Towards a multi-objective optimization approach for improving energy efficiency in buildings.pdf

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Towards a multi-objective optimization approach for improving energy

efficiency in buildings

Christina Diakaki a,*, Evangelos Grigoroudis b, Dionyssia Kolokotsa a

a Technological Educational Institute of Crete, Department of Natural Resources and Environment, 3 Romanou str., 73133 Chania, Crete, Greeceb Technical University of Crete, Department of Production Engineering and Management, University Campus, Kounoupidiana, 73100 Chania, Greece

1. Introduction

The energy sector faces evidently significant challenges that

everyday become even more acute.The current energytrends raise

great concerns about the three Es that are the environment, the

energy security and the economic prosperity as defined by the

International Energy Agency (IEA) [1]. Among the greater energy

consumers is the building sector that uses large amounts of energy

and releases considerable amounts of CO2. In the European Union

(EU), for example, the building sector uses the 40% of the total final

energy consumed therein and releases about 40% of the total CO2emissions. The mean energy dependency of the EU has increased

up to 56% in 2006 [2] with an increase rate of 4.5% between 2004

and 2005. As a consequence, the cornerstone of the European

energy policy has an explicit orientation to the preservation and

rational use of energy in buildings as the Energy Performance of

Buildings Directive (EPBD) 2002/91/EC indicates [3]. This is not

however a concern of only the EU, since other organizations

worldwide put significant efforts towards the same direction.

The International Organization for Standardization (ISO)

provides another sound example through the related standards

that has published based on the work of its Technical Committee

(TC) 163 for the thermal performance and energy use in the built

environment (e.g. [4,5], etc.). Moreover, the Centre Europeen de

Normalisation (CEN) introduced, recently, several new CEN

standards in relation to the Energy Performance of Buildings

Directive (EPBD) (e.g. [6,7], etc.).

As innovative technologies and energy efficiency measures are

nowadays well known and widely spread, the main issue is to

identify those that will be proven to be the more effective and

reliable in the long term. With such a variety of proposed

measures, the decision maker has to compensate environmental,

energy, financial and social factors in order to reach the best

possible solution that will ensure the maximization of the energy

efficiency of a building satisfying at the same time the final user/

occupant/owner needs.

The state-of-the-art approach to this problem is performed via

two approaches. According to the first approach, an energy analysis

of the building under study is carried out, and several alternative

scenarios, predefined by the energy expert, are developed and

evaluated [8]. These specific scenarios, which may vary according to

buildings characteristics, type, use, climatic conditions, etc., are

pinpointed by the building expert and are then evaluated mainly

through simulation (see e.g. [9]). The selection of the alternative

scenarios, energy efficiency measures and actions thatwill be finally

employed is largely based on the energy experts experience.

The second approach includes decision supporting techniques,

such as multicriteria-based decision making methods that are

Energy and Buildings 40 (2008) 17471754

A R T I C L E I N F O

Article history:

Received 14 January 2008

Received in revised form 5 March 2008

Accepted 8 March 2008

Keywords:

Building

Energy efficiency

Energy improvement

Multi-objective optimization

A B S T R A C T

The energy sector worldwide faces evidently significant challenges that everyday become even more

acute. Innovative technologies and energy efficiency measures are nowadays well known and widely

spread, and the main issue is to identify those that will be proven to be the more effective and reliable in

the long term. With such a variety of proposed measures, the decision maker has to compensate

environmental, energy, financial and social factors in order to reach the best possible solution that will

ensure the maximization of the energy efficiency of a building satisfying at the same time the buildings

final user/occupant/owner needs. This paper investigates the feasibility of the application of multi-

objective optimization techniques to the problem of the improvement of the energy efficiency in

buildings, so that the maximum possible number of alternative solutions and energy efficiency measures

may be considered. It further shows that no optimal solution exists for this problem due to the

competitiveness of the involved decision criteria. A simple example is used to identify the potential

strengths and weaknesses of the proposed approach, and highlight potential problems that may arise.

2008 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +30 28210 23045; fax: +30 28210 23003.

E-mail address: [email protected] (C. Diakaki).

C o n t e n t s l i s t s a v a i l a b l e a t S c i e n c e D i r e c t

Energy and Buildings

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n b u i l d

0378-7788/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2008.03.002
mailto:[email protected]://www.sciencedirect.com/science/journal/03787788http://dx.doi.org/10.1016/j.enbuild.2008.03.002http://dx.doi.org/10.1016/j.enbuild.2008.03.002http://www.sciencedirect.com/science/journal/03787788mailto:[email protected]


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employed toassist thereaching of a final decision (e.g. [10]),amonga

set of pre-defined by the building expert alternative actions.

In both of the aforementioned approaches, therefore, the whole

process as well as the final decisions are significantly affected by

the experience and the knowledge of the corresponding building

expert that from now on will be mentioned herein as decision

maker (DM). Although this experience and knowledge are

certainly significant and irreplaceable elements to the whole

process, it is however, necessary to develop practical tools that will

assist him/her in taking into account as much feasible alternatives

and decision criteria as possible, without the restrictions imposed

by the predefined scenarios. Such tools may be developed based

upon the concepts of multi-objective optimization techniques.

Multi-objective optimization is a scientific area that offers a

wide variety of methods with great potential for the solution of

complicated decision problems. The particular scientific area has

also a wide variety of applications in energy and environmental

problems as well as in issues that are related to the sustainable

development in general ([1113]).

It is the aim of this paper to investigate the feasibility of the

application of multi-objective optimization techniques to the

problem of the improvement of the energy efficiency in buildings,

so that the alternative solutions and energy efficiency measureswill not be predefined and limited to a discrete state space. It will

further show that no optimal solution exists for the examined

problem due to the competitiveness of the involved decision

criteria. To this end, a simple example is used to identify the

potential strengths and weaknesses of the approach, and to

highlight potential problems that may arise.

More specifically, the paper is structured in four more sections.

Section 2 provides a short review to the background of the subject

matterand to other systematic efforts towards theimprovementof

energy efficiency in buildings. Section 3 presents an introduction

to the proposed approach. Section 4 provides through an example

case study, a multi-objective modeling and solution process, and a

discussion of the main findings. Finally, Section 4 summarizes the

conclusions and discusses issues for future consideration, researchand development.

2. Overview and background

The various measures that may be considered for the impro-

vement of the energy efficiency in buildings may be distinguished

in the following basic categories:

Measures for the improvement of the buildings envelope

(addition or improvement of insulation, change of color,

placement of heat-insulating door and window frames, increase

of thermal mass, building shaping, superinsulated building

envelopes, etc.).

Measures for reducing the heating and cooling loads (exploita-tion of the principles of bioclimatic architecture, incorporation of

passive heating and cooling techniques, i.e. cool coatings [14],

control of solar gains, electrochromic glazing, etc.).

Use of renewables (solar thermal systems, buildings integrated

photovolatics, hybrid systems, etc.).

Use of intelligent energy management, i.e. advanced sensors,

energy control (zone heating and cooling) and monitoring

systems [15].

Measures for the improvement of the indoor comfort conditions

in parallel with minimization of the energy requirements

(increase of the ventilation rate, use of mechanical ventilation

with heat recovery, improvement of boilers and air-conditioning,

efficiency use of multi-functional equipment, i.e. integrated

water heating with space cooling, etc.) [16].

Use of energy efficient appliances and compact fluorescent

lighting.

With such a variety of proposed measures, the DM has to

compensate environmental, energy, financial and social factors in

order to reach the best possible and feasible solution.

As mentioned in the introduction, the most widely used

approach to this problem is the energy analysis of the building

under study via simulation, while the final decision is sometimes

assisted through multicriteria decision making techniques that are

performed upon a set of predefined alternative solutions. Gero

et al. [10] were among the first to propose a multicriteria model in

order to explore the trade-offs between the building thermal

performance and other criteria such as capital cost, and usable are

of the building to be used at the process of building design. More

recently, other researchers have also employed multicriteria

techniques to similar problems. Jaggs and Palmar [17], Flourentzou

and Roulet [18], and Rey [19] proposed multicriteria-based

approaches for the evaluation of retrofitting scenarios. Blondeau

et al. [20] used multicriteria analysis to determine the most

suitable, among a set of possible actions, ventilation strategy on a

university building. The aim of their approach was to ensure the

best possible indoor air quality and thermal comfort of theoccupants and the lower energy consumption. In the same

direction moved also the efforts of Wright et al. [21] that aimed

to optimize the thermal design and control of buildings employing

multicriteria genetic algorithms. Chen et al. [22] proposed a

multicriteria decision making model for a lifespan energy

efficiency assessment of intelligent buildings. Alanne [23] pro-

posed a multicriteria knapsack model to help designers to select

the most feasible renovation actions in the conceptual phase of a

renovation project. According to this approach, a set of renovation

actions is developed and for each of them, a utility score is defined

according to specific criteria. The obtained utility scores of all

actions are then used as weights in a knapsack optimization model

to identify which actions should be undertaken. Al-Homoud [24]

provides a review of such systematic approaches, which, however,are mainly based on the principles of multicriteria decision making.

Althoughthese approaches allow, to some extent, the consideration

of many alternatives, they are still based upon a set of actions or

scenarios that should be predefined and pre-evaluated.

The problem when employing multicriteria techniques is that

they are applied upon a set of predefined and pre-evaluated

alternative solutions. In case that a limited number of such

solutionshave been defined, there is no guaranteethat thesolution

finally reached is the optimal. Also, the selection of a representa-

tive set of alternatives is usually a difficult problem [25], while the

final solution is heavily affected by these predefined alternatives.

On theopposite case, i.e. when numerous solutionsare defined, the

required evaluation and selection process may become extremely

difficult to handle. In any case, however, the multicriteriaapproach, limits the study to a potentially large but certainly

finite number of alternative scenarios and actions, when the real

opportunities are enormous considering all the available improve-

ment measures that may be employed.

The problem of the DM, is in fact a multi-objective optimization

problem [26], that is a problem characterized by the existence of

multiple and in several cases competitive objectives (e.g. the

employment of energy efficiency improvement solutions is usually

accommodated by a cost increase) each of which should be

optimized against a set of feasible and available solutions that are

not predefined but are prescribed by a set of parameters and

constraints (e.g. available materials, maximum acceptable cost,

etc.) that should be taken into account in order to reach the best

possible solution.

C. Diakaki et al./ Energy and Buildings 40 (2008) 174717541748


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In multi-objective optimization problems, the searching of the

optimal solution is worthless, since the objectives are competitive.

Instead, a feasible intermediary solution that will satisfy his/her

preferences is peered, through an interactive procedure with

the DM.

The following sections investigate the feasibility of applying

this approach to the problem of energy efficiency improvement in

buildings.

3. The proposed multi-objective approach

3.1. Basic principles

The development of a multi-objective optimization methodol-

ogy to the problem of energy efficiency improvement in buildings

requires the definition of appropriate decision variables, criteria

and constraints, and the selection of an appropriate solution

technique.

The decision variables, discrete and/or continuous, should

reflect the total set of alternative measures that are available for

the improvement of energy efficiency (e.g. insulation, production

of electric energy, etc., see Section 2).

The objectives to be achieved (e.g. improvement of indoorcomfort, low energy consumption, etc.) should be identified and

formulated into appropriate linear and/or non-linear mathema-

tical expressions.

The set of the feasible solutions should be delimitated through

the identification of linear and/or non-linear constraints concern-

ingeither thedecisionvariables andtheirintermediary relationsor

the objectives of the problem. Natural and logical constraints may

also be considered as necessary.

Finally, an appropriate solution method should be identified

that will be able to handle the continuous as well as discrete

decision variables and linear and non-linear objective functions

and constraints.

The example case study developed in the following sections

aims to provide an insight in this process and highlight anypotential benefits and problems.

3.2. Problem definition

To investigate the feasibility of the proposed approach, a simple

case is considered. The problem under study concerns the

construction of the simple building displayed in Fig. 1. Moreover,

Fig. 2 displays the structure of the walls that is assumed for this

building. The structure consists of the following sequence of layers

from outside to inside: concrete, insulation and gypsum board.

The decisions regarding the building under study concern

appropriate choices for the window type, the walls insulation

material, and the thickness of the walls insulation layer. The aims

are to reduce the acquisition costs, which correspond to the initial

investment, and to increase the resulting energy savings.

The pursued goals are competitive, since materials with low

thermal conductivity,thus leading to lower building loadcoefficient

and consequently to higher energy savings, are usually more

expensive.

3.3. Decision model

To allow for the application of multi-objective optimization

techniques, an appropriate model should be developed in

accordance to the principles mentioned in Section 3.1. The

following sub-sections describe the development of the multi-

objective optimization model, and a few solution approaches.

3.3.1. Decision variables

As mentioned earlier, the decisions considered in this example

case study concern three choices. For these three choices, three

types of decision variables are defined respectively:

decision variables to reflect the alternative choices regarding the

window type, decision variables to reflect the alternative choices regarding the

walls insulation material, and

decision variables to reflect the alternative choices regarding the

thickness of the walls insulation material.

Assuming that Idifferent types of windows may be considered,

binary variables x1i with i = 1, 2 ,. . ., I may be defined as follows:

x1i 1 if windowof type i is selected0 else

(1)

Obviously, Eq. (2) holds for these decision variables, since only

one window type may be selected

XIi1

x1i 1: (2)

Assuming that J different insulation materials may be considered

for the walls insulation, binary variables x2j with j = 1, 2 ,. . ., Jmay

also be defined as follows:

x2j 1 if insulation materialj is selected0 else

(3)

Similarly with the previous ones, Eq. (4) holds for these latter

decision variables, since only one insulation material may be

selected

XJ

j1

x2j 1: (4)

Fig. 1. The building under study.

Fig. 2. Construction layers of the buildings walls.

C. Diakaki et al./ Energy and Buildings 40 (2008) 17471754 1749


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The existence of constraints (2) and (4) gives the optimization

problem a combinatorial nature that resembles to the well known

Knapsack problem [27], an NP-hard combinatorial optimization

problem.

As far as the insulation materials are concerned, they are

usually available in the market at layers of standard thickness.

Assuming that for all materials, there is only one standard

thickness ds (e.g. ds = 1 cm) available in the market, an integer

variable x3, with x3 = 1, 2, 3, . . ., may be defined to denote the

number of insulation layers to be used. Obviously, if d2max is the

thickness of the available space for insulation (see Fig. 2), the

following constraint applies:

x3ds d2max: (5)

3.3.2. Decision criteria

In the considered case study the aim is twofold. On the one

hand, the acquisition costs are to be reduced, while on the other

hand, the energy savings are to be increased.

The costs involved in the considered study concern only the

acquisition of materials. Therefore,the total cost Cmaybe obtained

simply, by adding the cost of the windows CWIN and the cost of the

walls insulation CWAL.Assuming thatAWIN is the windowsurface (in m

2), C1i is the cost

(ins/m2) forwindow type i, with i = 1,2, . . ., I,AWALis the surface of

the walls to be insulated (in m2), C2j is the cost (in s/m3) for

insulation material j, with j = 1, 2, . . ., J, the total cost is obtained

through the following equation:

C CWIN CWAL AWINXIi1

C1ix1i AWALx3dsXJj1

C2jx2j: (6)

Concerning the second aim of the study, that is the increase of

energy savings, several options are available (see discussion in

Section 2). For the purpose of this study, the second aim is

approached through the choice of materials with low thermal

conductivity. Therefore the corresponding decision criterion shouldbe developed so as to allow for such a choice. An appropriate

decision criterion in this respect is the building load coefficient.

Generally, the building load coefficient BLC is calculated

according to the following formula [8]:

BLC X

e

AeUe (7)

where e is the considered building envelope component

(with one unique U-value), Ae is the surface are of e (in m2),

and Ue is the thermal transmittance of construction part e (in W/

m2 K).

Applying Eq. (7) in the building under study, the following

equation results:

BLC AWINUWIN AWALUWAL (8)where UWINand UWALarethe thermal transmittance of thewindow

and the wall, respectively. In Eq. (8), the thermal transmittance of

the door has been omitted forsimplicity,since it has been assumed

that there is no choice regarding this construction part of the

building.

The thermal transmittance of the window is simply calculated

through the following formula:

UWIN XIi1

U1ix1i (9)

while, in the case of the walls, the necessary calculations are more

complex. Assuming that the wall is constructed from several

homogeneous layer parts, its thermal transmittance is calculated

through the following formula [8]:

UWAL 1

RT

1PnRn

1P

ndn=kn(10)

where RT is the overall thermal resistance of the construction (in

m2K/W), Rn is thethermal resistance (R-value) of the homogeneous

layer part n of the construction of the wall (in m2K/W), dn is the

thickness of layer part n (in m), kn is the thermal conductivity oflayer part n (in W/mK), and n with n = 1, 2, . . ., is the index of the

construction layer.

In the considered case study, the wall is assumed to be

constructed from three layers (see Fig. 2). From the three layers,

theconstruction of thetwo is assumed known andgiven,therefore,

the choice concerns only the third layer that is the insulation layer

(i.e. the intermediate layer in Fig. 2). Introducing this knowledge in

Eq. (10) the following formula results

UWAL 1P

n1dn1=kn1 d2=k2

(11)

where n1, with n1 = 1, 2, is an index to the two known construction

layers of the wall for which the thicknesses as well as the thermal

conductivities are known, while the parameters d2

and k2

correspond to the undefined insulation layer. More specifically,

d2 is the total thickness of the insulation layer and depends upon

the numberx3 of the insulation layers of standard thickness d3 that

will be used, i.e. d2 = x3ds, while k2 is the thermal conductivity of

the insulation layer and depends upon the choice of the insulation

material (i.e. x2j). Introducing these in Eq. (11), the following

formula results that allows for the calculation of the thermal

transmittance of the wall

UWAL 1P

n1dk1 =kn1 x3ds=

PJj1

k2jx2j(12)

Introducing Eqs. (9) and (12) in (8), the following formula results

that describes the load coefficient of the building under study

BLC AWINXIi1

U1ix1i AWALP

n1dn1=kn1 x3ds=

PJj1

k2jx2j(13)

3.3.3. The decision model and the solution approaches

The decision variables and criteria developed in the previous

sub-sections, lead to the formulation of the following multi-

objective decision problem:

ming1x C AWINXIi1

C1ix1i AWALx3dsXJj1

C2jx2j

ming2x BLC

AWIN

XI

i1

U1ix1i AWAL

Pn1 dn1 =kn1 x3ds=PJj1k2jx2js:t:

XIi1

x1i 1

XJj1

x2j 1

x1i 2 f0; 1g 8 i 2 f1; 2; . . . ; Ig

x2j 2 f0; 1g 8 j 2 f1; 2; . . . ;Jg

x3 2N f0g

x3 d2max=ds (14)

where the data described below have been assumed.

For the window, four types with the characteristics displayed in

Table 1 have been assumed available in the market. For the

insulation, it has been assumed that the layers available in the



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market have a standard thickness of ds = 0.05 m. It has also been

assumed that the available insulation materials include four types,

the characteristics of which are displayed in Table 2. The values of

the thermal transmittance and the thermal conductivity in Tables 1

and 2 have been chosen from the ASHRAE database [28], while the

values for the costs have been set so as to reflect the cost increase

that accompanies the decrease of thermal conductivity.

Moreover, considering the building dimensions, as displayed in

Figs. 1 and 2,AWIN is consideredto be6 m2,AWALis considered to be

388 m2, the maximum permissible thickness d2max for the

insulation layer of the wall is considered 0.10 m, while for the

layers 1 and 3 of the wall construction, thicknesses have been

assumed d1 = 0.10 m and d3 = 0.01 m, respectively, and thermal

conductivities k1 = 0.75 W/mK and k3 = 0.48 W/mK, respectively.The developed decisions problem (14) is a mixed-integer multi-

objective combinatorial optimization problem ([29] and [30]).

Moreover, as already mentioned, the two criteria that have been

considered are competitive, since any decrease of the one, leads to

an increase of the other. Table 3, that displays the payoff matrix

when each criterion is optimized independently from the other,

demonstrates this competitiveness. When the cost criterion is

optimized, low-cost decision choices are made that may, however,

lead to decreased energy savings since thebuilding load coefficient

takes high values and vice versa.

The problem has been programmed in the LINGO software [31]

and three well known multi-objective optimization techniques

have been used to solve it ([29,30]): the compromise program-

ming, the global criterion method, and the goal programming.To apply compromise programming, the decision model is

modified so as to include one only criterion. The aim in this

technique is to minimize the distance of the criterion values from

their optimum values. Considering this, the decision problem is

formulated as follows:

minz ls:t:all constraints of multiobjective problem14l! g1x g1minp1=g1minl! g2x g2minp2=g2minl! 0

(15)

where, l corresponds to the Tchebyshev distance, g1min and g2minare the optimum (minimum) values of the two criteria when

optimized independently (see Table 3), and p1 and p2 are

corresponding weight coefficients reflecting the relative impor-

tance of the two criteria. The weight coefficients allow the DM to

express his/her preferences regarding the criteria, and must satisfy

the following constraint

p1 p2 1: (16)

The solution of problem (15) for different values of the

weight coefficients, leads to the results summarized in Table 4.

Obviously, as the weight coefficient of the cost criterion

increases, the solution of problem (15) approaches and finally

reaches (when p1 = 1 and p2 = 0) the optimum solution when

only this criterion is optimized (see Table 3). At the same time,

when the weight coefficient of the building load coefficient

criterion increases, the solution approaches and finally reaches

(when p2 = 1 and p1 = 0) the optimum solution when this

criterion is optimized in isolation (see Table 3). For intermediary

values of the weight coefficients, several solutions may be

obtained that favor the criteria at higher or lower levels

depending upon the specific values, which have been chosen.

This behavior is also demonstrated through Fig. 3 that displayshow the criteria values change depending upon the specific

values of the weights. In addition, Table 4 provides information

on how the decision choices change with the modification of the

DMs preferences that are expressed through the weight

coefficients, e.g. as long as p1 ! 0.5, which means that the DM

pays more attention in the cost criterion than the building load

coefficient, the model suggests as insulation material the

cellular glass (x21 = 1, x22 = x23 = x24 = 0) that is the cheapest

solution (see Table 2), while for values of p1 less than 0.5, more

expensive but at the same time more energy efficient solutions

are suggested (x21 = 0).

To apply the global criterion method, the two criteria of the

initial problem (14) are integrated into one single criterion under

the rationale that the best choice may be obtained through thedecrease of a single criterion that will lead to decision choices

Table 1

Characteristics and data for different window types

Window types Thermal transmittance

(W/m2K)

Cost (s/m2)

Single pane windows U11 = 6.0 C11 = 50

Double pane windows of 6 mm space U12 = 3.4 C12 = 100

Double pane windows of 13 mm space U13 = 2.8 C13 = 150

Double low-e windows U14 = 1.8 C14 = 200

Table 2

Characteristics and data for different insulation materials

Insulation types Density (kg/m3) Thermal conductivity

(W/mK)

Cost (s/m3)

Cellular glass 136.000 U21 = 0.050 C21 = 50

Expanded polystyrene

molded beads

16.000 U22 = 0.029 C22 = 100

Cellular polyourethan 24.000 U23 = 0.023 C23 = 150

Polysocynaurale 0.020 U24 = 0.020 C24 = 200

Table 3

Payoff matrix

Type of solution C (s) BLC (W/K) x11 x12 x13 x14 x21 x22 x23 x24 x3

[min]g1(x) 1270 372.22 1 0 0 0 1 0 0 0 1

[min]g2(x) 8960 86.08 0 0 0 1 0 0 0 1 2

Table 4

Indicative problem solutions when applying compromise programming

z p1 p2 C (s) BLC (W/K) x11 x12 x13 x14 x21 x22 x23 x24 x3

0.00 1.0 0.0 1270 372.22 1 0 0 0 1 0 0 0 1

0.31 0.9 0.1 1570 356.62 0 1 0 0 1 0 0 0 1

0.61 0.8 0.2 2170 347.02 0 0 0 1 1 0 0 0 1

0.54 0.7 0.3 2240 216.13 1 0 0 0 1 0 0 0 2

0.60 0.6 0.4 2540 200.53 0 1 0 0 1 0 0 0 2

0.64 0.5 0.5 2840 196.93 0 0 1 0 1 0 0 0 20.71 0.4 0.6 3510 187.07 0 1 0 0 0 0 1 0 1

0.69 0.3 0.7 4180 143.71 1 0 0 0 0 1 0 0 2

0.51 0.2 0.8 4480 128.11 0 1 0 0 0 1 0 0 2

0.34 0.1 0.9 5080 118.51 0 0 0 1 0 1 0 0 2

0.00 0.0 1.0 8960 86.08 0 0 0 1 0 0 0 1 2

Fig. 3. Criteria value changes when applying compromise programming.



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as close as possible to those that would have been obtained, if

the two criteria had been optimized independently from one

another.

This rationale, leads to theformulationof thefollowing decision

problem:

minug1x;g2x p1g1x g1min

g1min

p2

g2x g2ming2min

s:t:all constraints of multiobjective problem14

(17)

where again, p1 and p2 correspond to the weight coefficients of the

two criteria of the initial problem (14), for which Eq. (16) holds, as

well as all the comments that were mentioned earlier in the case of

the compromise programming.

The application of the global criterion method to the decision

problem (17), leads to similar results as those of the compromise

programming. Table 5 displays the corresponding results that

demonstrate the change of the criteria values when different

values of the weight coefficients are used.

To apply finally the goal programming method, it is assumed

that specific goals are to be achieved for each criterion. These

goals, from which the variation is minimized through goalprogramming, are translated to two upper limits, g1max and

g2max, for the first and second criterion, respectively. In the

examined case study, these limits have been assumed to be equal

to theoptimum values of the twocriteria, obtained when they are

optimized independently from one another (see Table 3). More-

over, according to the principles of goalprogramming, weights are

defined representing the trade-off between the considered

criteria and not relative importance, as the weights did in the

previously considered methods. To define the weights, one

criterion is considered as the reference criterion taking weight

equal to 1, while for the other considered criteria weights are

definedso as toreflecttheir trade-off withthe reference one.In the

examinedcase, weights p01 and p02 areprovidedforthecostandthe

building load coefficient criteria, respectively. Considering cost asthe reference criterion, it takes weight p01 equal to 1, while the

weight p02 of theother criterion represents theeurothat theDM is

willing to pay, in order to reduce the building load coefficient by

1 W/K. In simple words, if the DM is willing to pay 5 s to reduce

the building load coefficient by 1 W/K, p02 should be chosen equal

to 5s

per W/K.

Given the above, the following decision problem is formulated

for the application of the goal programming method:

miny p01d1 p

02d

2

s:t:all constraints of multiobjective problem14g1x d

1 d

1 g1max

g2x d2 d

2 g2max

d

1 ; d

1 ; d

2 ; d

2 ! 0

(18)

where d1 and d1 represent the surplus and the deficit, respectively,

as faras thegoal for thefirst criterionis concerned, while d2 and d2

represent the surplus and the deficit, respectively, as far as the goal

for the second criterion is concerned.

Table 6 summarizes the results from the application of goal

programming for different values of the criteria weights. The

results make obvious again that the more a criterion is considered

important by the DM, the more the final decision is in favor of this

criterion.

3.4. Analysis of results and discussion

The results of all three multi-objective optimization techniques

employed forthe solution of theproblem under study demonstratethe feasibility as well as the strengths of applying such techniques

to the problem of energy efficiency improvement. The application

of this systematic approach allowed for the simultaneous

consideration, without having to prescribe any particular set of

choices, all possible combinations of alternative actions. The

approach, allows also the consideration of any logical, physical,

technical or other constraints that may apply and permits the DM

to guide the solutions according to his/her own preferences.

However, the case study examined in the previous sections is a

rather nave one. In reality, the corresponding decision models are

expected to be far more complicated and farmore difficult to solve.

To make this statement more apparent, consider a few simple

extensions of thepreviously studied case. Assumethat thedecision

choices include also the type of the door, and the size of both thedoor and the window frames.

In such a case, the decision problem expands as follows:

where

l is an index for the door and window frames, the sizes of which

maybe altered, n1 is an index to theknownconstruction layers of

ming1x C XLl1

x4lx5lXIlil 1

C1ilx1il

0@

1A APER XL

l1

x4lx5l

!XN2n21

x3n2 dsn2XJn

jn21

C2jn2x2jn2

0@

1A

ming2x BLC XLl1

x4lx5lXIlil1

U1ilx1il

0@

1A APER PLl1x4lx5lPN1

n1dk1=kn1

PN2n21

x3n2 dsn2 =PJn2

jn21k2jn2

x2jn2

s:t:XIlil 1

x1il 1 8 il 2 f1; 2; . . . ; Ilg

XJn2jn2

1

x2jn2 1 8 jn2 2 f1; 2; . . . ; In2 g

x1il 2 0; 1f g 8 il 2 f1; 2; . . . ; Ilgx2jn2

2 0; 1f g 8 jn2 2 f1; 2; . . . ;Jn2 g

x3n2 2N 0f g 8 n2 2 f1; 2; . . . ; N2gx3n2 dn2 max=dsn2 8 n2 2 f1; 2; . . . ; N2ghlmin x4l hlmax 8 l 2 f1; 2; . . . ; Lgwlmin x5l wlmax 8 l 2 f1; 2; . . . ; Lg

(19)



7/8

the walls, n2 is an index to the construction layers of the walls for

which the thicknesses as well as the materials to be used are to

be found, il is an index to the available types for frame l, and jn2 is

an index to the available materials for the walls construction

layer n2.

x1il are decision variables corresponding to the available types for

frame l, x2jn2are decision variables corresponding to available

materials for the walls construction layer n2, x3n2 are decision

variables used to identify the thicknesses of the unknown walls

construction layers n2, and x4l and x5l are decision variables that

correspond to the choices for the height and width of frame l. C1il is the cost of frame type il, C2jn2

is the cost of material jn2 , dsn2is a standard market thickness formaterial jn2 , hlmin and hlmax are

the minimum and maximum permissible heights for frame l,

wlmin and wlmax are the minimum and maximum permissible

widths for frame l, and APER is the total surface of the walls

including the door and window surfaces.

It is obvious, that even simple extensions of the considered

issues may increase greatly both the size and the complexity of the

decision problem.

Consider now all the criteria that the DM may wish to optimize

(indoor comfort, environmental andsocial criteria, etc.), andall the

possibilities that the DM has available in order to improve the

energy efficiency of a building (other choices may involve theelectrical systems, theheating andcooling options, etc., see Section

2). A final issue that should not be ignored in the frame of energy

efficiency improvement is the time dimension. In the examined

case study, the problem was of a static nature. However, the

problem under study has a dynamic nature since any present

choices have consequences that extend in time, and it is possible

that alternatives, which seem promising today, are not such good

through a long-term perspective. This means that in order to get

results valid in the long term, all the choices should be examined

for a time period that covers their lifetime (e.g. the cost criterion

should include, beside the initial investment, any future opera-

tional, maintenance and replacement costs that may emerge

during the use of the building as well as any resulting savings).

Without any doubt, the resulting decision problem, although finite,

may increase dramatically, thus making the solving procedure

extremely difficult. In such case, techniques other and more

sophisticated than those presented herein may become necessary,

like ([29,30]) aggregated approaches (e-constraint method, Tche-

byshev scalarisation, etc.), interactive techniques (interactive

surrogate worth trade-off method, GDF method, STEM, Light

Beam Search, etc.) or other methods (GUESS, NIMBUS, reference

point approach, etc.).

All these concerns, however, raise directions for a future study.

The particular investigations presented herein, despite the high-

lighted problem of complexity, demonstrate the feasibility and the

potential of a multi-objective optimization approach to the

problem of energy efficiency improvement in buildings.

4. Conclusions and future work

The improvement of energy efficiency of buildings is among the

first priorities of the energy policy in the EU and worldwide as

indicatedby the published directives and the promotion of ISO and

other related standards.

For the improvement of the energy efficiency of the buildings

and the quality of their indoor environment, several measures are

available, and the DM has to compensate environmental, energy,financial and social factors in order to make a selection among

them.

The problem of the DM is characterized by the existence of

multiple and in several cases competitive objectives each of which

should be optimized against a set of feasible and available

solutions that is prescribed by a set of parameters and constraints

that should be taken into account. In simple words, the DM is

facing a multi-objective optimization problem that is usually,

however, approached through simulation and/or multicriteria

decision making techniques that focus on particular aspects of the

problem rather than a global confrontation.

The aim of this paper was to investigate the feasibility of

developing a stand-alone multi-objective optimization model for

the decision problem that will allow for the consideration of asmany available options as possible without the need to be

combined and/or complemented by any other method such as

simulation, multicriteria decision analysis techniques, etc.

A simple case study was investigated that demonstrated the

feasibility of the approach. However, it was also found out, that

when the energy efficiency improvement problem is faced in its

real-world dimensions, it possesses inherent difficulties that

complicate both the modeling and the solution approach.

It remains now to future, more detailed investigations, to prove

or debunk the ability of the multi-objective optimization approach

to handle the problem of improving energy efficiency in buildings

in its real dimensions.

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Table 5

Indicative problem solutions when applying the global criterion method

u p1 p2 C (s) BLC (W/K) x11 x12 x13 x14 x21 x22 x23 x24 x3

0.00 1.0 0.0 1270 372.22 1 0 0 0 1 0 0 0 1

0.33 0.9 0.1 1270 372.22 1 0 0 0 1 0 0 0 1

0.82 0.8 0.2 1570 356.62 0 1 0 0 1 0 0 0 1

0.99 0.7 0.3 2240 216.13 1 0 0 0 1 0 0 0 2

1.06 0.6 0.4 2240 216.13 1 0 0 0 1 0 0 0 2

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Table 6

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3558.97 1 20 2540 200.53 0 1 0 0 1 0 0 0 2

4470.85 1 30 4480 128.11 0 1 0 0 0 1 0 0 2

5917.84 1 65 5080 118.51 0 0 0 1 0 1 0 0 26840.59 1 100 7020 96.99 0 0 0 1 0 0 1 0 2

7690.00 1 180 8960 86.08 0 0 0 1 0 0 0 1 2



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