ANT COLONY OPTIMISATION APPLIED TO JOB SHOP SCHEDULING PROBLEM

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQIUREMENTS

FOR THE DEGREE OF

Bachelor of Technology

in

Mechanical Engineering

By DEBASISH DAS

Under the Guidance of Prof. B.B. Biswal

Department of Mechanical Engineering National Institute of Technology

Rourkela

2009

National Institute of Technology Rourkela

CERTIFICATE

This is to certify that the thesis entitled.”ACO applied to job shop scheduling problem”

submitted by Mr. Debasish Das in partial fulfillment of the requirements for the award of

Bachelor of technology Degree in Mechanical Engineering at National Institute of Technology,

Rourkela (Deemed University) is an authentic work carried out by him under my guidance.

To the best of my knowledge the matter embodied in the thesis has not been submitted to any

University /Institute for the award of any Degree or Diploma.

Date: Prof. B.B. Biswal

Dept. of Mechanical Engg.

National Institute of Technology

Rourkela-769008

Acknowledgement

I would like to express my deep sense of gratitude and respect to our supervisor Prof.

B.B.Biswal, for his excellent guidance, suggestions and constructive criticism. I consider

ourselves extremely lucky to be able to work under the guidance of such a dynamic personality.

I am also thankful to Prof K.P. Maity and Prof. P.J. Rath (Project Coordinators) for smooth

completion of the project curriculum. I extend my gratitude to all staff members of Department

of Mechanical Engineering and other departments of NIT Rourkela.

Lastly we would like to render heartiest thanks to our M.Tech students(ME) whose ever helping

nature and suggestion has helped us to complete this present work.

Debasish Das

CONTENTS

Sl.No Topic Page

1. Certificate i

2. Acknowledgement ii

3. Contents iii

4. Abstract iv

5. Chapter 1: General Introduction

1-2

6. Chapter 2: Literature Survey 4 - 6

7. Chapter 3: Present Work and Problem Formulation

7 - 10

8. Results and Discussion 11-13

9. Conclusion 14-16

10. References 17-24

ABSTRACT

The problem of efficiently scheduling production jobs on several machines is an important

consideration when attempting to make effective use of a multimachines system such as a

flexible job shop scheduling production system (FJSP). In most of its practical formulations, the

FJSP is known to be NP-hard,so exact solution methods are unfeasible for most problem

instances and heuristic approaches must therefore be employed to find good solutions with

reasonable search time. In this paper, two closely related approaches to the resolution of the

flexible job shop scheduling production system are described. These approaches combine the

Ant system optimisation meta-heuristic (AS) with local search methods, including tabu search.

The efficiency of the developed method is compared with others.

CHAPTER 1GENERAL INTRODUCTION

Ant Colony Optimization (ACO) is a metaheuristic inspired by the foraging behavior of ants,

which has been used to solve combinatorial optimization problems and the Ant System (AS) was

the first algorithm within this class.

In the classical Job Shop Scheduling Problem, a finite number of jobs are to be processed by a

finite number of machines. Each job consists of a predetermined sequence of operations, which

will be processed without interruptions by a period of time in each machine. The operations that

correspond to the same job will be processed according to their technological sequence and none

of them will be able to begin its processing before the precedent operation has finished. A

feasible schedule is an assignment of operations in time on a machine without violation of the

job shop constraints. A makespan is defined as the maximum completion time of all jobs. The

objective of JSSP is to find a schedule that minimizes the makespan.

Modern hybrid heuristics are by their nature non-exhaustive, and so there is often scope for

different approaches to better previous solution methods according to the execution speed or the

quality of feasible solutions. Traditional approaches to resolve the FJSP are as varied as the

different formulations of the problem, but include fast, simple heuristics [2][12], tabu search

[15], evolutionary approaches [5] and modern hybrid meta-heuristics that consolidate the

advantages of various different approaches [1][13]. The ant colony optimisation (ACO) was

described by Dorigo in his PhD thesis [6] and was inspired by the ability and the organisation of

real ant colony using external chemical pheromone trails acting as a means of communication.

Ant system algorithms have since been widely employed on the NP-hard combinatorial

Optimisation problems including problems related to Continuous Design Spaces research [4],

and job shop scheduling [16]. However, they have not previously been applied to the FJSP

described in what follows. Local search methods encompass many optimisation approaches and

have been shown that the efficiency of their use with an ant system approach [7]. The approach

described in this paper for the FJSP shows the quality of solutions found, using benchmark

problems. The performances of the proposed approach are evaluated and compared with the

results obtained from other methods. In this paper, an application of the ant system algorithms

combined by the tabu search heuristic is proposed for solving the FJSP. Thus, The FJSP is

described and formulated in section 2. Then, in section 3, The suggested approach by ACO with

the tabu search is described. An illustrative example is given in section 4. The last section will be

devoted to the presentation of some results and some conclusions relating to this research work.

CHAPTER 2LITERATURE SURVEY

Ant Colony Optimization (ACO) is a paradigm for designing metaheuristic algorithms for

combinatorial optimization problems. The first algorithm which can be classified within this

framework was presented in 1991 and, since then, many diverse variants of the basic principle

have been reported in the literature. The essential trait of ACO algorithms is the combination of a

priori information about the structure of a promising solution with a posteriori information about

the structure of previously obtained good solutions. Metaheuristic algorithms are algorithms

which, in order to escape from local optima, drive some basic heuristic: either a constructive

heuristic starting from a null solution and adding elements to build a good complete one, or a

local search heuristic starting from a complete solution and iteratively modifying some of its

elements in order to achieve a better one. The metaheuristic part permits the lowlevel

heuristic to obtain solutions better than those it could have achieved alone, even if iterated.

Usually, the controlling mechanism is achieved either by constraining or by randomizing the set

of local neighbor solutions to consider in local search (as is the case of simulated annealing or

tabu search), or by combining elements taken by different solutions (as is the case of evolution

strategies and genetic or bionomic algorithms). The characteristic of ACO algorithms is their

explicit use of elements of previous solutions. In fact, they drive a constructive low-level

solution, as GRASP [30] does, but including it in a population framework and randomizing the

construction in a Monte Carlo way. A Monte Carlo combination of different solution elements is

suggested also by Genetic Algorithms, but in the case of ACO the probability distribution is

explicitly defined by previously obtained solution components.

The particular way of defining components and associated probabilities is problem- specific, and

can be designed in different ways, facing a trade-off between the specificity of the information

used for the conditioning and the number of solutions which need to be constructed before

effectively biasing the probability distribution to favor the emergence of good solutions.

Different applications have favored either the use of conditioning at the level of decision

variables, thus requiring a huge number of iterations before getting a precise distribution, or the

computational efficiency, thus using very coarse conditioning information. The chapter is

structured as follows. Section 2 describes the common elements of the heuristics following the

ACO paradigm and outlines some of the variants proposed. Section 3 presents the application of

ACO algorithms to a number of different combinatorial optimization problems and it ends with a

wider overview of the problem attacked by means of ACO up to now. Section 4 outlines the

most significant theoretical results so far published about convergence properties of ACO

variants.

5.2.1 Ant System

The importance of the original Ant System (AS) resides mainly in being the prototype of a

number of ant algorithms which collectively implement the ACO paradigm. AS already follows

the outline presented in the previous subsection, specifying its elements as follows. The move

probability distribution defines probabilities pιψk to be equal to 0 for all moves which are

infeasible (i.e., they are in the tabu list of ant k, that is a list containing all moves which are

infeasible for ants k starting from state ι), otherwise they are computed by means of formula

(5.1), where α and β are userdefined parameters (0 ≤ α,β ≤ 1):

In formula 5.1, tabuk is the tabu list of ant k, while parameters α and β specify the impact of trail

and attractiveness, respectively. After each iteration t of the algorithm, i.e., when all ants have

completed a solution, trails are updated by means of formula (5.2):

where Δτιψ represents the sum of the contributions of all ants that used move (ιψ) to construct

their solution, ρ, 0 ≤ ρ ≤ 1, is a user-defined parameter called evaporation coefficient, and Δτιψ

represents the sum of the contributions of all ants that used move (ιψ) to construct their solution.

The ants’ contributions are proportional to the quality of the solutions achieved, i.e., the better

solution is, the higher will be the trail contributions added to the moves it used. For example, in

the case of the TSP, moves correspond to arcs of the graph, thus state ι could correspond to a

path ending in node i, the state ψ to the same path but with the arc (ij) added at the end and the

move would be the traversal of arc (ij). The quality of the solution of ant k would be the length

Lk of the tour found by the ant and formula (5.2) would become τij(t)=ρ τij(t-1)+Δτij , with

where m is the number of ants and k Δτ ij is the amount of trail laid on edge (ij) by ant k, which

can be computed as

Q being a constant parameter.

The ant system simply iterates a main loop where m ants construct in parallel their solutions,

thereafter updating the trail levels. The performance of the algorithm depends on the correct

tuning of several parameters, namely: α, β, relative importance of trail and attractiveness, ρ, trail

persistence, τij(0), initial trail level, m, number of ants, and Q, used for defining to be of high

quality solutions with low cost. The algorithm is the following.

1. {Initialization}

Initialize τιψ and ηιψ, ∀(ιψ).

2. {Construction}

For each ant k (currently in state ι) do

repeat

choose in probability the state to move into.

append the chosen move to the k-th ant's set tabuk.

until ant k has completed its solution.

end for

3. {Trail update}

For each ant move (ιψ) do

compute Δτιψ

update the trail matrix.

end for

4. {Terminating condition}

If not(end test) go to step 2

5.2.2 Ant Colony System

AS was the first algorithm inspired by real ants behavior. AS was initially applied to the solution

of the traveling salesman problem but was not able to compete against the state-of-the art

algorithms in the field. On the other hand he has the merit to introduce ACO algorithms and to

show the potentiality of using artificial pheromone and artificial ants to drive the search of

always better solutions for complex optimization problems. The next researches were motivated

by two goals: the first was to improve the performance of the algorithm and the second was to

investigate and better explain its behavior. Gambardella and Dorigo proposed in 1995 the Ant-Q

algorithm, an extension of AS which integrates some ideas from Q-learning, and in 1996 Ant

Colony System (ACS) a simplified version of Ant-Q which maintained approximately the same

level of performance, measured by algorithm complexity and by computational results.

Since ACS is the base of many algorithms defined in the following years we focus the attention

on ACS other than Ant-Q. ACS differs from the previous AS because of three main aspects:

Pheromone

In ACS once all ants have computed their tour (i.e. at the end of each iteration) AS updates the

pheromone trail using all the solutions produced by the ant colony. Each edge belonging to one

of the computed solutions is modified by an amount of pheromone proportional to its solution

value. At the end of this phase the pheromone of the entire system evaporates and the process of

construction and update is iterated. On the contrary, in ACS only the best solution computed

since the beginning of the computation is used to globally update the pheromone. As was the

case in AS, global updating is intended to increase the attractiveness of promising route but ACS

mechanism is more effective since it avoids long convergence time by directly concentrate the

search in a neighborhood of the best tour found up to the current iteration of the algorithm.

In ACS, the final evaporation phase is substituted by a local updating of the pheromone applied

during the construction phase. Each time an ant moves from the current city to the next the

pheromone associated to the edge is modified in the following way:

τ ij (t) = ρ τ ⋅ ij (t -1)+ (1− ρ )

τ ⋅ 0 where 0 ≤ ρ ≤ 1 is a parameter (usually set at 0.9) and τ0 is the initial pheromone value. τ0 is

defined as τ0=(n·Lnn)-1, where Lnn is the tour length produced by the execution of one ACS

iteration without the pheromone component (this is equivalent to a probabilistic nearest neighbor

heuristic). The effect of local-updating is to make the desirability of edges change dynamically:

every time an ant uses an edge this becomes slightly less desirable and only for the edges which

never belonged to a global best tour the pheromone remains τ0. An interesting property of these

local and global updating mechanisms is that the pheromone τij(t) of each edge is inferior limited

by τ0. A similar approach was proposed with the Max-Min-AS that explicitly introduces lower

and upper bounds to the value of the pheromone trials.

State Transition Rule

During the construction of a new solution the state transition rule is the phase where each ant

decides which is the next state to move to. In ACS a new state transition rule called pseudo-

random-proportional is introduced. The pseudorandom- proportional rule is a compromise

between the pseudo-random state choice rule typically used in Q-learning [76] and the random-

proportional action choice rule typically used in Ant System. With the pseudo-random rule the

chosen state is the best with probability q0 (exploitation) while a random state is chosen with

probability 1-q0 (exploration). Using the AS random-proportional rule the next state is chosen

randomly with a probability distribution depending on ηij and τij. The ACS pseudo-random-

proportional state transition rule provides a direct way to balance between exploration of new

states and exploitation of a priori and accumulated knowledge. The best state is chosen with

probability q0 (that is a parameter 0 ≤ q0 ≤ 1 usually fixed to 0.9) and with probability (1-q0) the

next state is chosen randomly with a probability distribution based on ηij and τ ij weighted by

α (usually equal to 1) and β (usually equal to 2) .

5.2.3 ANTS

ANTS is an extension of the AS, which specifies some underdefined elements of the general

algorithm, such as the attractiveness function to use or the initialization of the trail distribution.

This turns out to be a variation of the general ACO framework that makes the resulting algorithm

similar in structure to tree search algorithms. In fact, the essential trait which distinguishes

ANTS from a tree search algorithm is the lack of a complete backtracking mechanism, which is

substituted by a probabilistic (Non-deterministic) choice of the state to move into and by an

incomplete (Approximate) exploration of the search tree: this is the rationale behind the name

ANTS, which is an acronym of Approximated Nondeterministic Tree Search. In the following,

we will outline two distinctive elements of the ANTS algorithm within the ACO framework,

namely the attractiveness function and the trail updating mechanism.

Attractiveness

The attractiveness of a move can be effectively estimated by means of lower bounds (upper

bounds in the case of maximization problems) on the cost of the completion of a partial solution.

In fact, if a state ι corresponds to a partial problem solution it is possible to compute a lower

bound on the cost of a complete solution containing ι. Therefore, for each feasible move ι,ψ, it is

possible to compute the lower bound on the cost of a complete solution containing ψ: the lower

the bound the better the move. Since a large part of research in ACO is devoted to the

identification of tight lower bounds for the different problems of interest, good lower bounds are

usually available. When the bound value becomes greater than the current upper bound, it is

obvious that the considered move leads to a partial solution which cannot be completed into a

solution better than the current best one. The move can therefore be discarded from further

analysis. A further advantage of lower bounds is that in many cases the values of the decision

variables, as appearing in the bound solution, can be used as an indication of whether each

variable will appear in good solutions. This provides an effective way of initializing the trail

values.

Trail update

A good trail updating mechanism avoids stagnation, the undesirable situation in which all ants

repeatedly construct the same solutions making any further exploration in the search process

impossible. Stagnation derives from an excessive trail level on the moves of one solution, and

can be observed in advanced phases of the search process, if parameters are not well tuned to the

problem. The trail updating procedure evaluates each solution against the last k solutions

globally constructed by ANTS. As soon as k solutions are available, their moving average z is

computed; each new solution zcurr is compared with z (and then used to compute the new

moving average value). If zcurr is lower than z , the trail level of the last solution's moves is

increased, otherwise it is decreased. Formula (5.6) specifies how this is implemented:

where z is the average of the last k solutions and LB is a lower bound on the optimal problem

solution cost. The use of a dynamic scaling procedure permits discrimination of a small

achievement in the latest stage of search, while avoiding focusing the search only around good

achievement in the earliest stages. One of the most difficult aspects to be considered in

metaheuristic algorithms is the trade-off between exploration and exploitation. To obtain good

results, an agent should prefer actions that it has tried in the past and found to be effective in

producing desirable solutions (exploitation); but to discover them, it has to try actions not

previously selected (exploration). Neither exploration nor exploitation can be pursued

exclusively without failing in the task: for this reason, the ANTS algorithm integrates the

stagnation avoidance procedure to facilitate exploration with the probability definition

mechanism based on attractiveness and trails to determine the desirability of moves.

Based on the elements described, the ANTS algorithm is as follows.

1. Compute a (linear) lower bound LB to the problem

Initialize τιψ ( ι∀ ,ψ) with the primal variable values

2. For k=1,m (m= number of ants) do

repeat

2.1 compute ηιψ ∀(ιψ)

2.2 choose in probability the state to move into

2.3 append the chosen move to the k-th ant’s tabu list

until ant k has completed its solution

2.4 carry the solution to its local optimum

end for

3. For each ant move (ιψ),

compute Δτιψ and update trails by means of (5.6)

4. If not(end_test) goto step 2.

It can be noted that the general structure of the ANTS algorithm is closely akin to that of a

standard tree search procedure. At each stage we have in fact a partial solution which is

expanded by branching on all possible offspring; a bound is then computed for each offspring,

possibly fathoming dominated ones, and the current partial solution is selected from among those

associated to the surviving offspring on the basis of lower bound considerations. By simply

adding backtracking and eliminating the MonteCarlo choice of the node to move to, we revert to

a standard branch and bound procedure. An ANTS code can therefore be easily turned into

an exact procedure.

Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem

The state transition rule used by ant system, called a random-proportional rule, is given by (1),

which gives the probability with which ant K in city R chooses to move to the city S.

where is the pheromone, is the inverse of the distance , is the

set of cities that remain to be visited by ant k positioned on city r (to make the solution feasible),

and is a parameter which determines the relative importance of pheromone versus distance .

In (1) we multiply the pheromone on edge (r,s) by the corresponding heuristic value .

In this way we favor the choice of edges which are shorter and which have a greater amount of

pheromone. In ant system, the global updating rule is implemented as follows. Once all ants have

built their tours, pheromone is updated on all edges according to

is a pheromone decay parameter, is the length of the tour performed by ant k ,

and m is the number of ants. Pheromone updating is intended to allocate a greater amount of

pheromone to shorter tours. In a sense, this is similar to a reinforcement learning scheme , in

which better solutions get a higher reinforcement (as happens, for example, in genetic algorithms

under proportional selection). The pheromone updating formula was meant to simulate the

change in the amount of pheromone due to both the addition of new pheromone deposited by

ants on the visited edges and to pheromone evaporation. Pheromone placed on the edges plays

the role of a distributed long-term memory: this memory is not stored locally within the

individual ants, but is distributed on the edges of the graph. This allows an indirect form of

communication called stigmergy.

JOB SHOP SCHEDULING PROBLEM (JSSP)

The classic JSSP is composed of n-jobs and m-machines and it is denoted by

n/m/T /Cmax, where the parameter n represents the number of jobs, m is the

number of machines, T is the technological sequence of the jobs in each

machine, and Cmax indicates the performance measure which should be

minimized (i.e., maximum time taken to complete all jobs). An instance of

the JSSP can be represented by a matrix as it is shown inTable I.

In the example of Table I, we have 3 jobs, 3 machines and a technological

sequence represented in each row of the jobs. In the case of job 1 in Table I,

we can see that it should be processed in machine 1 first with a processing

time of 3 (in the matrix, this time is represented between parentheses). After

that, this job 1 is processed in machine 2 with processing time of 3 and

finishes in machine 3 with a processing time of 3. This description is called

technological sequence of job 1. When a job i is processed in a machine j, it

is called as “operation (i,j)”.

To apply the AS algorithm for JSSP we will use the graph representation G =

(V,C _D) described in [11] where:

• V is a set of nodes representing operations of the jobs together with two

special nodes: a start (0) node and an end (*) node, representing the

beginning and the end of the schedule, respectively.

• C is a set of conjunctive arcs representing technological sequences of the

operations.

• D is a set of disjunctive arcs representing pairs of operations which must

be processed on the same machine.

Figure 1 shows the corresponding graph for the instance of the JSSP

described in Table I, whose nodes represent each operation (i, j) where i is

the current job and j its corresponding machine (except for the nodes

marked with (0) and (*) because they indicate the start and end of the

graph). The processing time of each operation is denoted by tij on each

node. The conjunctive arcs give the technological sequence connecting all

operations of the same job and disjunctive arcs indicate pairs of operations

in the same machine.

ANT SYSTEM (AS)

In this section, we describe the operation of the classical AS for the JSSP proposed in, in which a

population of m artificial ants builds solutions by iteratively applying n times a probabilistic

decision policy until obtaining a solution for the problem. In order to communicate the individual

search experience to the colony, the ants mark the corresponding paths with some amount of

pheromone according to the type of solutions found. This amount is inversely proportional to the

cost of the path generated (i.e., if the path found is long, the amount of pheromone deposited is

low; otherwise, the amount of pheromone deposited is high). Therefore, in the following

iterations more ants will be attracted to the most promising paths. Besides the pheromone, the

ants are guided by a heuristic value in order to help them in the construction process. All the

decisions taken by the ant (the path found or solution), are stored in a tabu list (TL). As it was

indicated above, to apply the AS algorithm, the instance of the problem must be first constructed

in a graphical representation G. The AS starts with a small amount of pheromone c along each

edge on G. Each ant is then assigned a starting position, which is added to its tabu list. The initial

ant position is usually chosen at random.

Once the initialization phase is completed, each ant will independently construct a solution by

using equation (1) at each decision point until a complete solution has been found. After every

ant’s tabu list is full, the cost Cmax of the obtained solution is calculated.

The pheromone amount along each edge (i,j) is calculated according to equation(2). Finally, all

tabu lists are emptied. If the stopping criterion has not been reached, the algorithm will continue

with a new iteration.

The decision of each ant is based, not only the amount of pheromone τij , located along edge (i,j),

but also on the heuristic value ηij along this edge. The transition probability to move from node i

to node j for the kth ant at iteration t is defined as:

where α and β are parameters which allow the user to balance the importance given to the

heuristic (parameter β) with respect to the pheromone trails (parameter α). Setting β = 0 will

result in only considering the pheromone information in the ant’s decision, whereas if α = 0, only

the heuristic information will be used for the ant.

The pheromone trail levels to be used in the next iteration of the algorithm are given by the

formula:

τij(t + 1) = ρ × τij (t) + Δτij (2)

where ρ is a coefficient, such that (1−ρ) can be interpreted as a trail evaporation coefficient; that

is, (1 − ρ) × τij (t) represents the amount of trail which evaporates on each edge (i,j) in the period

between iteration t and t+ 1. The total amount of pheromone laid by the m ants Δτij , is

calculated by:

where Δτk ij is calculated as:

where Q is a positive real valued constant and Cmax is the cost of the solution of the kth ant,

while Q/Ck max gives the quantity of pheromone per unit of time. It is important to note that

pheromone evaporation causes the amount of pheromone on each edge of G to decrease over

time. The evaporation process is important because it prevents AS from prematurely converging

to a sub-optimal solution. In this way, the AS has the capability of forgetting bad (or even

partially good) solutions, which favors a more in-depth exploration of the search space.

CHAPTER 3PROBLEM FORMULATION

The FJSP may be formulated as follows. Consider a set of n independent jobs, noted Á =

fJ1;J2; :::;Jn; 1 · j · Jg, which are carried out by K machines Mk, M = fM1;M2; :::;Mk; 1 · k ·

Kg. Each job Jj consists of a sequence of nj operations Oi; j, i = 1;2; :::nj. Each routing has to be

performed to achieve a job. The execution of each operation i of a job Jj requires one ressource

selected from a set of available machines. The assignment of the operation Oi; j to the machine

Mk µM entails the occupation of the latter one during a processing time, noted pi; j;k. The

problem is thus to both determine an assignment scheme and a sequence of the operations on all

machines that minimize some criteria.

• A set of J independent jobs.

• Each job is characterized by the earliest starting time r j and the latest finishing time dj.

• Denote by pti; j and ri; j respectively the processing time and the ready date of the operation

Oi; j.

The pi; j;k represent the processing time pti; j with the machine Mk.

• A started operation can not be interrupted.

• Each machine can not perform more than one operation at the same time.

• The objective is to find an operation ordering set satisfying a cost function under problem

constraints.

The considered objective is to minimize the makespan Cmax.

ACO and Tabu search for FJSP Scheduling

In this stage, the application of the combined ant systems with tabu search techniques in the

resolution of FJSP problem are described.

Construction Graph and Constraints

Generally, the FJSP can be represented by a bipartite graph with two categories of nodes: Oi; j

and Mk. A task is mapped to a Oi; j node; a machine is mapped to a Mk. There is an edge

between the Oi; j node and the Mk node if and only if the corresponding task can be assigned to

the corresponding machine while respecting the availability of the machine and the precedence

constraints among the operations of different jobs. The cost of assignment is directly related to

the processing time of the task upon the machine.

To model the process in a more straightforward manner, we use the construction graph that is

derived from the utilization matrix. Below is a sample construction graph.

Table 1: Construction graph of 4 machines and 7 tasks.

With this construction graph, we can transform the FJSP into a traveling ant problem.

Specifically, given the representative table of n rows and m columns, and each of its cells is

associated with pi; j;k, representing this one distance among Oi; j and Mk. An ant seeks to travel

across the table in such a way that all of the following constraints will be satisfied: one and only

one cell is visited for each of the rows. In the rest of this paper, "tour" and "solution" are used

interchangeably; a pair of (operation, machine) means: operation is assigned to machine, table 2.

Table 2: Solution of Construction graph table 1

Ant systems scheduling

The Ant system approach was inspired by the behaviour of the real ants. The ants depose the

chemical pheromone when they move in their environment, they are also able to detect and to

follow pheromone trails. In our case, the pheromone trail describes how the ant systems build the

solution of the FJSP problem. The probability of choosing a branch at a certain time depends on

the total amount of pheromone on the branch, which in turn is proportional to the number of ants

that used the branch until that time. The probability Pf i jk that an ant will assign an operation Oi;

j of job Jj to an available machine Mk. Each of the ants builds a solution using a combination of

the information provided by the pheromone trail ti jk and by the heuristic function defined by hi

jk = pi; j;k.

Formally, the probability of picking that an ant f th will assign an operation Oi; j of job Jj to the

machine Mk is given in equation 1.

In this equation, D denotes the set of available non-executed operations set and where a and b are

parameters that control the relative importance of trail versus visibility. Therefore the transition

probability is a trade-off between visibility and trail intensity at the given time.

Updating the pheromone trail

To allow the ants to share information about good solutions, the updating of the pheromone trail

must be established. After each iteration of the ant systems algorithm, equation 2 describes in

detail the pheromone update used when all ants have completed an own scheduling solution

denote Lants, that represent the length of ant tour. In order to guide the ant systems towards good

solutions, a mechanism is required to assess the quality of the best solution. The obvious choice

would be to use the best makespan Lmin =Cmax of all solutions given by a set of ant.

After all of the ants have completed their tours, the trail levels on all of the arcs need to be

updated. The evaporation factor r ensures that pheromone is not accumulated infinitely and

denotes the proportion of ´SoldŠ pheromone that is carried over to the next iteration of the

algorithm. Then for each edge the pheromone deposited by each ant that used this edge are added

up, resulting in the following pheromone level-update equation:

where NBA defines the number of ants to use in the colony.Tabu search optimization

A simple tabu search was also implemented for this optimisation FJSP problem. The proposed

approach is to allow the ants to build their solutions and then the resulting solutions are taken to

a local optimum by the local search mechanism.

Each of these ant solutions is then used in the pheromone update stage. The local search is

performed on every ant solution, every iteration, so it needs to be fairly fast. In the case of the

FJSP problem, the method is to pick the machine responsible to the Cmax and check if any

operations Oi; j could be swapped between other machines which would result in a lower

makespan. Following their concept, the local search considers one problem machine at a time

and attempts to swap one operation from the problem machine with any other (non-problem)

machine in the solution (non-problem operations). Then the ants are used to generate promising

scheduling production solutions and the tabu search algorithm is used to try to improve these

solutions. The tabu search is performed on each problem machine and continues until there is no

further improvement in the makspean value of the solution.

The set up parameter values

The set up parameter values used in the ant system scheduling algorithms are often very

important in getting good results, however the appropriate values are very often entirely problem

dependent, and cannot always be derived from features of the problem itself:

• α determines the degree to which pheromone trail is used as the ants build their solution. The

lower the value, the less ‘attention’ the ants pay to the pheromone trail, but the higher values

implicate the ants then perform too little exploration, after testing values in the range 0.1-0.75

this algorithm works well with relatively high values (around 0.5-0.75).

• β determines the extent to which heuristic information is used by the ants. Again, values

between 0.1-0.75 were tested, and a value around 0.5 appeared to offer the best trade-off

between following the heuristic and allowing the ants to explore the research space.

• Γ is the value to which the pheromone trail values are initialized. Initially the value of the

parameter should be moderately high to encourage initial exploration, while the pheromone

evaporation procedure will gradually stabilise the pheromone trail.

• ρ is the pheromone evaporation parameter and is always set to be in the range [0 < r < x]. It

defines how quickly the ants ‘forget’ past solutions. A higher value makes for a more aggressive

search; it tests a value of around 0.5-0.75 to find good solutions.

• NBA defines the number of ants to use in the colony, a low value speeds the algorithm up

because less search is done, a high value slows the search down, as more ants run before each

pheromone update is performed. A value of 10 appeared to be a good compromise between

execution speed and the quality of the solution achieved. It is interesting to note that for each

value of parameters the ant systems scheduling meta-heuristics yields a good solution. Moreover,

its convergence speed depends essentially on the number of used ants NBA.

Building a solution stepsThe main steps in the strategy of the FJSP system by ant systems and tabu search algorithm are

given below.

• Initialize parameters NBA, a, b , t0, r.

• Create an initial solution and an empty tabu list of a given size.

In order to generate feasible and diverse solutions, initial ants are represented by solutions issued

from heuristic rules SPT, DL, FIFO, etc) and a random method. Heuristics are used to

approximate an optima solution as near as possible.

• Repeat the following steps until the termination criteria are met:

– Find new solution by ant systems procedure scheduling given in section 3.2.

– Evaluate the quality of the new solution.

– If a new solution is improved then the current best solution becomes new solution

– else If no new solution was improved then apply the tabu search optimisation given in section

3.4.

– Add solution to the tabu list, if the tabu list is full then delete the oldest entry in the list.

– Apply the updating pheromone trail procedure given in section 3.3.

• END Repeat

Illustration exampleLet us consider a flexible job shop scheduling problem, this example is to execute three jobs Jj

(j=1,2,3) and six machines Mk (k = 1; : : :;6) described in table 1.

Applying the ant systems meta-heuristic, the simulation propose four different scheduling with

Cmax = 19 ut (unit of time), shown in table 2 to 7.

The solution given in the table 7 has a makespan equal to 19 ut. The machine M5 is the cause of

this value of makespan. To solve this problem, the tabu search optimisation is applied for this

solution. Indeed, this method finds the operation O2;2 for job J2 on M2 that can be swapped

with other machines which will reduce makespan to 18 ut. And this method finds that the

operation O1;3 for the job J1 executed by M2 and can be swapped with M5 who will execute the

operation O2;2 for the job J2. Finally, the obtained solution by the tabu search is better than

before, table 8.

RESULTS AND DISCUSSIONS

All ant systems and tabu search optimisation results presented are for 1000 iterations with 10 the

number of ants, and each run was performed 10 times. The algorithms have been coded in

Matlab and C++ and tested using a P4 Pentium processor 2.4 GHz and Windows XP system.

To illustrate the effectiveness and performance of the algorithm proposed in this paper, six

representative benchmark FJSP instances (represented by problem n£m) based on practical data

have been selected to compute.

Concerning the FJSP instances, the different results show that the solutions obtained are

generally acceptable and satisfactory. The values of the different objective functions show the

efficiency of the suggested approach, table 9. Moreover, the proposed method enables us to

obtain good results in a polynomial computation time. In fact, the efficiency of this approach can

be explained by the quality of the ant system algorithms combined by the tabu search heuristic to

the optimization of solutions.

CONCLUSION

In this paper, a new approach based on the combination of the ant system with tabu search

algorithm for solving flexible job-shop scheduling problems, is presented. The results for the

reformulated problems show that the ant systems with local search meta-heuristic can find

optimal solutions for different problems that can be adapted to deal with the FJSP problem. The

performances of the new approach are evaluated and compared with the results obtained from

other methods. The obtained results show the effectiveness of the proposed method. Ant system

algorithms and the tabu search techniques described are very effective and they alone can

outperform all the alternative techniques.

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