AN ANT COLONY OPTIMIZATION ALGORITHM FOR JOB SHOP SCHEDULING PROBLEM

International Journal of Artificial Intelligence & Applications (IJAIA), Vol. 4, No. 4, July 2013
DOI : 10.5121/ijaia.2013.4406 53
AN ANT COLONY OPTIMIZATION ALGORITHM FOR
JOB SHOP SCHEDULING PROBLEM
Edson Flórez1
, Wilfredo Gómez2
and MSc. Lola Bautista3
Universidad Industrial de Santander, Bucaramanga, Colombia
1
Systems Engineering Student
edson.florez@correo.uis.edu.co
2
Systems Engineer, member of the Research Group in Biomedical Engineering
wilfredo.gomez@correo.uis edu.co
3
Director of the Research Group in Biomedical Engineering
lxbautis@uis.edu.co
ABSTRACT
The nature has inspired several metaheuristics, outstanding among these is Ant Colony Optimization
(ACO), which have proved to be very effective and efficient in problems of high complexity (NP-hard) in
combinatorial optimization. This paper describes the implementation of an ACO model algorithm known as
Elitist Ant System (EAS), applied to a combinatorial optimization problem called Job Shop Scheduling
Problem (JSSP). We propose a method that seeks to reduce delays designating the operation immediately
available, but considering the operations that lack little to be available and have a greater amount of
pheromone. The performance of the algorithm was evaluated for problems of JSSP reference, comparing
the quality of the solutions obtained regarding the best known solution of the most effective methods. The
solutions were of good quality and obtained with a remarkable efficiency by having to make a very low
number of objective function evaluations.
KEYWORDS
Metaheuristics, Ant Colony Optimization, Swarm intelligence, Combinatorial Optimization, Job Shop
Scheduling Problem.
1. INTRODUCTION
ACO is a metaheuristic that brings together concepts from fields such as Artificial Intelligence
and Biology, inspired in the collective behavior of ants. These social insects form colonies of
ants, which are self-organizing systems and decentralized which are considered as a Swarm
Intelligence [12]. Thanks to that intelligence emerging from simple relationships between ants, a
colony can solve complex problems in their environment, such as the problem of finding the
shortest path between the colony and the food, which can be used to find the best solution for
combinatorial optimization problems.
In this paper, we apply the collective intelligence of many simple agents to the problem of Job
Shop Scheduling [22], which consists of finding an optimal plan that minimizes the makespan,
which is the time required to perform a finite number of tasks in a finite number of machines [13].
Each task is a sequence of operations, each one with a determined machine and processing time.
Feasible solutions must comply with the restrictions that apply to the problem of Job Shop
Scheduling, as respecting the precedence between operations determining the technological

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sequence without interrupting any operations until completion [21]. The operations conform the
graph nodes that represent the problem, united by edges in which ants are moving. Each
individual only has local information of the system that shares through a hormone called
pheromone.
The update of the pheromone trail deposited on the edges can be done globally or locally. Ants
build roads that represent feasible solutions, guided by the pheromone trails and the heuristic
information of each edge [1]. For this reason the ant population performs a stochastic search,
selecting the next node to visit only based on information available locally, used on a
probabilistic approach where initially the ant decisions are completely random in the absence of
pheromone trails.
In the literature, several algorithms have been proposed following the ACO probabilistic
technique for finding approximate solutions to complex optimization problems. The first ACO
algorithm was Ant System (AS), proposed by Marco Dorigo in 1991 [26], and completed with the
contributions of Maniezzo and Colorni [1]. New developments gave better results, like Ant
Colony System (ACS) [2], the Max-Min Ant System (MMAS) [7], the Rank-based Ant System
(ASrank) [8], among others. This article presents a variant of Elitist Ant System, also proposed by
Dorigo as an improvement to SH [1], applied in JSSP instances widely used known as LA
instances, that were raised by Lawrence [11].
2. JSSP PROBLEM FORMULATION
The JSSP or resource planning problem (or jobs) consists in "accommodate resources over time
to perform a set of jobs" [6], building plan or execution sequence of jobs j in a set of m machines
[13], where an operation is every job that is processed in each machine (Operation(j, m)) and is
assigned a specific processing time.
This problem is presented in multiple human activities, taking applications to tasks such as
scheduling for packet delivery (eg airway), computer networks (networking), computers
(multitasking and multiprocessing), project management (agenda or plan), production and
administrative processes (eg assembly lines, etc.) [20].
JSSP must comply with certain restrictions in the execution of jobs and the goal is to complete
them in the shortest possible time. This time to optimize is known as makespan (CMAX) or
Maximum Workflow which forms the objective function to minimize given as
, where is the job completion time. It is a combinatorial
optimization problem because the number of candidate solutions is combinatorial in size with
variables of discrete nature, therefore the representation of the solutions are permutations over the
operations of each job, making it impossible to determine all possible solutions in a reasonable
time.
2.1. Computational complexity of JSSP
In 1976 Michael Garey [17] provided evidence that this problem is NP-hard for m> 2, ie cannot
be quickly found (polynomial-time) an optimal solution for JSSP with more than two machines.
Along with David Johnson in 1979, they finished demonstrating that JSSP is NP-hard [18], unless
in Computational Complexity Theory is proved that P = NP, if so, any problem that can be
checked quickly by a computer, it could also be quickly resolved by that computer.
The NP-hard complexity of JSSP lies in the vast number of possible combinations that arise
because each sequence of operations on a machine can be permuted independently of the

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sequence of operations on another machine, so with a few jobs and machines can have
possible solutions which corresponds to the search space (S) of the problem.
2.2. Formal definition of Job Shop Scheduling Problem
Having [5]:
: Set of n jobs to be processed.
: Set of m machines or resources.
: Operation of the job that must be processed in the machine by .
: Uninterrupted period of processing time for each operation.
Objective function: Minimize
Subject to:
Start times restriction for each operation
Precedence constraint if preceding
Disjunctive restriction if preceding ,
in another case.
Where: , with .
The previous set of constraints of the JSSP is explained of this way [21]:
Start restriction: The time when an operation starts are not specified, so work can start at any
point in time as long as the required machine is available.
Restriction of precedence: Each job must go through a particular sequence of operations that is
predefined, so that operations cannot begin until the end of its predecessor, preventing the
processing of two operations of the same job simultaneously.
Restrictions disjunctive: A machine can process only one job at a time. Each operation must be
fully processed on a single machine and cannot be interrupted even if there are jobs waiting for
that machine to be available, for instance, no work may be processed more than once on the same
machine.
In addition to the above restrictions, we have determined that all operations have the same
priority of processing, and all machines are the same and can be idle at any time. The fulfillment
of these restrictions can be seen clearly by a Gantt chart (Figure 1), which shows an instance of
JSSP (Table 1) matrix defined by [15], in which it has an additional column to indicate that each
row of the matrix corresponds to a job (J1, J2 and J3).
Table 1. An instance of JSSP 3×3
Job (J) Machine (time)
Sequence: S1 S2 S3
J1 3 (4) 2 (3) 1 (3)
J2 2 (1) 3 (2) 1 (4)
J3 2 (3) 1 (2) 3 (3)

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Figure 1. Gantt diagram of a 3×3 instance of JSSP
The JSSP is usually represented as a disjunctive graph G = (V, C ∪ D) [14], where V is
the set of nodes (Figure 2) representing the Operations (job, machine) with the exception
of starting node (I) and ending nodes (F) of the graph, C is a set of directed graphs (→)
linking operations corresponding to the same job (technological sequence), and D is a set
of undirected graphs connecting operations running on a same machine. In addition
the processing time of each operation is placed in the upper part of node.
Figure 2. Graph of a 3×3 instance of JSSP
The problem of Job Shop Scheduling has been tackled with methods that can only solve
instances of a limited number of operations, because they perform exhaustive searches to
find the exact solution, as Branch and Bound (B&B) proposed in 1960 [23], can solve
only up to 15 x 14, ie up to 220 operations [24]. So it must use approximate methods
(Table 2 [9]) like simulated annealing (SA), Tabu Search (TS) [10], Iterative Local
Search (ILS), GRASP, ACO, Evolutionary Algorithms (EA) as Artificial Immune System
(AIS) and Cultural algorithm (CULT), etc.

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Table 2. The main features of metaheuristics
Metaheuristic Features
SA
Acceptance criteria
Cooling Time
TS
Choosing neighbor (tabu list)
Suction Criterion
EC
Recombination
Mutation
Selection
ILS
Local search
Initial movement
Acceptance criteria
ACO
Construction probabilistic
Update pheromone
GRASP
Local Search
Restricted Candidate List (RCL)
3. ANT COLONY OPTIMIZATION
This bioinspired algorithm is based on a population of ants that perform a cooperative search. In
an experiment of the self-organization of Argentine ants made in 1989 [12], we observed the
feeding behavior of a colony of ants, that were able to find the shorter branches of a bridge
between the nest and the food (Figure 3), through the pheromone trail they leave behind when
moving.
Figure 3. Picture of a colony of ants that find the shortest path to the food [12]
The ants initially move randomly in search of food and along the way back to the colony the
pheromone is deposited. If another ant finds this trail, probably it will follow it increasing the
amount of pheromone, which further stimulates other ants to follow this path (Figure 4). But over
time the pheromone trail starts to evaporate and reduces its attractiveness, making more attractive

58
only the most used trajectories, causing convergence to an optimal solution that is the only path
that eventually most ants will follow. By the long road less pheromone accumulates because of
the low passing frequency of the ants when they spend more time completing their road.
Figure 4. A. ants in a pheromone trail between nest and food; B. an obstacle interrupts the trail; C. ants find
two paths to go around the obstacle; D. a new pheromone trail is formed along the shorter path [19]
In the ACO algorithms family, ant’s behavior is simulated with a virtual agent that has the
capacity to explore a limited search space and obtain information about the surrounding
environment. The artificial ant (k) moves from one node to another (from source node i to
destination node j), building step by step solution to be written to the Tabuk memory (that stores
information about the nodes sequence or route taken until time t), that ends when it reaches one of
the accepting states defined by the objective of the problem.
Thus, the ants can construct approximate solutions to complex problems such as sequencing,
assigning, planning or programming. Each edge of the graph has two types of associated
information that guide the movement of the ant [4] and whose values are modified by ants at each
iteration:
ηij Heuristic information that measures the heuristics preference of moving from node i to node j,
when touring the edge aij. Ants do not change this information during the execution of the
algorithm.
τij Information of artificial pheromone trails, that measures the "desirability learned" of the i to j
movement. This information is modified during the execution of the algorithm depending on the
solutions found by the ants to reflect the experience gained by these agents.
Pseudocode of the ACO metaheuristic [3]:
ACO procedure
Set parameters,Initialize the pheromone trails
scheduled activities
Construction of solutions by ants
Server of actions (Optional)
Updating pheromone
End-Scheduled activities
End-procedure
The metaheuristic consists of a parameter initialization step and three algorithmic procedures
whose activation is regulated by the builder Scheduled activities, in which is repeated until a

59
termination condition is met, such as reaching a maximum number of iterations or a maximum
CPU time. The three algorithmic procedures submitted to the Scheduled activities consist of [25]:
Construction of solutions by ants is the probabilistic construction of solutions by all the ants in a
colony, which visit the adjacent states of the considered problem. The ants can move by applying
a stochastic decision policy using information from the pheromone trails and the heuristic
information, with which ants incrementally construct a solution to the problem.
Server of actions are centralized actions that modify the behavior of the algorithm and cannot be
developed by ants individually. The most common is the local optimization or improvement of
the solutions with the application of a local search algorithm. The locally optimized solutions are
then used to set the values of the pheromone to update.
Updating pheromone is the process that updates the pheromone trails on each aij edge, called
posteriori online update or offline because it is performed at the end of a road. The amount of
pheromone that deposits each ant at the edges depends on the total length of the path (equation 3).
It also can perform a step by step online update of the pheromone trails, that is a local update or
in "real time" of the pheromone, performed when an ant moves from node i to node j. The
pheromone trail value is reduced by a constant evaporation of pheromone, which prevents
premature convergence of the algorithm by discarding the less frequented corners.
4. ELITIST ANT SYSTEM (EAS)
This version of the ACO implements a simple change to the Ant System that improves the results,
simply reinforcing the pheromone trail of the best path that is found in each iteration. At the
edges of the best generated solution by an ant, more pheromone is deposited through all the other
ants.
In this algorithm artificial ants perform a probabilistic construction of solutions in each cycle, for
which they require represent the problem by means of a graph in which the ants move along each
edge from one node to another to build roads that represent solutions from a randomly chosen
initial node, the following choice is the next node in this path is done according to the state
transition rule (equation 1).
Equation 1
Where α and β parameters determine the influence of the values of the pheromone information
and from the heuristic information ( ) respectively, over the decision of each ant (k). It seeks
that the edges with large amount of pheromone to be the most visible, having a higher transition
probability to the edges of the other nodes of the set of achievable operations. To have a balanced
algorithm (with an appropriate adjustment), α and β parameters must have appropriate values,
avoiding close to zero values, because if α = 0, only the heuristic information would indicate that
possible elements of the solution will have a higher probability of being selected, which
corresponds to a stochastic greedy algorithm (greedy), and if β = 0, will only be relevant the
amount of pheromone. In both cases the ants might get stuck in a local optimum, generating the
same solution in each iteration, without opportunities to find a better solution which could be the

60
global optimum solution. These parameters are normally set to integer values between 1 and 5,
but in this case we will relate them as follows with .
The amount of pheromone present at each edge of the road in the generation is given by
the equation 2.
Equation 2
Where is the contribution of the ant to the total pheromone of the generation and is
the evaporation rate of the pheromone. The reason for including the evaporation rate is that old
pheromone should not have much influence on future decisions of the ants. The amount of
pheromone that each ant is contributing depends on the quality of the solution obtained which is
inversely proportional to the cost of the solution of the objective function (equation 3).
Equation 3
Where is a constant and is the length of the makespan of the solution obtained by the ant.
To accelerate the convergence of the algorithm, increasing the visibility of the pheromone trail on
all edges of the shortest path, passing all elitists ants (e) of the system. Therefore, the equation 3
for the best path built in each cycle is replaced by the equation 2.
Equation 4
5. EAS IMPLEMENTATION FOR JSSP
The rapid convergence of this algorithm can reduce the scanning capability since the ants soon
will end in a single way, which can be a local optimum. To compensate this, is allowed to include
in the set of achievable operations (point 3.3 of pseudocode), operations that makes the machines
wait (on pause) some units of time to begin execution because the corresponding job is still active
on another machine. But this operation that delay or retards the onset of the machines will only be
selected if the edge that reaches the node, has enough pheromone to make the probability to be
greater than the operations that have immediately available jobs. That will only be given with
large amounts of pheromone, because having idle machines is not adequate and is penalized
lowering the visibility of the operation.
This method further explores the search space in order to obtain many solutions, from which it
can be obtained solutions that exceed the local optima found in the first iterations. These optimal
are the ones limiting the search, stopping it on solutions distant up to a 5% the global optimum.
The initial diversity of the algorithm is the one that ensures that the ants move towards the search
space where the path corresponding to the overall optimal solution is found. The following is the
pseudocode implemented to solve the JSSP:

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Pseudocode Description:
1. The best results were obtained with the parameters initialized in ,
the number of cycles (or iterations) is fixed at 1000 and the amount of ants (K) is calculated
according to the number of jobs, which is the amount elements that the J set has, thereby:
Equation 5
2. The pheromone trail of all edges is started in a small positive constant.
3. The Probabilistic Construction Phase of solutions begins by K ants.
3.1 The first operation is selected randomly between nodes initially visited according to the
constraints of the problem.

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3.2 The selection of the decidability rule is done randomly, with equal probability between the
rule with the shortest processing time SPT (Shortest Processing Time) or the rule with the longest
processing time LPT (Longest Processing Time) of the operations [30].
3.3 While tabúk memory has not finished filling, it means that the ant has not completed the plan
generation therefore it continuous traveling the graph until completing the total operations
( ). The tabuk list restricts the choice of operations to prevent a return to recently
visited nodes. In the set of visited operations are included operations that generate a delay in the
machines less or equal to five time units. To maintain the balance affected by the delay generated,
visibility of the node is reduced on a percentage point per unit of time lost.
3.4 Once each ant has built a solution, the pheromone actualization process is started, reviewing
the traveled path to add the appropriate amount of pheromone according to equation 3 or 4, to the
pheromone accumulator of the current cycle. If the makespan of the solution is expensive, less
importance is given to the way, thus depositing few pheromone on edges.
3.5 Update pheromone trails of the visited edges using a process known as posteriori online
update, which is a global update performed offline, that is, after the execution of each cycle of the
algorithm. It is deposited in the pheromone trails of each of the edges of the graph, what the ants
have been added in the respective pheromone accumulator. Then the actual pheromone
accumulator is restarted at zero for not to redeposit this pheromone in the next cycle.
3.6 The best quality plan of the current cycle is saved with its respective makespan.
3.7 Memory (tabuk) is erased on each ant to start building new plans in the next cycle.
4. Shows the best plan of all cycles performed by the algorithm.
6. ANALYSIS AND COMPARISON OF RESULTS
Results shown in Table 3 were obtained in 30 executions of the algorithm (1000 iterations) for
each of the 40 JSSP instances raised by Lawrence [11], that are of different sizes and difficulty,
and because of its wide use, we can compare the results with other techniques that generate the
best known solution (BKS) taken from [13] and [27] The table shows, the name of the instance of
Lawrence, its size and BKS, the best makespan found and their percentage relative error respect
al BKS, the makespan average, standard deviation, and finally the average number of evaluations
of the objective function.
Table 3. Experimental results
Instance Size BKS
Best
Cmax
Relative
Error (%)
Cmax
Average
Standard
deviation
#Eval.
Average
LA01 10 x 5 666 666 0 667.8 2.3 2375
LA02 10 x 5 655 669 2.13 689.7 6.2 2809
LA03 10 x 5 597 623 4.36 644.8 8.0 2230
LA04 10 x 5 590 611 3.56 617.7 5.0 2257
LA05 10 x 5 593 593 0 593.0 0.0 101
LA06 15 x 5 926 926 0 926.0 0.0 531
LA07 15 x 5 890 890 0 898.2 5.6 3443
LA08 15 x 5 863 863 0 863.1 0.4 2251
LA09 15 x 5 951 951 0 951.0 0.0 391
LA10 15 x 5 958 958 0 958.0 0.0 637
LA11 20 x 5 1222 1222 0 1222.0 0.0 1504

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LA12 20 x 5 1039 1039 0 1039.0 0.0 1752
LA13 20 x 5 1150 1150 0 1150.0 0.2 2952
LA14 20 x 5 1292 1292 0 1292.0 0.0 471
LA15 20 x 5 1207 1212 0.41 1245.6 9.9 3836
LA16 10 x 10 945 1005 6.35 1020.1 10.1 2700
LA17 10 x 10 784 812 3.57 836.1 10.9 2401
LA18 10 x 10 848 885 4.36 904.8 9.8 2946
LA19 10 x 10 842 875 3.92 881.7 4.7 2394
LA20 10 x 10 902 912 1.11 936.8 9.6 2496
LA21 15 x 10 1046 1107 5.38 1162.3 16.4 3658
LA22 15 x 10 927 1018 9.82 1050.2 14.2 2938
LA23 15 x 10 1032 1051 1.84 1069.2 10.0 3826
LA24 15 x 10 935 1011 8.13 1033.5 8.3 3097
LA25 15 x 10 977 1062 8.7 1093.3 14.1 3632
LA26 20 x 10 1218 1296 6.4 1339.6 16.2 5955
LA27 20 x 10 1235 1362 10.28 1379.8 9.6 4450
LA28 20 x 10 1216 1330 9.38 1363.8 13.9 3938
LA29 20 x 10 1157 1339 15.73 1374.4 11.9 4532
LA30 20 x 10 1355 1410 4.06 1443.2 15.0 5186
LA31 30 x 10 1784 1798 0.78 1825.8 12.5 7098
LA32 30 x 10 1850 1868 0.97 1906.0 20.7 8016
LA33 30 x 10 1719 1731 0.7 1771.0 15.1 5796
LA34 30 x 10 1721 1788 3.89 1823.9 13.9 6811
LA35 30 x 10 1888 1913 1.32 1974.1 22.8 7357
LA36 15 x 15 1268 1396 10.09 1430.4 18.5 3405
LA37 15 x 15 1397 1517 8.59 1544.2 12.7 2142
LA38 15 x 15 1196 1315 9.95 1343.8 10.1 4051
LA39 15 x 15 1233 1304 5.76 1359.5 16.2 3266
LA40 15 x 15 1222 1307 6.96 1323.7 9.2 2655
Average: 3.96 9.09 3307.15
Although the efficacy of the algorithm to find the optimum isn't high, reaching the BKS in 27.5%
of the LA instances, the average relative error in the 40 instances is only 4%, which is a good
approximation to the optimal of JSSP. And the highlight is the low average number of objective
function evaluations, which is much lower compared to the methods that obtained the BKS (Table
4). The AIS on average takes 52 times more evaluations our algorithm and Cultural algorithm
(CULT) has 137 times more evaluations. Compared with Tabu Search (TS) the number of
evaluations is only three times lower, because it is not included the number of evaluations
performed by the INSA algorithm that gives the TS base solution [28]. So we can state that our
algorithm has a high computational efficiency, reducing costs in time and memory, something
very important in this type of problems where getting an "economic" solution is as important as
the quality of it. Furthermore, EAS is an algorithm stable because its standard deviation is very
low.
The following table compares the average number of objective function evaluations made by
EAS, with those made by TS, AIS and CULT [29]:

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Table 4. Number of evaluations of objective function
Algorithm N° of Average evaluations
EAS 3307
AIS 175058
CULT 454525
TS 11108
Table 3 shows that on the problems of size 10 x 5 did not have trouble finding the BKS, with the
exception of the instance LA04 until LA02l, where the best obtained result is close to the BKS
(less than 5%). Also for instances of size 15 x 5 and 20 x 5, with the exception of the LA15 that
only moves away from BKS in 5 units of time. In the other instances (size 10 x 10, 15 x 10, 20 x
10, 30 x 10 and 15 x 15), which have 5 or 10 machines more than the previous, complexity is
quite high because of the considerable number of operations to be performed, this means lower
quality solutions obtained. For example, to instances of size 30 x 10, 300 operations must be
performed, and the total number of possible combinations is (30!)10
, that is approximately 2.65 x
1042
. However, the algorithm achieves to present high quality solutions on instances of 30 x 10
(Figure 5). In general, 65% of executed instances approaches less than 5% of BKS and 47.5%
deviate by less than 3% of the BKS.
Figure 5. Relative Error average by instance size
7. CONCLUSIONS
The Ant Colony Optimization is a technique of swarm intelligence, which is applied for
combinatorial optimization problems as JSSP. The algorithm implemented, Elitist Ant System,
has proven to be competitive by find good quality solutions to JSSP in a low number of objective
function evaluations, although requires improvements to obtain the best known solution in all LA
instances. Therefore, ACO is a metaheuristic that has the potential to obtain efficiently solutions
of scheduling problems, with minimal cost of time and computational resources.

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AUTHORS
Edson Flórez was born at San Gil (Colombia) in 1991, is currently an engineer student at
School of Engineering and Computing Systems of the Universidad Industrial de Santander.
His research areas are on Swarm Intelligence algorithms, Operations Research and
Distributed systems.
Wilfredo Gómez was born at Bucaramanga(Colombia) in 1984, is currently an Magister
candidate in system engineering at School of Engineering and Computing Systems of
the Universidad Industrial de Santander. His research areas are on Bioinspired
Computation, Operations Research and Education.
Lola Bautista is MSc. in Computer Engineering University of Puerto Rico she is currently
the Director of the Research Group in Biomedical Engineering. His research areas are on
Software Design in Cardiology and Electrocardiography, Digital Signal Processing and
Graphics, Data Mining.

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