A vibration damping optimization algorithm for a

Applied Mathematical Modelling xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times Esmaeil Mehdizadeh a,⇑, Reza Tavakkoli-Moghaddam b, Mehdi Yazdani a a b

Department of Industrial Engineering, Faculty of Industrial & Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 16 October 2013 Received in revised form 21 December 2014 Accepted 12 February 2015 Available online xxxx Keywords: Parallel machines scheduling problem Family setup times Vibration damping optimization Taguchi experimental design

a b s t r a c t Parallel machines scheduling problem is a branch of production scheduling, which is among the most difficult combinatorial optimization problems. This paper develops a meta-heuristic algorithm based on the concept of the vibration damping in mechanical vibration, called vibration damping optimization (VDO) algorithm for optimizing the identical parallel machine scheduling problem with sequence-independent family setup times. The objective function of this problem is to minimize the total weighted completion time. Furthermore, the Taguchi experimental design method is applied to set and estimate the appropriate values of the parameters required in our proposed VDO. We computationally compare the results obtained by the proposed VDO with the results of the genetic algorithm (GA) and branch-and-bound method. Consequently, the computational results validate the quality of the proposed algorithm. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction The main elements of machine scheduling problems are machine configuration, job characteristics, and the objective function. The machine configuration can be classified into single and multi-machine problems in a broad sense. A parallel machines scheduling problem can be referred as a class of scheduling problems that origins from the multi-machine scheduling problems [1]. Machines may be identical, uniform, or completely unrelated and have different speeds. Each job can be performed on any of the machines. The aim of this problem is to sequence a set of n jobs on m parallel machines in order to minimize the performance indicator. In parallel machines scheduling problems, jobs can be partitioned into F families (F P 1) according to their similarity. In this condition and when setup time constraint is considered, the problem has family setup time, in which a machine should be set up when switching from one family to others and there is no setup time between two jobs from the same family. The family setup time can be sequence-dependent or sequence-independent. It is a sequence-dependent family setup time if its duration depends on the families of both the current and the immediately preceding batches (i.e., set of jobs of the same family). It is a sequence-independent family setup time if its duration depends solely on the family of the current batch to be processed [2].

⇑ Corresponding author. Tel./fax: +98 28 33675784. E-mail address: [email protected] (E. Mehdizadeh). http://dx.doi.org/10.1016/j.apm.2015.02.027 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.

Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027

2

E. Mehdizadeh et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

Many researchers have studied parallel machines scheduling problems with family or non-family setup times over the last two decades. Allahverdi et al. [2] conducted a comprehensive review on these problems. To solve these problems, various approaches are presented in the literature. Some exact methods can be studied in Schutten and Leussink [3], Cheng and Kovalyov [4], Webster and Azizoglu [5], Blazewicz and Kovalyov [6], Azizoglu and Webster [7], Chen and Powell [8], Lin and Jeng [9], Dunstall and Wirth [10], Nessah et al. [11] and Shim and Kim [12]. Additionally, many studies have applied heuristic and meta-heuristic algorithms to solve parallel machines scheduling problems with family or non-family setup times. Park et al. [13] applied a neural network approach for a parallel machines scheduling problem with sequence-dependent setup times to minimize the sum of the weighted tardiness. Weng et al. [14] propounded seven heuristic algorithms for a problem of scheduling a set of independent jobs on unrelated parallel machines with job sequence-dependent setup times in such a way that the total weighted completion time is minimized. Yi and Wang [15] proposed a tabu search (TS) algorithm for minimizing the total flow time in a parallel machines scheduling problem with sequence-independent family setup times. Mendes et al. [16] addressed the parallel machine scheduling problem with sequence-dependent setup times, in which the minimization of the completion time was considered as objective function. They proposed two meta-heuristic algorithms for solving this problem. The first algorithm was a TS-based method and the second one was a memetic algorithm, which combined a population-based method with local search procedures. Eom et al. [17] presented a three-phase heuristic for parallel machines scheduling problem with sequence-dependent family setup times to minimize the total weighted tardiness. TS algorithm is used in the final phase of the algorithm. Kim et al. [18] proposed a simulated annealing (SA) algorithm for unrelated parallel machines scheduling problem with sequence-dependent setup times to minimize the total tardiness. Monch et al. [19] attempted to minimize the total weighted tardiness on parallel batch machines with incompatible job families and unequal ready times of jobs. They proposed two different decomposition approaches and applied genetic algorithm (GA) in both approaches. Abdekhodaee et al. [20] proposed greedy heuristics and a GA for solving a parallel machines scheduling problem with sequence-independent setup times where the minimization of the completion time was considered as objective function. Armentano and Filho [21] applied GRASP versions that incorporate adaptive memory principles for the problem of scheduling jobs on uniform parallel machines with sequence-dependent setup times to minimize the total tardiness related to job due dates. Logendran et al. [22] propounded six different search algorithms based on TS for minimizing the weighted tardiness of jobs in the unrelated parallel machines scheduling problem with sequence-dependent setups. Tavakkoli-Moghaddam and Mehdizadeh [23] presented an integer linear programming (ILP) model for an identical parallel machines scheduling problem with sequence-independent family setup times that minimizes the total weighted flow time. A meta-heuristic algorithm based on GA is applied to the given problem. Tavakkoli-Moghaddam et al. [24] proposed a GA for solving bi-objective (i.e., the number of tardy jobs and the total completion time of all the jobs) unrelated parallel machines scheduling problem with sequence-dependent setup times and precedence constraints. Behnamian et al. [25] presented a hybrid meta-heuristic algorithm to minimize the completion time in scheduling problems with parallel machines and sequence-dependent setup times and comprised three components, namely an initial population generation method based on an ant colony optimization (ACO), SA for solution evolution, and a variable neighborhood search (VNS) involving three local search procedures to improve the population. Driessel and Monsh [26] presented VNS approaches for scheduling jobs on parallel machines with sequence-dependent setup times, precedence constraints and ready times to minimize the total weighted tardiness. Chang and Chen [27] addressed the parallel machines scheduling problem with machine-dependent and sequence-dependent setup times where the minimization of the completion time was considered as objective function. They propounded a set of dominance properties including inter-machine (i.e., adjacent and non-adjacent interchange) and intra-machine switching properties as necessary conditions of job sequencing orders in an optimal schedule. In addition, a new meta-heuristic algorithm was introduced by integrating the dominance properties with GA to further improve the solution quality for larger problems. Vallada and Ruiz [28] presented a genetic algorithm for the unrelated parallel machine scheduling problem in which machine and job sequence dependent setup times are considered. The proposed genetic algorithm includes a fast local search and a local search enhanced crossover operator. Sarıcicek and Celik [29] proposed tabu search (TS) and simulated annealing (SA) meta-heuristics for identical parallel machine scheduling problem with the aim of minimizing the total tardiness of the jobs considering a job splitting property. Li et al. [30] presented a simulated annealing algorithm, named LPDT-SA for solving uniform parallel machine scheduling problem which is to minimize the maximum lateness. A heuristic algorithm LPDT is built to generate initial solutions. Lin et al. [31] considered an ant colony optimization (ACO) algorithm incorporating a number of new ideas (heuristic initial solution, machine reselection step, and local search procedure) to solve the problem of scheduling unrelated parallel machines to minimize total weighted tardiness. Ruiz-Torres et al. [32] propounded a set of list scheduling algorithms and simulated annealing meta-heuristics to solve a new unrelated parallel machine scheduling problem with deteriorating effect and the objective of makespan minimization. Wang et al. [33] investigated into the parallel machine scheduling problem with splitting jobs and objective of makespan minimization. Differential evolution was employed as a solution approach due to its distinctive feature, and a new crossover method and a new mutation method was brought forward in the global search procedure, according to the job splitting constraint. A specific local search method was further designed to gain a better performance, based on the analytical result from the single product problem. In this paper, we study the identical parallel machines scheduling problem with sequence-independent family setup times. The objective function of this problem is to minimize the total weighted completion time. This problem is noted P by three fields as P=ST si;b = W i C i where these fields describe the shop environment (i.e., P = parallel machines), setup time Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027


3

information (i.e., ST si;b = sequence-independent batch or family setup times) and objective to be minimized (i.e., P W i C i = total weighted completion time), respectively. Blazewicz and Kovalyov [6] proved the strong NP-hardness of the P P P=ST si;b = C i problem, where C i is the total completion time of jobs. Based on this proof, it can be said in strong sense that P the P=ST si;b = W i C i problem is NP-hard. The high level of the complexity of the mentioned problem demonstrates a cogent reason for using heuristic or meta-heuristic algorithms. In this paper, we develop a vibration damping optimization (VDO) algorithm for solving the studied problem. Furthermore, we attempt to prove the efficiency and effectiveness of our proposed VDO by comparing its performance with the branch-and-bound method and GA. Vibration damping optimization (VDO) algorithm is a new meta-heuristic algorithm and stochastic search method is inspired by the SA algorithm and created based on the concept of the vibration damping in mechanical vibration introduced by Mehdizadeh and Tavakkoli-Moghaddam [34]. They briefly reviewed the central constructs in combinatorial optimization and vibration damping and then developed the similarities between the two fields. Mehdizadeh et al. [35] presented a hybrid algorithm based on two meta-heuristic algorithms, VDO and GA for solving the stochastic capacitated locationallocation problem. For solving the model more efficiently, the simplex algorithm, stochastic simulation and the proposed hybrid algorithm are integrated in order to design a powerful hybrid intelligent algorithm. Mehdizadeh et al. [36] proposed a new mathematical model for the capacitated multi-facility location-allocation problem with probabilistic customer’s locations and fuzzy customer’s demands under the Hurwicz criterion. Their proposed model is formulated as the a-cost minimization model according to different criteria. A new hybrid intelligent algorithm was presented to solve the stochastic-fuzzy model. The proposed algorithm was based on the VDO, which is combined with the simplex algorithm and fuzzy simulation (SFVDO). Mousavi et al. [37] applied the VDO algorithm to solve a new mathematical model for the capacitated multi-facility location–allocation problem with probabilistic customers’ locations and demands into the frameworks of the expected value model (EVM) and the chance-constrained programming (CCP) based on two different distance measures. In order to solve the model, two hybrid intelligent algorithms were proposed, where the simplex algorithm and stochastic simulation are the bases for both algorithms. However, in the first algorithm, named SSGA, a special type of GA is combined and in the second one, SSVDO is united. Aliabadi et al. [38] developed a mixed-integer non-linear programming (MINLP) model for a flow shop production planning problem with a basic period policy and sequence-dependent setup times. Three meta-heuristics including hybrid particle swarm optimization (PSO), hybrid VDO algorithm and hybrid GA are used to determine the sequence and economic lot sizes of each item. Vibration damping optimization (VDO) was proposed by Mehdizadeh and Nezhad-Dadgar [39] to solve Resource constrained project scheduling problem with weighted earliness-tardiness penalties which is an interval due dates for activities. Finally, the proposed VDO algorithm is computationally compared with simulated annealing (SA) algorithm. In this paper, an identical parallel machines scheduling problem with sequence-independent family setup times is presented and a vibration damping optimization (VDO) algorithm is proposed for solving the model. The proposed VDO has used a simple neighborhood structure for creating a neighboring solution. To improve the performances of the VDO parameters [40], the Taguchi [41] experimental design method as a useful and new approach [42] has been employed [43,44]. Furthermore, the result of proposed VDO is compared with those of branch-and-bound method and GA on generated test problems. The rest of this paper is organized as follows. Section 2 presents the problem definition and the integer programming model for the given problem. Section 3 explains vibration damping natural phenomenon. In Section 4, a vibration damping optimization (VDO) algorithm for solving the studied problem is described. In Section 5, the Taguchi experimental design method is applied for tuning the parameters of the proposed VDO. Section 6 reports the computational results. Finally, the conclusion and some further research suggestions are presented in Section 7. 2. Mathematical model 2.1. Problem definition This ILP model is based on the ILP model that was presented by Tavakkoli-Moghaddam and Mehdizadeh [23]. The problem considered in this paper can be stated as follows: There are a set of identical machines M ¼ fM 1 ; M 2 ; . . . ; M m g in parallel, and a set of jobs J ¼ fJ 1 ; J 2 ; . . . ; J n g. All jobs are available at time zero with known integer-processing times, setup times and weights. Jobs are partitioned into F families (F P 1) according to their similarity, and each job is related to a family. For each P family f, there are nf jobs to be scheduled so that Ff¼1 nf ¼ n: A setup is required between two jobs from different families, and the family setup time is independent of the preceding family. A setup is also required prior to the processing of the first job on a machine. This is a typical environment during scheduling at the beginning of a new shift after machine downtime. The given problem is to schedule the identical parallel machines with sequence-independent family setup times in order to minimize the total weighted completion time. For a given schedule, the weighted completion time of a particular job is the product of its weight and job completion time, and the total weighted completion time of a schedule is the sum of the weighted completion time of all the jobs.


4


2.2. Integer programming model In this section, we present the integer linear programming (ILP) model for the identical parallel machines scheduling problem with sequence-independent family setup times. Minimizing the total weighted completion time is considered as objective function of this model. Following parameters and decision variables are used in the presented model. TWCT i, j k f, g n m o nf M Pi,f Sf Wi,f

Total weighted completion time Job indices, where job 0 is a dummy job that is always at the first position on a machine (i, j = 0, 1, . . ., n) Machine index (k = 1, 2, . . ., m) Family indices Number of jobs Number of identical parallel machines Number of families, (o 6 n) Number of jobs of family f Large positive number Processing time of job i from family f (f = 1, 2, . . ., F) Setup time of family f Weight of job i from family f 1 if f – g; and 0, otherwise Completion time of job i from family f 1, if job i from family f is assigned to machine k; and 0, otherwise 1 if job j from family g immediately follows job i from family f on machine k; and 0, otherwise 1 if job i from family f on machine k is the first in the queue; and 0, otherwise

ci;f ;j;g Ci,f Yi,f,k Xi,f,j,g,k X0,i,f,k

The proposed mathematical model is as follows:

Min TWCT ¼

n X W i;f C i;f ;

ð1Þ

i¼1

s.t. m X Y i;f ;k ¼ 1 8i; f

ð2Þ

k¼1

C i;f P Sf Y i;f ;k þ Pi;f Y i;f ;k

8i; f ; k

ð3Þ

C j;g P C i;f þ Pj;g þ Sg ci;f ;j;g Mð1 X i;f ;j;g;k Þ;

8i; j; f ; g; k; i – j; f – g

nf m X X X i;f ;j;g;k ¼ 1 8f ; j; g i¼1 j – i

ð4Þ

ð5Þ

k¼1

nf X X i;f ;j;g;k ¼ Y j;g;k

8f ; j; g; k

ð6Þ

8i; f ; g; k

ð7Þ

i¼1 j – i

ng X X i;f ;j;g;k 6 Y i;f ;k j¼1 j – i

nf X X 0;i;f ;k 6 1 8f ; k

ð8Þ

i¼1

Y i;f ;k ; X i;f ;j;g;k ; X 0;i;f ;k in f0; 1g; C i P 0;

8i; j; f ; g; k

ð9Þ

Eq. (1) represents the objective function minimizing the total weighted completion time. Eq. (2) states that each job from each family should be assigned to exactly one machine. Eq. (3) ensures that the completion time of a job from a family should be later or equal to its processing time and setup time. Eq. (4) guarantees that the completion time of a job should be later or equal to the completion time of its direct predecessor job in the sequence, and its processing and setup time (if setup is Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027


5

necessary). This constraint becomes redundant if jobs i and j are assigned to different machines. Eq. (5) ensures that a job should be processed at one and only one position on a machine. Eq. (6) states that job j should immediately follow other job on machine k if it is placed on this machine. Eq. (7) states that if job i, i – 0, is processed on machine k, it will be immediately followed by at most one another job on this machine. Eq. (8) ensures that only at most one job immediately follows the dummy job 0 on each machine. Eq. (9) states the properties of the decision variables. 3. Vibration damping Vibration can be considered to be the oscillation or repetitive motion of an object around an equilibrium position [45]. The equilibrium position is the object that position will attain when the force acting on it is zero. The vibration of an object is always caused by an excitation force. This force may be externally applied to the object, or it may originate inside the object. Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external forces are absent. Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances [46]. The reduction process of amplitude, which is tending to zero with elapsed time, is known as vibration damping process. The calculation of the natural frequencies is of major importance in the study of vibrations. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation of the natural frequencies is nearly made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance. All bodies possessing mass and elasticity are capable of vibration. When the motion is repeated in equal intervals of time s, it is called period motion. The simplest form of the periodic motion is the harmonic motion, which is often represented as the projection on a straight line of a point moving on a circle at constant speed. The displacement x can be written by:

x ¼ A sin xt;

ð10Þ

where A is the amplitude of oscillation and the quantity x is referred to as the angular frequency. From the free-body diagram, the Newton’s equation of motion is seen to be:

m€x þ cx_ þ kx ¼ FðtÞ:

ð11Þ

With the homogeneous equation, we have following equations:

€x þ cx_ þ x20 x ¼ 0;

ð12Þ

pffiffiffiffiffiffiffiffiffiffi where parameter c ¼ is the damping coefficient and x0 ¼ k=m is referred to as the natural angular frequency of oscillatory system. Upon the solution of the above equation, we obtain the following equation: c m

t

x ¼ Aec2 Cosðxt þ UÞ;

ð13Þ

where x is the amplitude of oscillation and U is the phase of the displacement with respect to the exciting force. Fig. 1 shows a damped single degree of the freedom system. The loss of energy from the oscillatory system results in the decay of amplitude of free vibration. The lost energy per cycle due to a damping force F d is computed from the general equation:

Ed ¼

Z

T

F d :dt

ð14Þ

0

Fig. 1. Impulse response of a damped single degree of the freedom system.


6


_ With the In this section, we consider the simplest case of energy dissipation. The damping force in this case is F d ¼ cx. steady-state displacement and velocity, we have

x ¼ A sinðxt UÞ;

ð15Þ

x_ ¼ xA cosðxt UÞ:

ð16Þ

The dissipated energy per cycle from Eq. (15) becomes:

Ed ¼

Z

T

2

cx2 A2 sin ðxt þ /Þdt ) Ed ¼ pcxA2 :

ð17Þ

0

The total energy of an oscillation system can be computed by the following summation:

ET ¼ EP þ EK ¼

1 2 2 kx þ 1=2mV 2 ¼ 1=2kA ; 2

ð18Þ

where Ep and Ek are the potential and kinetic energy of the system, respectively. Thus, we can obtain the total energy of an oscillation system with damping as follows:

E ¼ ET Ed ¼ 1=2 KA2 pcxA2 :

ð19Þ

A random vibration is one whose absolute value is not predictable at any point in time. As opposed to sinusoidal vibration, there is not a well-defined point at any time. The lack of periodicity is apparent. The instantaneous amplitude of a random vibration cannot be expressed mathematically as an exact function of time; it is possible to determine the probability of occurrence of particular amplitude on a statistical basis. Certain distributions that frequently occur in nature are the Gaussian (or Normal) distribution and the Rayleigh distribution [40], both of which can be expressed mathematically. Random variables restricted to positive value, such as the absolute value A of the amplitude, often tend to follow the Rayleigh distribution, which is defined by:

pðAÞ ¼

A

r2

2

eA

=2r2

;

A>0

ð20Þ

The probability density pðAÞ for A < 0 is zero. 4. Vibration damping optimization Vibration damping optimization (VDO) was initially proposed by Mehdizadeh and Tavakkoli-Moghaddam [34] which is based on the vibration damping process. A briefly review on similarities between the combinatorial optimization and vibration damping presented in [39]. Table 1 indicates analogy between the optimization problem and the vibration damping process. In order to implement the VDO algorithm for solving a problem, there are several principal choices that must be made. In this section, we describe our proposed VDO algorithm that is developed for optimizing the identical parallel machines with sequence-independent family setup times in order to minimize the total weighted completion time. Section 4 is subdivided into the following three subdivisions. Section 4.1 provides a solution representation method, and the neighborhood structure employed in the proposed VDO is explained in Section 4.2. In Section 4.3, a comprehensive description of the proposed VDO is given. 4.1. Solution representation To be dealt with all types of multiple machine scheduling problems, there are two essential issues, namely partition of jobs to machines and sequence jobs for each machine. In this paper, an extended solution representation method is suggested to encode the partition of jobs to machines and sequence jobs for each machine into a solution with n (i.e., number of jobs) columns and two rows. The first row represents all the possible permutations of jobs (or sequence of jobs). In each cell of this row, the first number signifies the number of

Table 1 Analogy between optimization problem and vibration damping process. Vibration damping process

Optimization problem

System states Energy Change of state Amplitude Vibration damping

Feasible solution Cost Neighboring solution Control parameter Heuristic solution



7

job and the second number characterizes a number of job families. The second row indicates the partition of jobs to machines. Let us consider a simple example with three jobs (J1, J2, J3) two families (f1, f2) and two machines (M1, M2) where jobs J1 and J2 belonging to family f1, and job J3 belonging to family f2. One schedule for this example is provided in Fig. 2. A solution representation of this schedule is also shown in Fig. 3. 4.2. Neighborhood structure The main purpose of applying a neighborhood structure is to produce a neighboring solution from the current solution by making some changes in it. A variety of neighborhood structures have been applied to scheduling problems. These neighborhood structures should work so that they prevent any infeasible solution. In this paper, we use a simple neighborhood structure for creating a neighboring solution. This operator selects one job among the existing jobs and then produces the random number in range (0, 1). If the produced random number is less than 0.5, the operator changes the position of the selected job on the assigned machine (i.e., changing in the sequence of jobs. Two random positions are selected and then changed the position of the selected job on the assigned machine. In this case, the operator alters the job order for the machine as shown in Fig. 4(a).). Otherwise, the operator changes the assigned machine of the selected job (i.e., changing in the partition of jobs to machines. In this case, the operator alters both job order and job partition to machines for the solution representation as shown in Fig. 4(b).). 4.3. Proposed VDO algorithm The VDO algorithm consists of a sequence of iterations. Each iteration consists a randomly changing the current solution to create a new solution in the neighborhood of the current solution. The neighborhood is defined by the choice of the generation mechanism. Once a new solution is created the corresponding change in the cost function is computed to decide whether the newly produced solution can be accepted as the current solution. If the change in the cost function is negative, the newly produced solution is directly taken as the current solution. Otherwise, it is accepted according to Raleigh’s probability (pðAÞ). If the difference between the cost function value of the current and the newly produced solutions is equal to or larger than zero, a random number r in (0, 1) is generated from a uniform distribution and if r 6 pðAÞ, then the newly produced solution is accepted as the current solution. If not, the current solution is unchanged. The general process of our proposed VDO algorithm is presented in Fig. 5.

M2

(J1 , f1)

M1

(J3 , f2)

(J2 , f1)

Fig. 2. One schedule for the example with three jobs, two families and two machines.

(3 2) 1

(2 1) 1

(1 1) 2

Fig. 3. Solution representation for the schedule of Fig. 2.

(1 3) 1

(1 5) 1

(3 4) 1

(1 5) 1

Solution (3 4) (2 1) (3 6) 1 2 2 New solution (1 3) (2 1) (3 6) 1 2 2

(2 2) 3

(1 7) 3

(2 2) 3

(1 7) 3

(a) (1 3) 1

(1 5) 1

(1 3) 1

(3 6) 1

Solution (3 4) (2 1) (3 6) 1 2 2 New solution (3 4) (2 1) (1 5) 1 2 2

(2 2) 3

(1 7) 3

(2 2) 3

(1 7) 3

(b) Fig. 4. (a) The operator changes the position of the selected job on the assigned machine; (b) the operator changes the assigned machine of the selected job.


8


Step 1: Generating a feasible initial solution. Step 2: Initializing the algorithm parameters, which consist of: initial amplitude (A0), maximum iteration at each amplitude (L), damping coefficient (γ), and standard deviation (σ). Finally, parameter t is set in one (t=1) Step 3: Calculating the objective value U0 for the initial solution. Step 4: Initializing the internal loop. In this step, the internal loop is carried out for l =1 and repeat while l < L. Step 5: Neighborhood generation. In this paper, we use mutation. Step 6: Accepting the new solution Set Δ = U − U 0 Now, if ∆ < 0, accept the new solution, else if ∆ > 0 generate a random number r between (0, 1); If

⎛ −A 2 ⎞ , then accept a new solution; otherwise, reject the new solution and accept the r < 1 − exp ⎜ t ⎟ ⎜ ⎟ ⎜ 2σ 2 ⎟ ⎝ ⎠

previous solution. If l > L, then t +1 → t and go to Step 7; otherwise l +1 → l and go back to Step 5. Step 7: Adjusting the amplitude. In this step, A = A exp( −γ t ) is used for reducing amplitude at each t 0 2 iteration of the outer cycle of the algorithm. If At = 0 return to Step 8; otherwise, go back to Step 4. Step 8: Stopping criteria. In this step, the proposed algorithm will be stopped after pre-specified minimum amplitude

Amin

is achieved. At the end, the best solution is obtained.

Fig. 5. The proposed VDO algorithm.

There are many rules for the stopping condition in meta-heuristic algorithms that depends on the problem at hand. Some of them for the VDO algorithm can be as follows: – Maximum number of iterations. – Reaching to the point that the iteration will not improve the quality solution afterwards. – Reaching to minimum amplitude Amin . In this paper, the VDO algorithm continues until pre-specified minimum amplitude Amin is achieved. The minimum amplitude is set to be 106. 5. Parameter tuning The effectiveness of meta-heuristic algorithms greatly depends on the correct choice of parameters. In this section, we are going to study the behavior of the different parameters of the proposed VDO algorithm. To calibrate the algorithms, there are several methods to statistically design experimental investigation. The full factorial design is one of the present methods that are widely used in the most researches [47]. This method tests all possible combinations of factors. However, when the number of factors significantly increases, this method is not economical. Because, number of required experiments for tuning is increasing to a lot of number; time and cost are going up as well. In this condition, several experimental designs have been suggested to reduce the number of experiments [48,49]. Among several experimental design techniques, the Taguchi experimental design method has been successfully applied for optimization problems [50,51]. In this method, the orthogonal arrays are used to study a large number of decision variables with a small number of experiments. The Taguchi method separates the factors into two main groups: controllable and noise factors. Controllable factors will be placed in the inner orthogonal array and noise factors in the outer orthogonal array [42]. Due to unpractical and often impossible omission of the noise factors, the Taguchi method tends to both minimize the impact of noise and also find the best level of the influential controllable factors on the basis of robustness [52,53]. A transformation of the repetition data to another value which is the measure of variation is developed by Taguchi. The transformation is the signal-to-noise (S/N) ratio, which explains why this type of parameter design is called a robust design [41,43,44,50,51,53]. Here, the term ‘‘signal’’ denotes the desirable value (i.e., mean response variable) and ‘‘noise’’ denotes the undesirable value (i.e., standard deviation). So the S/N ratio indicates the amount of variation present in the response variable. The aim is to maximize the signal-to-noise ratio. In the Taguchi method, the S/N ratio of the minimization objectives is given below [40,42]: 2

S=N ratio ¼ 10log10 ðobjectiv e functionÞ

ð21Þ

In this paper, we apply the Taguchi method in parameter setting to achieve better robustness of the VDO algorithm whose control factors are as follows: Initial amplitude ðA0 Þ, sigma of Rayleigh distribution ðrÞ, number of neighborhood searches in the search loop for each amplitude ðLÞ and damping coefficient ðcÞ: Levels of these factors are illustrated in Table 2. Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027

9

E. Mehdizadeh et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 2 Factors and their levels for the VDO algorithm. Factors

Symbols

Levels

Initial amplitude ðA0 Þ

A

A(1) A(2) A(3) A(4)

– – – –

6 8 10 12

Sigma of Rayleigh distribution ðrÞ

B

B(1) B(2) B(3) B(4)

– – – –

0.5 1 1.5 2

Number of neighborhood searches in the search loop for each amplitude ðLÞ

C

C(1) C(2) C(3) C(4)

– – – –

40 60 80 100

Damping coefficient ðcÞ

D

D(1) D(2) D(3) D(4)

– – – –

0.05 0.1 0.15 0.2

Table 3 The modified orthogonal array L16. Trial

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Levels of control factors A

B

C

D

A(1) A(1) A(1) A(1) A(2) A(2) A(2) A(2) A(3) A(3) A(3) A(3) A(4) A(4) A(4) A(4)

B(1) B(2) B(3) B(4) B(1) B(2) B(3) B(4) B(1) B(2) B(3) B(4) B(1) B(2) B(3) B(4)

C(1) C(2) C(3) C(4) C(2) C(1) C(4) C(3) C(3) C(4) C(1) C(2) C(4) C(3) C(2) C(1)

D(1) D(2) D(3) D(4) D(3) D(4) D(1) D(2) D(4) D(3) D(2) D(1) D(2) D(1) D(4) D(3)

On the other hand, to select the appropriate orthogonal array, it is necessary to calculate the total degree of freedom. The proper array should contain a degree of freedom for the total mean and three degrees of freedom for each factor with four levels (4 3 = 12). Thus, the sum of the required degrees of freedom is 1 + 4 3 = 13. Therefore, the appropriate array should have at least 13 rows. From the standard table of orthogonal arrays, L16 is selected as the fittest orthogonal array design that fulfills our all minimum requirements. But this orthogonal array still entails some modifications to adapt itself to our experimental design. The modified orthogonal array L16 is presented in Table 3, in which control factors are assigned to the columns of the orthogonal array and the corresponding integers in these columns indicate the actual levels of these factors. The experiments on the parameters of the VDO algorithm are based on the L16 orthogonal array, therefore 16 different combinations of control factors (trials) are considered. For parameter tuning, 15 test problems with different sizes and specifications are generated. To yield more reliable information and because of having a stochastic nature of the VDO algorithm, we tackle each test problem three times. Therefore, we have 45 results for each trial to set parameters. To conduct the experiment, we implement the proposed algorithm in the Visual Basic language, and run on a PC with 2.0 GHz and 1 GB of RAM memory. After obtaining the results of the test problems in different trials, the results of each trial are transformed into the S/N ratio. The S/N ratios of trials are averaged in each level, and its value is plotted against each control factor in Fig. 6. As indicated in Fig. 5, better robustness of the algorithm is achieved when the parameters are set as follows:

A0 ¼ Að1Þ ¼ 6;

r ¼ Bð3Þ ¼ 1:5; L ¼ Cð1Þ ¼ 40; c ¼ Dð4Þ ¼ 0:2:

For understanding about adjustment factors, another measure, namely the average of relative percentage deviations (RPD), is used. The RPD value is defined by:

RPD ¼

Alg sol minsol 100; minsol

ð22Þ


10


Fig. 6. Average S/N ratio plot at each level of the factors for objective function values.

Fig. 7. Average RPD plot for each level of the factors.

where Alg sol is the objective function value obtained for each replication of trial in a given problem and minsol is the best solutions obtained for the same problem. After converting the objective values to RPDs, the average RPD is calculated for each parameter level. The results for each level are shown in Fig. 7. As depicted in this figure, the RPD plot illustrates the best parameters of factors A, B, C and D as A(1), B(3), C(1) and D(4), respectively. This analysis strongly supports our decision on the optimal level of factors A, B, C, D. 6. Computational results This section describes the computational tests used to evaluate the effectiveness and efficiency of the proposed VDO in finding good quality schedules. We implement the algorithm in the Visual Basic language and run on a PC with 2.0 GHz and 1 GB of RAM memory. Based on the previous section, the parameters of VDO algorithm are set as follows: Initial amplitude = 6; sigma of Rayleigh distribution = 1.5; number of neighborhood searches in the search loop for each amplitude = 40 and damping coefficient = 0.2. The stopping condition of the algorithm equaling to the minimum amplitude is set to be 106. The non-deterministic nature of the proposed algorithm made it necessary to carry out multiple runs on the same problem in order to obtain meaningful results. Therefore, the best solution is selected for each problem after ten runs of the VDO algorithm. In this section, we test the VDO algorithm versus the branch-and-bound method and GA. To measure the quality of solutions, the total weighted completion time is considered as objective function value. In the first subsection, the results of the VDO algorithm are compared with the results of the branch-and-bound method; and in the next subsection, the results of the VDO algorithm are compared with the results of the GA. 6.1. The proposed VDO algorithm versus the branch and bound method To compare the VDO algorithm versus the branch-and-bound method, we first generate 14 test problems in different sizes with the processing times, setup times and weights uniformly distributed on (1, 100), (1, 100) and (1, 10), respectively. Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027

11


Then, these test problems are formulated by the integer liner programming (ILP) model described in Section 2 and attained mathematical models are coded by the Lingo software. This software solves the model by the branch-and-bound method that is a traditional optimization technique. Also, we apply the VDO for solving these test problems. Table 4 compares the results of the proposed VDO with the results of the branch-and-bound method. The first column signifies the number of test problems. The second to fourth columns represent the number of jobs, number of machines and number of families for each test problem, respectively. The number of model variables and the number of constraints related to the mathematical model are shown in the fifth and sixth columns, respectively. The seventh to ninth columns signify the ILP bound (i.e., a bound on the best possible value of the objective can be attained), the best solution found by Lingo (i.e., the objective value of the best integer solution found by the branch-and-bound method), and the CPU time of the Lingo software, respectively. The tenth column represents the best value of objective obtained from ten runs of the VDO, and the eleventh column reports the average of the obtained results of ten runs. The twelfth column presents the average CPU times for ten runs of the proposed algorithm in terms of seconds. A review of the results in Table 4 shows that the VDO algorithm and branch-and-bound method are capable to attain the optimal solutions for test problems 1–5. But in other nine test problems, the proposed algorithm works better than the branch-and-bound method. In addition, the CPU times of the proposed VDO are significantly smaller than the CPU times of branch-and-bound method for all 14 test problems. As shown in Table 4, the CPU times of the proposed algorithm for all test problems are smaller than one second. It should be said that program of the VDO and the Lingo software are run on the same PC. 6.2. The proposed VDO algorithm versus the GA To compare the VDO algorithm versus the genetic algorithm (GA) presented by Tavakkoli-Moghaddam and Mehdizadeh [23], at first, 27 test problems are generated as follows: the set of test problems comprises 27 combinations of number of jobs (n), number of machines (m), number of families (F) being n = {10, 30, 50}, m = {3, 5, 8} and F = {3, 6, 9} with the processing times, setup times and weights uniformly distributed on (1, 100), (1, 50) and (1, 10), respectively. The parameters of the GA are tuned by the Taguchi method and are set as follows: population size = 100, crossover probability or rate = 75%, mutation probability = 10% and reproduction probability = 15%. The stopping condition of the GA equaling to the maximum number of generations is set to be 150. We implement the VDO and GA in the Visual Basic language and run on a PC with above-mentioned characterizes. All of the test problems should be solved ten times by these two algorithms. The results are shown in Table 5. Test problems specifications are represented in the first to fourth columns. The fifth column signifies the best result obtained from ten runs of the VDO, and the sixth column reports the average of the obtained results of ten runs. The seventh column represents the average CPU times for ten runs of the proposed algorithm in terms of seconds. The eighth column shows the best result obtained from ten runs of the GA, and the ninth column reports the average of the obtained results of ten runs. The tenth column represents the average CPU times for ten runs of the GA in terms of seconds. The results reveal that our proposed algorithm outperforms the GA in all 27 test problems. Also, the average CPU times of the proposed VDO are considerably smaller than the average CPU times of the GA for all 27 test problems. We use the RPD measure to compare the performances of GA and VDO. Table 6 shows the average RPD of these algorithms for each problem size, 10 data per average. The best performance is obtained by the VDO with the RPD of 1.751. To verify the statistical validity of the results, we perform an analysis of variance (ANOVA) method to accurately analyze the results. The results demonstrate that there is a clear statistically significant difference between performances of the algorithms, with respect to the RPD measure. The means plot and LSD intervals (at the 95% confidence level) for two algorithms are shown in Fig. 8. It can be inferred from Fig. 8 that the proposed VDO statistically works better than the GA. Table 4 Comparison of the proposed VDO algorithm with the branch-and-bound method in Lingo. Problem

Result of Lingo

Result of VDO algorithm

No.

n

m

F

Var

Cons

ILP obj. bound

Best obj. found by Lingo

CPU time (S)

Best found by VDO

Average of solutions

Average of CPU time (S)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3 4 5 7 10 10 15 15 20 25 30 40 45 50

2 3 2 3 3 5 3 5 5 5 8 10 15 15

2 2 3 3 5 5 5 7 5 8 8 10 10 15

27 64 65 175 340 560 735 1215 2120 3275 7470 16,440 31,095 38,300

39 84 83 207 384 626 799 1311 2246 3431 7749 16,891 31,831 39,116

430 477 1088 1669 4452 3294.3 9193.9 2240.5 2452.8 3296.8 2491.3 2227.4 1765.8 1265.5

430 477 1088 1669 4452 7262 18,141 18,438 29,604 93,734 50,666 72,006 N/A N/A

1 1 2 13 3863 3809 3681 3884 3894 3816 3740 3945 3935 3743

430 477 1088 1669 4452 7136 17,207 10,116 17,342 28,900 19,871 33,132 32,341 24,372

430 477 1088 1669 4452 7202 17435.4 10160.4 17874.6 29831.8 20148.2 33792.8 32,590 25,003

0.1335 0.1404 0.1678 0.1919 0.2356 0.2541 0.3011 0.3374 0.4066 0.4868 0.5789 0.7321 0.8306 0.9582


12


Table 5 Comparison of the presented VDO with the GA. Problem

Result of our proposed VDO

Result of GA

Number

n

m

F

Best VDO


Average of CPU-times (S)

Best GA


Average of CPU-times (S)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

10 10 10 10 10 10 10 10 10 30 30 30 30 30 30 30 30 30 50 50 50 50 50 50 50 50 50

3 3 3 5 5 5 8 8 8 3 3 3 5 5 5 8 8 8 3 3 3 5 5 5 8 8 8

3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9

5311 4116 3978 4770 5189 3601 6508 3523 3814 26,805 22,156 13,089 36,932 18,579 16,808 29,961 19,320 17,723 70,012 42,613 29,608 79,845 55,648 32,950 85,820 36,246 28,280

5329.6 4120.8 3996 4834.4 5193.8 3607.4 6529.6 3555.8 3820.6 27057.2 22,558 13360.8 38649.6 19457.2 17159.6 30942.6 19465.2 18094.4 70756.6 43,059 30040.4 81248.2 57426.6 33,206 88571.4 37323.8 29806.2

0.2212 0.2097 0.2371 0.2544 0.2719 0.2656 0.2813 0.3096 0.3281 0.4311 0.4765 0.4991 0.4854 0.5278 0.5098 0.55346 0.5633 0.60124 0.71756 0.73722 0.77174 0.75122 0.81038 0.7957 0.81062 0.83352 0.88692

5345 4656 4082 4770 5201 3635 6521 3559 3831 28,805 25,466 14,660 39,811 20,773 19,954 40,481 24,868 22,886 82,168 53,569 40,127 105,165 70,269 41,207 102,505 41,112 38,290

5458.4 4781.8 4112.6 4896.8 5237.6 3673.8 6556.2 3714.8 3872 29821.4 26118.4 15,037 41790.8 21252.6 20,282 42384.4 26,165 24106.6 85136.4 54242.2 41882.4 107,937 73053.2 43267.8 103866.2 42571.2 39819.4

0.4552 0.4349 0.4993 0.5150 0.5356 0.5713 0.5874 0.6225 0.6508 0.8447 0.8932 0.9168 0.9296 0.9780 0.9635 1.04108 1.1560 1.23212 1.4100 1.59998 1.72764 1.65286 1.77474 1.74722 1.8368 1.9048 1.9700

Table 6 Average relative percentage deviation (RPD) for the VDO and GA. Problem number

Algorithm VDO

GA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

0.3502 0.1166 0.4524 1.350 0.0925 0.1777 0.3318 0.9310 0.1730 0.9408 1.8144 2.0765 4.6507 4.7268 2.0918 3.2762 0.7515 2.0955 1.0635 1.0466 1.4604 1.7574 3.1961 0.7769 3.2060 2.9735 5.39674

1.6824 13.5711 3.3647 2.6582 0.9327 2.0083 0.7395 5.2685 1.5203 11.0334 17.4957 14.5811 12.6285 14.0416 19.6934 39.6402 32.2888 34.2105 21.0888 26.2348 39.3812 33.6561 30.1839 30.2490 20.3729 16.8958 38.4284

Average

1.7510

17.9204



13

Fig. 8. Means plot and LSD intervals for the VDO and GA, regarding the RPD criterion.

Table 7 Average relative deviation index (RDI) for the VDO and GA. Problem number

Algorithm VDO

GA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

6.6906 0.7500 8.4112 23.3333 6.6666 4.9230 23.2258 9.4797 5.4098 5.7461 9.0134 10.0221 26.4043 27.3157 8.4845 6.6012 1.5576 4.6746 4.1451 3.6587 3.1263 4.2794 9.0592 2.0045 14.2331 13.8463 11.3980

47.2661 89.2500 62.8972 45.9420 67.5000 56.0000 51.8279 55.4335 47.5409 68.7263 88.8430 71.8289 74.6933 83.1601 83.8320 83.5467 73.4284 80.3473 84.1975 95.3995 88.7455 85.6750 88.6527 80.7908 93.3536 81.2589 86.1792

Average

9.4244

74.6784

For more investigations in the computational results, another measure, namely relative deviation index (RDI), is used in this paper. The RDI is obtained by:

RDI ¼

Algsol minsol 100: maxsol minsol

ð23Þ

With this measure, an index between 0 and 100 is obtained for each method. The more the RDI is closer to zero, the more the algorithm is preferable. Note that if the worst and the best solutions take the same value, all the methods provide the best (same) solution, and hence, the index value will be 0 (i.e., best index value) for all methods. Table 7 shows the average RDI of the VDO and GA for each problem size, 10 data per average. The best performance is obtained by the VDO with the RDI of 9.424. For further accurate studying, ANOVA test is applied again. The results demonstrate that there is a clear statistically significant difference between performances of the algorithms, with respect to the RDI measure. The means plot and LSD intervals (at the 95% confidence level) for two algorithms are shown in Fig. 9. As depicted in this figure, the performance of the presented VDO statistically supersedes the GA. Please cite this article in press as: E. Mehdizadeh et al., A vibration damping optimization algorithm for a parallel machines scheduling problem with sequence-independent family setup times, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.02.027

14


Fig. 9. Means plot and LSD intervals for the VDO and GA, regarding the RDI criterion.

Regarding the obtained results from the computational study, it seems that the proposed VDO algorithm can be an effective approach for an identical parallel machines scheduling problem with sequence-dependent family setup times when the objective function is the minimization of the total weighted completion time. 7. Conclusions and future research In this paper, a vibration damping optimization (VDO) algorithm was investigated for solving the identical parallel machines scheduling problem with sequence-independent family setup times. The objective function of the problem was considered as minimization of the total weighted completion time. We have used an extended solution representation method to encode the partition of jobs to machines and sequence jobs for each machine into a solution with n (i.e., number of jobs) columns and two rows. The proposed VDO has used a simple neighborhood structure for creating a neighboring solution. The amplitude in the proposed algorithm had a control parameter role. This factor has controlled the possibility of the acceptance of the worse solution in various steps of the algorithm. The algorithm escapement from local optimum has been reduced in low amplitude, and the more accessibility to global optimization has been increased in higher amplitude. To adjust the parameters of the proposed VDO, the Taguchi parameter design method has been employed. The robustness of the algorithm might be improved by fine-tuning the VDO parameters, related to the initial amplitude ðA0 Þ, sigma of Rayleigh distribution ðrÞ, number of neighborhood searches in the search loop for each amplitude ðLÞ, damping coefficient ðcÞ: Furthermore, we have compared the result of our proposed VDO with those of branch-and-bound method and GA on generated test problems. The computational results have revealed the competent performance of the proposed algorithm to optimize the give problems. The following suggestions are offered for future work: Implementing effective parallelization strategies in the VDO algorithm to improve the solution quality in parallel machine scheduling problems. Developing hybrid algorithms, such as VDO-GA and VDO-PSO for parallel machines scheduling problems. Considering other constraints, such as job due dates and penalties for completing both early and tardy jobs.

References [1] B.H. Ahn, J.H. Hyun, Single facility multi-class job scheduling, Comput. Oper. Res. 17 (3) (1990) 265–272. [2] A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.Y. Kovalyov, A survey of scheduling problems with setup times or costs, Eur. J. Oper. Res. 187 (3) (2008) 985– 1032. [3] J.M.J. Schutten, R.A.M. Leussink, Parallel machine scheduling with release dates, due dates, and family setup times, Int. J. Prod. Econ. 46–47 (1996) 119– 125. [4] T.C.E. Cheng, M.Y. Kovalyov, Parallel machine batching and scheduling with deadlines, J. Sched. 3 (2) (2000) 109–123. [5] S. Webster, M. Azizoglu, Dynamic programming algorithms for scheduling parallel machines with family setup times, Comput. Oper. Res. 28 (2) (2001) 127–137. [6] J. Blazewicz, M.Y. Kovalyov, The complexity of two group scheduling problems, J. Sched. 5 (6) (2002) 477–485. [7] M. Azizoglu, S. Webster, Scheduling parallel machines to minimize weighted flowtime with family set-up times, Int. J. Prod. Res. 41 (6) (2003) 1199– 1215. [8] Z.L. Chen, W.B. Powell, Exact algorithms for scheduling multiple families of jobs on parallel machines, Nav. Res. Logist. 50 (7) (2003) 823–840. [9] B.M.T. Lin, A.A.K. Jeng, Parallel-machine batch scheduling to minimize the maximum lateness and the number of tardy jobs, Int. J. Prod. Econ. 91 (2) (2004) 121–134. [10] S. Dunstall, A. Wirth, A comparison of branch-and-bound algorithms for a family scheduling problem with identical parallel machines, Eur. J. Oper. Res. 167 (2) (2005) 283–296. P [11] R. Nessah, C. Chu, F. Yalaoui, An exact method for pm =sds; ri = ni¼1 C i problem, Comput. Oper. Res. 34 (9) (2007) 2840–2848. [12] S.H. Shim, Y.D. Kim, A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property, Comput. Oper. Res. 35 (3) (2008) 863–875. [13] Y. Park, S. Kim, Y.H. Lee, Scheduling jobs on parallel machines applying neural network and heuristic rules, Comput. Ind. Eng. 38 (1) (2000) 189–202.



15

[14] M.X. Weng, J. Lu, H. Ren, Unrelated parallel machine scheduling with setup consideration and a total weighted completion time objective, Int. J. Prod. Econ. 70 (3) (2001) 215–226. [15] Y. Yi, D.W. Wang, Tabu search for scheduling grouped jobs on parallel machines, J. Northeastern Univ. 22 (2001) 188–191. [16] A.S. Mendes, F.M. Muller, P.M. Franca, P. Moscato, Comparing meta-heuristic approaches for parallel machine scheduling problems, Prod. Plan. Control 13 (2002) 143–154. [17] D.H. Eom, H.J. Shin, I.H. Kwun, J.K. Shim, S.S. Kim, Scheduling jobs on parallel machines with sequence-dependent family set-up times, Int. J. Adv. Manuf. Technol. 19 (12) (2002) 926–932. [18] D.W. Kim, K.H. Kim, W. Jang, F.F. Chen, Unrelated parallel machine scheduling with setup times using simulated annealing, Robot. Comput. Integr. Manuf. 18 (3/4) (2002) 223–231. [19] L. Monch, H. Balasubramanian, J.W. Fowler, M.E. Pfund, Heuristic scheduling of jobs on parallel batch machines with incompatible job families and unequal ready times, Comput. Oper. Res. 32 (11) (2005) 2731–2750. [20] A.H. Abdekhodaee, A. Wirth, H.S. Gan, Scheduling two parallel machines with a single server: the general case, Comput. Oper. Res. 33 (4) (2006) 994– 1009. [21] V.A. Armentano, M.F.d. Filho, Minimizing total tardiness in parallel machine scheduling with setup times: an adaptive memory-based GRASP approach, Eur. J. Oper. Res. 183 (1) (2007) 100–114. [22] R. Logendran, B. Mcdonell, B. Smucker, Scheduling unrelated parallel machines with sequence-dependent setups, Comput. Oper. Res. 34 (11) (2007) 3420–3438. [23] R. Tavakkoli-Moghaddam, E. Mehdizadeh, A new ILP model for identical parallel machine scheduling with family setup times minimizing the total weighted flow time by a GA, Int. J. Eng. Trans. A Basic 20 (2) (2007) 183–194. [24] R. Tavakkoli-Moghaddam, F. Taheri, M. Bazzazi, M. Izadi, F. Sassani, Design of a GA for bi-objective unrelated parallel machines scheduling with sequence-dependent setup times and precedence constraints, Comput. Oper. Res. 36 (12) (2009) 3224–3230. [25] J. Behnamian, M. Zandieh, S.M.T. Fatemi Ghomi, Parallel machine scheduling problems with sequence-dependent setup times using an ACO, SA and VNS hybrid algorithm, Expert Syst. Appl. 36 (6) (2009) 9637–9644. [26] R. Driessel, L. Mnch, Variable neighborhood search approaches for scheduling jobs on parallel machines with sequence-dependent setup times, precedence constraints, and ready times, Comput. Ind. Eng. 61 (2) (2011) 336–345. [27] P.C. Chang, S.H. Chen, Integrating dominance properties with GAs for parallel machine scheduling problems with setup times, Appl. Soft Comput. 11 (1) (2011) 1263–1274. [28] E. Vallada, R. Ruiz, A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times, Eur. J. Oper. Res. 211 (3) (2011) 612–622. [29] I. Sarıcicek, C. Celik, Two meta-heuristics for parallel machine scheduling with job splitting to minimize total tardiness, Appl. Math. Model. 35 (8) (2011) 4117–4126. [30] K. Li, S.L. Yang, H.W. Ma, A simulated annealing approach to minimize the maximum lateness on uniform parallel machines, Math. Comput. Model. 53 (5–6) (2011) 854–860. [31] W.L. Wang, H.Y. Wang, Y.W. Zhao, L.P. Zhang, X.L. Xu, Parallel machine scheduling with splitting jobs by a hybrid differential evolution algorithm, Comput. Oper. Res. 40 (5) (2013) 1196–1206. [32] A.J. Ruiz-Torres, G. Paletta, E. Pérez, Parallel machine scheduling to minimize the makespan with sequence dependent deteriorating effects, Comput. Oper. Res. 40 (8) (2013) 2051–2061. [33] C.W. Lin, Y.K. Lin, H.T. Hsieh, Ant colony optimization for unrelated parallel machine Scheduling, Int. J. Adv. Manuf. Technol. 67 (1–4) (2013) 35–45. [34] E. Mehdizadeh, R. Tavakkoli-Moghaddam, Vibration damping optimization, in: Proc. of the Int. Conf. Operations Research 2008 – OR and Global Business, Germany, 3–5 September 2008. [35] E. Mehdizadeh, M.R. Tavarroth, S.M. Mousavi, Solving the stochastic capacitated location-allocation problem by using a new hybrid algorithm, in: The Proc. of the Int. Conf. of Recent Researches in Applied Mathematics, Greece, 2010, pp. 27–32. [36] E. Mehdizadeh, M.R. Tavarroth, V. Hajipour, A new hybrid algorithm to optimize stochastic-fuzzy capacitated multi-facility location-allocation problem, J. Ind. Eng. 7 (2011) 77–86. [37] S.M. Mousavi, S.T.A. Niaki, E. Mehdizadeh, M.R. Tavarroth, The capacitated multi-facility location–allocation problem with probabilistic customer location and demand: two hybrid meta-heuristic algorithms, Int. J. Syst. Sci. 44 (10) (2013) 1897–1912. [38] M. Aliabadi, F. Jolai, E. Mehdizadeh, M. Jenabi, A flow shop production planning problem with basic period policy and sequence dependent set up times, J. Ind. Syst. Eng. 5 (1) (2011) 286–304. [39] E. Mehdizadeh, S. Nezhad-Dadgar, Using vibration damping optimization algorithm for resource constrained project scheduling problem with weighted earliness-tardiness penalties and interval due dates, Econ. Comput. Econ. Cybern. Stud. Res. (ECECSR) 48 (2014) 331–343. [40] S. Molla-Alizadeh-Zavardehi, M. Hajiaghaei-Keshteli, R. Tavakkoli-Moghaddam, Solving a capacitated fixed-charge transportation problem by artificial immune and genetic algorithms with a Prüfer number representation, Expert Syst. Appl. 38 (2011) 10462–10474. [41] G. Taguchi, Introduction to quality engineering, White Plains: Asian Productivity Organization/UNIPUB, 1986. [42] S. Molla-Alizadeh-Zavardehi, S. Sadi Nezhad, R. Tavakkoli-Moghaddam, M. Yazdani, Solving a fuzzy fixed charge solid transportation problem by metaheuristics, Math. Comput. Model. 57 (2013) 1543–1558. [43] M. Rostamian Delavar, M. Hajiaghaei-Keshteli, S. Molla-Alizadeh-Zavardehi, Genetic algorithms for coordinated scheduling of production and air transportation, Expert Syst. Appl. 37 (2010) 8255–8266. [44] B. Naderi, M. Zandieh, V. Roshanaei, Scheduling hybrid flow shops with sequence dependent setup times to minimize makespan and maximum tardiness, Int. J. Adv. Manuf. Technol. 41 (11/12) (2008) 1186–1198. [45] W.J. Palm, Mechanical Vibration, John Wiley & Sons Inc., New York, 2007. [46] W.T. Thomson, Theory of Vibration with Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1988. [47] D.C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, New York, 2000. [48] N.K. Jerne, Towards a network theory of the immune system, Annu. Immunol. 125 (1974) 373–389. [49] R. Al-Aomar, A. Al-Okaily, A GA-based parameter design for single machine turning process with high-volume production, Comput. Ind. Eng. 50 (2006) 317–337. [50] R.J. Ross, Taguchi Techniques for Quality Engineering, McGraw-Hill, New York, 1989. [51] M.S. Phadke, Quality Engineering using Robust Design, Prentice-Hall, Sarmiento, NJ, 1989. [52] J.T. Tsai, W.H. Ho, T.K. Liu, J.H. Chou, Improved immune algorithm for global numerical optimization and job shop scheduling problems, Appl. Math. Comput. 194 (2007) 406–424. [53] B. Naderi, M. Zandieh, A.K. Ghoshe Balagh, V. Roshanaei, An improved simulated annealing for hybrid flowshops with sequence-dependent setup and transportation times to minimize total completion time and total tardiness, Expert Syst. Appl. 36 (2009) 9625–9633.


A vibration damping optimization algorithm for a

A vibration damping optimization algorithm for a

Suggest Documents

OptimizatiOn Of ROll VibRatiOn Damping Of a Vehicle

A Vibration Damping Setpoint Controller for Continuum Robots

Active vibration control modules for damping ...

DAMPING EVALUATION FOR FREE VIBRATION OF SPHERICAL ...

SELF-TUNING DAMPING SYSTEM FOR CABLE VIBRATION

Damping Models for Structural Vibration - PhD - G.eng.cam.ac.uk

New tools for teaching vibration damping concepts

DAMPING EVALUATION FOR FREE VIBRATION OF SPHERICAL ...

Damping Models for Structural Vibration - Swansea University

Damping Models for Structural Vibration

Flax fibre composite for vibration damping design

DAMPING EVALUATION FOR FREE VIBRATION OF SPHERICAL

A Distributed Optimization Algorithm for Stochastic

A Bacterial Swarming Algorithm For Global Optimization

A Modified BFGS Algorithm for Unconstrained Optimization

A stochastic algorithm for deterministic multistage optimization

a new multiobjective optimization algorithm for ...

A MODIFIED TAGUCHI'S OPTIMIZATION ALGORITHM FOR

A Multiobjective Optimization Algorithm for Discovering Driving ...

A Reliability Optimization Algorithm for Wireless

A Mean-Variance Optimization Algorithm

Vibration Dampers Damping – is the ability of a ... - [email protected]

passive vibration damping in a truss telecommunication tower

Free Vibration and Damping of Rotating Composite Shaft with a