Enumerating a Diverse Set of 

Building Designs Using Discrete 

Optimization 

Preprint 

Elaine Hale and Nicholas Long 

Presented at SimBuild 2010 

New York, New York 

August 15–19, 2010 

Conference Paper 

NREL/CP-550-48386 

August 2010 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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ENUMERATING A DIVERSE SET OF BUILDING DESIGNS USING DISCRETE
 

OPTIMIZATION
 

Elaine T. Hale1 and Nicholas L. Long1
 

1National Renewable Energy Laboratory, Golden, CO
 

ABSTRACT 

Numerical optimization is a powerful method for identifying energy-efficient building designs. Automating the 

search process facilitates the evaluation of many more options than is possible with one-off parametric simulation 

runs. However, input data uncertainties and qualitative aspects of building design work against standard optimization formulations that return a single, so-called optimal 

design. 

This paper presents a method for harnessing a discrete 

optimization algorithm to obtain significantly different, 

economically viable building designs that satisfy an energy efficiency goal. The method is demonstrated using 

NREL’s first-generation building analysis platform, OptE-Plus, and two example problems. We discuss the information content of the results, and the computational effort 

required by the algorithm. 

INTRODUCTION 

Building design is a multidisciplinary endeavor with 

qualitative and quantitative aspects. Input data, especially costs and long-term characteristics of equipment 

use and performance, are highly uncertain. Nonetheless, 

professional and governmental organizations provide design guidance (through standards and voluntary guides) to 

improve energy efficiency cost effectively. 

Because of ambitious energy efficiency goals, this guidance is often informed by sophisticated design and analysis methods that rely on simulation engines. One approach, which has been used to provide guidance for Advanced Energy Design Guides at the 50% energy savings 

level, is numerical optimization, which automatically generates numerous candidate designs in search of a single 

optimal design (Hale et al. 2009; Leach et al. 2009). Candidate designs are evaluated using EnergyPlus, because it 

can be used in parallel on Linux cluster computers and has 

a comprehensive set of modeling features (Crawley et al. 

2008). 

However, it is widely appreciated that there is no single 

best design for a given building type in a given location. 

Different owners and occupants have different values and 

resources (Papamichael and Protzen 1993). Thus, the purpose of this paper is to present a method to harness numerical optimization to identify a number of good designs that 

can be evaluated by a project’s stakeholders. 

The core idea is to run multiple optimization searches, 

each with a significantly modified search space. The modifications are designed to extract information about which 

design strategies are necessary to reach the energy efficiency goal, which are optional, and which can be used 

to compensate for others that have been excluded from a 

given search. For instance, if plug and process loads cannot be reduced from their baseline values, perhaps a more 

efficient HVAC system can make up the gap. An important property of the generated solutions is that they are 

significantly different from one another. 

The optimization literature, including the subfield of 

optimal building design, focuses primarily on searches 

for a single optimal solution (Bouchlaghem and Letherman 1990; Christensen, Barker, and Horowitz 2004; Wetter 2001; Wright, Loosemore, and Farmani 2002). Significant deviations that come closer to what this work is 

trying to do include algorithms for finding feasible solutions (Bonami et al. 2009), and combinatorial optimization algorithms that find the k-best solutions to a given 

problem, that is, in the course of identifying the optimal 

solution, they keep track of and report the k−1 runners-up 

as measured by objective function value (Hamacher and 

Queryranne 1985; Piper and Zoltners 1976). The work 

described in this paper differs from feasible solution algorithms in that solutions with better objective values are 

preferred, and from the k-best solution algorithms in that 

it further requires the solutions found to be significantly 

different from one another. 

The algorithm presented in this paper, and the information that can be extracted from it, are highly influenced 

by the building energy analysis platform used by the authors. The optimization problem formulation embedded 

in the platform is described in the next section. An algorithm for finding multiple, diverse designs that all satisfy 

an energy efficiency (or other goal) follows in the ALGORITHM section. Numerical results, discussion, and 

conclusions follow. 

SETTING 

The analysis platform solves bi-objective building design problems over a set of discrete options called energy 

design measures (EDMs) (Ellis et al. 2006). Most often, 

the two objectives are to minimize an energy metric and an 

economic metric. The search starts at a baseline building 

and proceeds by constructing numerous candidate designs 

through the addition and removal of EDMs. By default, all 

designs are plotted on a graph of the two objective functions. The minimum cost designs at every level of en1
ergy use comprise the minimum cost curve. The portion 

of the curve containing designs that are more energy efficient than the minimum cost building is the Pareto front, 

which is the set of designs for which one objective cannot 

be improved without reducing performance with respect 

to the other objective. 

Credit: Elaine T. Hale/NREL 

Figure 1: EDM selection and editing 

The set of EDMs to optimize over is chosen through a 

graphical user interface (GUI), a portion of which is depicted in Figure 1. The EDMs are grouped into functional 

categories that implicitly define a disjunctive constraint, 

that is, exactly one EDM from each category must be included in each design (although some EDMs may have 

no meaning in a given design, for instance, skylight construction in a design with no skylights installed). Every 

category includes a baseline selection that is either defined explicitly in the EDM tree or in the baseline building 

model, or is implicitly defined as null. Figure 1 shows the 

EDM selections for skylight amounts used in the full scale 

example. The baseline selection is 0% skylight coverage. 

The design optimization problem embodied in the analysis platform can be expressed mathematically as: 

∗x = arg{min f (x),g(x)|xi ∈ {0,...,ni} ⊂ Z} (1) x 

where xi represents the EDM selection for category i, 

xi = 0 is the baseline selection for category i, ni is the 

number of non-baseline selections in category i, f (x) is 

the economic objective function, and g(x) is the energy 

efficiency objective function. 

In what follows, it sometimes makes sense to group 

multiple EDM categories together to form a superset we 

call a design strategy. For instance, daylighting controls 

and skylights could together comprise a daylighting strategy. Such supersets fit into the framework of Equation 1 

once all combinations of the EDMs in the individual categories are enumerated. There is still one baseline option 

with all categories set to baseline. All other possibilities 

(for instance, 2% skylight coverage and no daylighting 

controls) are assigned a nonzero integer. 

ALGORITHM 

We now present an algorithm that uses a discrete optimization solver to enumerate solutions that all (a) meet a 

single quantitative goal, and (b) are qualitatively different. 

The algorithm relies on the assumptions: 

1. The decision variables are discrete, 

xi ∈ {0,1,...,ni} ⊂ Z,i = 1,...,N, (2) 

with x = 0 representing a baseline decision. To 

obtain diverse solutions (qualitatively different), we 

further require each variable to represent a distinct 

type of decision. 

2. There is a hard constraint 

g(x) ≤ 0,g(x) ∈ R, (3) 

whose satisfaction indicates feasibility (for instance, 

meeting an energy efficiency goal). 

3. There is a search algorithm 

0 ∗ P(x ,x , J,K) ∈ D ⊂ ZN (4) 

as described in Algorithm 1. Any combinatorial or 

discrete optimization solver that can handle inequality constraints and/or bicriteria problems should be 

satisfactory. 

In our setting, solutions are building designs defined by 

EDM selections, g(x) represents an energy savings goal, 

and our secondary objective f (x) is a cost metric, usually 

a lifecycle cost that accounts for capital, maintenance, and 

energy costs accumulated over an analysis period. 

0 Algorithm 1 Search Algorithm P(x , x ∗,J,K)

Require: x0, starting point 

Require: x ∗, reference feasible point 

Require: J,K ⊂ {1,...,N}, J ∩K = 0/

0 0 Require: x = 0, x = x ∗ = 0 J K K

return x 

Ensure: xJ = 0, xK = x ∗

K 

Ensure: g(x) ≤ 0 if possible, otherwise, x = argming(x)

Ensure: x = argmin f (x) s.t. g(x) ≤ max{0,ming(x)}, 

that is, x minimizes a secondary objective 

The enumeration of alternative designs begins after running the first search, P(0,0,0/,0/), and proceeds recursively. In short, for a given design x ∗ that satisfies the hard ∗ constraint, each strategy used in that design ({i|xi > 0}) is 

iteratively removed from consideration (set to zero), and 

the search algorithm is warm-started from the resulting 

point. We thereby generate new candidate designs that 

2
hopefully satisfy the hard constraint (energy efficiency 

goal) while minimizing a secondary objective (lifecycle 

cost) over a reduced search space, and are qualitatively 

different from the original feasible point in that the new 

point has xi = 0 but the original point has xi > 0. This 

idea is expressed in Algorithm 2. 

Algorithm 2 runIteration(x ∗,J,K,P) 

Require: x ∗ = P(·,·,J,K), x ∗

J = 0, x ∗K > 0 

for j ∈ {i|x ∗ > 0} −K do i
 

Jˆ= J ∪ { j}

x0 = x ∗
 

0
 x = 0 j 

0 xˆ = P(x , x ∗, Jˆ,K)
 

if g(xˆ) ≤ 0 then
 

queueIteration( ˆx,Jˆ,P)
 

else 

K = K ∪ { j}

end if
 

end for
 

setUninitializedKsInQueue(K)
 

(x ∗,J,K,P) = popIterationQueue()
 

runIteration(x ∗,J,K,P)
 

Algorithm 2 naturally leads to a tree structure of results, 

with the optimal decision over the full search space (the 

solution returned by P(0,0,0/,0/)) as the root node. Each 

level in the tree is associated with a certain number of 

strategies turned off, that is, a certain cardinality in the set 

J. The root node has J = 0/, its children have |J| = 1, etc. 

However, there can be multiple paths to the same search, 

so our implementation does not queue searches whose J, 

K, x ∗

K signature matches an existing search. This breaks 

the tree structure; see Figure 2. Future work includes restructuring the algorithm to more accurately reflect the 

structure of the enumerated set of solutions. 

The solution returned by P(0,0,0/,0/) (Node 0 in Figure 2) is the optimal solution over the full search space, 

since for that search all elements are allowed to vary over 

their full domain. It is therefore guaranteed to have the 

best objective function value compared to all other solutions found by the algorithm, as subsequent searches explicitly reduce the size of the search space. Recall, however, that the objective function is not the absolute measure of goodness for a given design. Building designs are 

generally subject to multiple quantitative and qualitative 

criteria that are difficult, if not impossible, to capture in a 

single quantitative metric. 

Running a search with decision variable xi set to 0 necessarily leads to one of two conclusions (in the context of 

Algorithm 2): either strategy i is not required to reach the 

goal, and we can continue to look for more designs that 

have xi = 0; or strategy i is required to meet the goal. In 

the latter case, to make sure that each search results in a 

reduction of the subsequent search spaces, we add index 

i to the set K. Then grandchildren of the point x ∗ have xi 

set equal to the value of that index in the parent search. 

This convention is overly rigid—the optimal decision for 

xi will certainly be nonzero, but may not be equal to that 

of its parent. Nonetheless, this is the heuristic used in our 

current code. 

Our implementation of Algorithm 2 saves all the items 

0 listed (x ∗,x , xˆ,J,K) plus the objective function values and 

a pointer to even more information about each saved design. Keeping x0 enables us to report perturbation information about individual strategies. In particular, we report the difference in key metrics between feasible designs 

(x ∗), and those designs with one strategy removed (x0). 

This provides the information needed to answer questions 

like: What happens if we take an energy efficient design 

and remove the daylighting infrastructure? What is the 

quantitative difference in energy efficiency? cost? This 

type of data tells us which strategies are valuable with regards to energy, power demand, and water savings per investment dollar. Example perturbation data are provided 

in Table 2. 

NUMERICAL RESULTS 

We now demonstrate Algorithm 2 in the context of 

energy-efficient building design. 

ILLUSTRATIVE EXAMPLE 

The first example is a 100 m2 office building located in 

St. Louis, Missouri, subject to a few design strategies. 

The baseline building was generated using Opt-E-Plus, 

with ASHRAE Standards 90.1-2004, and 62.1-2004 automatically applied (ASHRAE 2004a; ASHRAE 2004b). 

A design is considered feasible if it has a net site energy 

savings of 20%. The net site energy use of the baseline 

building is 770 MJ/m2y, and the analysis period used to 

calculate life cycle cost is 10 years. We use Opt-E-Plus’s 

sequential search algorithm as P (Andersen, Christensen, 

and Horowitz 2006; Ellis et al. 2006). 

The strategies available to meet the energy efficiency 

goal are: 

Plug load density reduction (PLD). One EDM category. Baseline and reduced value, respectively: 8.07 

W/m2, 2.69 W/m2. 

Lighting power density reduction (LPD). One EDM 

category. Baseline and reduced value, respectively: 

14.0 W/m2, 11.0 W/m2. 

Daylighting (DL). Two EDM categories. Daylighting 

controls can be installed with a set point of 400 lux. 

Tubular daylighting devices (TDDs) can be installed 

at a density of 18.58 m2/device. 

3
Table 1: Alternative illustrative example designs. Strategies used and summary metrics. A strategy is “included” if at 

least one EDM category in the strategy is set to a non-baseline value. 

PV Lifetime Capital Peak Energy 

Included Strategies Energy Cost Cost Demand Savings 

Node PLD LPD DL AR MJ/m2y $/m2 $/m2 kW % 

0 X X X 0.0 552 454 4.70 21.3 

1 X X 59.0 681 579 4.79 20.0 

2 X X 0.2 555 455 4.82 20.0 

3 X X 49.9 662 561 5.18 20.0 

4 X 70.7 707 605 4.82 20.0 

5 X X 126.2 814 715 4.95 20.0 

6 X X 84.1 725 627 4.91 20.0 

7 149.4 877 772 5.61 19.4 

8 149.4 877 772 5.61 19.4 

9 X 122.9 818 715 5.37 20.0 

10 X 78.9 724 623 5.34 20.0 

x  ̄ 81.0 727 625 5.10 

100σ/x  ̄ 65 16 18 6.6 

Aspect ratio (AR). One EDM category. The building 

can be stretched to an aspect ratio of 1:4 along either axis. The baseline building has an aspect ratio 

of 1:1. 

In addition, up to 30% of the roof area not covered by 

TDDs can be covered with photovoltaic (PV) panels (10% 

efficient, 90% inverter efficiency). During sequential 

search, 30% PV coverage is simply another EDM. However, when alternative designs are being enumerated, if 

PV is required to reach the energy savings goal, the actual 

amount of PV is dialed back so that the resulting building just meets the goal. Also, PV is always available as 

a search option—it is not enabled and disabled like the 

other strategies. 

The alternative designs found by the enumeration algorithm are depicted in Figure 2 and listed in Table 1. 

The initial point found by the search over the full space is 

represented by node 00. Three strategies are used in that 

point: plug load density reduction, lighting power density reduction, and daylighting. The algorithm removes 

those strategies in turn, and finds that none is necessary 

to reach the efficiency goal. However, reaching the goal 

now requires some PV. Going further down the tree, we 

can see, for instance, that with the lighting power density 

at baseline value, and no daylighting infrastructure available, Node 06 is able to meet the goal after it changes the 

aspect ratio and includes PV. The goal cannot be met if 

all three strategies used in the optimal point are removed 

from consideration. 

Table 1 provides additional summary information about 

the alternative designs. In addition to a diversity of strategies used to meet the energy efficiency goal, other metrics 

vary significantly. For instance, the standard deviation of 

the capital cost intensities is 18% of the mean value, and 

the standard deviation of peak electricity demand is 6.6% 

of the mean value. Echoing comments made in the ALGORITHM section, node 00 has the lowest lifetime cost, 

that is, the lowest objective function value, but a given decision maker may be just as interested in reducing peak 

demand, or in keeping plug load levels at their baseline 

value. 

For this small example, the computational impact of 

enumerating multiple feasible designs is modest. The 

original search required 71 EnergyPlus simulations, at 

about 30 s each on a Dell Latitude XT and a Linux cluster computer. Subsequent searches required fewer simulations total, and were able to reuse simulations from previous searches. On average, there were 14.7 total simulations and 6.4 new simulations per alternative design 

search. However, there were 11 additional searches, so 

the extra computational effort was about the same as doing the original search over the full space. In return, the 

enumeration provided nine new designs (nodes 7 and 8 are 

identical), of which eight met the energy efficiency goal. 

FULL SCALE EXAMPLE 

A partial run of Algorithm 2 was completed for an 

example taken from recent work on grocery store design (Leach et al. 2009). A fourteen zone prototypical 

grocery store model was instantiated in sixteen locations 

by applying ASHRAE Standard 90.1-2004 and ASHRAE 

4
Credit: Elaine T. Hale/NREL 

Figure 2: Diverse solutions for the illustrative example. 

Nodes represent designs, arcs represent the paths to get 

from the optimal design to the alternative designs. Designs marked in gold include photovoltaic (PV) panels. 

Designs marked with octagons were not able to reach the 

20% energy efficiency goal, even with 30% of the roof covered with PV. 

Standards 62.1-1999 (ASHRAE 2004a; ASHRAE 1999). 

Those baseline models were then optimized with an overall goal of 50% net site energy savings at minimum life 

cycle cost with an analysis period of five years. Here we 

optimize and then enumerate alternative designs for the 

baseline building in San Francisco, California. The EDMs 

are grouped into fourteen strategies: 

Infiltration (IN). One EDM category that combines air 

barrier and vestibule EDMs. Four options including 

baseline. 

Elec. Lighting (LPD). One EDM category that addresses lighting power density. Three options including baseline. 

Daylighting (DL). Two EDM categories: daylighting 

controls and skylights. One daylighting control set 

point (500 lux) and three skylight amounts (2%, 3%, 

and 4%) available, for a total of eight options including baseline. 

Window Area (WA). One EDM category that adjusts the 

south-facing glazing amount. Two options: baseline 

and 50% less glazing than baseline. 

Wall Insulation (WI). One EDM category. Eight options including baseline. 

Roof Insulation (RI). One EDM category. Fourteen options including baseline. Four options include a cool 

roof membrane as the top layer. 

Fenestration Types (FT). Two EDM categories. One for 

south-facing glazing type; the other for skylight type. 

There are eight options for south-facing glazing, and 

ten for skylights. 

HVAC. One EDM category. Twelve options including 

baseline arise from varying the coefficient of performance (COP), fan efficiency, and economizers. 

Demand Control Ventilation (DCV). One EDM category. Two options including baseline. 

Energy Recovery Ventilation (ERV). One EDM category. Three options including baseline: no ERV, 

50% effective ERV, and 70% effective ERV. 

Frozen Food Cases (FFC). One EDM category providing six types of frozen food case, including baseline. 

Ice Cream Cases (ICC). One EDM category providing 

six types of ice cream case, including baseline. 

Meat Cases (MC). One EDM category providing eight 

types of refrigerated meat cases, including baseline. 

Dairy/Deli Cases (DDC). One EDM category providing 

five types of refrigerated dairy/deli cases, including 

baseline. 

As in the illustrative example, PV is treated separately 

from the other EDMs—it is always available, and is used 

to make up the difference to the energy efficiency goal 

when necessary. 

For this large-scale example, the additional computational burden of enumerating a diverse set of designs was 

significant. Each EnergyPlus run required about 12 min of 

simulation time (averaged over runs completed on a Dell 

Precision, 4-core desktop and a Linux cluster computer), 

and 5.5 MB of hard drive space. We were unable to run 

Algorithm 2 to completion because the memory requirements became too large for one desktop computer as we 

approached 75,000 EnergyPlus simulations. The original 

search (to find the overall optimal point) required 2,938 

EnergyPlus simulations. The enumeration algorithm finished ten iterations, started another, and ran 73 searches to 

completion. Eleven searches derived from the root node; 

63 searches had two strategies set to zero; just one search 

5
in the third level (|J| = 3) ran to completion. On average, each additional search required 962 new simulations. 

Thus, the effort of one full-space search is approximately 

equivalent to that of three reduced-space searches, a result significantly worse than the ratio of 11:1 seen in the 

previous example. 

The goal of design enumeration was largely accomplished despite the computational drawbacks just described. A total of 69 feasible designs were generated. 

The optimal point over the full search space used eleven of 

the fourteen energy efficiency strategies. The remaining 

three (roof insulation, fenestration type, and DCV) came 

into play in subsequent searches, once one or more of the 

original eleven strategies were eliminated. 

Table 2 summarizes the energy efficiency strategies 

used in this example from the perturbation perspective. 

Each search for an alternative design produces a starting point x0 that is the same as the initial feasible point 

x ∗ except that one strategy has been removed (set to the 

baseline value). Therefore, differences in the performance of x0 and x ∗ shows the value or lack thereof of 

the strategy. The metrics reported in Table 2 are calcu0 lated by subtracting the metric for x ∗ from that of x . 

In other words, we report metric(without the strategy) −

metric(with the strategy). The first entry in the table, 

Dairy/Deli Cases, is therefore a strategy that always saves 

energy and lifetime cost, but requires more up front capital. On the other hand, Electric Lighting is a winner all 

around—it cost less up front and saves energy. 

The last two columns of Table 2 are an estimate of how 

much PV is required to match the energy savings of the 

given strategy. In one sense it is a simple unit conversion 

of the EUI Savings columns. However, it does provide a 

way to visualize the value of an energy efficiency measure, as in, “I can daylight my store”, or “I could add 24 

to 369 m2 of PV panels to achieve similar carbon dioxide 

emission reductions.” 

DISCUSSION 

Overall, the proposed algorithm accomplishes its goal: 

it is able to identify multiple energy-efficient building designs that all satisfy a specific energy efficiency goal, and 

are significantly different from one another. In addition, 

perturbation data help identify high-value design strategies. Research continues on how to communicate these 

results to potential users. 

One communication problem that is tied up with the 

organization of the algorithm is EDM hiding. In particular, because the algorithm operates on EDM categories 

as a whole (or even bundled with other categories) it is 

not immediately clear which EDMs are present in each 

design. In some sense it is an easy matter to remedy 

this situation—simply create larger tables that list all the 

EDMs. On the other hand, such verbose information becomes hard to absorb. Ideally, decisions in a given category would be grouped into meaningful levels. Then a 

strategy could be implemented at the baseline level, level 

1, level 2, etc. We expect to investigate this idea after a 

more natural data structure for the algorithm as a whole is 

identified. 

The computational issues encountered for the fullscale example could be mitigated by stopping sequential 

searches soon after the energy efficiency goal is met. For 

illustration purposes, see Figure 3, which depicts the full 

space search’s Pareto front in black, and the reduced space 

searches’ Pareto fronts in purple. The search highlighted 

by the orange circle didn’t really need to run—the starting 

point meets the 50% energy savings goal. In other cases, 

when some search is required because the starting point 

is not feasible, it may be worth the computational time 

and memory savings to stop the search early even though 

this may preclude finding the actual optimal point for that 

search. An even better solution to this problem would be 

to use an optimization algorithm that directly solves the 

problem at hand, {min f (x)|g(x) ≤ 0}, rather than adapting a bi-objective solver for the purpose, as such an algorithm should be equipped with better stopping criteria. 

CONCLUSION 

An algorithm for generating a number of feasible solutions to a combinatorial optimization problem in approximate rank order of objective function value was presented. 

If each decision variable represents a distinct aspect of the 

problem, the set of decisions so generated will be diverse 

in that they will be qualitatively different from one another. One-off perturbations of feasible points are also 

generated, and provide valuable information concerning 

the individual strategies that compose the overall problem. 

Numerical results for energy-efficient building design 

problems suggest that the additional computational effort 

is modest for small problems. For larger problems, the 

additional effort can be considerable. Stopping searches 

short once a feasible solution has been found, or using 

a more appropriate search algorithm, should remedy the 

situation to a large extent. 

ACKNOWLEDGMENTS 

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with 

the National Renewable Energy Laboratory. 

Thanks are also due to our colleagues in the Commercial Buildings Research Group at the National Renewable Energy Laboratory, especially Daniel L. Macumber, 

Shanti D. Pless, Matt M. Leach, Adam I. Hirsch, and Paul 

A. Torcellini; and to our reviewers, who provided valuable 

comments and suggestions. 

6
Table 2: Grocery store perturbation data 

EUI Savings Lifetime Cost Capital Cost Equivalent PV 

(MJ/m2yr) Savings ($/m2) Savings ($/m2) (m2) 

Strategy Count min max min max min max min max 

Dairy/Deli Cases 1 626.2 626.2 40.56 40.56 −9.87 −9.87 4501 4501 

Daylighting 8 3.4 51.4 −14.90 1.27 −21.12 0.11 24 369 

DCV 1 33.0 33.0 −5.37 −5.37 −4.66 −4.66 238 238 

Elec. Lighting 5 30.7 64.9 9.94 14.6 0.97 2.31 221 467 

ERV 9 118.6 193.8 −1.60 3.94 −7.42 −7.27 853 1393 

Fenestration Type 2 4.3 10.1 −2.33 −0.48 −1.12 −0.92 31 72 

Frozen Food Cases 1 232.3 232.3 13.75 13.75 −12.23 −12.23 1670 1670 

HVAC 9 46.9 193.1 3.82 14.88 −3.68 −0.87 337 1388 

Ice Cream Cases 9 57.5 60.7 8.91 9.24 2.87 2.9 413 436 

Infiltration 4 24.6 40.7 0.95 5.27 −0.75 2.1 177 292 

Meat Cases 9 118.5 132.2 0.09 1.01 −8.14 −8.08 852 950 

Wall Insulation 8 −0.1 42.6 −0.34 1.33 −1.23 0 −1 306 

Window Area 7 −3.6 35.3 5.25 10.06 5.58 8.41 −26 254 

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Proceedings of 29th National Passive Solar Conference. Portland, Oregon: American Solar Energy Society (ASES), 877–882. 

Crawley, Drury B., Jon W. Hand, Michal Kummert, and 

Brent T. Griffith. 2008. “Contrasting the capabilities of building energy performance simulation programs.” Building and Environment 43 (4): 661–673 

(April). 

Ellis, P. G., B. Griffith, N. Long, P. Torcellini, and 

D. Crawley. 2006. “Automated Multivariate Optimization Tool for Energy Analysis.” Conference paper NREL/CP-550-40353, National Renewable Energy Laboratory, Golden, CO. 

Hale, Elaine T., Matt Leach, Adam Hirsch, and Paul A. 

Torcellini. 2009. “General Merchandise 50% Energy Savings Technical Support Document.” Technical Report NREL/TP-550-46100, National Renewable Energy Laboratory, Golden, CO. 

Hamacher, H. W., and M. Queryranne. 1985. “K best solutions to combinatorial optimization problems.” Annals of Operations Research 4 (1): 123–143. 

Leach, Matt, Elaine T Hale, Adam Hirsch, and Paul A 

Torcellini. 2009. “Grocery Store 50% Energy Savings Technical Support Document.” Technical Re7
Credit: Elaine T. Hale/NREL 

Figure 3: Demonstration of possible computational savings. The search to which the highlighted starting point belongs 

did not need to run since the initial point already meets the energy savings goal. The cross hairs are x∗ designs that 

include PV. The red points denote search start points (x0’s). 

port NREL/TP-550-46101, National Renewable Energy Laboratory, Golden, CO. 

Papamichael, Konstantinos, and Jean-Pierre Protzen. 

1993. “The Limits of Intelligence in Design.” Technical report LBL-31742, BS-306, Lawrence Berkeley Laboratory, Berkeley, California. 

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“Some Easy Postoptimality Analysis for Zero-One 

Programming.” Management Science 22 (7): 759– 

765. 

Wetter, Michael. 2001. “GenOpt – A Generic Optimization Program.” Proceedings of IBPSA’s Building Simulation 2001 Conference. Rio de Janeiro, 

Brazil. 

Wright, Jonathan A., Heather A. Loosemore, and 

Raziyeh Farmani. 2002. “Optimization of building 

thermal design and control by multi-criterion genetic 

algorithm.” Energy and Buildings 34:959–972. 

NOMENCLATURE 

0/ the empty set 

∈ the entity on the left is an element of 

the set on the right 

⊂ the set on the left is a subset of the set 

on the right 

R the set of all real numbers 

Z the set of all integers 

f (x) ∈ R, objective function over x 

g(x) ∈ R, sometimes objective function, 

sometimes hard constraint over x 

ni ∈ Z, number of non-baseline decisions 

available for variable xi 

x ∈ D ⊂ ZN , vector of integer decision 

variables 

x ∗ feasible point used as the seed for an 

alternative design iteration 

xˆ best point found by a reduced space 

search, may or may not be feasible 

x0 starting point for a reduced space 

search 

xi i

th element of x 

xJ components of x whose indices are in J 

COP coefficient of performance 

EDM energy design measure 

D the domain of x, D1 ×D2 ×··· ×DN 

Di the domain of xi, {0,1,...,ni}

GUI graphical user interface 

J set of indices, a subset of {1,...,N}

|J| the cardinality of J 

K set of indices, a subset of {1,...,N}

N number of integer decision variables 

P search algorithm 

PV photovoltaics 

T DD tubular daylight device 

8
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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1. REPORT DATE (DD-MM-YYYY) 

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4. TITLE AND SUBTITLE 

Enumerating a Diverse Set of Building Designs Using Discrete 

Optimization: Preprint 

5a. CONTRACT NUMBER 

DE-AC36-08-GO28308 

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6. AUTHOR(S) 

E. Hale and N. Long 

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NREL/CP-550-48386 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT (Maximum 200 Words) 

Numerical optimization is a powerful method for identifying energy-efficient building designs. Automating the search 

process facilitates the evaluation of many more options than is possible with one-off parametric simulation runs. 

However, input data uncertainties and qualitative aspects of building design work against standard optimization 

formulations that return a single, so-called optimal design. This paper presents a method for harnessing a discrete 

optimization algorithm to obtain significantly different, economically viable building designs that satisfy an energy 

efficiency goal. The method is demonstrated using NREL’s first-generation building analysis platform, Opt- E-Plus, 

and two example problems. We discuss the information content of the results, and the computational effort required 

by the algorithm. 

15. SUBJECT TERMS 

numerical optimization; opt-e-plus; discrete optimization 

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