasing Time Slot Cost

3-Partition Instance: n=3q integers a1, a2, …, a3q integer b with b/4<aj<b/2 sum(aj)=q*bScheduling Instance: n=3q jobs processing time aj, weight ε*aj 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

Open Problems

J. Kulkarni and K. Munagala, Algorithms for cost aware scheduling, in Proceedings of 10th International Workshop on Approximation and Online Algorithms (WAOA 2012), 201-214.

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) 159-171.

Reference

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