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Pergamon

PII: SO360-1323(96)000654

Building and Environmew, Vol. 32, No. 4, pp. 331-339, 1997

6% 1997 Elsevier Science Ltd. All riehts reserved

Prmted m &eat Britain

036&1323/97 $17.00+0.00

Multicriteria Optimisation of Shape ofEnergy-Saving Buildings

WOJCIECH MARKS* (Receiv ed 18 December 1995; revised 6 Ap ril 1996; accepted 7 No vem ber 1996)

The object is to determine the optimum dimensions of the shape of a building of volume V and

height h, based on the follo w ing criteria: (1) minimum building costs, including the cost of the

mat erials and construction; (2) minimum y early heating costs. The solution to the problem is

presented in two ways. In the first, it is assumed that the shape of the plan of t he building is defined

by t wo arbitrary curves bounding the south and north faces and that the window s on the southern

side are defined by a continuous function as a percentage of the t otal w aN area. In t he second, i t is

assum ed that the buildin g is ofprism ati c shape on polygonalpl an, and using non-linear programm ingmethods the proportions of wal l lengths, w ail angles and buildi ng height are determined. This

problem was solved numerically by means of the CAMO S computer program. It is not the object

of the paper t o obtain a practical design. The results constit ute inf ormat ion for designers on the

optimum proportions of w all lengths, their angles and glazing parameters, taking int o account the

above-ment ioned criteria. The degree t o w hich these result s can be applied in practice depends on

many other requirements present in the design of buildings. 0 1991 Elsevier Science Ltd.

INTRODUCTION

The architectural design of a building is influenced by the

cost of the energy that will be needed during its service

life. It is necessary to find a compromise between the

classical elements of design (form, structure and function)and the requirements resulting from the introduction of

an additional criterion [11.

The optimisation problems of the shape of buildings

with regard to the economical use of energy have been

the subject of many publications. Those concerning uni-

criteria optimisation of building shape include, among

others, references [2%5].

In Fokin [2], the shape of a building of given volume

was optimised taking the minimum heat energy loss as

the criterion. The solution was a spherical shape. Intro-

ducing an additional constraint that the building must be

a rectangular prism, a cube was obtained. The problem

of choosing the optimum dimensions of a building on

rectangular plan with minimum heat requirement per m3

of volume was solved by Gadomski [3]. Heat gain due to

insulation was not taken into account.

The geometry of building shape was analysed in

Menkhoff et al. [4]. The notion of geometric compactness

was introduced as the quotient of the area of external

walls to the volume of the building. Buildings of various

shapes were set up using four identical cubes, giving geo-

metrical compactness coefficients between 4/a and 14.1/a.

A building having the shape of a rectangular prism

was optimised in Petzold [5], taking into account heat

gains due to insulation through transparent and opaque

partitions. Applying the criterion of minimum heat

*Institute of Fundamental Technological Research, Polish

Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw,

Poland.

requirement, the optimum relation between the lengths

of building walls and the optimum number of floors were

determined.

Examples of the application of multicriteria opti-

misation in the solution of architectural problems can be

found in references [&l 11.An illustration of the work in

these papers is given by the example of optimisation of a

prism-shaped multistorey office building in Australia [1 l]

applying the following criteria:

l minimum thermal load ratio (the ratio of the total

heating and/or cooling loads predicted for a building

to that of a model building in Sydney’s climate);

l minimum capital cost;

l maximum net usable area (the floor area within the

external walls less the area taken up by lifts and stair-

cases, circulation and toilet facilities).

The design variables with their range of values were:

0 aspect ratio ~ 1, 2 or 3;

l orientation - north, or 30” or 60” east of north;

l number of storeys - 1, 2, 3, 4 or 5;

l glazing fractions on each facade - 0.4, 0.5 or 0.6;

l glazing type-clear, heat reflection or heat absorbing.

The optimum solution was the building oriented towards

the north (in Australia), with a relative window area of

0.4, heat reflection glazing and an aspect ratio of unity.

The number of storeys depends on the method of choos-

ing the preferred solution.

The problem of optimisation of energy-saving build-

ings was most comprehensively presented by Owczarek

[121. He presented a model of solar radiation heat energygained across the windows. In this paper, the thermal

resistance of outer walls and the percentage of glazing of

walls for the fixed geometry of building shape, linear

dimensions of the hexagonal building perimeter at the

331



332 W. Mark s

imposed angles and wall angles were optimised. A bi-

criteria1 optimisation problem for a building of hex-

agonal shape in plan was also solved for the criteria

of minimum cost of the materials and minimum yearly

heating energy costs for the building.

A multicriteria optimisation problem [13, 141 was pre-sented for buildings of a given volume and octagonal

plan. Minimum construction costs and minimum yearly

running costs were adopted as optimisation criteria. The

decision variables were wall lengths. building height. wall

angles, window sizes and thermal resistance of individual

external partitions. The influence of the number of service

years of the building and the parameter defining the shape

of its plan on utility function were also investigated [ 141.

Adamski [151 formulated and solved a particular case

of optimisation of a building with vertical walls and the

plan defined by two arbitrary curves, adopting the same

criteria as in [ 131.

There are many review papers highlighting the need tofind a solution to the problems of shape definition of

energy-saving buildings (e.g. [161).

In this paper the problem of optimisaton of the shape

of buildings is discussed, taking into account (1) the mini-

mum building costs including materials and construction

costs, and (2) the minimum yearly heating costs. In order

to ensure that the building shapes so defined can form a

basis for further design, it is necessary to take into

account ~ in the form of constraints - functional and

construction requirements of these buildings, or, alter-

natively, to search for optimum solutions depending on

certain parameters which shall be defined subsequently

in such a way that those requirements are met.

The object of this paper is to determine the optimum

dimensions of a building of the volume V. The optimum

values of thermal resistances of walls, roof and floor can

be determined independently [12, 131.

The solution to the problem is presented in two ways.

In the first, it is assumed that the shape of the plan of the

building is defined by two arbitrary curves defining south

and north faces and that the south facade windows are

defined by a continuous function as the percentage of

wall area. This problem was solved using calculus of

variations. In the second it is assumed that the building

is of rectangular prism shape and polygonal in plan, and

the proportions of individual wall lengths, their angles

and the building height are determined using non-linear

programming methods.

It is not the object of this paper to obtain a practical

design. Its results present information for design engin-

eers on optimum proportions of wall lengths, their angles

and glazing parameters, in view of the above criteria. The

degree to which these results can be applied in practice

depends on many other requirements of building design.

THE BASICS OF THE MULTICRITERIA

OPTIMISATION PROBLEM

The basic notions in the formulation of a multicriteria

optimisation problem are decision variables, constraintsand optimisation criteria, also called objective functions.

Decision variables are quantities describing structures

or buildings subjected to variations during the opti-

misation process. Decision variables are usually ex-

pressed in the form of a vector xT = (u,. x2 .,... u,J in an

n-dimensional space called the decision space. Every

point in that space corresponds to a building with it

decision variables.

In the optimisation of structures or buildings, uncon-

strained extrema of the objective function are seldomlooked for. A great number of constraints are usually

imposed, defining the allowable solution space (feasible

region).

Constraints imposed on decision variables determine

the boundary of the feasible region. The feasible region

usually constitutes a part of the n-dimensional decision

space. The constraints occur in the form of equalities or

inequalities describing certain conditions to be satisfied

by a structure or building, imposed directly on the

decision variables by limiting certain quantities which

depend on the decision variables. They are of the form:

h,(x)=0 i= 1,2 ,.... r.

g,(x)<0 j = 1, 2 __.. s.

In an n-dimensional decision space. the constraints

form a hypersurface containing the points fulfilling these

constraints in the form of equalities.

In optimisation it is accepted to call a mathematical

expression describing a certain property of the structure

or the building ‘*the objective function”. The property

under examination can be described in the form of a

function or a functional, depending on the way the opti-

misation problem is formulated. The assessment of the

objective function is the basis of the selection of the

structure or material from many other possible solutions

u 7 1 .

Multicriteria optimisation consists of choosing the best

solution from many possible variants on the basis of

many criteria. i.e. an objective function vector fT = (,f;,

f;,...,,/i). A multicriteria optimisation problem can there-

fore be treated as an optimisation problem of the objec-

tive function vector, which is different from the single

criterion optimisation, which can be considered as the

optimisation of a scalar objective function.

The objective functions’ space is k-dimensional. Every

point in that space corresponds to one objective function

vector ,/;(s,). In that space, the feasible region R is rep-

resented by the region f(O) (Fig. 1).

The solution xl“ which makes every objective function

reach its extremum independently of the remaining func-

tions is called the ideal solution of multicriteria opti-

misation. In the case of the search for the minimum

f(x), xld is therefore the ideal solution of a multicriteria

problem if x& and f(x“‘) 5 f(x) for every x&I. As objec-

tive functions are usually in conflict, the ideal solution

does not exist in most cases.

The solution in which none of the objective functions

can be improved without simultaneous deterioration of

at least one of the remaining objective functions is called

the non-dominated solution. x* is a non-dominated solu-

tion when no x exists such thatJ;(x)<.f;(x*) at igK = { 1,

7_,.__, ) and,f;(x)<f;(x*) for at least one ~EK.The search for non-dominated solutions is called opti-

misation in Pareto sense. In general, the Pareto solution

is not unique. Many x* vectors usually exist, forming an

effective curve of decision variables in the Q space to



Optimisation of Shape of Energy-Saving Buildings 333

X

feasibleregion

CT

n

X*

idealpoint

\ the setof compromises

Fig. 1. Transformation of the feasible region into the objective region.

which the vector P = f(x*) corresponds, constituting the

set of compromises (Fig. 1).

In view of the great number of non-dominated solu-

tions, it is necessary to select the best solution on the

basis of an additional criterion. Such a solution is called

the preferred solution. Thus, the preferred solution x P’ is

a non-dominated solution selected on the basis of an

additional criterion. It corresponds to the values of f(xld)

contained within the objective region and is considered

to be the best solution.

A solution of a multicriteria optimisation problem

therefore includes objective quantities - to which belong

the set of compromises and the ideal point ~ and the

quantities which depend on additional preferences - the

preferred solution, i.e. the vector of objective functions

fP’ and the corresponding vector of decision variables

xpr. If there are no additional preferences, the preferred

solution is assumed to be the point belonging to the set

of compromises situated nearest to the ideal point and

the corresponding vector of decision variables [8].

The problem of multicriteria optimisation can be for-

mulated as follows:

yEig f(x),

where f: R=t=Rks the objective function vector given by

IT(x) = {fi(x),f2(x),...,fk(x)},

and R = { x d t ' l h ( x ) = 0 , g ( x ) < O} is the feasible domain

defined by the equality and inequality constraints. The

componentsJ: R-R, i = 1, 2,..., k, are called the partial

criteria of optimisation, and x is the vector of design

variables.

The problem of multicriteria optimisation can be

solved in two stages. The first stage consists of deter-

mining the compromise set. In the second stage the pre-

ferred solution will be found. A number of methods exist

allowing the compromise set to be generated. They are

discussed in various publications (e.g. [18-201). The

weighted objective method, the min-max method and the

constrained objective functions method are often used to

generate the compromise set. A few methods for the

selection of the preferred solution are discussed in ref-

erences [l8-201. Utility functions or matrix functions

methods, constrained objective functions methods and

lexicographic methods are often used to select the pre-

ferred solution.

HEAT LOSSES AND GAINS THROUGH

EXTERNAL WALLS OF BUILDINGS

The physical environment surrounding the building is

defined by the following values:

SD - number of degree-days in a year (day K);

Q,, 0, - average sums of the total solar radiation on

the vertical east, south or west walls during the heating

period (kWh/m’) [12];

CC, conductivity coefficient for the transmission of

heat from the outside to the inside (W/(m’ K));

CQ conductivity coefficient for the transmission of

heat from the inside to the outside (W/(m’K)).

Heat losses in a building include losses through the outer

walls and those resulting from ventilation. Heat gainsare due to solar radiation through the windows. The

difference between losses and gains constitutes the part

of energy that has to be supplied by the heating system

installed. In the problem under consideration, only those

constituent parts have been included which have an

important bearing on the solution, i.e. heat losses through

walls, floor slabs and roof, as well as heat losses and gains

through the transparent partitions.

Yearly losses through the walls, roof and floor slabs

have been obtained from the formula:

E, = Es+ E,+E, = 24SD.

The following notation is used: A, = area of walls;

A, = area of windows; Ad = roof area; A, = floor

area; R,, R,, Rd and R, are thermal resistances of the

walls, windows, roof and floor, respectively;

(Pd = ctdr - tdw)/ (fsz - kv>; ‘pp = t&z - tpnv)/ (t,, - tzw); t, t,,

and t,, are average temperatures of the external surface

of the roof, walls and floor, respectively; and fdw, t,, and

t,, are average temperatures of the internal surface of the

roof, walls and floor, respectively.

Daily heat gains through a vertical window of an area

A, inclined at an angle c(, to the N-S axis, due to solar

radiation of intensity J at an angle CC,o the N-S direction

are equal to

E, = AJcos(cr,- a,).

Annual solar gains that may be obtained by 1 mz of

vertical window have the following form:



334 W . Marks

where p is the inclination angle of the south-east wall

relative to the N-S direction.

OPTIMISATION OF A BUILDING OF ANARBITRARY SHAPE

Formulation qf the problem

The subject under consideration is a building with ver-

tical walls, of constant volume V and height h. The base

of the building is described by two arbitrary curves y,(x)

and y*(x) (Fig. 2). Heat gains through the north-facing

windows will be disregarded and the building assumed to

be symmetric along the N-S axis. The aim of the present

considerations is to determine the form of the curves y,(x)

and y2(x) using two criteria: (1) minimum building cost;

(2) minimum annual cost of heating.

The construction cost is defined by the following

relation:

F, =2~;:{[I-p(x)]c,+p(x)c,)hdl,+2~;;hcqdl,+D,;

where c, and c, are the cost per m* of a wall and window,

respectively; p(x) is the ratio of window area to the total

area of the wall; dl, and dl, are the lengths of elements of

they, and y2 curves, respectively; and D, represents other

costs, independent of the decision variables.

The function expressing the annual heating cost is

assumed to be of the form:

F2 = 48SDc,

where R, and R, are thermal resistances of the wall and

window, respectively; SD = number of degree-days in a

year; c, = unit cost of energy; and Dz represents other

costs independent of the decision variables.

1

r Y

tY 1B,(O>Y,(O))

It is assumed that the heat gain due to the solar radi-

ation during the heating period will be calculated from

the relation:

4= 0, - ~(0, - t 3,) arctan’ y,‘. (l)

Since dl = m dx, dl co@ = dx and dl si$ = dy ,

we obtain

F, = 2s

lid 1 -p(x)]c, +p(x)c,)h,/m dx0

‘.,+2

chc,JFjpdx+D,, (2)

0

Fz = 48SDc,I

xh,/wdx+

‘a

2c, 11QI $ (0, - t3,) arctan’ y’I

x p(x)hJl+y;2 dx + Dz . (3)

The decision variables in this problem are the functions

y,(x), y*(x). It is assumed that [I 51:

1. v,(x) and y>(x) are continuous functions of C class

within the range [0, x,];

2. the shape of the building is symmetric in relation to

the 0 Y axis, i.e.

Y!‘(O) = 0, (4)

y2’(0) = 0; (5)

3. the functions J?,(X), y2(x) bound a region of area V/h,

i.e.

,oP)

s,.s,= sB2(O>y2KV) O=O

Fig. 2. The form of the building.




rS:.[,.,(*_)-l.,(x+/h; (6)

the functions y,(x) and y*(x) equal zero at the point,

with the abscissa x,:

Yl(X.) = Y*(x,) = 0; (7)

the ratio of window area in the south wall ~(1,) = con-

stant.

Solution of the optimisation problem

A set of compromises can be determined by the method

of weight coefficients. We are seeking the minimum of

the objective function

F=iF,+(l-1)F,, (8)

where IL < 0, 1> , with the assumption that condition (6)

is satisfied. On the basis of equation (8), we obtain the

substitute functional

F* = As

:‘~~dx+B~arctan2y,‘vi+)i;idx

+C ;Jmdx+21,[;[y,(x)-yz(x)]dxs

-21,V/h+D,

where

(9)

A = 2/2h[( 1 -p)cs +pc,]

+ 2( 1 l)h24SDc, $1 -p)+ $I

-2h(l -i)pc,Q,,s 0

B = 2(1 -i)hc,p&‘, -Q,),

C = 2hic, + 2( 1 /l)h24SDc, ;,s

D = iD,+(l-1)D,.

This is an isoperimetric problem of variational calculus

[21]. The conditions (4)-(7) enable us to determine the

integration constants and the constant 2,.

The functional [equation (9)] reaches its extreme value

if Euler’s equations [21] are satisfied, that is

(A +2B+Barctan’y,‘)1

(J=.Y%y,“+2& = 0,

(10)

c1

(JG%y,“+2E,, = 0. (11)

Equation (4) can be reduced, by substituting

1u(x) = (Ay,+2Barctar1y,‘+By,‘arctan~y,‘)~

Jm’

(12)

W-4__ = (A+2B+ Barctan’y,‘)

1

dx (.&+“r

(13)

to the form

dv(x)----2/I, = 0.

dx(14)

On integrating equation (14) we obtain

v(x) = 21,x+ K. (15)

From condition (4) we find the integration constant

K = 0. Hence

1x = f (a_~,+2Barctany, +2Barctan2y,‘)-----

1 Jl+v;z’

(16)

Since equation (16) cannot be solved with regard to y,‘,

we will proceed in the following way. We denote

x = $(Yl’) = 4(P),

dY, =pdx,

dx = f(p) dp,

I Y;dYl = (A+2B+Barctan2YI )(,ll+y;2)1.

1y, = I(--A+2Barctany,‘+Barctan’y,‘)

1 Mx------f-.

J_ 2A

The integration constant M should be determined from

condition (7). Similarly, on integrating equation (11) wehave

Y2’W I1 x

Jm- C’(17)

that is

I I’(x) = J&p

Since y2’(x) > 0 for x@O, x, > , we have

y2’(x) = J&.

Hence, on integrating, we have

YZ(X) = - J_+ cz.

From condition (6) we find the constant

c, = JW.

Hence

(18)

(19)

(20)

yz(x) = -J_+J_. (21)

We obtain:

[y&Y- (c/2n,)‘-x,‘]‘+x’ = (A/U,)‘.

This is an equation of a circle with its centre at the point

O2 (0, C,) and radius Rz = C/2L,.

The area of the segment of that circle bounded by the

OX axis is, for y2(x) < 0,



336 W. Marks

= 2 ‘“[J,m-Jm]dxi

(22)II

= (C/21,)’ arcsin (x.‘i,/C) - x~J~.

The curve y,(x) was determined in parametric form. A

suitable segment of the area has been determined numeri-

cally. The constant I”, has been determined from equation

(4) solving the system of equations

2 ;Y.+Yt( 12x*4

n + (C/U,)’ arcsin 7( 1-x,Jm-1 = 0, (23)

1,x, = (Ay,,‘+2Barctany,,‘-By,,‘arctan2y,,’)

x &, i =O,l,...,n, (24)

where

1J:,, = I(-A+2B.y,,‘arctanyl,‘-Barctan’y,,’)

I

xJ&-;I

x (-A +2By,,’ arctanyln’- Barctan’~,~‘)1

J_’

The system of non-linear algebraic equations (23) and

(24) has been solved by the CAMOS computer system.

Numerical example

The data used in the computation were as follows:

c, = 40 PLZ/m’, c, = 400 PLZ/m’, c, = 0.135 PLZ/kWh,

SD = 4000, l/R, = 0.72 W/m2 K, l/R0 = 1.6 W/m2 K,

p = 0.3, 0, = 250kWh/m’, 0, = lOOkWh/m’, V = 1m3,

h = 1 m, x, = 0.8 m. The results are shown in Fig. 3, i.e.

the compromise set AB on the normalised coordinate

system, where the functions 4, and 42 are

4, =AF,(i = 0)’

42 = FzF,(3” = 1)

@)20.9

0.81

Fig. 3. The compromise set, the ideal point, the preferable pointand the corresponding shapes of the plans of buildings.

The ideal point C and the preferred solution D have

been found as the point within the set of compromises,

nearest to the ideal point [20].

The shapes of building plans corresponding to various

values & 0 <i 5 1, are shown in Fig. 3. The coefficient i

can be interpreted as the service life of the building. ). = 1corresponds to point zero of the service life, i.e. it takes

into account the construction cost only. i. = 0 cor-

responds to an infinitely long service life.

SHAPE OPTIMISATION OF BUILDINGS ON

POLYGONAL PLANS

Formulation qf‘the problem

The subject of our consideration is a building with

vertical walls, constant volume V and height h. The base

of the building is octagonal (Fig. 4). It is assumed that

the building is symmetric along the N-S axis. Heat gains

through the windows in the south, south-east, east, south-west and west walls are taken into account.

The aim is to determine the relationship between the

lengths of the walls and their angles. The shapes of the

building obtained in Owczarek [12] and Adamski and

Marks [131 cannot in many cases form a basis for further

design work, because of the contradiction with other

functional and structural requirements. In view of this,

an additional constraint was imposed, concerning the

minimum area of the rectangle inscribed in the octagon

(Fig. 4).

Yearly heat gains have been calculated from the for-

mula

0 = 0,cosb+&sinfi

The two following optimisation criteria have been

adapted: minimum construction cost,

F, = i I,h(l -p ,)c,i+ i pJ,hc,,+B,;/= I ,= I

(25)

and minimum yearly heating cost,

F2 = i +l,h(l -p,) + i $p,l,h 24SDc,/=I .,i i=1 sz 1

+ ~P,M~zI~, + B2, (26)

where c, = unit cost of energy; c, and c, are costs/m2 of

wall and window, respectively; B, and B, are other costs,

independent of the decision variables adopted. The

decision variables are: lengths of walls, I,, i = 1,...,5;

Fig. 4. Shape of the building and notation adopted.




ratios of window areas to wall areas,p,; inclination angles,

Jl and y, of south-east and north-west walls, respectively,

to the N-S direction (Fig. 4).

The variables must fulfil the following constraints: con-

dition of closure of the polygon,

I, + 21, sin b - I, +- 21, sin y = 0; (27)

condition of the constant volume of the building,

h[(l, + I2sin /I)& cos /?+ (Ii + 21, sin /I)&

+(Z,+I,siny)l,cosy]- V= 0; (28)

condition concerning the magnitude of the rectangular

area inscribed in the octagon,

(I, + 21, sin p)13 = I>; (29)

constraints regarding the linear and angular dimensions

1,20, O<Br71/2, oIyI7c/2; (30)

constraints concerning the size of the windows

pisp,<g,,i = I,...) 5. (31)

Solution of the optimisation problem.

The set of compromises can be derived analytically,

e.g. using the method of weighted coefficients [13, 221,

creating a substitute objective function (equation (32))

or by numerical methods, e.g. using the optimisation

program CAMOS [23].

The substitute objective function, taking account of

the equality constraints (27)-(29) and the inequality con-

straints (30) and (3 l), has the following form:

F= nF,+(l-1)F,+~,h(l,+21,sinP-1,-21,siny)

+ pz{ [(I, + Iz sin P)& cos B + (I, + 21, in BY3

+ (& + l4 sin y)& cos y]h - V}

+ P~[(& + 21, sin 8Y3 - Dl + W - 742) V,(Y - 742)

+ dpl -/d-t V‘dPI --I%)+ v,(p, -P*)

+ VdP, -a21 + Gp3 -Pd + VdP, -831, (32)

The necessary conditions for the minimum objective

function (32) resulting from the Kuhn-Tucker theorem

(221 have the following form:

3Fx--o,

1a x ,x , LPI, P>Y? PI. P2> o, ,

i=l,..., 5, u=l,..., 8, (33)

l?Fz >O, x,, = LP,> P>Y>

I

8Fax-, CO, x. = vu,

l,>O, i = l,..., 5, /I>O, y>O, v,>O, u = l,..., 8.

From the form of relations (32), it follows that the

ratios of window areas to wall areas are

PI = el orpi = Pi,

p2 = p2 orp 2 = B2,

p3 = p3 , o rb = p3

Altering the weight coefficiens /I within the range 0 to

1, we obtain the compromise set. The coordinates of the

optimum point within the objective space are

Fl = Fi,inr

F2 = F2mm

In order to find the preferred solution, the utility func-

tion

F= F,+NF2

may be formed, in which N denotes the modified number

of years of service life of the building, which is the number

of years multiplied by a coefficient taking account of the

interest rate and inflation. The following is the relation-

ship between the modified number of the service life years

and the weight coefficient:

1-nN=IZ.

The preferred solution corresponds to the minimum

construction and heating cost during N years.

In this paper the problem was solved numerically using

the computer system CAMOS [23]. It is an interactive

system for the solution of single criterion and mul-

ticriteria optimisation problems. The system allows the

solution of problems of the following dimensions: 10

objective functions, 20 decision variables, 100 inequality

constraints and 15 equality constraints.

NumericaI example

The numerical example of the optimisation of a build-

ing was solved using the CAMOS computer system. The

following numerical data were used: c,, = 27 PLZ/m’,

C o, = 20 PLZ/m’, c, = 0.025 PLZ/kWh, SD = 4000,

k,, = 1/R,, = 0.75W/m2K, k,, = 2.6W/m2K, p, = 0.1,

pi = 0.4, 0, = 350 kWh/m*, e2 = 120 kWh/m2, V = 1 m3,

h=lm.

The problem was solved assuming, in turn, N = 0 (con-

struction cost only), N = 10, 25, 50 and N (heating costs

only). For each N, two cases of the ratio of the rectangle

area D to the plan area of the building A (Fig. 4) were

considered: D/A = 0.7 and 1.0. For each of these two

cases the set of compromises and the optimum point were

defined (Figs 5 and 6). The corresponding shapes of the

Fig. 1a

a a

5. The compromise set, the ideal point and the

responding shapes of building plans for D/A = 0.



338 W . Mark s

0.8 - Al

Fig. 6. The compromise set, the ideal point and the cor-

responding shapes of building plans for D/A = 1 O.

plans of buildings are also shown in these figures. Dimen-

sionless wall lengths q( = l,/,,&.

CONCLUSIONS

The optimisation problem of the shape of a building

with an arbitrary base has been solved by variational

methods. The solution obtained is composed of a circular

segment bounding the northern part of the building and

a curve described in parametric form describing its sou-

thern part. The ratio of the area of the southern part to

the northern part depends on the size of the windows,

the density of solar radiation energy and the ratio of unit

costs of the windows and walls. They increase with the

number N, which determines the modified service time of

the building.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

The relationships between the particular partitions in

a building depend on the unit prices of non-transparent

partitions, windows, heating energy, number of degree-

days, total sum of the solar radiations and the period

of service of the building. Introducing the factor of the

cost/m’ of construction and heating of the partition Win the form

Y = [(I -~,k,+wo,l+ NC,24SD

(1-P,)~V

~ -U,cosfl,-&sin/?, ,

it can be stated that the size of the,jth partition should

increase with the diminishing factor IV,.

Depending on the data input, the plan of the optimum

shape of the building can be an octagon or a polygon

with a fewer number of sides. In the case of short heating

periods, construction costs are of great importance and

the shape of the building approaches a regular octagon.

Depending on the unit prices of walls and windows,

and on total solar radiation, the ratios of window areas

to the areas of the relevant walls assume the highest or

the lowest allowable values,

As a result of the optimisation of the shape of the

building, construction and heating costs for the N year

period can be lowered by several to several dozen per

cent. In an area close to the minimum, the function of

the increase of costs is flat and the differences are of the

order of several per cent. The gradient of the cost function

becomes steeper as the distance from the minimum

increases.

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00rad-multicriteria optimisation

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