1 / 20

Earliness and Tardiness Penalties

Earliness and Tardiness Penalties. Chapter 5 Elements of Sequencing and Scheduling by Kenneth R. Baker Byung-Hyun Ha. R1. Outline. Introduction Minimizing deviations from a common due date Four basic results Due date as decisions The restricted version

cade-newman
Download Presentation

Earliness and Tardiness Penalties

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. 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

  3. 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 • Ej = max{0, dj – Cj} = (dj – Cj)+ • Tj = max{0, Cj – dj} = (Cj – dj)+ • Linear penalty function with unit earliness (tardiness) penalty j (j) • f(S) = j=1n(j(dj – Cj)+ + j(Cj – dj)+) = j=1n(jEj + jTj) • Nonregular measure

  4. 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 (dj = 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

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

  6. 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.

  7. 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 Ci– pi d  Ci . • 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 = Ci – 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 – (Ci – pi). • 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.

  8. 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-shapedschedule • 2n 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 -- EBi = pB(i+1) + pB(i+2) + ... + pBb • Total penalty for the jobs in B • CB = i=1bEBi = i=1b(pB(i+1) + pB(i+2) + ... + pBb) = 0pB1 + 1pB2 + ... + (b – 2)pB(b–1) + (b – 1)pBb. • Total penalty for the jobs in A • CA = apA1 + (a – 1)pA2 + ... + 2pA(a–1) + 1pAa. • f(S) = CA + CB  minimized by assigning jobs regarding processing times

  9. 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.

  10. 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 •  = pn + pn–2 + pn–4 + ... • For unrestricted problem, Algorithm 1* will produce optimal schedule • Exercise: When d = 18, is it unrestricted? When d = 17?

  11. 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 

  12. 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 p1 = 1, p2 = 1, p3 = 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

  13. Restricted Version • An effective heuristic: S-A heuristic (Sundararaghavan and Ahmed, 1984) • Assuming p1 p2 ...  pn. 1. Let L = d and R = i=1npi – d. Let k = 1. 2. If L R, assign job k to the first available position in sequence and decrease L by pk. Otherwise, assign job k to the last available position in sequence and decrease R by pk. 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.

  14. 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 6-3-2-1-4-5 of jobs below with d = 90

  15. Different Earliness and Tardiness Penalties • A generalization of basic model • Minimize f(S) = j=1n(Ej + Tj) where    •  -- holding cost (endogenous),  -- tardiness penalty (exogenous) • Properties of optimal solution • Theorem 1, 2, and 3 hold • Components of objective function • CB = 0pB1 + 1pB2 + ... + (b – 2)pB(b–1) + (b – 1)pBb. • CA = apA1 + (a – 1)pA2 + ... + 2pA(a–1) + 1pAa. • 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.

  16. 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 •  = pB1 + pB2 + ... + pB(b–1) + pBb • Condition for delaying start of schedule • e n/( + ) • Effectiveness of modified S-A heuristic • Tested by randomly generated problems

  17. Quadratic Penalties • Avoiding large deviations from due date • Minimize f(S) = j=1n(Cj – d)2 = j=1n(Ej2 + Tj2) • Due date d as decision variable • d = = j=1nCj /n • Quadratic E/T problem, unrestricted • f(S) = j=1n(Cj – )2 • Problem of minimizing variance of completion times, but not easily solvable • A heuristic solution (Vani and Raghavachari, 1987) • Neighborhood search using pairwise interchanges

  18. Job Dependent Penalties • Permitting each job to have its own penalties • f(S) = j=1n(jEj + jTj) • 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 pj /j, and jobs that start late can be sequenced in non-decreasing order of the ratio pj /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=1nj

  19. Distinct Due Dates • Different due dates in job set • f(S) = j=1n(j(dj – Cj)+ + j(Cj – dj)+) = j=1n(jEj + jTj) • 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)

  20. 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-shapedschedule • 2n 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 • ...

More Related