chapter 6 improved sheep flock heredity algorithm...

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CHAPTER 6

IMPROVED SHEEP FLOCK HEREDITY ALGORITHM

6.1 SHEEP FLOCK HEREDITY ALGORITHM

6.1.1 Introduction

The Sheep Flock Heredity Algorithm was initially designed by

Nara et al (1999) and was applied to scheduling problems. The algorithm was

based on the natural evolution of sheep in a flock. The chromosomes

represent machine schedules.

6.1.2 Algorithm Description

Consider the several separated flocks of sheep in a field as shown in

Figure 6.1. Normally, sheep in each flock are living within their own flock

under the control of shepherds. So, the genetic inheritance only occurs within

the flock. In other words, some special characteristics in one flock develop

only within the flock by heredity, and the sheep with high fitness

characteristics to their environment breed in the flock. In such a world, let us

assume that two sheep flocks were occasionally mixed in a moment when

shepherds looked aside as shown in Figure 6.2. Then, shepherd of the

corresponding flocks run into the mixed flock, and separate the sheep as

before. However, shepherds cannot distinguish their originally owned sheep

because the appearance of any sheep is the same. Therefore, several sheep of

one flock are inevitably mixed with the other flocks as shown in Figure 6.3,

namely, the characteristics of the sheep in the neighboring flocks can be

inherent to the sheep in other flocks in this occasion. Then, in the field, the

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flock of the sheep which has better fitness characteristics to the field

environment breeds most.

Figure 6.1 Flocks of sheep in a field

Figure 6.2 Mix of two flocks of sheep

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Figure 6.3 New flock of sheep in field

The above natural evolution phenomenon of flocks can be

corresponded to the genetic operations of this type of string. For this kind of

string, we can define two kinds of operations. (i) Normal genetic operations

between strings. (ii) Genetic operations between sub-strings within one string.

In SFHA, special string structure and hierarchical genetic operations

(crossover and mutation) are introduced. They are (i) sub-chromosome level

genetic operation and (ii) chromosome (global) level genetic operation. This

hierarchical operation is referred to as “multi-stage-genetic operation”.

6.2 IMPROVED SHEEP FLOCK HEREDITY ALGORITHM

6.2.1 Introduction

Based on the mechanics of natural selection and genetics,

evolutionary algorithms such as genetic algorithms combine the concept of

survival of the fittest among solutions with a structured yet randomized

information exchange and offspring creation. These algorithms are naturally

parallel and exhibit implicit parallelism, which do not evaluate and improve a

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single solution but analyze and modify a set of solutions simultaneously. The

possibility of the solution being trapped in a local optimum is considerably

reduced by the ability of the algorithm to operate on many solutions

simultaneously and gather information from all current solutions to direct the

search.

Moreover, these evolutionary algorithms can retain useful redundant

information about what is learned from previous searches by their

representation of individuals in the population. Critical components of past

good solutions can be captured, which can be combined via crossover to form

high quality solutions. Unfortunately, they may lose solutions and

substructures because of disruptive effects of genetic operators, and it is not

easy to regulate the algorithm’s convergence so that a pure SFHA may easily

produce premature and poor results. To enhance the performance of genetic

searches and improve convergence, an improved SFHA (ISFHA) is proposed.

In contrast to the classical SFHA (CSFHA), the ISFHA has superior features.

First, the pair wise mutation process is replaced by a single mutation process.

In a CSFHA, the pair wise mutation process performs a limited local search

and introduces some diversity, but it is hard to control its behavior whereas in

an ISFHA, the mutation rate of the single point mutation is much easier to

control thereby enhancing the local search behavior.

A new robust-replace (R-R) heuristic is introduced to enrich the

neighbor search templates so that a more powerful exploring ability and a

larger exploring region can be attained. This heuristic helps in improving the

solution search of the algorithm in the global domain. Thus, by the

introduction of these two processes into the algorithm, the limited local search

of mutation is enhanced as well as a constant search of solution between local

and global search domains is established. Such a procedure can also be

executed in a parallel mode to reduce serial search time. In addition, since the

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ISFHA retains the generality of SFHA, it can be implemented easily and

applied to any combinatorial or functional optimization problems.

6.3 PROCESSES IN ISFHA

The ISFHA consists of the following processes.

i) Sub chromosomal level crossover

ii) Sub chromosomal level mutation

a. Inverse mutation

b. Single point mutation

iii) Chromosomal level crossover

iv) Chromosomal level mutation

v) Robust-Replace (R-R) heuristic

6.3.1 Sub chromosomal level crossover

In sub chromosomal level crossover, the entire chromosome is split

into sub chromosomes of equal length. These sub chromosomes are moved

randomly within the chromosome to new positions, thus forming a new

chromosome.

6.3.2 Sub chromosomal level mutation

The mutation consists of two steps, inverse mutation and single

point mutation.

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(a) Inverse mutation

In a sequence, two random positions are selected. The portion of the

sequence between these two positions is inverted to get a new mutated

sequence.

(b) Single point mutation

A random position is selected in the sequence and moved to another

random position in the sequence. As the mutation occurs with respect to a

single point, the process is called single point mutation.

6.3.3 Chromosomal level crossover

The chromosome level crossover is performed among all the

chromosomes in the population.

6.3.4 Chromosomal level mutation

The chromosome level mutation is performed among all the

chromosomes in the population.

6.3.5 Robust-Replace (R-R) heuristic

Each schedule (chromosome) has a makespan value that refers to

the amount of robustness of that chromosome. The robustness of each

schedule is calculated from the robust factor. The robust factor is defined as

given in equation (6.1).

Robust Factor = (6.1)

From this relation, a lower makespan value gives a higher robust

factor value. The average robust factor is calculated for the population. The

schedules with robust factor values less than the average robust factor value

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are replaced by new schedules. This mechanism allows finding new schedules

that correspond to new search regions in the total search space.

6.4 GENERAL SCHEMA OF ISFHA

Input: Set of initial job sequences

Output: Value of makespan and total flow time

Set of generated Job sequences

Set gen. = 1 and go to Start.

Start: For (gen. = 1 to gen.; max gen.; gen. + 1)

Generate random number between 0 and 1

If random number ≤ 0.7

Perform sub chromosome level crossover

End If



Perform sub chromosome level mutation

End If



Perform chromosome level crossover

End If



Perform chromosome level mutation

End If

Calculate makespan and total flow time

Evaluate robust factor of job sequences

Update and maintain best job sequences for gen. + 1

End for

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6.5 FLOW CHART OF ISFHA

The frame work of the proposed ISFHA is illustrated in Figure 6.4.

The various steps involved in ISFHA are explained in this section.

Figure 6.4 Flow chart of ISFHA

Use old sequence and corresponding

makespan

Sub chromosomal mutation

Use old sequence and corresponding makespan

Chromosomal crossover

Chromosomal mutation

No

Yes

Start

Initialize Population

Sub chromosomal Crossover

Use new sequence and corresponding makespan

R-R heuristics

Is Termination Criteria

Satisfied?

Stop

New makespan <

Old makespan

No

Yes

New makespan <

Old makespan

No

Yes

Use new sequence and corresponding makespan

Sequence for next generation

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6.6 NUMERICAL ILLUSTRTION OF ISFHA (EQUAL SUB

LOTS)

The algorithm is illustrated with an example problem (5 jobs- 2

machines) in this section. Table 4.1 provides the data of the example

problem.

6.6.1 ISFHA for makespan criterion

The illustration of the algorithm is shown in Table 6.1 and 6.2.

Table 6.1 ISFHA - Sub chromosome Level Operations (for equal sub lots)

Seed sequence

Sub chromosomal

Crossover Probability

≤ 0.7

Sub chromosomal

Crossover

Sub chromosomal

mutation Probability

≤ 0.8

Inverse mutation

Single point

mutation

RN Y/N RN Y/N

3 4 1 2 5 0.65 Y 4 3 1 2 5 0.6 Y 5 2 1 3 4 5 2 1 4 3

5 3 1 4 2 0.5 Y 5 3 2 4 1 0.7 Y 1 4 2 3 5 1 4 2 5 3

1 2 5 3 4 0.8 N 1 2 5 3 4 0.9 N 1 2 5 3 4 1 2 5 3 4

2 3 5 1 4 0.3 Y 3 2 5 1 4 0.8 Y 1 5 2 3 4 1 2 5 3 4

2 1 5 3 4 0.6 Y 4 1 5 3 2 0.5 Y 2 3 5 1 4 2 3 1 5 4

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Table 6.2 ISFHA - Chromosome Level Operations (for equal sub lots)

Sequence after single point

mutation

Chromosomal Crossover

Probability ≤ 0.8

Chromosomal Crossover


Probability ≤ 0.9


makespan (z)

Robust Factor (1/z)

RN Y/N RN Y/N 5 2 1 4 3 0.75 Y 5 2 1 3 4 0.8 Y 1 3 5 2 4 44* 0.022727 1 4 2 5 3 0.5 Y 1 2 4 5 3 0.7 Y 5 3 4 1 2 45 0.022222 1 2 5 3 4 0.85 N 1 2 5 3 4 0.95 N 1 2 5 3 4 45 0.022222 1 2 5 3 4 0.3 Y 1 2 5 4 3 0.55 Y 1 5 2 3 4 45 0.022222 2 3 1 5 4 0.6 Y 1 3 2 5 4 0.85 Y 5 4 2 1 3 46 0.021739

Avg.= 0.022226

6.6.2 Calculation of makespan criterion

For the 2- machine, 5- job problem provided in Ttable 4.1, the

Calculation of makespan for the sequence 1-3-5-2-4 is explained below:

Using equation (3.1),

(푠 + 푛 × 푝 ) (푓표푟푚푎푘푒푠푝푎푛)

Machine (i=1) → (for j=1) → 2+ (2×2) = 2+4 = 6

(for j=2) → 6+ 4 + (3×3) = 6+13 = 19

(for j=3) → 19+ 3 + (2×2) = 19+7 = 26

(for j=4) → 26+ 3 + (2×3) = 26+9 = 35

(for j=5) → 35+ 2 + (2×2) = 35+6 = 41

Machine (i=2) → (for j=1) → 4+2+ (2×3) = 4+8 = 12

(for j=2) → 13+ 2 + (3×3) = 13+11 = 24

(for j=3) → 24+ 4 + (2×2) = 24+8 = 32

(for j=4) → 32+ 3 + (2×2) = 32+7 = 39

(for j=5) → 39+ 3 + (2×1) = 39+5 = 44

So, the makespan value= 44

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The final job sequence is 1-3-5-2-4, the corresponding makespan

time is 44 and it is shown in Figure 6.5.

Figure6.5 Schedule for the sequence 1-3-5-2-4 in ISFHA (makespan

objective for equal sub lots)

6.6.3 ISFHA for total flow time criterion

The method to find total flow time is illustrated in this section. The

final result is 4-5-1-3-2 and the corresponding total flow time is 140. It is

shown in Figure 6.6. Total flow time calculation for the sequence

4-5-1-3-2 = 11+19+27+38+45 = 140. The following figure shows the time

schedule of the optimal solution.

Figure 6.6 Schedule for the sequence 4-5-1-3-2 in ISFHA (total flow

time objective for equal sub lots)

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6.7 NUMERICAL ILLUSTRTION OF ISFHA (VARIABLE SUB

LOTS)

The algorithm is illustrated with an example problem (5 jobs- 2

machines) in this section. Table 4.4 provides the data of the example problem.

6.7.1 ISFHA for makespan criterion

The illustration of the algorithm is shown in Table 6.3 and 6.4.

Table 6.3 ISFHA - Sub chromosome Level Operations (for variable

sub lots)

Seed sequence

Sub chromosome

Level Crossover

Probability ≤ 0.7

Sub chromosomal

crossover

Sub chromosome

Level Mutation

Probability ≤ 0.8

Sub chromosomal

mutation

RN Y/N RN Y/N 4 2 5 3 1 0.6 Y 4 5 2 1 3 0.6 Y 2 1 4 5 3 2 4 5 1 3 0.5 Y 5 4 2 1 3 0.8 Y 2 1 3 5 4 4 3 5 2 1 0.9 N 4 3 5 2 1 0.9 N 4 3 5 2 1 1 3 2 4 5 0.3 Y 1 2 3 5 4 0.65 Y 2 3 1 5 4 3 2 4 5 1 0.95 N 3 2 4 5 1 0.85 N 3 2 4 5 1

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Table 6.4 ISFHA - Chromosome Level Operations (for variable sub lots)

Seed sequence

Chromosome Level

Crossover Probability

≤ 0.8

chromosomal crossover

Chromosome Level

Mutation Probability

≤ 0.9

chromosomal mutation

makespan (z)

Robust Factor (1/z)

Sub lot size

RN Y/N RN Y/N 2 1 4 5 3 0.6 Y 1 4 2 3 5 0.8 Y 3 5 1 4 2 87 0.01149 {354}{321}{46}{2313}{2321} 2 1 3 5 4 0.5 Y 2 3 1 4 5 0.72 Y 1 4 5 2 3 88 0.01136 {46}{3132}{213}{1232}{543} 4 3 5 2 1 0.4 Y 3 5 2 1 4 0.9 N 3 5 2 1 4 89 0.01124 {543}{321}{2123}{46}{1332} 2 3 1 5 4 0.3 Y 3 2 1 5 4 0.55 Y 5 4 3 2 1 90 0.01111 {321}{3231}{543}{2312}{46} 3 2 4 5 1 0.1 Y 5 2 4 3 1 0.85 Y 5 3 1 2 4 85* 0.01177 {132}{543}{46}{3212}{1233}

Average = 0.011394

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The final result is 5-3-1-2-4, the corresponding makespan time is 85,

sub lot size is {132}{543}{46}{3212}{1233} and it is shown in Figure 6.7.

Figure 6.7 Schedule for the Sequence 5-3-1-2-4 in ISFHA (makespan

objective for variable sub lots)

6.7.2 ISFHA for total flow time criterion

The method to find total flow time is illustrated in this section. The

final result is 1-2-3-5-4 and the corresponding total flow time is 276, sub lot size

is {46} {2312} {435} {213} {3132} and it is shown in Figure 6.8. Total flow

time calculation for the Sequence 1-2-3-5-4= 28+39+53+68+88 = 276.

The following figure shows the time schedule of the optimal solution.

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Figure 6.8 Schedule for the Sequence 1-2-3-5-4 in ISFHA (total flow time

objective for variable sub lots)

6.8 PERFORMANCE EVALUATION

6.8.1 Proposed methodology illustrated with makespan criterion

The makespan results of ISFHA algorithm for equal sub lots (Table 6.5

- Table 6.8) and for variable sub lots (Table 6.12- Table 6.14) is given in this

section.

6.8.2 Proposed methodology illustrated with total flow time criterion

The total flow time results of ISFHA algorithm for equal sub lots

(Table 6.9 – Table 6.11) and for variable sub lots (Table 6.15 - Table 6.17) is

given in this section.

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Table 6.5 Comparison with ACO, TA algorithm for two - machine cases in ISFHA (equal sub lots)

Prob

lem

set N

o. Makespan

Number of jobs

15 25 35 50

ISFHA ACO TAP TAR TAI ISFHA ACO TAP TAR TAI ISFHA ACO TAP TAR TAI ISFHA ACO TAP TAR TAI

1 429 429 429 429 429 648 648 648 648 648 914 914 914 914 914 1258 1258 1258 1258 1258

2 363 363 363 363 363 581 581 581 581 581 772 772 772 772 772 1153 1153 1153 1153 1153

3 249 249 249 249 249 436 436 436 436 436 709 709 709 709 709 1013 1013 1013 1013 1013

4 291 291 291 291 291 489 489 489 489 489 720 720 720 720 720 1035 1035 1035 1035 1035

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Table 6.6 Makespan value for three – machine cases in ISFHA (equal

sub lots)

Problem set number

Makespan Number of jobs

5 10 15 20 25 30 1 162 382 415 572 635 794 2 170 390 415 572 650 691 3 158 380 406 568 648 620 4 168 388 416 575 642 618

Table 6.7 Makespan value for five – machine cases in ISFHA (equal

sub lots)

Problem set number


5 10 15 20 25 30 1 182 385 415 655 702 845 2 190 390 430 672 706 747 3 182 382 422 670 705 628 4 178 380 418 668 704 672

Table 6.8 Makespan value for seven – machine cases in ISFHA (equal

sub lots)

Problem set number


5 10 15 20 25 30 1 200 205 462 592 714 873 2 202 208 508 604 720 745 3 204 210 510 610 728 700 4 200 200 500 598 730 689

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Table 6.9 Total flow time value for three – machine cases in ISFHA

(equal sub lots)

Problem set number

Total flow time Number of jobs

5 10 15 20 25 30 1 436 540 3045 5725 7508 10974 2 452 558 3198 5876 7555 8545 3 450 554 3127 5542 7625 8522 4 442 550 3152 5527 7619 8161

Table 6.10 Total flow time value for five – machine cases in ISFHA

(equal sub lots)

Problem set number


5 10 15 20 25 30 1 480 2252 3156 11409 11496 12833 2 462 2298 3208 9643 11532 9730 3 500 2002 3114 9006 11512 9434 4 504 2096 3045 9400 11477 9774

Table 6.11 Total flow time value for seven – machine cases in ISFHA

(equal sub lots)

Problem set number


5 10 15 20 25 30 1 570 2204 4052 9154 9842 13541 2 600 2518 4018 9398 10043 10948 3 594 2414 4009 9332 9827 10804 4 590 2216 4025 9376 10009 10339

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Table 6.12 Makespan value for three – machine cases in ISFHA

(variable sub lots)

Problem set number


5 10 15 20 25 30 1 620 708 832 927 1015 1023 2 1048 3354 3545 4025 4198 4916 3 1006 3308 3445 4009 4175 5002 4 998 3434 3505 3998 4127 5008

Table 6.13 Makespan value for five – machine cases in ISFHA (variable

sub lots)

Problem set number


5 10 15 20 25 30 1 708 854 1127 1309 1517 1709 2 3548 4009 4815 7198 7823 8302 3 4009 4152 5012 7004 7415 8334 4 4027 4109 5025 7323 7725 8125

Table 6.14 Makespan value for seven – machine cases in ISFHA

(variable sub lots)

Problem set number


5 10 15 20 25 30 1 794 1018 1237 1424 1912 1934 2 4145 4934 6365 7356 9834 9989 3 4227 4917 6309 7434 10398 10509 4 4023 4952 6229 7089 10527 10424

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Table 6.15 Total flow time value for three – machine cases in ISFHA

(variable sub lots)

Problem set number


5 10 15 20 25 30 1 3488 3972 8423 10023 24309 28002 2 15155 16152 27089 70104 92267 145236 3 15545 17409 27198 73325 145008 173878 4 15543 16434 26406 72288 132052 168104

Table 6.16 Total flow time value for five – machine cases in ISFHA

(variable sub lots)

Problem set number


5 10 15 20 25 30 1 3423 4398 8255 13009 24089 30023 2 15298 18023 25458 70034 96045 175643 3 15112 19525 27098 73508 155132 193992 4 15102 19324 28045 74302 141324 176528

Table 6.17 Total flow time value for seven – machine cases in ISFHA

(variable sub lots)

Problem set number


5 10 15 20 25 30 1 4325 5032 9312 14027 26255 28098 2 15119 20027 29398 72052 96212 165234 3 15552 21008 30155 73215 157524 165654 4 15217 19875 30009 74008 163498 167998

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6.9 SUMMARY

The numerical illustration and computational results of proposed

ISFHA for equal size sub lots (from Table 6.5 – Table 6.11) and variable size

sub lots (from Table 6.12 – Table 6.17) are discussed.

chapter 6 improved sheep flock heredity algorithm...

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