Improved Bacteria Foraging Optimization Algorithm for Solving

Flexible Job-Shop Scheduling Problem

Xingang Wang

, Pengfei Yi

Qilu university of technology, Jinan , China

wxg@spu.edu.cn, yipengfeiok@163.com

Keywords: Bacterial foraging algorithm; Flexible shop scheduling; Bacteria motor step; Stop condition.

Abstract: Bacterial foraging algorithm (BFO) is an emerging algorithm, which has been widely applied in many

fields by researchers . This paper designed an improved adaptive step and stop condition for solving local-

optimal and premature problems, and applied this improved algorithm to the flexible job-shop scheduling

Problem(FJSP). According to the changes of crowding lever between bacteria, step’s evaluation are divided

into three stages. Numerical simulation shows that the improved algorithm has avoided local optimal and

premature problems，and is superior to standard BFOA and genetic algorithm．

1 INTRODUCTION

Flexible job-shop Scheduling Problem (FJSP) is the

extension of JSP. It is a typical NP - hard problem to

optimize the allocation of production resources

(Jingjing Cui, Yanming Sun, Lanxiu Che, 2011). In

addition to take unprocessed process into account,

allows a process to choose on multiple machines

which have the ability to process. FJSP more

complex than JSP in combinatorial optimization

problem and more in line with the actual production

environment, so it has important theoretical

significance and practical value to study FJSP.

In recent years, there are a variety of intelligent

optimization algorithm be used to solve FJSP at

home and abroad, such as simulated annealing, ant

colony optimization and genetic algorithm, Waligora

G. propose the simulated annealing algorithm for

solving multi-objective scheduling problem

(Waligora G., 2014), achieved good effect. Qin Zhao

studied queuing theory in the application of shop

scheduling, and presents a detailed of theoretical

basis(Qin Zhao, Fuqing Zhao, 2013). Hongjun Liu

proposed a kind of new optimization genetic

algorithm

(Hongjun Liu, Shuai Zhao, 2011), blend

the idea of annealing strategy and tabu search into

genetic algorithm, combined with the simulated

annealing mechanism and tabu search mutation

mechanism, it is suitable for solving the workshop

scheduling problem. Moslehi G. Proposed improved

particle swarm optimization algorithm to solve

multi-objective characteristic on flexible job shop

scheduling problem (Moslehi G., Mahnam M.,

2011). The algorithm through fitness function to

judge the individual or group’s location weather is

the best, Bacteria foraging process can be described

as three processes: tendency, replication and

migration.

In this paper, based on original algorithm,

through analyzing foraging behavior of individuals

and populations, improved movement step and the

iteration stop condition, effectively improve

convergence rate and accuracy. Through compare

with the original algorithm, the progressive of the

algorithm be verified.

2 FLEXIBLE JOB SHOP

SCHEDULING

Suppose processing workshop with W artifacts and

M machines, each artifact contains one or more

processes, each working procedure can be performed

on one or more machine, the processing times on

each machine can be predetermined. Express

the ith procedure of

artifact

, M

said optional

machine set of process

FJSP problem can be

simplified into two sub problems which are machine

Wang X. and Yi P.

Improved Bacteria Foraging Optimization Algorithm for Solving Flexible Job-Shop Scheduling Problem.

DOI: 10.5220/0006443800630067

In ISME 2016 - Information Science and Management Engineering IV (ISME 2016), pages 63-67

ISBN: 978-989-758-208-0

allocation and process order (Xiuli Wu, Zhiqiang

Zhang and Yanhua Du, 2015): Machine allocation

problem point that the process choose qualified

processing machine basis on the optimization goal;

Process scheduling problem refers to every process

allocated on each machine in what order processing.

In flexible manufacturing, there are many types of

the optimization goal, Tasks, for example, the

maximum completion time, the total load of the

machine and the total processing cost, it is unlikely

to achieve the optimal state of multiple targets at the

same time, the utmost pursuit of most businesses is

efficiency, namely, efficiently complete the general

assignment in the shortest possible time. This paper

set the maximum completion time as the

optimization goal. The objectives function as

follows:

))

(max(min

min

ijmax

= (1)

max

refers to the maximum completion time of

scheduling task,

refers to completion time of

Process

. In addition, during processing need

meeting some assumptions, as follows

：

0)1

)(

(

)

(

)(

(

)

(

hgij

ijkhgk

hgk

ijhg

hgij

ijkhgk

ijk

hgij

≥+−−

+−−−

(2)

Formula 2 means there are two artifacts are not

processing on the same machine. Among them,

ijk

refers to the processing time of process

machine k,

ijk

refers to the decision variable of the

machine allocation,

hgij

refers to the decision

variable of the order on process. Formula (4)

qualified procedure can only be completed in one

machine; (5), (6) limit the scope of the decision

variables.

⎪

⎩

⎪

⎨

⎧

elsewhere

machineonduneis

if,1

ijk

(3)

⎪

⎩

⎪

⎨

⎧

−=

otherwhere

adjacentandbehind

adjacentandbefore

ijhg

hgij

(4)

∑

∀∈= j,i,

k,1

ijijk

(5)

}{ 1,0

ijk

∈

(6)

}{ 1,0,1

hgij

−

∈

(7)

3 IMPROVED BACTERIAL

FORAGING ALGORITHM

3.1 Bacterial Foraging Optimization

Algorithm

BFOA abstracts from the biological foraging

behavior of bacterial. The foraging process was

described as three operation process, including

tendency, replication and migration. The optimal

performance is largely decided by numerous

parameters, such as bacteria population size, bacteria

movement step, tendency operations, the maximum

times of copy and transfer operation. Due to these

parameters have no adaptability, Its iterative times is

completely determined by the maximum times of

various operation, and no convergence criterion are

introduced, so it is difficult to ensure accuracy and

adding unnecessary iterative process. In this paper,

The improvement point on two aspect, there are

bacteria movement step and stop condition (Dalian

Yang, Xuejun Li and Lingli Jiang, 2012).

Tendency operations: Mainly simulate the

motor processes, including move forward and move

towards. Assuming that flora size is S, With D

vector

),...,,(

pppp

i 21

= represent the ith a

bacteria,

),,( lkjp

said the chemotactic operations

of bacteria i in the jth. The position after the kth

times copy operation and the ith times migration

operation means that a candidate solution of bacteria

i in the search space. The location updating formula

of the bacteria i after chemotactic operations as

follows:

)()()()( jicl,k,jpl,k,1jp

ϕ+=+ (8)

Among them:

)(ic represent chemotaxis step length

of bacterial i ;

)( j

represent a random direction

vector after bacteria i overturn

21T

iiij

)]()()/[()( θθθ=ϕ .

Replication operations: Bacteria will be

divided after completed the sated tropism times, this

operation is mainly simulates the breeding process

of bacteria individual evolution.

Assumes that the

size of bacterial population is N,

),,( lkjF

as the

ith individual fitness value, first, for the fitness of

the whole population in descending order, The first

2N individuals will be survived which order by

ISME 2016 - Information Science and Management Engineering IV

ISME 2016 - International Conference on Information System and Management Engineering

fitness value, And make no difference to split in two,

the lower

2N individuals are eliminated, when

completed such a copy operation, the bacteria

population will remain the same size.

Migratory operation: In order to improve

the global search ability of the algorithm, When a

problem's solution space have multiple extremum

points, its sociality make it easy to trap in local

extremum, the purpose of this process is to make the

new individual to replace the original individual,

Different from the copy operation, the occurrence of

migration is according to certain probability p,

When some bacteria can meet the conditions of the

migration, it will be randomly assigned to the

solution space.

3.2 Improved Bacteria Movement Step

Traditional algorithm using fixed chemotactic step

length, but BFOA is a dynamic process. The bacteria

activity will have a certain percentage of the decline

with each tendency operation. The greater step size

is, the more likely miss the optimal solution, it will

soon fall into local optimum. With the increase of

number of chemotactic, step length should make

appropriate adjustments, to maintain the searching

efficiency of the algorithm.

Through studying found that step length affected

by the number of nearby companions, namely

crowding level of population,

the more crowded the

higher demand for food, food consumption faster,

just search for this position, the individual fitness

may increase, but as more and more individual

learning to this position, the individual fitness

declines quickly, which requires quick to flee the

area, sat with adaptive step length to control the

residence time of an area, continue to look for a

better position. For above, this paper presents an

improved bacteria movement step for this problem.

congestion level of the region which Bacteria i in,

determined by the individual number and interval

length in this region,

lenncrowd = ;

Δ said the

difference of the fitness value between

)(

lkj

ics

and

)1(

)(

lkj

ics

−

, represent the fitness

value of the bacteria i after the ith tendency

operation, the kth copy operation, the lth replication

operations.

)()(

l,k,jl,k,1j

icsics

−=Δ

−

(9)

Step length have three changes, the early stage of the

search in global optimization , Few bacteria around,

that within a certain range inadequate nutrition, sets

the chemotactic compensation to a larger value, be

helpful for global optimization, when looking for the

crowded degree of large scope, into local

optimization, set small step length, so as not to miss

the optimum solution, when the congestion is too

large and the growth rate of individual fitness

gradually smaller, step length will gradually increase,

so as not to fall into local optimization.

⎪

⎩

⎪

⎨

⎧

Δ<Δ∂>−

−

∂<

∂>

−

crowd,

j1jminmax

max

min

max

且 ) (

)(

(10)

Among them,

N represent the tendency times

have been executed in current, N represent total

times,

max

represent the maximum step length,

min

represent minimum step length. ∂ is the preset

threshold.

3.3 Improved Termination Condition

Appropriate termination condition can ensure to

obtain optimization results with reasonable accuracy,

and, reduce unnecessary computation. BFOA is

actually a process with constant iterations and

convergence, its goal is to improve convergence

speed and computational precision (J Q Li, Q K Pan

and K Z Gao, 2012). In practical applications, it is

unknown about the optimal solution. we can’t

determine whether to stop the iteration by judging

the magnitude of the error between current solution

and real solution, through above mentioned

analyzing, Population movement step can is the

basis of iterative stop. The step size smaller, the

bunching of bacteria more concentrated. The

iteration will stop until the following expression was

founded (ε is a smaller number) :

εMlkj(FlkjF

−

))],,min(()),,([max(( (11)

3.4 Algorithm Implementation Steps

The execution steps of the IBFO to solve the optimal

value as follows:

Step 1: Initialize parameter.

Step 2: Initialize bacteria position. Generate initial

solution randomly in the feasible region, The

according to formula (2) and (3) calculate individual

adaptive value. Always keep the ethnicity in a

reasonable range.

Step 3: Perform tendency operation. According to

the formula (3), (4), (5) adaptively adjust tendency

step. When the adaptive value rise, keep moving

direction. When adaptive value falls, rotate to select

Improved Bacteria Foraging Optimization Algorithm for Solving Flexible Job-Shop Scheduling Problem

new direction. If reach to the upper limit of tendency

operation, then step out the tendency, and perform

step 5.

Step 4: Perform replication operation.

Step 5: Perform tendency operation.

Step 6: Whether algorithm meet the convergence

conditions. If convergence, output the run results

and terminates the running process; otherwise, return

to step 3.

4 APPLICATION OF

EXPERIMENT

4.1 Experimental Environment

All experiments were paper processor for Intel (R)

Core (TM) i5-34700 CPU @ 3.20GHz, 4.00GB

RAM, Windows 7 systems under the.

4.2 Computational Results

To BFO algorithm, The most important impact of

the efficiency is the times of tendency, the more

times, the optimization ability in local is more subtle.

At the same time, with increasing complexity,

followed by the size of the bacteria, the more

bacteria involved in foraging individuals, the faster

to find the optimal solution, but the computational

cost will increase; Again is the influence of

migration dispel, this factor determines the global

search ability of the algorithm, it is easy to fall into

local optimization if the times is too small, therefore,

too many will increase the complexity of the

algorithm(Shiv P, Deo PV., 2014).Through

integrated balance, through many experiments to

obtain the optimal experimental parameters: Flora

scale

S = 100, Maximum Iterations N

iter

200,tendency times N

=50, maximum swimming

times

= 10,copy times N

= 10, migration times

= 15.

Choose two instances make simulation

experiment: The 8 x 8 is part of the flexible

scheduling problem (P-FJSP), 10 x 10 is fully

flexible scheduling problem (T-FJSP)(Bagheri A.,

Zandieh M. and Mahdavi I.,2010).Compared IBFO

algorithm with standard BFO algorithm and the

improved genetic algorithm (IGA). Three algorithms

run 10 times respectively, the running results are

shown in table 2 and table 3.

Table 1 8 × 8 results by comparison

Algorithm Optimal Average

standard

deviation

BFO 17 19.0 2.31

IGA 13 16.1 1.23

IBFO 13 14.8 0.80

Table 2 10 ×10 results by comparison

Algorithm Optimal Average

standard

deviation

BFO 9 11.2 1.88

IGA 8 7.9 1.02

IBFO 8 7.4 0.71

Moreover, The experiment choose MK01

experiment(Brandimarte Paolo, 1993) to verified

which involved in Benchmark examples, 6 machine,

10 artifacts. In the number of 200 iterations the

experimental analysis was carried out on the three

algorithms respectively. Figure 1 is the convergence

curves about the three algorithm, can be seen from

the diagram, In this paper, optimization algorithm

and the IGA algorithm can search to the optimal

solution 38, standard BFOA algorithm search to a

solution 41, although the two algorithms can get the

optimal solution, but the convergence speed of IGA

algorithm significantly slower than the improved

algorithm. Improved algorithm would find optimal

solution after 30 times in the iteration, and, IGA find

the optimal solution after 70 iterative times, so the

algorithm can more quickly find the optimal solution.

Figure 2 is the Gantt diagram of improved BFOA;

represent the scheduling scheme in the process of

convergence. The Y-axis means machine number.

The X-axis means execution time.

0 255075100125150175200

Improved BFOA

Standerd BFOA

IGA

Figure 1 Convergence curves

ISME 2016 - Information Science and Management Engineering IV

ISME 2016 - International Conference on Information System and Management Engineering

0 5 10 15 20 25 30 35 40

Processing time

Machine number

p(3,1)=6p(3,3)=2

p(2,2)=3p(2,4)=6

p(3,2)=4

p(2,6)=3p(2,2)=8

p(3,1)=5

p(5,6)=5

p(6,5)=3 p(6,5)=4

p(4,1)=5

p(1,3)=8

p(3,5)=5

p(1,6)=4 p(1,3)=8

p(4,2)=10

p(1,5)=3

p(4,4)=9

p(1,2)=3

p(6,1)=5

p(3,4)=5p(3,3)=6

p(1,5)=9

p(2,1)=5

p(1,6)=9

p(6,2)=8

p(3,1)=6p(3,3)=2

p(2,2)=3p(2,4)=6

p(3,2)=4

p(2,6)=3p(2,2)=8

p(3,1)=5

p(5,6)=5

p(6,5)=4

p(4,1)=5

p(1,3)=8

p(3,5)=5

p(1,6)=4 p(1,3)=8

p(4,2)=10

p(1,5)=3

p(4,4)=9

p(1,2)=3

p(6,1)=5

p(3,4)=5p(3,3)=6

p(1,5)=9

p(2,1)=5

p(1,6)=9

p(6,2)=8

p(2,6)=3

p(4,1)=5

p(1,5)=3p(1,2)=3

Figure 2 Optimal scheduling of Mk01 (The shortest

completion time is 38））

5 CONCLUSION

An improved bacterial foraging algorithm is

proposed in this paper,

and applied to search for the

optimal solution on FJSP. Compared with the

traditional algorithm, the optimization ability of this

method is more accurate. Compared with the

improved genetic algorithm, the improved method

can reduce the iterative time and greatly reduce the

solving time. But in view of the BFOA program is

running slightly slow, and have numerous

parameters, the present study mainly focus on the

single objective optimization, need to strengthen the

research for solving combinatorial optimization

problem.

ACKNOWLEDGMENT

I would like to express my gratitude to all those who

helped me during the writing of this thesis. My

deepest gratitude goes first and foremost to

Professor, my supervisor XinGang Wang, for his

constant encouragement and guidance. He has

walked me through all the stages of the writing of

this thesis. Without his consistent and illuminating

instruction, this thesis could not have reached its

present form.

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Improved Bacteria Foraging Optimization Algorithm for Solving Flexible Job-Shop Scheduling Problem