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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

Earliness and Tardiness Penalties

Chapter 5

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha


Outline
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


Introduction
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


Introduction1
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

        • Corresponding cost factors are substantially different


Introduction2
Introduction

  • 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


Minimizing deviations from a common due date
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 unrestriced
Basic E/T Problem, Unrestriced

  • 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.


Basic e t problem unrestriced1
Basic E/T Problem, Unrestriced

  • 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 unrestriced2
Basic E/T Problem, Unrestriced

  • 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


Basic e t problem unrestriced3
Basic E/T Problem, Unrestriced

  • 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.


Basic e t problem unrestriced4
Basic E/T Problem, Unrestriced

  • 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 unrestriced5
Basic E/T Problem, Unrestriced

  • Due dates as decision

    • One way of finding an optimal solution

      • Set d =  and utilize algorithm 1*

Optimal

Total

Penalty

due date


Restricted version
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?

    • 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


Restricted version1
Restricted Version

  • 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.


Restricted version2
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
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 penalties1
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 the heuristic

    • Tested by randomly generated problems


Quadratic penalties
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
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
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

  • 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)


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