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ONE-MACHINE SEQUENCING TOMINIMIZE CERTAIN FUNCTIONS OF JOB TARDINESS

Hamilton EmmonsIntroductionSequencing a set of n jobs on one machineAll jobs are available at time 0Processing times and due dates are known in advanceSetup times are neglected; may be included in processing timesFlow time is a constant

LatenessTardiness and total tardinessIntroductionSpecial cases of minimizing total tardinessSPT schedule minimizes total tardiness if all jobs have positive tardinessEDD schedule minimizes total tardiness if at most job has positive tardiness

General objectiveMinimize J g(Ti) where g(.) is any convex, nondecreasing penalty functionPropertiesExistentialThere exists an optimal schedule with property A

UniversalAll optimal schedules have property AProperties can be accumulated to narrow down the subset of feasible pointsTheorems

Theorem 1

Bk = Theorems

Ak = Theorems

The Algorithm

The Algorithm

Examples

Sequencing to minimize J g(Ti)

ConclusionFirst theoretical development for SMTTP

Necessary conditions establish precedence (SPT/EDD) relations in an optimal sequence

Necessary conditions for a certain job to be either first or last in an optimal sequence

First theorem gives the necessary conditions for a shorter job to precede a longer one in an optimal sequence

Second theorem gives the necessary conditions for a longer job to precede a shorter one in an optimal sequence

Third theorem for a shorter job to precede a longer one; found limited use in applications

Precedence relations obtained by implementing these theorems can be accumulated

These theorems were generalized by Kanet (2007), and applied to the single-machine weighted total tardiness problem with arbitrary weights

Kanet, J.J., 2007. New precedence theorems for one-machine weighted tardiness. Mathematics of Operations Research 32, 579588.AppendixSet of n jobs: J = {J1, J2, . . . , Jn}Processing time of Ji: piDue date of Ji: diCompletion time of Ji: CiTotal completion time: p = J piLateness of Ji: Li = Ci - di Total lateness: L = J LiTardiness of Ji: Ti = max(0, Li)Total tardiness: T = J Ti

Jj precedes Jk: j kSet of jobs that follow Ji{Jj, Jk, . . . }Ai = {j, k, . . . }i j, k, . . .

j < k: pj < pk or pj = pk and dj < dk Appendix