Journal of Cleaner Production

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Evolutionary many-objective optimization for retro ﬁ t planning in public buildings: A comparative study

Hyojoo Son, Changwan Kim

^*

Department of Architectural Engineering, Chung-Aug University, Seoul, 06974, South Korea

a r t i c l e i n f o

Article history:

Available online 12 April 2018

Keywords:

Energy-efﬁcient retroﬁt CO2 emissions Energy consumption

Evolutionary many-objective optimization Public buildings

a b s t r a c t

There has been an increasing movement toward retrofitting existing (in-use) buildings to achieve a significant reduction in energy consumption and greenhouse gas emissions in the building sector. When planning retrofits for public buildings, decision-makers are required to make rational decisions that will achieve four critical objectives: minimize energy consumption, reduce CO2emissions, mitigate retrofit costs, and maximize thermal comfort. This study aims to solve this four-objective optimization problem (so-called the problem of many-objective optimization) for retrofit planning in public buildings via an evolutionary many-objective optimization (EO) algorithm that handles these objectives at the same time.

This study involves the application of EO algorithms (NSGA-II, MOPSO, MOEA/D, and NSGA-III) and the evaluation of their performance. A description of these algorithms is presented, and each algorithm is implemented in a public-building retrofit project. The algorithms’performances were analyzed, and the results were compared based on two aspects: diversity and convergence. The results indicated that NSGA-III can be used to derive a comprehensive set of trade-off alternatives from possible retrofit scenarios, thereby serving as a useful reference for retrofit planners. These decision-makers can then utilize the provided references to select optimal retrofit strategies and satisfy stakeholders.

1. Introduction

The building sector accounts for two-ﬁfths of the world's total annual energy consumption, and that proportion is increasing every year (EIA, 2017). The International Energy Agency pointed out that if there is no effort to improve energy efﬁciency, the energy demand will reach 50% by 2050 in the building sector (IEA, 2013a).

In addition, 40% of all greenhouse gases are released by this sectorda main contributor to global warming (IEA, 2013b).

Accordingly, several countries have implemented various policies to decrease the energy that buildings consume for the purpose of reducing greenhouse gas emissions. For example, the United States implemented the Energy Independence and Security Act (2007) to make all commercial buildings zero energy by 2050 (U.S. Congress, 2007). Europe implemented a policy that all new buildings must be nearly zero energy by 2020 (public buildings must meet this regulation after 2018) to reduce the energy consumption of buildings (Groezinger et al., 2014). Through these efforts, the European

Union has aimed to reduce greenhouse gas emissions in the building sector by 88e91% by 2050 compared to 1990 (COM, 2011).

South Korea created a policy that all new buildings must be nearly zero energy by 2025 (public buildings after 2020), aiming at reducing greenhouse gas emissions by 18.1% by 2030 in the building sector (MOLITT, 2016). To effectively accomplish the goals of these energy-saving building-sector policies, it is essential that each building is highly energy efﬁcient (BPIE, 2011; World Green Building Council, 2017).

The energy-efficiency requirements have been strengthened in recent years, so older buildings built that were constructed under less-strict regulatory requirements now have poor energy efficiency compared to those that are newly constructed (IEA, 2008; Li and Shui, 2015). Generally, buildings constructed within the past 10 years are defined as newly constructed (European Commission, 2010); in 2012, 18% of U.S. buildings were newly constructed (EIA, 2015), and there were 5e10 times as many older buildings. In the building sector in the European Union, 75% of buildings were re- ported as energy inefficient (World Green Building Council, 2017).

Overall, according to the report by theGABC (2016), the existing buildings (which were not recently constructed) would account for more than two-thirds of the entire building stock. For these existing

*Corresponding author.

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

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buildings, there is a way to reduce the energy consumption and minimize the greenhouse gas emissions by improving the building envelope and upgrading the mechanical systems while maintaining the maximum level of thermal comfort; which is referred to as retrofitting (Xu et al., 2015). It has been considered the most cost- effective and feasible option for improving energy efficiency and reducing greenhouse gas emissions rather than demolishing and rebuilding those buildings (UNEP, 2009; Pombo et al., 2016). Acting from this perspective, government organizations around the world have supported such retrofitting (Ma et al., 2011). In the United States, when retrofits reduce energy consumption, the costs are exempt from relevant taxes (IRS, 2012). In Europe, the European Union has partially supported such retrofit expenses (Bertoldi et al., 2010). In China, government organizations have funded and made financial supports, such as subsidies and interest exoneration, to retrofitting projects of existing government and large public buildings for energy efficiency (Li and Shui, 2015; Yuan et al., 2016).

Despite these supportive policies, decision-makers who under- take retrofitting have difficulty planning such projects (Rahmat and Ali, 2010) because hundreds and thousands of retrofit alternatives exist, and various objectives need to be achieved. Therefore, it is difficult to determine to what extent the various retrofit alternatives satisfy the various objectives (Kaklauskas et al., 2005). As the number of alternatives increases, it becomes more difficult to compare them and select the optimal retrofit scenarios (Penna et al., 2015). In particular, when planning retrofits in public buildings, decision-makers need to consider critical objectives to make rational decisions. Each tries to establish a plan to minimize energy consumption, CO2emissions, and expenses while maintaining the maximum level of thermal comfort in the retrofitted buildings within a limited budget (Bojic et al., 2012; Xu et al., 2015). However, these objectives contradict each other and have a trade-off relationship; it is difficult tofind an alternative that satisfies all of them (Chantrelle et al., 2011). For this reason, each decision-maker sub- jectively chooses a limited number of alternatives and compares them in general terms (Diakaki et al., 2008). Alternatively, the decision-maker can exclude some of the alternatives and then select a scenario intuitively (Shao et al., 2014). In such a process, the decision-maker intentionally considers only a few potential alternatives, making it difficult to choose the best option (Asadi et al., 2012a).

To solve this problem, researchers in previous studies employed the concept of multi-objective optimization (MOO) using evolutionary algorithms (Ma et al., 2012). Multi-objective optimization is a process for finding a solution that satisfies multiple objectives simultaneously (Abbass et al., 2001). It can obtain a Pareto-optimal solution (POS) that comprises a set of complementary alternatives (Marler and Arora, 2004). In earlier studies, before selecting an alternative, the decision-makers had to first define their preferences for the objectives so they could select one scenario from among the alternatives that would best satisfy all the objectives (Branke et al., 2001). However, as each decision-maker had different preferences and as the objectives could not be compared under equivalent conditions, it is difficult to determine accurate preferences in the context of real-world problems (Krettek et al., 2010). Thus, to overcome these limitations, researchers have proposed other methods to derive a set of complementary alternatives that satisfy multiple objectives and allow the decision-maker to find an optimal solution through a posteriori articulation of preferences. These methods have the advantage of not requiring pre- defined preferences from various decision-makers. The most well- known among these methods is the evolutionary algorithm (Jaimes and Coello, 2012), which is designed to evaluate numerous alternatives simultaneously through a global search and thus has a high possibility of obtaining an actual optimal solution (Goel and Deb,

2002; Saravanan et al., 2009). In a few previous studies (Chantrelle et al., 2011; Asadi et al., 2014; Shao et al., 2014; Penna et al., 2015; Fan and Xia, 2017) on the problem of MOO in building retroﬁt planning, a non-dominated sorting genetic algorithm (NSGA-II) was employed. In addition, none of these studies considered more than three objectives, which is referred to as the problem of many-objective optimization.

This study aims to solve the problem of many-objective optimization for retroﬁt planning in public buildings via EO algorithms that consider four objectives: minimizing energy consumption, CO2

emissions, and retrofit costs, and maximizing thermal comfort. This study involves the application of EO algorithms and the evaluation of their performance. Because these algorithms can handle four objectives at a time, they are suitable for the context of retrofit planning in public buildings. Recently, the multi-objective evolutionary algorithm based on decomposition (MOEA/D), the multi- objective particle-swarm-optimization algorithm (MOPSO), and the reference-point-based NSGA (NSGA-III) have demonstrated superior performance in solving the problem of MOO with more than three objectives when compared to the previously investi- gated NSGA-II (Bechikh et al., 2014; Svensson, 2015). Hence, in this study, the NSGA-III is compared with NSGA-II in the optimization of building retrofit planning. The Related Works section presents a comprehensive review of the related studies. Building Retrofit Planning via Evolutionary Many-Objective Optimization focuses on the methodology. The Experiments section provides the experi- mental results and discussion, and the Conclusion contains conclusions and suggestions for future research.

2. Related works

Multi-objective optimization is a process for deriving several complementary sets of alternatives that satisfy two or more contradicting objectives simultaneously (Abbass et al., 2001; Marler and Arora, 2004). To select the scenario that best satisﬁes all the objectives, decision-makers need to deﬁne their preferences (Branke et al., 2001). Multi-objective optimization methods can articulate those preferences as either a priori or a posteriori (Fonseca and Fleming, 1998).

The a priori methods are as follows.Asadi et al. (2012a; 2012b) used Tchebycheff programming to derive sets of complementary alternatives comprising 11 and 66 alternatives, respectively.

Diakaki et al. (2013)used compromise programming to derive a set of complementary alternatives with five alternatives. Antipova et al. (2014) used the ε-constraint method, a form of mixed- integer linear programming, to derive a set of complementary alternatives. Tchebycheff programming, compromise programming, and theε-constraint method all use mathematical programming to derive sets of complementary alternatives according to the weighted values of the objective functions (in Tchebycheff programming and compromise programming) or theε-value (in the ε-constraint method). The weighted values and theε-value differ depending on the decision-maker's preferences; no definite condition for weighing them exists. This makes it difficult to get an accurate preference (Krettek et al., 2010). Therefore, to optimize retrofit planning, it is more efficient tofirst derive a set of complementary alternatives and then to apply the methods using a posteriori articulation of preferences.

The most widely used algorithms for this approach are the evolutionary algorithms (Veldhuizen and Lamont, 2000). These algorithms use search methods that originated from the survival of theﬁttest. They simultaneously evaluate many alternatives through a global search, which helps themﬁnd optimal solutions (Goel and Deb, 2002; Saravanan et al., 2009). The most popular of these algorithms are the multi-objective genetic algorithm (MOGA)

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(Fonseca and Fleming, 1995) and NSGA-II (Deb, 2002), both of which are based on the genetic algorithm.

Ascione et al. (2015)used MOGA to assess the cost-optimal solution for retroﬁt planning in buildings. Previous studies (Chantrelle et al., 2011; Asadi et al., 2014; Shao et al., 2014; Penna et al., 2015; Fan and Xia, 2017) on retroﬁt planning primarily used NSGA-II or a variant of it. NSGA-II emphasizes the elitism of a solution and uses a less complicated calculation that does not require a sharing parameter to be selected (Panda and Yegireddy, 2013).

Chantrelle et al. (2011)were thefirst to use the energy simulation tool and NSGA-II tofind a set of complementary alternatives; they developed the alternative-selection tool called MultiOpt to establish retrofit plans for schools.Shao et al. (2014)used NSGA-II tofind a set of complementary alternatives suitable for retrofit planning in office buildings.Asadi et al. (2014)used a variant of NSGA-II with an artificial neural network to reduce the analysis time in school retrofitting scenarios.Penna et al. (2015)used NSGA-II to plan zero- energy buildings. Fan and Xia (2017) used NSGA-II for retrofit planning of building envelopes.

These studies considered no more than three objectives at a time. For example,Asadi et al. (2014)optimized three objectives:

minimizing energy consumption, minimizing retrofit costs, and maximizing thermal comfort.Ascione et al. (2015)optimized two objectives: minimizing the primary energy demand of the air- conditioning system and maximizing thermal comfort. Then, to find cost-optimal solutions, they considered the global cost during the building's life cycle.Fan and Xia (2017)optimized three objectives: minimizing energy consumption and the payback period of the investment and maximizing net present value. However, the performance of the NSGA-II algorithm is diminished when solving many-objective optimization problems (those that require consideration of four or more objectives) (Deb et al., 2001). For this reason, several EO algorithms have been developed tofind the right bal- ance among more than three objectives and to thereby improve upon the performance of NSGA-II. These include MOEA/D, MOPSO, and NSGA-III (Deb and Jain, 2014; Tavana et al., 2016).

2.1. Building retroﬁt planning via evolutionary many-objective optimization

Solving the problem of MOO is the process ofﬁnding a set of complementary alternatives that maximize the decision variable by adjusting a series of constraints so that the given objectives are maximized or minimized. The MOO formula, which uses n(the objective functions' number) andm(the decision variables’number), is shown as follows:

min=maxy¼fðxÞ ¼ ðf₁ðxÞ;f₂ðxÞ; :::;f_nðxÞÞ subject tox¼ ðx₁;x₂; :::;xmÞ2X

y¼ ðy₁;y₂; :::;ynÞ2Y ; (1)

where x represents the decision vector and y represents the objective vector, and whereXandYare the parameter space and objective space respectively (Steuer, 1986; Ringuest, 1992; Srinivas and Deb, 1994; Fonseca and Fleming, 1995; Zitzler and Thieie, 1999).

Generally, objective functions have a relationship that is contradicting on the decision variable. For this reason, it is almost impossible to derive an optimization solution that satisﬁes objective functions simultaneously (Jones et al., 2002; Konak et al., 2006). Therefore, solving the problem of MOO requires deriving a set of alternatives that satisfy a certain level for each objective function (Reyes-Sierra and Coello, 2006). To do so, a deﬁnition of the Pareto dominance on the decision vectors ofaandb(a;b2X) is required:

c_i2f1;2; :::;ng:f_iðaÞ f_iðbÞ∧d_i2f1;2; :::;ng:f_iðaÞ f_iðbÞ (2) When this formula is satisfied, it is defined as“adominatesb.” Here, if the formula is satisfied,acan be considered the Pareto- optimal front (POF) or non-dominated solution. This is the trade- off of the objective function and the final result of the MOO (Konak et al., 2006).

2.2. Building-retroﬁt elements

The elements considered in a building retrofit are largely cate- gorized into the exterior; the heating, ventilating, and air- conditioning (HVAC) system; and the building operations (Ascione et al., 2015). In the process of retrofitting, building operations cannot be selected, but the user can change them according to the use condition; therefore, this element was excluded from this study. Although the details of building-retrofit elements considered in previous studies (Chantrelle et al., 2011; Asadi et al., 2014; Shao et al., 2014; Penna et al., 2015; Fan and Xia, 2017) differ depending on the targeted buildings’characteristics, the following elements were commonly considered: the insulation materials in the walls, floors, roofs, and ceilings; the windows; and the HVAC system types.

Based on the selected building project's situation at the time of the retrofit, this study considered the following as selectable elements: the insulation materials in the external and internal walls, groundfloor, intermediate floor, roof, and ceiling; the windows;

and the HVAC system. The building-retroﬁt elements we considered are summarized inTable 1.

2.3. Objective functions

In this study, four objective functions (energy consumption, CO2

emissions, retroﬁt costs, and thermal comfort) were used to consider the three main aspects of sustainable building retroﬁts:

environmental, economic, and social (Gervasio et al., 2014). The building's total energy consumption,FEC;is the sum of the energy used for coolingðE_CoolingÞ;the energy used for heatingðE_HeatingÞ;the energy used for lightingðE_LightingÞ;and the energy used for appli- ance use ðEApplianceUseÞ: This sum is calculated using the energy simulation program of EnergyPlus 8.1 as follows:

F_EC¼E_CoolingþE_HeatingþE_LightingþEApplianceUse: (3) Second, the Inventory of Carbon&Energy database (Hammond and Jones, 2006) was used to calculate the CO2emissionsðFCEÞof the materials and equipment used. This database provides the amount of CO2emissions that occur during a building's life cycle when particular materials or pieces of equipment are used. In previous studies, the CO₂emissions generated during operations

Table 1

Summary of the building-retroﬁt elements.

Element's name Number of elements

Insulation materials* External wallðEWÞ 9 Internal wallðIWÞ 9 IntermediateﬂoorðIFÞ 7

GroundﬂoorðGFÞ 7

RoofðRFÞ 4

CeilingðCLÞ 2

Window types (glazing and gas)ðWNÞ 13

HVAC system typesðHVACÞ 4

Note. * Thermal conductivity, thickness, and density are different depending on the insulation materials.

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after a retrofit were considered to be the objective function. In this case, the CO₂emissions generated from the use of energy is pro- portional to the energy used, so the MOO of contradicting objective functions could not be performed. Therefore, CO2emissions during the operation stage were excluded to lead to a proper MOO. Instead, the CO2 emissions during the building's life cycle for particular materials or pieces of equipment were calculated by considering these stages: raw-material collection, production, delivery, and disposal. The relevant objective function was calculated by multiplying the coefficient of CO2 emissions per kgðEACⁱðxÞÞfor each retrofit elementi(from the Inventory of Carbon&Energy database) via the weight of the materials and equipment consideredðW_iðxÞÞ for that building element:

F_CE¼X

EACⁱðxÞ W_iðxÞ;

i¼EW;IW;IF;GF;CL;RF;WN;HVAC: (4) The retrofit costsðF_RCÞare the sum of the material and equipment costs as well as the construction costs. The price information regarding facility construction from the Public Procurement Service (Public Procurement Service, 2014) was used to calculate the costs of the materials and equipment. For the installation and construction costs ðCÎnstallðxÞÞ of an optimized retrofit alternative x; the market wage in the construction sector was determined based on the Construction Association of Korea's database (2014). The costs of the insulation materials and windows were calculated by multiplying the unit price per m²ðUPⁱðxÞÞfor each retrofit elementi by the areaðA_iðxÞÞfor each retrofit elementi:

F_RC¼n X

UPⁱðxÞ A_iðxÞo

þC^InstallðxÞ;

i¼EW;IW;IF;GF;CL;RF;WN;HVAC: (5) The thermal comfort timeðF_TCÞis the total time during which most occupants feel discomfort. The deﬁnition of this value is based on the 55e2004 standard of the American Society of Heating, Refrigerating and Air-Conditioning Engineers. The occupants’ thermal sensations are shown in the predicted mean votePMV; thermal environment evaluation index, for which a value between0.5 and 0.5 is deﬁned as“comfortable”:

F_TC¼^TimeS jPMVj<0:5: (6)

The sum of time (not included in this scale) was calculated by using the EnergyPlus 8.1. With this software, each EO is implemented toﬁnd a set of optimized retroﬁt alternatives by considering four objectives simultaneously.Fig. 1illustrates the proposed process.

3. EO algorithms 3.1. NSGA-II

Srinivas and Deb (1994)proposed theﬁrst NSGA. Although this was a powerful tool for solving the problem of MOO, it had several disadvantages: a lack of consideration of the elitist approach (Zitzler and Thieie, 1999; Deb et al., 2001), a high complexity of computations (Deb, 2002), and the need to specify the sharing parameter (Panda and Yegireddy, 2013). To complement the disadvantages,Deb (2002)proposed NSGA-II, which is different from the previous version of NSGA only in terms of how the selection mechanism is incorporated. The main operating steps of NSGA-II are population initialization; non-dominated sorting; the selection of a well-distributed set of points using the measures of crowding distance, selection, and recombination; crossover; and

mutation. The details of the NSGA-II process can be found inDeb (2002), Panda (2011), andPanda and Yegireddy (2013).

3.2. MOPSO

MOPSO (Reyes-Sierra and Coello, 2006) is based on the particle- swarm-optimization (PSO) algorithm thatKennedy and Eberhart (1995)developed. The PSO algorithm is a population-based sto- chastic optimization technique. Once a group of particles (potential solutions) is randomly selected, the PSO algorithm updates the generations to search for optima. For any particle in the population, particles are updated at each generation using the solution and value determined as the best up to that point. When a particle considers the entire population, the best value is global, but when it considers only a part of that population, the best value is local. Once these two values are found, the velocity and position of the particle are updated (Reyes-Sierra and Coello, 2006; Stoppato et al., 2014).

The particle's next position,x_iðtÞ, is determined by its current position and velocity,viðtÞ:

x_iðtÞ ¼x_iðt1Þ þviðtÞ; (7)

where the particle's velocity vector is deﬁned as

viðtÞ ¼w,viðt1Þ þC₁r₁½P_bestx_iðtÞ þC₂r₂½G_bestx_iðtÞ;

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wherewis the inertial weight used to control the effect that the particle's previous velocity has on its current velocity. Here,C₁(a

Initiate Population

Calculate Retrofit Costs

Evaluate Energy Consumption, CO2Emissions, Retrofit Costs, and Thermal Comfort

Calculate CO2

Emissions Run Energy

Simulation

Maximum Iteration or Convergence Reached?

Start

Building-Retrofit Elements

Optimization Completion Yes

Decode Building-Retrofit Elements and Generate Input File for Energy Simulation

Define New Generation

No

Fig. 1.A process for solving the multi-objective optimization problem of retroﬁt planning in public buildings, with four objectives.

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factor of cognitive learning) andC2(a factor of social learning) are deﬁned as positive constants (Stoppato et al., 2014).C₁indicates the particle's attraction toward its own success, andC₂indicates the particle's attraction toward its neighbors' success. The variables of r₁andr₂(in the range of [0, 1]) represent random functions. They are used to allow for a small percentage of particles to be diverted toward greater exploration in the search space and thereby to prevent entrapment in local minima (Eberhart and Shi, 2001;

Stoppato et al., 2014). P_best and G_best indicate the particle's optimal positions at the personal and global levels respectively. In the PSO algorithm, once a neighborhood topology is formulated, the leader is determined; each particle then updates its position based on the leader's position. In the MOPSO, by contrast, a set of different leaders is assigned to each particle, only one of which is used to update the position of particle. This leaders' set is retained as an external archive, the contents of which make up theﬁnal output. The details for MOPSO can be found inEberhart and Shi (2001), Reyes-Sierra and Coello (2006), Tripathi et al. (2007), Chaudhary and Dua (2012), andStoppato et al. (2014).

3.3. MOEA/D

MOEA/D (Zhang and Hui, 2007) is based on scalarizing functions (SF). To facilitate various search directions, a number of SF are used with weight vectors that are uniformly distributed in the objective space. The generation of those weight vectorsw_iis as follows:

w₁þw₂þ,,,w_k¼1; (9)

w_i2

0;1 H;2

H; :::;H H

; i¼1;2; :::;k; (10)

whereHrepresents a positive integer andkrepresents the objective functions' number. Here, the positive integer can be regarded as the granularity of discretization of the weight value. This study used a cellular version of MOEA/D (Ishibuchi et al., 2009a). In k-dimensional weight space, MOEA/D has a neighborhood struc- ture. Each cell has its own SF with a different weight vector. Each cell's number of neighbors is prespecifiedddefined via the Euclidian distance. A single cell that includes a single particle is placed at the same location as each of the weight vectors. Thus, the weight vectors' number is equal to the size of the population. It specifies that the size of the population varies depending on positive integerH in Equation(10) and on the number of objective functions used to evaluate each neighbor (using SF with the current cell's weight vector). Once an offspring is generated for a cell via crossover and mutation, it is compared with each neighbor and the current individual using SF. When the offspring is better than the current individual, it replaces that individual, and when it is superior to all the neighbors, it replaces those neighbors. This study uses the following Equation(11)because it has advantages over the weighted Tchebycheff function for the problems of many-objective optimization (Ishibuchi et al., 2009b, 2009c):

fðxÞ ¼w₁f₁ðxÞ þw₂f₂ðxÞ þ,,,þw_kf_kðxÞ: (11)

3.4. NSGA-III

The basic concept of NSGA-III (Deb and Jain, 2012) is not much different from that of NSGA-II (Deb, 2002). However, NSGA-III is based on a set of reference points for selecting a set of points that is well distributed, unlike NSGA-II (Jain and Deb, 2013; Deb and Jain, 2014). At the initiation of the algorithm, the population's individuals (“solutions”) are initialized randomly with the initial

(“parent”) population. The offspring is then created using binary tournament selection and an arithmetic crossover operator, as Michalewicz (1996) proposed. Once the offspring is created, the mutation operator is used to generate a new offspring. In this operation, the mutation operator is assumed to be Gaussian so that the algorithm has more mutations at the initiation than at the end of the algorithm. During this process, each solution needs to comply with the problem's constraints. Thereafter, the combining and sorting of the offspring and the parent population are completed based on Pareto dominance. After identifying the various levels of non-dominance, the best alternatives are chosen among the population, which is combined. The purpose of this process is to deﬁne the next new generation and to thus ensure that the best alternatives from the combined population become the parent population. This process continues until the size of the population reaches a predeﬁned value. The NSGA-III's operation principle and details can be found in Deb and Jain (2014) and Tavana et al. (2016).

4. Experiments

In this study, a set of alternatives for a retroﬁt plan was derived by applying the NSGA-II, MOEA/D, MOPSO, and NSGA-III to an existing public school building as a case study. The case study was performed by selecting a building with characteristics that can represent the subject to demonstrate the applicability of the proposed approach in a real-world situation (Asadi et al., 2012a, 2014, 2012b; Diakaki et al., 2013; Antipova et al., 2014; Shao et al., 2014).

The selected building is a main building of Chung-Ang University in Seoul, South Korea, with three stories above ground, single un- derground levels, and a totalﬂoor area of 3,287 m². This building was constructed in the late 1960s and is more than 50 years old;

thus appropriate retroﬁtting is necessary. Seoul weather data provided by the Korea Meteorological Administration was used in energy simulation. The performances of the four algorithms were compared in the optimization of four objectives. In implementing the four algorithms, the maximum number of generations and the population was set to 50 and 100, respectively; the crossover and mutation values were also matched.

4.1. Performance measures

For the problem of optimization, one can derive an optimal solution under a single objective function. However, the problem of optimization in this study has four objective functions that are opposed or conﬂicting, so a set of multiple optimal alternatives was derived as a solution. Therefore, for the MOO, performance measures were required to evaluate the properties of the derived alternatives so that the algorithms' performance could be compared.

The properties included the proximity of the generated non- dominated solution to the true POF, the diversity among generated non-dominated solutions, and convergence to the POS (Kollat and Reed, 2006; Wu et al., 2010). In practice, it is inefﬁcient to calculate the true POF by calculating the objective function of all possible alternatives (Chutima and Chimklai, 2012). For this reason, the performances of MOO algorithms can be evaluated and compared in terms of diversity and convergence among these properties (Deb et al., 2001; Chambari et al., 2012; Chutima and Chimklai, 2012). In this study, the algorithms’performances were compared in terms of diversity and convergence. The diversity among the optimized solutions indicates how equal the distribution is in the POS, and the convergence indicates how close the derived set of optimal alternatives is to the global POF (Panda and Yegireddy, 2013). Two metrics were used to compare the diversity and convergence properties of the MOO algorithms: the

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spacing of non-dominated solutions and the average distance between the POF and the solutions.

The spacing, whichSchott (1995)proposed, measures the POS's distribution. The value of spacing for a non-dominated solution is 0, which indicates that all POS are ideally distributed with equal in- tervals. The fact that the POS is ideally distributed is important in the sense that bias in the algorithm can be prevented by taking only the extreme values as solutions for a particular objective function.

An algorithm with a smaller spacing value is superior to one with a larger spacing value. When an algorithm has a smaller spacing value, the decision-maker can obtain alternatives that are more evenly distributed than the biased values for a particular objective function. Therefore, the decision-maker can consider alternatives with various values.

The average distance measures how close each solution is to its nearest POF (Behnamian et al., 2009). In this study, the average distance was measured by calculating each solution's average Euclidian distance from its nearest POF in the objective space.

Because the true POF is unknown in this problem, a POF was arti- ﬁcially generated by merging the non-dominated solutions with the results of the four algorithms. Therefore, the distances between the solutions for all individuals and the nearest solutions on the artiﬁcial POF were averaged in a set of non-dominated solutions.

Because the POF was artiﬁcially generated and is assumed to be representative of a true POF, the average distance can be used to compare the closeness for each algorithm's derived set of non-

dominated solutions. An algorithm with a smaller average distance is superior to one with a larger average distance.

5. Results and discussion

The results from applying the NSGA-II, MOEA/D, MOPSO, and NSGA-III to the selection of optimized retrofit alternatives are summarized inTable 2. Of the four algorithms, NSGA-III has the best diversity and convergence. In terms of the spacing of non- dominated solutions, NSGA-III has the best diversity, followed by MOPSO, MOEA/D, and NSGA-II, in that order. The smaller value of the spacing means that the derived set of alternatives has more even distribution in the objective space and a higher diversity (Deb and Jain, 2012). It implies that the values of the objective functions among derived alternatives vary when the decision-makers utilize NSGA-III to select retrofit alternatives. In terms of the average distance, NSGA-III has the best convergence, followed by MOEA/D, MOPSO, and NSGA-II, in that order. The smaller value of the average distance means that the solutions are closer to the true POF and has better convergence performance than others when optimizing with the same number of iterations (Chambari et al., 2012). In other words, NSGA-III's smaller average distance between the assumed POF and its solutions means that it approaches that front more quickly than the other algorithms do when performing an optimization with the given number of generations. In summary, NSGA-III rankedfirst in both metrics (diversity and convergence) and can provide more appropriate alternatives to decision-makers for public-building retrofit planning. Overall, this means that a decision-maker can utilize NSGA-III tofind a comprehensive set of trade-off retrofit alternatives that satisfy the objective function better in terms of reducing energy consumption, CO2 emissions, and retrofit costs while improving thermal comfort.

Fig. 2presents the optimization results being visually analyzed Table 2

Comparison of the algorithms’performances.

Performance metric NSGA-II MOEA/D MOPSO NSGA-III

Spacing 2.40Eþ07 2.14Eþ07 1.33Eþ07 1.01Eþ07

Average distance 2.49Eþ09 1.25Eþ09 2.28Eþ09 1.23Eþ09

(a) (b)

(c) (d)

Fig. 2.Optimization results obtained by using four many-objective optimization algorithms

(a) Three dimensional visualizationðF_EC;FRC;andFTCÞ, (b) Three dimensional visualizationðF_TC;FCE;andFRCÞ, (c) Two dimensional projectionðF_ECandFRCÞ, (d) Two dimensional projection.ðFTCandFCEÞ

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in the objective space. Although four objectives were dealt with in this study, the optimization results were plotted with three of these objectives as it is only possible to plot them in three dimensions or less. By comparing multiple visualizations in three and two dimensions, the derived set of alternatives from NSGA-III are located at the outermost position, which means that there are more solutions that better satisfy the four objective functions. The NSGA-III results show relatively small values for the four objective functions, closer to the ideal point, and a larger dominated volume, and the convergence performance is better.

In previous studies on the problem of MOO in building retrofit planning (e.g.,Chantrelle et al., 2011; Asadi et al., 2014; Shao et al., 2014; Penna et al., 2015; Fan and Xia, 2017), the NSGA-II has been employed. Remarkably, the NSGA-III showed better performance than the NSGA-II, and for the problem of many-objective optimization for public-building retrofit planning, the NSGA-II showed the worst results of two performance measures among four algorithms, with four objectives. After analyzing the derived set of alternatives, NSGA-III improved the results for each of the four objective functions; it proved to have better convergence performance and to be closer to the assumed POF than the other algorithms. This means that when a decision-maker uses NSGA-III to select a retrofit alternative, the objective function values of each alternative are diverse within the same objective function scale. Based on the results of this study, when deriving optimal alternatives, the NSGA-III has good convergence performance and will provide appropriate alternatives for the selection of the retrofit scenario.

6. Conclusion

When considering the ownership, operation, and maintenance of public buildings, there are four critical objectives in retrofit planning: minimizing energy consumption, CO2 emissions, and retrofit costs, and maximizing thermal comfort. These concerns need to be considered together. This study evaluated the performance of four EO algorithms (NSGA-II, MOEA/D, MOPSO, and NSGA-III) to determine which is most suitable to solving optimization problems in public-building retrofit planning.

Based on the four objectives, NSGA-III obtained the best performance of the four algorithms in terms of the spacing of non- dominated solutions and average distance. Using these two performance metrics, this study showed that NSGA-III resulted in better diversity and convergence than did the conventional algorithm (NSGA-II) for the problem of many-objective optimization.

Many-objective optimization via the NSGA-III can be expected to contribute to the derivation of a comprehensive set of trade-off alternatives for possible retrofit scenarios, thereby providing references that retrofit planners can use to select optimal retrofit strategies for public buildings that satisfy all stakeholders.

This study involved a public-building retrofit project as a case study. To reach more definitive conclusions, future work will be devoted to experiments on various retrofit projects. Additionally, when considering the practical aspect of solving many-objective optimization problems, relatively few alternatives are manageable when an a posteriori articulation of preferences is used to make decisions. This study can be extended to develop a procedure that will efficiently select optimal retrofit strategies based on a derived set of trade-off alternatives using an a posteriori articulation of stakeholders’preferences.

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2A10058175).

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