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# Earliness and Tardiness Penalties - PowerPoint PPT Presentation

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

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### Earliness and Tardiness Penalties

Chapter 5

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha

• Introduction

• Minimizing deviations from a common due date

• Four basic results

• Due date as decisions

• The restricted version

• Different earliness and tardiness penalties

• Job dependent penalties

• Distinct due dates

• 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

• 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

• Corresponding cost factors are substantially different

• Primary role of penalty functions

• Guiding solutions toward the target of meeting all due date exactly

• Ideal schedule

• Different penalty functions

• Suggestions for measuring suboptimal performance of nonideal schedules

• 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

• 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.

• Exercise

• Prove Theorem 1 using proof by contradiction.

• Prove Theorem 2 using proof by contradiction.

• 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.

• 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

• 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

• 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 T/T problem with jobs below and d = 24.

• 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?

• Due dates as decision

• One way of finding an optimal solution

• Set d =  and utilize algorithm 1*

Optimal

Total

Penalty

due date

• 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?

• Two jobs with p1 = p2 = 3 and d = 2

• Three jobs with p1 = 1, p2 = 1, p3 = 10, and d = 2

• NP-hardness

• A dynamic programming technique (Hall et al., 1991)

• Solving problems with several hundreds of jobs

• An effective 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.

• 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

• 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.

• 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 the heuristic

• Tested by randomly generated problems

• 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

• 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

• 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

• Different due dates in job set

• f(S) = j=1n(j(dj – Cj)+ + j(Cj – dj)+) = j=1n(jEj + jTj)

• NP-hardness

• 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, as well

• A neighborhood search (Armstrong and Blackstone, 1987)

• A branch-and-bound procedure (Darby-Dowman and Armstrong, 1986)