Earliness and Tardiness Penalties

Earliness and Tardiness Penalties Chapter 5 Elements of Sequencing and Schedulingby 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 • Different earliness and tardiness penalties • Quadratic penalties • Job dependent penalties • Distinct due dates • Summary

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

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

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

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.

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.

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

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.

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?

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 

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

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.

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

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.

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

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

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

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)

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 • ...

Earliness and Tardiness Penalties