Parallel-machine scheduling to minimize tardiness penalty and power cost

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Parallel-machine scheduling to minimize tardiness penalty and power cost

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Kuei-Tang Fang, Bertrand M.T. Lin

⇑

Department of Information and Finance Management, Institute of Information Management, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 2 March 2012

Received in revised form 3 October 2012 Accepted 10 October 2012

Available online 17 October 2012

Keywords:

Parallel-machine scheduling Total weighted tardiness Dynamic voltage scaling Particle swarm optimization

a b s t r a c t

Traditional research on machine scheduling focuses on job allocation and sequencing to optimize certain objective functions that are defined in terms of job completion times. With regard to environmental con-cerns, energy consumption becomes another critical issue in high-performance systems. This paper addresses a scheduling problem in a multiple-machine system where the computing speeds of the machines are allowed to be adjusted during the course of execution. The CPU adjustment capability enables the flexibility for minimizing electricity cost from the energy saving aspect by sacrificing job completion times. The decision of the studied problem is to dispatch the jobs to the machines as well as to determine the job sequence and processing speed of each machine with the objective function com-prising of the total weighted job tardiness and the power cost. We give a formal formulation, propose two heuristic algorithms, and develop a particle swarm optimization (PSO) algorithm to effectively tackle the problem. Since the existing solution representations do not befittingly encode the decisions involved in the studied problem into the PSO algorithm, we design a tailored encoding scheme which can embed all decisional information in a particle. A computational study is conducted to investigate the performances of the proposed heuristics and the PSO algorithm.

1. Introduction

In traditional scheduling research, the processing speeds of ma-chines are assumed to be static. This paper investigates a schedul-ing problem with heterogeneous parallel machines that exhibits the ﬂexibility to adjust their processing speeds for each individual job. The ﬂexibility of machine speed adjustment comes from real world applications that invoke intense computing tasks on a clus-ter of compuclus-ters. The clock-rate of a processor is declus-termined by the frequency of an oscillator crystal. It was commonly accepted that a computer runs at its highest speed so as to attain the best performances. When a computer runs at a higher speed/frequency, it will not only consume more electricity but also produce more heat. To cool down the high temperature, the computer needs to activate extra devices, like fans, thus causing more energy con-sumption. In scheduling problems, most of the objective functions are regular, i.e. non-decreasing in terms of job completion times. No ground seems to exist for slowing down the machine speeds to defer the processing of jobs. Nevertheless, if the delivery is not urgent or the implied cost of deference is offset by the reduction of energy cost, it is reasonable to reduce the processing speeds of the processors.

Nowadays more and more industrials not only pursuit better work efficiency, but also focus on energy-saving issues. Slower paces are attracting more considerations in different industries. For example, in maritimes, ships with lower voyage speeds are much preferred due to a lower consumption of energy. Demands for lower electricity costs also arise in the IT industry. Cloud com-puting has emerged as a booming business model. While some ser-vice providers start developing a wide variety of serser-vices to attract more users or customers, others on the other hand focus on how to optimize work schedules for efficiency. Cloud computing service providers host a larger number of processors that are deployed to fulfill the orders of computing tasks placed by clients. In addi-tion to customer satisfacaddi-tion and efficient utilizaaddi-tion of facilities, lower electricity consumption is also crucial to the service provid-ers.Koomey (2007)showed that the electricity used for servers worldwide, including their associated cooling and auxiliary equip-ment, cost 7.2 billion US dollars in 2005. The cost in that year had doubled when compared with consumption in 2000. Therefore, the electricity issue is not negligible but critical because energy con-sumption plays one of the key roles in the overall cost structure. To attain energy saving objectives, the CPU frequency can be ad-justed through the dynamic voltage scaling technique. The CPU adjustment technique allows for flexibility to seek for potential balance between job completion times and energy cost.

This paper considers the objective function with a weighted sum of the total job tardiness penalty and the total power cost. The former one is an efﬁciency indicator in traditional schedule

http://dx.doi.org/10.1016/j.cie.2012.10.002

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This manuscript was processed by Area Editor T.C. Edwin Cheng.

⇑Corresponding author. Tel.: +886 3 5131472; fax: +886 3 5723792.

E-mail addresses:[email protected](K.-T. Fang),[email protected]

(B.M.T. Lin).

Contents lists available atSciVerse ScienceDirect

Computers & Industrial Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c a i e

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problems to reflect both customer service (concerning average waiting time) and internal inventory management (concerning work-in-process parts). This weighted combinatorial objective function can significantly close to the business considerations in schedule planning. The machines exhibit the possibility to allow the shop floor manager to select their processing speeds by adjust-ing CPU frequencies. Higher speeds permit a shorter processadjust-ing makespan of jobs, yet more power cost will be incurred. The sched-uling policy thus rests in the trade-off between the performance measures defined in job completion times and the total power cost. To be more precisely, we have to (a) dispatch the jobs to the ma-chines; (b) sequence the jobs on each machine; and (c) select the processing speed for each job.

The remainder of the paper is organized as follows: In Section2, we ﬁrst give a formal deﬁnition of the studied problem and present the notations that will be used throughout this paper. An integer linear programming (ILP) model follows. Literature review on re-lated scheduling problems and the concept of energy saving is also presented. Since the studied problem is strongly NP-hard, we adopt approximation approaches to produce approximate sched-ules in a reasonable time. Two heuristic algorithms are proposed in Section3. Then, in Section4, we introduce the meta-heuristic particle swarm optimization (PSO). The development of a PSO algo-rithm, especially the tailor-designed encoding scheme, for the studied problem is introduced in Section5. Section6is dedicated to a computational study for appraising the effectiveness of the proposed algorithms. Concluding remarks and suggestions for sev-eral future research subjects are given in Section7.

2. Problem statements and literature review

This section presents a formal deﬁnition of the allocation and scheduling of jobs from the aspect of parallel-machine scheduling. Following the problem deﬁnition is an integer linear programming (ILP) model that describes the studied problem.

2.1. Problem deﬁnition and ILP

Consider a set of n jobs fJ1;J2; . . . ;Jng to process on a set of m

heterogeneous machines fM1;M2; . . . ;Mmg. The jobs can be

pro-cessed on any machine. Each job Jjis characterized by a processing

load ‘jand a due date dj, before which the job is assumed to be

ﬁn-ished, and a weight indicating the penalty of unit-time tardiness wj. Due to the operating characteristics, we can select the machine

frequency to adjust the processing speed for the jobs. A higher fre-quency allows the machines to run faster at the cost of more en-ergy consumption. The actual processing time of a job Jj thus

depends on the speed selected for the machine in charge of it.

The decision is to assign and schedule the jobs to the machines as well as to select appropriate execution speeds for processing the jobs. The objective function to minimize is the weighted sum of the tardiness penalties of jobs and the energy consumption of the machines. Total weighted tardiness and energy consumption cost may not be added up in a natural way for they are not of the same type of measures. Yet it is possible to adopt a normaliza-tion through a common measurement, say a monetary base. The weighted sum of two criteria adopted in this paper is for demon-strating the signiﬁcance of the proposed models and solution approaches.

In the following, we introduce the notations that will be used throughout the paper and the integer programming formulation.

In the following, we present an integer linear programming model of the studied problem, which is denoted as the WTPC (Weighted Tardiness and Power Consumption) problem.

Problem WTPC: Minimize X n i¼1 wjTjþ Xm i¼1 Xn j¼1 Xn l¼1 Xli s¼1 tjiseisxjils ð1Þ subject to X m i¼1 Xn l¼1 Xli s¼1 xjils¼ 1; 1 6 j 6 n; ð2Þ Xn j¼1 Xli s¼1 xjils61; 1 6 i 6 m; 1 6 l 6 n; ð3Þ bi;lþ1 bi;l¼ Xn j¼1 Xli s¼1 xjilstjis; 1 6 i 6 m; 1 6 l 6 n 1; TjPbi;lþ tjisxjils ð1 xjilsÞM dj; 1 6 i 6 m; 1 6 s 6

l

i; 1 6 j; l 6 n; ð5Þ

TjP0; 1 6 j 6 n; ð6Þ

xjils2 f0; 1g; 1 6 i 6 m; 1 6 s 6

l

i;

1 6 j; l 6 n: ð7Þ

The objective function of Eq.(1)consists of two parts. The first part is concerned about the total tardiness penalty, and the second part gives the total power consumption resulted from the dispatching decision. Constraints(2)confine each job to be processed on exactly one position of some machine running at a specific speed. Con-straints(3)reflect the fact that each position of any machine can accommodate at most one job. Constraints(4)define the starting time of the job in each position on the machines. The computation of the tardiness Tj¼ fCj dj;0g of each job is given in constraints

(5) and (6). We would elaborate on constraints(5)in more details. If job Jjis not scheduled at the lth position on machine i with speed

s, then xjils¼ 0 and the inequality is automatically satisﬁed due to

the negative value M. On the other hand, when xjils¼ 1, the value

Nomenclature Notation

J set of jobs fJ1;J2; . . . ;Jng to be scheduled

j index 1 6 j 6 n of jobs

M set of machines fM1;M2; . . . ;Mmg for scheduling

i index 1 6 i 6 m of machines

l index 1 6 l 6 n of positions on the machines

s index 1 6 s 6

l

i of speeds of machine Mi, where

l

i is

the number of different speeds of machine Mi

‘j processing load of job Jj

dj due date of job Jj

Cj completion time of job Jj

Tj tardiness of job Jj, i.e. maxfCj dj;0g

wj penalty of unit-time tardiness

Sis s-th processing speed of machine Mi

eis energy consumption (per time unit) of machine Mi at

speed Sis

tjis execution time of job Jj if assigned to machine Mi at

speed s; tjis¼ ‘j=Sis

pjis actual power consumption of job Jjif it is executed by

machine Miat speed Sis, i.e. tjis eis

Decision variables

xjils if job Jjis assigned to the lth position on machine Miat

speed Sis, then xjils¼ 1; otherwise, xjils¼ 0

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of Tjwill reﬂect the difference between the job completion time and

the job due date.

2.2. Literature review

As aforementioned in Section1, the subject of energy saving techniques draws considerable research attention. Adjusting CPU speeds, by sacriﬁcing job completion times, has been proven to be one of the most signiﬁcant and effective approaches (Bunde, 2006). The main idea of CPU speed adjustment is supported by dy-namic voltage scaling (DVS), which equips the machines with the ability to change their computing states or clock frequencies. With the possibility to change the processing speeds of machines, many of the traditional scheduling problems can be discussed again. Some researchers have combined the idea of energy saving and DVS with traditional scheduling problems.Zhu, Melhem, and Chil-ders (2003)proposed two algorithms for task scheduling with and without precedence constraints on multiprocessor systems.Zhu,

Mosse, and Melhem (2004) designed a greedy slack stealing

scheme to solve the AND/OR model of real-time applications and showed that their scheme outperforms other existing ones. Ge, Feng, and Cameron (2005)proposed distributed performance-di-rected DVS scheduling strategies for use in scalable power-aware HPC clusters and obtained signiﬁcant energy savings (36%) without increasing the required execution time.Kumar and Palani (2012) applied the DVS technique to reduce the energy consumption of the embedded system technology to mobile systems. They used genetic algorithm for solving the problem of minimizing schedule length subject to an energy consumption constraint and the prob-lem of minimizing energy consumption subject to a schedule length constraint. Rizvandi, Zomaya, Lee, Boloori, and Taheri (2012, chap. 17)addressed the energy issue with task scheduling and presented the MFS-DVFS algorithm to reduce energy con-sumption. Akgul, Puschini, Lesecq, Miro-Panades, and Benoit (2012)applied DVFS techniques to mobile computing platforms where performance constraints, such as task deadlines, are given. They recast the problem in a linear program and solve the problem by the simplex algorithm.

The objective function considered in this paper comprises the minimization of total weighted tardiness and power cost. For the single-machine scheduling problem, Lawler (1977) designed a pseudo-polynomial time algorithm for solving the 1jjPTj

prob-lem. The complexity status remained open until Du and Leung (1990)gave an NP-hardness proof to conﬁrm the intractability of the 1jjPTj problem. The weighted version of this problem,

1jjPwjTj, is known to be NP-hard in the strong sense (Lenstra,

Rinnooy Kan, & Brucker, 1977). In the case of parallel machines, Lenstra et al. (1977)showed the problem with two machines to be ordinary NP-hard. For the case with a ﬁxed number of machines, Garey and Johnson (1978)gave an proof of ordinary NP-hard. The case with an arbitrary number of machines turns to be NP-hard in the strong sense (Garey & Johnson, 1978). Hence, the complexity of the total weighted tardiness of parallel machines is also NP-hard in the strong sense. Some well-known meta-heuristics are applied to solve parallel-machine total weighted tardiness problems ( Anghin-oﬁ & Paolucci, 2007; Runwei, Mitsuo, & Tatsumi, 1995).

The studied WTPC problem is concerned about (a) assigning the jobs to the machines, (b) determining the processing sequence of the jobs on each machine, and (c) selecting the processing speed for each job, such that the weighted sum of tardiness penalty and power cost in minimum. From the intrinsic complexity of the scheduling problem with the objective function of total tardi-ness, WTPC is obviously strongly NP-hard. As a consequence, it is very unlikely to design polynomial or pseudo-polynomial time algorithms to solve the problem to optimality. This paper develops

two heuristics and a meta-heuristic algorithm to derive approxi-mate solutions.

3. Heuristics

This section introduces two construction heuristics, based upon the earliest due date (EDD) rule, and the weighted shortest pro-cessing time (WSPT) rule, respectively. The jobs are then dis-patched to the machines by the list scheduling algorithm (Graham, 1966), which assigns the ﬁrst unscheduled job to the machine with the shortest completion time until all jobs are dispatched.

The EDD rule, proposed byJackson (1955), guarantees the opti-mality in the minimization of the maximum lateness on a single machine. It has been widely adopted to deal with due-date related objective functions. For example,Baker (1974)showed that based on the EDD rule, if there is at most one tardy job, the minimum to-tal tardiness is guaranteed. Therefore, we adopt the EDD rule and expect that it will work reasonably well when the number of tardy jobs is small. This ﬁrst heuristic algorithm is outlined in the following:

Due-date-based heuristic HEDD.

Step 1. Sort the jobs in non-decreasing order of due dates. Step 2. Remove the ﬁrst job, say Jj, from the list and assign it to the

machine which has the shortest completion time. Step 3. Select the machine speed that minimizes the contributions

(weighted tardiness and power cost) that Jj will make to

the objective function.

Step 4. Execute job Jjby using the speed selected in Step 3. Update

the completion time of the machine. Step 5. Repeat Steps 2–4 until all jobs are processed.

To solve the single-machine scheduling problem of minimizing the total weighted completion time, Smith (1956)proposed the WSPT rule that sequences the jobs in non-decreasing order of the ratio between job processing time and job weight. In view of a spe-cial case where all jobs have a common due date, the total weighted tardiness is minimized by the WSPT rule. This stimulates the adoption of the WSPT rule for dealing with the problem of min-imizing the total weighted tardiness, especially when most jobs tend to be tardy.

WSPT-based heuristic HWSPT

Step 1. Sort the jobs in non-decreasing order of the ratio between processing load and weight, ‘j=wj.

Step 2. Remove the ﬁrst job, say Jj, from the list and assign it to the

machine which has the shortest completion time. Step 3. Select the machine speed that minimizes the contributions

(weighted tardiness and power cost) that Jj will make to

the objective function.

Step 4. Execute job J_jby using the speed selected in Step 3. Update the completion time of the machine.

Step 5. Repeat Steps 2–4 until all jobs are processed.

The comparison between the two heuristics would be carried out in Section6. In the next section, we develop a meta-heuristic to produce better solutions at the cost of longer computing time. 4. Particle swarm optimization

In the nature, some animals exhibit social behavior, like a flock of birds find food sources or a school of fish avoid predators. These kinds of animals can transfer their own messages to each other and achieve some goals in a collective way. Particle swarm optimization

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(PSO) is a swarm-based intelligence algorithm.Kennedy and Eber-hart (1995)studied the above-mentioned behavior of swarms in the nature, such as birds, and ﬁsh and developed the PSO frame-work for dealing with hard optimization problems. PSO has been widely applied to many scheduling problems. Dye and Ouyang (2011)deployed PSO algorithms for solving joint pricing and lot-sizing problem.Önüt, Tuzkaya, and Dog¨aç (2008)studied multi-ple-level warehouse layout design problem by applying PSO algo-rithms.Ai and Kachitvichyanukul (2009)used PSO algorithms for solving the capacitated vehicle routing problem.

In PSO algorithms, each solution is represented as a particle, which is analogous to a bird flying through the solution space. Each particle is associated with two key parameters, current position and current velocity. The current position is used to evaluate the dis-tance of the current solution away from the best position of any swarm particle so far. The current velocity, represented a vector, dictates the direction and the speed at which the bird is flying. At any time, each particle position is influenced by the best posi-tion that the particle has ever visited. The performances of the par-ticles in different positions are measured by their fitness values, which reflect the objective function values of the studied problem. The general framework of the PSO algorithm is described below.

Let xt

jdenote the current position of particle j at iteration t, and

v

tj

the velocity of particle j at iteration t. The velocity and position of solu-tion j at the next iterasolu-tion are computed by the following equasolu-tions:

v

tþ1 j ¼

x

v

t jþ c1 rand1 ðpbestj xtjÞ þ c2 rand2 ðgbest xtjÞ ð8Þ xtþ1 j ¼ x t jþ

v

tþ1j ð9Þ where:

x

: The inertia weight is a constant deﬁned by the user to regulate the impact of the current velocity on the velocity of the next iteration.

ck: The acceleration coefﬁcients for k ¼ 1; 2, where c1 is

the acceleration coefﬁcient indicating the attraction to the previous best position of the current particle, c2is the acceleration coefﬁcient indicating the

attrac-tion to the previous best posiattrac-tion of the swarm, and rand1and rand2are random numbers generated from

the uniform interval [0, 1].

pbesti: The best position ever visited in the historical course of

particle i.

gbest: The best position ever visited by the swarm.

As in Eq.(8), the current velocity of a particle is a linear combina-tion of three types of velocities: (a) the inertia velocity which is related to its previous velocity; (b) the velocity to the best location found by the particle and (c) the velocity to the best location found by the swarm. Eq.(9)indicates that the current position of a parti-cle is the previous position plus the adjustment induced by the cur-rent velocity. A larger

x

value implies that the current velocity encounters a higher traction force from the previous velocity. Empirical studies of PSO with inertia weights have shown that a relatively larger

x

value implies a more global search ability and on the other hand a relatively smaller

x

value results in faster con-vergence. Larger c1 values exploit more personal behavior, while

larger c2 values exploit more the global swarm behavior. The

trade-off among the inertia weight (i.e. the previous velocity), the explorations (i.e. the global search) and the exploitations (i.e. local search) of search space is crucial to the efﬁciency and effec-tiveness of PSO algorithms. Through the adjustment of the position and the velocity of each particle iteratively, PSO algorithms could obtain near-optimal solutions when it reaches the convergence conditions or the stop criteria.

Most PSO algorithms start with a population of random solu-tions, each of which is associated a random velocity. Each particle keeps track of its potential best solution ðpbestÞ, which is the best solution in its movement history, and the global best solution ðgbestÞ, which is the best solution ever visited thus far by all parti-cles of this swarm.

PSO ALGORITHM

Step 1. Set the particle dimension equal to the number of jobs n. Step 2. Randomly generate a set of particle positions xjand

veloc-ities

v

jfor initialization.

Step 3. For each generated particle, calculate its ﬁtness value. Step 4. Check if the ﬁtness value is better than pbest or not. If yes,

update pbest.

Step 5. If the minimum pbest in the swarm is better than gbest, then replace gbest with the minimum pbest.

Step 6. For each particle, use Eq.(8)to calculate the new velocity and Eq.(9)to update the position.

Step 7. Repeat from Step 3, until the stopping criteria are satisfied. One of the key designs to the success of PSO algorithms is con-cerned about the representation of solutions. A proposed represen-tation must reflect the characteristics of the studied problem as well as permit easy manipulation in the course of PSO execution. The WTPC problem involves decisions of allocation, sequencing and state tuning. It is not intuitive to construct a mapping between the particles of PSO and the solutions of WTPC. In general PSO algo-rithms, a particle can completely express the action of a solution, including the relative position in the solution space and the motion from the previous position to the current position. In the context of the WTPC problem, a solution should clearly indicate the machine each job is assigned to and the position and speed at which each job is processed on the machine it resides. The challenge is the necessity to encode at least four types of information, namely jobs, machines, positions and speeds in the representation of solutions. A straightforward resolution approach is to construct a 4-dimensional array for encoding solutions and let each dimension denote one type of information. Nevertheless, there is no straight-forward 4-dimensional velocity array befitting for the structure and inherent information of particles. The above-mentioned 4-dimensional array would not work when applied to Eq.(9). Another alternative design is to use a 4-level linked list to encode the solu-tions. Unfortunately, we cannot interpret the physical meaning of the acceleration coefficient times a linked list (Eq.(8)). Therefore, we need to design a more flexible solution encoding method, which is feasible and meaningful to the PSO algorithm, and also conve-nient for computing objective function values. In the next section, we will elaborate on the PSO solution encoding scheme.

5. The framework of PSO for WTPC

This section describes how PSO is adapted to solve the WTPC problem. As aforementioned, how the solutions are represented for further manipulations is crucial to the success of PSO deploy-ment. Some solution representations applied to PSO for parallel machine scheduling problems are studied. Then, we analyze the solution structure of the WTPC problem. Since the existing solution representations are not necessarily suitable for WTPC, we need to design a new encoding method. An example solution is given for illustrating the design.

5.1. Existing solution representations

There are several types of solution representations in the deployment of PSO for coping with parallel-machine scheduling

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problems.Kashan and Karimi (2009)proposed a discrete encoding approach to solve the parallel-machine scheduling problem of makespan minimization. An array with a length equal to the num-ber of jobs is adopted to represent the solutions. The value stored in the jth position of the array corresponds to the machine to which the jth job is assigned.

Niu, Zhou, and Wang (2010)used a similar array structure and proposed a real number encoding scheme to tackle parallel-ma-chine scheduling of minimizing the total tardiness. In each real number used, the integer part denotes the machine that the job is allocated to, and the fractional part stands for a relative execu-tion order of the jobs on that machine.

5.2. The solution structure of WTPC problem

In the WTPC problem, besides for job dispatching and job sequencing, an additional solution encoding mechanism to de-scribe the decision of machine speed selection. In previous papers, the integer part and the fractional part of real numbers in the solu-tion encoding scheme are both used. One possibility for the speed selection is to use the positive sign and negative sign to express additional solution information. Yet it could be limited to binary choices, low speed or high speed. Therefore, we would like to ex-tend the use of integer part and fractional part of real numbers to denote the selection of multiple machine speeds. Unlike the method ofNiu et al. (2010)for encoding job sequencing, we need to know the exact execution speed of each job. Moreover, each ma-chine could have distinct abilities to adjust its speeds. In other words,

l

i, the number of different speeds of machine Mi, varies

across the machines. This ﬂexibility demands an encoding mecha-nism which is ﬂexible enough for precisely indicating the execu-tion speed for each job.

5.3. New encoding method of problem WTPC

With reference to the above mentioned solution representa-tions and difficulties encountered in the studied problem, we de-sign a new solution encoding representation to overcome the difficulties. We first randomly generate n real numbers from a uni-form distribution on the interval ½1; m þ 1Þ with a 3-digit fractional part. For example, if m ¼ 6 machines are available, then we may generate such real numbers as 2:783; 3:513; 6:987, and so forth. Such a real number is used to encode the information of the three decisions: job dispatching, job sequencing, and machine speed selection. The integer part stores the index of the machine to which the job is assigned. The fractional part represents the processing order of the job on its machine. For example, the job Jj1annotated

by the real number 2:415 and the job Jj2 annotated by the real

number 2:873 are both processed on machine M2, and the job Jj1

precedes the job Jj2because :415 < :873. Moreover, the last 2 digits

of the fractional part are also used to indicate the speed selected for the job execution. Suppose the last 2 digits of the fractional part are q1and q2. For machine Mithat permits

l

i different speeds, if

b100 ðr 1Þ=

l

ic 6 q1 10 þ q2<d100 r=

l

ie, then the job is

executed on machine Mi at speed Sir for 1 6 r 6

l

i. For example,

if the last two digits of the factional part are 4 and 5 and there are three different speeds on the machines, then this job will be processed at the second speed.

5.4. Example of the new encoding method

The following numerical example solution is given to illustrate the proposed encoding scheme. Suppose that there are 6 jobs, 3 machines, and 3 different speeds on each machine. The detail inter-pretation of the solution

p

¼ ½2:554; 1:263; 1:711; 3:487; 2:822; 2:176 is as follows:

Job J1is executed on machine M2at speed S22.

Job J₄is executed on machine M3at speed S33.

The job sequence on machine M1is ðJ2;J3Þ.

The job sequence on machine M2is ðJ6;J1;J5Þ.

The job sequence on machine M3is ðJ4Þ.

The proposed solution representation approach makes a good use of all the digits of the real numbers. Another advantage is that a PSO algorithm can easily compile the stored information of a real number to realize the corresponding schedule and to calculate its objective function value. For a given solution, we sort the array in non-decreasing order of the real numbers. Knowing the integer part of a job, we can assign the job to the corresponding machine. Based on the preprocessing calculation of the processing time of each job on each machine under distinct speeds, the completion times of all jobs on all machines can be computed in OðnÞ time. Hence, the objective function values, total weighted tardiness and total power cost, of a solution in the proposed representation approach can be computed in Oðn log nÞ time.

As mentioned above, PSO algorithms iterate for exploring solu-tion space until the defined convergence condisolu-tions or the stopping criteria are met. Considering the efficiency without sacrificing solution quality, we use limits on the elapsed CPU time and the number of iterations as the stopping criteria in the PSO algorithm.

Table 1

Comparison of solutions produced by CPLEX and two heuristics.

R Time CPLEX HEDD Dev. (%) HWSPT Dev. (%)

m ¼ 3; n ¼ 5 0.1 195.32 1387.44 1871.75 25.87 1932.96 28.22 0.2 45.86 872.74 1193.90 26.90 943.95 7.54 0.3 4.08 915.89 1088.31 15.84 1242.29 26.27 0.4 24.22 647.95 869.33 25.47 764.51 15.25 0.5 283.40 893.68 1091.66 18.14 1157.01 22.76 0.6 12.30 558.32 590.45 5.44 623.17 10.41 0.7 7.08 505.32 544.85 7.26 811.75 37.75 0.8 5.47 1060.59 1241.56 14.58 1252.18 15.30 0.9 13.69 596.64 672.91 11.33 709.08 15.86 1.0 19.22 993.15 1094.14 9.23 1246.23 20.31 m ¼ 3; n ¼ 10 0.1 60 min 1877.00 2327.45 19.35 3096.92 39.39 0.2 60 min 1757.94 2437.20 27.87 3049.67 42.36 0.3 60 min 1536.94 2274.49 32.43 2786.02 44.83 0.4 60 min 1315.97 1778.45 26.00 2465.03 46.61 0.5 60 min 1640.72 2876.04 42.95 3149.07 47.90 0.6 60 min 1982.68 2567.91 22.79 3836.15 48.32 0.7 60 min 2118.40 2984.72 29.03 3733.19 43.25 0.8 60 min 2004.27 3042.41 34.12 2819.48 28.91 0.9 60 min 1810.99 2386.82 24.13 3408.94 46.88 1 60 min 1623.79 2031.01 20.05 3006.34 45.99 m ¼ 5; n ¼ 10 0.1 60 min 1429.50 1786.19 19.97 2218.51 35.56 0.2 60 min 1441.12 1935.80 25.55 2376.91 39.37 0.3 60 min 1472.24 2676.17 44.99 2400.67 38.67 0.4 60 min 1297.86 2101.38 38.24 2224.99 41.67 0.5 60 min 1144.63 1805.87 36.62 2164.71 47.12 0.6 60 min 1660.90 2297.67 27.71 3066.80 45.84 0.7 60 min 1974.28 3387.65 41.72 3794.63 47.97 0.8 60 min 1422.05 1936.61 26.57 2225.21 36.09 0.9 60 min 1112.78 1586.97 29.88 2052.13 45.77 1.0 60 min 1454.65 2093.35 30.51 2786.00 47.79

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That is, the PSO algorithm will keep searching for optimal solutions until the speciﬁed limits are reached.

6. Computational experiments

This section presents a computational study for examining the performances of the designed solution algorithms. The test in-stances were generated as follows. The processing loads lj of the

jobs were randomly generated by the uniform distribution Uð1; 100Þ. The tardiness penalty weights wjof the jobs were drawn

from Uð1; 10Þ. The generation of due dates followed the scheme

proposed byHall and Posner (2000). A parameter R was to adjust the period of due dates to reﬂect relative tightness. The mean value of due dates is equal to a half of the average total processing load on each machine divided by the average machine processing speed, namely D ¼1 2 P jlj m P i P sSis P i

l

i :

Therefore, D is an estimate of the makespan of the jobs and machine characteristics in the generated instance. The due dates were drawn from the uniform distribution U½Dð1 RÞ; Dð1 þ RÞ, where R is a

Table 2

Summary of the experiments setting.

ðc1;c2Þ n m x;R Solution Approaches E1 (1, 1) E2 (1, 2) {30, 50} {3, 5} f0:1; 0:2; . . . ; 1:0g PSO E3 (2, 1) E4 min of fE1;E2;E3g {30, 50} {3, 5} HEDD;HWSPT E5 max of fE1;E2;E3g {30, 50} {3, 5} HEDD;HWSPT Table 3 Results of experiment E1;ðc1;c2Þ ¼ ð1; 1Þ. R x¼ 0:2 x¼ 0:4 x¼ 0:6 x¼ 0:8 x¼ 1:0

Obj. Dev. (%) Obj. Dev. (%) Obj. Dev. (%) Obj. Dev. (%) Obj. Dev. (%)

30-3 0.1 8120.4 1.95 8066.7 1.28 7964.7 0.00 8305.9 4.28 10432.2 30.98 0.2 8779.2 17.27 7901.0 5.54 7486.1 0.00 8235.7 10.01 9927.5 32.61 0.3 7996.3 8.02 8053.1 8.79 7402.4 0.00 8928.9 20.62 11394.7 53.93 0.4 8490.9 21.05 8432.3 20.22 7014.3 0.00 8568.3 22.15 10416.5 48.50 0.5 8938.5 22.68 7716.4 5.90 7286.2 0.00 8722.5 19.71 10605.9 45.56 0.6 9044.5 35.29 8414.6 25.87 6685.3 0.00 8877.2 32.79 10287.3 53.88 0.7 9110.4 31.65 8995.9 29.99 6920.3 0.00 10535.1 52.23 10271.6 48.43 0.8 8521.5 11.51 8068.8 5.58 7642.2 0.00 8348.3 9.24 10367.4 35.66 0.9 9028.0 23.85 8427.2 15.61 7289.2 0.00 8480.9 16.35 10487.6 43.88 1.0 8549.9 20.13 8770.3 23.22 7117.4 0.00 9923.2 39.42 11140.0 56.52 30-5 0.1 19868.8 12.84 18852.4 7.07 17607.4 0.00 22801.9 29.50 27369.0 55.44 0.2 18939.4 7.40 18485.8 4.83 17633.7 0.00 23799.4 34.97 29842.0 69.23 0.3 20991.3 15.30 18760.4 3.05 18205.3 0.00 22856.9 25.55 29639.2 62.81 0.4 19318.3 8.33 18248.4 2.33 17832.3 0.00 23391.0 31.17 27584.0 54.69 0.5 20594.9 6.29 19375.7 0.00 19598.6 1.15 24707.6 27.52 29014.3 49.75 0.6 20655.9 17.02 17923.1 1.53 17652.2 0.00 24426.1 38.37 27609.9 56.41 0.7 20721.7 20.78 18269.8 6.49 17157.1 0.00 23803.5 38.74 27689.0 61.39 0.8 20068.1 22.37 19129.0 16.64 16399.7 0.00 23371.2 42.51 26896.4 64.01 0.9 20405.6 12.35 18162.2 0.00 20780.6 14.42 21725.7 19.62 28124.9 54.85 1.0 19948.1 15.96 18701.5 8.71 17203.1 0.00 24334.6 41.45 28532.0 65.85 50-3 0.1 7555.9 13.62 6810.7 2.41 6650.3 0.00 8626.2 29.71 10655.7 60.23 0.2 7655.4 13.11 7822.2 15.57 6768.3 0.00 8306.3 22.72 9701.1 43.33 0.3 7756.7 14.11 7422.7 9.20 6797.3 0.00 8430.8 24.03 11174.8 64.40 0.4 7888.0 27.44 6818.9 10.17 6189.5 0.00 7291.3 17.80 9450.2 52.68 0.5 8021.5 15.71 6932.6 0.00 7071.5 2.00 9186.9 32.52 9529.0 37.45 0.6 7826.4 19.90 7117.9 9.05 6527.2 0.00 7707.0 18.08 10197.6 56.23 0.7 7309.6 8.89 6986.0 4.07 6712.6 0.00 7910.2 17.84 10624.2 58.27 0.8 7578.9 7.55 7046.7 0.00 7325.1 3.95 8478.9 20.32 10281.9 45.91 0.9 7974.0 21.25 7017.5 6.70 6576.6 0.00 7833.9 19.12 9568.2 45.49 1.0 7204.7 5.53 6827.0 0.00 6933.3 1.56 9068.8 32.84 10787.4 58.01 50-5 0.1 17011.5 19.60 15200.0 6.86 14224.1 0.00 20242.7 42.31 21936.7 54.22 0.2 15241.5 2.81 14825.3 0.00 15462.5 4.30 21361.2 44.09 20901.2 40.98 0.3 15959.8 1.49 15725.5 0.00 16063.0 2.15 21862.5 39.03 22804.7 45.02 0.4 16790.9 13.97 14732.2 0.00 15872.8 7.74 20498.7 39.14 21537.0 46.19 0.5 15719.0 8.73 14457.0 0.00 15327.9 6.02 22554.5 56.01 23236.4 60.73 0.6 15941.2 14.22 14810.8 6.12 13956.7 0.00 22144.2 58.66 20698.5 48.31 0.7 17484.1 17.28 14907.4 0.00 16353.4 9.70 19592.2 31.43 21991.1 47.52 0.8 16311.8 13.73 15781.3 10.03 14342.8 0.00 21014.7 46.52 23523.7 64.01 0.9 18199.5 24.69 15653.0 7.25 14595.3 0.00 23087.1 58.18 23057.5 57.98 1.0 17588.7 29.20 13613.5 0.00 15900.3 16.80 22640.6 66.31 23003.1 68.97

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control parameter of the due-date ranges and selected from f0:1; 0:2; . . . ; 1:0g. Regarding the feature of CPU clock frequency adjustment, we assumed three distinct CPU frequencies. The CPU frequencies on distinct machines were assigned from the uniform distribution Uð1; 2Þ. The energy consumption of each machine was given to the third power of the corresponding machine frequency. It is expected that using a higher execution speed incurs a higher power consumption per time unit.

All the proposed algorithms were coded in C and executed on a personal computer equipped with an Intel Core 2 Quad Q8200 2.33 GHz CPU and 2 GB of RAM. The commercial software ILOG CPLEX 7.0 was used to solve the integer linear programming model for small-size problems. The derived solutions are used for com-parisons with the heuristic solutions. The experiment results are shown inTable 1. We set the time limit as 60 min in the CPLEX runs. Once the CPLEX solution-ﬁnding process reached the time limit, it aborted and output the incumbent best solution. The ﬁrst column of the table contains different due-date parameters, R. We show the computational times and solutions obtained from CPLEX in the next two columns. The two heuristics, HEDD and

HWSPT, took less than 0.01s to get the objective function values.

Therefore, we only list the output objective values. We also show

the percentage deviation from the solutions obtained by CPLEX and two heuristics in the row (dev.%). The percentage deviations (dev.%) are deﬁned as

objðHEDDÞ objðCPLEXÞ objðHEDDÞ 100%: and objðHWSPTÞ objðCPLEXÞ objðHWSPTÞ 100%:

With regard to the required computing time, the deviation between CPLEX and two proposed heuristics varies in a great range. In the case of ðm; nÞ ¼ ð5; 10Þ, the deviation even grows up to more than 40% in some instances. For the cases with a higher deviation be-tween the heuristic solutions and the CPLEX solutions, we examine the derived schedules and ﬁnd that there is one machine totally va-cant. This is due to the fact that all scalable processing speeds of that machine are relatively higher than those on other machines. If jobs are assigned to that machine with relatively higher process-ing speeds, the power cost of that machine is much larger than other machines. Thus, that machine would not be selected by the

Table 4

Results of experiment E2;ðc1;c2Þ ¼ ð1; 2Þ.

R x¼ 0:2 x¼ 0:4 x¼ 0:6 x¼ 0:8 x¼ 1:0

30-3 0.1 7004.1 0.00 8292.3 18.39 10227.4 46.02 10925.4 55.99 11543.5 64.81 0.2 7344.8 0.00 8030.8 9.34 9399.6 27.98 9624.7 31.04 10565.7 43.85 0.3 6878.1 0.00 7964.1 15.79 9001.5 30.87 10312.6 49.93 12537.8 82.29 0.4 7393.3 0.00 8347.7 12.91 8689.0 17.53 10157.8 37.39 11313.5 53.02 0.5 7667.5 5.93 7238.5 0.00 9084.9 25.51 10901.8 50.61 11726.7 62.00 0.6 7411.5 0.00 8237.0 11.14 8572.0 15.66 11401.3 53.83 11353.7 53.19 0.7 7507.7 0.00 8708.0 15.99 8982.1 19.64 11097.0 47.81 12250.4 63.17 0.8 6940.9 0.00 7190.1 3.59 9391.4 35.31 10102.7 45.55 11349.2 63.51 0.9 6451.5 0.00 8098.3 25.53 9605.7 48.89 11053.9 71.34 11965.1 85.46 1.0 6554.6 0.00 8308.4 26.76 9341.4 42.52 11788.6 79.85 12755.7 94.61 30-5 0.1 15631.9 0.00 19519.1 24.87 23384.6 49.60 26942.7 72.36 28996.9 85.50 0.2 14489.1 0.00 20964.6 44.69 25485.9 75.90 27033.3 86.58 30910.3 113.33 0.3 15856.9 0.00 20082.0 26.65 24023.4 51.50 25578.4 61.31 28779.5 81.50 0.4 14450.6 0.00 19530.8 35.16 23556.5 63.01 25928.2 79.43 27525.2 90.48 0.5 15058.5 0.00 20405.0 35.50 26020.1 72.79 27937.3 85.53 28226.4 87.44 0.6 15638.4 0.00 19226.7 22.95 25513.1 63.14 26869.5 71.82 27360.0 74.95 0.7 14821.6 0.00 19803.0 33.61 25679.2 73.26 27168.1 83.30 27453.0 85.22 0.8 13840.1 0.00 21160.5 52.89 25424.3 83.70 27358.1 97.67 28878.3 108.66 0.9 14448.9 0.00 20326.2 40.68 27525.3 90.50 25796.1 78.53 27316.5 89.06 1.0 13744.5 0.00 19803.7 44.08 25054.6 82.29 28484.6 107.24 28881.6 110.13 50-3 0.1 6690.4 0.00 7200.2 7.62 8737.5 30.60 9578.4 43.17 9757.2 45.84 0.2 6387.2 0.00 8507.7 33.20 8577.7 34.30 9183.7 43.78 10457.8 63.73 0.3 6427.6 0.00 7809.4 21.50 8820.4 37.23 9914.2 54.24 10804.4 68.09 0.4 6599.7 0.00 7240.7 9.71 8193.9 24.16 8863.9 34.31 9675.1 46.60 0.5 6689.2 0.00 7153.4 6.94 9473.5 41.62 9486.1 41.81 9504.0 42.08 0.6 6416.2 0.00 7898.9 23.11 8534.9 33.02 9563.8 49.06 9805.0 52.82 0.7 6205.4 0.00 7249.6 16.83 8793.1 41.70 9598.8 54.68 10807.6 74.16 0.8 6341.8 0.00 7464.0 17.70 9532.1 50.31 9960.6 57.06 10063.2 58.68 0.9 6466.0 0.00 7443.1 15.11 8773.5 35.69 9533.1 47.43 10263.9 58.74 1.0 6133.2 0.00 7392.1 20.53 9411.1 53.45 10302.8 67.98 10999.1 79.34 50-5 0.1 12909.4 0.00 17090.5 32.39 20412.5 58.12 20098.9 55.69 22119.9 71.35 0.2 11858.5 0.00 17010.7 43.45 21849.0 84.25 20806.2 75.45 20585.5 73.59 0.3 12102.3 0.00 18053.2 49.17 21703.2 79.33 22062.0 82.30 22342.5 84.61 0.4 12201.6 0.00 17088.0 40.05 20431.0 67.45 21992.1 80.24 22200.1 81.94 0.5 12215.9 0.00 16781.2 37.37 20972.5 71.68 21730.3 77.89 22606.8 85.06 0.6 11976.6 0.00 16974.4 41.73 20446.1 70.72 23537.7 96.53 20274.2 69.28 0.7 14023.5 0.00 17116.6 22.06 21060.3 50.18 21405.2 52.64 20786.2 48.22 0.8 12676.9 0.00 17965.1 41.72 20511.4 61.80 21489.8 69.52 24135.3 90.39 0.9 12742.9 0.00 18299.5 43.61 20205.1 58.56 21917.0 71.99 21499.3 68.72 1.0 12817.4 0.00 15848.9 23.65 20933.3 63.32 23612.7 84.22 23207.1 81.06

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heuristics. However, the proposed heuristics, which is based upon list scheduling, would anyhow allocate at least one job on each machine. Thus, encountering such special instance patterns, the proposed heuristic will deteriorate due to the processing with high-er powhigh-er consumption costs. The hardness of the studied problem lies in job scheduling, job dispatching issues as well as in speed selection. The EDD rule and the WSPT rule reﬂect the intuitive ideas behind the scheduling and dispatching parts. Since the speed selec-tion issue is addressed in the heuristic without retrospective correc-tions, the performances could be then less favorable. This inspires the necessity in the development of meta-heuristics that address all decision issues and provide iterative correction mechanisms.

For the large-size data, we only compared the proposed heuris-tics and the PSO algorithm because CPLEX failed to report a decent solution before the time limit is elapsed. This experiment consists of two parts. The ﬁrst part is carried out by considering the relative values of the acceleration coefﬁcients used in PSO. Three settings are of interest, namely, ðc1;c2Þ ¼ ð1; 1Þ; ðc1;c2Þ ¼ ð1; 2Þ, and

ðc1;c2Þ ¼ ð2; 1Þ. The three settings address the relative importance

of the two parameters. In each of the three settings, another PSO coefﬁcient

x

varies within f0:1; 0:2; . . . ; 1:0g for all tests. The second part is designed to compare the solutions provided by the proposed PSO algorithm and the two heuristics.

The settings of the computational experiments are summarized inTable 2. Settings E1;E2and E3are designed to examine the

per-formances of the proposed PSO under the three combinations of the values of ðc1;c2Þ. For each of the setting, instances were

gener-ated with different inertia weights, different numbers of machines and different numbers of jobs. In the setting E4, we compare the

best solution yielded in among E1;E2and E3with the solutions

pro-duced by the two heuristics. For further contrast, we conducted setting E5to compare the worst solutions yielded in among E1;E2

and E3with the solutions produced by the two heuristics.

Since a certain kind of randomness exists in the execution of the PSO algorithm, in each scenario of the experiment we invoked the PSO Algorithm 10 times and computed the average values.

Tables 3–5summarize the computational results. The tables are headlined with the number of jobs and the number of machines. For example, ‘‘30-3’’ indicates that n ¼ 30 jobs were to be dis-patched to m ¼ 3 machines. The number in each cell was obtained through 10 repetitions of the PSO algorithm. The effectiveness of the PSO algorithm is measured by the average objective function values reported. Each row corresponds to a speciﬁc value of the due-date range control parameter, R. The performance of the PSO algorithm under a speciﬁc inertia

x

value is indicated by two val-ues, namely, the average objective function values (obj.) and the

Table 5

Results of experiment E3;ðc1;c2Þ ¼ ð2; 1Þ.

R x¼ 0:2 x¼ 0:4 x¼ 0:6 x¼ 0:8 x¼ 1:0

30-3 0.1 6944.8 0.00 8709.4 25.41 9984.5 43.77 10402.1 49.78 11691.4 68.35 0.2 7108.9 0.00 7853.6 10.48 9737.0 36.97 9760.8 37.30 10896.9 53.29 0.3 6500.5 0.00 8287.5 27.49 9046.6 39.17 10248.0 57.65 12068.8 85.66 0.4 7127.1 0.00 9075.5 27.34 8627.4 21.05 10246.0 43.76 11741.1 64.74 0.5 7300.0 0.00 7602.3 4.14 9027.4 23.66 10722.4 46.88 11864.6 62.53 0.6 6856.8 0.00 8723.8 27.23 8802.9 28.38 10901.3 58.99 11466.3 67.23 0.7 7454.4 0.00 9084.7 21.87 9081.8 21.83 11310.0 51.72 11530.0 54.67 0.8 6773.0 0.00 8152.6 20.37 9251.9 36.60 10260.7 51.49 11481.0 69.51 0.9 6285.8 0.00 8241.5 31.11 9578.0 52.38 11194.7 78.10 12039.4 91.53 1.0 6415.2 0.00 8440.8 31.58 9601.7 49.67 11708.2 82.51 12431.0 93.77 30-5 0.1 15285.3 0.00 20242.0 32.43 22697.3 48.49 25301.6 65.53 28799.5 88.41 0.2 14189.8 0.00 20958.2 47.70 24526.9 72.85 27800.2 95.92 30344.8 113.85 0.3 14892.6 0.00 21962.9 47.48 24431.5 64.05 25342.4 70.17 30147.6 102.43 0.4 14788.2 0.00 19996.9 35.22 22495.5 52.12 25410.3 71.83 28171.3 90.50 0.5 14537.0 0.00 20785.7 42.98 25759.3 77.20 28312.1 94.76 29065.0 99.94 0.6 14654.6 0.00 19832.1 35.33 24195.9 65.11 26564.4 81.27 28412.8 93.88 0.7 14585.6 0.00 19123.8 31.11 24409.8 67.36 26309.6 80.38 27080.5 85.67 0.8 13251.8 0.00 21694.6 63.71 25629.5 93.40 26922.8 103.16 27723.1 109.20 0.9 14816.4 0.00 19963.3 34.74 27867.8 88.09 25025.5 68.90 28547.5 92.68 1.0 13620.7 0.00 19826.2 45.56 23538.0 72.81 27699.4 103.36 27597.0 102.61 50-3 0.1 6744.7 0.00 7268.5 7.77 8462.6 25.47 9552.8 41.63 10091.9 49.63 0.2 6384.0 0.00 8718.3 36.56 8547.4 33.89 9126.5 42.96 9957.6 55.98 0.3 6428.4 0.00 8028.8 24.90 8591.2 33.64 9592.9 49.23 10847.6 68.74 0.4 6425.0 0.00 7425.1 15.57 8125.1 26.46 8395.7 30.67 9001.5 40.10 0.5 6617.3 0.00 7421.1 12.15 9238.1 39.61 9647.7 45.80 9671.5 46.15 0.6 6553.7 0.00 7851.6 19.80 8202.1 25.15 9010.1 37.48 10232.0 56.13 0.7 5977.5 0.00 7376.5 23.40 8376.6 40.14 9448.2 58.06 11014.5 84.27 0.8 6400.3 0.00 7559.9 18.12 9270.7 44.85 9870.7 54.22 9988.8 56.07 0.9 6529.6 0.00 7313.7 12.01 8683.1 32.98 9401.2 43.98 9416.8 44.22 1.0 5736.9 0.00 7196.6 25.44 9193.6 60.25 9986.1 74.07 10565.8 84.17 50-5 0.1 12988.5 0.00 16884.1 29.99 19685.3 51.56 20124.6 54.94 21914.0 68.72 0.2 12449.2 0.00 16390.5 31.66 21152.1 69.91 21149.4 69.89 21785.2 74.99 0.3 12044.0 0.00 18162.3 50.80 21117.5 75.34 20646.5 71.43 20846.6 73.09 0.4 12314.6 0.00 16696.2 35.58 19767.7 60.52 20158.5 63.70 22120.2 79.63 0.5 12833.9 0.00 17019.1 32.61 20917.3 62.98 22050.7 71.82 23112.4 80.09 0.6 11383.3 0.00 17036.0 49.66 20099.1 76.57 22960.8 101.71 20466.7 79.80 0.7 13439.8 0.00 17052.6 26.88 21425.4 59.42 21653.8 61.12 20896.2 55.48 0.8 12306.6 0.00 17830.2 44.88 19534.2 58.73 21405.2 73.93 22155.1 80.03 0.9 13089.4 0.00 17928.2 36.97 19512.7 49.07 23614.3 80.41 22567.5 72.41 1.0 12612.8 0.00 16006.3 26.91 20281.5 60.80 21972.3 74.21 23446.8 85.90

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percentage deviation from the best objective function values in this row (dev.%). The percentage deviation (dev.%) of the PSO algorithm with the inertia weight

x

0_{2 f0:2; 0:3; . . . ; 1:0g is deﬁned as} objðPSOx0Þ min_x_{2f0:2;...;1:0g}fobjðPSO_xÞg

objðPSOx0Þ 100%:

The execution of the PSO algorithm was terminated once either 1,000 iterations were reached or the speciﬁed convergence status was encountered.

The results of the setting ðc1;c2Þ ¼ ð1; 1Þ are summarized in

Ta-ble 3. Through each combination of due date variable R, the number of jobs and the number of machines, we investigate the inﬂuence caused by the different values of the PSO coefﬁcient

x

. In terms of the average objective function values, we can ﬁnd that in most of the runs, the best solutions are obtained at

x

¼ 0:6. In two runs of 30-5, three runs of 50-3 and six runs of 50-5, the PSO algorithm with

x

¼ 0:4 delivered the minimum objective function values. This suggests that setting the inertia weight

x

around 0:5 can better exhibit the strength of the proposed PSO algorithm. On the other hand, for most of the cases, the inferior solutions were encountered when the inertia weight

x

is 1:0. Moreover, in one run of 30-3 and three runs of 50-5, the setting

x

¼ 0:8 leads to the maximum objec-tive function values. Larger inertia weight indicates larger the adop-tion of previous velocity when determining the new velocity. This highlights the phenomenon that each particle moves indepen-dently by it own velocity, resulting in a type of straight-line routes. Diversity required for probing different areas is diminished and thus the solution quality is unfavorable.

Table 4reports the results of the experiment framework E2. The

PSO coefﬁcients ðc1;c2Þ ¼ ð1; 2Þ reﬂect that the global best

solu-tions have double impact on the new velocity of each particle than the local optimal solution of the particle itself. From the numerical values, we can easily see that the setting

x

¼ 0:2 leads to favorable solutions with the minimum objective function values, yet the set-ting

x

¼ 1:0 again results in the worst performance in most of the tests. Especially, for the cases of 30-3, 30-5 and 50-3, a clear trend shows that the average objective function values increase as

x

val-ues become larger.

The results of setting E3are summarized inTable 5. The PSO

coefﬁcients are ðc1;c2Þ ¼ ð2; 1Þ to reenforce the impact of local

optimal solutions of the particles. For all test cases, the best solu-tions were reported by the setting with

x

¼ 0:2. In most of the cases, the setting

x

¼ 1:0 still leads to the worst solutions. In the runs of 50-5 with R ¼ 0:6, the results of

x

¼ 0:8 and

x

¼ 1:0 are rather inferior.

The second part of the computational study is to investigate the relative performances of the PSO algorithm and the two heuristics. InTable 6, each row shows the objective function values derived by the two heuristics and the best results delivered by the PSO algo-rithm under three different settings of c1and c2. The numerical

re-sults clearly show the superiority of heuristic HEDD over heuristic

HWSPT. The PSO algorithm under different settings can produce

schedules whose objective function values are much smaller than those produced by the heuristics. One of the most major reasons behind the results could be the following observation: In the heu-ristics, once a job is assigned to a machine at a favorable processing speed, no adjustment is allowed during the remaining execution course of the heuristics. For the jobs that are prone to be early, they will be processed at the lowest possible speeds so as to save the power cost. This policy on the other hand may induce inevitable tardiness to the unscheduled jobs.

InTable 7, we show a comparison among two heuristics and the worst results delivered by the PSO algorithm under three different settings of c1and c2. In the test of instances with 30 jobs and 50

jobs, the PSO algorithm even performed at least 11.27% and

43.37% better than two heuristics. Also the parameter setting ðc1;c2Þ ¼ ð2; 1Þ indices a higher probability of producing inferior

solutions in the 30-job instances. However, for the 50-job problem instances, the parameter settings, ðc1;c2Þ ¼ ð1; 1Þ and

ðc1;c2Þ ¼ ð1; 2Þ are prone to be unfavorable.

For each job-machine combination as displayed by columns of Table 8, we list the best objective function values and the param-eter settings of ðc1;c2;

x

Þ corresponding to these results. The

experiment setting with

x

¼ 0:2, which performs better than oth-ers in most of experiments, is suggested to applied as the number of machines and jobs increase. Due to the fact that the random va-lue of the coefﬁcients of parameters c1and c2may affect the

pro-cess of evolutionary solution signiﬁcantly. The setting of parameters c1and c2are hard to justify which one is better than

another. However, the experiment results suggest that using differ-ent values of c1and c2can obtain better solutions than using the

same values.

Only under the case of 30 jobs with 3 machines, the PSO algorithm achieves convergence with less than 1,000 iterations. For the rest of job-machine combinations, the objective function values are the incumbent best solutions while achieving 1000

Table 6

Comparison of the best solutions of PSO with the solutions of two heuristics. ðc1;c2Þ (1, 1) (1, 2) (2, 1) HEDD HWSPT

R min min min Obj. Dev.

(%) Obj. Dev. (%) 30-3 0.1 7964.7 7004.1 6944.8 16085.6 131.62 24478.6 252.47 0.2 7486.1 7344.8 7108.9 14739.4 107.34 22374.9 214.74 0.3 7402.4 6878.1 6500.5 14045.1 116.06 23886.4 267.45 0.4 7014.3 7393.3 7127.1 13922.5 98.49 23505.2 235.10 0.5 7286.2 7238.5 7300.0 13937.5 92.55 24019.1 231.82 0.6 6685.3 7411.5 6856.8 14479.6 116.59 24459.2 265.87 0.7 6920.3 7507.7 7454.4 15684.6 126.65 24715.5 257.14 0.8 7642.2 6940.9 6773.0 14750.3 117.78 23147.9 241.77 0.9 7289.2 6451.5 6285.8 14942.8 137.72 25299.5 302.49 1.0 7117.4 6554.6 6415.2 15122.2 135.72 24785.1 286.35 30-5 0.1 7964.7 7004.1 6944.8 36350.9 423.43 59237.7 752.98 0.2 7486.1 7344.8 7108.9 34392.9 383.80 59390.6 735.44 0.3 7402.4 6878.1 6500.5 39033.6 500.47 62421.5 860.26 0.4 7014.3 7393.3 7127.1 33507.6 377.70 59850.1 753.26 0.5 7286.2 7238.5 7300.0 34395.3 375.17 59654.2 724.12 0.6 6685.3 7411.5 6856.8 36756.6 449.81 59566.6 791.01 0.7 6920.3 7507.7 7454.4 36260.4 423.97 63760.2 821.35 0.8 7642.2 6940.9 6773.0 35852.2 429.34 57949.2 755.59 0.9 7289.2 6451.5 6285.8 38218.3 508.01 57064.7 807.84 1.0 7117.4 6554.6 6415.2 34093.6 431.45 61600.0 860.22 50-3 0.1 7964.7 7004.1 6944.8 17224.2 148.02 25808.6 271.62 0.2 7486.1 7344.8 7108.9 19863.2 179.41 29136.8 309.86 0.3 7402.4 6878.1 6500.5 17520.9 169.53 28814.5 343.27 0.4 7014.3 7393.3 7127.1 16830.5 139.95 27462.5 291.52 0.5 7286.2 7238.5 7300.0 18825.5 160.07 26371.7 264.33 0.6 6685.3 7411.5 6856.8 18963.6 183.66 28098.1 320.30 0.7 6920.3 7507.7 7454.4 15791.2 128.19 26303.3 280.09 0.8 7642.2 6940.9 6773.0 18136.4 167.77 27139.1 300.70 0.9 7289.2 6451.5 6285.8 17398.3 176.79 27118.7 331.43 1.0 7117.4 6554.6 6415.2 16978.0 164.65 27588.2 330.04 50-5 0.1 7964.7 7004.1 6944.8 51831.3 646.33 83221.2 1098.32 0.2 7486.1 7344.8 7108.9 52488.1 638.34 77014.6 983.35 0.3 7402.4 6878.1 6500.5 48300.9 643.03 73473.5 1030.27 0.4 7014.3 7393.3 7127.1 50368.2 618.08 80980.7 1054.51 0.5 7286.2 7238.5 7300.0 45309.8 525.96 73926.1 921.29 0.6 6685.3 7411.5 6856.8 49253.7 636.75 78208.4 1069.86 0.7 6920.3 7507.7 7454.4 50374.8 627.93 80297.6 1060.32 0.8 7642.2 6940.9 6773.0 50327.1 643.05 80561.8 1089.46 0.9 7289.2 6451.5 6285.8 51171.0 714.07 79596.4 1166.29 1.0 7117.4 6554.6 6415.2 50185.4 682.29 81320.5 1167.62

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iterations. The PSO algorithm produced better solutions when

x

¼ 0:2 (34 of the 40 cases as shownTable 7). Moreover, for the cases of 30 jobs on 5 machines, 50 jobs on 3 machines and 50 jobs on 5 machines, the PSO parameters ðc1;c2;

x

Þ are either (1, 2, 0.2) or

(2, 1, 0.2), which implies that distinct values of c1and c2achieves

better solutions than the setting with identical values of c1and c2.

Here we summarize the experiment results. The comparison results between CPLEX and the two heuristics on small-size

Table 7

Comparison of the worst solutions of PSO with the solutions of two heuristics.

ðc1;c2Þ (1, 1) (1, 2) (2, 1) HEDD HWSPT

R max max max Obj. Dev. (%) Obj. Dev. (%)

30-3 0.1 10432.2 11543.5 11691.4 16085.6 37.58 24478.6 109.37 0.2 9927.5 10565.7 10896.9 14739.4 35.26 22374.9 105.33 0.3 11394.7 12537.8 12068.8 14045.1 12.02 23886.4 90.52 0.4 10416.5 11313.5 11741.1 13922.5 18.58 23505.2 100.20 0.5 10605.9 11726.7 11864.6 13937.5 17.47 24019.1 102.44 0.6 10287.3 11401.3 11466.3 14479.6 26.28 24459.2 113.31 0.7 10535.1 12250.4 11530.0 15684.6 28.03 24715.5 101.75 0.8 10367.4 11349.2 11481.0 14750.3 28.48 23147.9 101.62 0.9 10487.6 11965.1 12039.4 14942.8 24.12 25299.5 110.14 1.0 11140.0 12755.7 12431.0 15122.2 18.55 24785.1 94.31 30-5 0.1 27369.0 28996.9 28799.5 36350.9 25.36 59237.7 104.29 0.2 29842.0 30910.3 30344.8 34392.9 11.27 59390.6 92.14 0.3 29639.2 28779.5 30147.6 39033.6 29.47 62421.5 107.05 0.4 27584.0 27525.2 28171.3 33507.6 18.94 59850.1 112.45 0.5 29014.3 28226.4 29065.0 34395.3 18.34 59654.2 105.24 0.6 27609.9 27360.0 28412.8 36756.6 29.37 59566.6 109.65 0.7 27689.0 27453.0 27080.5 36260.4 30.96 63760.2 130.27 0.8 26896.4 28878.3 27723.1 35852.2 24.15 57949.2 100.67 0.9 28124.9 27525.3 28547.5 38218.3 33.88 57064.7 99.89 1.0 28532.0 28881.6 27699.4 34093.6 18.05 61600.0 113.28 50-3 0.1 10655.7 9757.2 10091.9 17224.2 61.64 25808.6 142.20 0.2 9701.1 10457.8 9957.6 19863.2 89.94 29136.8 178.61 0.3 11174.8 10804.4 10847.6 17520.9 56.79 28814.5 157.85 0.4 9450.2 9675.1 9001.5 16830.5 73.96 27462.5 183.85 0.5 9529.0 9504.0 9671.5 18825.5 94.65 26371.7 172.67 0.6 10197.6 9805.0 10232.0 18963.6 85.34 28098.1 174.61 0.7 10624.2 10807.6 11014.5 15791.2 43.37 26303.3 138.81 0.8 10281.9 10063.2 9988.8 18136.4 76.39 27139.1 163.95 0.9 9568.2 10263.9 9416.8 17398.3 69.51 27118.7 164.21 1.0 10787.4 10999.1 10565.8 16978.0 54.36 27588.2 150.82 50-5 0.1 21936.7 22119.9 21914.0 51831.3 134.32 83221.2 276.23 0.2 21361.2 21849.0 21785.2 52488.1 140.23 77014.6 252.49 0.3 22804.7 22342.5 21117.5 48300.9 111.80 73473.5 222.19 0.4 21537.0 22200.1 22120.2 50368.2 126.88 80980.7 264.78 0.5 23236.4 22606.8 23112.4 45309.8 94.99 73926.1 218.15 0.6 22144.2 23537.7 22960.8 49253.7 109.25 78208.4 232.27 0.7 21991.1 21405.2 21653.8 50374.8 129.07 80297.6 265.14 0.8 23523.7 24135.3 22155.1 50327.1 108.52 80561.8 233.79 0.9 23087.1 21917.0 23614.3 51171.0 116.69 79596.4 237.07 1.0 23003.1 23612.7 23446.8 50185.4 112.54 81320.5 244.39 Table 8

Best parameter settings of PSO.

R 30-3 30-5 50-3 50-3

Obj. ðc1;c2;xÞ Obj. ðc1;c2;xÞ Obj. ðc1;c2;xÞ Obj. ðc1;c2;xÞ

0.1 6944.8 (2, 1, 0.2) 15285.3 (2, 1, 0.2) 6650.3 (1, 1, 0.6) 12909.4 (1, 2, 0.2) 0.2 7108.9 (2, 1, 0.2) 14189.8 (2, 1, 0.2) 6384.0 (2, 1, 0.2) 11858.5 (1, 2, 0.2) 0.3 6500.5 (2, 1, 0.2) 14892.6 (2, 1, 0.2) 6427.6 (1, 2, 0.2) 12044.0 (2, 1, 0.2) 0.4 7014.3 (1, 1, 0.6) 14450.6 (1, 2, 0.2) 6189.5 (1, 1, 0.6) 12201.6 (1, 2, 0.2) 0.5 7238.5 (1, 2, 0.4) 14537.0 (2, 1, 0.2) 6617.3 (2, 1, 0.2) 12215.9 (1, 2, 0.2) 0.6 6685.3 (1, 1, 0.6) 14654.6 (2, 1, 0.2) 6416.2 (1, 2, 0.2) 11383.3 (2, 1, 0.2) 0.7 6920.3 (1, 1, 0.6) 14585.6 (2, 1, 0.2) 5977.5 (2, 1, 0.2) 13439.8 (2, 1, 0.2) 0.8 6773.0 (2, 1, 0.2) 13251.8 (2, 1, 0.2) 6341.8 (1, 2, 0.2) 12306.6 (2, 1, 0.2) 0.9 6285.8 (2, 1, 0.2) 14448.9 (1, 2, 0.2) 6466.0 (1, 2, 0.2) 12742.9 (1, 2, 0.2) 1.0 6415.2 (2, 1, 0.2) 13620.7 (2, 1, 0.2) 5736.9 (2, 1, 0.2) 12612.8 (2, 1, 0.2)

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instances show that the performance of the heuristics is highly re-lated to the machine speeds of the test instances. CPLEX needs more than 60 min to solve even the small-size problem instances. The experiment has already demonstrated the proposed PSO algorithm could obtain quality results but took more computing times, when compared to the heuristics. The computing time re-quired by PSO algorithms could be acceptable if the beneﬁts of less tardiness penalty and lower power cost can be realized. Another important point to highlight is that cloud computing systems usually have more than thousands of computers in the real world. Taking the effectiveness and the efﬁciency of scheduling simulta-neously in the context of cloud computing, the proposed PSO algo-rithm can be considered as a viable solution approach.

7. Conclusions

This paper investigated a scheduling problem with heteroge-neous unrelated parallel machines to minimize the weighted sum of tardiness penalty and power consumption cost. The power consumption cost varies due to the ﬂexibility of adjustment of CPU frequencies (dynamic voltage scaling) for processing the jobs. Jobs executed at a higher machine speed for time saving incurs more energy cost. We formulated an integer linear programming model. Since the studied scheduling problem is NP-hard in the strong sense, we designed two heuristics and a PSO algorithm to produce approximate solutions. A major contribution of the PSO design is the encoding scheme proposed to represent a schedule as a parti-cle. We carried out a computational study to assess the perfor-mances of the proposed algorithms and using CPLEX to solve the studied problem. The two heuristics and the PSO algorithm under distinct settings of PSO coefﬁcients were tested for larger-size data of the problems. Statistics showed that the PSO algorithm can pro-vide quality solutions in an acceptable time. The proposed encod-ing scheme of the PSO algorithm can be easily applied to other processing contexts with adjustable machine speeds.

For further research, it could be interesting to extend the pro-posed approach to dealing with other performance criteria. Further attempts to use Pareto front based method to evaluate the fitness. Some interesting metrics as hyper-volume are very suitable for multi-objective optimization based on Pareto front (Zitzler & Thi-ele, 1998). We can also introduce the concept of CPU frequency adjustment to the traditional machine scheduling problems by allowing speed adjustment during the execution of each individual job. Moreover, applying the proposed encoding method to more complex machine environments might be an interesting and chal-lenging direction. Machine eligibility is an example in which not all but only specific machines are eligible for processing each specific job. This may introduce a type of complexity in the manipulation of particles. It is also worthwhile to design other versions of PSO algo-rithms to continue pursuing the best performance in solving flow-shop scheduling problems. Hybrid PSO algorithms with other metaheuristics (Xia & Wu, 2005; Zhang, Shao, Li, & Gao, 2009) or integrating PSO algorithms with discrete Lagrange multipliers (Nezhad & Mahlooji, 2011) or designing new PSO features, e.g. dynamically adjusting acceleration coefficients as the course of iterations continues, are other possible research directions. Acknowledgements

This research was partially supported by the National Science Council of Taiwan under Grant NSC-100-2410-H-009-MY2. The authors are also grateful to the reviewers for their valuable comments.

References

Ai, J., & Kachitvichyanukul, V. (2009). Particle swarm optimization and two solution representations for solving the capacitated vehicle routing problem. Computers & Industrial Engineering, 56(1), 380–387.

Akgul, Y., Puschini, D., Lesecq, S., Miro-Panades, I., Benoit, P., & Torres L. et al. (2012). Power mode selection in embedded systems with performance constraints. In Proceedings of the faible tension faible consommation conference. IEEE, June 6–8 (pp. 1–4).

Anghinoﬁ, D., & Paolucci, M. (2007). Parallel machine total tardiness scheduling with a new hybrid metaheuristic approach. Computers & Operations Research, 34(11), 3471–3490.

Baker, KR. (1974). Introduction to sequencing and scheduling. New York: Wiley. Bunde, DP. (2006). Power-aware scheduling for makespan and ﬂow. Journal of

Scheduling, 12(5), 489–500.

Du, J., & Leung, JY. (1990). Minimizing total tardiness on one machine is NP-hard. Mathematics of Operations Research, 15(3), 483–495.

Dye, CY., & Ouyang, LY. (2011). A particle swarm optimization for solving joint pricing and lot-sizing problem with ﬂuctuating demand and trade credit ﬁnancing. Computers & Industrial Engineering, 60(1), 127–137.

Garey, MR., & Johnson, DS. (1978). ‘‘Strong’’ NP-completeness results: Motivation, examples, and implications. Journal of the Association for Computing Machinery, 25(3), 499–508.

Ge, R., Feng, X., & Cameron, K. W. (2005). Performance-constrained distributed DVS scheduling for scientiﬁc applications on power-aware clusters. In Proceedings of the ACM/IEEE conference on supercomputing, November 12–18, Seattle, Washington (pp. 34–44).

Graham, RL. (1966). Bounds for certain multiprocessing anomalies. Bell System Technical Journal, 45, 1563–1581.

Hall, NG., & Posner, ME. (2000). Generating experimental data for computational testing with machine scheduling applications. Operations Research, 49(6), 854–865.

Jackson, J. R. (1955). Scheduling a production line to minimize maximum tardiness. Research report 43. Management Science Research Project, UCLA.

Kashan, A., & Karimi, B. (2009). A discrete particle swarm optimization algorithm for scheduling parallel machines. Computers & Industrial Engineering, 56(1), 216–223.

Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings of IEEE international conference on neural networks (pp. 1942–1948).

Koomey, J. G. (2007). Estimating total power consumption by servers in the US and the world. <http://enterprise.amd.com/Downloads/svrpwrusecompleteﬁnal.pdf>. Kumar, P. R., & Palani, S. (2012). A dynamic voltage scaling with single power supply

and varying speed factor for multiprocessor system using genetic algorithm. In Proceedings of the international conference on pattern recognition, informatics and medical engineering, March 21–23 (pp. 342–346).

Lawler, E. L. (1977). A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness. Annals of Discrete Mathematics, 1, 331–342.

Lenstra, J. K., Rinnooy Kan, A. H. G., & Brucker, P. (1977). Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1, 343–362.

Nezhad, A. M., & Mahlooji, H. (2011). A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems. Computers & Operations Research, 38(8), 1164–1174.

Niu, Q., Zhou, T., & Wang, L. (2010). A hybrid particle swarm optimization for parallel machine total tardiness scheduling. International Journal of Advanced Manufacturing Technology, 49, 723–739.

Önüt, S., Tuzkaya, R. U., & Dog¨aç, B. (2008). A particle swarm optimization algorithm for the multiple level warehouse layout design problem. Computers & Industrial Engineering, 54(4), 783–799.

Rizvandi, N. B., Zomaya, A. Y., Lee, Y. C., Boloori, A. J., & Taheri, J. (2012). Multiple frequency selection in DVFS-enabled processors to minimize energy consumption. arXiv:1203.5160v1, ed.

Runwei, C., Mitsuo, G., & Tatsumi, T. (1995). Minmax earliness/tardiness scheduling in identical parallel machine system using genetic algorithms. Computers & Industrial Engineering, 29(1), 513–517.

Smith, WE. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3, 59–66.

Xia, W., & Wu, Z. (2005). An effective hybrid optimization approach for multi-objective ﬂexible job-shop scheduling problems. Computers & Industrial Engineering, 48(2), 409–425.

Zhang, GH., Shao, XY., Li, PG., & Gao, L. (2009). An effective hybrid particle swarm optimization algorithm for multi-objective ﬂexible job-shop scheduling problem. Computers & Industrial Engineering, 56(4), 1309V1318.

Zhu, D., Melhem, R., & Childers, BR. (2003). Scheduling with dynamic voltage/speed adjustment using slack reclamation in multiprocessor real-time systems. IEEE Transaction on Parallel and Distributed Systems, 14(7), 686–700.

Zhu, D., Mosse, D., & Melhem, R. (2004). Power-aware scheduling for AND/OR graphs in real-time systems. IEEE Transactions on Parallel and Distributed Systems, 15(9), 849–864.

Zitzler, E., & Thiele, L. (1998). Multiobjective optimization using evolutionary algorithms – A comparative case study. In Proceedings of the 5th conference on parallel problem solving from nature (PPSN5). Lecture notes in computer science (Vol. 1498, pp. 292–301). Springer Verlag.