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Ann Oper Res (2015) 229:451–474
DOI 10.1007/s10479-014-1770-8
Using the gravitational emulation local search algorithm
to solve the multi-objective flexible dynamic job shop
scheduling problem in Small and Medium Enterprises
Ali Asghar Rahmani Hosseinabadi ·Hajar Siar ·
Shahaboddin Shamshirband ·Mohammad Shojafar ·
Mohd Hairul Nizam Md. Nasir
Published online: 19 December 2014
© Springer Science+Business Media New York 2014
Abstract Scheduling problems are naturally dynamic. Increasing flexibility will help solve
bottleneck issues, increase production, and improve performance and competitive advantage
of Small Medium Enterprises (SMEs). Maximum make span, as well as the average work-
flow time and latency time of parts are considered the objectives of scheduling, which are
compatible with the philosophy of on-time production and supply chain management goals.
In this study, these objectives were selected to optimize the resource utilization, minimize
inventory turnover, and improve commitment to customers; simultaneously controlling these
objectives improved system performance. In the job-shop scheduling problem considered in
this paper, the three objectives were to find the best total weight of the objectives, maximize
the number of reserved jobs and improve job-shop performance. To realize these targets, a
multi-parametric objective function was introduced with dynamic and flexible parameters.
The other key accomplishment is the development of a new method called TIME_GELS
that uses the gravitational emulation local search algorithm (GELS) for solving the multi-
objective flexible dynamic job-shop scheduling problem. The proposed algorithm used two
A. A. R. Hosseinabadi
Young Research Club, Behshahr Branch, Islamic Azad University, Behshahr, Iran
H. Siar
Department of Electrical and Computer Engineering, Semnan University, Semnan, Iran
S. Shamshirband (B)
Department of Computer System and Technology, Faculty of Computer Science and Information
Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia
e-mail: shamshirband@um.edu.my
M. Shojafar (B)
Department of Information Engineering Electronics and Telecommunications (DIET),
Sapienza University of Rome, Via Eudossiana 18, 00184, Rome, Italy
e-mail: m.shojafar@yahoo.com; shojafar@diet.uniroma1.it
M. H. N. M. Nasir
Department of Software Engineering, Faculty of Computer Science and Information Technology,
University of Malaya (UM), 50603 Kuala Lumpur, Malaysia
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452 Ann Oper Res (2015) 229:451–474
of the four parameters, namely velocity and gravity. The searching agents in this algorithm
are a set of masses that interact with each other based on Newton’s laws of gravity and
motion. The results of the proposed method are presented for slight, mediocre and complete
flexibility stages. These provided average improvements of 6.61, 6.5 and 6.54 %. The results
supported the efficiency of the proposed method for solving the job-shop scheduling problem
particularly in improving SME’s productivity.
Keywords Flexible job-shop ·Scheduling ·Makespan ·GELS Algorithm ·Newton’s law ·
Small Medium Enterprises
1 Introduction
The nation recognises the contribution of Small and Medium Enterprises (SMEs) to the
national economy, and they are a common form of employment, seedbeds for innovation and
entrepreneurship and continuously generate national revenue. As such an important sector,
the SMEs, especially those in the production line, face many challenges, including market
globalization, economic change, shortened product development lifecycles, changing con-
sumer needs and increased competition. Time-to-market element is particularly critical in
make-to-order production industries, where flexibility and reduction of process flow-time
determine the rate of success of the SMEs. This would require the SMEs to carry out pro-
duction planning and scheduling in order to meet the deliv-ery dates and in making quality
decision. SMEs have limited production capabilities and they are frequently at a disadvantage
relative to their larger counterparts with respect to their ability to have technical expertise to
tackle production-scheduling problems. This is due to many optimization problems in pro-
duction systems, particularly job-shop scheduling optimization problems, are very complex
and solving them is not feasible using ordinary optimization methods.
The allocation of n jobs to m machines by considering objectives such as minimizing turned
around time, maximizing production and improving the quality of the service is called jobs
shop scheduling. In real manufacturing environments, the set of jobs that must be scheduled
varies over time, something that is indicative of the dynamics of scheduling problem (Luu and
Tang 2014;Chunlin et al. 2009;Mansouri et al. 2013;Saffari-Aman et al. 2008). On the other
hand, considering different type of flexibilities in such problems can lead to finding solutions
for bottleneck problems, increasing production, and enhancing performance and competitive
advantages. Although flexibility in scheduling increases the complexity of a problem, finding
approximate optimal solutions is significantly more difficult. Several scheduling researchers
believe that scheduling problems in manufacturing environments are multi-objective and
should take into consideration basic scheduling objectives unique to these environments, and
optimize the use of resources, minimize inventory turnover, increase workflow and com-
mitment to customers while reducing turn-around-times. Other scheduling researchers have
suggested that genetic algorithms is provide an appropriate approach for solving optimiza-
tion problems (Goldberg 1989;Wong et al. 2013;Bennett and Parrado-Hernández 2006;
González-Mendoza et al. 2014;Son 2014); however, one of the disadvantages of classical
genetic algorithms is their premature convergence (Shamshirband and Anuar 2013).
Job shop scheduling is an NP-hard problem, there are no polynomial solutions that solve
every dimensions of this problem. By increasing the size of a problem, the computational time
for solving different instances is increased exponentially (Brandimarte 1993;Frutos et al.
2010). Many heuristics algorithms have introduced to solve job shop scheduling problems
(Brandimarte 1993;Frutos et al. 2010;Li and Yin 2013;Pooranian et al. 2013;Ventura
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Ann Oper Res (2015) 229:451–474 453
and Yoon 2013). Gravitational emulation local search algorithm (GELS) has the ability
to effectively search the problem space and generate near optimal and quality solutions
(Voudouris and Tsang 1995;Webster 2004). In this paper, GELS was to solve job shop
scheduling problems.
In the current study, the dilemma of multi-objective flexible job-shop scheduling was
studied using parallel machines in a dynamic job-shop environment. In real manufacturing
environments, scheduling algorithms are naturally dynamic. However, flexibility can be an
effective solution for enhancing system performance with regards to scheduling problems.
On the other hand, to achieve objectives compatible with supply chain philosophy related to
on-time production and management goals, operation criteria compatible with these concepts
must be controlled while simultaneously solving the problem.
In addition to considering the dynamic nature of the manufacturing environment (owing
to the input of jobs in the job-shop in a non-zero time period) and the capability of operation
flexibility and flexibility attributable to parallel machines, the present paper includes a multi-
objective target function to achieve the objectives compatible with on-time production and
the management goals of supply chains. In this paper, besides the problem space extension,
a novel algorithm was proposed as a solution. Its structure was based on MO-FDJSPM
characteristics. To overcome constraints, two strategies were used in the proposed algorithm.
The first strategy attempted to enhance the algorithm’s search space diversity, and the second
strategy assisted with intelligently performing the algorithm search. The results of running
the proposed algorithm and a comparison with an existing ultra-innovative method were
indicative of its efficiency.
In Sect. 2a literature review is presented. A definition of the job-shop scheduling problem
and a mathematical model will be discussed in Sect. 3. Section 4includes an explanation
of the GELS algorithm and the proposed algorithm is described in Sect. 5. The numerical
experiment design and simulation results are represented in Sects. 7and 8, respectively, and
finally, Sect. 8concludes the paper.
2 Literature review
In the typical job-shop scheduling matter, job paths are fixed and specified and it is not
necessary to perform all jobs from the same path. In this problem, it is assumed there is only
one processing path for each job, which indicates a lack of flexibility in the manufacturing
system. Flexible job-shop production is the developed state of a job-shop problem, whereby
each operation can be processed by a set of machines. This challenge was first studied by Xia
and Wu (2005). The flexible job-shop production problem is comprised two sub-problems,
namely routing and operation scheduling (Scrich et al. 2004).
The Flexible Job-Shop with Parallel Machines (FJSPM) problem was initially studied (Su
et al. 1998) with the objective function of minimizing the highest levels of tardiness. It was the
developed form of the flexible job-shop production and parallel machine problems. In each
stage of the flexible job-shop problem, there is only one machine whereas in each FJSPM
problem phase, a set of parallel machines are placed together so that each has the ability
to process the operation allocated to the respective stage. Therefore, in each stage, various
paths can be considered for job processing. The principal objectives of using flexibility and
increase in, production increases, bottleneck problems are solved. Furthermore, flexibility
can serve as a competitive advantage in economic environments.
In a study conducted by Riane et al. (1998), the hybrid three-stage flow-shop problem
was studied. This problem was structured with one machine in the first and third stages and
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454 Ann Oper Res (2015) 229:451–474
two dedicated machines in stage two, with the aim of minimizing the maximum makespan.
A Tabu search algorithm was proposed (Nowicki and Smutniciki 1998)tosolvetheflow-
shop problem with parallel machines also to minimize the maximum makespan. Kyparisis
and Koulamas (2004) investigated the flexible flow-shop scheduling problem with parallel
machines and extended a set of existing heuristic algorithms to solve the flexible flow shop
problem. They demonstrated that the Weighted Shortest Processing Time Heuristic (WSPT)
was asymptotically optimal. Low (2005) solved the multi-stage flow-shop scheduling prob-
lem with different parallel machines using a heuristic algorithm based on the simulated
annealing algorithm, where the objective was to minimize the total flow time in the sys-
tem. The authors investigated flexible flow-shop scheduling with uniform parallel machines
(Kyparisis and Koulamas 2006), developed the existing heuristics for this problem, and indi-
cated that the heuristic based on vector summation techniques when asymptotically optimal
when the number of jobs is too great.
In Paternina-Arboleda et al. (2008) a heuristic algorithm based on the identification and
exploitation of bottleneck stages was proposed for minimizing the makespan on a flexible
flow shop with k stages and msmachines. The performance of this algorithm was, on average,
comparable with other bottleneck based algorithms and the required computational effort was
less. Grobler et al. (2010) studied the application of PSO on the multi-objective flexible job
shop scheduling problem with sequence dependent set-up times, auxiliary resources, and
machine down time . Four PSO-based heuristics were developed that were all different in
terms of particle representation and problem mapping mechanisms. The paper concluded
that when the quality of the solution and computational complexity was considered, the
priority based PSO (P-PSO) algorithm provided the best performance. Frutos et al. (2010)
used a genetic algorithm, namely NSGAII, integrated with simulated annealing to solve the
flexible job shop scheduling problem. The algorithm was formulated so that non-feasible
outcomes did not arise. Their results showed that the introduced memtic algorithm provided
fast convergence to areas close to the solution.
3 Presenting the mathematical model
In this section, following a definition of flexibility, the MO-FDJSPM problem statement,
variables, and mathematical sets employed will be described. The mathematical model of
the problem will be introduced to demonstrate that MO-FDJSPM is an NP-hard problem.
3.1 Definition of flexibility
Manufacturing flexibility is widely known as a fundamental factor in achieving a competitive
advantage. The term manufacturing flexibility does not refer to a single variable but to a
general category of variables. Different categories and definitions have been offered for
flexibility in manufacturing systems (Baker 1974;Kyparisis and Koulamas 2004;Scrich et
al. 2004;Tkindt and Billaut 2002). Tvay and Ho (2007) described three kinds of flexibility,
namely complete, mediocre, and slight and these were used as FJSP-100, FJSP-50, and FJSP-
20. They demonstrated that in C% flexibility only up to C% of existing machines in the job
shop can be processed in all operations of various jobs.
3.2 Problem statement
Many scheduling researchers believe that the nature of scheduling problems in manufacturing
environments is multi-objective (Tkindt and Billaut 2002). To achieve their objectives, it was
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Ann Oper Res (2015) 229:451–474 455
necessary for different operation factors compatible with scheduling concepts to be controlled
simultaneously during the problem solving process.
Generally, the MO-FDJSPM problem is defined as follows. There are mprocessing stages
and njobs. As a result, the performance of each job in the dynamic job shop requires a set of
distinctive operations. Every ith part enters the job shop in non-zero time. The Jipart consists
of o1,j,o2,j,...,onj,joperations, and without loss of generality it can be assumed that
the order of processing operations in the job shop is the same.
The number of operations required for complementing each job can be less than or equal to
the number of processing stages in the job shop. In each process phase, for instance Mk,there
are lkversions of parallel machines with different speeds, and each version has the ability to
process the allocated operations; that stage and the amount of parameter lkis greater than 1
in at least one flexible workstation of the job shop. Every operation can be processed by at
least one of the workstations and there is at least one operation that can be processed in more
than one workstation (because of the flexibility of operations in the job shop). At most, the
operations of each job can be processed by one machine at the same time.
The processing time of operation Oi,jon a machine with speed Sk,pm =1andinstagek
is considered equal to Ppm,j,k. If processing this operation is done on a machine with speed
Sk,pm >1, the processing time of the mentioned operation will decrease to Pi,j,k/Sk,pm.Itis
assumed that the decision maker has sufficient information about the weight of the objectives
or their priorities.
The purpose behind solving this problem is to identify the best total weight of the objectives
and present this information to the decision maker with the following assumptions:
•Each machine can process only one operation at any given moment.
•Each job at any moment can be processed by only one machine.
•All machines are available from time zero and will not break down.
•Interrupting the operations is not permitted.
•The processing time of operations is definite and specified.
•The preparation times between two operations are inappreciable, or include processing
times and transfer time that are negligible.
Some of these assumptions are practically unrealistic and only considered for simplicity and
to model the problem.
3.3 Definitions of the variables and sets
To solve the job-shop scheduling problem, a few variables and sets were defined as input
parameters as show below:
–j=l,...,m: The operation index of every input part.
–i=l,...,n: job index
–k=l,...,m: stage index
–M
k: kth stage of processing
–n
i: number of operations of job i
–M
k,r: rth parallel machine in stage k
–r
i: entry time of part iinto job shop
–O
i,j:jth operation of job i.
–S
k,r: speed of machine Mk,r
–St
i,j: process stage of operation Oi,j
–P
i,j: processing time of operation Oi,jon a machine with unit speed
–l
k: number of parallel machines in stage k
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456 Ann Oper Res (2015) 229:451–474
–L
max: maximum of lk(s)
– pm: index of parallel machines in the workstation
–c
i,j: earliest completion time of operations
–Ft(i,j)
k,pmoi,j: completion time of operation Oi,jon machine Mk,pm
–a(i,j)
k=1,if the machines of k stage are part of alternate machines for Oi,j
0 Otherwise
–X(i,j)
k,pm =1,if operation Oij is processed in machine of Mk,pm
0,Otherwise
–R(i,j)( p,q)
k,pm =1,if operation Oa,bin Mk,pm precedes operation Oi,jin Mk,pm
0,Otherwise
3.4 Mathematical model
3.4.1 Objective functions
The objective functions of the flexible job shop scheduling problem, considered in this paper
are as follows:
F=α1F1+α2F2+α3F3(1)
F1=Cmax =Max{Ci|i=1,...,N}(2)
F2=¯
F=1
N
N
i=1
Max{Ci−ri}(3)
F3=¯
T=1
N
N
i=1
Max{βi(Ci−di),0}(4)
Equation 1is the objective function of the MO-FDJSPM problem and shows the minimization
of the total weight of the three objective functions estimated in Eqs. 2–4with the given
coefficients α1,α2and α3. The penalty for each tardiness unit related to delivering parts was
assumedtobe(
βi=1) for all jobs.
AccordingtoEq.2Function F1calculates the make-span, F2 estimates turnover of the jobs
and F3 calculates the tardiness unit related to delivering parts. Considering these objective
functions is very essential in the performance of the system. These functions were selected
to optimize the resource utilization, minimize inventory turnover, and improve commitment
to customers. Function F as the total weight of these functions tries to find the optimal
combination of them.
3.4.2 Constraints of job-shop scheduling problem
To solve the job-shop scheduling problem, a set of constraints were used to validate the
solution. These constraints are illustrated in this section. Inequality 5 guarantees that the
sequence set of job operations has no time interference. The sum of constraints 6 and 7
simultaneously assures that the operations process on a machine has no time interference.
Equation 8and inequality 9 guarantee that each job operation can be allocated to only one
of the available alternate machines.
Inequality 10 indicates that if no machine in stage kis allocated to operation Oi,j,the
completion time of this operation over all machines at this stage must be considered to be zero.
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Ann Oper Res (2015) 229:451–474 457
Since the objective functions of the problem are based on ordinary job makespan criteria, the
maximum makespan was calculated in inequality 11. The set of inequality 12 is considered
to represent the model so that the completion time of the first operation of jobs will be greater
than or equal to the sum of the processing time for the operations and the job’s presence time
in the job shop.
Ft(i,j+l)
k,pm −Ft(i,j)
k,pm+L×1−a(i,j+1)
k×X(i,j,l)
k,pm
≥Pi,j+l,k/Sk,pm
∀j:l:i:n;∀i:l:j:(ni−l);
∀k,k:l≤k,k≤m;
∀pm,pm:l≤pm,pm≤lk:
(5)
Ft(i,j)
k,pm −Ft(p,q)
k,pm ×X(i,j)
k,pm +L×R(i,j)( p,q)
k,pm
≥X(i,j)
k,pm ×Pi,j,kSk,pm
∀i,q:i=1,...,n−1&q=i−1,...,n
∀i:1≤j≤nj;∀P:1≤p≤nq;
∀k:1≤k≤m;
∀pm :1≤pm ≤1k;
(6)
Ft(p,q)
k.pm −Ft(i,j)
k,pm ×Xt(p,q)
k,pm +L×1−R(i,j)( p,q)
k,pm ≥X(p,q)
k,pm ×Pp,q,k/Sk,pm (7)
m
k=1
lk
pm=l
a(i,j)
k×X(i,j)
k,pm =1∀j:1≤i≤n;
∀i:1≤j≤,j;(8)
X(i,j)
k,pm ≤a(i,j)
k
∀:1≤i≤n;∀i:1≤j≤nj;
∀k:1≤k≤m;
∀pm :1≤pm ≤lk;
(9)
Ft(i,j)
k,pm ≤L×X(i,j)
k,pm
∀j:1≤i≤n;∀i:1≤j≤nj;
∀k:1≤k≤m;
∀pm :1≤pm ≤1k;
(10)
Ci≥
m
k=1
lk
pm=1
Ft(i,nj)
k,pm ∀j:1≤i≤n;(11)
Ft(i,1)
k,pm ≥X(i,l)
k,pm ×Pi,l,k/Sk,pm +ri
∀j:1≤i≤n;
∀k:1≤k≤m;
∀pm :1≤pm ≤1k;
(12)
3.5 Complexity of the problem
The MO-FDJSPM problem is the developed form of the flexible job-shop manufacturing
model created by the flexibility of parallel machines. Thus, this problem will contain all the
complexities and difficulties of that model. Considering only the flexibility of operations
in such problems dramatically increases the intricacy of finding approximately optimum
solutions (Su et al. 1998). Since the flexible job-shop manufacturing problem is a NP-hard
problem (Tvay and Ho 2007), the flexible operations found in MO-FDJSPM will be consid-
ered a NP-hard problem when the flexibility of the parallel machines is also considered.
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458 Ann Oper Res (2015) 229:451–474
3.6 Solution to the Problem
The intent of solving the MO-FDJSPM problem is to find the best total weight and present
it to the decision maker. In this paper the GELS algorithm was used to design a method
for solving this problem. In the next section, the proposed algorithm and its structure are
illustrated with respect to problem solving.
4 The GELS algorithm
In 1995, Voudouris and Tesang (Voudouris and Tsang 1995) introduced the GLS algorithm
for seeking and solving NP-complete problems. In 2004, Barry Webster (Webster 2004)
presented it powerful algorithm known as the GravitationalEmulation L ocal Search algorithm
(GELS). GELS is an intelligent search scheme for combinatorial optimization problems. The
main feature of this algorithm is the iterative use of a local search. This is a novel heuristic
method that is the same as several popular and mature heuristics found in many famous
problems, In many cases, it outperforms those heuristics in terms of the quality of solutions
it produces and the effectiveness of its searches (Voudouris and Tsang 1995;Webster 2004).
Because the efficient features of GELS, this algorithm is used in many different fields such
as human computer, and civil engineering (Bagrezai et al. 2013;Balachandar and Kannan
2010;Voudouris and Tsang 1995;Rezaeian et al. 2013).
GELS is based on a randomization concept and two parameters (velocity and power), and
it uses random numbers from a local search algorithm to prevent local optimum solutions.
The notion behind this algorithm is related to gravity, which causes objects to be attracted
to each other; heavier objects have more gravitational pull that causes lighter object to be
attracted to them. The distance between two objects affects the strength of their gravitational
pull. As a result, in a situation where there are two objects with the same weight but at
different distances from a lighter object, the object closer to a lighter object will have more
gravitational pull on the lighter object. Newton’s laws are essential and fundamental laws in
physics. These laws have many applications for technology and in engineering. The regnant
principles in industry, construction, and astronautics all depend on Newton’s laws. Force is
the reason behind the movement of an object. Newton’s laws describe the relation between
force and motion, they are more properly known as Newton’s Laws of Motion.. The motion
of an object is determine by the nature and arrangement of other objects in the object’s space.
There is a force between two objects which attracts them to each other. This force is weak
for smaller objects, but if one of the two objects is a planet, then the force is very strong
(Halliday et al. 2010). This force is known as gravity and it can be expressed as shown in Eq.
(13), which is Newton’s law of universal gravitation (Voudouris and Tsang 1995).
F=GM
1M2
R2(13)
Where m1and m2indicate the mass of Objects 1 and 2, respectively, Gis the gravitational
constant and Ris the radius or distance between two objects.
GELS emulates this natural process in the search space. In this algorithm, the search
space is imagined as the universe and the objects are seen as possible solutions. Each object
or solution has a mass, which indicates the performance or search index. Thus, the best
solution has the highest mass and no object has zero mass (Webster 2004).
In this scheme, the possible solutions in the search space are categories in several segments
based on a measure that is dependent on the type of problem; each segment is known as a
dimension of the solution, and for each dimension a value spots as the initial velocity.
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Ann Oper Res (2015) 229:451–474 459
The GELS estimates the effect of gravity on solutions in using two different methods. In
the first method, a solution for the local neighborhood is selected as the current solution and
the gravity between current and candidate solutions will be estimated. The second technique
applies the formula to all solutions in the neighborhood and calculates the gravity of the cur-
rent solution for all of them. GELS can use two schemes to search the search space. In the first
method, GELS searches only possible solutions in the neighborhood space. Each of methods
can either method to estimate gravity; therefore, modeling four schemes of GELS is possible.
GELS contain a vector whose size determines the number of solution dimensions. The
entries for this vector reflect the relative velocity in each dimension. The algorithm needs an
initial solution, initial velocity vector, and direction before it can begin. A random number
is chosen between one and the maximum velocity in each dimension, which is the value for
every component in each dimension. The initial solution, as of current solution, is generated
by the user or randomly. For each dimension in the initial velocity vector, a direction is
selected based on the initial velocity vector of the solution dimensions. This direction is
equal to the solution dimension with the greatest initial velocity in the initial velocity vector.
The algorithm involves a pointer object that can move over the search space, which has a
fixed mass in all calculations. This object always points to the solution with the most mass.
The algorithm terminates if one of the following conditions is satisfied: all components of the
initial velocity vector equals zero or the number of iterations reaches a determined maximum
iteration number.
In every iteration of the algorithm a candidate solution for the local neighborhood space for
the current solution will be selected according to the current direction. The gravity between
the current and candidate solutions is estimated, and the corresponding velocity vector will be
updated. For the next iteration, the velocity vector is checked and a new direction is selected.
The second method is very similar to the first with respect to algorithm iterations, with the
only difference being in the update step of the second method. In this update step, the gravity
and initial velocity of all candidate solutions is estimated to find the candidate solution from
the current one. The gravity between two objects in Eq. (13) is calculated by replacing the
masses with a difference in cost value between the candidate and current solutions as shown
in Eq. (14)(Voudouris and Tsang 1995):
F=G(Cu −CA)
R2(14)
Where CU and CA indicate the cost values of the current and candidate solutions, respectively.
The value of Eq. (14) is positive if the current solution for cost value is greater than the
candidate value, otherwise it is negative. Then, this force value, whether positive or negative,
will be added to the velocity vector in the current path. If this addition causes the velocity
value to deviate from the maximum arrangement, it will obtain a maximum value. If the
update step generates a negative value for velocity, it will bring it to zero. The available
parameters in GELS are as follows (Hosseinabadi et al. 2013):
Maximum velocity: The maximum value that can be assigned to each entry of the initial
velocity vector, which prevents the further expansion of these entries.
Radius: The radius (R) employed in the gravity calculation formula.
Iteration: The maximum number of algorithm iterations, which ensures that the algo-
rithm will terminate (Hosseinabadi et al. 2013).
Algorithm 1 shows the GELS algorithm. This algorithm shows that an initial response to
a problem is created, and each mass is evaluated. Next, the problem is updated as G,orBest,
and/or Wors t ,andthemand aparameters are calculated for each mass. Then velocity and the
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460 Ann Oper Res (2015) 229:451–474
location of each mass are also updated. Finally, the algorithm is concluded if the maximum
number of iterations meets or all the initial velocity vector elements become zero. Otherwise,
the algorithm goes back to Step 2 and continues until an optimal answer is reached.
Algorithm1. The GELS algorithm
1. Generate initial population
2. While (termination condition is not satisfied)
3. Evaluate the fitness for each agent
4. Update the G, Best and Worst of the population
5. Calculate mand a parameters for each agent
6. Update velocity and position
7. End While
8. Return best solution
5 The proposed algorithm
In the proposed method, the GELS algorithm serves as a strategy for solving the multi-
objective flexible dynamic job-shop scheduling problem. The goal of this algorithm is to
find the optimal total weight, to increase the number of reserved jobs, and to improve the
effectiveness of machinery. Essentially, a group of jobs necessitates that a group of machines
held in reserve for the future. Solving this problem would be difficult even if there were a small
number of jobs and machines. For this reason, the GELS algorithm is an appropriate solution.
To solve the MO-FDJSPM problem, first three matrixes (distance, initial velocity and
mass) were considered, so that the initial distance and velocity matrices were initialized
randomly. In the velocity matrix, an initial velocity was assigned to every job that was
considered to be a mass, after which the velocity was changed in the following iteration. The
mass matrix was estimated based on the distance and velocity matrices as:
M=(YB−YA)2+(XB−XA)2
Vin A,B
(15)
The first step in solving an optimization problem with the help of the GELS algorithm is
to represent the solutions as chromosomes (CU). The MO-FDJSPM was divided into two
sub-problems related to allocation and operation sequence detection. GELS is designed to
solve both sub-problems in an integrated manner and simultaneously. A two-dimensional
chromosome was used. In this representation, the length of each chromosome was equal to
the total number of job operations for scheduling, and the width was equal to three. Thus, each
solution to the problem was represented as a two-dimensional array. This method is similar
to the method presented by Lee et al. (2002), in which the allocation string was assumed to
be two separate strings and were identified as the workstation and machine allocation strings,
owing to the presence of a parallel machine at each station.
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Ann Oper Res (2015) 229:451–474 461
In GELS, the first string of chromosomes indicates the operation processing workstation
under examination and the second string of chromosomes represents the number of machines
that handle the operation in the corresponding workstation. The third string in this represen-
tation indicates the priority assigned to each operation. Every element in the third row is a
number between one and the total number of operations, and with this constraint, the priority
of two operations cannot be the same. Therefore, the solutions are always warranted and
there will never be an unwarranted chromosome (CU) during the algorithm.
An appropriate reservation factor must be defined using the GELS algorithm under certain
conditions. The reservation factor refers to the number of jobs reserved for machines in the
future. The set of job allocations to the machines is a solution. In the form of two n ×n
matrices (representative of the number of jobs and machines), this solution can be utilized
for response representation. Each column and row of the matrix represents one of the jobs
in the group and each entry shows the number of machines allocated to the job. When the
algorithm terminates, the solution for each job will denote the reserved machines along with
their reservation factors.
The steps of the proposed algorithm are:
Step 1. In the simulation system, the number of iterations, machine matrix, jobs matrix,
jobs processing time matrix, job turn-around time matrix and population number are
initialized.
Step 2. The idle time for the machines and execution time for every job will be initialized
by a random number generator.
Step 3. The velocity matrix values are selected randomly from zero to the total number
of jobs.
Step 4. In each algorithm iteration, the maximum value of each row and column is selected
in the velocity matrix, after which the machines allocated to that dimension are changed.
In fact, a new object is created and then the current solution cost is calculated. If the
current solution cost is higher than the optimum estimated solution cost, that solution
cost is selected as the best object up to this iteration.
Step 5. Gravity is calculated based on the solution cost of the current and candidate
objects.
Step 6.The Gravity obtained in Step 5 will be added to the value of the velocity vector in
each index.
Steps 1 through 5 are continued until all elements of the velocity vector become zero or
the number of iterations is equal to the maximum number.
Algorithm 2 shows the steps in the proposed algorithm. The system environment is deter-
mined first and initialization takes place. Then, a dynamic two-dimensional chromosome is
generated and CA is created by applying gravity to CU. As a result, the next step of the
algorithm will continue according to the gravity algorithm and finally, the algorithm returns
the optimal answer.
6 Designing the numerical experiments
6.1 Random problem generation
To generate random problems, eleven parameters were detected from problems as shown in
Tabl e 1. The data provided by Tvay and Ho (2007) was utilized to distribute the first eight
parameters, and the remaining parameters were distributed according to data provided by
Kurz and Askin (2003).
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462 Ann Oper Res (2015) 229:451–474
Algorithm2. The modified GELS algorithm
1. Determine the system space and initialization
2. Initialize the two-dimensional dynamic
chromosome
3. Generate the CA from CU by applying the
gravity
4. Order the solutions according to their masses
5. While (stopping condition is not satisfied)
6. Choose the first solution as the best solution
7. Evaluate the bodies
8. Calculate G(t)
9. Calculate the imposed gravitational force on
each body
10. Calculate the acceleration, time and
velocity of each body
11. Update the V and T parameters
12. Displace the solutions in each dimension
based on the force imposed on them in different
dimensions
13. End While
14. Return best solution
The job turn-around time was determined using an equation developed by Baker (1974)
that used the TWK method. In this equation, parameter cis the “hardness of delivery”
index. The higher the value of parameter cfor any job, the weaker the turn-around time. All
combinations at this level were tested. As demonstrated in Table 1, there were 12 experimental
scenarios and 10 data sets were generated for each job combination and stage. Each algorithm
was run using 120 data sets. According to Tvay and Ho (2007), the formula can be adapted
to the MO-FJSP problem and with the flexibility of their own operation, the researchers
replaced the processing time parameter pi,jsuggested by Baker (1974) with their proposed
parameter pi,jwhich is calculated in Eq. (16).
Pij =n(f(oij )) Pi,j,k
nfoij (16)
Where n(f(oij)) shows the number of flexible job stages needed to perform the operations of
job i. In this paper, a new formula was proposed to complete Eq. (16) with the MO-FDJSPM
problem, as follows:
Pij =m
k=la(i,j)
k×li
pm=lPi,j,k/Sk,pm
m
k=la(i,j)
k×lk
(17)
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Ann Oper Res (2015) 229:451–474 463
Tab l e 1 The operational stages of the MO-FDJSPM problem
Parameter Values States
Flexibility 100 % (FJSP −100);50%(FJSP −50);20%(FJSP −20)1
[1×m];[0.5×m];[0.2×m]
n Jobs ×n Machines 10 ×5,20 ×5,50 ×5,20 ×10,50 ×10,100 ×10 1
50 ×15,100 ×15 and 200 ×15
Processing time (pi,j,k)U[(nMachines)/2,(nMachines)×2]1
dev(pi,j,k,pi,j,k)51
# of Operation U[m/2,m]1
Release Date (ri)If nj obs ≥50 :U[0,40];Otherwise,U[0,20]1
Tightness factor of Due Date (C) 1.2(tight),1.5(moderat e),2(l oose)3
Due Date (di)ri+c×mi
i=1pij 1
Machine distribution Constant V ari able 2
# of machines (Li)2−nU
[1,4]−U[1,n]2
Speed of machines U[1,3]1
Number of scenarios 12
where a(i,j)
kis a Boolean variable. If the machines in the kth processing stage are among
the conceivable alternate machines of Oi,jthen a(i,j)
k=1, otherwise a(i,j)
k=0. Sk,pm and
lkindicate the speed of machine Mk,pm and number of parallel machines in workstation k,
respectively. For further calculation simplification, the penalty of each tardiness unit for all
parts was considered to be equal to one unit.
7 Simulation results
The proposed job-shop scheduling system was implemented using C++ programming lan-
guage and ran on a personal computer with 2.4 GHz Pentium-IV processor and 1 GB RAM.
The proposed algorithm TIME_GELS was compared against the GP1and ACO2algorithms
(Dorigo and Stützle 2004;Koza 1992) using the values of the total weight function. As
explained in Sects. 3–4, the objective function is the total weight of three parameters, make-
span, turnover of the jobs and the tardiness unit related to delivering parts. The multi-
parametric objective function of the proposed algorithm tries optimizing the resource uti-
lization, minimizing inventory turnover, and improving commitment to customers. To study
the performance of the solutions obtained from the comparison of two algorithms, the best
estimated value for the total weight objective function and average weight objective function
for each category were considered. Each of the algorithms (the proposed algorithm, GP and
ACO) was run using 120 generated data sets. The results of executing the algorithms for three
flexibility stages were slight, mediocre and complete and they are represented discussed in
Tabl e s 2,3,4,5,6and 7(Tay and Wibowo 2004).
The results of studying the algorithms in the slight, mediocre and complete flexibility
stages are discussed below.
1Genetic Programming.
2Ant Colony Optimization.
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464 Ann Oper Res (2015) 229:451–474
Tab l e 2 Comparing the results of the algorithms on the FJSP-20 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
Objective
Function
TIME_GELS Best 12.12 20.40 43.34 25.28 105.26 300.32 549.04 318.20 542.72 1006.63 1754.1 542.72 292.57
Average 16.51 21.24 56.09 31.28 115.66 305.47 561.87 327.66 542.50 1027.48 1804.63 1124.87 494.60
GP Best 12.53 21.35 46.53 26.80 105.68 315.23 578.82 333.24 580.82 1067.48 1854.89 580.82 313.62
Average 16.90 23.33 59.93 33.38 116.02 323.01 594.38 344.47 598.38 1087.37 1906.42 1197.42 525.09
ACO Bes t 14.73 24.56 48.12 29.13 109.45 317.98 581.39 336.27 584.97 1072.34 1859.21 1172.17 512.52
Average 18.64 26.82 63.59 36.35 120.98 327.76 599.43 349.39 603.49 1092.31 1912.88 1202.89 529.54
Improvement
rate (%)
Best 7.96 12.50 9.19 10.62 2.19 2.77 0.56 0.52 7.46.28 5.87 61.537.95
Average 7.63 18.04 10.08 11.46 2.45 0.65 0.63 0.59 10.77 6.07 5.82 6.69 6.61
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Tab l e 3 Comparing the results of the algorithms for the FJSP-50 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
Objective
function
TIME_GELS Best 10.66 21.20 53.09 28.31 94.20 227.31 514.93 94.20 443.29 781.86 1627.94 443.29 188.60
Average 12.32 23.64 54.61 30.19 97.92 243.04 525.47 288.81 454.33 829.03 1686.4 989.94 436.31
GP Best 10.75 22.41 57.12 30.09 95.77 238.05 545.93 95.77 472.09 829.82 1731.01 472.09 199.31
Average 12.46 26.05 59.79 32.76 100.84 257.73 559.43 305.93 484.95 865.01 1796.44 1048.80 462.49
ACO Bes t 12.53 25.98 61.24 33.25 98.32 242.74 547.09 296.05 477.93 833.22 1739.87 1017.00 488.76
Average 15.67 28.41 63.57 35.88 103.56 262.43 563.81 309.93 491.46 869.83 1801.78 1054.35 466.72
Improvement
rate (%)
Best 9.29 14.13 11.47 11.87 3.02 5.76 6.14 107.98 7.15 6.35 6.61 69.96 82.41
Average 10.66 21.20 53.09 28.31 94.20 227.31 514.93 94.20 443.29 781.86 1627.94 443.29 188.60
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Tab l e 4 Comparing the results of the algorithms for the FJSP-100 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
Objective
function
TIME_GELS Best 8.61 20.43 47.11 25.38 102.29 228.18 489.14 102.29 407.51 773.51 1529.05 407.51 178.39
Average 12.78 22.34 51.05 28.72 104.88 243.42 511.18 286.49 412.27 803.21 1562.26 925.91 413.70
GP Best 8.73 21.76 49.96 26.81 102.93 242.00 519.92 102.93 432.37 816.40 1650.60 442.47 190.73
Average 12.99 23.69 55.08 30.58 105.05 260.11 545.39 303.51 442.46 844.51 1659.08 982.20 438.76
ACO Bes t 10.53 24.36 54.73 29.87 106.54 247.32 523.64 292.50 436.91 819.74 1656.57 475.40 265.92
Average 14.25 27.98 59.17 33.80 108.21 265.48 549.86 307.85 446.93 849.14 1663.35 986.47 442.70
Improvement
rate (%)
Best 11.85 12.76 11.11 11.66 2.39 1.16 6.67 93.29 6.66 5.76 8.15 12.62 28.01
Average 6.57 15.65 11.90 12.09 1.67 7.96 7.14 6.70 7.86 5.43 6.34 6.31 6.54
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Tab l e 5 Average execution times of the algorithms (in second) for the FJSP-20 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
TIME_GELS 0.02 0.02 0.03 0.02 0.02 0.03 0.04 0.03 0.03 0.03 0.05 0.03 0.03
GP 0.35 0.88 8.61 3.28 0.71 12.09 69.21 27.34 28.18 170.89 521.18 240.08 90.23
ACO 0.38 0.93 10.79 4.03 5.44 4.32 15.76 8.50 32.67 177.43 536.94 249.01 87.18
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Tab l e 6 Average execution times of the algorithms (in second) on the FJSP-50 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
TIME_GELS 0.02 0.02 0.03 0.02 0.02 0.03 0.04 0.03 0.02 0.03 0.05 0.03 0.05
GP 0.36 0.90 8.78 3.44 0.72 12.33 70.59 27.88 28.74 174.31 53160 244.88 92.04
ACO 0.42 1.14 10.32 3.96 0.96 15.52 74.38 30.28 0.02 182.26 535.49 249.85 94.69
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Tab l e 7 Average execution times of the algorithms (in second) on the FJSP-100 problem
Number of machines 5 10 15 Average of all
dimensions
Number of jobs 10 20 50 Total 20 50 100 Total 50 100 200 Total
TIME_GELS 0.02 0.03 0.04 0.03 0.02 0.06 0.08 0.05 0.00 0.09 0.11 0.07 0.05
GP 0.36 0.91 8.90 3.39 0.73 12.49 71.51 28.24 29.11 175.57 538.52 247.73 93.12
ACO 0.46 1.64 11.92 4.67 0.91 15.27 75.38 30.52 32.65 178.99 542.75 251.46 95.55
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The “least value of the objective function for minimizing the maximum total weight of
the problem’s performance criteria” index for the proposed TIME_GELS algorithm was
compared to and the GP and ACO algorithms were compared. More than 5 runs for each one
of the problem scenarios (low, medium and high) and in the three flexibility stages (slight,
mediocre and complete) were completed. For the slight flexibility stage, the TIME_GELS
was superior to the compared algorithms with improvement rates that ranged from 0.52 to
61.5 %. The average improvement rate of TIME_GELS at this stage was 37.95%. In the
mediocre flexibility stage, the improvements of the proposed algorithm were between 11.87
and 107.98 % and the average improvement rate was 82.4%. In the complete flexibility
stage, the improvements of the TIME_GELS ran from 11.66 to 93.29% and the average
improvement rate was 28.01%. These results demonstrate that the proposed TIME_GELS
outperformed GP and ACO in terms of the least value of the objective function for minimizing
the maximum total weight of the problem’s performance criteria.
For 5 runs of each one of the problem scenarios, the “averaged objective function” was
the best criterion for studying the algorithm’s performance. In the slight flexibility stage, the
proposed TIME_GELS in the low, medium and high dimensions had improvement rates of
11.46, 0.59 and 6.69 % respectively, and an average improvement rate of 6.61%, demon-
strating its superior performance compared to the GP and ACO algorithms. In terms of this
index, in the mediocre flexibility stage, the TIME_GELS demonstrated improvements of
13.68, 6.62 and 6.23 % and obtained an average improvement rate of 6.5%. In the complete
flexibility stage, the improvements gained by the proposed algorithm compared to the other
algorithms, were 12.09, 6.7, 6.31 % and the average improvement rate was 6.54%. These
results confirm the superior performance of the proposed algorithm in terms of the averaged
objective function index, against the GP and ACO heuristic in all three flexibility stages.
The average execution times for the algorithms in the slight, mediocre and complete
flexibility stages are shown in Tables 5,6and 7. These results shown that the execution
time for the proposed TIME_GELS algorithm for all three flexibility stages was less than the
execution time for the GP and ACO algorithms. The execution time indicated the convergence
speed of the proposed algorithm to the optimal scheduling.
The results recorded in Tables 5,6and 7indicate that the convergence speed of the proposed
algorithm was greater than the convergence speed for the other methods and the algorithm
converged to the optimal solution in less time. According to the results, GP performed better
than ACO, after the proposed scheme. Additionally, Tables2,3and 4reveal that the proposed
algorithm achieved a better outcome then the GP and ACO algorithms in terms of the best
and average values achieved for the objective function. These results reveal the power of the
proposed algorithm to reach the optimal solution in less time.
In Figs. 1,2and 3, the difference in averaged objective functions of the proposed
TIME_GELS algorithm and the GP and ACO meta-heuristic methods for different flexibility
stages and problem dimensions with 5 number of machines in the system, are compared. The
results show improvements for low, mediocre and complete dimensions and that with increas-
ing problem dimensions, the amount of improvement for the proposed method increased in
contrast to the meta-heuristic methods in the flexibility stages.
As demonstrated in Figs. 1,2and 3, by increasing the number of jobs entered into the
system, the proposed scheme preformed significantly better compared to the GP and ACO
schemes, for all three flexibility stages. These results indicate that the superiority of the
proposed scheme was sustained when the dimensions of the problem were increased.
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Ann Oper Res (2015) 229:451–474 471
10 15 20 25 30 35 40 45 50
15
20
25
30
35
40
45
50
55
60
65
Number of Jobs
Averaged Objective Function
TIME-GELS
GP
ACO
Fig. 1 Average objective functions in the slight flexibility stage
10 15 20 25 30 35 40 45 50
10
20
30
40
50
60
70
Number of Jobs
Averaged Objective Function
TIME-GELS
GP
ACO
Fig. 2 Average objective functions in the mediocre flexibility stage
8Conclusion
In real scheduling problems, considering the flexibility of parallel machines in addition to
the flexibility of the operations are effective solutions for improving system performance. In
this article, the multi-objective flexible dynamic job-shop scheduling problem with parallel
machines was studied with intent to improve the SMEs’s productivity. Problem parameters
make mathematical model analysis using traditional methods very difficult and even impracti-
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472 Ann Oper Res (2015) 229:451–474
10 15 20 25 30 35 40 45 50
10
15
20
25
30
35
40
45
50
55
60
Number of Jobs
Averaged Objective Function
TIME-GELS
GP
ACO
Fig. 3 Average objective functions in the complete flexibility stage
cal. Thus, the capabilities of meta-heuristic algorithms are used in these situations, especially
the gravitational emulation local search algorithm, which is specifically applied for solving
time and scheduling problems.
The performance of the proposed algorithm’s enhanced version was compared with an
existing evolutionary algorithm from the literature. The results showed improvement rates
of 37.95, 82.4 and 28.01 % for the three flexibility stages, (slight, mediocre and complete),
respectively, and 6.61, 6.5 and 6.54 % improvement in the index of the mean values for the
solutions.
The results indicated improvement and the superiority of TIME_GELS over the compared
algorithm in terms of the optimality of the solution and execution time of the algorithm. These
improvements in the proposed TIME_GELS algorithm are more evident in large systems. The
multi-parametric objective function introduced in this paper tries to maximize the number of
reserved jobs and improve job-shop performance. Simulation results are shown the solution
of the TIME_GELS is averagely achieved in less time to the more optimal fitness against
compared popular evolutionary algorithms. For future research, combining TIME_GELS
with the proposed algorithm to achieve a hybrid scheduling algorithm should be studied.
Acknowledgments This research was supported financially by the University of Malaya Grant (no. RG316-
14AFR).
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