Branch-and-bound algorithms for scheduling in permutation flowshops to minimize the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime, weighted tardiness and weighted earliness of jobs

Abstract

The problem of scheduling in permutation flowshops is considered in this paper with the objectives of minimizing the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime, weighted tardiness and weighted earliness of jobs, with each objective considered separately. Lower bounds on the given objective (corresponding to a node generated in the scheduling tree) are developed by solving an assignment problem. Branch-and-bound algorithms are developed to obtain the best permutation sequence in each case. Our algorithm incorporates a job-based lower bound (integrated with machine-based bounds) with respect to the weighted flowtime/weighted tardiness/weighted flowtime and weighted tardiness, and a machine-based lower bound with respect to the weighted earliness of jobs. The proposed algorithms are evaluated by solving many randomly generated problems of different problem sizes. The results of an extensive computational investigation for various problem sizes are presented. In addition, one of the proposed branch-and-bound algorithms is compared with a related existing branch-and-bound algorithm.

Appendices

Appendix A: Problem formulation (with respect to job i * being appended to σ; and the computation of weighted flowtime, weighted tardiness and weighted earliness)

Suppose we have σ, and job i * is appended to σ. We have q(σ i *,1)=q(σ,1)+t _{i *1}, and q(σ i *,j)=max{q(σ,j);q(σ i *,j−1)}+t _{i * j} , for j=2,3,…,m, where q(∅,j)=0 with ∅ denoting a null set.

We have a _{i *} =q(σ i *,m)×h _{i *} , if the objective is related to weighted flowtime; a _{i *} =max{q(σ i *,m)−D _{i *} ;0}×r _{i *} , if the objective is related to weighted tardiness; a _{i *} =(q(σ i *,m)×h _{i *} )+(max{q(σ i *,m)−D _{i *} ;0}×r _{i *} ), if the objective is related to sum of weighted flowtime and weighted tardiness; and a _{i *} =(q(σ i *,m)×h _{i *} )+(max{q(σ i *,m)−D _{i *} ;0}×r _{i *} )+(max{D _{i *} −q(σ i *,m);0}×e _{i *} ), if the objective is related to sum of weighted flowtime, weighted tardiness and weighted earliness of jobs.

Appendix B: Numerical illustrations

We present numerical illustrations for the mechanics of the proposed branch-and-bound algorithms (with weights of jobs assumed to be 1, for the sake of ease of understanding the bounds).

Consider the following five-job, three-machine flowshop scheduling problem with processing times and due date for each job, given in Table B1.

Table b1 A flowshop problem of size (5×3)

In this problem n=5, m=3. The computations of lower bounds for node {1} (ie, with job 1 present in σ) are as follows.

Node {1}:

Here, Π={2345} with n ^″=4, and σ={1}.

We have q(1,1)=5, q(1,2)=15, and q(1,3)=30.

Let job i=job 2.

The job-based lower bound computation using JOB_LB_ALGO (with the consideration of job 2) is now presented.

For job 2 following σ, we have the lower bound on the completion times computed as follows, by using Equation (1):

LBq(12,1)=5+30=35;

LBq(12,2)=max{15;35}+5=40; and

LBq(12,3)=max{30;40}+10=50.

For job 2 to be placed in any position (except 1), we first compute the lower bound on the earliest start time of any job found in position 1 following σ (see Equation (2), as follows: LBST(1[1],1)=5; LBST(1[1],2)=max{5+15;15}=20; and

LBST (1[1],3)=max{max{5+35; 20+10}; 30}=40.

With respect to the placement of job 2 in the third position following σ, we do the following computations:

{

First, compute the lower bound on the completion time of job found in position 2

following σ, on machine 1:

LBq(1[1][2],1)=5+15+20=40.

Next, we have (with respect to machine 2):

T _[1]1′=35; T _[1]2′=10; T _[2]1′=35; τ _[1]1 ^″=10 (as per Equations (4) and (5));

τ _[1]2 ^″=10 and τ _[2]2 ^″=15 (as per Equation (5); and hence we have

LBq(1[1][2],2)=max{5+35+10;20+10+15;40+10;20+35;40+10}=55.

Similarly, we have with respect to machine 3:

T _[1]1′=40; T _[1]2′=15; T _[1]3′=5; T _[2]1′=55; τ _[1]1 ^″=5; T _[2]2′=35; τ _[1]2 ^″=5; τ _[1]3 ^″=5

and τ _[2]3 ^″=20, and hence

LBq(1[1][2],3)=max{5+55+5;20+35+5;40+5+20;40+15;55+5;20+40;40+15;45+5}=65.

We finally have

LBq(1[1][2]2,1)=40+30=70;

LBq(1[1][2]2,2)=max{70;55}+5=75 and

LBq(1[1][2]2,3)=max{75;65}+10=85.

}

Similarly, the lower bounds on the completion time of every job in Π, placed in all possible positions, are computed. Therefore, the sum of flowtime, sum of tardiness, sum of flowtime and tardiness, and the sum of flowtime, tardiness and earliness of jobs are computed.

Appendix C: A counter example to the machine-based bounds proposed by Chung et al (2006)

Consider the four-job, two-machine permutation flowshop scheduling problem with the processing times and due-dates for jobs, given in Table C1.

Table c1 A (4×2) flowshop problem

In this problem n=4, m=2. Let us consider, σ={1}. For the sequence 1–2–4–3, the completion times are as follows:

The above computation of completion times represents the tardiness of the sequence considered.

The computations of lower bounds (as given in Chung et al, 2006) are as follows (we follow the same nomenclature presented in Chung et al, 2006). Since the due dates of the jobs are assumed to be zero, T _tk (which actually represents the lower bound on the tardiness of the job found in position t on machine k) now represents the lower bound on the completion time of job found in position t on machine k.

Now, compute the lower bounds on the tardiness:

Therefore, the lower bound on the tardiness of job found in position 4 following σ is 216 and it is >123 (see Equation (22)).

As per our proposed branch-and-bound algorithm, the job-based lower bound on the completion time of job found in position 4 following σ, is given as follows (see Equation (6)):