A multiobjective parallel machine problem considering eligibility and release and delivery times
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A multiobjective parallel machine problem considering eligibility and release and delivery times. Manuel Mateo [email protected] Departament Organització d’Empreses, Universitat Politècnica Catalunya. Barcelona (Spain). HAROSA, Barcelona (14/06/12). Summary. Introduction: the real case

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A multiobjective parallel machine problem considering eligibility and release and delivery times

A multiobjective parallel machine problem considering eligibility and release and delivery times

Manuel Mateo

[email protected]

Departament Organització d’Empreses, Universitat Politècnica Catalunya. Barcelona (Spain)

HAROSA, Barcelona (14/06/12)


Summary
Summary eligibility and release and delivery times

  • Introduction: the real case

  • Scheduling of jobs The multiobjective problem

  • Problem Pm/rj,qj,Mj/(Cmax,W)

  • State of the art

    • A biobjective problem, the Pareto front

    • Algorithm

    • Computational experiments

  • Computational experience

  • Conclusions


Introduction the real case i
Introduction: the real case (I) eligibility and release and delivery times

  • The manufacturing of products is usually divided in operations or phases of transformation.

  • Usually one of them becomes the bottleneck of the process.

  • In the presented problem, we suppose this bottleneck is an intermediate phase.

  • Therefore, some operations are done before (the total time to work them out leads to a release time) and some others are done after (their total time is called delivery or queue time).

release times

bottleneck

queue times


Introduction the real case ii
Introduction: the real case (II) eligibility and release and delivery times

  • Manufacturing plants usually have several machines or assembly lines.

  • There are several products to be manufactured.

  • A usual situation is a product is assigned to a machine (line) and will be only manufactured in that machine.

Line 1

INITIAL SITUATION

Line 2

Line 3

2 l

1 l

50cl

33cl

Line 4


Introduction the real case iii
Introduction: the real case (III) eligibility and release and delivery times

  • Nevertheless, there could be more product-machine assignments according to capabilities of the machines.

High-level

machines

1 l

50cl

33cl

Line 1

2 l

1 l

50cl

33cl

Medium-level

machines

Line 2

CONSIDERED SITUATION

2 l

Med.-

level

High-

level

Low-

level

1 l

50cl

33cl

Line 3

2 l

Low-level

machines

1 l

50cl

33cl

Line 4

2 l


Introduction the real case iv
Introduction: the real case (IV) eligibility and release and delivery times

  • The managers prefer the use of the most modern resources (high-level machines). But if all the jobs were done in these machines, the makespan would be very high. The rest of machines would be completely free.

  • Some works from machines of the high-level are moved to the other machines.

  • Machinery for reduced products is considered in the low-level.

  • The medium-level machines can work the reduced products and also others. They are preferred to the low-level machines. 

  • We define a penalty or weight for job j:

  • wj=1 if a job (of medium-level or low-level) is scheduled in a medium-level machine;

  • wj=2 if a job (of low-level) is scheduled in a low-level machine.


Scheduling of jobs i
Scheduling of jobs (I) eligibility and release and delivery times

A set of n jobs (j=1,…,n) to be scheduled on m parallel machines (i=1,…,m)

Given a job j, it is known:

  • the processing time pjfor the operation,

  • the release time rj(also called head times),

  • the delivery or queue time qj(also tail times),

  • the associated level lj.

    The machines are distributed among p groups or levels (k=1,…,p).

    • Particularly, we propose an algorithm for p=3 (high-level, medium-level and low-level).

  • Any machine i and job j is classified into one of the levels.

  • A machine associated to a level k can produce jobs of its own level and from a lower level.

  • The processing time of a job is the same for any machine.


0 eligibility and release and delivery times

t

Scheduling of jobs (II)

For a job of any level: high (h), medium (m) or low (l)

  • The release times (rj) and the delivery times (qj) are considered due to the initial availability of the job and the necessary subsequent tasks.

  • After the delivery time, the job is considered finished (completion time, cj).

  • Setup times and pre-emption are not considered.

cj


Problem pm r j q j m j c max w tot
Problem eligibility and release and delivery timesPm | rj,qj,Mj | (Cmax ,Wtot)

  • Objective: find a feasible schedule  (where j includes the assigned machine and the start time of job j) of minimum completion time (cmax) and minimum total penalization (W).

  • Given tjthe starting time for the job j: ;

    the makespan is determined:

  • Given xjthe machine level where the job j is assigned,

    the weight for the job j is: wj=xj-1

    and the total weight is determined:

  • A schedule is feasible if the next conditions are accomplished:

    • Each machine processes at most one job at a time.

    • A job is only processed in a single machine.

    • Pre-emption is not allowed.

    • Starting time is not lower than the release time:

    • A job of level k is processed in a machine of the same level or a higher level.


HL eligibility and release and delivery times

HL

ML

ML

LL

LL

t

0

t

0

Problem Pm | rj,qj,Mj | (Cmax ,Wtot)

Machines

Jobs

High-level

High-level

Medium-level

Medium-level

Low-level

Low-level

Wtot=0

Wtot=8


Example 1 eligibility and release and delivery times

Wtot

(14;6)

Max Wtot

(36;0)

Min Wtot

Cmax

Min Cmax

Max Cmax


Basis for the main algorithm
Basis for the main algorithm eligibility and release and delivery times

  • Gharbi, A; Haouari, M. (2002). Minimizing makespan on parallel machines subject to release dates and delivery times. Journal of Scheduling; vol. 5; pp. 329-355.

  • It considers release and delivery times.

  • Problem without considering eligibility

  • It is the base for solving the problem (for mk>1)

Methodology

  • Classification of the jobs into 3 groups.

  • Scheduling of the 3 groups of jobs:

    • Scheduling of jobs with medium qj and rj values ( )

    • Scheduling of jobs with low qj values ( JQ )

    • Scheduling of jobs with low rjvalues ( JR )

Condition 1:

Condition 2:


Basis for the main algorithm (II)

Scheduling at each level

Selection of a job to be moved between levels

  • Which one to select? According to processing, release, queue times or an addition of these?

Sequence of changes between levels

  • Initial tests: High-medium, high-low, medium-low


The best of both solutions eligibility and release and delivery times

Scheduling Algorithm (I)

Generating a sub-solution per level

Heuristic 1

Heuristic 2

is a job of the set J such that or corresponds to

is a job of the set J such that or corresponds to

1

1

release time  first position; else, last position

release time  last position; else, first position

2

2

3

3

If , end; else, Step 1

If , end; else, Step 1


m eligibility and release and delivery timesl

t

0

10

20

25

30

5

15

Scheduling Algorithm (II)

Example 1: ml=1

j10

j9

j7

j11

j8


Step 0.1: eligibility and release and delivery times

Initialization

Step 0.2:

Condition 1.

If there is no job;

Step 0.3.

If not:

Step 2.1:

Assignment of jobs such that j

Update u0

Step 2.3:

Assignment of jobs such that j

Update u0

Step 2.2:

Invert scheduling

Step 2.4:

Invert scheduling

Step 0.3:

Condition 2.

If there is no job,

Step 0.4.

If not:

Step 1:

Assignment of jobs such that j

Scheduling Algorithm (III)

Generating a sub-solution per level

Step 0.4:

If there are no changes in Step 0.3,

STOP.

If not,

Step 0.2.

Process Gharbi & Haouari (2002) Pre-process


m eligibility and release and delivery timesl1

j7

ml2

j9

j11

j8

0

10

20

5

15

Scheduling Algorithm (IV)

Example 2 (algorithm of Gharbi & Haouari): ml=2

Step 1

Step 1

Step 2

10


How to face the multiobjective problem
How to face the multiobjective problem? eligibility and release and delivery times

  • At the beginning, all the jobs are scheduled in the high-level machine(s), as they are the preferred machines to manufacture any product.

  • This can induce a relative high Cmax.

  • Then, in order to improve this value, some of the jobs are moved from a level to another lower level. In this way, the orders can be finished in a lower Cmax.

  • The changes between adjacent levels will be denoted by a weight =1, i.e. between high and medium level and between medium and low level, and a weight = 2, i.e. between high and low level.

  • Therefore, the algorithm gives different solutions, characterized by two objectives:

    • Min {Cmax}

    • Min {W}


Main algorithm
Main Algorithm eligibility and release and delivery times

  • Basically the algorithm is divided in the following phases:

    • PHASE I.

      All the jobs are scheduled in the high-level machines.

      If mh=1, apply Heuristic 1 for all the jobs.

      Otherwise, apply Heuristic 2 for all the jobs.

      Compute Cmaxº

      This implies the first solution s1=(Cmax0; Wtot0=0)

    • PHASE II.

      While Cmax can be reduced:

      a) select a job to be changed to a different level (only a subset of jobs are available);

      b) the origin level of the movement is predetermined; the destination level should be selected.

      Machines in both levels will be re-scheduled.

      This leads to a new solution sz=(Cmaxz; Wtotz)


Main algorithm ii
Main Algorithm (II) eligibility and release and delivery times

PHASE II: Movement of jobs between different levels

  • II.1. Job movement from high to medium level

    Compute the new solutions (Cmax, CmaxH, CmaxM)

  • II.2. Job movement from high to low level

    Compute the new solutions (Cmax, CmaxH, CmaxL) 

  • II.3. Job movement from medium to low level

    Compute the new solutions (Cmax, CmaxM, CmaxL)


Main algorithm iii
Main Algorithm (III) eligibility and release and delivery times

II. Job movement from a level to a lower level

Given the previous solution s (Cmaxs; Wtots) and Cmaxº= Cmaxs

Briefly, the substeps in this phase are: 

  • Select the origin level of the movement is given (here is prefixed).

  • Select a job to be changed to a different level (only a subset of jobs is available):

    Search for job candidates: JC

    Select a job between the candidates according to a prefixed rule (j*JC)

  • Select the destination level of the movement (here is prefixed).

  • Reschedule jobs in both levels.

  • Compute both objectives of the new solution (Cmax; Wtot).

  • IF Cmax < Cmaxº 

    save the new solution (Cmaxs+1=Cmax ; Wtots+1=Wtot)

    Cmaxº=Cmax


Computational experience
Computational experience eligibility and release and delivery times

  • To check the efficiency of the algorithm, a set of instances similar to those used by Gharbi & Haouari (2002) are created:

  • # jobs (n = 20)

  • 100 instances

  • Jobs of high level, 20-30% of the total number; medium level, 20-50% of the total; low level, the rest (20-60% of the total)

  • # machines (m = 4, 5, 6)

  • Processing time: discrete uniform distribution .

  • Release and delivery times: discrete distribution .

    K=3,5


Computational experience rules
Computational experience. Rules eligibility and release and delivery times

  • Given the subset of candidates (considering the origin and destination levels), different rules are used to select the job to be reassigned:

    Consider the processing time and the other times (release and queue)

    j = argminj (rj+pj, qj+pj) min(rp,qp)

    j = argmaxj (rj+pj, qj+pj) max(rp,qp)

    Consider the other times (release and queue)

    j = argminj (rj, qj) min(r,q)

    j = argmaxj (rj, qj) max(r,q)

    Consider only the processing time

    j = argminj (pj) min(p)

    j = argmaxj (pj) max(p)


Comparison of results v1 rule selection
Comparison of results v1 (rule selection) eligibility and release and delivery times

  • Given the solutions in the Pareto front with two different rules A and B for the job selection:

  • Proportion of non-dominated solutions (overall 13 distributions)


Results v1 analysis by machines
Results v1 (analysis by machines) eligibility and release and delivery times

  • Proportion of non-dominated solutions for max(p) rule depending on the number of machines per level


Conclusions
Conclusions eligibility and release and delivery times

  • In the first situation all the jobs are assigned to machines of the high level; this solution shows a great cmax with wtot=0.

  • The rest of solutions in the Pareto front have a decreasing cmax while wtot increases.

  • Different rules are tested to select a job to be moved from a level to another.

  • The best results are achieved with the max(pj), although the rule max(rj+pj,qj+pj) has also a good performance.

  • About the current and future research:

    • Select the origin level of job movement (level in which there is the machine with the highest Cmax).

    • Use of metaheuristics.


A multiobjective parallel machine problem considering eligibility and release and delivery times1

A eligibility and release and delivery timesmultiobjective parallel machine problem considering eligibility and release and delivery times

Thank you

for your attention!


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