A single-machine bi-criterion scheduling problem with two
agents
Wen-Chiung Lee
a,⇑, Yu-Hsiang Chung
b, Zong-Ren Huang
aa
Department of Statistics, Feng Chia University, Taichung, Taiwan b
Department of Industrial & Engineering Management, National Chiao Tung University, Hsinchu, Taiwan
a r t i c l e
i n f o
Keywords: Scheduling
Total completion time Maximum tardiness Two-agent Single-machine
a b s t r a c t
The multiple-agent scheduling problems have received increasing attention recently. How-ever, most of the research focuses on studying the computational complexity of the intrac-table cases or examining problems with a single criterion. Often a decision maker has to decide the schedule based on multiple criteria. In this paper, we consider a single machine problem where the objective is to minimize a linear combination of the total completion time and the maximum tardiness of jobs from the first agent given that no tardy jobs are allowed for the second agent. We develop a branch-and-bound algorithm and several simulated annealing algorithms to search for the optimal solution and near-optimal solu-tions for the problem, respectively. Computational experiments show that the proposed branch-and-bound algorithm could solve problems of up to 24 jobs in a reasonable amount of time and the performance of the combined simulated annealing algorithm is very good with an average error percentage of less than 0.5% for all the tested cases.
Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
In most scheduling models, there is a common goal to minimize for all the jobs[1–6]. However, jobs might be from
dif-ferent customers which have their own goals to pursue. For instance, Peha[7]gave a telecommunication service example
where various types of packets and service compete for the use of a commercial satellite, and the problem is to satisfy
the service requirements of individual agents to transfer voice, image and text files for their clients. Kim et al.[8]pointed
out that in project scheduling the problem is concerned with negotiation to resolve conflicts whenever the agents find their
own schedules unacceptable. Kubzin and Strusevich[9]presented another example that maintenance operations complete
with the real jobs for machine occupancy on maintenance planning. Balasubramanian et al.[10]provided the manufacturing
examples where the agents belong to different subjects competing for the usage of machines.
Agnetis et al.[11]and Baker and Smith[12]were the pioneers that brought the multi-agent problems into scheduling
field. For more articles with multiple-agent scheduling, readers can refer to[13–23].Since then, many researchers have
de-voted efforts to this area. Recently, Lee et al.[24]studied a two-agent problem with deteriorating jobs on a single machine to
minimize the total weighted completion time of jobs from agent 1 given that no tardy jobs are allowed for agent 2. In their
model, the actual job processing time is
a
jþ bt if its starting time is t wherea
jis the normal processing time of job j, and b isthe common deterioration rate. They provided the branch-and-bound algorithm to search for the optimal solution for prob-lems with up to 20 jobs. Moreover, they provided three heuristic algorithms based on linear combinations of processing times and weights for jobs from agent 1 and linear combinations of processing times and due dates for jobs from agent 2.
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.025
⇑Corresponding author.
E-mail address:[email protected](W.-C. Lee).
Contents lists available atSciVerse ScienceDirect
Applied Mathematics and Computation
Wu et al.[25]considered a single-machine scheduling problem with learning effects where the objective is to minimize the total tardiness of jobs from the first agent with the constraint that no tardy job is allowed for the second agent. The actual
processing time of job j under the model is pjraif it is scheduled in the rth position where pjis the processing time of job j, a is
learning effect and a 0. The proposed branch-and-bound algorithm could solve problems with up to 24 jobs optimally, and the proposed heuristic algorithms sort jobs according to the linear combination of the job processing times and the due
dates. They presented the computational experiment for their heuristics with 50 jobs. Lee et al.[26]studied a two-agent
scheduling problem on two-machine permutation flowshop where the objective is to minimize the total tardiness of jobs from the first agent given that no tardy job of the second agent is allowed. They provided a branch-and-bound algorithm to solve problems with 12 jobs optimally. In addition, they presented the computational results of two simulated annealing
algorithms for problems with 25, 50 and 75 jobs. Cheng et al.[27]considered a two-agent single-machine scheduling
prob-lem with deteriorating jobs and learning effects simultaneously. The objective is to minimize the total weighted completion time of jobs from the first agent with the restriction that no tardy job is allowed for the second agent. They assumed that jobs from the first agent have a position-based learning effect, and jobs from the second agent have a position-based deteriorating effect. Their branch-and-bound algorithm could present the optimal solutions for problems with up to 16 jobs. Moreover, they conducted a computational experiment for the proposed simulated algorithm for problems with 40 and 50 jobs.
Most of the research on multiple-agent focuses on studying the computational complexity of the intractable cases or examining problems of minimizing a single criterion of one agent given a bound of a single criterion of the other agent. How-ever, a decision maker has to decide the schedule based on many aspects. For instance, the manufacturer might be interested in reducing the inventory cost as well as reducing the impact of late delivery. To the best of our knowledge, multi-agent scheduling problem with the consideration of multiple objectives for the same agent is never been discussed in literature. Since the total completion time is commonly used to represent the internal efficiency while the maximum tardiness is com-monly used to represent the external efficiency. We consider these two criteria in this paper. That is, we study a two-agent scheduling problem on a single machine where the objective is to minimize the weighted combination of the total comple-tion time and the maximum tardiness of jobs from one agent given that the number of tardy jobs from the other agent is zero. The rest of the paper is organized as follows. In the next section we describe the formulation of our problem. In Sec-tion 3, we construct a branch-and-bound algorithm incorporating several eliminaSec-tion rules and a lower bound to speed up the search for the optimal solution. In Section 4, several simulated annealing algorithms are proposed to solve this problem. In Section 5, a computational experiment is conducted to evaluate the efficiency of the branch-and-bound algorithm and the performance of the proposed heuristic algorithms. A conclusion is given in the last section.
2. Problem description
The problem description is given as follows. There are n jobs ready to be processed on a single machine. Each job belongs
to either one of the two agents AG0or AG1. For each job j, there is a processing time pj, a due date dj, and an agent code Ij,
where Ij¼ 0 if j 2 AG0 or Ij¼ 1 if j 2 AG1. Under a schedule S, let CjðSÞ be the completion time of job j,
TjðSÞ ¼ maxf0; CjðSÞ djg be the tardiness of job j, TmaxðSÞ ¼ max1jnfTjðSÞð1 IjÞg be the maximum tardiness of jobs from
agent AG0under schedule S, and UjðSÞ ¼ 1 if TjðSÞ > 0 and zero otherwise. The objective of this paper is to find a schedule that
minimizes the weight combination of the total completion time and the maximum tardiness of jobs from AG0 with the
restriction that no tardy jobs from AG1are allowed. Using the conventional three fields notation, this problem is denoted
as 1jPUjIj¼ 0j
a
PCjð1 IjÞ þ ð1a
ÞTmaxwhere 0 <a
<1.3. A branch-and-bound algorithm
Agnetis[28]pointed out that the two-agent problem is NP-hard if both the agents have the sum-type objectives. Thus, the
branch-and-bound algorithm is developed to obtain the optimal solution. 3.1. Dominance properties
In this subsection, we first provide several adjacent dominance properties. Suppose that S and S0are two job schedules
and the difference between S and S0 is a pairwise interchange of two adjacent jobs i and j. That is, S ¼ ð
p
;i; j;p
0Þ andS0
¼ ð
p
;j; i;p
0Þ, wherep
andp
0 each denote a partial sequence. In addition, let t denote the completion time of the lastjob in
p
and B be the maximum tardiness of jobs from agent AG0under partial sequencep
, that is, B ¼ maxk2p\AG0fTkðSÞg.The completion times of jobs i and j in S are
CiðSÞ ¼ t þ pi ð1Þ
and
CjðSÞ ¼ t þ piþ pj; ð2Þ
whereas the completion times of jobs j and i in S0are
and
CiðS0Þ ¼ t þ pjþ pi: ð4Þ
Property 1. If jobs i; j 2 AG0, pi<pj, di<djand t B > maxfdi pi;dj pjg, then S dominates S 0
.
Proof. Since jobs in
p
are processed in the same order in both S and S0, we have from Eqs.(2) and (4)thatCkðSÞ ¼ CkðS0Þ if job k 2
p
orp
0:Thus, to show S dominates S0, it suffices to show
a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTmaxðSÞ <a
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTmaxðS0Þ: ð5ÞSince t B > maxfdi pi;dj pjg, we have
TiðSÞ ¼ t þ pi di; ð6Þ
TjðSÞ ¼ t þ piþ pj dj; ð7Þ
TjðS0Þ ¼ t þ pj dj; ð8Þ
and
TiðS0Þ ¼ t þ pjþ pi di: ð9Þ
To show Eq.(5)hold, we divide it into the following three cases:
Case I: TmaxðSÞ ¼ TiðSÞ. From pi<pjand Eqs.(6) and (9), we have
a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTmaxðSÞ ¼a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTiðSÞ <a
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTiðS0Þa
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTmaxðS0Þ:Case II: TmaxðSÞ ¼ TjðSÞ. From pi<pj, di<djand Eqs.(7) and (9), we have
a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTmaxðSÞ ¼a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTjðSÞ <a
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTiðS0Þa
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTmaxðS0Þ:Case III: TmaxðSÞ ¼ TkðSÞ for some k 2
p
orp
0. From pi<pj, we havea
CiðSÞ þa
CjðSÞ þ ð1a
ÞTmaxðSÞ ¼a
CiðSÞ þa
CjðSÞ þ ð1a
ÞTkðSÞ <a
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTkðS0Þa
CjðS0Þ þa
CiðS0Þ þ ð1a
ÞTmaxðS0Þ:Thus, S dominates S0since Eq.(5)holds for all the three cases.
Property 2. If jobs i; j 2 AG0, pi<pj, and di<t þ pi B dj pj, then S dominates S0.
Property 3. If jobs i; j 2 AG0, pi<pj, t þ pi B di, and t þ piþ pj B dj, then S dominates S0.
Property 4. If jobs i; j 2 AG0, pi<pj,
a
ðpi pjÞ þ ð1a
Þðt þ piþ pj dj BÞ < 0, and dj pi<t þ pj B minfdi pi;djg,then S dominates S0.
Property 5. If jobs i; j 2 AG0, pi<
a
pjand dj<t þ pj B di pi, then S dominates S 0.Property 6. If jobs i; j 2 AG0, pi<pj, di<dj, 0 B ðt þ pj djÞ < pi, and t þ pi B > di, then S dominates S0.
Property 7. If jobs i; j 2 AG0, pi<
a
pj, and t B > maxfdi pi;dj pjg, then S dominates S0.Property 8. If jobs i; j 2 AG0, pi<pj,
a
ðpi pjÞ þ ð1a
Þðdi djÞ < 0, t þ pj B > dj, and 0 B ðt þ pi diÞ < pj, then Sdom-inates S0.
Property 10. If jobs i; j 2 AG0, pi<pj, t þ piþ pj B > dj, t B minfdi pi;dj pjg, and di<dj, then S dominates S0.
Property 11. If job i 2 AG0, job j 2 AG1, and t þ piþ pj dj 0, then S dominates S0.
To further facilitate the search process, we provide a proposition to determine the feasibility of a partial schedule and a proposition to determine the ordering of the remaining unscheduled jobs. Assume that ðPS; USÞ is a sequence of jobs where PS is the scheduled part with k jobs and US is the unscheduled part with (n–k) jobs. It is assumed among the unscheduled
jobs, there are n0jobs from agent AG0and n1jobs from agent AG1where n0þ n1¼ n k. Let S¼ ðPS;USÞ be a sequence in
which n0jobs from agent AG0are scheduled first in the shortest processing time (SPT) rule, followed by n1jobs from agent
AG1in the earliest due date (EDD) rule. In addition, let d1ð1Þ d
1
ð2Þ . . . d 1
ðn1Þdenote the due dates of the remaining n1jobs
from agent AG1when they are arranged in the EDD rule. Moreover, let p1ð1Þ;p1ð2Þ; . . . ;p1ðn1Þdenote their corresponding
process-ing times and C½kbe the completion times of the last job in PS.
Proposition 1. If there is a unscheduled job j from agent AG1 such that C½kþ
Pj
i¼1p1ðiÞ d 1
ðjÞ>0, then sequence (PS; US) is
infeasible.
Proposition 2. If maxk2PS\AG
0fTkðS Þg maxk2PS\AG0fTkðS Þg andPk2PS\AG 1UkðS Þ ¼ 0, then S¼ ðPS;US Þ dominates sequences of the type ðPS; USÞ.
3.2. A lower bound
The performance of the branch-and-bound algorithm also depends on the efficiency of the lower bound. Assume that S ¼ ðPS; USÞ be a schedule in which PS is the scheduled part with k jobs and US is the unscheduled part with (n–k) jobs.
Let t be the completion time of the last job in partial schedule PS, and TPS
maxbe the maximum tardiness of jobs from agent
AG0in PS. That is, TPSmax¼ maxj2AG0\PSf0; CjðSÞ djg. In addition, it is assumed that among the unscheduled jobs, there are
n0jobs from agent AG0and n1jobs from agent AG1. Moreover, let d1ð1Þ d
1
ð2Þ . . . d 1
ðn1Þdenote the due dates of the
remain-ing n1jobs from agent AG1when they are arranged in the EDD rule and p1ð1Þ;p1ð2Þ; . . . ;p1ðn1Þdenote their associated processing
times. In addition, let p0
ð1Þ p0ð2Þ . . . p0ðn0Þdenote the processing times of the remaining n0jobs from agent AG0when they
are arranged in the SPT rule and d0ð1Þ d0ð2Þ . . . d
0
ðn0Þdenote the due dates of the remaining n0jobs from agent AG0when
they are arranged in the EDD rule. Note that p1
ðjÞand d
1
ðjÞare from the same jobs, but p0ðjÞand d
0
ðjÞare not necessarily from the
same job. We then construct n0pseudo jobs from agent AG0such that job j from agent AG0has processing time p0ðjÞand due
date d0ðjÞfor j = 1 , . . . , n0. The basic idea to develop the lower bound is (1) to complete jobs from agent AG
1as late as possible
without violating the assumption of no tardy jobs from agent AG1are allowed, and (2) to insert n0pseudo jobs from agent
AG0into the machine available periods where job preemption is allowed for jobs from agent AG0. The procedures are divided
into two phases, and given as follows. Phase I:
Step 1: Set i ¼ n1, st ¼ 1, and tn1þ1¼ 1.
Step 2: If d1ðiÞ<st, set ti¼ d 1
ðiÞ p1ðiÞ. Otherwise, set ti¼ st p1ðiÞ.
Step 3: Set st ¼ tiand i ¼ i 1. If i 1, go to Step 2.
Step 4: Output ðt1;t2; . . . ;tn1;tn1þ1Þ.
Phase II:
Step 1: Set i ¼ 1, and k ¼ 0. Step 2: Set k ¼ k þ 1.
Step 3: If t þ p0
ðiÞ>tk, set p0ðiÞ¼ p 0
ðiÞ ðtk tÞ, t ¼ tkþ p1ðkÞand go to Step 2. Otherwise, set t ¼ t þ p
0
ðiÞand C
0 ðiÞ¼ t.
Step 4: If i < n0, set i ¼ i þ 1 and go to Step 3.
Step 5: Output C0ð1Þ;C0ð2Þ; . . . ;C0ðn
0Þ
.
The purpose of Phase I is to calculate tj, the latest time to start processing job j from agent AG1without violating the
no-tardy jobs assumptions, whereas the main goal of Phase II is to schedule jobs from agent AG0into the machine available
peri-ods, and the output C0
ðjÞis the completion time for pseudo job j. Thus, the lower bound of the weighted combination of the
total completion time and the maximum tardiness for sequence S is LBðSÞ ¼
a
Pki¼1C½iðPSÞð1 I½iÞ þ
Pn0 i¼1C 0 ðiÞ h i þ ð1
a
Þ maxfTPSmax;max1in0
fC0ðiÞ d 0 ðiÞgg.
3.3. Description of the branch-and-bound algorithm
A depth-first search is adopted in the branching procedure. In this paper, the algorithm assigns jobs in a forward manner starting from the first position. We first implement the proposed SA algorithms (discussed in the next section) to obtain a sequence as the initial incumbent solution. In the branching tree, we systematically work down the tree. We apply Propo-sition 1, Properties 1 to 11 to eliminate the dominated partial sequence. For the non-dominated nodes, we apply PropoPropo-sition 2 to determine the sequence of the unscheduled jobs or compute the lower bound on the weighted combination of the total
completion time and the maximum tardiness of jobs from agent AG0. If the lower bound on the objective function for the
partial sequence is greater than the incumbent solution, eliminate that node. If the objective function of the completed se-quence is less than the incumbent solution, replace it as the new solution. The procedure is repeated until we explore the whole tree.
4. The simulated annealing algorithm
Evolutionary heuristic algorithms have been successfully applied to solve many combinatory optimization problems[29–
33]. In this paper, the simulated annealing (SA) algorithm[34]is utilized to derive a near-optimal solution for the proposed
problem. The essential elements of the SA algorithm included:
(1) Initial sequence: In this study, two initial sequences are used. In the first initial sequence, jobs from agent AG1are first
placed according to the EDD rule, followed by jobs from agent AG0according to the SPT first rule, denoted as SAEDDþSPTin later
analysis. In the second initial sequence, jobs from agent AG1are first placed according to the EDD rule, followed by jobs from
agent AG0in the EDD rule, denoted as SAEDDþEDD.
(2) Neighborhood generation: Three neighborhood generation methods are analyzed in this study. They are the pairwise interchange (PI), the extraction and forward-shifted reinsertion (EFSR), and the extraction and backward-shifted reinsertion (EBSR) movements. Depending on the initial sequences and the neighborhood generation movements, there are six SA
algorithms considered in this study, and denoted as SAPIEDDþSPT, SA
EFSR EDDþSPT, SA EBSR EDDþSPT, SA PI EDDþEDD, SA EFSR EDDþEDD, and SA EBSR EDDþEDD.
(3) Acceptance probability: The probability of acceptance is generated from an exponential distribution, P acceptð Þ ¼ exp k ð
D
TC=hÞ;whereDTC is the change in the objective function[35]. After some pretests, we choose h ¼ 5000. If the weight combination of
the total completion time and the maximum tardiness of jobs from agent AG0increases, the new sequence is accepted with
probability r, where r is a uniform random number between 0 and 1.
(4) Stopping condition: According to our pretests, the procedure is stopped after 300n iterations, where n is the number of jobs.
5. Computational experiments
In this section, the computational experiments were conducted to evaluate the performance of the branch-and-bound and the SA algorithms. They were coded in Fortran 90 and run on a personal computer with 3.20 GHz Intel(R) Core(TM) i5 CPU 650 3.21 GHz and 3.45 GB RAM under Windows XP. The job processing times were generated from a uniform distri-bution over the integers 1–100 and the due dates of jobs were generated from another uniform distridistri-bution over the integers
between Tð1
s
R=2Þ and Tð1s
þ R=2Þ, where R is the due date range factor,s
is the tardiness factor, and T is the totalprocessing times of all the jobs.
The computational experiment consisted of five parts. In the first part of the experiment, the job size was fixed at 10, the
weighted combination of the total completion time and the maximum tardiness
a
was 0.5, and the proportion of jobs fromagent AG1was fixed at 50% to test the efficiency of the dominance properties and the lower bound separately, and the results
are compared with the enumeration method. Eight combinations of ð
s
;RÞ values were tested, i.e. (0.25, 0.25), (0.25, 0.50),Table 1
Results of the branch-and-bound algorithm and enumeration method with n = 10, P = 0.5,a=0.5.
s R BB_L Number of nodes BB_P Number of nodes BB_P + L Number of nodes Enumeration Number of nodes
Mean SD Mean SD Mean SD
0.25 0.25 111755.8 85379.2 299.5 134.5 295.1 129.5 3628800 0.50 79758.3 61617.9 299.6 132.8 273.9 104.5 3628800 0.75 51875.5 62046.2 310.1 389.7 247.4 229.5 3628800 0.50 0.25 10078.9 14995.6 107.7 74.7 104.1 72.1 3628800 0.50 9408.5 16062.1 142.6 156.9 136.2 151.5 3628800 0.75 12953.7 29669.8 164.0 167.9 156.5 152.9 3628800 0.75 0.25 812.5 1635.7 41.6 24.0 41.5 24.0 3628800 0.50 772.8 1155.4 45.5 33.7 44.8 32.8 3628800
(0.25, 0.75), (0.5, 0.25), (0.5, 0.50), (0.5, 0.75), (0.75, 0.25), and (0.75, 0.50). The branch-and-bound algorithm with the
dom-inance properties is denoted by BB_P, and the branch-and-bound algorithm with only the lower bound is denoted by BB_L,
while the branch-and-bound algorithm with the properties and the lower bound is denoted by BB_P + L. The average and standard deviation of the number of nodes were reported for the branch-and-bound algorithms, while the number of nodes, 10!, was given for the enumeration method. 100 replications were randomly generated for each condition and the results
were presented inTable 1. It is seen that the dominance properties and the lower bound benefit the searching process in
terms of the number of nodes explored. Thus, the branch-and-bound algorithm with the properties and the lower bound was used in later analysis.
In the second part of the experiment, the effects of the due date factors
s
and R to the performance of thebranch-and-bound algorithm were studied. The values of parameters were the same as those inTable 1, except the job size was fixed at
12. The mean and the standard deviation of the number of nodes and the mean and the standard deviation of the CPU time (in seconds) were reported for the branch-and-bound algorithm. 100 replications were randomly generated for each case
and the results were presented in Table 2. A one-way analysis of variance (ANOVA) on the number of nodes of the
branch-and-bound algorithm was constructed and given inTable 3. The resulting F-value was 38.122 with a p-value of close
to 0, which indicated that the due date factors
s
or R have statistically significant effects on the difficulty of the problem. Tofurther analyze the impact of
s
and R, a two-way ANOVA on the number of nodes for the first six cases was conducted andthe results were presented inTable 4. It indicated that
s
is a significant factor since its resulting F-value is 97.664 and ap-value of close to 0. In addition, the impact of R is significant with a resulting F-p-value of 4.457 and a p-p-value of 0.012. However,
there is no significant indication of the interaction effects between
s
and R since its resulting F-value is 1.415 and itsasso-ciated p-value is 0.244. A closer look atTable 2revealed that the problems are harder to solve as the value of
s
is smaller. Themain reason is that Proposition 1 and the lower bound are less powerful in that case. In addition, it was noted that the case
ð
s
;RÞ ¼ ð0:25; 0:75Þ has the most number of nodes with an average of 1239.66 among the eight cases. Thus, it was used in thethird part of the experiment.
Similar to the second part of the experiment, the third part was to test the effects of the coefficient
a
and the proportion ofjobs from the second agent P. The job size was fixed at 12, and ð
s
;RÞ was (0.25, 0.75). Three different values ofa
(0.25, 0.5,0.75), and of P (0.25, 0.50, 0.75) were used. 100 replications were randomly generated for each case and the results were
presented inTable 5. A two-way ANOVA on the number of nodes was utilized to test the effects of the parameters to the
performance of the branch-and-bound algorithm, and the result was reported inTable 6. The resulting F-value of
a
wasTable 2
The performance of the branch-and-bound algorithm with n = 12, P = 0.5,a=0.5.
s R Number of nodes CPU time
Mean SD Mean SD 0.25 0.25 952.26 443.83 0.006 0.008 0.5 961.62 442.47 0.005 0.007 0.75 1239.66 1704.13 0.008 0.011 0.5 0.25 300.98 193.29 0.001 0.004 0.5 465.16 447.05 0.003 0.006 0.75 479.26 440.50 0.003 0.006 0.75 0.25 110.99 81.45 0.001 0.003 0.5 117.65 159.42 0.001 0.004 Table 3
One-way ANOVA table for the number of nodes with n = 12, P = 0.5,a=0.5.
Source SS DF MS F p-value
Due date factors 125427020 7 17918146 38.122 0.000
Error 372253902 792 470018
Total 497680922 799
Table 4
Two-way ANOVA table for the number of nodes with n = 12, P = 0.5,a=0.5.
Source SS DF MS F p-value Factor (s) 60683304 1 60683304 97.664 0.000 Factor (R) 5538663 2 2769331 4.457 0.012 Interaction 1759012 2 879506 1.415 0.244 Error 369081068 594 621349 Total 437062047 599
Table 5
The performance of the branch-and-bound algorithm with n = 12 ands= 0.25, R = 0.75.
P a Number of nodes CPU time
Mean SD Mean SD 0.25 0.25 566.6 667.1 0.004 0.007 0.5 474.3 392.7 0.002 0.006 0.75 453.9 374.0 0.003 0.006 0.5 0.25 1136.4 1113.0 0.006 0.009 0.5 1239.7 1704.1 0.008 0.011 0.75 1234.9 1576.9 0.007 0.011 0.75 0.25 958.7 1324.8 0.005 0.009 0.5 737.6 945.6 0.005 0.008 0.75 1109.8 1765.0 0.006 0.012 Table 6
ANOVA table for the number of nodes with n = 12 ands= 0.25, R = 0.75.
Source SS DF MS F p-value Factor (P) 76059350 2 38029675 26.061 0.000 Factor (a) 2037967 2 1018983 0.698 0.498 Interaction 6371767 4 1592942 1.092 0.359 Error 1300175652 891 1459232 Total 1384644736 899 Table 7
The performance of the branch-and-bound algorithm (a=0.5).
n s R P Number of nodes CPU time
Mean SD Mean SD 16 0.25 0.50 0.25 3654.0 7230.1 0.03 0.06 0.50 14300.7 27645.3 0.12 0.22 0.75 28873.3 43994.4 0.27 0.37 0.75 0.25 5045.4 7508.4 0.04 0.06 0.50 15669.7 18501.6 0.13 0.15 0.75 12879.9 31172.2 0.12 0.27 0.50 0.50 0.25 15844.5 44713.8 0.12 0.35 0.50 5679.1 6499.3 0.05 0.05 0.75 2137.1 2936.6 0.02 0.03 0.75 0.25 18269.0 56922.6 0.13 0.42 0.50 15293.9 23433.1 0.12 0.18 0.75 7006.8 16149.6 0.06 0.14 20 0.25 0.50 0.25 16499.3 21739.4 0.19 0.24 0.50 187894.2 436495.2 2.17 4.93 0.75 653635.3 1258141.5 8.85 15.88 0.75 0.25 111090.4 511446.1 1.13 5.11 0.50 272520.9 560226.2 3.09 6.23 0.75 317470.9 805000.9 4.35 10.93 0.50 0.50 0.25 479755.2 3290233.5 5.30 37.44 0.50 200706.0 539656.7 2.21 5.82 0.75 36039.8 93098.6 0.50 1.30 0.75 0.25 383251.0 1124434.6 4.08 12.01 0.50 843572.2 3247301.9 9.12 36.53 0.75 112184.1 395111.2 1.41 4.95 24 0.25 0.50 0.25 252867.7 730185.0 3.58 10.20 0.50 2182444.9 5099456.3 32.73 73.43 0.75 10881767.2 20926257.4 198.34 362.17 0.75 0.25 739842.0 1664482.6 10.44 23.52 0.50 12731415.3 28723857.3 186.32 412.87 0.75 28187420.7 110415273.5 486.52 1862.78 0.50 0.50 0.25 21904170.1 94524057.3 299.96 1299.99 0.50 7246913.0 19021221.9 100.86 266.49 0.75 519383.6 1598286.6 8.94 25.94 0.75 0.25 7114554.9 17944690.0 95.45 245.22 0.50 22155286.8 75160846.0 321.92 1102.85 0.75 5764492.3 22341614.3 100.18 375.64
0.698 with a p-value of 0.498, which implied that
a
does not affect the performance of the branch-and-bound algorithm. On the other hand, the statistical test showed that the proportion of jobs from the second agent P is a significant factor with an F-value of 26.061 and a p-value of close to 0. There was also an indication of no interaction effects between these two factorssince its corresponding F-value and p-value were 1.092 and 0.359. It was seen fromTable 5that the problems are easier to
solve when the value of P is smaller. The main reason was that the properties are more powerful in that case.
The main purpose of the fourth part of the experiment was to study the impact of the number of jobs to the performance of the branch-and-bound algorithms, and the accuracy of the proposed simulated annealing algorithms. Three different job
sizes (n = 16, 20 and 24) were tested. Since problems were harder to solve if the due date factor
s
was smaller or R was larger,and the proportion of jobs from the second agent P were also found to be the significant factors in the previous experiments,
they were also considered in the experiment. Two different values of
s
(0.25, 0.50), two values of R (0.5, 0.75) and threeval-ues of P (0.25, 0.50, 0.75) were used. We recorded the mean and standard deviation of the number of nodes and of the CPU time (in seconds) for the branch-and-bound algorithm, while only recording the mean and standard deviation of the error percentages of the SA algorithms, where the last one, denoted as the combined one, is the minimum value of the first six proposed SA algorithms. The execution time of SA algorithm was not recorded since they were finished within a second.
For each condition, 100 replications were generated and the results were given inTables 7 and 8. It was seen that the number
of nodes and execution time grows exponentially as the number of jobs increases, and the standard deviations of the num-bers of nodes are many times of their mean numnum-bers of nodes since the problem is NP-hard. It was also observed that the branch-and-bound algorithm could solve problems of up to 24 jobs in a reasonable amount of time. The most
time-consum-ing case took an average of 486 s when ð
s
;R; PÞ = (0.25, 0.75, 0.75). As to the performance of the SA algorithms, it was seenthat SAPIEDDþSPTand SA PI
EDDþEDDperform better, followed by SA EFSR EDDþSPTand SA EFSR EDDþEDD, and SA EBSR EDDþSPTand SA EBSR
EDDþEDDhave the worst
performance. It implied that the PI movement yields a better result, and the EBSR yields the worst result overall. However, there was no clear dominance relation between these six algorithms, since the error percentages of the combined SA were much smaller than those of the six SA algorithms. Moreover, it was noted that average error percentage the combined SA
Table 8
The error percentages of the proposed simulated annealing algorithms.
n s R P SAPI
EDDþSPT SAEFSREDDþSPT SAEBSREDDþSPT SAPIEDDþEDD SAEFSREDDþEDD SAEBSREDDþEDD CombinedSA
Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
16 0.25 0.50 0.25 0.25 0.87 1.60 2.76 8.09 7.07 0.13 0.32 1.28 2.09 8.22 7.89 0.04 0.07 0.50 0.25 1.06 5.38 8.57 14.57 14.99 0.32 1.46 4.09 7.01 11.21 13.38 0.02 0.06 0.75 0.98 5.00 5.78 9.04 14.62 29.03 1.50 6.48 6.18 12.68 12.54 27.20 0.02 0.20 0.75 0.25 0.14 0.26 0.69 1.45 7.76 9.19 0.12 0.26 0.74 1.29 5.64 5.08 0.04 0.12 0.50 0.20 0.67 2.06 4.10 12.15 20.18 0.17 0.66 2.42 5.13 8.15 9.89 0.03 0.09 0.75 0.12 0.54 1.63 6.26 4.62 12.04 0.58 4.70 0.62 3.32 5.65 18.90 0.00 0.00 0.50 0.50 0.25 0.08 0.13 0.61 1.73 9.68 10.16 0.10 0.15 0.80 1.90 4.91 5.57 0.03 0.06 0.50 0.16 1.11 1.20 2.94 9.10 12.99 0.21 1.39 1.04 2.71 5.22 8.00 0.02 0.06 0.75 0.17 1.66 0.18 1.65 0.81 3.36 0.17 1.65 0.33 2.16 0.81 3.63 0.16 1.65 0.75 0.25 0.14 0.28 0.79 1.28 6.71 9.95 0.10 0.21 0.62 1.02 4.59 4.51 0.03 0.11 0.50 0.29 1.14 1.62 3.26 7.26 11.31 0.20 0.82 1.85 3.07 6.04 9.06 0.01 0.04 0.75 0.49 2.65 1.42 6.23 3.99 7.08 0.49 2.64 1.22 4.18 2.93 5.98 0.10 0.95 20 0.25 0.50 0.25 0.13 0.27 1.30 2.13 10.34 8.06 0.16 0.40 1.49 2.27 10.00 10.01 0.04 0.08 0.50 0.74 2.51 3.71 5.36 9.00 8.35 0.65 2.56 4.84 5.73 10.28 10.64 0.13 1.13 0.75 1.63 6.52 7.69 11.29 10.34 14.33 1.82 7.34 4.14 7.60 11.97 19.44 0.02 0.12 0.75 0.25 0.10 0.17 1.07 1.64 10.19 8.91 0.11 0.20 0.94 1.62 8.85 8.45 0.04 0.07 0.50 0.58 2.43 2.66 5.74 8.38 8.82 0.34 1.26 2.66 4.66 9.97 13.18 0.12 0.79 0.75 0.10 0.38 1.01 4.05 9.05 35.58 0.13 0.39 0.94 3.29 7.28 19.98 0.00 0.02 0.50 0.50 0.25 0.09 0.20 1.14 1.75 12.70 12.11 0.12 0.24 0.87 1.48 8.38 8.42 0.03 0.05 0.50 0.10 0.33 1.47 2.79 13.48 11.05 0.12 0.75 2.14 3.87 9.01 11.05 0.02 0.03 0.75 0.12 1.08 0.25 1.23 1.26 4.31 0.12 1.08 0.44 2.21 1.16 4.24 0.11 1.08 0.75 0.25 0.11 0.20 0.66 0.91 10.60 8.18 0.17 0.34 0.90 1.30 7.43 6.16 0.06 0.16 0.50 0.33 1.32 2.77 4.08 9.75 9.53 0.36 1.36 2.76 3.86 9.45 10.19 0.10 0.65 0.75 0.24 1.72 1.01 3.39 6.37 11.37 0.30 1.76 2.72 9.07 5.62 13.62 0.03 0.24 24 0.25 0.50 0.25 0.26 0.84 1.58 2.11 13.84 10.22 0.16 0.68 1.63 2.30 10.38 8.43 0.04 0.11 0.50 0.68 2.17 4.38 6.01 8.45 8.41 0.70 2.30 4.87 7.75 13.12 14.02 0.14 0.64 0.75 1.57 5.79 7.15 10.05 9.97 11.91 0.88 4.59 5.66 13.40 14.58 22.88 0.45 3.51 0.75 0.25 0.17 0.33 1.20 1.50 12.53 8.45 0.12 0.33 1.07 1.39 9.90 8.53 0.06 0.22 0.50 0.56 1.87 4.05 6.44 8.08 7.77 0.54 1.95 3.66 5.11 9.10 8.39 0.08 0.37 0.75 0.55 2.71 1.98 6.33 9.50 21.49 0.52 2.75 1.84 6.15 9.73 25.50 0.31 2.21 0.50 0.50 0.25 0.13 0.40 1.20 1.50 15.18 8.21 0.11 0.31 1.34 1.73 11.18 8.06 0.04 0.14 0.50 0.34 1.57 2.36 3.75 18.33 12.91 0.10 0.74 2.00 3.56 10.19 9.37 0.08 0.73 0.75 0.02 0.05 0.76 3.42 2.43 5.37 0.15 1.27 0.19 1.47 2.01 5.50 0.00 0.04 0.75 0.25 0.14 0.30 0.93 1.01 11.57 7.35 0.12 0.26 1.10 1.62 9.55 7.19 0.06 0.22 0.50 0.28 1.06 2.69 3.93 12.04 9.52 0.28 1.25 2.34 3.01 10.34 9.73 0.04 0.22 0.75 0.47 2.29 2.74 5.80 7.93 10.49 0.95 3.66 2.56 5.73 8.07 10.62 0.17 1.48
algorithm was less than 0.5% for all the tested cases, thus, it was recommended as the ultimate algorithm since all the SA can be easily implemented.
The last part of the computational experiments was used to test the performance of the proposed SA algorithm when the number of jobs is large. We tested two job sizes, i.e., n = 100 and 200. We randomly generated 100 instances for each
situ-ation and we reported the results inTable 9. We recorded the mean and standard deviation of the relative deviation
percent-age (RDP). For instance, the RDP of the solution produced by SAPIEDDþSPTis calculated as
Table 9
The RDP and nTof the proposed simulated annealing algorithms for large job-sized problems.
n s R P SAPI
EDDþSPT SAEFSREDDþSPT SAEBSREDDþSPT SAPIEDDþEDD SAEFSREDDþEDD SAEBSREDDþEDD
RDP RDP RDP RDP RDP RDP
Mean SD nT Mean SD nT Mean SD nT Mean SD nT Mean SD nT Mean SD nT 100 0.25 0.50 0.25 0.15 0.21 39 1.21 0.77 2 3.67 1.78 0 0.08 0.17 59 2.85 1.74 0 5.28 4.12 0 0.50 1.73 1.79 23 0.83 0.98 35 12.07 5.80 0 1.53 1.60 24 2.83 3.35 20 15.32 8.81 0 0.75 5.22 9.33 37 3.83 4.01 25 33.40 27.77 3 4.42 8.12 40 2.35 5.42 49 33.05 33.92 2 0.75 0.25 0.17 0.21 28 1.55 0.71 0 4.61 2.06 0 0.04 0.09 73 2.39 1.32 0 6.74 3.88 0 0.50 0.91 1.04 17 0.52 0.73 41 10.84 5.13 0 0.53 0.85 40 5.76 4.24 2 14.52 7.75 0 0.75 0.48 2.67 62 0.01 0.07 94 11.44 21.89 39 0.48 2.70 61 0.01 0.08 94 10.00 17.43 35 0.50 0.50 0.25 0.04 0.06 42 1.67 0.81 0 6.99 2.53 0 0.02 0.05 60 1.93 1.07 0 14.69 4.61 0 0.50 0.22 0.56 40 0.81 1.08 15 24.34 6.04 0 0.30 0.67 45 3.21 2.88 5 27.90 8.56 0 0.75 0.11 0.63 48 0.30 1.05 16 8.23 5.79 3 0.09 0.52 57 0.56 1.54 24 7.35 6.70 2 0.75 0.25 0.03 0.06 56 2.10 0.89 0 10.11 2.54 0 0.03 0.06 54 2.19 1.00 0 16.64 5.13 0 0.50 0.13 0.21 44 0.99 0.81 2 20.87 5.39 0 0.11 0.30 56 7.01 3.33 0 24.65 8.41 0 0.75 0.60 1.65 46 1.10 1.78 15 23.14 8.67 0 0.94 2.27 42 4.54 3.92 6 25.71 11.96 0 200 0.25 0.50 0.25 0.53 0.32 1 0.11 0.15 48 2.68 1.13 0 0.08 0.12 51 4.12 1.67 0 3.01 1.45 0 0.50 3.96 1.93 1 0.16 0.42 83 13.99 7.47 1 2.75 1.66 3 3.44 3.44 12 12.77 6.93 0 0.75 7.01 9.93 16 3.45 4.03 18 25.56 23.28 2 6.02 8.71 31 2.58 4.59 33 27.58 22.33 1 0.75 0.25 0.45 0.36 6 0.65 0.27 1 3.53 1.34 0 0.01 0.04 93 3.46 1.25 0 3.75 1.99 0 0.50 3.15 1.57 1 0.04 0.13 89 16.54 6.78 0 1.38 0.99 10 10.69 4.51 0 15.55 8.11 0 0.75 0.27 1.50 4 0.00 0.00 47 1.60 9.52 69 0.18 1.17 9 0.00 0.01 34 1.05 4.85 60 0.50 0.50 0.25 0.09 0.12 26 0.69 0.20 0 5.81 1.70 0 0.02 0.05 74 3.27 0.93 0 11.94 4.24 0 0.50 0.59 0.56 13 0.21 0.49 61 27.69 5.18 0 0.60 0.60 25 2.43 1.64 1 35.95 7.40 0 0.75 0.12 0.48 24 0.24 0.64 38 10.65 6.66 0 0.05 0.27 40 0.87 1.52 12 9.97 6.58 1 0.75 0.25 0.08 0.08 16 0.75 0.13 0 9.26 2.02 0 0.00 0.01 84 3.24 0.94 0 15.19 4.23 0 0.50 0.34 0.34 19 0.16 0.19 37 25.69 4.22 0 0.16 0.23 44 10.78 3.10 0 33.12 5.90 0 0.75 1.16 1.41 18 0.21 0.46 53 31.28 8.76 0 1.22 1.42 22 6.15 4.57 7 39.13 11.25 0 Table 10
The CPU times (s) of the proposed simulated annealing algorithms for large job-sized problems.
n s R P SAPI
EDDþSPT SAEFSREDDþSPT SAEBSREDDþSPT SAPIEDDþEDD SAEFSREDDþEDD SAEBSREDDþEDD
Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
100 0.25 0.50 0.25 0.12 0.01 0.09 0.01 0.16 0.01 0.12 0.01 0.09 0.01 0.16 0.01 0.50 0.11 0.01 0.09 0.01 0.16 0.01 0.11 0.01 0.09 0.01 0.17 0.02 0.75 0.12 0.01 0.11 0.01 0.15 0.01 0.12 0.01 0.11 0.01 0.15 0.01 0.75 0.25 0.13 0.01 0.09 0.01 0.19 0.02 0.13 0.01 0.09 0.01 0.20 0.02 0.50 0.13 0.01 0.09 0.01 0.23 0.03 0.13 0.01 0.10 0.01 0.25 0.03 0.75 0.13 0.01 0.11 0.01 0.20 0.02 0.13 0.01 0.11 0.01 0.20 0.02 0.50 0.50 0.25 0.16 0.01 0.09 0.01 0.27 0.03 0.17 0.01 0.10 0.01 0.30 0.04 0.50 0.18 0.02 0.12 0.01 0.29 0.03 0.18 0.02 0.12 0.01 0.30 0.03 0.75 0.21 0.03 0.21 0.03 0.26 0.03 0.22 0.03 0.22 0.03 0.26 0.02 0.75 0.25 0.20 0.02 0.10 0.01 0.47 0.08 0.20 0.02 0.10 0.01 0.51 0.09 0.50 0.24 0.03 0.13 0.01 0.63 0.12 0.25 0.03 0.13 0.01 0.64 0.11 0.75 0.31 0.06 0.20 0.03 0.52 0.10 0.31 0.06 0.21 0.04 0.54 0.09 200 0.25 0.50 0.25 0.45 0.01 0.32 0.01 0.60 0.03 0.45 0.02 0.33 0.01 0.62 0.03 0.50 0.43 0.02 0.35 0.01 0.68 0.04 0.44 0.02 0.36 0.01 0.70 0.05 0.75 0.45 0.02 0.41 0.02 0.60 0.03 0.45 0.02 0.41 0.02 0.59 0.03 0.75 0.25 0.51 0.02 0.33 0.01 0.77 0.05 0.51 0.02 0.34 0.01 0.78 0.05 0.50 0.52 0.02 0.37 0.01 1.05 0.08 0.52 0.02 0.38 0.02 1.07 0.10 0.75 0.49 0.03 0.42 0.02 0.82 0.05 0.49 0.03 0.42 0.02 0.84 0.06 0.50 0.50 0.25 0.66 0.03 0.36 0.01 1.06 0.09 0.68 0.03 0.37 0.01 1.20 0.11 0.50 0.69 0.04 0.44 0.02 1.21 0.10 0.70 0.04 0.44 0.02 1.23 0.11 0.75 0.82 0.07 0.81 0.09 1.03 0.08 0.84 0.07 0.81 0.08 1.03 0.07 0.75 0.25 0.82 0.05 0.37 0.01 2.17 0.34 0.83 0.05 0.37 0.01 2.47 0.43 0.50 0.95 0.06 0.46 0.02 3.12 0.43 0.95 0.06 0.50 0.03 3.39 0.46 0.75 1.22 0.22 0.75 0.10 2.42 0.33 1.23 0.22 0.80 0.12 2.42 0.33
VPI EDDþSPT V =V 100%
where VPIEDDþSPTis the value of the weighted combination of the completion time and the maximum tardiness of jobs from the
first agent generated by SAPI
EDDþSPTand V
is the minimum value obtained from the six algorithms. In addition, we recorded
the number of times (nT) it yields the minimum value. The result was presented inTable 9. We also recorded the mean and
standard deviation of the execution time (in seconds) inTable 10. It was seen fromTable 9that SAPIEDDþEDDhas the best
per-formance in terms of mean RDP and the number of times it yields the minimum value. However, it did not perform well for
all the cases. For instance, the mean RDP and nTof SAPIEDDþEDDwere 2.75% and 3 when ðn;
s
;R; PÞ ¼ ð200; 0:25; 0:5; 0:5Þ, in thiscase SAEFSREDDþSPTis the best with a mean RDP of 0.16% and nT¼ 83. Moreover, it was seen fromTable 10that the execution time
of the algorithms was only a few seconds. Thus, the combined simulated annealing algorithm is recommended when the number of jobs is large.
6. Conclusions
In this paper we studied a two-agent single-machine scheduling problem where the objective is to minimize the weighted combination of the completion time and maximum tardiness of the jobs of the first agent, given that no tardy jobs are allowed for the second agent. We proposed a branch-and-bound algorithm to solve the problem, and a combined sim-ulated annealing algorithm to find near-optimal solutions. We conducted computational experiments to evaluate the perfor-mance of the proposed algorithms. The computational results showed that the branch-and-bound algorithm can solve problems with up to 24 jobs in a reasonable amount of time. It also showed that the performance of the combined SA algo-rithm is very good, yielding an average error percentage error of less than 0.5% for all the tested cases.
Acknowledgements
The authors are grateful to the editor and the referees, whose constructive comments have led to a substantial improve-ment in the presentation of the paper. This work was supported by the NSC of Taiwan, under NSC 101-2221-E-035-021. References
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