Earliness and Tardiness Penalties Chapter 5 Elements of Sequencing and Scheduling by Kenneth R. Baker Byung-Hyun Ha R1

1 Outline  Introduction  Minimizing deviations from a common due date  Four basic results  Due date as decisions  The restricted version  Different earliness and tardiness penalties  Quadratic penalties  Job dependent penalties  Distinct due dates  Summary

2 Introduction  Until now  Basic single-machine model with regular measures of performance, which are nondecreasing in job completion times  Among regular measures, total tardiness criterion has been a standard way of measuring conformance to due dates The measure does not penalize jobs completed early  Just-In-Time (JIT) production  “Inventory is evil”  Earliness, as well as tardiness, should be discouraged  E/T criterion in basic single-machine model  Earliness and tardiness E j = max{0, d j – C j } = (d j – C j ) + T j = max{0, C j – d j } = (C j – d j ) +  Linear penalty function with unit earliness (tardiness) penalty  j (  j ) f(S) =  j=1 n (  j (d j – C j ) + +  j (C j – d j ) + ) =  j=1 n (  j E j +  j T j )  Nonregular measure

3 Introduction  Variations in E/T criterion  Decision variables Job sequence with due dates given Due dates and job sequence  Setting due dates internally, as targets to guide the progress of shop floor activities  Due dates Common due dates (d j = d)  Several items constitute a single customer’s order  Assembly environment where components should all be ready at the same time Distinct due dates  Penalties Common penalties (  j = ,  j =  ) Distinct penalties  Role of penalty functions Guiding solutions toward the target of meeting all due date exactly Measuring suboptimal performance of nonideal schedules

4 Minimizing Deviations from a Common Due Date  Basic E/T problem  Minimizing sum of absolute deviations of job completion times from common due date (d j = d,  j =  j = 1)  f(S) =  j=1 n |C j – d j | =  j=1 n (E j + T j )  Due date can be in the middle of jobs?  Tightness of due date d  Restricted version vs. unrestricted version d d

5 Basic E/T Problem, Unrestricted  Theorem 1  In the basic E/T model, schedules without inserted idle time constitute a dominant set.  Theorem 2  In the basic E/T model, jobs that complete on or before the due date can be sequenced in LPT order, while jobs that start late can be sequenced in SPT order.  V-shaped schedule  Exercise  Prove Theorem 1 using proof by contradiction.  Prove Theorem 2 using proof by contradiction.

6 Basic E/T Problem, Unrestricted  Theorem 3  In the basic E/T model, there is an optimal schedule in which some job completes exactly at the due date.  Proof sketch of Theorem 3 (proof by contradiction)  Suppose S is an optimal schedule where C i – p i  d  C i.  Let b (a) denote the number of early (tardy) jobs in sequence.  Case 1 (a  b) Consider S' where S is shifted earlier by  t = C i – d. Increase in earliness (decrease in lateness) penalty is b  t (a  t). Hence, f(S)  f(S'), because a  t  b  t.  Case 2 (a  b) Consider S' where S is shifted later by  t = d – (C i – p i ). Decrease in earliness (increase in lateness) penalty is b  t (a  t). Hence, f(S)  f(S'), because a  t  b  t.  Therefore, in either case a schedule with the property of the theorem is at least as good as S.

7 Basic E/T Problem, Unrestricted  Properties of optimal schedule by Theorem 1, 2, 3  Optimum is describable by a sequence of jobs and a start time of 1st job  V-shaped schedule  2 n candidates instead of n! candidates  Analysis on optimal schedule  Notations A (B) -- set of jobs completing after (on or before) the due date a = |A|, b = |B| Ai (Bi) -- ith job in A (B)  Earliness penalty for job Bi -- E Bi = p B(i+1) + p B(i+2) p Bb  Total penalty for the jobs in B C B =  i=1 b E Bi =  i=1 b (p B(i+1) + p B(i+2) p Bb ) = 0p B1 + 1p B (b – 2)p B(b–1) + (b – 1)p Bb.  Total penalty for the jobs in A C A = ap A1 + (a – 1)p A p A(a–1) + 1p Aa.  f(S) = C A + C B  minimized by assigning jobs regarding processing times

8 Basic E/T Problem, Unrestricted  Algorithm 1: Solving the Basic E/T Problem 1.Assign the longest job to set B. 2.Find the next two longest jobs. Assign one to B and one to A. 3.Repeat Step 2 until there are no jobs left, or until there is one job left, in which case assign this job to either A or B. Finally, order the jobs in B by LPT and the jobs in A by SPT.  Exercise: solve basic E/T problem with jobs below and d = 24. Job j pjpj

9 Basic E/T Problem, Unrestricted  Algorithm 1*  Considering secondary measure: minimum total completion time  Same as Algorithm 1 except that, in Step 2, shorter job is assigned to B and, in Step 3, if n is even, assign the shortest job in A  Theorem 4  In the basic E/T model, there is an optimal schedule in which the bth job in sequence completes at time d, where b is the smallest integer greater than or equal to n/2.  Due date for unrestricted version  Supposing jobs are indexed SPT order  The problem is unrestricted for d  , where  = p n + p n–2 + p n–  For unrestricted problem, Algorithm 1* will produce optimal schedule  Exercise: When d = 18, is it unrestricted? When d = 17? Job j pjpj

10 Basic E/T Problem, Unrestricted  Due dates as decision  One way of finding an optimal solution Set d =  and utilize algorithm 1*  optimal total penalty f(S) common due date d

11 Restricted Version  Basic E/T problem, restricted (d   )  Optimal solution may contain a straddling job  Theorem 1 and 2 hold, but Theorem 3 does not V-shaped schedules still constitute a dominant set  Should optimal schedule start at time zero always?  Three jobs with p 1 = 1, p 2 = 1, p 3 = 10, and d = 5  Optimal schedule, in which either the schedule starts at time zero, or some job completes exactly at the due date  NP-hardness  A dynamic programming technique (Hall et al., 1991) Solving problems with several hundreds of jobs

12 Restricted Version  An effective heuristic: S-A heuristic (Sundararaghavan and Ahmed, 1984)  Assuming p 1  p 2 ...  p n. 1.Let L = d and R =  i=1 n p i – d. Let k = 1. 2.If L  R, assign job k to the first available position in sequence and decrease L by p k. Otherwise, assign job k to the last available position in sequence and decrease R by p k. 3.If k  n, increase k by 1 and go to Step 2. Otherwise, stop.  Exercise  Find good sequence for the jobs below with d = 90. Job j pjpj

13 Restricted Version  Adjustment of start time  Delay of start time leads to reduction in total penalty, when e  n/2 where e is number of jobs that finish before due date  Schedule of jobs below with d = 90 Job j pjpj

14 Different Earliness and Tardiness Penalties  A generalization of basic model  Minimize f(S) =  j=1 n (  E j +  T j ) where      -- holding cost (endogenous),  -- tardiness penalty (exogenous)  Properties of optimal solution  Theorem 1, 2, and 3 hold  Components of objective function  C B = 0  p B1 + 1  p B (b – 2)  p B(b–1) + (b – 1)  p Bb.  C A = a  p A1 + (a – 1)  p A  p A(a–1) + 1  p Aa.  Algorithm 2: E/T with different earliness and tardiness penalties 1.Initially, sets B and A are empty, and jobs are in LPT order. 2.If  |B|   (1 + |A|), then assign the next job to B; otherwise, assign the next job to A. 3.Repeat Step 2 until all jobs have been scheduled.  Exercise: consider jobs below with  = 5,  = 2, and d = 24. Job j pjpj

15 Different Earliness and Tardiness Penalties  Generalization of Theorem 4  In the basic E/T model with earliness penalty  and tardiness penalty , there is an optimal schedule in which the bth job in the sequence completes at time d, where b is the smallest integer greater than or equal to n  /(  +  ).  Criterion for unrestricted version   = p B1 + p B p B(b–1) + p Bb  Condition for delaying start of schedule  e  n  /(  +  )  Effectiveness of modified S-A heuristic  Tested by randomly generated problems  =      Problem SizeAverage ErrorNo. of OptimaAverage ErrorNo. of Optima n = 8 n = 10 n = 12 n = % 0.24% 0.26% 0.32% % 0.84% 0.66% 0.07%

16 Quadratic Penalties  Avoiding large deviations from due date  Minimize f(S) =  j=1 n (C j – d) 2 =  j=1 n (E j 2 + T j 2 )  Due date d as decision variable  d =  =  j=1 n C j /n  Quadratic E/T problem, unrestricted  f(S) =  j=1 n (C j –  ) 2  Problem of minimizing variance of completion times, but not easily solvable  A heuristic solution (Vani and Raghavachari, 1987) Neighborhood search using pairwise interchanges

17 Job Dependent Penalties  Permitting each job to have its own penalties  f(S) =  j=1 n (  j E j +  j T j )  NP-hardness  A dynamic programming technique (Hall and Posner, 1991) Solving problems with hundreds of jobs in modest run times  Generalization of Theorem 1–4 1.There is no inserted idle time. 2.Jobs that complete on or before the due date can be sequenced in non- increasing order of the ratio p j /  j, and jobs that start late can be sequenced in non-decreasing order of the ratio p j /  j. 3.One job completes at time d. 4.In an optimal schedule the bth job in sequence completes at time d, where b is the smallest integer satisfying the inequality  i  B (  j +  j )   j=1 n  j

18 Distinct Due Dates  Different due dates in job set  f(S) =  j=1 n (  j (d j – C j ) + +  j (C j – d j ) + ) =  j=1 n (  j E j +  j T j )  NP-hardness T-problem reduces to this problem  A solution technique  Decomposing into two subproblems Finding a good job sequence Scheduling inserted idle time Solvable in polynomial time  Refer to p. 74 of Pinedo, 2009  A neighborhood search (Armstrong and Blackstone, 1987)  A branch-and-bound procedure (Darby-Dowman and Armstrong, 1986)

19 Summary  Earliness/tardiness problem  From JIT concepts  Nonregular performance measure  Properties  Optimum is describable by a sequence of jobs and a start time of 1st job  V-shaped schedule  2 n candidates instead of n! candidates  Restricted vs. unrestricted versions  Difficulties in finding good schedules with tight due date  Extended models  Job-dependent penalty and due dates ...