Computer methods in preliminary ship design

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L.K. KUPRAS

COMPUTER

,

.

METHODS IN

PREUMINARY

SHIP

DESIGN

,~

(2)

("l -.I .... 0 0 IJl 0 0(;1)

eno

Computer Methods in

Prelim

,

inary Ship Design

BIBLIOTHEEK TU Delft P 1718 5298

1111111111111

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Computer Methods in

Preliminary Ship Design

L.K Kupras

1983

(4)

..

Delft University Press

Mijnbouwplein 11

2628 RT Delft

The Netherlands

Tel. (0)15-783254

Copyright

©

1983 by Delft University Press, Delft,

The Netherlands

No part of this book may be reproduced in any form

by print, photoprint, microfilm or any other means

without written permission from the publisher:

Delft University Press, Mijnbouwplein 11,

2628 RT Delft, The Netherlands.

Printed in the Netherlands by Princo BV, Culemborg.

ISBN 90 6275 106 7

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CONTENTS A. METHODS

Solving Equality Constraints ln Preliminary Ship Design A Parametric Study

1.1 Introduction 1 .2 Defini t ions

1.3 The Method of Solution 1.4 Numerical Examples 1.5 Procedure COSOL 1.6 Final Remarks 1.7 References Appendix Appendix 2 Appendix 3 Appendix 4

Procedures Used in Numerical Examples

COSOL - Procedure for Equality Constraints Solving Ship Design Model Construction. Procedure DESIGN Construction of the Program for Parametric Study

1 3 5

9

9 9 12 12 14 15 2 Better Point Algorithm - An Optimisation Method for Preliminary Ship 17

Design Studies

2. 1 Introduction 17

2.2 Description of the Method 17

2.3 Test Cases 21 2.4 Remarks 22 2.5 References 22 Appendix 1 Appendix 2 Appendix 3 Appendix

4

Appendix

5

Computer Program. Design Model as for Test Cases 5,

6

and 7 Examples of Input and Output for Test Case 5

Optimisation Procedure Input Data Form

Input Data Form, Test Case

5

B. APPLICATIONS

3 Chemical Carrier - Parametric Study Program 3.1 Program for an Analysis of Basis Ship Data 3.2 Sample of Input - Output

3.3 Chemical Carrier - Parametric Study Design Program 3.4 Sample of Input as for Test Case

3.5 Sample of Output as for Test Case Appendix 1

Appendix 2

Damage Stability Criterion Seakeeping Criteria

3 .

6

Referenc es

4

Bulkcarrier - Parametric Study Design Program 4.1 Parametric Study Program. Analysis and Synthesis 4.2 Input Data Form. Sample of Input

4.3 Sample of Output

4 .

4

References 31 32 34 53 54 57 57 57

64

68

8

7

89

96

98

101 101 101 129 138 144

v

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5 Bulkcarrier Preliminary Design and Optimisation Program 145

5.1 Introduction 145

5.2 Free Variables 145

5.3 Method of Solution 147

5.4 Program Construction 148

5.5 Longitudinal and Vertical Subdivision 151

5.6 Numerical Procedures 151

5.7 Termination Conditions 151

5.8 Bulkcarrier Preliminary Design and Optimisation. Computer Program 155 Listing

5.9 Input Data Form. Sample of Output 184

5.10 Sample of Output 195

5.11 Referenc es 200

6 Procedures 201

6.1 POWKEL - Main Engine Power Calculation. Method of Auf'm KeIler 201 6.2 VOLCRE - Cargo Tanks Capacity Calculation for Chemical Carrier 206 6.3 WEICHE - Light Weight Calculation for Chemical Carrier 207 6.4 FREBTA - Freeboard for "A" Ships 209 6.5 KGCHE - Vertical Centre of Gravity of Fully Loaded Chemical Carrier 210 6.6 LMF15 - Maximum Permissible Floodable Lenght of Wing Tanks 211 6.7 PERIOD - Periods of Roll, Pitch and Heave 212 6.8 FREB60 - Freeboard for "B-6o" Ships 212 6.9 VOLBUL - Cargo Holds Capacity for Bulkcarriers 213 6.10 WEIBUL - Light Weight of Bulkcarriers 215 6.11 STOBUL - Weights and Volumes of Supplies for Bulkcarriers 216 6.12KGBUL - Deadweight, Payload and Vertical Centre of Gravity of Fully 216

Loaded Bulkcarrier

6.13 BMKB - BM and KB Calculation 217

6.14INSTAB - GM Calculation for BUlkcarriers, Fully Loaded 217 6.15 COSBUL - Building Costs of Bulkcarriers 218 6.16 PANKUP - Stability Cross Curves of Series 60 218 6.17 RFRBUL - Required Freight Rate for Bulkcarriers 219

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Introduct ion

In the last decade a continuous progress has been observed in the application of computers and computer based methods to solving various ship design problems. The experience which designers have gathered during that time reveals the st rong impact exerted upon the process of design by the employment of computers. As a result, through heated discussions, with many pro and con arguments, a new approach to ship design is being developed. Perhaps in the clearest r'orm it occurs in preliminary design.

Preliminary design is the first and fundament al stage of ship design process. Primarily it consists in determining the main ship particulars in a way which is optimal for the fulfilment of the requirements imposed upon the design. The solution to such a problem can be achieved by applying one of the following techniques: the trial and error approach, an optimization method and a para -metric study. Since the first of them is well known and has been described by many authors only the ot her two are discussed in the book. The techniques are first described and next illustrated by means of simple numerical examples in Chapter 1 and 2. In that context the construction of numerical design models and design algorithms is explained. The Chapters 3,

4

and

5

follow with the presentation of two sample design programs, one for a bulkcarrier and the other for a chemicalcarrier. Besides showing at work the ideas introduced in the preceding chapters, the sample programs provide scope for further explanations and comments. The ending part of the book, Chapter 6, contains listings of procedures which allow to carry out estimations of cargo holds capacity, free -board, main engine power, weights and stability.

Computer programs presented in the book are written in Algol-60. Due to the modular structure, the translation of them into other programming languages

should not create major problems.

The book should be accessible to naval architects and software specialists concerned with ship design, as well as to senior and graduate students of naval architecture. It is based on lectures delivered to the fourth year students and papers read at various symposia and professional meetings.

The manuscript was compiled of computer printed output and is reproduced without retyping. Consequently possible printing errors have been avoided and the price of a single copy has been pressed down . .

The author would like to express his sincere thanks to Prof.Dr.lng. C. Gallin, Ing. A.P. de Zwaan and Ing. W.B. Tinbergen who assisted him with criticism and discussion.

Department of Shipbuilding and Shipping Delft University of Technology

September, 1982

L.K. Kupras

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A. MET H 0 D S

l' SOLVING' EQUALITY CONSTRAINTS IN PRELIMINARY SHIP DESIGN. A PARAMETRIC STUDY

1.1 INTRODUCTION

IN MOST CASES ÜP THE PRELIHINARY SHIP DESIGN THE tUIN DIMEN-SIONS HTLL BE DETERMINATED FROM AN ITERATION PROCESS.THE TRIAL AND ERROR BETRaD IS VERY POPDLAR IN PRACTICE WHEN THE DESIGNER DECIDES ON THE DIRECTION OF 1HE TRIAL STEPS.

IF A COMPUTER IS ENGAGED THE TECHNIQUE

OP

SOLVING BY LOOPS IS VERY OFTEN OSED. THE BELOW DESCRIBED TECHNIQUE,BECAUSE OF lTS SIMPLICITY MAY BE USED BOTH WHEN EITHER A COMPUTER OR A MODERN POCKET CALCULATOR IS AVAILABLE. IT IS BASED ON THE PRINCIPLE CF PROBLEM LINEARISATION. IT REQUIRES A MINIMAL NDMBEB OF EVALUATIONS SO IT CAN BE VERY ATTRACTIVE TO USE IT IN PRACTICE. TO SUPPORT THAT STATE~ENT, TVO NUMERICAL EXA-MPLES ARE PRESENTED, ONE POR A TANKER AND ONE FOR A BULKCAR-RIER TO ESTI!'!!TE SRIP DIMENSIONS.

FURTHERMORE THE PROCEDURE COSOL IS LISTED. IT ALLOWS Ta PIND A SOLUBLE SET OF MAIN DIMENSIONS SATISFYING THE EQUALITY CON-STRAINT REQUIREMENTS. TRIS PROCEDURE MAY BE APPLIED IN PARA-METRIC STUDIES WHEN SEARCHING FOR EQUAL-LEVEL-CONSTRAINT-CONTOURS.

1 ~2 DEFINITIONS

TO SIHPLIFY FURTHEB DESCRIPTION THE POLLOVING DEFINITIONS ARE ASSUMED :

CONSTRAINTS: FUNCTICNAL AND ENVIRONMENTAL REQUIREMENTS NAMED LATER DESIGN REQUIREMENTS,MAY BE EXPRESSED IN AN ANALYTICAL WAY AS EQUALITY AND INEQUILITY CON-STRAINTS. EXAMPLES: MINIMAL REQUIRED GM VALUE, RE-QUIRED VALUES POR DEADWEIGHT OR FOR CARGO CIPACITY ETC.

FREE VARIABLES : FREE VARIABLES OR DECISION VARIA-BLES ARE QUANTITIES THAT REMA.IN UNDER THE DESIGNER, S CONTROL, E.G. HAIN DI~ENSIONS iITHOUT RESTRICTIONS.

PARAMETERS : PARAMETERS OR FIXED VIRIABLES ARE QUAN-TITIES NOT UNDER THE DESIGNER,S CONTROL, E.G. HAXI-MAL RESTBICTED BREADTH OR RESTRICTED DRAUGET.

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CONSTANTS : CONSTANTS ARE SUCH AS WATER DENSITY,

WAG! RATES, ETC.

THERE DOES NOT EXIST INY

PER~ANENT

DELIMITATION BETWEEN PREE

VIRIABLES AND PARAM

.

ETE

.

RS.

POR

EXAMPLE IN SO!!E CISES BREADTH

CAN BE CONSIDERED AS A FREE VARIABLE AND IN OTHERS BREADTH

~UST

BE KEFT AS A CONSTANT VALUE. THERE EXISTS AN ANALYTICAL

BELATIONSHIP BETWEEN CONSTRAINTS

(~,

PREE VARIABLES

(0)

AND

PARUETEBS (P) :

H

=

F{D,P)

THIS RELATIONSHIP MAY BE EXPRESSED VEBY SELDOM BY THE AID OF

AN EQUATION IND DAT HER OrrEN BY THE lID OF A NUI'IEBICAL

PRO-CEDURE.

EQUALITY CONSTRAINTS COM

.

E PROM

.

FUlICTIONAL RE

.

LATIONSHIPS AMONG

N

VARIABLES VHICH MUST BE STRICTLY SATISFIED E.G. :

WHERE

FBMIN - FBA

DWR - DWA

CAPE - CAPA

FBlUM, FBA

DW

'

R, DWI

CAPE, CAPA

o

- I'lINlaAL IND AVAILABLE VALUES

OP

l"REEBOARD,

-

~EQUIBED

AND AVAILABLE VILUES OF

DEADWEIGHT,

- REQUIRED AND lVAILIBLE CARGO

HOLD CAPACITIES.

INEQUALITY CONSTRAINTS USUALLY CO!'!E FROI'! SO!'!E SPECIFIED DESIGN

LII'!ITITIONS OF THE FORI'! AS FOB EXAI'IPLE

2

FB

t

UM -

FBI

<=

0 DESIGN MODEL : THE NUI'IERICAL DESIGN KODEL ALLORS TO

COMPUTE

COMSTRAINT VILUES WHEM FREE 'ARlABtE

VI LUES ARE KNOWN. IT I'!AY BE REPRESENTED EITHER BI

A

SINGLE PROCEDURE OR BY A BLOCK OF PROCEDURES

(SUBROUTINES) PUT IN SUCB A SEQUENCE TBAT OUTPUT

FRO~

A

FOREGOING PROCEDURE SERVES AS INPUT FOT THE NEXT

ONE.

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DESIGN(X,G,H); DESOPT(X.G,H,O.FV); DESCH E (X, H) ; DESCHE;

WHERE :

DESIGN,DESOPT AND DESCHE - NAHES OF THE DESIGN PROCEDURES

X,G,H,OFV - F~RMAL PARAMETERS X - FREE VARIABLES(INPUT)

G,H - INEQUALITY AND EQUALITY CONSTRAINTS (OUTPUT)

O~V - OBJECT FUNCTION (OUTPUT)

1.3 THE METHOn OF SOLUTION

1.3.1

ASSUMPTIONS

THE ~ETHOD CAN BE APPLIED UNDER THE ASSUMPTIONS : - ONLY EQUALITY CONSTRAINTS ARE CONSIDEiED,

- NUMBER OF FREE VARIABLES IS EQUAL Ta THE NUMBER OF EQUALTTY CONS~RAINTS,

- ONLI ONE SOLUTION EXISTS IN THE NEIGHBOURHOOD OF THE START POINT.

r .3.2

'

SOLUTION

THE EQUALITI CCNSTRAINTS WILL BE APPROXIMATED BY A SET OF LINEAR EQUATIONS : TNEC H(J)

=L

A(J,I) • XI') • HP) J=1 WBEEE : X{l) - FREE VARIABLE, (1)

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I

-

1,2, •••••• TNEC -

SUCCESSIVE

NU~BER

OF

PREE VARIABLE

J

-

1,2, •••••• TNEC - SUCCESSIVE

NU~BER

OF

CONSTRlINTS

TNEC - TOTAL

NU~BER

OF EQUALITY CONSTRAINTS

THE ITERATION PBOCE55 WILL START PRO! AN! POINT NAMED

START

POINT ( X(S)

=

X(',S), X(2,5), ••

~l(I,S) , ••

X(TBEC,S) ).

THE NUMERICAL DESIGN MODEL WILL BE EXECUTED AND CONSTRAINT

IA LUES CALCULATED (H(l,S), H(2,S), ••• H(J,S), ••• H(TNEC,S) ).

THE CALCULATIONS WILL BE REPEATED FOR TNEC TEST POINTS WHERE

FOR EACH POINT ONE SOCCESSIVE PREE VARIABLE VALUE

X«I)

WILL BE INCREASED BI STEP WIDTH

DELP(I) :

START

POINT

T E 5 T

POl

N T 5

S

1 ••••••••••••••• I ••••••••••••••••• THEe

XP,5)

X(1,5)+DELP(1) ••

1(1,5) •••••••••••

l(l,S)

X(I,S)

X(I,S) ••••••••••

X(I,S)+DELP(I) •••

X(I,S)

X(TNEC,S)

X(TNEC,S) •••••••

X(TNEC,S) ••••••••• X(TNEC,5)+DELP(TNBC)

AMD PROPER CONSTR!INTS V!LUES :

H (',5)

H(1,l) . . . .

..

.

H(1,I) •••••••••••• H(l,TNEC)

H (J, S)

H(J,1) ••••••••••

H(J,I) •••••••••••• H(J,TNEC)

H (TNEC, S)

H

(TNEC,

1) H (TNEC,I) ••••••••• H (TNEC, THEC)

THE COEFFICIENTD A(J,I) AND B(J) WILL BE POUND PROM THE

EQUATIONS

A (J, I) B (J) 4

(H(J,I)-H(J,S»/DELP(I)

H (J, S)

TNEC

L

A(J,I)

*

X(I,S)

1=1 ( 2)

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NOW VIllEN A (J, I) AND D (J) ARE KNOWN THE POLLOWING SET OP EQUATIONS CAN BE SOLVED :

TNEC

L

A(J,I)

*

XCI) .+ B(J) 1=1

AND THE SOLUTICN IS :

o

x

= X(1), X(2), ••• 1(I), ••• X(TNEC)

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POR TH!T SET OF PREE VARIABLE VALUES THE CONSTRAINTS WILL BE CALCULATED AND COMPARED TO THE ALLORED TOLERANCES. lP NECE-SSARY THE ITERATION PROCESS IILL BE BEPEATED OSING SMALLER STEP WIDTHS.

1.4 NUMERI CAL EXAMPLES

POR A BETTER ILLUSTRATION OF THAT TECHNIQOE TWO PRACTICAL EXA~PLES ARE WORKED OOT AND COMPARED.

1.4.1" EXAMPLE 1

MAIN DI8ENSIONS ESTI~ATION FOR A TANKER. (2-VARIABLES DESIGN !IODE.L) •

DESIGN REQUIBEMENTS

DEADWEIGHT - 250000 T

RESTRICTED DRAUGBT - 19.30 M

PREEBOARD - MINIMAL

BAIN ENGINE - STEAft TURBINE PROPELLER REVOLUTIONS -

85

'/MIN

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ASSUMPTIONS :

IN FURTHER CALCULATIONS WE ~ILL MAKE USE OF DATA ANALYSIS OF A BASIC (SIMILAR) SHIP • THE BASIC SHIP SELECTED WILt BE SUCH THAT THE TECHNICAL CHARACTERISTICS HAVE Ta BE RATHER GO aD AND ARE CLOSE TO THE REQUIRE~ENTS OF THE SHIP TO BE DESIGNED. THEY iILL BE RE-CALCULATED WITH THE AID OF THE SAME PROCEDURES AS WILL BE USED LATER FOR A NEW DESIGN. THE EXPERIENCE CCEFFICIENTS ARE CALCOLATED WHICR REPRESENT THE RATlOS OF ACTUALLY TO CALCULATED VALUES. DESIGNING A NEW SHIP. THE CALCULATED VALUES WILL BE MULTIPLIED BY PROPER EXPERIENCE COEP.FICIENTS.

SOLUTlCN :

FROM TABLE 1 THE SEQUENCE OP THE CALCULATION MAY BE LISTED. IN THE 4-TH COLunN. BASIC SHIP DATA ARE SROiN AND IN THE 6-TH COLOMN PROPER EXPERlENCE COEFlICIENTS.

PROH THE SET OF MAlN DIMENSICNS AND FORM COEFFICIENTS,THE POLLOWlNG ARE ASSUMED TO BE FREE VARIABLES OR PARAMETERS

FREE VARIABLES .PARAHETERS

L,D

K=CB+0.5*V/SQRT(L/0.3048) =1.08846 AS FOR SIMILAR SHIP T=19.3.o M

L/B=6.5836 AS POR SIMILAR SRIP

TWO EQUALlTY CONSTRAINTS MUST BE SATISFIED : FOR DEADWEIGHT

FOR FREEBOARD

H(1) DliR - DiA 0

: H(2) = FBMIN - (D-T+S) = 0 'WHERE

S - STRINGER PLATE THlCKNESS.

THB COLUMN 7 SHOWS DESIGN REQUIREMENTS. TRB CALCULATIONS CAN START FROPI ANY POINT, SEE COLUPIN 8 (L=312.00 M,.o=24.60 ft). THE COLUMN 9 CONTAINS THB BESULTS iHEN LENGTH IS INCREASED WITH 10.00 MAND COLUMN 10 ~HEN DEPTH IS INCREASED WITH

0.60

M.

PINALLY CONSTRAINT

VA

LUES ARE CAtCULATED POR THOSE

3

POINTS. NOli COEFFlClENTS A (J, I) AND B (J) POR EQUATIONS {3)

ARE CALCULATED : 6 A(l,1)=(H(l,l)-S(1.S»/DELP{1)=(20626-J51Q8)/10= -1452.2 A ( 1 , 2)

=

(H (1 , 2) - H ( 1 • S) ) / D EL P (2)

=

(35 2 25- 3 514 8) /0 • 6

=

1 28. 3 B ( 1) = H ( 1 , S) - (A (1 , 1) * x ( 1, S) + A (1. 2) *X (2, S) ) =35148-(-1452.2*312+128.3*24.6)= 485077

·1

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A(2,l')=(H(2,1)-H(2,S»jDELP(1)=(0.158-0.251)j10= -0.0093 A(2,2)=(H(2,2)-H(2,S»jDELP(2)=(-0.199-0.251)jO.6= -0.75

B (2) =H (2, S) - (A (2, 1) *X (1, S) + A (2, 2) *X (2, S) )

=0.251-(-0.0093.312-0.75*24.60)= 21.6026

SOL VING TBE SET OF EQUITIONS (3) WE OBTAIN

1(2)= D =-(B{1)jA(1,1)-B(2)jA(2,1»/

(A(1,2)

JA

(1,

1) -A (2,2)

JA (2, 1»

=

24.635

X(1)= L =-(A(1,2>*X.(2)+B(1»/1(1,1)= 336.206

THE COLUMN 11 CONTAlNS THE CHECK OP TBE SOLUTION. TBE DIPPERENCES ARE: POR FREEBOARD O. 005~, !ND FOR DEADWE.IGHT

747

TON.

lP

RECESSARY THE ~TERATION PBOCESS CAB BE BEPEATED WITS S~ALLER STEP iIDTHS.

1.4.2 EXAMPLE 2

MAIN DIMENSIONS ESTIMATION FOB A BULKCARRIER. (DESIGN MODEL WITH 3 VARIABLES)

DESIGN REQUIREMENTS :

DEADWEIGHT - 8QOOO T

CA.RGO HOLDS CAPACITY - 97000 M**3

FREEBOARD - ~INI!lL

RESTRICTED BREADTH - 32.23 M

SERVICE SPEED - 15.50 KB

~AIN ENGINE - SLOi RUNNING DIESEL ENGINE PROPELLER REVOLUT.IONS - 115 Vl'1IN

ASSUMPTION: WING TANKS DI"ENSICN-PROPORTIONS ARE TBE SAME AS FOB BAS~S SRIP.

SOL UT ION :

TBE RESULTS FROM ITERATION PROCESS .ARE LISTED IB TASLE 2. TBE FOLLOWING PREE VARIASnES AND PARAMETERS ARE ASSOMED : PREE VARIABLES

PARAMETERS L, T ,_K .D _{1.12479, AS POR BASIC SHIP}

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THREE EQUALIT~ CONSTRAINTS HAVE TO BE SATISFIED :

FOR DEAD~EIGHT

FOR CARGO HOLDS CAP.

FOR FREEBOARD H (1) H (2) II (3)

DWR -

D~A CAPE - CAP! FBI1IN - (D-T+S)

o

COEF.FICIENTS A (J, I) AND B (J) ARE CALCULATED AS FOLLOWS :

A(1,1) =(H(1,1)-H(1,S»jDELP(1)=(5876-7333)/4.4= 331.U6 A (1 , 2) = (H ( 1 , 2) - H ( 1 , S) ) / DEL P (2) = (6 77 5- 7 3 3 3) jO. 1 0= - 5 580 • 0 !(l,3)=(H(1,3)-S(1,S)jDELP(3)=(7376-7333)jO.30= 143.333

WHERE :

DELP(l)=STEP FOR LENGTH DELP(2)=STEP POR DRAUGHT DELP(3)=STEP FOR DEPTH

4.40

M

0.10 M 0.30 ft B (1) =H (1, S) - (A (1, 1) d ( 1, S) + A (1,2)

*

X (2, S) + A ( 1,3)

*

X (3, S) )

=7333-(-331.136.218-558014.026+143.33319.2)= 155033.807

A (2, 1) = (H (2, 1) -H (2,S» jDELP (1)= -524.772 A(2,2)=(H(2,2)-H(2,S»jDELP(2)= 870.0 A(2,3)=(H(2,3)-H{2,S»jDELP(3)= -5806.67 B (2) =H (2, S) - (A (2, 1) * X ( 1, S) + A (2,2) *X (2, S) + A (2, l) * X (3, S) ) = 223179.83 A(3,1)= -0.00022727 A{3,2)= 0.99 A{l,3)= -0.736667 B (3)

=

0.307805

SOLVING TH! SET OF EQUATIONS (3) WE OBUIN

X(3)= D -(I1(3)jM{1)-N(3)/N(1»j

(1'I(2'jM{1)-N(2)jN(1» 20.98 M

X(2)= T -(.M{2)*X(3)+I1(J»jl1(1) 15.35 11

x

(1) L - (A (1,2)

*

X (2)+.A (1,3)

*

X (3) + B (1) ) jA (1,1) 218.58 l'\

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WHERE : M(1)=A(1,2)/A(1,1)-A(2,2)/A(2,1) M (2) = A ( 1 ,3) / A ( 1. 1 ) -A (2,3) / A (2, 1) ti (3) =B (1) /A (1,1)-13 (2) IA (2,1) N (1) =A (2,2) IA (2,1) -A (3,2) IA P,1) N (2) =A (2,3) IA (2, 1) -A (3,3) /A (3, 1) N (3) =B (2) /A (2,1) -B (3) /A (3, 1)

THE COLonN 12 CONTAINS THE CHECK OF SOLUTlON. THE DIFFERENCES ARE : POR DEADWEIGHT 743 TON, CARGO HOLDS CAPACITY 24 M •• 3 AND FREEBOARD 0.012 M. lP NECESSARY TBE ITERATION PROCESS CAN BE BEPEATED IIT8 SMALLER STEP WIDTBS.

1.5 PROCEDURE COSOL

THE ABOVE DESCRIBED METHOD HAS BEEN PROGBAMtlED.

POR COMPUTER LISTING SEE THE PROCEDURE COSOL AS SHOWN IN APPENDIX 2.

1.6 FINAL REMARKS

BECAUSE OP lTS SIHPLICITI, THE TECHNIQUE CAN BE OSED IN PRELIMINARY SRIP DESIGN OPTIMISATION PROBLE~S, PABAMETRIC STUDIES AND SINGLE APPLICATION, AS

FOB

MIIN DIHENSIONS

ESTIMATION. TBE ITERATION PROCESS REQUIRES THE LOWEST NUHBER OP EVALOATIONS (TNEC+1 CALCULATIONS PRO 1 CICLE), SO IT IS

VERY EPFICIENT.

1.7 REFERENCES

(1) HAGEN E.,JOHNSON J.,OVREBO B., flHULL STEEL WEIGHT OP LARGE alL TANKER AND BULKCARRIERS", EUROPEAN SHIPBUILDING, Nb. 6, 1967.

(2) ERICKSEN S., "OPTIMIZING CONTAINERSHIPS IND THE~B TERMINALS", DOCTOR THESIS, 1912.

(3) AUP,M KELLER W.H., "EXTENDED DIAGRAMS POR DETERMINING TRE RES'IS'l'ANCE .AND REQUIRED POWER P~B SINGLE SCREW SHIPS", S.&W., NO.24, 1914.

(ij) KUPRAS L.K., "PROCEDURES IN PEELIMlNARY BULKCARRIER DESIGN", T.H. DELPT REPORT, 1975.

(5) KUPRAS L.K., "OPTIMISATION ME!HOD AND PARAMETRIC STUDY IN P.RECONTRACTED SHIP DESIGN", INTERNATIONAL SHIPBUILDING PROGRESS , NO.261, MAl, 1976.

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T1BLE 1 . ~AIN DI~ENSIONS POR TANKER (2 VARIABLES) L B T D CB B 0.85D B~IN WA WF RPM PB WSM DW LIB Hl .HERE 10 o lilT KlI 3 EXPLANATION L ENGHT BREADT? DRAUGHT

-

---

4

--

[-

---

---J---

5

---

6 -------- ---

---ANALYSIS OP BASIC SHIP DA~l --- DESIGN TRUE V1CUES 310.535 47.168 18.96

CALCULATE EXPER. REQUIRE-VALUES COE". ~ENTS

19.30 DEPTH 24.50 BLOCKCOE' -FICIEIIT =CB+0.5,VI SQRT (Lo3. 28) CB POR PREEBOARD MINIeAL FREEBOARD SHEEB AT AP SHEER A'i' PP TRIAL SPEED 0.8457 1.08846 0.8517 5.573 0.70 2.50 15.50 ,

---0.8472 1.00578 5.582 ---~---J---[---8

l

9 10 11 ------- --- ---

---START POINT S 312.00 SOLunON

;J-~~~~;

322.00 312.00 47.391 48.909 47.391 19.30 19.30 19.30 0.8431 0.8470 0.8431 SOLOTION E CHECK 336.206 51.067 19.30 0.8521 1.08846 1.08846 1.08846 1.03846

•

0.8488 0.8520 0.8506 0.8571 5.584 5.491 5.740 5.363 0.70.. 0.70 * 0 .. 70 * 0.70 • 2.50. 2.50. 2.50 * 2.50 • 15.70 15.70 15.70 15.70 11"Ill PROPELLER REVOLUnONS 85 85 85 85 85 HP TON TON TOII "AlN ENGINE 28000 28039 0.9986 29424 30405 29424 32871 POWER, TRI AL CON LIGHT SUIP 32260 31684 0.95764 32718 35521 32795 39795 WEIGHT DEADWEIGHT 2094CO 250000 214852 229374 214775 250747 6.5836 6.5836 6.5836 6.5836 6.5836 6.5836 STRINGER PLATE 0.033 THICKNESS 0.033 0.033 • 0.033 0 0.033. 0.033. COllSTRAINT PCP DEADWEIGHT H (1. S) H (1.1) H (1,2) Hl = DWR - DWA 35148 20626 35225 -7q7 CClISTRAIIiT PCR .REEBOARD ij (2,S) H (2, 1) H (2,2) H2 = FBMIlI - (D-T+S) 0.251 0.158 -0.199 -0.005 • - ASSUMED VALUES

(18)

TABU: SYMBOL CB f1srlIU SEReOtl RP" PO iS!'t ow CAP H (1) H (2) H (J)

!1AIN DI!1E!lSIONS ES'rI!1ATIOII peR A BULKCA RRIER ( ) VARIABLE5)

ANALYSIS CF BASIC SHIP CATA DESIGN Ul/IT Kil EY.PLAIlATIC!! LENGTH BREADTH DRAUGHT DEPTH BLOCK COEfFICIEllT =CB+0.5,.V/ SQRT (L.3. 28) !lIUIMAL fREEBOARD P.ATIO OF ACT -VAL SHEER TC tHE STANDARD CIIE

SERVICE SPEEl::

SERV.CC!ID I-TION FACTOP l/"IN PROPELLER REVCLUTICNS TRur VALUES 218.00 32.23 H. 026 19.20 0.835 1.1247Q 5.200 0.1 B 15.50 1.20 115 HP eREAK POWH AT 20000 SERV.COtW. TON LIGII! S:HP 12090 WEIGHT Ton DEADWEIGHT 12661 ~~~~~~;~~i~;;;; ~-VALUES COErp. --- - ---------- ----- ----------- ----- --5.199 20611 0.96011 12628 0.9574 REQUIRE -"ENTS 32.2] PB" lil 0.18 15.50 1.20 115 80000 /"Ju ) CARGO HOLDS 87506 '.0066 97000 TON CAPACITY STR!NGER PLATE 0.026 TIUC!(NESS

CCIISTFAItITS FOF DEADWEIGHT:

H(1) = DWR - OWA

CCUSTRA lilT FeR CARGO !-iOLDS CAPACITY :

H (2) = (APR - CAPA CCtlSTrtAIIIT PCR PREEEOARD :

11 (3) = FBtHN - (0-1+ S)

WHERE ASSU:1F.D VALUES

0.026 START POl NT S 218.00 32.2] 14.026 19.20 0.8]5 L+4.4 POIN'I 1 222.4 32.23 14.026 19.20 0.83188 1.12479 1.12479 5.20 0.18 15.5 1.20 115 20000 12090 12661 87506 0.026 11 (1, S) 113] 11 (2, S) 9494 5.199 0.18 15.5 1. 20 11~ 20229 12642 fl9B15 0.026 11(1,1) 5816 JI (2,1) 71115 R (J, 1) -0.001 SOLUTION T+O.l POIliT 2 218.00 32.23 14.126 0.835 1.12479

.

'i.199 '0.18 15.'i 1.20 115 20089 12136 73225 97419 0.026 n (1,2) 6115 11(2,2) 9581 H (J, 2) 0.099 0+0.3 SOLU'ItON

POIrIT 3 & CHECK

218.00 218.58 32.23 J2.2J 15.35 19.50 20.98 0.835 0.8350 1.12479 1.12479 5.219 5.644 0.18 0.18 lS.5 15.5 ,.20 115 115 20000 21210 12133 12309 12624 A0740 89248 96916 0.026 0.026 H (1, J) ---------- -740 1316 11 (2, J) 1752 20 11 (3, J) -0.012 -0.221

(19)

APPENDIX 1 PROCEDURES USED IN NUMERICAL EXAMPLES

THE FOLLOWING PROCEDURES AND FORMULAE ARE

USED IN

CALCULATIONS :

iE1GHT OF STEEL -

METHOD OF E.HAGEN,J.JOHNSCN, B.OVREBO(1),

IUCH~NEBY

AND OUTFIT WE.1GHT -

P>ORl'IULAE O

.

F H. NOWACKI (2),

RESISTANCE AND PROPULSION -

METHOD OF W.H. AUF,M KELLER (3),

CARGO HOLDS CAPACITY -

PROCEDURE OF T.B.DELFT (4),

FREEBOARD -

REGULATIONS OF ILLC -

1966.

APPENDIX

2. COSOL -

PROCEDURE FOR EQU1L1TY CONSTRAINTS SOLVING

LISTING OF TBE PROCEDURE

COSOL

7 ' PROCEDURE' COSOL;

8

'BEGIN'

9

'COMlrENT' PROCEDURE COSOL SEARCHES FOB SOLUBLE SET OF FREE

9 VARIABLES(E.G. MllN DlMENSIONS) SATISPYING THE

9 EQUALlTY CONSTRAINT

REQU1REMENTS;

9

II:AT:=O;

10

'BEGIN'

11

'COMMENT' CONSTRAINT

VALUES CALCULATION FOR THE START POINT;

11

DESCHE;

12 EVAL:=EV1L+1;

13

SOL

1:

13

'FOR'IC:=l'STEP'l'UN~IL'TNEC'DO'HHCVIC/)

:=H(/IC/);

16

'FOB'LK:=l'STEP'l'UNTIL'TNEC'DO'

17 'BEGIN'

18

'COMMENT' CONSTRAINTS VALUES CALCULATION FOR SUCCESSIVE

18 TEST POINTS;

18

X(/L~/):=X(/LK/)+DELP(/LK/);

19 DESCHE;

20 EVAL:=EV1L+l;

21 X{/LK/):=X(/LK/)-DELP(/LK/);

22 'FOR'IC:=l'STEP'l'UNTIL'TNE€'DO'

23 'BEGIN'

24 lP(/IC,LK/):=-(HHC(/IC/)-H(/IC/)/DELP(/LK/);

25 'END';

26'END'i

27 ·POR'IC:=1'STEp

.

1'UNTIL'TNEC'DO'

28 'BEG.IN'

29

CK

VIC/)

:=HHC

VIC/) i 30

'FOR'LK:=l'STEP'1'UNTIL'TNEC'DC'

31 'BEG.IN'

32 CK(/IC/):=CK(/IC/)-AP{/IC,LK/)*X(/LK/);

33 'BND';

34 'END';

35

'POR'IC:=l'STEP'l'UNTIL'TNEC'DO'

36

'BEGIN'

37 'FOB'LK:=1'STEP'1'UNTIL'TNFC'DO'

12 - --- -- -- -- - --

(20)

-38 AFF(/IC,LK/) :=AF(/IC,LK/); 39 'END'; 40 'FOR'IC:=l'STEP'l'ONTIL'TNEC'DC' 41 CL (lIC/) :=-CK (lIC/) ; 42 'FOR'IC:=1'STEP'1'UHTIL'TNEC'DC' 43 AFFC/-IC,TNEC+'/) :=CL(lIC/);

44 'CC~~ENT' SOLUTION OF LINEAR EQUATIONS;

44 LINORT(AFF,CL,TllEC,LIN1); 45 'FOR'LK:=1'STEP'1'UNTIL'TNEC'DO' 46 X (lLK/) :=CL (lLK/) ; 47 DESCRE; 48 EVAL:=EVAL+1; 49 KAT:=KA1+1; 50 'IF'KAT>4'THEN"GOTO'SOL2; 52 'FOB'IC:=1'STEP'1'UNTIL'TNEC'DC' 53 'BEGIN'

54 'CO~~ENT' CHECK WHETHER TOLERANCES OF EQUALITY CONSTRAINTS

54 WERE NOT VI0LATED;

54 'IF'ABS(H{/IC/»>GA~O(/IC/)'THEN' 55 'GOTO' SOL 1: 56 'END': 57 SOL2: 57 'GOTO' LIN2: 59 LIN1:

59

59 'CO~MENT' LINORT DOES NOT WORK. EITHER WRONG INPUT

60 OR NO SOLUTION:

60

60 'CO~MENT' GO TO THE END OF !HE PROGRAM WHEN NO SOLUTION;

60 'GOTO'KBAK; 61 LIM2:

61 'END'; 63 'END'COSOL;

DECLARATION OP TBE PARAMETERS AS CALLED BY COSOL

'COM~ENT' DECLARATION OF THE PARA~ETERS AS CALLED BY COSOL 'INTEGER'IC,TNEC,LK,K,KAT,EVAL;

'ARRAY' BHC (11: TNEC/) , AF

Cl

1: TNEC, 1: TNEC/) , CK (11: TNEC/) , CL(/1:TNEC+l/1 ,AFF(j1:TNEC,1:'INEC+l/),

DELP,X,H,GAMO(/l:TNEC/) ;

LINGRT - PROCEDURE POR LINEAR EQUATIONS SOLUT~ON

67 'PROCEDURE' tINORT (A, B, N, LAB) ;

68 'VALUE'N;'INTEGER'N;'ARRAY'A,B;'LABEL'LAB; 72 'BEGIH"INTEGER'K,J,I,L;

74 'REAL'W,T,D;

75 'ARRAY'C (/1: NI), V (/1 :N+1,': N+l/) ; 76 'FOB'I:=l'STEP"'UNTIL'N+l'DC'

7 7 ' BEG.I N" 'POR' K: = l' STEP' " U NT IL' N + l' DO' 79 V(lI,K/) :='IF'L-.=K'THEN'O'ELSE"; 80 'END';

81 K:=O;J:=l;D:=l; 84 LO: W:=O;T:=O;

87 'FOR'L:=1'STEP"'UNTIL'N+l'DC' 88 'BEGIN' W:=W+A (IJ, L/) *V (I' ,L/) ; 90 T:=A(/J,L/) *A(/J,L;) +T;

(21)

91 B(lL/):=V(ll,L/);

92 'END';

93 'IF'ABS(W/T)<'-9'THEN·'GOTO-L1'ELSE'K:=O;

97

D:=D*W;

98 'FOR'I:=1'STEP'1'UNTIL'N+1-J'DO'

99 'BEGIN'T:=O;

101 'FOR'L:=l'STEP'l'UNTIL'N+l'DO'

102 T: =T+A

(lJ,L/) *V (11+1 ,L/) ;

10.3

C ( l I l ) :=-T/w;

104 'FOR'L:=l'STEP' l'UNTIL'N+1'DO'

105 V

(lI,L/)

:=C

(lIl)

*B

(lL/) +v (lI+',L/) ;

106 'END';

107 J:=J+l;

108

'IF'J~=N+1'THEN"GOTO'LO; 110 B(lN+1/) :=D;

111 'FOR'L:=l'STEP' l'ONTIL'N'DO'

.

B(jL/) :=-V(l1,L/);

113 'GOTO'L2;

314 L1:

'IF'K=N-J'THEN"GOTO'LAB'ELSE'K:=K+l;

119 D:=-D;

120 'FOR'L:=l'STEP'l'UNTIL'N+l'DO'

121 'BEGIN'V(lN+l,L/) :=A(lJ,L/);

123 A{/J,L/}:=A(/J+K,L/);

124 A (/J+K,L/l :=V(JN+l,L/);

125 V(/N+l,L/):=O;

126 'END';

127 V(/N+l,N+1/):=li

128 'GOTO'LO;

129 L2: 'END';

APPENDIX

3. SHIP DESIGN ftODEL CONSTRUCTION.

PROCEDURE DESIGN

'PROCEDURE'DESIGN;

'BEGIN'

'CC~MENT'

IDENTIPICATION OF PREE VARIABLES;

L: =X (/'/): B:=X (/2/) ; ••• __ . . . .

'COMMENT' NOW FOLLOW TEE DESIGN CALCULATIONS

WEIGHTS

&

THEIR C. OF G.

RESISTANCE

&

PROPULSION,

CAPACITIES,

DEADWEIGHT,

FREEBOARD,STABILITY, •••••• ETC.;

'COMMENT' EQUALITY CONSTRAINTS;

H(/'/):=.···;

H (12/) : = ••••••••••• ;

(22)

, END' ;

APPENDIX 4. CONSTRUCTI0N CF THE PROGRAM FOR PARAMETRIC STUDY

'BEGIN'

'COMMENT' DECLARATION OF INPUT PARAMETERS;

'CO~~ENT' DECLARATION OF OUTPUT PARAMETERS;

'COMMENT' DECLARATION

OF

THE PARAMETERS

AS

CALLED

BY

COSOL;

'COMMENT' THE FCLLOWING PReeEDURES ARE DECLARBD AND LISTED - LINORT - PROCEDURE FOR LINBAR EQUATIONS SOLUTION, - COSOL - PROCEDURE POE PARAMETRIC STUDY,

- DESIGN - PROCEDURE FCE SBIP DESIGN MODEL;

'COMMENT' START OP TBE PROGRAn EXECUTION;

'CO!MENT' READ OF INPUT DATA:

/

- EXPERIENCE COEPFICIENTS AS OBTAINED FROM TH! ANALYSIS OF A BASIS (SIIHLAR) SHIP,

- DESIGNED SHIP REQUIREMENTS,

- CONSTANT VALUES(E.G. STANDABD-RESISTANCE DATA,

HYDROSTATIC DATA, COEPPICIENTS FOR REGRES ION EQUATIONS, ETC.) ,

- TOLERANCES POR CONSTRAINTS(E.G. 100 TON -FOR DEADWEIGHT, 0.01 M FOR GM-VALUE, ETC.),

- START VALUES OF FREE VARIABLES AS NEEDED FOR INITIATION OP THE ITERATION PROCESS (E. G. L=I

V

1/)

=

120.00,

(23)

'COMMENT' CALL CF THE PROCEDURE CCSOL COSOL;

'COMMENT' STATEMENTS FOR PRINTING OF INPUT AND OUTPUT

'CCMMENT' KRAK 15 THE EXIT LABEL FOR C050L IF DO SOLUTION; KRAK:

'COMMENT' END OF THE PROGRAM 'END'

(24)

I

2 BETTER POINT ALGORITHM - AN OPTIMISATION METHOD FOR PRELIMINARY SHIP DESIGN STUDIES

2.1 INTRODUCTION

THE METHOD PRESENTED HERE ALLOWS NONL INE.AR CONSTBAINED 1'IINIl'lISATION PROBLERS TO BE SOLVED UNDER THE ASSDMPTION THAT ONLI VILOES OF ONIMODAL OBJECT FUNCTION CAN BE CALCULATED AND NOT DERIVATIVES.

MAKING DSE OF THE TRIAL AND ERROR TECHNIQUE OF HOOKE AND JEEVES

AND OF THE DEFINITION OP THE BETTER POINT ALGORITSM TBE SEARCH FOLLOWS IN THE DIRECTlOl OF THE FRASIBLE REGION, IND AFTER THIS REGION IS

REAC~ED

,IN

THE DIRECTION OF THE ~INIMUM.

ACCORDING TO THE DEFINITION OF TEE BETTER POINT ALGDRITHH, THE ~ETHOD

COMPARES TVO CALCULATED POINTS BY COMPARISON OF THEIR OBJECT FUNCTION VILUES AND OF TEE CONSTRAINTS, AND CHOSES A BETTER POINT.

THE BETTER POINT ALGORITHM I CCNTRARI TO THE SUMT METHODS DOES NOT MAKE USE OF ANI PENALTY PUNCTIONS. THI5 IS THE FlaST MAIN DIFFERENCE BETWEEN PENALTY FUNCTIOK METHODS AND THAT ~p BETTER POINT ALGORITHM. THE SECOND DIFfERENCE IS TEAT IN THE BETTER POINT ALGORITHM , EACE SEARCR CYCLE IS TEBMINATED BY FIND~NG THE TEMPOHARY MINIMUM IN THE fEASIBLE SPACE, SHILE IN TH! PENALTY PONCTION METHODS TEMPORARY MINIMA DAl LIE IN THE INFEASIBLE SPICE.

THE 'METHOD ALLOWS MIHIMISATICN PROBLEMS TO BE ~ANAGED WITH BaTH EQUALITY AND INEQUALITY CONSTRAINTS. DURING THE SEARCH THE EQUALITY CORSTRAIHTS ARE SOLVED EACB TIME WITH THE AID OF LINEARISATION AND ITERATION PROCEDURES. FURTHEMORE THE METHOD ASSUMES THAT THE ABSOLUTE ACCURACIES AND DESIGN MODEL ACCURACIES ~HICH ARE REQUIRED ARE KNORN

(I.E. NUMERICAL MODEL OF THE OBJECT FUNCTION AND CONSTRAINTS). THE APPLICABILITY OF THE DEVELOPED METHeD RITH REGARD TC DESIGN PROBLEMS HAS BEEN TESTED ON A BULK-CARRIER DESIGN MODEL.

2.2 ·DESCRIPTION OF THE METHOD 2.2.1 THE PROBLEM

THE PROBLEM IS FORMULATED AS FOLLOWS

MINIMISE THE OBJECT FUNCTION P(X) SUBJECT TO THE CONSTRAINTS,

EQU~LITY AND INEQUALITY WHERE Hl (X) = 0 , GJ (X)

<=

0 , 17

(25)

x ~'. X2, •••••••• XK - PREE VIRIIBLES,

R

= NUM3ER OF FREE VARIABLES,

I 1,2,3, •••••••••• TNEC,

J 1.2.3, •••••••••• 7NIC.

TNEC

=

nUMBER OF EQOILITI CCNSTRAI~TS,

TNrC = NUMBER OF INEQUALITI CONSTRAINTS,

TNEC( 'K - CONDITI03 THAT HORE THAN ONE SOLUTION CAN BE FOURD,

TNC

>

0 - CONDITION THAT THE PROBLEM IS CONSTRAINED, TNC = THEC + TNIC = TCTAL NU~BER OF CONSTRAIHTS.

2.2.2 ASSUMPTIONS

IT IS ASSUMED THAT THE OBJECT FUNCTION IS UNIMODAL IN THE RANGE Ta BE

TEST~D. THIS ~EANS THAT THI FUNCTION HAS ONLI ONE LOCIL MINIHUM IND ACCORDINGLY ONLY ONE SEARCH IS NEEDED Ta REICll TriIS MINII'!U~.

AS PREVIOUSLY STATED ONLt FUNCTION VALUES CAN BE CALCULATED, HITHIN THE METHaD CONSIDERED AND NO DERIVATIVES.

2.2.3 SOLUTION

2.2.3.1 BETTER POINT ALGORITHM

Ta SOLVE THE ABOVE STATED PROBLEM, TilE DIRECT SEARCH TECHNIQCE

OF HOOKE AND JFEVES IS ADOPTED(2). THE DIRECT SEAReH OF HOOKE AND JEEVES IS A TRIAL AND ERROR TEcnNIQUE DEVELOPED POR NON-CONSTRAINED PROBLEMS AND IS BASED ON TWC TYPES OF STEP-BI-STEP SEIRCHES

ALTERNATING IN TURN: A LOCAt SEARCR, MBIcn FOLLOWS PARALLEL Ta THE MAIN CO-ORDINATES, AND A PITTERN MOVE WHleR REPRESENTS A ROTATIon OF

THE SEARcn DIRECTION WHICH ACCELERATES THE SEARCH BI THE AID OF

INCREASING THE STEP WIDTHS. FOR FURTHER EXPLANATION SEB FIG.1.

aSING THE ORIGINAL nOOKE AND JEEVES TECHNIQUE, TNO SELECTED POINTS

ARE COBPARED OH THE BASIS CF TEEIR FU~CTION VALUES WHEREAS IN

THE BETTER POINT ALGORITHl'l T:IESE POINTS ARE COr.PARED ON THE BASIS

OP BOTH THEIR OBJECT FUNCTION AND CCHSTRAINTS VALUES. THE ALGORITHM

CONSIDERES THE BETTER POINT OUT OF THESE THO POINTS AS BEING TUIT

POINT WHICH

EITHER - LIES CLOSER TO THE FEASIBLE REGICR(I.E. HAS LCNER

VALUES OF THB CONSTRAINTS). IN CASE BOTH POINTS

LIE O:JTSIDE TUE FEASIBLE REGION (FIG. 2) ,

OR - LIES INSIDE THE PEASIRLE REGION iHILE TUE eTHER

POINT LUS OUTSIDE IT (FIG.3).

OR - HAS TH! LOKEST 09JECT rUNeTION VALUE WHEN BOTH

POINTS LIE INSIDE THE FEASIBLE REG lOM (PIG.4).

THE SEARcn CA~ START PRaM ANY GIVEN PCINT, EITHER THAT LYI~G IN THE

INFEASIBLE OR IN THE PEASIBLE SPACE. 1T THEN CONSISTS OF TNO PRASES

1- A SEARCH FOR FEASIBLE SPACE (FIG.S),

2- A SEIEcn POR TIIE ~INIMUM (FIG.6).

ONCE THE FEASIBLE SPACE HAS BEEN BEleRED, THE ALGO~ITHM CARES NOT TO

VIOLATE THE CONSTRAINTS AND MOVES FOR~AFD TCHARDS THE MINIUUM.

(26)

2.2.3.2 SEARCH ALONG

BOUND

ARIES

IN SOME CASES THE ADAPTED PROCEDURE OP HOOKE AND JEEVES MAJ EAIL. THIS IN EFPECT MEANS THAT THE SEADCE WILL BE TERMINATED AT A POINT OTHER THAN THE MINIMUM. SUCH CASES CAN OCCUR

EITHER WHEN - THE EQUAL -LEVEL-CONTOURS OF THE OBJECT

FUNCTION

FORM A

NARROW RIDGE WHleH IS NOT SITUATED PARALLEL TO ANY

OF THE

AXIS OF

THR

MAIN CO-ORDINATE SYSTEM, AND

THE

LAST BASE POINT LIES ON THE TOP-LINE OF TH! RIDGE(PIG.7), OR - THE ~INIMOM POINT LIES ON THE BOUNDARY,

THE EAST BASE POINT LIES EITHER ON THE SAME BOUNDARY OR CLOSE

TO

IT, AND THE SLOPE OF THIS BOONDAEY IN THE NEIGBOURHOOD OF THE LAST BASE POINT IS NOT SITUATED PARALLEL TO ANI OF THE MA IN AXES

(lIG.8).

IN CASES WHEN THE ADAPTED HOCKE AND JEEVES PROCEDURE FAILS ANCTHER PROCEDURE IS ACTIVEVATED. IN THE FIRST STEP THIS PROCEDURE CHECRS IF EITHER ANI OF THE BOUNDARIES LIES SUFFICIENTLY CLOSE TO THE LAST MET BASE POINT (ACTI'E BOUNDARY).

IF

T8IS IS

50, TH!

PROCEDURE RUNS THE NECESSARY CALCULATIONS TO PIND

THE

SLCPE-VALUE OF THE BODNDARY, AND ftAKES A SEARCH-STEP FROM THE BASE POINT, PARALLEL TO THAT

BOUNDAR! SLOPE. THE PROCEDURE THEM ACCEPTS THIS GIVEN DIRECTION TOWARDS THE MINIMUM, AND REMAlNS FURTHERMORE ACTIV! UNTIL THE POINT WITH THE LOWEST OBJECT FUNCTION VALDE HAS BEEN FOUND. A NE~ SEARCH THEN STARTS AGAIN, USING THE ADAPTED HOOKE AND JEEVES PROCEDURE. FIG.9

SCHEMATI-CALLY EIPLAINS THE ABOVE.

2 .

3.3 ROTATION

IN THE CASE HOREVER THAT THIS SEADCH UNDERTAKEN ALONG THE ACTIVE

BOUNDARY FAILS (I.E. DOES NOT PROMISE TO BE A SUCCESS), ANOTRER PROCEDURE RILL BE ACTIVATED WHICH ALLOWS A SIMPLE ROTATION ABOURD THE LAST BASE POINT, TO BE UNDERTAKEN IN STEPS OP 30 DEGREES

(FIG. 10 AND 11 ). T8IS PROCEDURE IS ALSO ACTIVATED IN THE CASE WHEN THE ADAPTED HOCKE AliD JEEVES PROCEDURE STOPS THE SEARCS AND NONE OF THE BOUNDARIES LIE CLOSE TO THE LAST BASE POINT.

SOLVING EQUALITY

CONSTRA

I

NTS

RHEBEVED THE CO-ORDINATES CF THE SEARCHING POINTS ARE ALTERATED,

THE EQUALITY CONSTRAIN~S ARE SOLVED WITH THE AID OF LINEARISATION AND ITERATION PROCEDURES. THE EQUALITY CONSTRAINTS ARE APPROXIMATED

BY LINEAR EQUATIONS

H (I)

E,J)

*X (J) + B (I)

I'rHERF: J 1,2 •••••••••••• K, I

=

1,2 ••••••••• TNEC ,

K NUHBER OF FREE VARIABLES

(27)

X

(oJ) -=

FREE VA

R

IABLES

,

USING TRIAL TES'l'S

AROUND

THE

SEARCHI~

POINT T

l

lE COEFFICIENTS

A(I,~

A

rD

B(I)

ARE CALCDLATED. TH! EQOALITY C

O

nSTRAINTS ARE

TUEN

CALC

U

LATED

BY SOLVING TUE FOLLOWING SET OF EQUATIONS

K-TNEC

E<J)

*X(

,

1)

J=1

+ K

~J)*X(J)

J=K-TNEC+1

WHERE IT IS ASSUMED

THAT

~"E

VARIABLES

+ B (I)

Y (J)

FO

P

J=K-TNEC+ 1 ,

K-TllEC+

2 •••••••••• K

WILL

BE

ELIMINATED BY THE AID OF EQUATION SOLUTIONS, AND

Y. (J)

FOR

J=1,2 ••••••••••••

K-T~!EC

WILL

REAMAIN AS

FREE

VARIABLES.

o

THE EQUALI'T'Y CON5TRAIN'T'S

ARE

TIiEN

CHECKED

AGAIN

FOR

V!OL

.

ATION Of

THEI

R

PER

M

ISSIBLE TOLERANCES (

GAMO(I)

), IN

W

HICH

CASE THE

WHOLE

PPOCESS IS AGIJN

REPEATED.

THB

LAST

SOLUBLE

POINT IS TUEN TAKEN

AS

A

NFW

START POINT FOR

THE NFXT

SET

OF C1LCULAT10N5.

GA

M

O(I)

-

TOLERANCES FOR EQUALITY

CO

N

STRATNTS.

2.2.3.5 ACCURACY OF' THE DESIGN MODEL·

7HO

KINDS

OF

TEE

DESIGN MODEL

ACC

D

RACIES HAVE AL6EADY BEEN

INTRODUCED

llH1ELY:

ABSOLUTE

ACCURACY

AND

H

EQ

U

I

R

ED ACCURACY

• -THE

ABSOLUTE

ACCUBICY IS REPRESEHTED EY

n

EAN E

R

RCRS CAOSED BY

COMPUTER CALCULATTONS

W

HEN

CONSTRAINTS

AND OBJECT lUNCTTON

VALUES

ARE

EVILUATED. THIS

ACCU

R

IC!

IS

PR!SENT~D

BY

TH!

ARRAY

ICeDR(l),

l

HB

R

E :

ACCUR(O) -

ACC

U

DACY

FOB

THE

O

BJ!CT FUNCTION

,

ACCUBIl)

-

ACCUBACY

FCP

T

H

E

CONS~RATNT

I ,

I 1, 2 •••••• > • • • T N

Ee

KNORLEDGE

OF

THE ABSOLUTE ACC

U

PACY

VAIUE IS

NECESSARY HHEn

CONSTRAINTS AMD

OBJECT

FUNCTION VALUES

OF

TVO

TESTING

POINTS ARE

~

COMPARED. IF

FO~

Teo TESTING PCINTS

TUE ABS0LUTE

VALDE

OF

~HE

DIFFERENCE BETiEEN TiO VALUES OF

T9E

SA~E

CONSTRATNT I,

OR OF TilE

OBJECT

FurCTI1N, IS

LOWE~ THA~

2*

ACCU

B

(I)

THEN THOSE

TWO VALUES

ARR ISSUMED AS

EQUAt. lP THE

ABSOLUTE ACCDRACY IS

NOT KNOWN

TAEN

TBE

KETHOD ALLCP5 THE

CALCULATIO

N

OF

IT

UNDE

R

THE AS5UMPTION THAT THE

ABSOLUTE

ftCCUqACY BE

A TIMES

GEEATEE THAn THE HE

Q

UrRED ACCURACI.

THE

RE

Q

UI

P

ED

~CCURACY

REP

R

ESENTS HEAN TOLERANC

E

S ALLOWED BY

TEC~NICAL

RECUIREMENTS TO THE SOLUTION OF TUE DESIGN

TASK. IN TEIS

METHOD THE

REQUIRED

ACCURACY

IS EXPRE

S

SED IN

TERMS

OF FINAL STEP

WIDT

H

S,

AS

DESCRIBED UNDER

2.2.3.6 .

(28)

STEP WIDTHS

IT IS ASSD!ED THAT THE FIRAL STEP WIDTHS OF THE FREE VARIABLES

IlEPRESENT THE FEQUIRED ACCUFACY OF THE DESIGN MODEL IN SUCR A W.AY

THAT IN THE NEIGHBOURROOD CF THE MINIMUM, AN INCREMEN! OF ONE VARIABLE X(I) BY FINAL STEP WlDTH DELTA XCI) WILL CAUSE

INCREMENTS OF THE OBJECT FONCTION AND OF CONSTRAINTS LOWER THAN,

OR EQUAL TO THE TOLERANCES ALLOiED BY TECHNICAL REQUIREMENTS Ta

THE DESIGN. FCR EXAilPLE IF lNITIAL STABILlTY GM HAS TO BE

CALCOLATED HIT"!N THE ACCnRACY OF + - 0.01 M THEN A SEPARATE

INCREASE OF THE MAIN DIMENSIONS L, B, D

BJ

FIIAL STEP WIDTHS DL,

DB, DD MAY NOT CAUSE AN INCREMENT OF INITIALSTABILITY , DGM

( IN lTS ABSOLUTE VALUE ) OP HIGH En THAN 0.01 ti. lIMAL STEP WIDTHS

SHOULD THEREFORE BE CHOSEN ~ITH CAREFUL ATTENTION.

INITIAL STEP HIDTHS AFE THE STEP WIDTHS USED IN THE FIRST SEARCH

CYCLE AND MUST BE LARGEN THAN FINAL STEP ~IDTHS. BY TAKING LARGE

ENOUGH INITIAL STEP HIDTHS, THE METHOD IS ALLOVED TO PIND A POINT

CLCSE TO TH! MINIMUMIN IN THE FIRST CYCLE APTER ONLY

A

S~ALL NUMBER

OP CALCULATlcns.

2.2.3.7 SEARCR CYCLES

THE SEARcn CONSISTS OF A CERTAIN NUMBER OF CYCLES WRICH NU9BER

~UST BE SPECIFIED IN TilE INPUT DATA. ONE CYCLE IS ONE RUN OP THE PROGRAh Ta A TE~PORARY OPTIMUM. THE PURPOSE OF USING MORE THAN

ONE CYCLE IS TO GIVE THE OPPORTUNITY Ta START WIT~ THE LARGE INITIAL STEP WIDTHS BI ~HICH THE SEinCH PROGBESS VERY FAST IN THE PIBST CICLE.

EACl FOLLOWING CYCLE USES SMALLER STEP i IDTHS BY iRICR THE OPTIMUM

IS REACHED WITH HIGHER ACCURACY. EACH CICLE WILL BE TERMINATED WHEH

THE METilOD CANNOT FIND A BETTER POINT USING THE STEP WIDTHS VALlD

POR TH AT CYCLE. THE NEXT CYCLE STARTS WITH DECREASING THE STEP WIDTHS.

IF IN TilE LAST SPECIFIED SEARCR CYCLE THE OPTIMUM WILL BE FOUND, BUT

AT LEAST ONE CONSTRAINT VIOLATED, THE ADDITIONAL SEARCH CYCLE WILL

BE ALLOWED RITH DECREASED STEP WIDTHS.

2.3 TEST CASES

THE ~ETHOD HAS TESTED CN ThE HIGHLY EXCENTRIC AND T~ISTED

FUNCTION, DERIVED BY FOX (1)

+

x

(1)

**

2 - 2 *.1: ( 1 ) + 5

WUIcn HAS BOUNDED BI 3 DIFFERENT SETS OF CONSTRAINTS APPLIED

IN TORN :

1-ST SET OF CONSTRAINTS : -X(1) - 2

<=

0

2-ND SET OF CONSTRAI~TS

-x

(1) - 2

<=

0

X(1) -X(2) 0

(29)

3-RD SET OF CONSTRAINTS

0.5 -

X (1)

<=

0 0.5 -

X(2)

<=

0 ,

-X(1) + 1/(O.9d~2)+O.1)

<=

0 ,

FOR ALL THESE CASES THE TRUE MINIMUM IS

ij.OO ( X(l)

=

1.0 ,X(2)

=

1.0).

THE RESULTS FR OM TRIAL TESTS ARE SBOWN IN TABLE 1 •

THE FUNCTION !MD )-RD SET CF CONSTRAIRTS ARE SROiN IN FIG. 12 •

2.4

REMARKS

APPLYING THE ME~HOD TO THE SRIP DESIGN OPTIMISATION PROBLEMS ONE HlS TO REMEMBER THAT THE SPEED

V

AID BLOCK COEPFICIENT CB ARE IN FACT LESS SENSITIVE VARIA BLES ~HEN CONSIDERING SUCH

EQUILITY CONSTRAINTS AS DEAD~EIGHT (OR CARGO HOLDS CAPACITY). STABILITY AND FREEBOARD. CONSEQUENTLY Ta THAT AND IN THE CASE

~HEN CB AND/OR V ARE ASSTI~ED TO BE FREE VARllBLES THE FOLLOWING CONDITlûN SHCULD BE FULFlLLED :

TNEC

<

K - P

WHERE

TNEC - NUBBER OF HQUALITY CONSTRAINTS K - NU~BER OP FREE VARIABLES

P - NUMBER OF LESS SENSITlV! VIRIABtES

THE SEQUENCE OF FREE VARIIBtES BAS Ta BE AS FOLLOiS : - FIRST TBE LESS SENSITIV! FREE VARIABLES - FCLLCWED BY MORE SENSITI'E PREE VARIABLES E.G.

- V,CB,L,B,T,D, - V,L,B,D, - CB,L,B,

2.5

REFERENCES

(1) FOX H.L.: OPTIMIZATION METHODS FOR ENGINEERING DESIG~,

ADDISON-WESELEY PUBLISHING CC., 19711 •

,

(2) HOOKE R, JEVEES T.A.: DIRECT SEARCH SOLUTION OF NUHERICAL AND STATISTICAL PROBLF~S , JOURNAL OF THE ASSOCIATION POR

COMPUTING MACHINES,VOL.8,APRIL,1962.

(30)

TABLE Results from optimisation Object function, F 10X 4 2 2 2 .

I - 20X2XI + IOX2 + XI - 2XI + 5 CASE START POINT STEP WIDTHS OBJECT TOLERANCY CYCLES NUMBER

NQ MINIM. POINT INITIAL FUNCTION OF NUMBER OF

FINAL VALUE EQUALITY FUNCTION CONSTRAINTS

CONSTRAINT

EVALUA-X( I) X(2) DEL( I) DEL(2) TIONS

I -0.53 _{1. 08} 0.17 _{1. 17} 0.10 _0.01 0.10 4.0065 2 7I 0.01 2 -0.53 _1.002 0.17 _1.005 0.10 _0.001 0.10 _0.001 4.0000 3 174 INEQUHITY: 3 0.0 _1.0000 0.0 _1.0000 0.10 _0.01 0.10 _0.01 4.0000 2 59 -X(I) - 2.0 ;'; 0 4 0.37 0.37 0.10 0.10 4.0010 3 100 1. 0317 1.0650 0.001 0.001 5 0.53 _1.010 0.00 _1.010 0.10 _0.01 0.10 _0.01 4.0011 _-+ 0.01 2 31 INEQUALITY: 2. IS 0.35 0.10 0.10 -X(I) - 2.0 ;; 0 6 0.95 0.95 0.01 0.01 4.0251 + 0;01 2 31 EQUALITY: 7 2.15 0.35 0.10 0.10 4.0000 + 0.001 3 38 -XCI) - X(2) = 0 1.005 1.005 0.001 0.001 -8 1. 45 0.27 0.10 0.10 4.0004 3 130 1.0105 1.0155 0.001 0.001 9 0.42 _1.0130 2.030 0.10 0.10 4.0002 3 98 INEQUALITY: 1.0280 0.001 0.001 0.5-XCI)~0 10 0.80 _1.0183 2.50 0.10 0.10 4.0005 3 113 0.5 - X(2) ~ 0 1.0405 0.00 I 0.001 -XC I) + 2. IS 0.00 0.10 0.10 1/(0.9

*

X(2) + 0.1) ;; 0 11

1.0208 1.0495 0.001 0.001 4.0010 3 lOS see Fig. 12

12 1.50 2.00 0.10 0.10 4.0004 3 117

(31)

X(

i

+1)

24

• ₅

MINIMUM

r

.LOCAL SEARCH

3

4 PATTERN MOVE

(32)

X2

P2 IS "BETTER"

X2

P2 IS "BETTER"

X1

FIG.2

B

ETTER

PO

INT ALGORITHM (BPA),

TWO POI

N

TS IN INFEASIBLE

(33)

X2

26

P2 IS "BETTER"

FIG.3 BPA, ONE POINT IN INFEASIBLE

THE OTHER ONE IN FEASIBLE

~

_,

'

_O""}

P2 IS "BETTER"

'0

:or

,7

X1

(34)

X(I

+

1)

FIG.5 BPA, PRASE l-SEARCR FOR FEASIBLE

FIG.6 BPA, PRASE 2-SEARCR FOR MINIMUM

(35)

28

XlI

+

1)

MIN

.

4 X

(I)

FIG.7 EXAMPLE WHEN HOOKE & JEEVES TECHNIQUE MAY FAIL

X(I+1)

4 X

(I)

FIG.8 EXAMPLE WHEN HOOKE & JEEVES TECHNIQUE MAY FAIL

(36)

X(I+1)

X

(I)

FIG.9 BPA,

SEARCH ALONG BOUNDARIES

X(I+ 1)

₃

₂

1

8

6

7 x

(I)

(37)

x

(1+1)

XII)

FIG.))

BPA,

SEARCH ALONG BOUNDARIES

X(2)

=4.0

X 1

FIG.12 SAMPLE OF TESTED FUNCTION

(38)

APPENDIX

Arm 7

COMPUTER PROGFAM. DESIGN MODEL AS FCR TEST CASES 5,6

• BEGIN'

, CC!1I'lENT'

TEST PROGRAM OF THE OPT.1l'IISATION PROCEDURE DOPT1D, DESIGN MODELS AS POR TEST CASES 5,6 AND 7

STANDARD DECLARATIONS AND DATA FOR OPTHI. PRoèEDURE; 'INTEGER'DSN1,DSN2,DSN3;

'INTEGES"ARRAY'IIA(/1:10/),OIA(/1:3/); 'REAL~!,OFV;

INTABR!! (O,lIA) ;

DSll1:=IIA (/1/) ;DSN2:=IIA{/2/1 ; DSNJ:=IIA(/3/) ; SETTING (DSN1, 132,60); 'IF'-DSN2=DSN1'THEN' SETTING{DSN2.132,60) ; 'IF' .... DSN3=DSN21-nSN3=DSN P T.HEN' SETTING (DSN3, 132,60) ; "BEGIN' 'ARRAY-IRA(/1:3.IIA(/6/)/),ORA(/1:II1V6/)/),GST{/1:IIA(/9/)/), H,GAMC(/1:'IF'IIA(/7/)=O'THEN'1'ELSE'IIA{/7/)/), G(/1:'IP'I1A(/8/)=O'THEN"'ELSE'IIA(/8/)/); 'ARRAY'X(/1: 100/); 'A~RAyIACCUR(/0:IIA(/9/)/) • INARRAY(O.IR~ ;INREAL(O,A); 'IF'IIA(/7/)=O'THEN'·GOTO'DA1. INARRAY (O, GArIC) ;

DAl: • BEGIN'

'COMMENT' DECLARATIONS FQR DESIGN ~ODEL

---_._---_._--- ---,-~ 'PROCEDURE'DESOPT(H4G,OFV,X) ; 'ARRAY'H,G,X; 'REAL'OPV; 'BEGIN' OFV:=10.0.X(/1/).X{/1/)*X(/1/)*X(/1/) - 20.0.1 (/2/) .1 (/l/).X (/1/) + 1 O.O •. X (/2/) d

V2/)

+X{/1/).X(/1/)-2.0.X(/1/}+5.0; H(/l/) :=X(/1/)-Xl/2/); G(/l/) :=-1(/1/)-2.0. 'END- DESOPT;

•

'COftMENT'---; 'PROCEDURE'DQPT1D

{IIA,IRA,OIA,CRA,A,OFV,X,H,GAMC,G,GST,ACCUR,DESOPT,RESULT~; 'INTEGER"ARRAY'IIA,OIA;

'A~RAY'IRA,ORA,X,H,GAMO,G,GST,ACCUR;

'REAL'A,OFV; 'LABEL'RESULTS;

'PROCEDURE' DESOPT; 'CODE';

(39)

DOP'l'1D

(IIA, IRA, OIA,ORA, A,OFV ,.X" H, GillO, G, GST" lCCUB,DESOPT,RESULTS) ;

RESU.L'l'S:

'END' :

'END';

, EIID'

APPENDIX 2 EXAMPLES

OF

INPUT/OUTPUT, TEST CASE

5 INPUT

DATA

FOR OPTIftIZATION

DSN1 DSN2 DSN3 DIR MAXEVAL

K TNEC TNIC TNC

TCl

+1

+2

+3

+1

+500

+2

+1

+2

K DEL DELTA PSI

+1

+.1000

+.0100

+.5300

+2

+.1000

+.0100

+.0000

N

'

RC

GAl'IO .1

+.0100

STEP lfIDTHS DECR.PACTOR FOR

DES. MODEL

ACCURACY

1

+.01000

K

ACCU~

(K)

+0

+.00057180

+1

+.00010000

+2

+.00010000

REsuns

THB LAST BEST POINT

epv

EVAL CY K

+1

+2

32

+IJ.0011

x

.'.01QO

+1.0100

+31

+2

DELF

+.0100

(40)

Ne GST +1 +.0000 +1 -3.0100 IlEe GAI1 +1 +.0100 BRTWEEN RESULTS Z CY

EVAL

OFV

X X 2 H 1 G 1

+'

+1 +7 +4.8414 +.5300 +.5300 +.0000 -2.5300 +2 +1 +8 +4.6803 +.6300 +.6300 +.0000 -2 .• 6300 +4 +1 +9 +4.4614 +.7300 +.7300 +.0000 -2.7300 +2 +1 +10 +4.2280 +.8300 +.8300 +.0000 -2.8300 +4 +1 +11 +4.0104 +1.0300 +1.0300 +.0000 -3.0300 +2 +1 +'2 +4.2327 +1.1300 +1.1300 +.0000 -3.1300 +2 +1 +13 +4.0473 +.9300 +.9300 +.0000 -2.9300 +4 +' +14 +4.8532 +1.2300 +1.2300 +.0000 -3.2300 +2 +1 +15 +6.0352 +1.3300 +1.3300 +.0000 -3.3300 +2 +1 +16 +4.2327 +1.1300 +1.1300 +.0000 -3.1300 +2 +1 +17 +4.0473 +.9300 +.9300 +.0000 -2.9300 +2 +1 +18 +4.2327 +1.1300 +1.1300 +.0000 - 3. 1300 +2 +2 +20 +4.0189 +1 .• 0400 +1.0400 +.0000 -3.0400 +2 +2 +21 +4.0046 +1.0200 +1.0200 +.0000 -3.0200

(41)

+4

+2

+22

+4.0011

+1.0100

+.0000

-3.0100

+2

+23

+4.0000

+1.0000

+.0000

-3.0000

+2

+24

+4.0046

+1.0200

+.0000

-3.0200

+4

+2

+25

+4.0000

+1.0000

.1.0000

+.0000

-3.0000

+2

+26

+4.0011

+1.0100

+.0000

-3.0100

+2

+27

+4.0011

+.9900

+.0000

-2.9900

+2

+28

+4.0000

+1.0000

+.0000

-3.0000

+2

+29

+4.0046

+1.0200

+.0000

-3.0200

APPENDIX 3 OPTIMISATION PROCEDURE

32

'PBOCED6BE'nOPT1D

33 (IIA,IRI.OIA.ORI.A.OP',X.H.GA~O.G.GST,ICCUB,DESOPT.RESULTS); 33 'INTEGE~"ARRAY'I1l,OI1;