Improving Energy Efficiency in Buildings Using an Interactive Mathematical Programming Approach

Abstract

Improving energy efficiency in buildings is a major priority and challenge worldwide. The employed measures vary in nature, and the decision analyst, who is typically the architect, the engineer, or the building expert that has undertaken the task to suggest energy efficient solutions, faces a complex decision problem comprising numerous decision variables and multiple, usually competitive objectives. The solution of such multi-objective problems typically involves some sort of objectives aggregation, which reflects the preferences of the involved final decision maker that is the building’s user, occupant, and/or owner. The preferences elicitation, however, is a difficult task, and this paper aims to provide an interactive framework that will allow their consideration in a relatively easy manner. More specifically, a mathematical programming approach is proposed herein, which allows the elicitation and incorporation of the decision maker’s preferences in the decision model via the assessment of his/her utility function with the assistance of the multicriteria decision aid method UTASTAR. To study the feasibility and efficiency of the proposed approach, the case of a simple building is examined as an application example. The study results suggest that the proposed approach is capable of helping the decision analyst to suggest energy measures that satisfy, as much as possible, the decision maker’s preferences, without having to precisely prescribe them beforehand.

1. Introduction

Despite the long-lasting research and development in the particular field, the problem of improving energy efficiency in buildings still remains under investigation, according to recent reviews [1,2], due to its inherent complexity. The complexity of the problem stems from the involvement of several, typically competitive objectives (e.g., cost versus energy consumption) and the availability of numerous alternative measures (e.g., addition of insulation, change of color, use of cool coatings and renewables, etc.) [3], based on which, a final choice has to be made.

In practice, the specific measures to be adopted are typically suggested by the architect, the engineer, or the building expert, who undertakes the task to study the problem, thus playing the role of the decision analyst (DA). However, for any suggestions to be accepted by the final decision maker (DM), who may be the building’s user, occupant, and/or owner, they have to satisfy his/her specific requirements and preferences. This further increases the complexity of the problem, and calls for solution approaches that allow the realistic comparative evaluation of all the available alternatives [4]. Such an approach has been proposed by Diakaki et al. [5], who developed a relevant multi-objective decision model based on the principles of mathematical programming.

The aforementioned model considers as objectives to minimize the primary energy consumption of a building and the released CO₂ emissions during operation, as well as the initial investment cost. The particular formulation lends itself for solution via mathematical optimization techniques [5], as well as evolutionary methods like genetic algorithms [6], should the problem complexity become such that a solution via analytic techniques is no longer feasible. Despite the reduced precision compared to the simulation models typically employed for the evaluation of alternative measures [2], the mathematical programming-based approach has been proved to allow for the realistic comparative evaluation of all the available, alternative measures [7], it has thus been adopted by several researchers in the field (see, e.g., [8,9,10,11,12,13]).

Irrespective of the particular technique that one may employ for the solution of a multi-objective mathematical programming problem, to reach a single, final solution, which will be satisfactory, thus acceptable by the corresponding DM, weights need to be assigned to the different objectives [2,14]. These weights reflect the relevant importance of each considered objective to the DM, and/or the trade-off that exists among them, due to their competitive nature. The determination of such weights is a difficult task, as it is very unlikely for a DM to be able to explicitly state his/her preferences and satisfaction levels for each and every considered objective. Thus, rather than trying to determine the criteria weights [14], the implicit elicitation and learning of the preferences and value system of the DM, and their incorporation and use in the decision making process, seems more convenient. The development of such an approach for the multi-objective decision problem of improving energy efficiency in buildings is the purpose of the work presented herein.

Specifically, it is the aim of this paper to present an approach, whereby the DA will manage to reach a single, final solution of maximum utility to the DM, as an outcome of an interactive process of individual inter-alternative preference modelling. To this end, the main principles and rationale of a two-phase, iterative procedure proposed by Siskos and Despotis [15] for similar decision problem settings have been adopted. The procedure starts with identifying an initial compromise solution for the energy efficiency improvement problem established in Diakaki et al. [5] (first phase), and then runs iteratively (second phase) as many times as necessary to extract the DM’s aspiration levels for each objective, and estimate a respective utility function, which is used in order to reach a single, final solution, which is as close as possible to the DM’s actual preferences and value system. Throughout the iterative procedure, interaction is offered at two levels: (i) interactive modification of the DM’s satisfaction levels on the different pursued objectives; and (ii) interactive assessment of the DM’s utility function via the development and use of the UTASTAR multicriteria decision aid model [16]. UTASTAR is a preference disaggregation approach, which aims at inferring the value or utility function(s) of a DM, given his/her expressed preferences over a reference set of alternatives.

Through the aforementioned interactive procedure, the proposed approach allows the DA to (a) develop the DM’s overall utility function for the considered problem; (b) solve the problem by optimizing the developed utility function, rather than aggregating the individual objective functions of the considered problem via potentially arbitrary weights, like in the original multi-objective problem formulation in [5]; and (c) reach a single, final solution of maximum utility to the DM.

To study the feasibility and efficiency of the proposed approach, the case of a simple building is examined as an application example. The study results suggest that the proposed approach is capable of helping the DA to select and suggest energy measures that satisfy, as much as possible, the DM’s spectrum of desires, without having to precisely prescribe them beforehand.

The rest of the paper is structured in three more sections. Section 2 introduces the proposed approach, while Section 3 presents the application example. Section 4 discusses the results and findings, and Section 5, finally, summarizes the conclusions of the study and highlights future research directions.

2. Materials and Methods

2.1. Overview

The approach proposed herein builds upon the mixed-integer, non-linear, multi-objective optimization problem developed by Diakaki et al. [5], which may be generally defined as follows:

$$\begin{array}{l}\min \left[g_1(x),g_2(x),\dots ,g_n(x)\right]\\ \mathrm{subject} \mathrm{to} x\in X,\end{array}$$

(1)

where $x=\left(x_1,x_2,\dots ,x_m\right)$ is the vector of $m$ binary or continuous decision variables reflecting alternative choices (e.g., doors and windows types that can be used in the building, structure of multi-layer components such as walls, ceilings, and floors, materials to be used for their construction, and systems that can be used for heating, cooling and hot water supply), $X\subseteq R^m$ is the feasible region or decision space of the problem under study, which is implicitly dictated by a set of constraints concerning the decision variables and their intermediary relations; and $g_1(x)$, $g_2(x)$, … $g_n(x)$ are the values of n considered objectives. In the problem defined in [5], $n=3$, as the considered objectives are the total annual primary energy consumption (MJ/year), the CO₂ emissions (kg CO₂/year) released to the environment by the operation of the heating, cooling, and/or hot water supply systems, and the investment cost for the construction or retrofit of the building envelope and the acquisition and installation of the aforementioned systems, respectively.

The decision model (1) is used herein in the following two-phase procedure:

• In the first phase, each individual objective of (1) is first minimized and then maximized over the set of the feasible solutions, thus providing lower and upper bounds for the objectives. Given that $g_i(x)$ are minimized in (1), the lower bounds represent the ideal values of the objectives, and remain the same throughout the whole process, while the upper bounds represent the anti-ideal ones, and are refined during the second phase of the procedure. In addition, an initial efficient solution, i.e., a solution, which is not dominated by any other acceptable solution in the decision space is estimated that is closest to the ideal one with respect to the weighted Tchebycheff norm [15].
• In the second phase, an iterative process is followed, which comprises three successive steps. The first step can be viewed as a learning process of the trade-offs among the objectives for the DM. Through questions and answers, this step refines the upper bounds, thus gradually reducing the feasible region of the decision problem. The second step can be viewed as a learning process of the DM’s preferences. During this step, the DM is asked to rank, according to his/her preferences, a reference set of fictitious non-dominated decision profiles. This subjective ranking is then used by a UTASTAR model to generate the DM’s utility function u, over the intervals created by the lower and upper limits of the objectives’ values, and use them in transforming the decision problem (1) in the following:

  $$\begin{array}{l}\max u\left[g(x)\right]\\ \mathrm{subject} \mathrm{to} x\in X,\end{array}$$

  (2)

  where, $g(x)=\left(g_1(x),g_2(x),\dots ,g_n(x)\right)$ is the vector of the values of the objectives of the initial Problem (1). The decision Problem (2) is solved in the third step of the process, the solution is presented to the DM, and the iterations restart until a solution is reached that will be sufficiently satisfactory for the DM, so that he/she will not wish to further improve it.

Figure 1 presents the flowchart of the aforementioned procedure, while the following subsection provides the details of its different phases.

2.2. The Interactive Mathematical Programming Approach

2.2.1. Phase One

As mentioned earlier, within the first phase of the proposed interactive mathematical programming approach, the individual objectives of decision Problem (1) are minimized and maximized to establish the initial lower and upper bounds of the objectives. More specifically, the lower bound $l_i$, which is the ideal solution for each objective i, with $i=1,2,\dots ,n$, is calculated as follows:

$$\begin{array}{l}{l}_i=\min \left[g_i(x)\right]\\ \mathrm{subject} \mathrm{to} x\in X,\end{array}$$

(3)

while for the upper bound $h_i$, which is the anti-ideal solution, the following problem is solved:

$$\begin{array}{l}{h}_i=\max \left[g_i(x)\right]\\ \mathrm{subject} \mathrm{to} x\in X.\end{array}$$

(4)

Then, an initial compromise solution is obtained via the solution of the following problem:

$$\begin{array}{l}\min z\\ \mathrm{subject} \mathrm{to} x\in X\\ z>m_i\left(g_i(x)-l_i\right), i=1,2,\dots ,n\\ z>0\end{array}$$

(5)

where

$$m_i=d_i/{\displaystyle \sum _i{d}_i}, i=1,2,\dots ,n,$$

(6)

and

$$d_i=(h_i-l_i)/h_i, i=1,2,\dots ,n.$$

(7)

The solution of Problem (5) is the closest one to the ideal values of the objectives calculated via (3) in the sense of the weighted Tchebycheff norm.

2.2.2. Phase Two

The second phase of the proposed interactive mathematical programming approach is the iterative one, so let q be the number of iteration. Let also $X^q$ be the feasible region, $h_i^q$ the upper bound of objective i, and $g^q$ the vector of the optimal values of the objectives reached in iteration q.

When entering for the first time in phase two, for the upper bounds and the objectives values, the following hold:

• The upper bound values $h_i^0$ are equal to the solutions of the corresponding problems (4), obtained in phase one;
• The optimal values of the objectives $g^0$ are equal to the values obtained via the solution of the multi-objective Problem (5) in phase one.

In addition, $X^0=X$ holds.

Given the above initial values, as well as the lower bounds $l_i$, i.e., the ideal solutions of the objectives, the three steps described below are successively executed.

Step 1

At the first step of phase two, interaction takes place in order to learn the trade-offs among the objectives for the DM. More specifically, the DM is asked to express his/her satisfaction with respect to the values of the objectives that have been reached so far, i.e., for the values in $g^{q-1}$.

If the DM does not find any objective value satisfactory, the multi-objective decision problem has no satisfactory solution. In such case, the problem should be reviewed and revised, and the procedure should restart from phase one. However, if some values in $g^{q-1}$ are satisfactory, the DM is asked to suggest the objectives, which he/she insists to further decrease, and the whole set of objectives G is split in the following two complementary sets:

• $G_D$: the subset of G, which comprises the objectives that the DM insists to decrease;
• ${\overline{G}}_D$: the complement of $G_D$ in G.

Given the split of G in the two subsets, the DM is asked again to suggest, if there are any objectives in ${\overline{G}}_D$, which could be increased to make room for the desired further decrease of the objectives in $G_D$. If the response to this question is no, there is no room for further improvement, the procedure stops, and the solution reached so far is the best compromise solution to the examined problem. If, however, the response of the DM is yes, the upper bounds of the objectives are updated as follows:

$$h_i^q=\left\{\begin{array}{l}{g}_i^{q-1} \mathrm{if} g_i\in G_D\\ h_i^{q-1} \mathrm{if} g_i\in {\overline{G}}_D\end{array}\right.$$

(8)

For each $g_j\in G_D$, the following problem is solved:

$$\begin{array}{l}\min g_j(x)\\ \mathrm{subject} \mathrm{to} x\in X\\ g_i(x)\le h_i^q i=1,2,\dots ,n \mathrm{and} i\ne j,\end{array}$$

(9)

and the feasible region is finally reduced as shown below:

$$X^q=X^{q-1}\cap \left\{x\in R^m/g_i(x)\le h_i^q,i=1,2,\dots ,n\right\}.$$

(10)

Step 2

The second step of phase two is also a learning process aiming at the DM’s preferences elicitation. To this end, for an arbitrary chosen integer s, $s+1$ reference alternative profiles $a_k$, with $k=0,1,\dots ,s$, are generated. Each profile comprises a coordinate $a_{ik}$ for each objective i, which is calculated as follows:

$$a_{ik}=l_i+\left(k/s\right)\left(h_i^q-l_i\right)$$

(11)

Apparently, any other number of alternative profiles, as well as profiles generation procedure, can be adopted, as long as the generated profiles are representative of the trade-off among the objectives, and do not dominate each other. As their purpose is not to be offered to the DM as possible problem solutions, the generated profiles do not need to be efficient or feasible. They are just presented to the DM, who is asked to rank order them. The ranked set of alternative profiles is then used in the UTASTAR method [16] to assess the DM’s utility function $u\left[g(x)\right]$, as described in Appendix A.

Step 3

The utility function assessed in Step 2 is maximized in this last step over the feasible region. In other words, the Problem (2) is modified as follows, to take into account the reduction of the decision space according to (10):

$$\begin{array}{l}\max u\left[g(x)\right]\\ \mathrm{subject} \mathrm{to} x\in X^q,\end{array}$$

(12)

and then solved.

The solution of Problem (12) marks the end of the current iteration, and the procedure restarts from step 1 with the new solution, feasible region, and upper objective bounds.

3. Application Example

3.1. Overview of the Decision Problem

To assess the feasibility and efficiency of the proposed approach in suggesting measures that satisfy the competitive objectives of the energy efficiency improvement problem in a way that is compatible with the preferences and value system of the DM, the case of a simple building is studied. The building, taken from the study of Diakaki et al. [5], assumes an envelope, which comprises a floor and ceiling area of 100 m², 2 walls of area 24 m², 2 walls of area 30 m², and a door and window area both of 6 m².

The decisions regarding the considered building concern appropriate choices for:

• The type of the building’s door and window among the alternatives of

  Table 1. Alternative door types.

  Type	Thermal Transmittance (W/m2 °C)	Cost (€/m2)
  1. Hollow-core flush door	2.7	800
  2. Solid-core flush door with single glazing (17% glass)	2.1	1000

  and

  Table 2. Alternative window types.

  Type	Subtype	Thermal Transmittance (W/m2 °C)	Effective Total Solar Energy Transmittance (%)	Cost (€/m2)
  1. Single glazing	1. Typical glazing	5.0	80	40
  2. Double glazing	1. 4-20-4, uncoated, air filled	2.6	72	55
  	2. 4-12-4, coated, argon filled	1.6	76	65

  , respectively;
• the structure of the building’s walls, ceiling, and floor among the alternatives of

  Table 3. Alternative wall structures.

  Structure	Layer	Material	Thickness (m)	Thermal Conductivity (W/m °C)	Cost (€/m3)
  1	1	Plaster	0.025	0.87	10
  	2	Brick (complex)	0.150	0.72	23
  	3	Plaster	0.025	0.87	10
  2	1	Plaster	0.025	0.87	10
  	2	Brick (complex)	0.060	0.72	6.2
  	3	Brick (complex)	0.060	0.72	6.2
  	4	Plaster	0.025	0.87	10

  ,

  Table 4. Alternative ceiling structures.

  Structure	Layer	Material	Thickness (m)	Thermal Conductivity (W/m °C)	Cost (€/m3)
  1	1	Tiles	0.02	1.00	55
  	2	Concrete	0.15	0.72	55
  2	1	Tiles	0.02	1.00	55
  	2	Wood	0.03	0.17	70

  and

  Table 5. Alternative floor structures.

  Structure	Layer	Material	Thickness (m)	Thermal Conductivity (W/m °C)	Cost (€/m3)
  1	1	Tiles	0.01	1.00	55
  	2	Concrete	0.15	0.72	55
  2	1	Wood	0.02	0.17	85
  	2	Concrete	0.15	0.72	55

  , respectively;
• the addition or not, in the building’s walls, ceiling, and floor, of an insulation layer of maximum permissible thickness 0.10 m and material chosen among the alternatives of

  Table 6. Alternative insulation materials.

  Material	Thermal Conductivity (W/m °C)	Cost (€/m3)
  1. Polystyrene	0.036	200
  2. Mineral fiber	0.042	180
  3. Plastic fiber	0.020	300

  ;
• the space heating system among the alternatives of

  Table 7. Alternative heating-only systems.

  Category	Type	Generation Efficiency (%)	Cost (€)
  Electrical systems
  1. Resistance-based	1. Dry core storage boiler type 1	100	5000
  	2. Dry core storage boiler type 2	85	4200
  Non-electrical systems
  1. Oil-based	1. Condensing	83	5300
  	2. Standard oil boiler	62	4700
  2. Natural-gas based	1. Condensing	85	5800
  	2. Floor mounted boiler	55	4500

  ,

  Table 8. Alternative heating–cooling systems.

  Category	Type	Generation Efficiency (%)	Cost (€)
  1. Water cooled electric	1. <12,000 BTU	200	500
  	2. <18,000 BTU	230	800
  	3. <24,000 BTU	250	1200

  and

  Table 9. Alternative heating–hot water supply systems.

  Category	Type	Generation Efficiency (%)	Cost (€)
  Electrical systems
  1. Resistance-based	1. Electric CPSU	100	7200
  	2. Water storage boiler	85	5800
  Non-electrical systems
  1. Oil-based	1. Condensing combi	81	6200
  	2. Combi	70	5800
  2. Natural-gas based	1. Condensing combi	84	7200
  	2. Combi	65	5700

  ;
• the space cooling system among the alternatives of Table 8;
• the hot water supply system(s) among the alternatives of Table 9 and

  Table 10. Alternative hot water supply-only systems.

  Category	Type	Generation Efficiency (%)	Cost (€)
  Electrical systems
  1. Resistance-based	1. Electric immersion	100	1200
  	2. Electric instantaneous at point of use	85	1000
  Non-electrical systems
  1. Oil-based	1. Oil boiler/circulator	80	1000
  	2. Oil single burner	60	800
  2. Natural-gas based	1. Circulator built into a gas warm air system type 1	73	850
  	2. Circulator built into a gas warm air system type 2	60	650

  ; and
• the addition or not of a solar collector system among the alternatives of

  Table 11. Alternative solar collector systems.

  Category	Type	Generation Efficiency (%)	Cost (€)
  1. Flat collector	1. Type 1	90	900
  	2. Type 2	80	600
  2. Vacuum hear pipe CPC collector	1. Type 1	72	780
  	2. Type 2	67	500

  .

The values of the thermal and solar transmittance, and the thermal conductivity of construction materials and components in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 have been taken from the ASHRAE database [17], while the cost values in all the aforementioned tables were obtained through a short, unofficial market survey that took place for the needs of the study described in [5].

The application of the multi-objective decision modelling approach to the particular decision problem leads to a mathematical model of the form (1), which includes 18 continuous and 57 binary variables. The model, which is summarized in Appendix B, aims at determining measures that minimize the following three objectives:

• The primary energy consumption $g_1(x)$;
• the release of CO₂ emissions $g_2(x)$; and
• the initial investment cost $g_3(x)$.

These objectives are competitive, since, typically, the cost-efficient solutions are less environmentally friendly and vice versa. Thus, the search for a globally optimal solution is infeasible, and the DA has to search for a feasible solution, which will comply as much as possible with the DM’s preferences and value system. To assist the DA in this search, the multi-phase iterative procedure described in Section 2 is applied.

3.2. Application of the Interactive Mathematical Programming Approach

3.2.1. Phase One

In the first phase of the proposed approach, the individual objectives of the examined decision problem are minimized and maximized, according to (3) and (4), respectively, in order to establish the ideal and anti-ideal solutions of the problem. In addition, an initial compromise solution is identified via the solution of Problem (5).

Table 12. Summary of phase one outcomes.

Decisions and Criteria	Type of Solution
	Minimize			Maximize			Compromise
	${\mathit{g}}_{\mathbf{1}}(x)$	${\mathit{g}}_{\mathbf{2}}(x)$	${\mathit{g}}_{\mathbf{3}}(x)$	${\mathit{g}}_{\mathbf{1}}(x)$	${\mathit{g}}_{\mathbf{2}}(x)$	${\mathit{g}}_{\mathbf{3}}(x)$
Door type	2	2	1	1	1	2	1
Window type/subtype	2/2	2/2	1/1	1/1	1/1	2/2	2/2
Wall structure	1	1	2	2	2	1	2
Wall insulation thickness (m)	0.10	0.10	0.00	0.00	0.00	0.10	0.07
Wall insulation material	3	3	-	-	-	3	3
Ceiling structure	1	1	2	2	2	1	2
Ceiling insulation thickness (m)	0.10	0.10	0.00	0.00	0.00	0.10	0.07
Ceiling insulation material	3	3	-	-	-	3	3
Floor structure	2	2	1	1	1	2	1
Floor insulation thickness (m)	0.10	0.10	0.00	0.00	0.00	0.10	0.07
Floor insulation material	3	3	-	-	-	3	3
Heating system type	EHC	NEH	EHC	EHW	EHW	EHW	EHC
Heating system category/type	1/3	2/1	1/1	1/2	1/2	1/1	1/3
Cooling system type	-	EC	-	EC	EC	EC	-
Cooling system category/type	-	1/3	-	1/1	1/1	1/3	-
Hot water supply system type	NEW	NEW	NEW	-	-	-	NEW
Hot water supply system category/type	1/1	2/1	2/2	-	1/2	-	2/2
Solar collector category/type	1/1	1/1	-	-	-	1/1	2/2
$g_1$: Primary energy consumption (MJ/year)	13,593	13,970	321,276	722,123	722,123	32,475	19,582
$g_2$: Release of CO2 emissions (kg CO2/year)	1400	810	32,758	74,559	74,559	3353	1986
$g_3$: Initial investment cost (€)	21,987	27,637	7524	12,674	12,674	28,187	15,540

EHC: electrical system that will be used for both heating and cooling (see Table 8); EHW: electrical system that will be used for both heating and hot water supply (see Table 9); EC: electrical system that will be used only for cooling (see Table 8); NEH: electrical system that will be used only for heating (see Table 7); NEW: non-electrical system that will be used only for hot water supply (see Table 10).

summarizes the outcomes of this phase. The outcomes clearly demonstrate that the choices made depend on the pursued objective(s). For example, when the objective is solely to minimize the primary energy consumption, the most energy efficient choices are made in contrast to the choices made when aiming solely at the reduction of the initial investment cost. In this latter case, the cheapest choices are made, which are the worst from the energy efficiency perspective. These two objectives are clearly competitive to each other, but also to the emissions objective. The release of CO₂ emissions does not depend solely on the generation efficiency of the heating, cooling, and hot water supply systems, but also on the utilized fuel. Thus, some energy efficient choices are no longer efficient when emissions come into the picture.

Table 13. Basic information generated in phase one (iteration $q=0$).

Information	Primary Energy Consumption (MJ/Year) i = 1	Release of CO2 Emissions (kg CO2/Year i = 2	Initial Investment Cost (€) i = 3
Ideal solution (lower bound) $l_i$	13,593	810	7524
Initial compromise solution $g_i^0$	19,582	1986	15,540
Anti-ideal solution $h_i$	722,123	74,559	28,187
Initial upper bound $h_i^0$	722,123	74,559	28,187
Rate of closeness to the ideal solution	0.85%	1.59%	38.80%

summarizes and highlights the basic information about the problem at hand, which has been generated by the proposed approach in phase one. More specifically, the table comprises the ideal and anti-ideal objective values, the initial upper bound for each objective, the initial compromise solution, as well as the rate of closeness of the objectives to their ideal values, being calculated as follows:

$$\mathrm{Rate} \mathrm{of} \mathrm{closeness} \mathrm{to} \mathrm{the} \mathrm{ideal} \mathrm{solution}=100\frac{g_i^q-l_i}{h_i-l_i}$$

(13)

with $q$ the number of iteration; for phase one, $q=0$ holds. Apparently, the lower the value of the rate, the better.

Table 13 makes clear that the initial compromise solution comprises choices that lead the objectives of primary energy consumption and release of CO₂ emissions very close to their ideal solutions (rates of closeness are 0.85% and 1.59%, respectively). The initial investment cost, on the other hand, is not similarly close to its ideal value (rate of closeness is 38.80%), and this may cause dissatisfaction to the DM. For this reason, the second phase of the proposed approach is activated, to examine the satisfaction level of the DM and refine, if necessary, the problem solution.

3.2.2. Phase Two-Iteration 1-Step 1

Entering in phase two, the basic information of Table 13 is presented to the DM. Assuming that he/she is satisfied by the performance on objectives 1 and 2, but asks for an improvement on objective 3, i.e., a further cost reduction, even at the expense of the other two objectives, the following sets are formed:

• $G=\left\{\mathrm{objective} 1, \mathrm{objective} 2, \mathrm{objective} 3\right\}$;
• $G_R=\left\{\mathrm{objective} 3\right\}$;
• ${\overline{G}}_R=\left\{\mathrm{objective} 1, \mathrm{objective} 2\right\}$;

and the upper bound of objective 3 is updated as follows:

$$h_3^1=h_3^0=15540.$$

(14)

Being members of set ${\overline{G}}_R$, the upper bounds of the other two objectives remain equal to their initial values, i.e.:

$$\begin{array}{l}{h}_1^1=h_1^0=722123,\\ h_2^1=h_2^0=74559.\end{array}$$

(15)

Then, the Problem (9) is solved for the third objective, which is the only member of set $G_R$:

$$\begin{array}{l}\min g_3(x)\\ \mathrm{subject} \mathrm{to} x\in X\\ g_1(x)\le h_1^1\\ g_2(x)\le h_2^1\end{array}$$

(16)

and the feasible region of the decision problem is reduced as follows:

$$X^1=X^0\cap \left\{x\in R^{75}/g_i(x)\le h_i^1,i=1,2,3\right\},$$

(17)

with $X^0$ being the decision space X of the initial problem.

3.2.3. Phase Two-Iteration 1-Step 2

On the basis of information from step 1 and assuming $s=9$, 10 alternative profiles $a_k$, $k=0,1,\dots ,9$, are generated, according to Equation (11), and presented to the DM in order to rank order them.

Table 14. Reference set of alternatives and DM’s ranking.

Profile	Primary Energy Consumption (MJ/Year)	Release of CO2 Emissions (kg CO2/Year)	Initial Investment Cost (€)	DM’s Ranking
$a_0$	13,593	810	15,540	3
$a_1$	92,319	9004	14,650	2
$a_2$	171,044	17,199	13,759	1
$a_3$	249,770	25,393	12,868	4
$a_4$	328,495	33,587	11,978	5
$a_5$	407,221	41,782	11,087	6
$a_6$	485,947	49,976	10,196	7
$a_7$	564,672	58,171	9306	8
$a_8$	643,398	66,365	8415	9
$a_9$	722,123	74,559	7524	10

presents the profiles of these alternatives for each objective, along with their assumed ranking r, $r=1,2,\dots ,10$.

The information of Table 14 is then used in UTASTAR, leading to the marginal utility functions graphically displayed in Figure 2, which define the global utility of the DM via the following additive function:

$$u\left[g(x)\right]=u_1\left(g_1(x)\right)+u_2\left(g_2(x)\right)+u_3\left(g_3(x)\right),$$

(18)

or the equivalent:

$$u\left[g(x)\right]=0.300{u^{\prime }}_1\left(g_1(x)\right)+0.525{u^{\prime }}_2\left(g_2(x)\right)+0.175{u^{\prime }}_3\left(g_3(x)\right),$$

(19)

where ${u^{\prime }}_i$, $i=1,2,3$, are the normalized, in the range [0, 1], values of the marginal utilities $u_i$, graphically displayed in Figure 3.

3.2.4. Phase Two-Iteration 1-Step 3

In this last step of phase two, the utility Function (18) or (19) is maximized over the decreased feasible solution space $X^1$, defined in (17). More specifically, the following problem:

$$\begin{array}{l}\max u\left[g(x)\right]=u_1\left(g_1(x)\right)+u_2\left(g_2(x)\right)+u_3\left(g_3(x)\right)\\ \mathrm{subject} \mathrm{to} x\in X^1\end{array}$$

(20)

or its equivalent:

$$\begin{array}{l}\max u\left[g(x)\right]=0.300{u^{\prime }}_1\left(g_1(x)\right)+0.525{u^{\prime }}_2\left(g_2(x)\right)+0.175{u^{\prime }}_3\left(g_3(x)\right)\\ \mathrm{subject} \mathrm{to} x\in X^1\end{array}$$

(21)

is solved.

The solution of any of the aforementioned problems generates the new compromise solution, displayed in

Table 15. Basic information generated in iteration $q=1$ of phase two.

Information	Primary Energy Consumption (MJ/Year) i = 1	Release of CO2 Emissions (kg CO2/Year) i = 2	Initial Investment Cost (€) i = 3
Ideal solution (lower limit) $l_i$	13,593	810	7524
New compromise solution $g_i^1$	166,640	17,199	11,147
Anti-ideal solution $h_i$	722,123	74,559	28187
New upper bound $h_i^1$	722,123	74,559	15,540
Rate of closeness to the ideal solution	21.60%	22.22%	17.53%

, the current iteration is terminated, and a new iteration starts from step 1.

3.2.5. Phase Two-Iteration 2-Step 1

The second iteration of phase two starts with the results of Table 15 being presented to the DM. Apparently, the cost objective has been reduced as desired, coming closer to its ideal value; a rate of closeness 17.53% has been achieved, which is also reduced compared to its previous value (38.80%). This improvement, however, has come at the expense of the other two objectives, the values of which, as well as their corresponding rates of closeness, present an increase.

If the consequences of the obtained solution are not satisfactory, the interaction with the DM should continue, like in the previous iteration, until reaching a satisfactory solution. Otherwise, the procedure stops here and the final choices made through this multi-phase procedure (see

Table 16. Initial and final compromise solutions.

Decisions and Criteria	Initial Compromise Solution	Final Compromise Solution
Door type	1	1
Window type/subtype	2/2	2/1
Wall structure	2	1
Wall insulation thickness (m)	0.07	0
Wall insulation material	3	-
Ceiling structure	2	2
Ceiling insulation thickness (m)	0.07	0.01
Ceiling insulation material	3	1
Floor structure	1	1
Floor insulation thickness (m)	0.07	0.07
Floor insulation material	3	1
Heating system type	EHC	EHC
Heating system category/type	1/3	1/1
Cooling system type	-	-
Cooling system category/type	-	-
Hot water supply system type	NEW	NEW
Hot water supply system category/type	2/2	2/2
Solar collector category/type	2/2	1/1
$g_1$: Primary energy consumption (MJ/year)	19,582	166,640
$g_2$: Release of CO2 emissions (kg CO2/year)	1986	17,199
$g_3$: Initial investment cost (€)	15,540	11,147

EHC: electrical system that will be used for both heating and cooling (see Table 8); NEW: non-electrical system that will be used only for hot water supply (see Table 10).

) are presented to the DM.

4. Discussion

The previous two sections presented an interactive mathematical programming approach to the problem of improving energy efficiency in buildings, and demonstrated its use via an example case study. The problem is difficult to solve as it involves multiple, competitive objectives, and a large number of decision variables, given the large number of available, alternative measures, which can be adopted in this respect. In addition, the solution of the problem requires the DM to express his/her preferences to the considered objectives, a fact that further increases the problem’s complexity.

The approach proposed herein exploits the mathematical programming model proposed by Diakaki et al. [5] and the UTASTAR value elicitation method proposed by Siskos and Yannacopoulos [16] under an interactive decision framework, which has been developed following the rationale and principles of the decision-oriented method for multi-objective linear programming problems proposed by Siskos and Despotis [15]. The proposed framework assists the decision making procedure so that decisions are made, which comply with the value system of the DM, without the need to prescribe it beforehand.

The proposed approach can be also adopted in other decision settings within, but also beyond, the field of energy and environment. A similar approach, for example, lies on the basis of ADELAIS, an interactive computer program developed to support the search for a satisfactory solution in multi-objective linear programming problems, which has been used as a tool for the selection of stock portfolios [18]. In contrast, however, to both the initial conception in [15] and the ADELAIS program, the decision framework developed herein concerns a mixed-integer nonlinear mathematical programming problem, which aims at minimizing rather than maximizing the considered multiple objectives. This means that the overall framework can be adopted to any possible decision settings, should adequate care be taken to consider any potential particularities; e.g., in a case where objectives with a positive preference direction (e.g., comfort) should also be considered, to incorporate them, preserving at the same time the required cohesiveness of all considered objectives, their preference direction should be reversed by changing the sign of their corresponding functions. In addition, the mathematical programming formulation is quite flexible, allowing the incorporation of additional DM’s objectives and preferences.

5. Conclusions

The study presented herein demonstrated the feasibility as well as the strengths of applying an interactive mathematical modelling approach to the problem of energy efficiency improvement. The application of such a systematic approach allows for the simultaneous consideration of all available combinations of alternative actions, the consideration of any logical, physical, technical, or other constraints that may apply, and the incorporation of the preferences and value system of the DM without having to explicitly prescribe them beforehand. In addition, the application of the proposed approach ensures that a single, final solution will be reached, which will be satisfactory, and thus acceptable by the corresponding DM.

The proposed approach addresses the problem of improving energy efficiency in buildings in a systematic way. Thus, it can provide the basis for the development of a corresponding decision support system (DSS), which could assist the respective DAs in their difficult task of identifying, among the large volume of available measures, those that will satisfy the needs, requirements, and preferences of the DMs. According to Li et al. [19], there is still plenty of room for the enhancement of the existing relevant toolkits and the development of new ones, and the proposed approach provides the ground in this direction.